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0904.1300 | {
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] | # Generalized Rejection Sampling Schemes and Applications in Signal Processing
Luca Martino and Joaquín Míguez
Department of Signal Theory and Communications, Universidad Carlos III de Madrid.
Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain.
E-mail: luca@tsc.uc3m.es, joaquin.miguez@uc3m.es
###### Abstract
Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters, have become very popular in signal processing over the last years. However, in many problems of practical interest these techniques demand procedures for sampling from probability distributions with non-standard forms, hence we are often brought back to the consideration of fundamental simulation algorithms, such as rejection sampling (RS). Unfortunately, the use of RS techniques demands the calculation of tight upper bounds for the ratio of the target probability density function (pdf) over the proposal density from which candidate samples are drawn. Except for the class of log-concave target pdf’s, for which an efficient algorithm exists, there are no general methods to analytically determine this bound, which has to be derived from scratch for each specific case. In this paper, we introduce new schemes for (a) obtaining upper bounds for likelihood functions and (b) adaptively computing proposal densities that approximate the target pdf closely. The former class of methods provides the tools to easily sample from _a posteriori_ probability distributions (that appear very often in signal processing problems) by drawing candidates from the prior distribution. However, they are even more useful when they are exploited to derive the generalized adaptive RS (GARS) algorithm introduced in the second part of the paper. The proposed GARS method yields a sequence of proposal densities that converge towards the target pdf and enable a very efficient sampling of a broad class of probability distributions, possibly with multiple modes and non-standard forms. We provide some simple numerical examples to illustrate the use of the proposed techniques, including an example of target localization using range measurements, often encountered in sensor network applications.
Rejection sampling; adaptive rejection sampling; Gibbs sampling; particle filtering; Monte Carlo integration; sensor networks; target localization.
## I Introduction
Bayesian methods have become very popular in signal processing during the past decades and, with them, there has been a surge of interest in the Monte Carlo techniques that are often necessary for the implementation of optimal _a posteriori_ estimators [6; 4; 13; 12]. Indeed, Monte Carlo statistical methods are powerful tools for numerical inference and optimization [13]. Currently, there exist several classes of MC techniques, including the popular Markov Chain Monte Carlo (MCMC) [6; 11] and particle filtering [3; 4] families of algorithms, which enjoy numerous applications. However, in many problems of practical interest these techniques demand procedures for sampling from probability distributions with non-standard forms, hence we are often brought back to the consideration of fundamental simulation algorithms, such as importance sampling [2], inversion procedures [13] and the accept/reject method, also known as _rejection sampling_ (RS).
The RS approach (13, Chapter 2) is a classical Monte Carlo technique for “universal sampling”. It can be used to generate samples from a target probability density function (pdf) by drawing from a possibly simpler proposal density. The sample is either accepted or rejected by an adequate test of the ratio of the two pdf’s, and it can be proved that accepted samples are actually distributed according to the target density. RS can be applied as a tool by itself, in problems where the goal is to approximate integrals with respect to (w.r.t.) the pdf of interest, but more often it is a useful building block for more sophisticated Monte Carlo procedures [8; 15; 10]. An important limitation of RS methods is the need to analytically establish a bound for the ratio of the target and proposal densities, since there is a lack of general methods for the computation of exact bounds.
One exception is the so-called _adaptive rejection sampling_ (ARS) method [8; 7; 13] which, given a target density, provides a procedure to obtain both a suitable proposal pdf (easy to draw from) and the upper bound for the ratio of the target density over this proposal. Unfortunately, this procedure is only valid when the target pdf is strictly log-concave, which is not the case in most practical cases. Although an extension has been proposed [9; 5] that enables the application of the ARS algorithm with \(T\)-concave distributions (where \(T\) is a monotonically increasing function, not necessarily the logarithm), it does not address the main limitations of the original method (e.g., the impossibility to draw from multimodal distributions) and is hard to apply, due to the difficulty to find adequate \(T\) transformations other than the logarithm. Another algorithm, called _adaptive rejection metropolis sampling_ (ARMS) [14], is an attempt to extend the ARS to multimodal densities by adding Metropolis-Hastings steps. However, the use of an MCMC procedure has two important consequences. First, the resulting samples are correlated (unlike in the original ARS method), and, second, for multimodal distributions the Markov Chain often tends to get trapped in a single mode.
In this paper we propose general procedures to apply RS when the target pdf is the posterior density of a signal of interest (SoI) given a collection of observations. Unlike the ARS technique, our methods can handle target pdf’s with several modes (hence non-log-concave) and, unlike the ARMS algorithm, they do not involve MCMC steps. Hence, the resulting samples are independent and come exactly from the target pdf.
We first tackle the problem of computing an upper bound for the likelihood of the SoI given fixed observations. The proposed solutions, that include both closed-form bounds and iterative procedures, are useful when we draw the candidate samples from the prior pdf.
In this second part of the paper, we extend our approach to devise a generalization of the ARS method that can be applied to a broad class of pdf’s, possibly multimodal. The generalized algorithm yields an efficient proposal density, tailored to the target density, that can attain a much better acceptance rate than the prior distribution. We remark that accepted samples from the target pdf are independent and identically distributed (i.i.d).
The remaining of the paper is organized as follows. We formally describe the signal model in Section II. Some useful definitions and basic assumptions are introduced in Section III. In Section IV, we propose a general procedure to compute upper bounds for a large family of likelihood functions. The ARS method is briefly reviewed in Section V, while the main contribution of the paper, the generalization of the ARS algorithm, is introduced in Section VI. Section VII is devoted to simple numerical examples and we conclude with a brief summary in Section VIII.
## II Model and Problem Statement
### _Notation_
Scalar magnitudes are denoted using regular face letters, e.g., \(x\), \(X\), while vectors are displayed as bold-face letters, e.g., **x**, **X**. We indicate random variates with upper-case letters, e.g., \(X\), **X**, while we use lower-case letters to denote the corresponding realizations, e.g., \(x\), **x**. We use letter \(p\) to denote the true probability density function (pdf) of a random variable or vector. This is an argument-wise notation, common in Bayesian analysis. For two random variables \(X\) and \(Y\), \(p(x)\) is the true pdf of \(X\) and \(p(y)\) is the true pdf of \(Y\), possibly different. The conditional pdf of \(X\) given \(Y=y\) is written \(p(x|y)\). Sets are denoted with calligraphic upper-case letters, e.g., \(\mathcal{R}\).
### _Signal Model_
Many problems in science and engineering involve the estimation of an unobserved SoI, \(\textbf{x}\in\mathbb{R}^{m}\), from a sequence of related observations. We assume an arbitrary prior probability density function (pdf) for the SoI, \(\textbf{X}\sim p(\textbf{x})\), and consider \(n\) scalar random observations, \(Y_{i}\in\mathbb{R}\), \(i=1,\ldots,n\), which are obtained through nonlinear transformations of the signal **X** contaminated with additive noise. Formally, we write
\[Y_{1}=g_{1}(\textbf{X})+\Theta_{1},\ldots,Y_{n}=g_{n}(\textbf{X})+\Theta_{n}\] (1)
where \(\textbf{Y}=[Y_{1},\ldots,Y_{n}]^{\top}\in\mathbb{R}^{n}\) is the random observation vector, \(g_{i}:\mathbb{R}^{m}\rightarrow\mathbb{R},\ \ i=1,\ldots,n\), are nonlinearities and \(\Theta_{i}\) are independent noise variables, possibly with different distributions for each \(i\). We write \(\textbf{y}=[y_{1},\ldots,y_{n}]^{\top}\in\mathbb{R}^{n}\) for the vector of available observations, i.e., a realization of **Y**.
We assume exponential-type noise pdf’s, of the form
\[\Theta_{i}\sim p(\vartheta_{i})=k_{i}\exp\left\{-\bar{V}_{i}( \vartheta_{i})\right\},\] (2)
where \(k_{i}>0\) is real constant and \(\bar{V}_{i}(\vartheta_{i})\) is a function, subsequently referred to as marginal potential, with the following properties:
1. It is real and non negative, i.e., \(\bar{V}_{i}:\mathbb{R}\rightarrow[0,+\infty)\).
2. It is increasing (\(\frac{d\bar{V}_{i}}{d\vartheta_{i}}>0\)) for \(\vartheta_{i}>0\) and decreasing (\(\frac{d\bar{V}_{i}}{d\vartheta_{i}}<0\)) for \(\vartheta_{i}<0\).
These conditions imply that \(\bar{V}_{i}(\vartheta_{i})\) has a unique minimum at \(\vartheta_{i}^{*}=0\) and, as a consequence \(p(\vartheta_{i})\) has only one maximum (mode) at \(\vartheta^{*}_{i}=0\). Since the noise variables are independent, the joint pdf \(p(\vartheta_{1},\vartheta_{2},\ldots,\vartheta_{n})=\prod_{i=1}^{n}p(\vartheta _{n})\) is easy to construct and we can define a joint potential function \(V^{(n)}:\mathbb{R}^{n}\rightarrow[0,+\infty)\) as
\[V^{(n)}(\vartheta_{1},\ldots,\vartheta_{n})\triangleq-\log\left[p(\vartheta_{1 },\ldots,\vartheta_{n})\right]=-\sum_{i=1}^{n}\log[p(\vartheta_{n})].\] (3)
Substituting (2) into (3) yields
\[V^{(n)}(\vartheta_{1},\ldots,\vartheta_{n})=c_{n}+\sum^{n}_{i=1}\bar{V}_{i}( \vartheta_{i})\] (4)
where \(c_{n}=-\sum^{n}_{i=1}\log{k_{i}}\) is a constant. In subsequent sections we will be interested in a particular class of joint potential functions denoted as
\[V_{l}^{(n)}(\vartheta_{1},\ldots,\vartheta_{n})=\sum^{n}_{i=1} \left|\vartheta_{i}\right|^{l},\ \ \ \ 0<l<+\infty,\] (5)
where the subscript \(l\) identifies the specific member of the class. In particular, the function obtained for \(l=2\), \(V_{2}^{(n)}(\vartheta_{1},\ldots,\vartheta_{n})=\sum^{n}_{i=1}\left|\vartheta_ {i}\right|^{2}\) is termed quadratic potential.
Let \(\textbf{g}=[g_{1},\ldots,g_{n}]^{\top}\) be the vector-valued nonlinearity defined as \(\textbf{g}(\textbf{x})\triangleq[g_{1}(\textbf{x}),\ldots,g_{n}(\textbf{x})]^{\top}\). The scalar observations are conditionally independent given a realization of the SoI, \(\textbf{X}=\textbf{x}\), hence the likelihood function\(\ell(\textbf{x};\textbf{y},\textbf{g})\triangleq p(\textbf{y}|\textbf{x})\), can be factorized as
\[\ell(\textbf{x};\textbf{y},\textbf{g})=\prod^{n}_{i=1}p(y_{i}| \textbf{x}),\] (6)
where \(p(y_{i}|\textbf{x})=k_{i}\exp\left\{-\bar{V}_{i}(y_{i}-g_{i}(\textbf{x}))\right\}\). The likelihood in (6) induces a system potential function \(V(\textbf{x};\textbf{y},\textbf{g}):\mathbb{R}^{m}\rightarrow[0,+\infty)\), defined as
\[V(\textbf{x};\textbf{y},\textbf{g})\triangleq-\log[\ell(\textbf{ x};\textbf{y},\textbf{g})]=-\sum^{n}_{i=1}\log[p(y_{i}|\textbf{x})],\] (7)
that depends on **x**, the observations **y**, and the function **g**. Using (4) and (7), we can write the system potential in terms of the joint potential,
\[\small V(\textbf{x};\textbf{y},\textbf{g})=c_{n}+\sum^{n}_{i=1}\bar{V}_{i}(y_{ i}-g_{i}(\textbf{x}))=V^{(n)}(y_{1}-g_{1}(\textbf{x}),\ldots,y_{n}-g_{n}( \textbf{x})).\] (8)
### _Rejection Sampling_
Assume that we wish to approximate, by sampling, some integral of the form \(I(f)=\int_{\mathbb{R}}f(\textbf{x})p(\textbf{x}|\textbf{y})d\textbf{x}\), where \(f\) is some measurable function of **x** and \(p(\textbf{x}|\textbf{y})\propto p(\textbf{x})\ell(\textbf{x};\textbf{y}, \textbf{g})\) is the posterior pdf of the SoI given the observations. Unfortunately, it may not be possible in general to draw directly from \(p(\textbf{x}|\textbf{y})\), so we need to apply simulation techniques to generate adequate samples. One appealing possibility is to perform RS using the prior, \(p(\textbf{x})\), as a proposal function. In such case, let \(\gamma\) be a lower bound for the system potential, \(\gamma\leq V(\textbf{x};\textbf{y},\textbf{g})\), so that \(L\triangleq\exp\{-\gamma\}\) is an upper bound for the likelihood, \(\ell(\textbf{x};\textbf{y},\textbf{g})\leq L\). We can generate \(N\) samples according to the standard RS algorithm.
1. Set \(i=1\).
2. Draw samples \(\textbf{x}^{\prime}\) from \(p(\textbf{x})\) and \(u^{\prime}\) from \(U(0,1)\), where \(U(0,1)\) is the uniform pdf in \([0,1]\).
3. If \(\frac{p(\textbf{x}^{\prime}|\textbf{y})}{Lp(\textbf{x}^{\prime})}\propto\frac{ \ell(\textbf{x}^{\prime};\textbf{y},\textbf{g})}{L}>u^{\prime}\) then \(\textbf{x}^{(i)}=\textbf{x}^{\prime}\), else discard \(\textbf{x}^{\prime}\) and go back to step 2.
4. Set \(i=i+1\). If \(i>N\) then stop, else go back to step 2.
Then, \(I(f)\) can be approximated as \(I(f)\approx\hat{I}(f)=\frac{1}{N}\sum_{i=1}^{N}f(\textbf{x}^{(i)})\). The fundamental figure of merit of a rejection sampler is the acceptance rate, i.e., the mean number of accepted samples over the total number of proposed candidates.
In Section IV, we address the problem of analytically calculating the bound \(L=\exp\{-\gamma\}\). Note that, since the \(\log\) function is monotonous, it is equivalent to maximize \(\ell\) w.r.t. **x** and to minimize the system potential \(V\) also w.r.t. **x**. As a consequence, we may focus on the calculation of a lower bound \(\gamma\) for \(V(\textbf{x};\textbf{y},\textbf{g})\). Note that this problem is far from trivial. Even for very simple marginal potentials, \(\bar{V}_{i}\), \(i=1,...,n\), the system potential can be highly multimodal w.r.t. **x**. See the example in the Section VII-A for an illustration.
## III Definitions and Assumptions
Hereafter, we restrict our attention to the case of a scalar SoI, \(x\in\mathbb{R}\). This is done for the sake of clarity, since dealing with the general case \(\textbf{x}\in\mathbb{R}^{m}\) requires additional definitions and notations. The techniques to be described in Sections IV-VI can be extended to the general case, although this extension is not trivial. The example in Section VII-C illustrates how the proposal methodology is also useful in higher dimensional spaces, though.
For a given vector of observations \(\textbf{Y}=\textbf{y}\), we define the set of _simple estimates_ of the SoI as
\[\mathcal{X}\triangleq\left\{x_{i}\in\mathbb{R}:\ \ y_{i}=g_{i}(x_{i})\ \ \mbox {for}\ i=1,\ldots,n\right\}.\] (9)
Each equation \(y_{i}=g_{i}(x_{i})\), in general, can yield zero, one or more simple estimates. We also introduce the maximum likelihood (ML) SoI estimator \(\hat{x}\), as
\[\hat{x}\in\arg\max\limits_{x\in\mathbb{R}}{\ell(x|\textbf{y},\textbf{g})}=\arg \min\limits_{x\in\mathbb{R}}{V(x;\textbf{y},\textbf{g})},\] (10)
not necessarily unique.
Let us use \(\mathcal{A}\subseteq\mathbb{R}\) to denote the support of the vector function **g**, i.e., \(\textbf{g}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R}^{n}\). We assume that there exists a partition \(\{\mathcal{B}_{j}\}_{j=1}^{q}\) of \(\mathcal{A}\) (i.e., \(\mathcal{A}=\cup_{j=1}^{q}\mathcal{B}_{j}\) and \(\mathcal{B}_{i}\cap\mathcal{B}_{j}=\emptyset\), \(\forall i\neq j\)) such that the subsets \(\mathcal{B}_{j}\) are intervals in \(\mathbb{R}\) and we can define functions \(g_{i,j}:\mathcal{B}_{j}\rightarrow\mathbb{R},\ \ \ j=1,\ldots,q\) and \(i=1,\ldots,n\), as
\[g_{i,j}(x)\triangleq g_{i}(x),\ \ \forall x\in\mathcal{B}_{j},\] (11)
i.e., \(g_{i,j}\) is the restriction of \(g_{i}\) to the interval \(\mathcal{B}_{j}\). We further assume that (a) every function \(g_{i,j}\) is invertible in \(\mathcal{B}_{j}\) and (b) every function \(g_{i,j}\) is either convex in \(\mathcal{B}_{j}\) or concave in \(\mathcal{B}_{j}\). Assumptions (a) and (b) together mean that, for every \(i\) and all \(x\in\mathcal{B}_{j}\), the first derivative \(\frac{dg_{i,j}}{dx}\) is either strictly positive or strictly negative and the second derivative \(\frac{d^{2}g_{i,j}}{dx^{2}}\) is either non-negative or non-positive. As a consequence, there are exactly \(n\) simple estimates (one per observation) in each subset of the partition, namely \(x_{i,j}=g_{i,j}^{-1}(y_{i})\) for \(i=1,\ldots,n\). We write the set of simple estimates in \(\mathcal{B}_{j}\) as \(\mathcal{X}_{j}=\{x_{1,j},\ldots,x_{n,j}\}\). Due to the additivity of the noise in (1), if \(g_{i,j}\) is bounded there may be a non-negligible probability that \(Y_{i}>\max\limits_{x\in[\mathcal{B}_{j}]}g_{i,j}(x)\) (or \(Y_{i}<\min\limits_{x\in[\mathcal{B}_{j}]}g_{i,j}(x)\)), where \([\mathcal{B}_{j}]\) denotes the closure of set \(\mathcal{B}_{j}\), hence \(g_{i,j}^{-1}(y_{i})\) may not exist for some realization \(Y_{i}=y_{i}\). In such case, we define \(x_{i,j}=\arg\max\limits_{x\in[\mathcal{B}_{j}]}g_{i,j}(x)\) (or \(x_{i,j}=\arg\min\limits_{x\in[\mathcal{B}_{j}]}g_{i,j}(x)\), respectively), and admit \(x_{i,j}=+\infty\) (respectively, \(x_{i,j}=-\infty\)) as valid simple estimates.
## IV Computation of upper bounds on the likelihood
### _Basic method_
Let **y** be an arbitrary but fixed realization of the observation vector **Y**. Our goal is to obtain an analytical method for the computation of a scalar \(\gamma(\textbf{y})\in\mathbb{R}\) such that \(\gamma(\textbf{y})\leq\inf\limits_{x\in\mathbb{R}}V(x;\textbf{y},\textbf{g})\). Hereafter, we omit the dependence on the observation vector and write simply \(\gamma\). The main difficulty to carry out this calculation is the nonlinearity **g**, which renders the problem not directly tractable. To circumvent this obstacle, we split the problem into \(q\) subproblems and address the computation of bounds for each set \(\mathcal{B}_{j}\), \(j=1,\ldots,q\), in the partition of \(\mathcal{A}\). Within \(\mathcal{B}_{j}\), we build adequate linear functions \(\left\{r_{i,j}\right\}_{i=1}^{n}\) in order to replace the nonlinearities \(\left\{g_{i,j}\right\}_{i=1}^{n}\). We require that, for every \(r_{i,j}\), the inequalities
\[\left|y_{i}-r_{i,j}(x)\right|\leq\left|y_{i}-g_{i,j}(x)\right|,\ \mbox{and}\] (12)
\[(y_{i}-r_{i,j}(x))(y_{i}-g_{i,j}(x))\geq 0\] (13)
hold jointly for all \(i=1,\ldots,n\), and all \(x\in\mathcal{I}_{j}\subset\mathcal{B}_{j}\), where \(\mathcal{I}_{j}\) is any closed interval in \(\mathcal{B}_{j}\) such that \(\hat{x}_{j}\in\arg\min\limits_{x\in[\mathcal{B}_{j}]}V(x;\textbf{y},\textbf{g})\) (i.e., any ML estimator of the SoI \(X\) restricted to \(\mathcal{B}_{j}\), possibly non unique) is contained in \(\mathcal{I}_{j}\). The latter requirement can be fulfilled if we choose \(\mathcal{I}_{j}\triangleq[\min(\mathcal{X}_{j}),\max(\mathcal{X}_{j})]\) (see the Appendix for a proof).
If (12) and (13) hold, we can write
\[\bar{V}_{i}(y_{i}-r_{i,j}(x))\leq\bar{V}_{i}(y_{i}-g_{i,j}(x)),\ \ \ \forall x \in\mathcal{I}_{j},\] (14)
which follows easily from the properties (P1) and (P2) of the marginal potential functions \(\bar{V}_{i}\) as described in Section II-B. Moreover, since \(V(x;\textbf{y},\textbf{g}_{j})=c_{n}+\sum_{i=1}^{n}\bar{V}_{i}(y_{i}-g_{i,j}(x))\) and \(V(x;\textbf{y},\textbf{r}_{j})=c_{n}+\sum_{i=1}^{n}\bar{V}_{i}(y_{i}-r_{i,j}(x))\) (this function will be subsequently referred as the modified system potential) where \(\textbf{g}_{j}(x)\triangleq[g_{1,j}(x),\ldots,g_{n,j}(x)]\) and \(\textbf{r}_{j}(x)\triangleq[r_{1,j}(x),\ldots,r_{n,j}(x)]\), Eq. (14) implies that \(V(x;\textbf{y},\textbf{r}_{j})\leq V(x;\textbf{y},\textbf{g}_{j})\), \(\forall x\in\mathcal{I}_{j}\), and, as a consequence,
\[\gamma_{j}=\inf\limits_{x\in\mathcal{I}_{j}}{V(x;\textbf{y}, \textbf{r}_{j})}\leq\inf\limits_{x\in\mathcal{I}_{j}}{V(x;\textbf{y},\textbf{g }_{j})}=\inf\limits_{x\in\mathcal{B}_{j}}{V(x;\textbf{y},\textbf{g})}.\] (15)
Therefore, it is possible to find a lower bound in \(\mathcal{B}_{j}\) for the system potential \(V(x;\textbf{y},\textbf{g}_{j})\), denoted \(\gamma_{j}\), by minimizing the modified potential \(V(x;\textbf{y},\textbf{r}_{j})\) in \(\mathcal{I}_{j}\).
All that remains is to actually build the linearities \(\left\{r_{i,j}\right\}_{i=1}^{n}\). This construction is straightforward and can be described graphically by splitting the problem into two cases. Case 1 corresponds to nonlinearities \(g_{i,j}\) such that \(\frac{dg_{i,j}(x)}{dx}\times\frac{d^{2}g_{i,j}(x)}{dx^{2}}\geq 0\) (i.e., \(g_{i,j}\) is either increasing and convex or decreasing and concave), while case 2 corresponds to functions that comply with \(\frac{dg_{i,j}(x)}{dx}\times\frac{d^{2}g_{i,j}(x)}{dx^{2}}\leq 0\) (i.e., \(g_{i,j}\) is either increasing and concave or decreasing and convex), when \(x\in\mathcal{B}_{j}\).
Figure 1 (a)-(b) depicts the construction of \(r_{i,j}\) in case 1. We choose a linear function \(r_{i,j}\) that connects the point \((\min{(\mathcal{X}_{j})},g(\min{(\mathcal{X}_{j})}))\) and the point corresponding to the simple estimate, \((x_{i,j},g(x_{i,j}))\). In the figure, \(d_{r}\) and \(d_{g}\) denote the distances \(\left|y_{i}-r_{i,j}(x)\right|\) and \(\left|y_{i}-g_{i,j}(x)\right|\), respectively. It is apparent that \(d_{r}\leq d_{g}\) for all \(x\in\mathcal{I}_{j}\), hence inequality (12) is granted. Inequality (13) also holds for all \(x\in\mathcal{I}_{j}\), since \(r_{i,j}(x)\) and \(g_{i,j}(x)\) are either simultaneously greater than (or equal to) \(y_{i}\), or simultaneously lesser than (or equal to) \(y_{i}\).
Figure 1 (c)-(d) depicts the construction of \(r_{i,j}\) in case 2. We choose a linear function \(r_{i,j}\) that connects the point \((\max{(\mathcal{X}_{j})},g(\max{(\mathcal{X}_{j})}))\) and the point corresponding to the simple estimate, \((x_{i,j},g(x_{i,j}))\). Again, \(d_{r}\) and \(d_{g}\) denote the distances \(\left|y_{i}-r_{i,j}(x)\right|\) and \(\left|y_{i}-g_{i,j}(x)\right|\), respectively. It is apparent from the two plots that inequalities (12) and (13) hold for all \(x\in\mathcal{I}_{j}\).
A special subcase of 1 (respectively, of 2) occurs when \(x_{i}=\min{(\mathcal{X}_{j})}\) (respectively, \(x_{i,j}=\max{(\mathcal{X}_{j})}\)). Then, \(r_{i,j}(x)\) is the tangent to \(g_{i,j}(x)\) at \(x_{i,j}\). If \(x_{i,j}=\pm\infty\) then \(r_{i,j}(x)\) is a horizontal asymptote of \(g_{i,j}(x)\).
It is often possible to find \(\gamma_{j}=\inf\limits_{x\in\mathcal{I}_{j}}V(x;\textbf{y},\textbf{r}_{j})\leq \inf\limits_{x\in\mathcal{I}_{j}}V(x;\textbf{y},\textbf{g}_{j})\) in closed-form. If we choose \(\gamma=\min\limits_{j}\gamma_{j}\), then \(\gamma\leq\inf\limits_{x\in\mathbb{R}}V(x,\textbf{y},\textbf{g})\) is a global lower bound of the system potential. Table I shows an outline of the proposed method, that will be subsequently referred to as bounding method 1 (BM1) for conciseness.
<figure><img src="content_image/0904.1300/figcase1.png"><figcaption>Fig. 1: Construction of the auxiliary linearities {ri,j}ni=1. We indicatedr=∣∣yi−ri,j(x)∣∣ and dg=∣∣yi−gi,j(x)∣∣, respectively. It is apparent thatdr≤dg and ri,j(x) and gi,j(x) are either simultaneously greater than (or equalto) yi, or simultaneously lesser than (or equal to) yi, for all x∈Ij. Hence,the inequalities (12) and (13) are satisfied ∀x∈Ij. (a) Function gi,j isincreasing and convex (case 1). (b) Function gi,j is decreasing and concave(case 1). (c) Function gi,j is decreasing and convex (case 2). (d) Functiongi,j is increasing and concave (case 2).</figcaption></figure>
1\. Find a partition {Bj}qj=1 of the space of the SoI.
---
2\. Compute the simple estimates Xj={x1,j,…,xn,j} for each Bj.
3\. Calculate Ij≜[min(Xj),max(Xj)] and build ri,j(x), for x∈Ij and i=1,…,n.
4\. Replace gj(x) with rj(x), and minimize V(x;y,rj) to find the lower bound
γj.
5\. Find γ=minjγj.
TABLE I: Bounding Method 1.
### _Iterative Implementation_
The quality of the bound \(\gamma_{j}\) depends, to a large extent, on the length of the interval \(\mathcal{I}_{j}\), denoted \(|\mathcal{I}_{j}|\). This is clear if we think of \(r_{i,j}(x)\) as a linear approximation on \(\mathcal{I}_{j}\) of the nonlinearity \(g_{i,j}(x)\). Since we have assumed \(g_{i,j}(x)\) is continuous and bounded in \(\mathcal{I}_{j}\), the procedure to build \(r_{i,j}(x)\) in BM1 implies that
(16)
for all \(x\in\mathcal{I}_{j}\). Therefore, if we consider intervals \(\mathcal{I}_{j}\) which are shorter and shorter, then the modified potential function \(V(x;\textbf{y},\textbf{r}_{j})\) will be closer and closer to the true potential function \(V(x;\textbf{y},\textbf{g}_{j})\), and hence the bound \(\gamma_{j}\leq V(x;\textbf{y},\textbf{r}_{j})\leq V(x;\textbf{y},\textbf{g}_{j})\) will be tighter.
The latter observation suggests a procedure to improve the bound \(\gamma_{j}\) for a given interval \(\mathcal{I}_{j}\). Indeed, let us subdivide \(\mathcal{I}_{j}\) into \(k\) subintervals denoted \(\mathcal{I}_{v,v+1}\triangleq[s_{v},s_{v+1}]\) where \(v=1,\ldots,k\) and \(s_{v},s_{v+1}\in\mathcal{I}_{j}\). We refer to the elements in the collection \(\mathcal{S}_{j,k}=\{s_{1},\ldots,s_{k+1}\}\), with \(s_{1}<s_{2}<\ldots<s_{k+1}\), as support points in the interval \(\mathcal{I}_{j}\). We can build linear functions \(\textbf{r}_{j}^{(v)}=[r_{1,j}^{(v)},\ldots,r_{n,j}^{(v)}]\) for every subinterval \(\mathcal{I}_{v,v+1}\), using the procedure described in Section IV-A. We recall that this procedure is graphically depicted in Fig. 1, where we simply need to
* substitute \(\mathcal{I}_{j}\) by \(\mathcal{I}_{v,v+1}\) and
* when the simple estimate \(x_{i,j}\notin\mathcal{I}_{v,v+1}\), substitute \(x_{i,j}\) by \(s_{v}\) (\(x_{i,j}\) by \(s_{v+1}\)) if \(x_{i,j}<s_{v}\) (if \(x_{i,j}>s_{v+1}\), respectively).
Using \(\textbf{r}_{j}^{(v)}\) we compute a bound \(\gamma_{j}^{(v)}\), \(v=1,\ldots,k\), and then select \(\gamma_{j,k}=\min\limits_{v\in\{1,\ldots,k\}}\gamma_{j}^{(v)}\). Note that the subscript \(k\) in \(\gamma_{j,k}\) indicates how many support points have been used to computed the bound in \(\mathcal{I}_{j}\) (which becomes tighter as \(k\) increases). Moreover if we take a new (arbitrary) support point \(s^{*}\) from the subinterval \({\mathcal{I}}_{v^{*},v^{*}+1}\) that contains \(\gamma_{j,k}\), and extend the set of support points with it, \(\mathcal{S}_{j,k+1}=\{s_{1},\ldots,s^{*},\ldots,s_{k+2}\}\) with \(s_{1}<s_{2}<\ldots<s^{*}<\ldots<s_{k+2}\), then we can iterate the proposed procedure and obtain a refined version of the bound, denoted \(\gamma_{j,k+1}\).
The proposed iterative algorithm is described, with detail, in Table II. Note that \(k\) is an iteration index that makes explicit the number of support points \(s_{v}\). If we plug this iterative procedure for the computation of \(\gamma_{j}\) into BM1 (specifically, replacing steps 3 and 4 of Table I), we obtain a new technique that we will hereafter term bounding method 2 (BM2).
As an illustration, Figure 2 shows four steps of the iterative algorithm. In Figure 2 (a) there are two support points \(\mathcal{S}_{j,1}=\{\min{(\mathcal{X}_{j})},\ \max{(\mathcal{X}_{j})}\}\), which yield a single interval \(\mathcal{I}_{1,2}=\mathcal{I}_{j}\). In Figures 2 (b)-(c)-(d), we successively add a point \(s^{*}\) chosen in the interval \(\hat{\mathcal{I}}_{v^{*},v^{*}+1}\) that contains the latest bound. In this example, the point \(s^{*}\) is chosen deterministically as the mean of the extremes of the interval \(\mathcal{I}_{v^{*},v^{*}+1}\).
1\. Start with I1,2=Ij, and Sj,1={min(Xj), max(Xj)}. Let v∗=1 and k=1.
---
2\. Choose an arbitrary interior point s∗ in Iv∗,v∗+1, and update the set of
support points Sj,k=Sj,k−1∪ {s∗}.
3\. Sort Sj,k in ascending order, so that Sj,k={s1,…,sk+1} where s1=min(Xj),
sk+1=(maxXj),
and k+1 is the number of elements of Sj,k.
4\. Build r(v)j(x) for each interval Iv,v+1=[sv, sv+1] with v=1,…,k.
5\. Find γ(v)j=minV(x;y,r(v)j), for v=1,…,k.
6\. Set the refined bound γj,k=minv∈{1,…,k}γ(v)j, and set v∗=argminvγ(v)j.
7\. To iterate, go back to step 2.
TABLE II: Iterative algorithm to improve γj.
<figure><img src="content_image/0904.1300/FirstTime.png"><figcaption>Fig. 2: Four steps of the iterative algorithm choosing s∗ as the middle pointof the subinterval Iv∗,v∗+1. The solid line shows the system potentialV(x;y,g)=(y1−exp(x))2−log(y2−exp(−x)+1)+(y2−exp(−x))+1 (see the example inSection VII-A), with y1=5 and y2=2, while the dashed line shows the modifiedpotential V(x;y,rj). We start in plot (a) with two points Sj,1={min(Xj),max(Xj)}. At each iteration, we add a new point chosen in the subintervalIv∗,v∗+1 that contains the latest bound. It is apparent that V(x;y,rj) becomesa better approximation of V(x;y,gj) each time we add a new support point.</figcaption></figure>
### _Lower bound \(\gamma_{2}\) for quadratic potentials_
Assume that the joint potential is quadratic, i.e., \(V^{(n)}_{2}(y_{1}-g_{1,j}(x),\ldots,y_{n}-g_{n,j}(x))=\sum_{i=1}^{n}(y_{i}-g_{ i,j}(x))^{2}\) for each \(j=1,\ldots,q\), and construct the set of linearities \(r_{i,j}(x)=a_{i,j}x+b_{i,j}\), for \(i=1,\ldots,n\) and \(j=1,\ldots,q\). The modified system potential in \(\mathcal{B}_{j}\) becomes
\[V_{2}(x;\textbf{y},\textbf{r}_{j})=\sum_{i=1}^{n}(y_{i}-r_{i,j}(x))^{2}=\sum_{ i=1}^{n}(y_{i}-a_{i,j}x-b_{i,j})^{2},\] (17)
and it turns out straightforward to compute \(\gamma_{2,j}=\min\limits_{x\in\mathcal{B}_{j}}V(x;\textbf{y},\textbf{r}_{j})\). Indeed, if we denote \(\textbf{a}_{j}=[a_{1,j},\ldots,a_{n,j}]^{\top}\) and \(\textbf{w}_{j}=[y_{1}-b_{1,j},\ldots,y_{n}-b_{n,j}]^{\top}\), then we can readily obtain
\[\tilde{x}_{j}=\arg\min_{x\in\mathcal{B}_{j}}V(x;\textbf{y}, \textbf{r}_{j})=\frac{\textbf{a}_{j}^{\top}\textbf{w}_{j}}{\textbf{a}_{j}^{ \top}\textbf{a}_{j}},\] (18)
and \(\gamma_{2,j}=V(\tilde{x}_{j};\textbf{y},\textbf{r}_{j})\). It is apparent that \(\gamma_{2}=\min\limits_{j}\gamma_{2,j}\leq V(x;\textbf{y},\textbf{g})\). Furthermore, \(\tilde{x}_{j}\) is an approximation of the ML estimator \(\hat{x}_{j}\) restricted to \(\mathcal{B}_{j}\).
### _Adaptation of \(\gamma_{2}\) for generic system potentials_
If the joint potential is not quadratic, in general it can still be difficult to minimize the modified function \(V(x;\textbf{y},\textbf{r})\), despite the replacement of the nonlinearities \(g_{i,j}(x)\) with the linear functions \(r_{i,j}(x)\). In this section, we propose a method to transform the bound for a quadratic potential, \(\gamma_{2}\), into a bound for some other, non-quadratic, potential function.
Consider an arbitrary joint potential \(V^{(n)}\) and assume the availability of an invertible increasing function \(R\) such that \(R\circ V^{(n)}\geq V^{(n)}_{2}\), where \(\circ\) denotes the composition of functions. Then, for the system potential we can write
\[\begin{split}(R\circ V)(x;\textbf{y},\textbf{g})& \geq V_{2}^{(n)}(y_{1}-g_{1}(x),\ldots,y_{n}-g_{n}(x))\\ &=\sum_{i=1}^{n}(y_{i}-g_{i}(x))^{2}\geq\gamma_{2}.\end{split}\] (19)
and, as consequence, \(V(x;\textbf{y},\textbf{g})\geq R^{-1}\left(\gamma_{2}\right)=\gamma\), hence \(\gamma\) is a lower bound for the non-quadratic system potential \(V(x;\textbf{y},\textbf{g})\) constructed from \(V^{(n)}\).
For instance, consider the family of joint potentials \(V^{(n)}_{p}\). Using the monotonicity of \(\mathcal{L}^{p}\) norms, it is possible to prove [16] that
\[\small\left(\sum_{i=1}^{n}\left|\vartheta_{i}\right|^{p}\right)^{\frac{1}{p}} \geq\left(\sum_{i=1}^{n}\vartheta_{i}^{2}\right)^{\frac{1}{2}},\ \mbox{for}\ \ 0\leq p\leq 2,\ \mbox{and}\] (20)
\[\small n^{\left(\frac{p-2}{2p}\right)}\left(\sum_{i=1}^{n}\left|\vartheta_{i} \right|^{p}\right)^{\frac{1}{p}}\geq\left(\sum_{i=1}^{n}\vartheta_{i}^{2} \right)^{\frac{1}{2}},\ \mbox{for}\ \ 2\leq p\leq+\infty.\] (21)
Let \(R_{1}(v)=v^{2/p}\). Since this function is, indeed, strictly increasing, we can transform the inequality (20) into
\[R_{1}\left(\sum_{i=1}^{n}|y_{i}-g_{i}(x)|^{p}\right)\geq\sum_{i=1}^{n}(y_{i}-g _{i}(x))^{2},\] (22)
which yields
(23)
hence the transformation \(\gamma_{2}^{p/2}\) of the quadratic bound \(\gamma_{2}\) is a lower bound for \(V_{p}^{(n)}\) with \(0<p\leq 2\). Similarly, if we let \(R_{2}(v)=\left(n^{\left(\frac{p-2}{2p}\right)}v^{1/p}\right)^{2}\), the inequality (21) yields
\[\sum_{i=1}^{n}|y_{i}-g_{i}(x)|^{p}\geq R_{2}^{-1}\left(\sum_{i=1}^{n}(y_{i}-g_ {i}(x))^{2}\right)=\left[n^{\left(-\frac{p-2}{2p}\right)}\left(\sum_{i=1}^{n}( y_{i}-g_{i}(x))^{2}\right)^{1/2}\right]^{p}\geq n^{\left(-\frac{p-2}{2}\right) }\gamma_{2}^{p/2},\] (24)
hence the transformation \(R_{2}^{-1}(\gamma_{2})=n^{-(p-2)/2}\gamma_{2}^{p/2}\) is a lower bound for \(V_{p}^{(n)}\) when \(2\leq p<+\infty\).
It is possible to devise a systematic procedure to find a suitable function \(R\) given an arbitrary joint potential \(V^{(n)}({\mbox{\boldmath$\vartheta$}})\), where \({\mbox{\boldmath$\vartheta$}}\triangleq[\vartheta_{1},\ldots,\vartheta_{n}]^{T}\). Let us define the manifold \(\Gamma_{v}\triangleq\left\{{\mbox{\boldmath$\vartheta$}}\in\mathbb{R}^{n}:\ \ V^{(n)}({\mbox{\boldmath$\vartheta$}})=v\right\}\). We can construct \(R\) by assigning \(R(v)\) with the maximum of the quadratic potential \(\sum_{i}^{n}\vartheta_{i}^{2}\) when \({\mbox{\boldmath$\vartheta$}}\in\Gamma_{v}\), i.e., we define
\[\small R(v)\triangleq\max_{{\mbox{\boldmath$\vartheta$}}\in\Gamma_{v}}\sum_{i= 1}^{n}\vartheta_{i}^{2}.\] (25)
Note that (25) is a constrained optimization problem that can be solved using, e.g., Lagrangian multipliers.
From the definition in (25) we obtain that, \(\forall\textbf{{\mbox{\boldmath$\vartheta$}}}\in\Gamma_{v}\), \(R(v)\geq\sum_{i=1}^{n}\vartheta_{i}^{2}\). In particular, since \(V^{(n)}({\mbox{\boldmath$\vartheta$}})=v\) from the definition of \(\Gamma_{v}\), we obtain the desired relationship,
\[\small R\left(V^{(n)}(\vartheta_{1},\ldots,\vartheta_{n})\right)\geq\sum_{i=1} ^{n}\vartheta_{i}^{2}.\] (26)
We additionally need to check whether \(R\) is a strictly increasing function of \(v\). The two functions in the earlier examples of this Section, \(R_{1}\) and \(R_{2}\), can be readily found using this method.
### _Convex marginal potentials \(\bar{V}_{i}\)_
Assume that \(\mathcal{A}=\left\{\mathcal{B}_{j}\right\}_{j=1}^{q}\) and that we have already found \(r_{i,j}(x)=a_{i,j}x+b_{i,j}\), \(i=1,\ldots,n\) and \(j=1,\ldots,q\), using the technique in Section IV-A. If a marginal potential \(\bar{V}_{i}(\vartheta_{i})\) is convex, the function \(\bar{V}_{i}(y_{i}-r_{i,j}(x))\) is also convex in \(\mathcal{B}_{j}\). Indeed, for all \(x\in\mathcal{B}_{j}\)
\[\frac{d^{2}\bar{V}_{i}(y_{i}-r_{i,j}(x))}{dx^{2}}=\frac{d^{2}r_{i,j}}{dx^{2}} \frac{d\bar{V}_{i}}{d\vartheta_{i}}+\left(\frac{dr_{i,j}}{dx}\right)^{2}\frac{ d^{2}\bar{V}_{i}}{d\vartheta_{i}^{2}}=0+a_{i}^{2}\frac{d^{2}\bar{V}_{i}}{d \vartheta_{i}^{2}}\geq 0\] (27)
where we have used that \(\frac{d^{2}r_{i,j}}{dx^{2}}=0\) (since \(r_{i,j}\) is linear).
As a consequence, if all marginal potentials \(\bar{V}_{i}(\vartheta_{i})\) are convex, then the modified system potential, \(V(x;\textbf{y},\textbf{r}_{j})=c_{n}+\sum_{i=1}^{n}\bar{V}_{i}(y_{i}-r_{i,j}(x))\), is also convex in \(\mathcal{B}_{j}\). This is easily shown using (27), to obtain
\[\frac{d^{2}V(x;\textbf{y},\textbf{r}_{j})}{dx^{2}}=\sum_{i=1}^{n}a_{i}^{2} \frac{d^{2}\bar{V}_{i}}{d\vartheta_{i}^{2}}\geq 0,\ \ \forall x\in\mathcal{B}_ {j}.\] (28)
Therefore, we can use the tangents to \(V(x;\textbf{y},\textbf{r}_{j})\) at the limit points of \(\mathcal{I}_{j}\) (i.e, \(\min(\mathcal{X}_{j})\) and \(\max(\mathcal{X}_{j})\)) to find a lower bound for the system potential \(V(x;\textbf{y},\textbf{g}_{j})\). Figure 3 (left) depicts a system potential \(V(x;\textbf{y},\textbf{g}_{j})\) (solid line), the corresponding modified potential \(V(x;\textbf{y},\textbf{r}_{j})\) (dotted line) and the two tangent lines at \(\min(\mathcal{X}_{j})\) and \(\max(\mathcal{X}_{j})\). It is apparent that the intersection of the two tangents yields a lower bound in \(\mathcal{B}_{j}\). Specifically, if we let \(W(x)\) be the piecewise-linear function composed of the two tangents, then the inequality \(V(x;\textbf{y},\textbf{g}_{j})\geq V(x;\textbf{y},\textbf{r}_{j})\geq W(x)\) is satisfied for all \(x\in\mathcal{I}_{j}\).
## V Adaptive Rejection Sampling
The adaptive rejection sampling (ARS) [8] algorithm enables the construction of a sequence of proposal densities, \(\left\{\pi_{t}(x)\right\}_{t\in\mathbb{N}}\), and bounds tailored to the target density. Its most appealing feature is that each time we draw a sample from a proposal \(\pi_{t}\) and it is rejected, we can use this sample to build an improved proposal, \(\pi_{t+1}\), with a higher mean acceptance rate.
Unfortunately, this attractive ARS method can only be applied with target pdf’s which are log-concave (hence, unimodal), which is a very stringent constraint for may practical applications. Next, we briefly review the ARS algorithm and then proceed to introduce its extension for non-log-concave and multimodal target densities.
Let \(p(x|\textbf{y})\) denote the target pdf¹. The ARS procedure can be applied when \(\log[p(x|\textbf{y})]\) is concave, i.e., when the potential function \(V(x;\textbf{y},\textbf{g})\triangleq-\log[p(x|\textbf{y})]\) is strictly convex. Let \(\mathcal{S}_{t}=\{s_{1},s_{2},\ldots,s_{k_{t}}\}\) be a set of support points in the domain \(D\) of \(V(x;\textbf{y},\textbf{g})\). From \(\mathcal{S}_{t}\) we build a piecewise-linear lower hull of \(V(x;\textbf{y},\textbf{g})\), denoted \(W_{t}(x)\), formed from segments of linear functions tangent to \(V(x;\textbf{y},\textbf{g})\) at the support points in \(\mathcal{S}_{t}\). Figure 3 (center) illustrates the construction of \(W_{t}(x)\) with three support points for a generic log-concave potential function \(V(x;\textbf{y},\textbf{g})\).
[FOOTNOTE:1][ENDFOOTNOTE]
Once \(W_{t}(x)\) is built, we can use it to obtain an exponential-family proposal density
\[\pi_{t}(x)=c_{t}\exp[-W_{t}(x)],\] (29)
where \(c_{t}\) is the proportionality constant. Therefore \(\pi_{t}(x)\) is piecewise-exponential and very easy to sample from. Since \(W_{t}(x)\leq V(x;\textbf{y},\textbf{g})\), we trivially obtain that \(\frac{1}{c_{t}}\pi(x)\geq p(x|\textbf{y})\) and we can apply the RS principle.
When a sample \(x^{\prime}\) from \(\pi_{t}(x)\) is rejected we can incorporate it into the set of support points, \(\mathcal{S}_{t+1}=S_{t}\cup\{x^{\prime}\}\) (and \(k_{t+1}=k_{t}+1\)). Then we compute a refined lower hull, \(W_{t+1}(x)\), and a new proposal density \(\pi_{t+1}(x)=c_{t+1}\exp\{-W_{t+1}(x)\}\). Table III summarizes the ARS algorithm.
1\. Start with t=0, S0={s1, s2} where s1<s2, and the derivatives of V(x,y,{g})
in s1,s2∈D having different signs.
---
2\. Build the piecewise-linear function Wt(x) as shown in Figure 3 (center),
using the tangent lines to V(x;y,g)
at the support points St.
3\. Sample x′ from πt(x)∝exp{−Wt(x)}, and u′ from U([0,1]).
4\. If u′≤p(x′|y)exp[−Wt(x′)] accept x′ and set St+1=St, kt+1=kt.
5\. Otherwise, if u′>p(x′|y)exp[−Wt(x′)], reject x′, set St+1=St∪{x′} and
update kt+1=kt+1.
6\. Sort St+1 in ascending order, increment t and go back to step 2.
TABLE III: Adaptive Rejection Sampling Algorithm.
## VI Generalization of the ARS Method
In this section we introduce a generalization of the standard ARS scheme that can cope with a broader class of target pdf’s, including many multimodal distributions. The standard algorithm of [8], described in Table III, is a special case of the method described below.
### _Generalized adaptive rejection sampling_
We wish to draw samples from the posterior \(p(x|\textbf{y})\). For this purpose, we assume that
* all marginal potential functions, \(\bar{V}_{i}(\vartheta_{i})\), \(i=1,\ldots,n\), are strictly convex,
* the prior pdf has the form \(p(x)\propto\exp\{-\bar{V}_{n+1}(\mu-x)\}\), where \(\bar{V}_{n+1}\) is also a convex marginal potential with its mode located at \(\mu\), and
* the nonlinearities \(g_{i}(x)\) are either convex or concave, not necessarily monotonic.
We incorporate the information of the prior by defining an extended observation vector, \(\tilde{\textbf{y}}\triangleq[y_{1},\ldots,y_{n},y_{n+1}=\mu]^{\top}\), and an extended vector of nonlinearities, \(\tilde{\textbf{g}}(x)\triangleq[g_{1}(x),\ldots,g_{n}(x),g_{n+1}(x)=x]^{\top}\). As a result, we introduce the extended system potential function
\[V(x;\tilde{\textbf{y}},\tilde{\textbf{g}})\triangleq V(x;\textbf{y},\textbf{g} )+\bar{V}_{n+1}(\mu-x)=-\log[p(x|\textbf{y})]+c_{0},\] (30)
where \(c_{0}\) accounts for the superposition of constant terms that do not depend on \(x\). We remark that the function \(V(x;\tilde{\textbf{y}},\tilde{\textbf{g}})\) constructed in this way is not necessarily convex. It can present several minima and, as a consequence, \(p(x|\textbf{y})\) can present several maxima.
Our technique is adaptive, i.e., it is aimed at the construction of a sequence of proposals, denoted \(\pi_{t}(x)\), \(t\in\mathbb{N}\), but relies on the same basic arguments already exploited to devise the BM1. To be specific, at the \(t\)-th iteration of the algorithm we seek to replace the nonlinearities \(\{g_{i}\}_{i=1}^{n+1}\) by piecewise-linear functions \(\{r_{i,t}\}_{i=1}^{n+1}\) in such a way that the inequalities
\[\left|y_{i}-r_{i,t}(x)\right|\leq\left|y_{i}-g_{i}(x)\right|\ \ \mbox{and}\] (31)
\[(y_{i}-r_{i,t}(x))(y_{i}-g_{i}(x))\geq 0\] (32)
are satisfied \(\forall x\in\mathbb{R}\). Therefore, we repeat the same conditions as in Eqs. (12)-(13) but the derivation of the generalized ARS (GARS) algorithm does not require the partition of the SoI space, as it was needed for the BM1.
We will show that it is possible to construct adequate piecewise-linear functions of the form
\[r_{i,t}(x)\triangleq\left\{\begin{array}[]{l}\max[\bar{r}_{i,1}(x),\ldots,\bar {r}_{i,K_{t}}(x)],\ \ \mbox{if}\ g_{i}\ \mbox{is convex}\\ \min[\bar{r}_{i,1}(x),\ldots,\bar{r}_{i,K_{t}}(x)],\ \ \mbox{if}\ g_{i}\ \mbox {is concave}\\ \end{array}\right.\] (33)
where \(i=1,\ldots,n\) and each \(\bar{r}_{i,j}(x)\), \(j=1,\ldots,K_{t}\), is a purely linear function. The number of linear functions involved in the construction of \(r_{i,t}(x)\) at the \(t\)-th iteration of the algorithm, denoted \(K_{t}\), determines how tightly \(\pi_{t}(x)\) approximates the true density \(p(x|\textbf{y})\) and, therefore, the higher \(K_{t}\), the higher expected acceptance rate of the sampler. In Section VI-B below, we explicitly describe how to choose the linearities \(\bar{r}_{i,j}(x)\), \(j=1,\ldots,K_{t}\), in order to ensure that (31) and (32) hold. We will also show that, when a proposed sample \(x^{\prime}\) is rejected, \(K_{t}\) can be increased (\(K_{t+1}=K_{t}+1\)) to improve the acceptance rate.
Let \(\tilde{\textbf{r}}_{t}\triangleq[r_{1,t}(x),\ldots,r_{n,t}(x),r_{n+1,t}(x)=x]^ {\top}\) be the extended vector of piecewise-linear functions, that yields the modified potential \(V(x;\tilde{\textbf{y}},\tilde{\textbf{r}}_{t})\). The same argument used in Section IV-A to derive the BM1 shows that, if (31) and (32) hold, then \(V(x;\tilde{\textbf{y}},\tilde{\textbf{r}}_{t})\leq V(x;\tilde{\textbf{y}}, \tilde{\textbf{g}})\), \(\forall x\in\mathbb{R}\). Finally, we build a piecewise-linear lower hull \(W_{t}(x)\) for the modified potential, as explained below, to obtain \(W_{t}(x)\leq V(x;\tilde{\textbf{y}},\tilde{\textbf{r}}_{t})\leq V(x;\tilde{ \textbf{y}},\tilde{\textbf{g}})\).
The definition of the piecewise-linear function \(r_{i,t}(x)\) in (33) can be rewritten in another form
\[r_{i,t}(x)\triangleq\bar{r}_{i,j}(x)\ \ \mbox{for}\ \ x\in[a,b]\] (34)
where \(a\) is the abscissa of the intersection between the linear functions \(\bar{r}_{i,j-1}(x)\) and \(\bar{r}_{i,j}(x)\), and \(b\) is the abscissa of the intersection between \(\bar{r}_{i,j}(x)\) and \(\bar{r}_{i,j+1}(x)\). Therefore, we can define the set of all abscissas of intersection points
\[\mathcal{E}_{t}=\{u\in\mathbb{R}:\ \bar{r}_{i,j}(u)=\bar{r}_{i,j+1}(u)\ \ \mbox{for}\ i=1,\ldots,n+1,\ j=1,\ldots,K_{t}-1\},\] (35)
and sort them in ascending order
\[u_{1}<u_{2}<\ldots<u_{Q}\] (36)
where \(Q\) is the total number of intersections. Then
* since we have assumed that the marginal potentials are convex, we can use Eq. (34) and the argument of Section IV-E to show that the modified function \(V(x;\tilde{\textbf{y}},\tilde{\textbf{r}}_{t})\) is convex in each interval \([u_{q},u_{q+1}]\), with \(q=1,\ldots,Q\), and,
* as a consequence, we can to build \(W_{t}(x)\) by taking the linear functions tangent to \(V(x;\tilde{\textbf{y}},\tilde{\textbf{r}}_{t})\) at every intersection point \(u_{q}\), \(q=1,\ldots,Q\).
Fig. 3 (right) depicts the relationship among \(V(x;\tilde{\textbf{y}},\tilde{\textbf{g}})\), \(V(x;\tilde{\textbf{y}},\tilde{\textbf{r}}_{t})\) and \(W_{t}(x)\). Since \(W_{t}(x)\) is piecewise linear, the corresponding pdf \(\pi_{t}(x)\propto\exp\{-W_{t}(x)\}\) is piecewise exponential and can be easily used in a rejection sampler (we remark that \(W_{t}(x)\leq V(x;\tilde{\textbf{y}},\tilde{\textbf{g}})\), hence \(\pi_{t}(x)\propto\exp\{-W_{t}(x)\}\geq\exp\{-V(x;\tilde{\textbf{y}},\tilde{ \textbf{g}})\}\propto p(x|\textbf{y})\)).
Next subsection is devoted to the derivation of the linear functions needed to construct \(\tilde{\textbf{r}}_{t}\). Then, we describe how the algorithm is iterated to obtain a sequence of improved proposal densities and provide a pseudo-code. Finally, we describe a limitation of the procedure, that yields improper proposals in a specific scenario.
<figure><img src="content_image/0904.1300/marginalConvex2.png"><figcaption>Fig. 3: Left: The intersection of the tangents to V(x;y,rj) (dashed line) atmin(Xj) and max(Xj) is a lower bound for V(x;y,gj) (solid line). Moreover,note that the resulting piecewise-linear function W(x) satisfies theinequality V(x;y,gj)≥V(x;y,rj)≥W(x), for all x∈Ij. Center: Example ofconstruction of the piecewise-linear function Wt(x) with 3 support pointsSt={s1,s2,s3}, as carried out in the ARS technique. The function Wt(x) isformed from segments of linear functions tangent to V(x;y,g) at the supportpoints in St. Right: Construction of the piecewise linear function Wt(x) astangent lines to the modified potential V(x;~y,~rt) at three intersectionspoints u1, u2 and u3, as carried out in the ARS technique.</figcaption></figure>
### _Construction of linear functions \(\bar{r}_{i,j}(x)\)_
A basic element in the description of the GARS algorithm in the previous section is the construction of the linear functions \(\bar{r}_{i,j}(x)\). This issue is addressed below. For clarity, we consider two cases corresponding to non-monotonic and monotonic nonlinearities, respectively. It is important to remark that the nonlinearities \(g_{i}(x)\), \(i=1,\ldots,n\) (remember that \(g_{n+1}(x)=x\) is linear), can belong to different cases.
#### Vi-B1 Non-monotonic nonlinearities
Assume \(g_{i}(x)\) is a non-monotonic, either concave or convex, function. We have three possible scenarios depending on the number of simple estimates for \(g_{i}(x)\): (a) there exist two simple estimates, \(x_{i,1}<x_{i,2}\), (b) there exists a single estimate, \(x_{i,1}=x_{i,2}\), or (c) there is no solution for the equation \(y_{i}=g_{i}(x)\).
Let us assume that \(x_{i,1}<x_{i,2}\) and denote \(\mathcal{J}_{i}\triangleq[x_{i,1},x_{i,2}]\). Let us also introduce a set of support points \(\mathcal{S}_{t}\triangleq\{s_{1},\ldots,s_{k_{t}}\}\) that contains at least the simple estimates and an arbitrary point \(s\in\mathcal{J}_{i}\), i.e., \(x_{i,1},x_{i,2}\in\mathcal{S}_{t}\). The number of support points, \(k_{t}\), determines the accuracy of the approximation of the nonlinearity \(g_{i}(x)\) that can be achieved with the piecewise-linear function \(r_{i,t}(x)\). In Section VI-C we show how this number increases as the GARS algorithm iterates. Now, we assume it is given and fixed.
Figure 4 illustrates the construction of \(\bar{r}_{i,j}(x)\), \(j=1,\ldots,K_{t}\) where \(K_{t}=k_{t}-1\), and \(r_{i,t}(x)\) for a convex nonlinearity \(g_{i}(x)\) (the procedure is completely analogous for concave \(g_{i}(x)\)). Assume that the two simple estimates \(x_{i,1}<x_{i,2}\) exist, hence \(|\mathcal{J}_{i}|>0\). For each \(j\in\{1,\ldots,k_{t}\}\), the linear function \(\bar{r}_{i,j}(x)\) is constructed in one out of two ways:
* if \([s_{j},s_{j+1}]\subseteq\mathcal{J}_{i}\), then \(\bar{r}_{i,j}(x)\) connects the points \((s_{j},g_{i}(s_{j}))\) and \((s_{j+1},g_{i}(s_{j+1}))\), else
* if \(s_{j}\notin\mathcal{J}_{i}\), then \(\bar{r}_{i,j}(x)\) is tangent to \(g_{i}(x)\) at \(x=s_{j}\).
From Fig. 4 (left and center) it is apparent that \(r_{i,t}(x)=\max[\bar{r}_{i,1}(x),\ldots,\bar{r}_{i,K_{t}}(x)]^{\top}\) built in this way satisfies the inequalities (31) and (32), as required. For concave \(g_{i}(x)\), (31) and (32) are satisfied if we choose \(r_{i,t}(x)=\min[\bar{r}_{i,1}(x),\ldots,\bar{r}_{i,K_{t}}(x)]^{\top}\).
When \(|\mathcal{J}_{i}|=0\) (i.e., \(x_{i,1}=x_{i,2}\) or there is no solution for the equation \(y_{i}=g_{i}(x)\)), then each \(\bar{r}_{i,j}(x)\) is tangent to \(g_{i}(x)\) at \(x=s_{j}\), \(\forall s_{j}\in\mathcal{S}_{t}\), and in order to satisfy (31) and (32), we need to select
\[r_{i,t}(x)\triangleq\left\{\begin{array}[]{l}\max[\bar{r}_{i,1}(x),\ldots,\bar {r}_{i,K_{t}}(x),y_{i}],\ \ \mbox{if}\ g_{i}\ \mbox{is convex}\\ \min[\bar{r}_{i,1}(x),\ldots,\bar{r}_{i,K_{t}}(x),y_{i}],\ \ \mbox{if}\ g_{i} \ \mbox{is concave}\\ \end{array}\right.\] (37)
as illustrated in Fig. 4 (right).
<figure><img src="content_image/0904.1300/LowerBFunction1change.png"><figcaption>Fig. 4: Construction of the piecewise linear function ri,t(x) for non-monotonic functions. The straight lines ¯ri,j(x) form a piecewise linearfunction that is closer to the observation value yi (dashed line) than thenonlinearity gi(x), i.e., |yi−ri,t(x)|≤|yi−gi(x)|. Moreover, ri,t(x) and gi(x)are either simultaneously greater than (or equal to) yi, or simultaneouslylesser than (or equal to) yi, i.e., (yi−ri,t(x))(yi−gi(x))≥0. Therefore, theinequalities (31) and (32) are satisfied. The point (sj,gi(sj)), correspondingto support point sj, is represented either by a square or a circle, dependingon whether it is a simple estimate or not, respectively. Left: construction ofri,t(x) with kt=4 support points when the nonlinearity gi(x) is convex,therefore ri,t(x)=max[¯ri,1(x),…,¯ri,3(x)] (Kt=kt−1=3). We use the tangent togi(x) at x=s4 because s4∉Ji=[s1,s3], where s1=xi,1 and s3=xi,2 are the simpleestimates (represented with squares). Center: since the nonlinearity gi(x) isconcave, ri,t(x)=min[¯ri,1(x),…,¯ri,3(x)]. We use the tangent to gi(x) at s4because s1∉Ji=[s2,s4], where s2=xi,1 and s4=xi,2 are the simple estimates(represented with squares). Right: construction of the ri,t(x), with twosupport points, when there are not simple estimates. We use the tangent lines,but we need a correction in the definition of ri,t(x) in order to satisfy theinequalities (31) and (32). Since gi(x) in the figure is convex, we takeri,t(x)=max[¯ri,1(x),…,¯ri,Kt(x),yi].</figcaption></figure>
#### Vi-B2 Monotonic nonlinearities
In this case \(g_{i}(x)\) is invertible and there are two possibilities: there exists a single estimate, \(x_{i}=g_{i}^{-1}(y_{i})\), or there is no solution for the equation \(y_{i}=g_{i}(x)\) (where \(y_{i}\) does not belong to the range of \(g_{i}(x)\)). Similarly to the construction in Section IV-A, we distinguish two cases:
* if \(\frac{dg_{i}(x)}{dx}\times\frac{d^{2}g_{i}(x)}{dx^{2}}\geq 0\), then we define \(\mathcal{J}_{i}\triangleq(-\infty,x_{i}]\), and
* if \(\frac{dg_{i}(x)}{dx}\times\frac{d^{2}g_{i}(x)}{dx^{2}}\leq 0\), then we define \(\mathcal{J}_{i}\triangleq[x_{i},+\infty)\).
The set of support points is \(\mathcal{S}_{t}\triangleq\{s_{1},\ldots,s_{k_{t}}\}\), with \(s_{1}<s_{2}\ldots<s_{k_{t}}\), and includes at least the simple estimate \(x_{i}\) and an arbitrary point \(s\in\mathcal{J}_{i}\), i.e., \(x_{i},s\in\mathcal{S}_{t}\).
The procedure to build \(\bar{r}_{i,j}(x)\), for \(j=1,\ldots,K_{t}\), with \(K_{t}=k_{t}\), is similar to Section VI-B1. Consider case (a) first. For each \(j\in\{2,\ldots,k_{t}\}\), if \([s_{j-1},s_{j}]\subset\mathcal{J}_{i}=(-\infty,x_{i}]\), then \(\bar{r}_{i,j}(x)\) is the linear function that connects the points \((s_{j-1},g_{i}(s_{j-1}))\) and \((s_{j},g_{i}(s_{j}))\). Otherwise, if \(s_{j}\notin\mathcal{J}_{i}=(-\infty,x_{i}]\), \(\bar{r}_{i,j}(x)\) is tangent to \(g_{i}(x)\) at \(x=s_{j}\). Finally, we set \(\bar{r}_{i,1}(x)=g_{i}(s_{1})\) for all \(x\in\mathbb{R}\). The piecewise linear function \(r_{i,t}\) is \(r_{i,t}(x)=\max[\bar{r}_{i,1}(x),\ldots,\bar{r}_{i,K_{t}}(x)]\). This construction is depicted in Fig. 5 (left).
Case (b) is similar. For each \(j\in\{1,\ldots,k_{t}\}\), if \([s_{j},s_{j+1}]\subset\mathcal{J}_{i}=[x_{i},+\infty)\), then \(\bar{r}_{i,j}(x)\) is the linear function that connects the points \((s_{j},g_{i}(s_{j}))\) and \((s_{j+1},g_{i}(s_{j+1}))\). Otherwise, if \(s_{j}\notin\hat{\mathcal{I}}_{i}=[x_{i},+\infty)\), \(\bar{r}_{i,j}(x)\) is tangent to \(g_{i}(x)\) at \(x=s_{j}\). Finally, we set \(\bar{r}_{i,k_{t}}(x)=g_{i}(s_{k_{t}})\) (remember that, in this case, \(K_{t}=k_{t}\)), for all \(x\in\mathbb{R}\). The piecewise linear function \(r_{i,t}\) will be \(r_{i,t}(x)=\min[\bar{r}_{i,1}(x),\ldots,\bar{r}_{i,K_{t}}(x)]\). This construction is depicted in Fig. 5 (right).
It is straightforward to check that the inequalities (31) and (32) are satisfied. Note that, in this case, the number of linear functions \(\bar{r}_{i,j}(x)\) coincides with the number of support points. If there is not solution for the equation \(y_{i}=g_{i}(x)\) (\(y_{i}\) does not belong to the range of \(g_{i}(x)\)), then (31) and (32) are satisfied if we use (37) to build \(r_{i,t}(x)\).
<figure><img src="content_image/0904.1300/LowerBFunction4.png"><figcaption>Fig. 5: Examples of construction of the piecewise-linear function ri,t(x) withkt=3 support points sj, for the two subcases. It is apparent that|yi−ri,t(x)|≤|yi−gi(x)| and that ri,t(x) and gi(x) are either simultaneouslygreater than (or equal to) yi, or simultaneously lesser than (or equal to) yi,i.e., (yi−ri,t(x))(yi−gi(x))≥0. The simple estimates are represented bysquares while all other support points are drawn as circles. Left: the figurecorresponds to the subcase 1 where ri,t(x)=max[¯ri,1(x),…,¯ri,3(x)] (Kt=kt=3).Right: the figure corresponds to to the subcase 2 whereri,t(x)=min[¯ri,1(x),…,¯ri,3(x)] (Kt=kt=3).</figcaption></figure>
### _Summary_
We can combine the elements described in Sections VI-B1 and VI-B2 into an adaptive algorithm that improves the proposal density \(\pi_{t}(x)\propto\exp\{-W_{t}(x)\}\) each time a sample is rejected.
Let \(\mathcal{S}_{t}\) denote the set of support points after the \(t\)-th iteration. We initialize the algorithm with \(\mathcal{S}_{0}\triangleq\left\{s_{j}\right\}_{j=1}^{k_{0}}\) such that
* all simple estimates are contained in \(\mathcal{S}_{0}\), and
* for each interval \(\mathcal{J}_{i}\), \(i=1,\ldots,n+1\) , with non-zero length (\(|\mathcal{J}_{i}|>0\)), there is at least one (arbitrary) support point contained in \(\mathcal{J}_{i}\).
The proposed GARS algorithm is described in Table IV. Note that every time a sample \(x^{\prime}\) drawn from \(\pi_{t}(x)\) is rejected, \(x^{\prime}\) is incorporated as a support point in the new set \(\mathcal{S}_{t+1}=\mathcal{S}_{t}\cup\{x^{\prime}\}\) and, as a consequence, a refined lower hull \(W_{t+1}(x)\) is constructed yielding a better approximation of the system potential function. In this way, \(\pi_{t+1}(x)\propto\exp\{-W_{t+1}(x)\}\) becomes closer to \(p(x|\textbf{y})\) and it can be expected that the acceptance rate be higher. This is specifically shown in the simulation example in Section VII-B.
1\. Start with t=0 set S0≜{sj}k0j=1.
---
2\. Build ¯ri,j(x) for i=1,…,n+1, j=1,…,Kt, where Kt=kt−1 or Kt=kt depending
on whether gi(x) is
non-monotonic or monotonic, respectively.
3\. Calculate the set of intersection points Et≜{u∈R: ¯ri,j(u)=¯ri,j+1(u) for
i=1,…,n+1, j=1,…,Kt−1}.
Let Q=|Et| be the number of elements in Et.
4\. Build Wt(x) using the tangent lines to V(x;~y,~rt) at the points uq∈Et,
q=1,…,Q.
5\. Draw a sample x′ from πt(x)∝exp[−Wt(x)].
6\. Sample u′ from U([0,1]).
7\. If u′≤p(x′|y)exp[−Wt(x′)] accept x′ and set St+1=St.
8\. Otherwise, if u′>p(x′|y)exp[−Wt(x′)] reject x′ and update St+1=St∪{x′}.
9\. Sort St+1 in ascending order, set t=t+1 and go back to step 2.
TABLE IV: Steps of Generalized Adaptive Rejection Sampling.
### _Improper proposals_
The GARS algorithm as described in Table IV breaks down when every \(g_{i}(x)\), \(i=1,\ldots,n+1\), is nonlinear and convex (or concave) monotonic. In this case, the proposed construction procedure yields a piecewise lower hull \(W_{t}(x)\) which is positive and constant in an interval of infinite length. Thus, the resulting proposal, \(\pi_{t}(x)\propto\exp\{-W_{t}(x)\}\) is improper (\(\int_{-\infty}^{+\infty}\pi_{t}(x)dx\rightarrow+\infty\)) and cannot be used for RS. One practical solution is to substitute the constant piece of \(W_{t}(x)\) by a linear function with a small slope. In that case, \(\pi_{t}(x)\) is proper but we cannot guarantee that the samples drawn using the GARS algorithm come exactly from the target pdf. Under the assumptions in this paper, however, \(g_{n+1}(x)=x\) is linear (due to our choice of the prior pdf), and this is enough to guarantee that \(\pi_{t}(x)\) be proper.
## VII Examples
### _Example 1: Calculation of upper bounds for the likelihood function_
Let \(X\) be a scalar SoI with prior density \(X\sim p(x)=N(x;0,2)\) and the random observations
\[Y_{1}=\exp{(X)}+\Theta_{1},\ \ Y_{2}=\exp{(-X)}+\Theta_{2},\] (38)
where \(\Theta_{1}\), \(\Theta_{2}\) are independent noise variables. Specifically, \(\Theta_{1}\) is Gaussian noise with \(N(\vartheta_{1};0,1/2)=k_{1}\exp\left\{-(\vartheta_{1})^{2}\right\}\), and \(\Theta_{2}\) has a gamma pdf, \(\Theta_{2}\sim\Gamma(\vartheta_{2};\theta,\lambda)=k_{2}\vartheta_{2}^{\theta- 1}\exp\left\{-\lambda\vartheta_{2}\right\}\), with parameters \(\theta=2,\lambda=1\).
The marginal potentials are \(\bar{V}_{1}(\vartheta_{1})=\vartheta_{1}^{2}\) and \(\bar{V}_{2}(\vartheta_{2})=-\log(\vartheta_{2})+\vartheta_{2}\). Since the minimum of \(\bar{V}_{2}(\vartheta_{2})\) occurs in \(\vartheta_{2}=1\), we replace \(Y_{2}\) with the shifted observation \(Y_{2}^{*}=\exp{(-X)}+\Theta_{2}^{*}\), where \(Y_{2}^{*}=Y_{2}-1\), \(\Theta_{2}^{*}=\Theta_{2}-1\). Hence, the marginal potential becomes \(\bar{V}_{2}(\vartheta_{2}^{*})=-\log(\vartheta_{2}^{*}+1)+\vartheta_{2}^{*}+1\), with a minimum at \(\vartheta_{2}^{*}=0\), the vector of observations is \(\textbf{Y}=[Y_{1},Y_{2}^{*}]^{\top}\) and the vector of nonlinearities is \(\textbf{g}(x)=[\exp{(x)},\exp{(-x)}]^{\top}\). Due to the monotonicity and convexity of \(g_{1}\) and \(g_{2}\), we can work with a partition of \(\mathbb{R}\) consisting of just one set, \(\mathcal{B}_{1}\equiv\mathbb{R}\). The joint potential is \(V^{(2)}(\vartheta_{1},\vartheta_{2}^{*})=\sum_{i=1}^{2}\bar{V}_{i}(\vartheta_{ i})=\vartheta_{1}^{2}-\ln(\vartheta_{2}^{*}+1)+\vartheta_{2}^{*}+1\) and the system potential is
\[\begin{split} V(x;\textbf{y},\textbf{g})& =V^{(2)}(y_{1}-\exp{(x)},y_{2}^{*}-\exp{(-x)})=\\ &=(y_{1}-\exp{(x)})^{2}-\log(y_{2}^{*}-\exp{(-x)}+1)+(y_{2}^{*}- \exp{(-x)})+1.\end{split}\] (39)
Assume that, \(\textbf{Y}=\textbf{y}=[2,5]^{\top}\). The simple estimates are \(\mathcal{X}=\{x_{1}=\log(2),x_{2}=-\log(5)\}\), and, therefore, we can restrict the search of the bound to the interval \(\mathcal{I}=[\min(\mathcal{X})=-\log(5),\max(\mathcal{X})=\log(2)]\) (note that we omit the subscript because we have just one set, \(\mathcal{B}_{1}\equiv\mathbb{R}\)). Using the BM1 technique in Section IV-A, we find the linear functions \(r_{1}(x)=0.78x+1.45\) and \(r_{2}(x)=-1.95x+1.85\).
In this case, we can analytically minimize the modified system potential, to obtain \(\tilde{x}=-0.4171=\arg\min\limits_{x\in\mathcal{I}}V(x,\textbf{y},\textbf{r})\). The associated lower bound is \(\gamma=V(\tilde{x},\textbf{y},\textbf{r})=2.89\) (the true global minimum of the system potential is \(3.78\)). We can also use the technique in Section IV-D with \(R^{-1}(v)=-\log(\sqrt{v}+1)+\sqrt{v}+1\). The lower bound for the quadratic potential is \(\gamma_{2}=2.79\) and we can readily compute a lower bound \(\gamma=R^{-1}(\gamma_{2})=1.68\) for \(V(x;\textbf{y},\textbf{g})\). Since the marginal potentials are both convex, we can also use the procedure described in Section IV-E, obtaining the lower bound \(\gamma=1.61\).
Figure 6 (a) depicts the system potential \(V(x;\textbf{y},\textbf{g})\), and the lower bounds obtained with the three methods. It is the standard BM1 algorithm that yields the best bound.
In order to improve the bound, we can use the iterative BM2 technique described in Section IV-B. With only \(3\) iterations of BM2, and minimizing analytically the modified potential \(V(x,\textbf{y},\textbf{r})\), we find a very tight lower bound \(\gamma=\min\limits_{x\in\mathcal{I}}(V(x,\textbf{y},\textbf{r}))=3.77\) (recall that the optimal bound is \(3.78\)). Table V summarizes the bounds computed with the different techniques.
Next, we implement a rejection sampler, using the prior pdf \(p(x)=N(x;0,2)\propto\exp\{-x^{2}/4\}\) as a proposal function and the upper bound for the likelihood \(L=\exp\{-3.77\}\). The posterior density has the form
\[p(x|\textbf{y})\propto p(\textbf{y}|x)p(x)=\exp\{-V(x;\textbf{y},\textbf{g})-x ^{2}/4\}.\] (40)
Figure 6 (b) shows the normalized histogram of \(N=10,000\) samples generated by the RS algorithm, together with the true target pdf \(p(x|\textbf{y})\) depicted as a dashed line. The histogram follows closely the shape of the true posterior pdf. Figure 6 (c) shows the acceptance rates (averaged over \(10,000\) simulations) as a function of the bound \(\gamma\). We start with the trivial lower bound \(\gamma=0\) and increase it progressively, up to the global minimum \(\gamma=3.78\). The resulting acceptance rates are \(1.1\%\) for the trivial bound \(\gamma=0\), \(18\%\) with \(\gamma=2.89\) (BM1) and approximately \(40\%\) with \(\gamma=3.77\) (BM2). Note that the acceptance rate is \(\approx 41\%\) for the optimal bound and we cannot improve it any further. This is an intrinsic drawback of a rejection sampler with constant bound \(L\) and the principal argument that suggests the use of adaptive procedures.
Method | BM1 | BM1 + trasformation R | BM1 + tangent lines | BM2 | Optimal Bound
---|---|---|---|---|---
Lower Bound γ | 2.89 | 1.68 | 1.61 | 3.77 | 3.78
TABLE V: Lower bounds of the system potential function.
<figure><img src="content_image/0904.1300/ejemploOtro.png"><figcaption>Fig. 6: (a) The system potential V(x,y,g) (solid), the modified systempotential V(x,y,r) (dashed), function (R−1∘V2)(x,y,r) (dot-dashed) and thepiecewise-linear function W(x) formed by the two tangent lines to V(x,y,r) atmin(X) and max(X) (dotted). The corresponding bounds are marked with darkcircles. (b) The target density p(x|y)∝p(y|x)p(x) (dashed) and the normalizedhistogram of N=10,000 samples using RS with the the calculated bound L. (c)The curve of acceptance rates (averaged over 10,000 simulations) as a functionof the lower bound γ. The acceptance rate is 1.1% for the trivial bound γ=0,18% with γ=2.89, approximately 40% with γ=3.77 and 41% with the optimal boundγ=3.78.</figcaption></figure>
### _Example 2: Comparison of ARMS and GARS techniques_
Consider the problem of sampling a scalar random variable \(X\) from a posterior bimodal density \(p(x|y)\propto p(y|x)p(x)\), where the likelihood function is \(p(y|x)\propto\exp\{-\cosh(y-x^{2})\}\) (note that we have a single observation \(Y=y_{1}\)) and prior pdf is \(p(x)\propto\exp\{-\alpha(\eta-\exp(|x|))^{2}\}\), with constant parameters \(\alpha>0\) and \(\eta\). Therefore, the posterior pdf is \(p(x|y)\propto\exp\left\{-V(x;\tilde{\textbf{y}},\tilde{\textbf{g}})\right\}\), where \(\tilde{\textbf{y}}=[y,\eta]^{\top}\), \(\tilde{\textbf{g}}(x)=[g_{1}(x),g_{2}(x)]^{\top}=[x^{2},\exp(|x|)]^{\top}\) and the extended system potential function becomes
\[V(x;\tilde{\textbf{y}},\tilde{\textbf{g}})=\cosh(y-x^{2})+\alpha(\eta-\exp(|x| ))^{2}.\] (41)
The marginal potentials are \(\bar{V}_{1}(\vartheta_{1})=\cosh(\vartheta_{1})\) and \(\bar{V}_{2}(\vartheta_{2})=\alpha\vartheta_{2}^{2}\). Note that the density \(p(x|y)\) is an even function, \(p(x|y)=p(-x|y)\), hence it has a zero mean, \(\mu=\int xp(x|y)dx=0\). The constant \(\alpha\) is a scale parameter that allows to control the variance of the random variable \(X\), both _a priori_ and _a posteriori_. The higher the value of \(\alpha\), the more skewed the modes of \(p(x|y)\) become.
There are no standard methods to sample directly from \(p(x|y)\). Moreover, since the posterior density \(p(x|y)\) is bimodal, the system potential is non-log-concave and the ARS technique cannot be applied. However, we can easily use the GARS technique. If, e.g., \(\tilde{\textbf{Y}}=\tilde{\textbf{y}}=[y=5,\eta=10]^{\top}\) the simple estimates corresponding to \(g_{1}(x)\) are \(x_{1,1}=-\sqrt{5}\) and \(x_{1,2}=\sqrt{5}\), so that \(\mathcal{J}_{1}=[-\sqrt{5},\sqrt{5}]\). In the same way, the simple estimates corresponding to \(g_{2}(x)\) are \(x_{2,1}=-\log(10)\) and \(x_{2,2}=\log(10)\), therefore \(\mathcal{J}_{2}=[-\log(10),\log(10)]\).
An alternative possibility to draw from this density is to use the ARMS method [14]. Therefore, in this section we compare the two algorithms. Specifically, we look into the accuracy in the approximation of the posterior mean \(\mu=0\) by way of the sample mean estimate, \(\hat{\mu}=\frac{1}{N}\sum_{i=1}^{N}x^{(i)}\), for different values of the scale parameter \(\alpha\).
In particular, we have considered ten equally spacial values of \(\alpha\) in the interval \([0.2,5]\) and then performed \(10,000\) independent simulations for each value of \(\alpha\), each simulation consisting of drawing \(5,000\) samples with the GARS method and the ARMS algorithm. Both techniques can be sensitive to their initialization. The ARMS technique starts with \(5\) points selected randomly in \([-3.5,3.5]\) (with uniform distribution). The GARS starts with the set of support points \(\mathcal{S}_{0}=\{x_{2,1},x_{1,1},s,x_{1,2},x_{2,2}\}\) sorted in ascending order, including all simple estimates and an arbitrary point \(s\) needed to enable the construction in Section VI-B. Point \(s\) is randomly chosen in each simulation, with uniform pdf in \(\mathcal{J}_{1}=[x_{1,1},x_{1,2}]\).
The simulation results show that the two techniques attain similar performance when \(\alpha\in[0.2,1]\) (the modes of \(p(x|y)\) are relatively flat). When \(\alpha\in[1,4]\) the modes become more skewed and Markov chain generated by the ARMS algorithm remains trapped at one of the two modes in \(\approx 10\%\) of the simulations. When \(\alpha\in[4,5]\) the same problem occurs in \(\approx 25\%\) of the simulations. The performance of the GARS algorithm, on the other hand, is comparatively insensitive to the value of \(\alpha\).
Figure 7 (a) shows the posterior density \(p(x|y)\propto\exp\left\{-\cosh(y_{1}-x^{2})-\alpha(\mu-\exp(|x|))^{2}\right\}\) with \(\alpha=0.2\) depicted as a dashed line, and the normalized histogram obtained with the GARS technique. Figure 7 (b) illustrates the acceptance rates (averaged over \(10,000\) simulations) for the first \(20\) accepted samples drawn with the GARS algorithm. Every time a sample \(x^{\prime}\) drawn from \(\pi_{t}(x)\) is rejected, it is incorporated as a support point. Then, the proposal pdf \(\pi_{t}(x)\) becomes closer to target pdf \(p(x|y)\) and, as a consequence, the acceptance rate becomes higher. For instance, the acceptance rate for the first sample is \(\approx 16\%\), but for the second sample, it is already \(\approx 53\%\). The acceptance rate for the \(20\)-th sample is \(\approx 90\%\).
Simulation | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
ARMS | -2.2981 | 0.0267 | 0.0635 | 0.0531 | 2.2994
GARS | 0.0772 | -0.0143 | 0.0029 | 0.0319 | 0.0709
TABLE VI: Estimated posterior mean, ^μ (for α=5).
<figure><img src="content_image/0904.1300/GARShist.png"><figcaption>Fig. 7: (a) The bimodal density p(x|y)∝exp{−V(x;~y,~g)} (dashed line) and thenormalized histogram of N=5000 samples obtained using GARS algorithm. (b) Thecurve of acceptance rates (averaged over 10,000 simulations) as a function ofthe accepted samples.</figcaption></figure>
### _Example 3: Target localization with a sensor network_
In order to show how the proposed techniques can be used to draw samples from a multivariate (non-scalar) SoI, we consider the problem of positioning a target in a \(2\)-dimensional space using range measurements. This is a problem that appears frequently in localization applications using sensor networks [1].
We use a random vector \(\textbf{X}=[X_{1},X_{2}]^{\top}\) to denote the target position in the plane \(\mathbb{R}^{2}\). The prior density of \(X\) is \(p(x_{1},x_{2})=p(x_{1})p(x_{2})\), where \(p(x_{i})=N(x_{i};0,1/2)=k\exp\left\{-(x_{i})^{2}\right\}\), \(i=1,2\), i.e., the coordinate \(X_{1}\) and \(X_{2}\) are i.i.d. Gaussian. The range measurements are obtained from two sensor located at \(\textbf{h}_{1}=[0,0]^{\top}\) and \(\textbf{h}_{2}=[2,2]^{\top}\), respectively. The effective observations are the (square) Euclidean distances from the target to the sensors, contaminated with Gaussian noise, i.e.,
\[\begin{split}& Y_{1}=X_{1}^{2}+X_{2}^{2}+\Theta_{1}, \\ & Y_{2}=(X_{1}-2)^{2}+(X_{2}-2)^{2}+\Theta_{2},\end{split}\] (42)
where \(\Theta_{i}\), \(i=1,2\), are independent Gaussian variables with identical pdf’s, \(N(\vartheta_{i};0,1/2)=k_{i}\exp\left\{-\vartheta_{i}^{2}\right\}\). Therefore, the marginal potentials are quadratic, \(\bar{V}_{i}(\vartheta_{i})=\vartheta_{i}^{2}\), \(i=1,2\). The random observation vector is denoted \(Y=[Y_{1},Y_{2}]^{\top}\). We note that one needs three range measurements to uniquely determine the position of a target in the plane, so the posterior pdf \(p(\textbf{x}|\textbf{y})\propto p(\textbf{y}|\textbf{x})p(\textbf{x})\) is bimodal.
We apply the Gibbs sampler to draw \(N\) particles \(\textbf{x}^{(i)}=[x_{1}^{(i)},x_{2}^{(i)}]^{\top}\), \(i=1,\ldots,N\), from the posterior density \(p(\textbf{x}|\textbf{y})\propto p(\textbf{y}|x_{1},x_{2})p(x_{1})p(x_{2})\). The algorithm can be summarized as follows:
1. Set \(i=1\), and draw \(x_{2}^{(1)}\) from the prior pdf \(p(x_{2})\).
2. Draw a sample \(x_{1}^{(i)}\) from the conditional pdf \(p(x_{1}|\textbf{y},x_{2}^{(i)})\), and set \(\textbf{x}^{(i)}=[x_{1}^{(i)},x_{2}^{(i)}]^{\top}\).
3. Draw a sample \(x_{2}^{(i+1)}\) from the conditional pdf \(p(x_{2}|\textbf{y},x_{1}^{(i)})\).
4. Increment \(i=i+1\). If \(i>N\) stop, else go back to step 2.
The Markov chain generated by the Gibbs sampler converges to a stationary distribution with pdf \(p(x_{1},x_{2}|\textbf{y})\).
In order to use Gibbs sampling, we have to be able to draw from the conditional densities \(p(x_{1}|\textbf{y},x_{2}^{(i)})\) and \(p(x_{2}|\textbf{y},x_{1}^{(i)})\). In general, these two conditional pdf’s can be non-log-concave and can have several modes. Specifically, the density \(p(x_{1}|\textbf{y},x_{2}^{(i)})\propto p(\textbf{y}|x_{1},x_{2}^{(i)})p(x_{1})\) can be expressed as \(p(x_{1}|\textbf{y},x_{2}^{(i)})\propto\exp\{-V(x_{1};\tilde{\textbf{y}}_{1}, \tilde{\textbf{g}}_{1})\}\) where \(\tilde{\textbf{y}}_{1}=[y_{1}-(x_{2}^{(i)})^{2},y_{2}-(x_{2}^{(i)}-2)^{2},0]^{\top}\), \(\tilde{\textbf{g}}_{1}(x)=[x^{2},(x-2)^{2},x]^{\top}\) and
\[V(x_{1};\tilde{\textbf{y}}_{1},\tilde{\textbf{g}}_{1})=\left[y_{1}-(x_{2}^{(i) })^{2}-x_{1}^{2}\right]^{2}+\left[y_{2}-(x_{2}^{(i)}-2)^{2}-(x_{1}-2)^{2} \right]^{2}+x_{1}^{2},\] (43)
while the pdf \(p(x_{2}|\textbf{y},x_{1}^{(i)})\propto p(\textbf{y}|x_{2},x_{1}^{(i)})p(x_{2})\) can be expressed as \(p(x_{2}|\textbf{y},x_{1}^{(i)})\propto\exp\{-V(x_{2};\tilde{\textbf{y}}_{2}, \tilde{\textbf{g}}_{2})\}\) where \(\tilde{\textbf{y}}_{1}=[y_{1}-(x_{1}^{(i)})^{2},y_{2}-(x_{1}^{(i)}-2)^{2},0]^{\top}\), \(\tilde{\textbf{g}}_{2}(x)=[x^{2},(x-2)^{2},x]^{\top}\) and
\[V(x_{2};\tilde{\textbf{y}}_{2},\tilde{\textbf{g}}_{2})=\left[y_{1}-(x_{1}^{(i) })^{2}-x_{2}^{2}\right]^{2}+\left[y_{2}-(x_{1}^{(i)}-2)^{2}-(x_{2}-2)^{2} \right]^{2}+x_{2}^{2}.\] (44)
Since the marginal potentials and the nonlinearities are convex, we can use the GARS technique to sample the conditional pdf’s.
We have generated \(N=10,000\) samples from the Markov chain, with fixed observations \(y_{1}=5\) and \(y_{2}=2\). The average acceptance rate of the GARS algorithm was \(\approx 30\%\) both for \(p(x_{1}|\textbf{y},x_{2})\) and \(p(x_{2}|\textbf{y},x_{1})\). Note that this rate is indeed as a average because, at each step of the chain, the target pdf’s are different (if, e.g., \(x_{1}^{(i)}\neq x_{1}^{(i-1)}\) then \(p(x_{2}|\textbf{y},x_{1}^{(i)})\neq p(x_{2}|\textbf{y},x_{1}^{(i-1)})\)).
Figure 8 (a) shows the shape of the true target density \(p(x_{1},x_{2}|\textbf{y})\), while Figure 8 (b) depicts the normalized histogram with \(N=10,000\) samples. We observe that it approximates closely the shape of target pdf.
Finally, it is illustrative to consider the computational savings attained by using the GARS method when compared with a rejection sampler with a fixed bound. Specifically, we have run again the Gibbs sampler to generate a chain of \(10,000\) samples but, when drawing from \(p(x_{1}|\textbf{y},x_{2})\) and \(p(x_{2}|\textbf{y},x_{1})\), we have used RS with prior proposals (\(p(x_{1})\) and \(p(x_{2})\), respectively) and a fixed bound computed (analytically) with the method in Section IV-C for quadratic potentials. The average acceptance rate for the rejection sampler was \(\approx 4\%\) and the time needed to generate the chain was approximately \(10\) times the time needed in the simulation with the GARS algorithm.
<figure><img src="content_image/0904.1300/Gibbsejemplo2.png"><figcaption>Fig. 8: (a) The target density p(x|y)=p(x1,x2|y)∝p(y|x1,x2)p(x1)p(x2). (b) Thenormalized histogram with N=10,000 samples, using the GARS algorithm within aGibbs sampler.</figcaption></figure>
## VIII Conclusions
We have proposed families of generalized rejection sampling schemes that are particularly, but not only, useful for efficiently drawing independent samples from _a posteriori_ probability distributions. The problem of drawing from posterior distributions appears very often in signal processing, e.g., see the target localization example in this paper or virtually any application that involves the estimation of a physical magnitude given a set of observations collected by a sensor network. We have introduced two classes of schemes. The procedures in the first class are aimed at the computation of upper bounds for the likelihood function of the signal of interest given the set of available observations. They provide the means to (quickly and easily) design sampling schemes for posterior densities using the prior pdf as a proposal function. Then, we have elaborated on the bound-calculation procedures to devise a generalized adaptive rejection sampling (GARS) algorithm. The latter is a method to construct a sequence of proposal pdf’s that converge towards the target density and, therefore, can attain very high acceptance rates. It should be noted that the method introduced in this paper includes the classical adaptive rejection sampling scheme of [8] as a particular case. We have provided some simple numerical examples to illustrate the use of the proposed techniques, including sampling from multimodal distributions (both with fixed and adaptive proposal functions) and an example of target localization using range measurements. The latter problem is often encountered in positioning applications of sensor networks.
## IX Acknowledgements
This work has been partially supported by the Ministry of Science and Innovation of Spain (project MONIN, ref. TEC-2006-13514-C02-01/TCM, and program Consolider-Ingenio 2010, project CSD2008-00010 COMONSENS) and the Autonomous Community of Madrid (project PROMULTIDIS-CM, ref. S-0505/TIC/0233).
## Appendix
**Proposition**: The state estimators \(\hat{x}_{j}\in\arg\max\limits_{x\in[{\mathcal{B}}_{j}]}{\ell(x|\textbf{y}, \textbf{g})}=\arg\min\limits_{x\in[{\mathcal{B}}_{j}]}{V(x;\textbf{y},\textbf{ g})}\) belong to the interval \(\mathcal{I}_{j}\), i.e.,
(45)
where \(\mathcal{X}_{j}\triangleq\{x_{1,j},\ldots,x_{n,j}\}\) is the set of all simple estimates in \(\mathcal{B}_{j}\) and \(\mathcal{I}_{j}\subseteq\mathcal{B}_{j}\).
**Proof**: We have to prove that the derivative of the system potential function is
\[\frac{dV}{dx}<0,\ \ \mbox{for all}\ \ x<\min{(\mathcal{X}_{j})}\quad(x\in[{ \mathcal{B}}_{j}]),\] (46)
and
\[\frac{dV}{dx}>0,\ \ \mbox{for all}\ \ x>\max{(\mathcal{X}_{j})}\quad(x\in[{ \mathcal{B}}_{j}]),\] (47)
so that all stationary points of \(V\) stay inside \(\mathcal{I}_{j}=[\min{(\mathcal{X}_{j})},\max{(\mathcal{X}_{j})}]\). Routine calculations yield the derivative
\[\frac{dV}{dx}=-\sum^{n}_{i=1}\frac{dg_{i}}{dx}\left[\frac{d\bar{V}_{i}}{d \vartheta_{i}}\right]_{\vartheta_{i}=y_{i}-g_{i}(x)}\] (48)
and we aim to evaluate it outside the interval \(\mathcal{I}_{j}\). To do it, let us denote \(x_{min}=\min(\mathcal{X}_{j})\) and \(x_{max}=\max(\mathcal{X}_{j})\) and consider the cases \(\frac{dg_{i}}{dx}>0\) and \(\frac{dg_{i}}{dx}<0\) separately (recall that we have assumed the sign of \(\frac{dg_{i}}{dx}\) to remain constant in \(\mathcal{B}_{j}\)).
When \(\frac{dg_{i}}{dx}>0\) and since, for every simple estimate, \(x_{i,j}\geq x_{min}\), we obtain that \(y_{i}=g_{i}(x_{i,j})\geq g_{i}(x_{min})>g_{i}(x)\)\(\forall x<x_{min}\). Then \(y_{i}-g_{i}(x)>0\), for all \(x<x_{min}\), and, due to properties (P1) and (P2) of marginal potential functions, \(\left[\frac{d\bar{V}_{i}}{d\vartheta_{i}}\right]_{\vartheta_{i}=y_{i}-g_{i}(x) >0}>0\) for all \(i\). As a consequence, \(\frac{dV}{dx}<0\)\(\forall x<x_{min}\), \(x\in[{\mathcal{B}}_{j}]\).
When \(\frac{dg_{i}}{dx}<0\) and \(x_{i,j}\geq x_{min}\), we obtain that \(y_{i}=g_{i}(x_{i,j})\leq g_{i}(x_{min})<g_{i}(x)\), \(\forall x<x_{min}\). Then \(y_{i}-g_{i}(x)<0\) for all \(x<x_{min}\) and \(\left[\frac{d\bar{V}_{i}}{d\vartheta_{i}}\right]_{\vartheta_{i}=y_{i}-g_{i}(x) <0}<0\), again because of (P1) and (P2). As a consequence, \(\frac{dV}{dx}<0\)\(\forall x<x_{min}\), \(x\in[{\mathcal{B}}_{j}]\).
A similar argument for \(x>x_{max}\) yields \(\frac{dV}{dx}>0\) for all \(x>x_{max}\) and completes the proof. \(\Box\)
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] | # Traces of exomoons in
computed flux and polarization phase curves
of starlight reflected by exoplanets
J. Berzosa Molina
Now at GMV AD, Calle de Isaac Newton, 11, 28760 Tres Cantos, Madrid, SpainFaculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
L. Rossi
Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
D. M. Stam
j.berzosamolina@gmail.com
Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
Received 30 April 2018; accepted 23 July, 2018
Key Words.:**methods: numerical – polarization – radiative transfer – stars: planetary systems – exomoons** †
[FOOTNOTE:†][ENDFOOTNOTE]
###### Abstract
Context:Detecting moons around exoplanets is a major goal of current and future observatories. Moons are suspected to influence rocky exoplanet habitability, and gaseous exoplanets in stellar habitable zones could harbor abundant and diverse moons to target in the search for extraterrestrial habitats. Exomoons will contribute to exoplanetary signals but are virtually undetectable with current methods.
Aims:We identify and analyze traces of exomoons in the temporal variation of total and polarized fluxes of starlight reflected by an Earth–like exoplanet and its spatially unresolved moon across all phase angles, with both orbits viewed in an edge–on geometry.
Methods:We compute the total and linearly polarized fluxes, and the degree of linear polarization \(P\) of starlight that is reflected by the exoplanet with its moon along their orbits, accounting for the temporal variation of the visibility of the planetary and lunar disks, and including effects of mutual transits and mutual eclipses. Our computations pertain to a wavelength of 450 nm.
Results:Total flux \(F\) shows regular dips due to planetary and lunar transits and eclipses. Polarization \(P\) shows regular peaks due to planetary transits and lunar eclipses, and \(P\) can increase and/or slightly decrease during lunar transits and planetary eclipses. Changes in \(F\) and \(P\) will depend on the radii of the planet and moon, on their reflective properties, and their orbits, and are about one magnitude smaller than the smooth background signals. The typical duration of a transit or an eclipse is a few hours.
Conclusions:Traces of an exomoon due to planetary and lunar transits and eclipses show up in \(F\) and \(P\) of sunlight reflected by planet–moon systems and could be searched for in exoplanet flux and/or polarization phase functions.
## 1Introduction
**Since the detection of the first planets beyond** **our Solar System** **(****Wolszczan & Frail****,** 1992; **Campbell et al.****,** 1988**)****, the number of discoveries has steadily increased, yielding almost 4000 confirmed exoplanets and 2500 unconfirmed, candidate exoplanets to this day** **(****Han et al.****,** 2014**)****. Exoplanet space telescopes, such as ESA’s CHEOPS (CHaracterising ExOPlanet Satellite) and Plato (PLAnetary Transits and Oscillations of stars), and NASA’s Transiting Exoplanet Survey Satellite (TESS), are dedicated to find exoplanets around bright, nearby stars. The relative small distances to these stars and their planets combined with the high sensitivity of these missions and the upcoming JWST/NASA (James Webb Space Telescope) and ARIEL/ESA** **(****Tinetti et al.****,** 2016**)** **missions will allow the search for lunar companions and planetary rings.**
**The continuous increase in instrument precision, stability and spatial resolution has enabled a new generation of ground–based instruments, such as the Gemini Planet Imager (GPI) instrument** **(see** **Macintosh et al.****,** 2014**)** **on the Gemini North telescope, the Spectro–Polarimetric High–contrast Exoplanet Research (SPHERE) instrument** **(see** **Beuzit et al.****,** 2006**)** **on ESO’s Very Large Telescope (VLT) and the proposed Exoplanet Imaging Camera and Spectrograph (EPICS)** **(see** **Keller et al.****,** 2010; **Gratton et al.****,** 2010**)** **on the European Extremely Large Telescope (E-ELT), which is under construction by ESO. These high–contrast instruments use direct imaging of planetary radiation to not only detect but also characterize exoplanetary systems through a combination of** **spectroscopy and polarimetry techniques. Using GPI and SPHERE, respectively,** **Macintosh et al.****(**2015**)** **and** **Wagner et al.****(**2016**)** **announced the discoveries of young, and thus hot Jovian planets, whose atmospheric properties and orbits were characterized using near–infrared spectroscopy.**
**Direct detection of extrasolar bodies presents a major challenge as their observed radiation, both emitted and reflected, is very weak compared to that of the host star. In addition, the angular distance from the planet to the star is extremely small. Consequently, the vast majority of exoplanets have only been detected indirectly. In contrast, direct imaging of the planet can reveal a wealth of information of the planet properties that cannot be obtained through other methods, such as lower atmospheric composition and, for rocky planets, their surface coverage.**
**Despite not having been exploited yet, great hope is placed in the polarimetry capabilities of current and future telescopes as powerful tools for detecting and characterizing exoplanets** **(see** **Snik & Keller****,** 2013; **Stam et al.****,** 2004; **Seager et al.****,** 2000; **Hough et al.****,** 2003; **Hough & Lucas****,** 2003; **Saar & Seager****,** 2003; **Stam****,** 2003, 2008**)****. Previous works in this field involve the modeling of stellar polarization during planetary transits** **(****Wiktorowicz & Laughlin****,** 2014; **Kostogryz et al.****,** 2011, 2015; **Sengupta****,** 2016; **Carciofi & Magalhães****,** 2005**)****,** **the modeling of light curves and polarization of starlight reflected signals in the visible range of Earth-like planets** **(****Rossi & Stam****,** 2017; **Karalidi et al.****,** 2012; **Stam****,** 2008**)****, and giant Jupiter-like planets** **(****Stam et al.****,** 2004; **Seager et al.****,** 2000**)****, as well as the modeling of exoplanetary atmospheres in the infrared** **(****De Kok et al.****,** 2011; **Marley & Sengupta****,** 2011**)****, which demonstrated the usefulness of direct observations on exoplanet characterization. More recently,** **Bott et al.****(**2016**)** **reported linear polarization observations of the hot Jupiter system HD 189733, and** **Ginski et al.****(**2018**)** **announced the detection of planetary thermal radiation that is polarized upon reflection by circumstellar dust. Indeed, polarization has also been proposed as a means for exomoon detection:** **Sengupta & Marley****(**2016**)** **studied the effects of a satellite transiting its hot host planet in the polarization signal of (infrared) thermally emitted radiation for the case of homogeneous, spherically symmetric cloudy planets. Studying exomoons can improve our understanding of in particular:**
1. **Planet formation: Solar System moons appear to support diverse formation histories. For instance, Titan might have formed from circumplanetary debris,** **while the Moon, Phobos and Deimos suggest a cumulative bombardment** **(****Rufu et al.****,** 2017; **Rosenblatt et al.****,** 2016**)****. Triton might have been captured by Neptune** **(****Agnor & Hamilton****,** 2006**)****, while collisions are thought to have altered the relative alignment between Uranus and its moons** **(****Morbidelli et al.****,** 2012**)****. Indeed, studying Solar System moons gives essential insights on formation mechanisms and evolution** **(see** **Heller****,** 2017**, and references therein)****. Exomoon research would allow refining planet formation theories in a way not achievable by studying exoplanets alone.**
2. **Extra-solar system characterization: studying exomoons will not only provide information on lunar orbits and physical properties, but will also allow constraining planet characteristics such as i.e. mass, oblateness, and rotation axis** **(****Barnes & Fortney****,** 2003; **Kipping et al.****,** 2009; **Schneider et al.****,** 2015**)****. A signal of a planet-moon system could be interpreted as that of a planet alone, resulting in e.g. an overestimation of the planet mass and effective temperature** **(****Williams & Knacke****,** 2004**)****, and/or an anomalous composition from spectroscopy** **(****Schneider et al.****,** 2015**)****. Extrasolar system characterization would indeed require analysis of all of its elements, i.e. planets, moons, rings, and exozodiacal dust.**
3. **Exoplanet and exomoon habitability: a moon may influence its planet’s habitability** **(****Benn****,** 2001**)****, and moons of giant exoplanets within the stellar habitable zone (HZ) might host habitable environments** **(****Canup & Ward****,** 2006**)****.** **Reynolds et al.****(**1987**)** **and** **Heller & Barnes****(**2015**)** **mention the role of a moon’s orbit on the presence of liquid, life–supporting water. Indeed, tidal heating could maintain surface temperatures compatible with life on large moons around cold giant planets** **(****Scharf****,** 2006**)****.** **Lehmer et al.****(**2017**)** **show that small moons could retain atmospheres over limited time periods, while Ganymede–sized moons in a stellar HZ could hold atmospheres and surface water indefinitely. Although radiation in a giant planet’s magnetic field and eclipses could threaten local conditions for life** **(****Heller & Barnes****,** 2013; **Heller****,** 2012; **Forgan & Yotov****,** 2014**)****, exomoons are interesting targets in the search for extraterrestrial life.**
**Led by Kipping’s** _Hunt for Exomoons with Kepler_**, which uses a combination of photometric transits, Transit Timing Variations (TTV) and Transit Duration Variations (TDV) data** **(****Kipping et al.****,** 2015; **Kipping****,** 2009; **Sartoretti & Schneider****,** 1999; **Szabó et al.****,** 2006; **Simon et al.****,** 2007, 2015**)****, and Hippke’s search using the Orbital Sampling Effect (OSE)** **(****Hippke****,** 2015; **Heller****,** 2014; **Heller et al.****,** 2016**)****, the search for exomoons is in its starting phase. Mars–sized and possibly even Ganymede-sized satellites could be traceable in archived Kepler data** **(****Heller et al.****,** 2014**)****. Unfortunately, as of yet no exomoons have been confirmed.**
**In this paper, we use numerical simulations to show how an exomoon could influence the flux and degree of polarization of the starlight that is reflected by an Earth-like exoplanet, using the following outline. In Sect.** 2**, we describe the numerical code to compute the various geometries of the exoplanet–exomoon system that are required for our radiative transfer computations and the radiative transfer code to compute the reflected fluxes and polarization for a given exoplanet–exomoon system. In Sect.** 3**, we present computed flux and polarization phase functions at 450 nm, for an Earth-like planet (with a Lambertian reflecting surface and a gaseous atmosphere) with a Moon–like satellite (with a Lambertian reflecting surface) in an edge–on geometry. Finally, in Sect.** 4**, we summarize and discuss our findings and their implications.**
## 2Computing the reflected starlight
### Stokes vectors and polarization
**We describe the flux and polarization of starlight that is reflected by a body, with a Stokes vector** **(see e.g.** **Hansen & Travis****,** 1974**)****:**
\[\mathbf{F}=\left[{\begin{array}[]{*{20}{c}}F\\ Q\\ U\\ V\end{array}}\right],\] (1)
**with** \(F\) **the total flux,** \(Q\) **and** \(U\) **the linearly polarized fluxes, and** \(V\) **the circularly polarized flux, all with dimensions W m**\({}^{-2}\)**. In principle, these fluxes are wavelength dependent. However, we will not explicitly include the wavelength in the dimensions, because we focus on a single wavelength region. Fluxes** \(Q\) **and** \(U\) **are defined with respect to a reference plane, for which we use the planetary (or lunar) scattering plane, which contains the observer, and the centers of the planet (or moon) and the** **star. We do not compute the circularly polarized flux** \(V\)**, because it is usually much smaller than the linearly polarized fluxes** **(see** **Rossi & Stam****,** 2018; **Kawata****,** 1978; **Hansen & Travis****,** 1974**)****, and because ignoring** \(V\) **does not lead to significant errors in the computation of** \(F\)**,** \(Q\)**, and** \(U\)******(see** **Stam & Hovenier****,** 2005**)****. The light of the star is assumed to be unpolarized** **(see** **Kemp et al.****,** 1987**)****, and is given by** \(\mathbf{F}_{\rm 0}=F_{0}\mathbf{1}\)**, with** \(\mathbf{1}\) **the unit column vector and** \(\pi F_{0}\) **the flux measured perpendicular to the light’s propagation direction. If the orbit of the barycenter of the planet–moon system around the star is eccentric, the incident flux varies along the orbit. Our standard stellar flux,** \(\pi F_{0}\)**, is defined with respect to the periapsis of the orbit of the system’s barycenter.**
**The degree of linear polarization,** \(P\)**, of vector** \(\mathbf{F}\) **is defined as**
\[P=\sqrt{Q^{2}+U^{2}}/F,\] (2)
**and the direction of polarization,** \(\chi\)**, with respect to the reference plane can be computed from**
\[\tan{2\chi}=U/Q,\] (3)
**where** \(\chi\) **is chosen such that** \(0\leq\chi<\pi\)**, while** \(\cos{2\chi}\) **and** \(Q\) **have the same sign** **(****Hansen & Travis****,** 1974; **Hovenier et al.****,** 2004**)****.**
### Disk–integrated reflected Stokes vectors
**We compute the reflected Stokes vector** \(\mathbf{F}\) **of the spatially unresolved planet–moon system as a summation of the reflected Stokes vectors** \(\mathbf{F}^{\rm p}\) **and** \(\mathbf{F}^{\rm m}\) **of, respectively, the planet and the moon (the pair is spatially resolved from the star):**
\[\mathbf{F}=\hskip 1.422638pt\mathbf{F}^{\rm p}+\frac{R^{2}_{\rm m}}{R^{2}_{\rm p }}\hskip 1.422638pt\mathbf{L}(\psi)\hskip 1.422638pt\mathbf{F}^{\rm m}.\] (4)
**Vectors** \(\mathbf{F}^{\rm p}\) **and** \(\mathbf{F}^{\rm m}\) **are disk–integrated vectors that include the effects of eclipses and transits. They are normalized such that the total fluxes reflected by the planet and moon at a phase angle** \(\alpha=0^{\circ}\) **and without shadows and/or eclipses on their disks,** **equal the planet’s and moon’s geometric albedo’s, respectively** **(see** **Stam et al.****,** 2006**)****. Furthermore,** \(R_{\rm p}\) **and** \(R_{\rm m}\) **are the radii of the (spherical) planet and moon, respectively.**
**Vectors** \(\mathbf{F}\) **and** \(\mathbf{F}^{\rm p}\) **in Eq.** 4 **are defined with respect to the planetary scattering plane, while** \(\mathbf{F}^{\rm m}\) **is defined with respect to the lunar scattering plane. Depending on the orientation of the lunar orbit, the lunar scattering plane can have a different orientation than the planetary scattering plane. Matrix** \({\bf L}\) **in Eq.** 4 **rotates** \(\mathbf{F}^{\rm m}\) **from the lunar to the planetary scattering plane. It is given by** **(see** **Hovenier & van der Mee****,** 1983**)******
\[\mathbf{L}(\psi)=\left[{\begin{array}[]{*{20}{c}}1&0&0&0\\ 0&{\cos{2\psi}}&{\sin{2\psi}}&0\\ 0&{\sin{2\psi}}&{\cos{2\psi}}&0\\ 0&0&0&1\end{array}}\right]\,,\] (5)
**with** \(\psi\) **the rotation angle measured in the clockwise direction from the lunar to the planetary scattering plane when looking towards the moon** **(**\(0\leq\psi<\pi\)**).**¹****
[FOOTNOTE:1][ENDFOOTNOTE]
<figure><img src="content_image/1807.10266/x1.png"><figcaption>Figure 1: 3D–view (left) and projection onto the x0−y0 plane (right) of thediscretized planet or moon. The z0–axis points towards the observer. Theorientation of the x0 and y0 axes with respect to the disk of the planet ormoon can be chosen arbitrarily.</figcaption></figure>
**To compute the disk–integrated vectors** \(\mathbf{F}^{\rm p}\) **and** \(\mathbf{F}^{\rm m}\)**, we divide the disks of the planet and the moon as seen by the observer, into a grid of equally sized, square pixels (see Fig.** 1**). The number of pixels on the planetary disk is** \(N_{\rm p}\)******and that on the lunar disk** \(N_{\rm m}\)**. A given pixel will contribute to a disk–signal when its center is within the disk–radius. Obviously, the larger the number of pixels (and the smaller each pixel), the better the approximation of the curved limb of the disk, the terminator, and the shadows, such as those due to eclipses (see App.** C **for insight into the effect of the number of pixels on the computed signals). The disk–integrated vectors are obtained by summing up the contributions of the individual pixels across the disk, fully taking into account shadows and eclipses, i.e.**
(6)
**where ’x’ is either ’p’ or ’m’. Factor** \(\pi/N_{\rm x}\) **is the surface area per pixel. Furthermore,** \({\mathbf{F}}^{\rm x}_{i}\) **is the reflected Stokes vector for the** \(i\)**-th pixel on the planet (x=p) or moon (x=m), the computation of which is described in Sect.** 2.3**. Matrix** \(\mathbf{L}\) **is a rotation matrix (see Eq.** 5**) that is used for rotating the local Stokes vector** \({\mathbf{F}}^{\rm x}_{i}\) **that is defined with respect to the local reference plane,** **to the planetary or lunar scattering plane. Factor** \(b_{i}\) **accounts for the visibility of pixel** \(i\)**: if** \(b_{i}=1\)**, the pixel is visible to the observer, and if** \(b_{i}=0\) **it is invisible due to a transiting body. Factor** \(c_{i}\) **accounts for the dimming of the local incident stellar flux due to a (partial) eclipse:** \(c_{i}=0.0\) **indicates that pixel** \(i\) **is eclipsed and receives no flux, and** \(c_{i}=1.0\) **indicates that the pixel is not eclipsed. For partial (penumbral) eclipses,** \(0.0<c_{i}<1.0\)**. The computation of factors** \(b_{i}\) **and** \(c_{i}\) **is described in Sect.** 2.4**. Factor** \(d_{i}\)**, finally, indicates the decrease of the standard incident stellar flux** \(\pi F_{0}\) **due to an increase of the distance to the star, according to**
\[d_{i}=\left(r_{\rm ref}/r_{i{\rm s}}\right)^{2},\] (7)
**where** \(r_{\rm ref}\) **is the reference distance at which the standard stellar flux is defined and** \(r_{i{\rm s}}\) **is the actual distance between pixel** \(i\) **and the star.**
### The locally reflected starlight
**The Stokes vector of the starlight that is reflected by pixel** \(i\) **on the planet or moon is computed using** **(see** **Hansen & Travis****,** 1974**)****:**
\[{\mathbf{F}_{i}^{\rm x}}(\theta_{i},\theta_{0i},\phi_{i}-\phi_{0i})=\cos{ \theta_{0i}}~{}{\bf\mathbf{R}}_{1i}^{\rm x}(\theta_{i},\theta_{0i},\phi_{i}- \phi_{0i})~{}F_{0},\] (8)
**with** \(\theta_{i}\) **the angle between the local zenith direction and the local direction to the observer,** \(\theta_{0i}\) **the angle between the local zenith direction and the local direction to the star, and** \(\phi_{i}-\phi_{0i}\) **the local azimuthal difference angle, i.e. the angle between the plane containing the local zenith direction and the local direction to the observer and the plane containing the local zenith direction and the local direction to the star** **(see** **Rossi et al.****,** 2018; **de Haan et al.****,** 1987**)****. Furthermore,** \({\mathbf{R}}^{\rm x}_{1i}\) **is the first column of the local reflection matrix of the planet or moon. Only the first column is needed because the incident starlight is assumed to be unpolarized (cf. Sect.** 2.1**). For a given pixel, the illumination and viewing angles, and thus** \(\mathbf{R}^{\rm x}_{1i}\)**, depend on the position of the planet or moon with respect to the star and to each other. Local reflection matrix** \(\mathbf{R}^{\rm x}_{i}\) **also depends on the local composition and structure of the atmosphere and/or surface of the reflecting body.** **We compute reflected starlight for an Earth–Moon–like planetary system, keeping the reflection models for the Earth and the moon simple to avoid introducing too many details that increase computational times while not adding insight into the observable signals.**
**Our model planet has a flat, Lambertian (i.e. isotropically and depolarizing) reflecting surface with an albedo,** \(a_{\rm surf}\)**, of 0.3. The surface is overlaid by an atmosphere that is assumed to consist of only gas. We compute the atmospheric optical thickness at a given wavelength** \(\lambda\)**, using a model atmosphere consisting of 32 layers, with the ambient pressure and temperature according to a mid-latitude summer profile** **McClatchey et al.****(**1972**)****. The surface pressure is 1.0 bars. The molecular scattering optical thickness** \(b^{\rm m}_{\rm sca}\) **of an atmospheric layer at wavelength** \(\lambda\) **is calculated according to**
\[b^{\rm m}_{\rm sca}(\lambda)=\sigma^{\rm m}_{\rm sca}(\lambda)\hskip 5.690551ptN,\] (9)
**with** \(\sigma^{\rm m}_{\rm sca}\) **the molecular scattering cross–section (in m**\({}^{2}\)**) and** \(N\) **the molecular column number density (in m**\({}^{-2}\)**) of the atmospheric layer. The molecular scattering cross–section is calculated according to**
\[\sigma^{\rm m}_{\rm sca}(\lambda)=\frac{24\pi^{3}}{N_{\rm L}^{2}}\frac{(n( \lambda)^{2}-1)^{2}}{(n(\lambda)^{2}+2)^{2}}\frac{6+3\delta(\lambda)}{6-7 \delta(\lambda)}\frac{1}{\lambda^{4}},\] (10)
**with** \(N_{\rm L}\)**, Loschmidt’s number at standard pressure and temperature,** \(n\)**, the wavelength dependent refractive index of dry air under standard pressure and temperature, and** \(\delta\)**, the depolarization factor of the atmospheric gas** **(see** **Stam****,** 2008**, and references therein for the values that have been chosen for the various parameters)****. To calculate the molecular column number density** \(N\)**, we assume hydrostatic equilibrium in each atmospheric layer, thus**
\[N=\frac{\delta p}{mg},\] (11)
**with** \(\Delta p\) **the difference between the pressure at the bottom and at the top of the atmospheric layer (in Pa),** \(m\) **the average molecular mass in the layer (in kg), and** \(g\) **the acceleration of gravity (in m s**\({}^{-2}\)**). The atmospheric optical thickness at a given wavelength** \(\lambda\) **is calculated by adding the values of** \(b^{\rm m}_{\rm sca}\) **for all atmospheric layers at that wavelength (note that for a model atmosphere containing only gas, the radiative transfer of incident sunlight only depends on the total optical thickness, not on the vertical** **distribution of the optical thickness). The total atmospheric optical thickness at 450 nm, the wavelength of our interest, is 0.14. At this wavelength, there is no significant absorption by atmospheric gases in the Earth’s atmosphere** **(see** **Stam****,** 2008**, for sample spectra)****. The single scattering albedo of the gaseous molecules can thus be assumed to equal 1.0. And, at this short wavelength, the horizontal inhomogeneities of the Earth’s surface and the contributions of clouds and aerosol to the reflected signal are relatively small** **(see** **Stam****,** 2008**, for simulations of the Earth’s signal at 440 nm)****. Our model moon has no atmosphere above its flat, Lambertian (i.e. isotropic and depolarizing) reflecting surface with** \(a_{\rm surf}=0.1\)******(****Williams****,** 2017**)****.**
**The computation of the local illumination and viewing geometries** \(\theta_{i}\)**,** \(\theta_{0i}\)**, and** \(\phi_{i}-\phi_{0i}\) **is described in Appendix** A**. Given these angles and the planet’s atmosphere--surface model, we use PyMieDAP**²******(****Rossi et al.****,** 2018**)****, an efficient radiative transfer code based on the adding–doubling algorithm described by** **de Haan et al.****(**1987**)****. PyMieDAP fully includes polarization for all orders of scattering, and assumes a locally plane–parallel atmosphere–surface model** **to compute** \({\mathbf{R}}_{1i}^{\rm p}\) **for every pixel on the planet. The computed locally reflected Stokes vector,** \({\mathbf{F}_{i}}^{\rm p}\)**, is defined with respect to the local meridian plane, i.e. the plane through the local zenith and the local direction towards the observer. For each illuminated pixel on the moon,** \({\mathbf{R}}_{1i}^{\rm m}=a_{\rm surf}\mathbf{1}\)**. A detailed description of PyMieDAP including benchmark results can be found in** **Rossi et al.****(**2018**)****.**
[FOOTNOTE:2][ENDFOOTNOTE]
**Results of our radiative transfer code have been compared against results presented in e.g.** **Stam****(**2008**)**; **Stam et al.****(**2006, 2004**)** **(who all used the same adding–doubling code, but an entirely different disk–integration algorithm), and** **Karalidi et al.****(**2012**)** **(who used their own version of an adding–doubling code and an independent disk–integration method).** **Buenzli & Schmid****(**2009**)** **and** **Stolker et al.****(**2017**)****, each compared their own, independently implemented Monte Carlo radiative transfer codes successfully against results from the code used by** **Stam et al.****(**2004**)** **and** **Stam et al.****(**2006**)****.**
<figure><img src="content_image/1807.10266/x2.png"><figcaption>Figure 2: Sketch illustrating the mutual events between a planet and its moon:(a) a lunar transit, (b) a planetary transit, (c) a planetary eclipse, and (d)a lunar eclipse. The positive z0–axis (in sub-figures a and b) points to theobserver. The white–black scale indicates full–null illumination of a body. Anarrow indicates the lunar motion around the planet. The distances and radiiare not to scale.</figcaption></figure>
### Computing transits and eclipses
**As described in Eq.** 6**, the contribution of the light reflected by a pixel** \(i\) **on the planet or the moon to the disk–integrated Stokes vector** \(\mathbf{F}\)**, depends on the factors** \(b_{i}\) **and** \(c_{i}\)**, that account for the pixel’s visibility and dimming, respectively. The values of these factors depend on** **so–called mutual events, specifically, transits, in which one body is (partially) blocking the light that is reflected towards the observer by another body, and eclipses, in which one body is casting a (partial) shadow on the illuminated and visible disk of another body. Limiting ourselves to systems in which a single star is orbited by a planet with a single moon, we distinguish the following four mutual events (cf. Fig.** 2**):**
1. **A planetary eclipse: the moon is between the star and the planet, casting its shadow on the planet, the extent of which depends on the planet–star and moon–star distances, on the stellar, planetary and lunar radii, and on their orbital positions.**
2. **A lunar eclipse: the planet is between the star and the moon, casting its shadow on the moon, the extent of which depends on the planet–star and moon–star distances, on the stellar, planetary and lunar radii, and on their orbital positions.**
3. **A planetary transit: the planet is between the moon and the observer, blocking the view of the moon, the extent of which depends on the planetary and lunar radii, and their orbital positions.**
4. **A lunar transit: the moon is between the planet and the observer, occulting a region of the planetary disk, the extent of which depends on the planetary and lunar radii,** **and their orbital positions.**
**We exclude planetary and lunar transits of the star, i.e. the epochs in which these bodies move in front or behind the star. Numerical simulations of transiting planets with moons have been published by** **Kipping****(**2011**)****. Modeling the transmission and scattering of starlight in the planetary atmosphere during those epochs (which is not included in the work by** **Kipping****(**2011**)****), requires a fully spherical atmosphere model instead of a locally plane-parallel one** **(****de Kok & Stam****,** 2012**)** **and falls outside the scope of this paper.**
**For our computation of the effects of transits of the planet in front of the moon and vice versa on the flux and polarization of the reflected starlight, we assume that the bodies are at ’infinite’ distance of the observer. For our computation of the effects of the eclipses, i.e. the shadow of one body darkening regions on the other body, on the reflected flux and polarization, we follow the mathematical description of eclipses in the Moon–Earth system as developed by** **Link****(**1969**)****, taking into account the sizes of the planet and the moon, their distances and positions with respect to the star, and the size of the** **stellar disk. The latter is crucial for the modeling of the umbra, antumbra en penumbra shadow regions (for an example of the umbra and penumbra, see Fig.** 2**). The contribution of the starlight reflected by pixels in the antumbral or penumbral region of the planet or moon to the total signal is weighted by the depth of the shadow (i.e. factor** \(c_{i}\) **in Eq.** 6**). We ignore stellar limb darkening and the transmission of starlight through the planetary atmosphere during a lunar eclipse.**
**A detailed description of our numerical computation of eclipses and the factor** \(c_{i}\) **in Eq.** 6 **is provided in Appendix** B**. This computation requires the positions of the planet and the moon with respect to the star across time, and thus the dynamics of the three–body system. The basics of this dynamics is outlined in the next section.**
### Computing the orbits of the planet & moon
<figure><img src="content_image/1807.10266/x3.png"><figcaption>Figure 3: Sketch of the reference frames and angles used to describe theKeplerian orbits of the planet–moon system barycenter around the parent star(Fig. a) and the Keplerian orbit of the moon around the planet–moon systembarycenter (Fig. b). Plane p1 (a) is the plane of the sky as seen by theobserver on the positive z1–axis. Plane p2 (Figs. a & b) is the barycenter’sorbital plane, and plane p3 (Fig. b) is the orbital plane of the moon aroundthe barycenter. Angle i is the orbital inclination angle, ω the argument ofperiastron, Ω the right ascension of the ascending node, and ν the trueanomaly. Subscript bs refers to the barycenter of the planet–moon systemaround the star, and mb to the moon around the barycenter. Vectors rbs and rmbare the position vectors of the barycenter and the moon, respectively. Thebarycenter and the moon are indicated by b and m, respectively.</figcaption></figure>
**We compute the position vectors of the planet and moon as functions of time for determining the factors** \(b_{i}\)**,** \(c_{i}\)**, and** \(d_{i}\) **of each pixel** \(i\) **and for evaluating the disk–integration according to Eq.** 6**. Both the motions of the planet and its moon around the star depend on their mutual gravitational interactions. Assuming each body attracts as a point mass and neglecting the gravity of other planets and/or moons in the system, our star–planet–moon system is a classical, generic three–body problem.**
**A precise computation of the orbital positions in the generic three–body problem requires the numerical propagation of a given set of initial conditions. Instead, we use the ’nested two–body’ approximation described by** **Kipping****(**2011, 2010**)****, which assumes that the orbits of the planet and moon around the planet–moon system barycenter, and the orbit of this barycenter around the star can all be described by Keplerian orbits. The advantages of the nested two–body approximation are: 1. the solution can be described analytically; 2. the computational time is significantly shorter than with numerical integrations; 3. it provides better insight in the computed orbits as the elements of all orbits can be specified; 4. unlike the circular restricted three–body problem simplification** **(****Wakker****,** 2015**, see e.g.)****,** **it can handle eccentric orbits.**
**As demonstrated by** **Kipping****(**2010**)****, the nested two–body approximation is excellent for the generic three–body problem provided** \(\Re\leq 0.531\)**, where** \(\Re\) **is the moon–planet separation in units of the planet’s Hill’s sphere radius** **(see e.g.** **De Pater & Lissauer****,** 2015**)****. As follows from** **Domingos et al.****(**2006**)** **and** **Kipping****(**2011**)****, stable, prograde orbiting moons should fulfill** \(\Re\leq 0.4895\)**, while retrograde orbiting moons can be stable up to** \(\Re\approx 0.9309\)**. The nested two–body approximation can thus be applied to all prograde orbiting moons, while retrograde orbiting moons are only partially covered, depending on** \(\Re\)**. We will limit ourselves to prograde orbiting moons, as we do not expect any influence of the moon’s orbital direction on the magnitude of reflected flux and polarization features, except on their timing.**
**Figure** 3 **shows the geometry of the planet–moon system with the reference frames describing the orbit of the planet–moon barycenter around the star, and the orbit of the moon around the barycenter. The nested two–body approximation assumes that the motions of the planet–moon barycenter around the star and that of the moon around the barycenter are independent.** **Orthonormal, right–handed coordinate system** \(S_{1}=\{x_{1},y_{1},z_{1}\}\) **is the reference frame for the observation of the planet–moon–star system, with the star at the origin, and the** _\(z_{1}\)_**–axis pointing towards the observer. Plane** \(p_{1}\)**, through** \(x_{1}\) **and** \(y_{1}\)**, is the plane on the observer’s sky, onto which the pixels (Fig.** 1**) are projected. Axes** _\(x_{1}\)_ **and** _\(y_{1}\)_ **have an arbitrary (but fixed) orientation. Coordinate system** \(S_{2}=\{x_{2},y_{2},z_{2}\}\) **is the reference frame for the orbit of the barycenter, which lies in plane** \(p_{2}\)**, through** \(x_{2}\) **and** \(y_{2}\)**. The lunar orbit itself lies in the** \(x_{3}\)**–**\(y_{3}\)**–plane of coordinate system** \(S_{3}=\{x_{3},y_{3},z_{3}\}\) **that is centered at the barycenter’s position. The various orbital elements in these coordinate systems are indicated as follows:**
\[\begin{array}[]{lp{0.8\linewidth}}a&\tabularcell@hbox{semi--major axis}\\ e&\tabularcell@hbox{eccentricity [0, 1]}\\ i&\tabularcell@hbox{inclination angle [0\degr, 180\degr]}\\ \omega&\tabularcell@hbox{argument of periastron [0\degr, 360\degr]}\\ \Omega&\tabularcell@hbox{right ascension of the ascending node [0\degr, 360 \degr]}\\ \nu&\tabularcell@hbox{true anomaly [0\degr, 360\degr]}\end{array}\]
**With the orbital parameters of the barycenter and the moon, we compute the true anomalies** \(\nu_{\rm bs}\) **of the barycenter and** \(\nu_{\rm mb}\) **of the moon around the barycenter, respectively, at any time** \(t\) **using Kepler’s equation for elliptical orbits, i.e.**
\[E(t)-e\sin{E(t)}=M(t),\] (12)
**where** \(E\) **is the eccentric anomaly and** \(M\) **the mean anomaly of the orbit. The true anomalies of the barycenter and the moon are related to the eccentric anomaly** \(E\) **through:**
\[\tan{\dfrac{\nu(t)}{2}}=\sqrt{\dfrac{1+e}{1-e}}\tan{\dfrac{E(t)}{2}}.\] (13)
**We solve for** \(\nu(t)\) **for each orbit in the appropriate reference system by applying the Newton–Raphson method** **(****Wakker****,** 2015**)** **to Eqs.** 12 **and** 13**.**
**We compute the position of the barycenter,** \(\mathbf{r}_{\rm bs}\)**, in coordinate system** \(S_{2}\)**, and the position of the moon around the barycenter,** \(\mathbf{r}_{\rm mb}\)**, in coordinate system** \(S_{3}\)**. The absolute position of the moon in coordinate system** \(S_{2}\) **is then obtained through:**
\[\mathbf{r}_{\rm ms}(t)=\mathbf{r}_{\rm mb}(t)+\mathbf{r}_{\rm bs}(t)\,.\] (14)
**As formulated by** **Murray & Correia****(**2010**)****, the position of the barycenter and the moon in** \(S_{2}\) **at time** \(t\) **can be put through a series of transformation matrices to yield the position of the barycenter and the moon in the observer’s coordinate system** \(S_{1}\)**. For further details on these transformation matrices, see** **Kipping****(**2011, 2010**)****.**
**After having computed the positions of the planet and the moon in** \(S_{1}\) **at time** \(t\)**, we compute the positions of the pixels across the planetary and lunar disks (see Fig.** 1**), and the angles** \(\beta_{i}\) **that are used to rotate locally computed Stokes vectors to the planetary and lunar scattering planes (Eq.** 6**), respectively, Then we calculate parameter** \(d_{i}\)**, which accounts for the change of the standard incident flux due to the changing distance to the star** **(see Eq.** 6**), and the local illumination and viewing angles required for the computation of the Stokes vector of reflected starlight for each pixel seen by the observer. Details on these computations can be found in App.** A**. For each** \(t\)**, we also compute angle** \(\psi\) **to rotate** \({\bf F}^{\rm m}\)**, the disk-integrated Stokes vector for the moon, to the planetary scattering plane (Eq.** 4**).**
### Our baseline planet–moon system
**In this paper, we focus on planet–moon systems in edge–on geometries, in which the inclination angle of the barycenter’s orbit is 90**\({}^{\circ}\)**, because exoplanets in (near) edge–on orbits are prime targets for space telescopes such as TESS, JWST, PLATO and CHEOPS, that all will employ the transit method to detect and/or characterize exoplanets, as well as for follow–up missions including telescopes aimed at directly detecting planet signals.**
<figure><img src="content_image/1807.10266/x4.png"><figcaption>Figure 4: Sketch illustrating the orbital geometry of our edge–on planet–moonsystem at time t=0 as seen from the positive y–axis (left) and from theobserver’s position at the positive z–axis (right). Indexes s, p, m, and brefer to the positions of the star, the planet, the moon, and the planet–moonsystem barycenter, respectively. Distances and radii are not to scale.</figcaption></figure>
**Table** 1 **lists the orbital elements of our baseline planet–moon system. Both the barycenter’s and the lunar orbit are assumed to be circular (**\(e=0.0\)**), and their semi–major axes match those of the Earth–Moon system** **(****Williams****,** 2017**)****. We neglect the Earth–barycenter distance, so that** \(a_{\rm bs}=1\) **AU. The semi–major axis of the lunar orbit,** \(a_{\rm mb}\)**, is computed from the Moon–Earth semi–major axis,** \(a_{\rm mp}=2.5696\cdot 10^{-3}\) **AU** **(****Williams****,** 2017**)****, as follows:**
\[a_{\rm mb}=a_{\rm mp}\hskip 1.422638pt\frac{m_{\rm p}}{m_{\rm p}+m_{\rm m}} \approx 2.54\cdot 10^{-3}\hskip 5.690551pt{\rm AU}\,,\] (15)
**with** \(m_{\rm p}\) **and** \(m_{\rm m}\) **the masses of the Earth and Moon, respectively. Because of the edge–on geometry, the right ascensions of the ascending nodes of the orbits of the barycenter and the moon are set to zero. Because both orbits are assumed to be circular, their perihelions are undefined. The barycenter’s argument of perihelion is chosen precisely behind the star at time** \(t=t_{0}=0\)**, i.e.** \(\omega_{\rm b}=270\degr\)**. For the moon,** \(\omega_{\rm m}\) **is set to zero. The observational and orbital geometry at** \(t=0\) **is sketched in Fig.** 4**. We use** \(R_{\rm p}=6371.0\) **km and** \(R_{\rm m}=1737.4\) **km for the baseline radii of the planet and the moon, respectively.**
| Barycenter | Moon
---|---|---
a [AU] | 1.0 | 0.00254
e [-] | 0.0 | 0.0
i [°] | 90.0 | 0.0
ω [°] | 270.0 | 0.0
Ω [°] | 0.0 | 0.0
t0 [sec] | 0.0 | 0.0
Table 1: The orbital elements of the barycenter of our planet–moon system and
of the orbit of the moon, with a the semi–major axis, e the eccentricity, i
the inclination angle, ω the argument of the perihelion, Ω the right ascension
of the ascending node, and t0 the time of perihelion passage. The inclination
angle of the lunar orbit is defined with respect to the normal on the
barycenter orbit.
<figure><img src="content_image/1807.10266/x5.png"><figcaption>Figure 5: Reflected fluxes F, Q, and U, and the direction of polarization χacross the planet and the moon at α=0\degr (top) and 50\degr (bottom). FluxesQ and U, and χ are defined with respect to the scattering planes of the planetand the moon, respectively. In order to facilitate the interpretation of thedegree of polarization, we plot 180\degr−χ. Fluxes Q and U are zero across themoon’s disk because of the Lambertian reflection. All fluxes have beennormalized such that the disk–integrated flux F at α=0\degr equals the body’sgeometric albedo. Absolute planetary fluxes per pixel are not comparable tothe absolute lunar fluxes per pixel because of the different number of pixelsacross each disk.</figcaption></figure>
## 3Numerical results
**Here, we present the computed total flux** \(F\)**, the linearly polarized fluxes** \(Q\) **and** \(U\)**, and degree of polarization** \(P\) **of starlight that is reflected by our model planet–moon system across time. As a trade–off between spatial resolution, radiometric and polarimetric accuracy, and computational time, we use 50 and 14 pixels along the equators of the planet and moon, respectively (see App.** C**), resulting in** \(N_{\rm p}=1956\) **and** \(N_{\rm m}=156\) **(Eq.** 6**). In Sect.** 3.1**, we analyze the individual contributions of the planet and the moon, and in Sect.** 3.2**, the results for the** **spatially unresolved planet–moon system. In Sect.** 3.3**, we take a closer look at particular transit and eclipse events.**
### Reflection by the spatially resolved planet & moon
**In order to understand the traces of eclipses and transits in the flux and polarization of starlight reflected by spatially unresolved planet–moon systems, we first discuss the disk–resolved signals of the planet and the moon separately. Figure** 5 **shows the elements of the locally reflected Stokes vectors** \(\mathbf{F}^{\rm p}\) **and** \(\mathbf{F}^{\rm m}\) **and the direction of polarization** \(\chi\)**, with respect to the planetary and lunar scattering planes, respectively, at phase angles** \(\alpha\) **of 0**\(\degr\) **and 50**\(\degr\)**.**
**At** \(\alpha=0\degr\) **(Fig.** 5**a), both the planet and the moon would be behind the star and thus invisible, but their disk–resolved signals give insight in the reflection processes. For both bodies, total flux** \(F\) **is maximum at the sub-stellar/sub-observer region and decreases towards the terminator (which coincides with the limb at this phase angle). Because of the Lambertian reflection of the lunar surface and the lack of atmosphere around the moon, the reflected** **flux is unpolarized and** \(\chi\) **undefined for the moon. The linearly polarized fluxes** \(Q\) **and** \(U\) **of the planet are due to Rayleigh scattering in the planet’s atmosphere. At the sub-stellar region, both** \(Q\) **and** \(U\) **are zero because of symmetry. The general increase of** \(Q\) **and** \(U\) **towards the limb is due to polarized second order scattered light, which is also apparent from the direction of polarization** \(\chi\)**. Because of its definition,** \(Q\) **(**\(U\)**) equals zero along the lines at angles of 45**\({}^{\circ}\) **(0**\({}^{\circ}\)**) and -45**\({}^{\circ}\) **(90**\({}^{\circ}\)**) with the horizontal. Integrated across the planetary disk,** \(P\) **would equal zero. Note that because of the Lambertian reflection of the surface of the planet,** \(Q\) **and** \(U\) **are independent of planetary surface albedo** \(a_{\rm surf}\)**, while** \(P\) **will generally decrease with increasing** \(a_{\rm surf}\) **because of the increasing flux** \(F\)******(see e.g.** **Stam****,** 2008**, for sample computations)****.**
**At** \(\alpha=50^{\circ}\) **(Fig.** 5**b), the total flux** \(F\) **of the moon is maximum at the sub-stellar region and decreases towards the terminator, due to the isotropic surface reflection and the absence of an atmosphere. The planet also shows a decrease of** \(F\) **towards the terminator, but the location of the flux maximum is more diffuse and more towards the limb than on the moon, because light that is incident on the planet is scattered in the atmosphere in addition to being reflected by the surface; the reflected flux thus also depends on the optical path–lengths through the atmosphere, which in turn depend on the local** **illumination and viewing angles. The planet’s polarized fluxes** \(Q\) **and** \(U\)**, and angle** \(\chi\) **are mostly determined by starlight that has been singly scattered by the atmospheric gas molecules. For our choice of reference plane,** \(U\) **is anti–symmetric with respect to this plane (and** \(U\) **would thus equal zero when integrated across the disk), and** \(Q\) **is symmetric. The negative values for** \(Q\) **in Fig.** 5**b indicate that the reflected light is polarized perpendicular to the reference plane, which is indeed also clear from the polarization angle** \(\chi\)**, and what is expected for a Rayleigh scattering atmosphere.**
<figure><img src="content_image/1807.10266/x6.png"><figcaption>Figure 6: Sketch illustrating the sequence of planetary (1, 4, 8, …) and lunar(2, 6, …) transits, as well as planetary (3, 7, …) and lunar (1, 5, 9, …)eclipses for part of the barycenter’s orbit for an edge-on system. Thepositive z0–axis points towards the observer. Position 1 corresponds to phaseangle α=0∘ and time t=0 s in our simulation (cf. Fig. 7).</figcaption></figure>
<figure><img src="content_image/1807.10266/x7.png"><figcaption>Figure 7: The total flux F, the linearly polarized fluxes Q and U, and thedegree of polarization P of the spatially unresolved, edge–on, base-lineplanet–moon system, as functions of the planet’s phase angle αp. Also includedare curves for the isolated planet (coinciding with Q and U of the planet–moonsystem) and the isolated moon (equal to zero in Q, U, and P). Fluxes have beennormalized such that at αp=0\degr, F equals the geometric albedo of theplanet–moon system or each of the isolated bodies. The labels in the plot forF refer to the illustrations in Fig. 6.</figcaption></figure>
### Reflection by the spatially unresolved planet & moon
**Figure** 7 **shows the disk–integrated** \(F\)**,** \(Q\)**,** \(U\)**, and** \(P\) **as functions of the planetary phase angle** \(\alpha_{\rm p}\)**. For our edge–on system, the curves cover only half of the barycenter’s orbital period. For comparison, we have also included curves for the planet and the moon as isolated bodies, thus without any mutual events. The total flux of the planet–moon system is lower than the sum of the total fluxes of the isolated planet and moon because the latter have not been scaled to the actual radii of the planet and the moon. Indeed the moon’s flux at** \(\alpha_{\rm p}=0^{\circ}\)**, equals the moon’s geometric albedo, i.e. 0.067, which matches the theoretical geometric albedo of a Lambertian reflecting body with surface albedo of 0.1** **(see** **Stam et al.****,** 2006**)****. The geometric albedo of the unresolved planet–moon system (with both bodies at** \(\alpha_{\rm p}=0^{\circ}\) **and next to each other) is about 0.33. Note that in Fig.** 7**, the** _observable_ **planet–moon flux at** \(\alpha_{\rm p}=0^{\circ}\) **is zero, because both bodies are then located behind the star as seen from the observer (in addition, the moon is located behind the planet,** **as can be seen from Fig.** 4**, and from situation 1 in Fig.** 6**).**
**The curves for** \(F\) **decrease smoothly with increasing** \(\alpha_{\rm p}\)**, apart from the occasional sharp dips due to eclipses and transits (to be discussed below) and reach zero close to** \(\alpha_{\rm p}=180^{\circ}\)**, where the planet and moon would both be in front of the star. The slightly different slope of the lunar flux phase function as compared to that of the planet is due to the scattering of light in the latter’s atmosphere. As can be seen in Fig.** 7**, without the sharp dips, the smooth flux phase function of the planet–moon system does not reveal the presence of a moon, especially not without accurate information on the planetary radius, orbital distance, atmospheric and surface properties.**
**Because the lunar surface is completely depolarizing, the moon’s polarized fluxes** \(Q\) **and** \(U\) **are zero at each** \(\alpha_{\rm p}\)**. The disk–integrated** \(U\) **of the light reflected by the planet is zero due to symmetry (see Fig.** 5**). Polarized flux** \(Q\) **and degree of polarization** \(P\) **of this light both show a smooth dependence on** \(\alpha_{\rm p}\)**, apart from the occasional sharp peaks that will be discussed below. The degree of polarization is maximum at phase angles between 90**\({}^{\circ}\) **and 100**\({}^{\circ}\) **due to the atmospheric Rayleigh scattering. At about 165**\({}^{\circ}\)**, the direction of polarization changes from perpendicular (**\(\chi=90^{\circ}\)**) to parallel (**\(\chi=0^{\circ}\)**) to the reference plane,** **and** \(P\) **equals zero. The degree of polarization of the unresolved planet–moon system is somewhat lower than that of the isolated planet, because of the added unpolarized lunar flux. If the planetary atmosphere would contain clouds, the shape of this continuum curve would depend on the optical thickness and altitude of the clouds, the microphysical properties of the particles, and the cloud coverage across the planetary disk** **(for sample curves, see** **Rossi & Stam****,** 2017; **Karalidi et al.****,** 2012**, and references therein)****.**
**While the smooth curves for the spatially unresolved planet–moon system shown in Fig.** 7**, do not provide direct evidence of the presence of a moon, the mutual events result in a series of dips and peaks in the reflected flux and polarization, respectively. Figure** 6 **illustrates the various events. Both the planet and the moon are initially (**\(\alpha_{\rm p}=0^{\circ}\)**) behind the star (position 1 in Fig.** 6**). Given the prograde lunar motion, the next event, when planet and moon are in view of the observer, is a lunar transit (position 2) and an eclipse of the star on the planet (3). After the first lunar period, the moon again disappears behind the planet (4), followed by an eclipse of the star on the moon (5). This sequence repeats along the barycenter’s orbit.**
**Both the planetary and lunar eclipses and transits temporarily reduce the flux** \(F\) **that the observer receives. Indeed, when the planet transits** **the moon, the system’s flux phase function equals that of the isolated planet. The dip in the system’s flux due to a lunar transit (moon in front of the planet) will depend on the radius of the moon as compared to that of the planet and on the lunar surface albedo: the lower the lunar surface albedo and/or the larger the lunar radius, the deeper the dip compared to the continuum. The depth of the dip in the system’s flux** \(F\) **due to an eclipse depends on the relative sizes of the moon and the planet, the reflection properties of the eclipsed body, and on the precise orbital geometry, especially because an eclipse shadow on the moon will not be completely black (cf. Fig.** 2**) (and the total flux** \(F\) **thus slightly larger) due to starlight that is refracted through the limb of the planetary atmosphere and reaches the moon. This refraction is not included in our code (due to the wavelength dependence of Rayleigh scattering, the contribution of refracted light would be larger in the (near) infrared region of the spectrum than at 450 nm).**
**Because the moon reflects unpolarized light, neither a planetary transit (planet in front of the moon) nor an eclipse on the moon leads to a reduction of the polarized fluxes, as can be seen in Fig.** 7**. Because the planet reflects polarized light, a transit of the moon and an eclipse on the planet will both decrease** \(Q\)******(given the geometry of our system). Because** \(P\) **depends on** \(F\) **and** \(Q\)**, the dips in** \(F\) **due to less (unpolarized) lunar light being observed yield peaks in** \(P\)**. The peak value of** \(P\) **that is due to the planet transiting the moon equals** \(P\) **of the isolated planet at that value of** \(\alpha_{\rm p}\)**. In our computational results, peaks in** \(P\) **that are due to an eclipse on the moon, would equal** \(P\) **of the isolated planet when the whole lunar disk would be in the planet’s umbra because we neglect refracted starlight through the limb of the planetary atmosphere. Changes in** \(P\) **that are due to the moon transiting the planet or due to the moon casting a shadow on the planet will depend on the total and polarized fluxes of the region of the planetary disk that is covered or darkened, and thus, for a given model planet and its atmosphere, on the relative sizes of the moon and the planet and the precise orbital geometry. This will be discussed further in Sect.** 3.3**.**
**The absolute depth of the dips in** \(F\) **and** \(Q\) **decreases with increasing** \(\alpha_{\rm p}\) **because the fraction of a body’s disk that is illuminated and visible decreases with increasing** \(\alpha_{\rm p}\)**. The amplitudes of features in** \(P\) **for our planet-moon model system are maximum when** \(\alpha_{\rm p}\approx 90\degr\)**. This is particularly convenient for exomoon detection with direct imaging techniques, because that is the phase** **angle range where the angular distance between the planet–moon system and the parent star will be largest.**
**As can be seen in Fig.** 7**, the phase angle gap between a lunar transit and the subsequent planetary eclipse (or a planetary transit and a subsequent lunar eclipse) increases with increasing** \(\alpha_{\rm p}\)**. As the orbital speed of both bodies is constant in our baseline system with circular orbits, this also applies in the time domain. Indeed, the lunar and planetary transits have a characteristic period because an observer–planet–moon alignment occurs twice per lunar orbit (see Fig.** 6**). The time gap between two consecutive transits and eclipses, however, increases with increasing** \(\alpha_{\rm p}\) **because of the movement of the barycenter along its orbit.**
### Analysis of the mutual events
**In this section, we analyze individual mutual events, i.e. their shape, symmetry, periodicity, magnitude, and duration. For this analysis, the change in flux** \(F\) **and degree of polarization** \(P\) **during an event are defined as follows**
\[\Delta F = F_{\rm event}-F_{\rm continuum},\] (16)
\[\Delta P = P_{\rm event}-P_{\rm continuum}.\] (17)
**First, we’ll discuss the lunar and planetary transits, then the lunar and planetary eclipses.**
<figure><img src="content_image/1807.10266/x8.png"><figcaption>Figure 8: Sketch of the ingress and egress of the moon during a lunar transitfor αp=70\degr and αp=110\degr. The arrow indicates the direction of motion ofthe moon across the planetary disk.</figcaption></figure>
**Lunar transits******
**Figures** 9**a sand b provide detailed views of** \(\Delta F\) **and** \(\Delta P\) **during the six lunar transits (moon in front of the planet) shown in Fig.** 7**, together with sketches of the geometry of the planet and the moon at the beginning and the end of the transit for** \(\alpha_{\rm p}\approx 80^{\circ}\)**. As expected with constant orbital speeds, the duration of a lunar transit event decreases with increasing** \(\alpha_{\rm p}\) **because of the decrease of the illuminated area on the planetary disk, and thus the shift of the time of ingress (see Fig.** 8**). Because egress takes place over the planetary limb, all curves in Figs.** 9**a and b have the same egress time. Also, the planet is relatively dark near the terminator, and thus yields a smooth flux decrease upon the lunar ingress, while it is bright near the limb (see Fig.** 5**), yielding a rapid increase of** \(F\) **upon the lunar egress.**
**The depth** \(\Delta F\) **depends strongly on** \(\alpha_{\rm p}\)**, because with increasing** \(\alpha_{\rm p}\)**, the illuminated area, and hence also the covered area on the planet decreases. The shape of** \(\Delta F\) **also depends on** \(\alpha_{\rm p}\)**. At** \(\alpha_{\rm p}=0^{\circ}\)**, the curve would be symmetric. At larger values of** \(\alpha_{\rm p}\)**, the trace of the lunar night–side starts to appear in the curve.** **Because of the moon’s prograde orbit, the lunar day–side ingresses before the night–side. In the curves for** \(\alpha_{\rm p}=13.4^{\circ}\)**, and 40.3**\({}^{\circ}\)**, the steeper decrease of** \(F\) **due to the ingress of the lunar night–side can be seen. The value of** \(\Delta F\) **that is reached within a transit at a given** \(\alpha_{\rm p}\) **depends on the lunar albedo and on the area of the planetary disk that is covered, thus on the lunar radius. The lower the lunar surface albedo and/or the larger the lunar radius as compared to the planetary radius, the larger** \(\Delta F\) **will be. As an example, Fig.** 11**a shows** \(F\) **at** \(\alpha_{\rm p}=67.2^{\circ}\) **for various values of the lunar radius expressed as fraction of the planetary radius (the value for the baseline model is approximately 0.3). As can be seen, the continuum flux increases with increasing lunar radius due to the increased amount of flux reflected by the moon, and, indeed the lowest flux during the transit decreases and** \(\Delta F\) **increases with increasing lunar radius.**
**Note that a change in the lunar radius implies a change in the lunar mass (assuming a similar composition) and, thus, a change in the lunar period around the planet. While the frequency of the events decreases non–linearly with increasing lunar radii, we have aligned the mutual events in Fig.** 11 **in time to facilitate a comparison. Mutual transits show up every half lunar sidereal period.** **Because** \(T\propto\sqrt{1/(M_{\rm m}+M_{\rm p})}\)**, relative timing variations of 1% to 13% are obtained for lunar–to–planet radius ratios from 0.1 to 0.7 (with our baseline value of approximately 0.3). In the case of eclipses, and assuming coplanar circular lunar and planetary orbits, the repetition period equals the lunar synodic period, for which timing variations of 1% to 14% are obtained for the same range of radii ratios.**
<figure><img src="content_image/1807.10266/x9.png"><figcaption>Figure 9: Changes in the total reflected flux ΔF (a and c), and the degree ofpolarization ΔP (b and d), as functions of the lunar true anomaly, νmb, andrelative time for the lunar transits (top) and planetary transits (bottom)shown in Fig. 7. The time–step of these simulations is 3 minutes.</figcaption></figure>
<figure><img src="content_image/1807.10266/x10.png"><figcaption>Figure 10: Similar to Fig. 9, except for the planetary eclipses (top) and thelunar eclipses (bottom), both as functions of angle φms (see Fig. 12).</figcaption></figure>
<figure><img src="content_image/1807.10266/x11.png"><figcaption>Figure 11: Changes in the total reflected flux ΔF (a and c) and ΔP (b and d)during lunar transits at α=67.2\degr (top) and planetary eclipses atα=72.5\degr (bottom) for various lunar–to–planetary radius ratios r. Thehorizontal axis shows the elapsed time since the concentric alignment of theplanet and moon as seen from the star in the case of an eclipse and as seenfrom the observer in the case of a transit. The time–step of these simulationsis 3 minutes. The baseline lunar–to–planetary radius ratio r is about 0.3.</figcaption></figure>
**Figure** 9**b shows** \(\Delta P\) **during lunar transits. It can be seen that** \(P\) **can also decrease during a transit, which is not apparent from the curves in Fig.** 7**. The curves exhibit a strong variation in shapes, and with increasing** \(\alpha_{\rm p}\)**, get increasingly asymmetric. The largest** \(\Delta P\) **is found around** \(\alpha_{\rm p}=90^{\circ}\)**, where** \(P\) **of our model planet is highest (see Fig.** 7**). The precise shapes of the curves depend on the properties of the planet and its moon and the path of the transit across the planetary disk.**
**In our planet–moon system, the lunar transit occurs along the planet’s equator, where the antisymmetry of** \(U\) **yields a null net contribution, and the shape of** \(\Delta P\) **thus depends on the variation of** \(Q\) **and** \(F\) **along the path (cf. Fig.** 5**). At** \(\alpha_{\rm p}=13.4^{\circ}\)**,** \(Q\) **is maximum near the planet’s limb and close to zero at the center of the body. The disk–integrated** \(P\) **is close to zero due to symmetry. During ingress and egress, the moon breaks the symmetry, and (slightly) increases the disk–integrated value of** \(P\) **(see Fig.** 9**b). With increasing** \(\alpha_{\rm p}\)**, the maximum** \(\Delta P\) **increases,** **to reach a maximum (in the figure) at** \(\alpha_{\rm p}=90.1^{\circ}\)**. The** \(\alpha_{\rm p}\) **where the maximum** \(P\) **is found, corresponds roughly with the** \(\alpha_{\rm p}\) **of the minimum** \(F\)**. This increase in** \(P\) **appears to be driven by the decrease of** \(F\)**.**
**The negative values of** \(\Delta P\) **in the curves for** \(\alpha_{\rm p}=67.2^{\circ}\)**,** \(90.1^{\circ}\) **and 120.9**\({}^{\circ}\)**, indicate that during that part of the transit, the decrease in** \(Q\) **is larger than that in** \(F\)**. This happens in particular when the illuminated part of the moon is transiting the illuminated part of the planetary disk, while the dark part of the moon is still transiting the dark part of the planet, and when the moon transits the limb of the planet, where** \(Q\) **is relatively large and where the transit thus strongly decreases** \(Q\)**.**
**As mentioned earlier, the maximum** \(\Delta P\) **(on the order of 2 % for our planet–moon system) should be observable when** \(\alpha_{\rm p}\) **is between about 70**\({}^{\circ}\) **and 150**\({}^{\circ}\)**, where the angular distance between the planet–moon and the star is relatively large, which would facilitate the detection of lunar transits with direct detection methods.**
**Figure** 11**b shows the change in** \(P\) **at** \(\alpha_{\rm p}=67.2^{\circ}\)**, for various values of** \(r\)**, the lunar radius expressed as a fraction of the planetary radius. With increasing** \(r\)**, the continuum** \(P\) **decreases because more** **unpolarized flux reflected by the moon is added to the total flux. During a transit,** \(P\) **can be seen to be very sensitive to the lunar radius. Indeed, with increasing** \(r\)**,** \(\Delta P\) **changes from positive (**\(P\) **during the event is higher than in the continuum) to negative (**\(P\) **during the event is lower than in the continuum) because apparently, the polarized flux reflected by the planet decreases more during the event than the total flux reflected by the planet with the moon in front of it.**
**Planetary transits******
**Figures** 9**c and d show the planetary transits (planet in front of the moon) from Fig.** 7 **in detail, both for** \(F\) **and** \(P\)**. At** \(\alpha_{\rm p}=0^{\circ}\)**, the planet and the moon are behind the star and would thus not be visible to the observer, but, similarly to the lunar transits, planetary transits would yield symmetric events in** \(\Delta F\) **and** \(\Delta P\)**. With increasing** \(\alpha_{\rm p}\)**, the events become more asymmetric.**
**With a planetary transit, the shapes of the** \(\Delta F\) **curves are very different from those of the lunar transits, because, for our orbital geometry,** **the moon will be completely covered during the planetary transit and while it is covered, the transit curve is flat. The depth of the transit depends on the total amount of reflected flux received from the (isolated) moon. For a given value of** \(\alpha_{\rm p}\)**,** \(\Delta F\) **will increase linearly with the surface area of the lunar disk (thus, with the lunar radius squared), and/or with the lunar surface albedo. The start time of the transit depends on** \(\alpha_{\rm p}\)**, as** \(\alpha_{\rm p}\) **determines the extent of the illuminated area on the moon, and thus when it will be covered. Like with the lunar transits, the end of the transit, over the bright limb of the moon, is independent of** \(\alpha_{\rm p}\)**; it only depends on the lunar true anomaly.**
**For the planetary transits,** \(\Delta P\) **is entirely due to the decrease in** \(F\)**, as the moon itself reflects only unpolarized light, and the planet thus blocks no polarized flux. As a result, the transits in** \(\Delta P\) **are flat as long as the illuminated part of the lunar disk is covered. The maximum** \(\Delta P\) **depends both on the** \(\Delta F\) **and on the planet’s polarized flux** \(Q\)**, and thus on** \(\alpha_{\rm p}\) **for a given planet-moon model. In Fig.** 9**d, a maximum** \(\Delta P\) **of about 2.5**\(~{}\%\) **occurs at** \(\alpha_{\rm p}=80.6\degr\)**. Through** \(\Delta F\)**,** \(\Delta P\) **will increase** **with the lunar surface albedo and/or the surface area of the lunar disk at a given** \(\alpha_{\rm p}\) **and for a given planet-moon model; a darker and/or smaller moon would yield a smaller** \(\Delta F\) **and hence a smaller** \(\Delta P\)**.**
**Planetary eclipses******
**Figures** 10**a and b show the curves of** \(\Delta F\) **and** \(\Delta P\) **during the planetary eclipse events shown before in Fig.** 7**. During these eclipses, the moon casts its shadow on the planet, and because in our geometry the lunar orbital plane coincides with the barycenter’s orbital plane, the shadow of the moon travels along the horizontal line crossing the center of the planetary disk. Figure** 12 **illustrates the geometries for the planetary eclipse, with** \(\varphi_{\rm ms}\) **the angle between the star and the moon measured positive in the counter clock–wise direction from the center of the planet. Angle** \(\varphi_{\rm ms}\) **is used as relative measure of eclipse events. Its relation with time is linear, given the circular orbital motion of the bodies.**
**For planetary eclipses, the explanation regarding the asymmetry of** \(\Delta F\) **and** \(\Delta P\) **is the same as for lunar transits, with one important difference: while the transit events start increasingly later with increasing** \(\alpha_{\rm p}\) **and end at the same (relative) time, the planetary eclipses start at the same (relative) time and end increasingly earlier with increasing** \(\alpha_{\rm p}\)**. The reason for this difference is that eclipses depend on the position of the star with respect to the planet–moon system, and not on the position of the observer. The observer’s position does influence the fraction of the eclipse that is captured, as it determines the phase angle and hence the fraction of the illuminated part of the planetary disk across which the eclipse travels. Thus, with increasing** \(\alpha_{\rm p}\)**, the duration of an eclipse decreases, as is visible in Figs.** 10**a and b. The depth** \(\Delta F\) **decreases with increasing** \(\alpha_{\rm p}\) **because less of the illuminated part of the planetary disk is visible.**
**The shape of the** \(\Delta F\) **curves for the planetary eclipses (where the moon’s shadow moves across the planetary disk) appears to be more gradual than that of the lunar transit curves (where the moon itself moves across the planetary disk) (cf. Fig.** 9**). This is because the planet first travels through the lunar penumbral shadow, before entering the deep, umbral shadow cone. Because we discuss only half of the barycenter’s orbit, the ingress of the lunar eclipse shadow on the illuminated part of the planetary disk is through the terminator and the egress through the bright limb, as illustrated** **in Figs.** 8 **and** 12**. However, because the spatial extent of the penumbral and umbral shadows across the disk are smallest halfway the total duration of the eclipse (as seen from a vantage point on the moon facing the planet), the egress of the lunar shadow yields a much smoother** \(\Delta F\) **curve than observed during the egress of a lunar transit. The difference in the maximum** \(\Delta F\) **during a lunar transit and a planetary eclipse is most apparent at the smaller phase angles, because with increasing** \(\alpha_{\rm p}\)**, the contribution of light reflected by the moon decreases. Note that differently than for lunar transits, the value of** \(\Delta F\) **during a planetary eclipse is independent of the lunar surface albedo. It will obviously increase with the radius of the moon relative to that of the planet. This can also be seen in Figs.** 11**c and d, which show** \(F\) **at** \(\alpha_{\rm p}=72.5^{\circ}\) **for various values of** \(r\)**, the lunar radius expressed as fraction of the planetary radius (for the baseline model,** \(r\) **is 0.3). Indeed, with increasing** \(r\)**, the continuum flux increases because of the added flux reflected by the moon, and the minimum flux during the event decreases because of the increasing extent of the lunar shadow.**
**The change in** \(P\) **during the planetary eclipses is shown in Fig.** 10**b. The increase in** \(P\) **during the ingress of the lunar shadow is due to the decrease in** \(F\) **and a decrease in** \(Q\)**. As the shadow progresses across the disk, its spatial** **extent decreases, and its influence on** \(P\) **decreases. The maximum of** \(\Delta P\) **appears to be** \(1-2~{}\%\) **for phase angles** \(\alpha_{\rm p}\approx 70\degr-160\degr\)**. For the largest phase angles, the corresponding value of** \(\Delta F\) **is relatively small, because only a narrow crescent of the planet is illuminated, so there** \(\Delta P\) **is mostly due to a change in** \(Q\)**.**
**Figure** 11**d shows** \(P\) **at** \(\alpha_{\rm p}=72.5^{\circ}\) **for various values of** \(r\) **(the lunar radius expressed as fraction of the planetary radius). With increasing** \(r\)**, the continuum** \(P\) **decreases because of the added unpolarized flux reflected by the moon, and** \(P\) **during the eclipse decreases, too, apparently because the polarized flux** \(Q\) **decreases more than the total flux** \(F\)**.**
<figure><img src="content_image/1807.10266/x12.png"><figcaption>Figure 12: The geometrical definition of angle φms as moon m passes betweenplanet p and star s. Seen from the top, the moon moves anti–clockwise aroundthe planet. A similar definition holds for angle φmp as the planet passesbetween the moon and the star. Distances between bodies and radii are not toscale.</figcaption></figure>
**Lunar eclipses******
**As can be seen in Figs.** 10**c and d, lunar eclipses, when the planet casts its shadow on the moon, show a similar symmetry as the planetary transits, where the planet moves in front of the moon (Fig.** 9**c and d). Because our model moon is small compared to the planetary shadow, both the** \(\Delta F\) **and** \(\Delta P\) **curves are flat except during ingress and egress. The flux changes during ingress and egress of the lunar eclipse, respectively, are smoother than with the planetary transits, due to the extended penumbral region of the planet’s shadow.**
**Like with the planetary eclipses discussed above, the moon’s ingress** **into the planetary shadow occurs at the same value of angle** \(\varphi_{\rm ms}\) **(cf. Fig.** 12**) independent of phase angle** \(\alpha_{\rm p}\)**. The duration of the eclipse decreases with increasing** \(\alpha_{\rm p}\) **because of the decrease of the illuminated area on the moon with increasing** \(\alpha_{\rm p}\)**. The change in** \(P\) **during lunar eclipses, shown in Fig.** 10**, is similar to that during planetary transits (Fig.** 9**), as in both cases** \(P\) **changes during the event because** \(F\) **decreases and because there is no actual change in the amount of polarized flux from the system, because our model moon reflects only unpolarized light. The maximum** \(\Delta P\) **value is about** \(2.7~{}\%\)**, attained at** \(\alpha_{\rm p}\approx 87.1^{\circ}\) **in Fig.** 10**.**
**With increasing lunar radius as compared to the planetary radius, and depending on the distance between the moon and the planet and the distance to the star, the planetary shadow might cover only part of the lunar disk. In that case,** \(F\) **and** \(P\) **will no longer be flat during the eclipse, because there will be a contribution of unpolarized lunar flux that will vary in time. Indeed, the curves will become asymmetric (more similar to those for the planetary eclipses). Because the moon only reflects unpolarized flux,** \(P\) **will always increase during the eclipses.**
## 4Conclusions
**We present numerically simulated flux and polarization phase functions of starlight that is reflected by an orbiting planet–moon system, including mutual events, such as transits and eclipses. Most results presented in this paper apply to a Moon–sized, Lambertian (i.e. isotropically and depolarizing) reflecting moon orbiting an Earth–sized exoplanet with an Earth–like, gaseous atmosphere on top of a Lambertian reflecting surface (the surface pressure is 1 bar), in an edge–on configuration. Our results show that the flux and polarization phase functions of starlight reflected by such a planet–moon system contain traces of the moon in the form of periodic changes in the total flux** \(F\) **and degree of polarization** \(P\) **as the bodies shadow each other (eclipses) and/or hide one another from the observer’s view (transits) along their orbit around the star. These changes in** \(F\) **and** \(P\) **are only one order of magnitude smaller than the system’s continuum phase functions. The magnitude, shape and duration of the obtained total flux signatures are comparable with the results by** **Cabrera & Schneider****(**2007**)****, except that they do not include the influence of the penumbra.**
**During events that darken the planet, i.e. the lunar transits and planetary eclipses, the shape of the dip in** \(F\) **depends on the reflection properties of the regions on the planet along the path of the shadow. The change in** \(P\) **during such events strongly depends on the ratio of polarized–to–total reflected flux across the disk and along the path of the shadow. Indeed,** \(\Delta P\approx 1\%-1.8\%\) **for** \(67\degr<\alpha_{\rm p}<121\degr\)**. For the planet–moon system used in this paper, we found the strongest changes in** \(P\) **during either the first (planetary eclipse) or the last (lunar transit) half of the event, compared to the duration of the event in** \(F\)**, in particular at intermediate phase angles. The asymmetry of the planet darkening events as imprinted on the change in polarization** \(\Delta P\) **is due to the variation of the polarization across the planetary disk with our model atmosphere–surface: the polarized flux at the limb is higher than at the terminator.**
**During lunar darkening events, i.e. planetary transits and lunar eclipses, the size difference between the planet and the moon** **yields a relatively symmetric change in** \(F\) **and, due to the non–polarizing lunar reflection, a similarly relatively symmetric change in** \(P\)**, as the latter is only due to the decrease in total flux, upon a lunar darkening event, not to a change in polarized flux. The curves for planetary transits have steeper slopes during the ingress and egress phases than the curves for the lunar eclipses, because with the latter, the moon travels through the penumbral shadow of the planet. The change in** \(P\) **depends on the size and albedo of the moon, and on the polarization signal of the planet, which itself depends on the atmosphere-surface model and the phase angle. Our simulations have been performed at 450 nm, i.e. in the blue, where the scattering by the gas in the Earth–like planetary atmosphere strongly contributes to the planet’s polarization signals. In particular at intermediate phase angles, the polarization signal of a gaseous atmosphere is strong, and the change in polarization during the lunar darkening events can reach a few percent. Indeed,** \(\Delta P\approx 1.25\%-2.66\%\) **for** \(54\degr<\alpha_{\rm p}<108\degr\) **during planetary transits. Note that at these phase angles, the angular distance between the planet–moon system and the parent star is relatively large, so these angles are favorable for the detection of reflected starlight. For a planet with clouds in its atmosphere, the continuum flux phase function will have a similar shape as that for** **our cloud–free planet, except the total amount of reflected flux will be larger (of course not at wavelengths where atmospheric gases absorb the light). The polarization curve of a cloudy planet will show angular features due to the scattering of light by the cloud particles, such as the rainbow for liquid water droplets** **(see e.g.** **Karalidi et al.****,** 2012; **Bailey****,** 2007**, and references therein)****.**
**The duration of a transit event depends on the orbital parameters, on the sizes of the planet and moon, and on the phase angle (the latter mostly for lunar transits). In our simulations, a typical planetary transit takes** \(\sim 4\) **hours both in flux and polarization. A lunar transit at an intermediate phase angle of 90**\({}^{\circ}\)**, takes about 2 hours in flux. In polarization, the change in** \(P\) **is apparent during a shorter period than the change in flux** \(F\)**, due to the distribution of polarized flux across the planetary disk. The duration of eclipse events is somewhat longer than that of transit events due to the diverging shape of the shadow cone. In our simulations, eclipse events can take up to 6 hours, where the polarization change in the planetary eclipses is only apparent during part of the time of the flux change.**
**The results presented in this paper correspond to half of the planetary orbit around the star. The results for the other half of the orbit will be similar, except that the curves will be mirrored with respect to the central event time, because transit and eclipse ingresses and egresses will happen** **over the other side of the darkened body.**
**Our results show that measuring the temporal variation in** \(F\) **and/or** \(P\) **during transits and eclipses could provide extra information on the properties of a planet and/or moon and their orbits. Extracting such information, however, requires not only detecting such events but also measuring the shape of the variations in** \(F\) **and/or** \(P\)**. For the interpretation of such measurements, numerical simulations to map in more detail the influence of the physical characteristics of the moon and the planet (radius, albedo, atmosphere-surface properties) and their orbital characteristics (inclination angles, ellipticity) on the temporal variation in** \(F\) **and** \(P\) **are required. Such simulations will be targeted in future research.**
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## Appendix ALocal illumination and viewing angles
**In this appendix, we describe the computation of the angles required to compute the starlight that is reflected by each pixel on the planet (cf. Eq.** 8**): the phase angle** \(\alpha\)**, the local viewing zenith angle** \(\theta_{i}\)**, the local illumination zenith angle** \(\theta_{0i}\)**, the local azimuthal difference angle** \(\phi_{i}-\phi_{0i}\)**, and the local rotation angle** \(\beta_{i}\)**. To add the computed Stokes vectors of the moon to those of the planet, we usually also need rotation angle** \(\psi\) **that redefines the lunar Stokes vector from the lunar scattering plane to the planetary scattering plane (cf. Eq.** 4**).**
<figure><img src="content_image/1807.10266/x13.png"><figcaption>Figure 13: Angular geometry of the spherical triangle centered at pixel i anddefined by the zenith direction unit vector, uzi, the observer’s directionunit vector, uob, and the star direction unit vector, us. The sides of thespherical triangle are: the observer–zenith angle θi , the star–zenith angleθ0i, and the pixel–based phase angle αi, all centered at pixel i. The anglebetween sides θ0i and θi is the azimuthal difference angle ϕi−ϕ0i.</figcaption></figure>
### Phase angle \(\alpha_{\rm x}\)
**Phase angle** \(\alpha\) **is the angle between the direction to the star and the observer, as measured from the center of a body (see Fig.** 13**). In principle its value ranges from 0**\({}^{\circ}\) **to** \(180\degr\)**, although the phase angle range accessible to an observer depends on the inclination angle of the orbit of a body. A body in an edge–on orbital geometry (inclination angle** \(i=90^{\circ}\)**) can attain phase angles between 0**\({}^{\circ}\) **(when it is located behind the star) and 180**\({}^{\circ}\)**, while a body in a face–on orbital geometry (**\(i=0^{\circ}\)**) can only be observed at** \(\alpha=90^{\circ}\)**. Generally, given an orbital inclination angle** \(i\)**, the phase angle range is given by**
\[90^{\circ}-i\leq\alpha\leq 90^{\circ}+i.\] (18)
**The phase angle of the planet or the moon at time** \(t\) **is computed as**
(19)
**where subscript ’x’ refers to either ’p’ (planet) or ’m’ (moon),** \(\mathbf{u}^{T}_{\rm z}=\left[0,0,1\right]\) **is the unit vector along the** \(z\)**–axis, pointing towards the observer, and vector** \({\bf r}_{\rm xs}\) **connects the center of the planet or moon with the center of the star. Given the small separation between the planet and moon compared to their distances to the star,** \(\alpha_{\rm p}\) **is virtually the same as** \(\alpha_{\rm m}\)**, yet our numerical model uses both values.**
### Local viewing zenith angle \(\theta_{i}\)
**The local viewing zenith angle** \(\theta_{i}\) **is the angle between the zenith direction of pixel** \(i\) **and the direction towards the observer (see Fig.** 13**). Angle** \(\theta_{i}\) **takes values between** \(0\degr\) **(at the sub-observer location) and** \(90\degr\) **(at the limb). It depends on the location of the pixel on the disk of the planet or moon and is thus time-independent. It is computed according to**
\[\theta_{i}=\arccos\left[\mathbf{u}^{T}_{\rm z}\cdot\frac{\mathbf{r}_{i{\rm x}} }{\left\lVert\mathbf{r}_{i{\rm x}}\right\rVert}\right]\,,\] (20)
**where subscript ’x’ refers to either ’p’ or ’m’,** \(\mathbf{u}_{\rm z}\) **is the unit vector along the** \(z\)**–axis that points towards the observer, and** \(\mathbf{r}_{i{\rm x}}\) **is the vector pointing to the center of the pixel from the center of either the planet or the moon.**
### Local illumination zenith angle \(\theta_{0i}\)
**The local illumination zenith angle** \(\theta_{0i}\) **is defined as the angle between the local zenith direction of pixel** \(i\) **and the direction towards the star (see Fig.** 13**). Angle** \(\theta_{0i}\) **takes values between** \(0^{\circ}\) **(at the sub-stellar location) and** \(90^{\circ}\) **(at the terminator). The position of the star changes in time, so that the time–dependent local illumination zenith angle can be computed as**
\[\theta_{0i}(t)=\arccos\left[\,\frac{\mathbf{r}^{T}_{i{\rm x}}(t)}{\left\lVert \mathbf{r}_{i{\rm x}}(t)\right\rVert}\cdot\left(-\frac{\mathbf{r}_{i{\rm s}}(t )}{\left\lVert\mathbf{r}_{i{\rm s}}(t)\right\rVert}\right)\right]\,,\] (21)
**where subscript ’x’ refers to either ’p’ or ’m’,** \(\mathbf{r}_{i{\rm x}}\) **is the vector from the center of the planet or moon to the center of the pixel, and** \(\mathbf{r}_{i{\rm s}}\) **the vector from the center of the star to the center of the pixel on the planet or moon.**
### Local azimuthal difference angle \(\phi_{i}-\phi_{0i}\)
**The azimuthal difference angle** \(\phi_{i}-\phi_{0i}\) **for pixel** \(i\) **on the planet or moon is the angle between the plane described by the local zenith direction and the direction towards the observer and the plane described by the local zenith direction and the direction towards the star**⁶**. As Fig.** 13 **shows,** \(\phi_{i}-\phi_{0i}\) **follows from**
[FOOTNOTE:6][ENDFOOTNOTE]
\[\phi_{i}-\phi_{0i}(t)=\arccos{\left(\frac{\cos{\alpha_{i}(t)}-\cos{\theta_{i}} \cos{\theta_{0i}(t)}}{\sin{\theta_{i}}\cos{\theta_{0i}(t)}}\right)}\,.\]
**where** \(\alpha_{i}\) **is the angle between the direction to the observer and the direction to the star measured from the center of pixel** \(i\)**. Given that** \(\left\lVert\mathbf{r}_{i{\rm x}}\right\rVert\ll\left\lVert\mathbf{r}_{\rm xs}\right\rVert\) **with ’x’ referring to either ’p’ or ’m’,** \(\alpha_{i}\) **can be approximated by the body’s phase angle** \(\alpha_{\rm x}\)**, and thus**
\[\phi_{i}-\phi_{0i}(t)=\arccos{\left(\frac{\cos{\alpha_{\rm p}(t)}-\cos{\theta_ {i}}\cos{\theta_{0i}(t)}}{\sin{\theta_{i}}\cos{\theta_{0i}(t)}}\right)}\,.\] (22)
### Local rotation angle \(\beta_{i}\)
**Angle** \(\beta_{i}\) **is used to rotate a locally computed vector** \({\bf F}^{\rm x}_{i}\) **(see Eq.** 6**) for pixel** \(i\) **on the planet or the moon from the local meridian plane to the scattering plane of the body, which is used as the reference plane for the disk–integrated signal of the body. The pixel grid across the planet is defined with respect to the planetary scattering plane, and** \(\beta_{i}\) **is thus time independent for the planetary pixels. For a pixel** \(i\)**,** \(\beta_{i}\) **is computed according to**
\[\beta_{i}=\arcsin{\dfrac{y_{i{\rm p}}}{x^{2}_{i{\rm p}}+y^{2}_{i{\rm p}}}}\,,\] (23)
**where** \(x_{i{\rm p}}\) **and** \(y_{i{\rm p}}\) **are the coordinates of the center of the pixel (recall that the** \(z\)**–axis points towards the observer).**
**For the lunar pixels, the alignment between the lunar scattering plane and the lunar grid, and hence angle** \(\beta_{i}\)**, is time-dependent and requires to redefine the pixel coordinates with respect to the lunar scattering plane. Indicating the redefined coordinates of lunar pixel** \(i\) **with subscript** \(j\)**, angle** \(\beta_{j}\) **is then computed as:**
\[\beta_{j}(t)=\arcsin{\frac{y_{j{\rm m}}(t)}{x^{2}_{j{\rm m}}(t)+y^{2}_{j{\rm m }}(t)}}.\] (24)
### Scattering plane rotation angle \(\psi\)
**Scattering plane rotation angle** \(\psi\) **is used to rotate a Stokes vector that is defined with respect to the lunar scattering plane to the planetary scattering plane, which we use as the reference plane for the** **planet–moon system. Angle** \(\psi\) **is measured in the clock–wise direction from the lunar scattering plane to the planetary scattering plane. For the results presented in this paper, the moon and the planet orbit in the same, edge–on plane, and angle** \(\psi\) **equals zero. In the general case, however, it is computed using**
\[\psi(t)=\arctan\left(-(\mathbf{u}^{T}_{\rm y}\cdot\mathbf{r}_{\rm ms}(t))/( \mathbf{u}^{T}_{\rm x}\cdot\mathbf{r}_{\rm ms}(t))\,\right)\,,\] (25)
**where** \(\mathbf{r}_{\rm ms}\) **is the vector from the star to the center of the moon, and** \(\mathbf{u}_{\rm x}\) **and** \(\mathbf{u}_{\rm y}\) **are the unit vectors along the** \(x\)**-axis and** \(y\)**-axis in coordinate system** \(S_{1}\)**.**
## Appendix BComputing eclipses
**An eclipse occurs when body** \(A\) **is between the star** \(S\) **and body** \(B\) **such that the shadow of** \(A\) **falls onto** \(B\)**.** **The effect of a planetary or lunar eclipse depends on the positions of the star, moon and planet, and, due to the extended size of the star, the size, shape, and depth of the shadow depend not only with the radii of the star and the eclipsing body, but also on the distances and angles involved. Computing eclipses has been discussed in great detail by** **Link****(**1969**)** **for the Moon–Earth system, which we apply to our exoplanetary system. We model the umbral, antumbral, and penumbral shadow regions. Figure** 14 **shows the geometries involved in the various types of eclipses. The equations used for computing the influence of eclipses are described here.**
**The flux arriving at a pixel** \(i\) **of eclipsed body** \(B\) **at time** \(t\) **depends on the fraction of the stellar disk and the local stellar surface brightness, as seen from the center of the pixel. In Eq.** 6**, this is accounted for by factor** \(c_{i}\)**, the ratio between the actual flux** \(e^{\prime}_{i}\) **on pixel** \(i\) **and** \(e_{i}\)**, the flux on the non–eclipsed pixel:**
\[c_{i}(t)=e^{\prime}_{i}(t)/e_{i}(t)=S^{\prime}_{Si}(t)/S_{Si}(t),\] (26)
**with** \(S^{\prime}_{Si}\) **and** \(S_{Si}\) **the stellar disk area as observed from pixel** \(i\) **when it is eclipsed and when it is non–eclipsed, respectively. Note that we ignore stellar limb darkening and stellar light that travels through the atmosphere of the eclipsing body (if present).**
**To determine** \(c_{i}\)**, we first have to identify whether or not pixel** \(i\) **is eclipsed. Obviously,** \(c_{i}=1\) **for a non--eclipsed pixel. If the pixel is (partly eclipsed), we have to determine the type of eclipse: umbral (i.e. total), antumbral, or annular.** ³**. For an umbral eclipse** \(c_{i}=0.0\)**, for an antumbral and annular eclipse, we have to compute** \(S^{\prime}_{Si}\) **in order to determine** \(c_{i}\)**.**
[FOOTNOTE:3][ENDFOOTNOTE]
<figure><img src="content_image/1807.10266/x14.png"><figcaption>Figure 14: Geometry of the umbral, antumbral and penumbral shadow cones, whenstar S is eclipsed by body A, casting a shadow on body B. The shadow casted byA into space is rotationally symmetric around the axis through the center ofthe star and body A. The radii of the star, body A and body B are denoted byRS, RA, and RB, respectively. Points O1, O2, O3, O4 and O5 denote auxiliarypoints: the umbral and antumbral cones have apex O1 and aperture 2Ψ and thepenumbral cone has apex O2 and aperture 2Ω. The lower figure also shows anglesζ, ρ, ωi, βi, and θ, that are used in the computation of the eclipse shadowdepth. Distances between bodies and radii are not to scale in order toemphasize the geometry of the system.</figcaption></figure>
**As can be seen in Fig.** 14**, a pixel on body** \(B\) **is eclipsed when it falls within the penumbral cone of body** \(A\)**. Opening angle** \(\Omega\) **of the penumbral cone is given by:**
\[\sin{\Omega}=\dfrac{R_{S}+R_{A}}{\left\lVert\mathbf{r}_{\rm AS}\right\rVert}\,.\] (27)
**Indeed, pixels in eclipse on the disk of body** \(B\) **can be found at times when**
\[\sin{\rho}(t)<\sin{\Omega}(t),\] (28)
**with angle** \(\rho\) **(**\(\left[0\degr,90\degr\right)\)**) given by (see Fig.** 14**)**
(29)
**Vector** \(\mathbf{AO_{4}}\) **is a function of the radii of the shadowed and eclipsing bodies and of** \(\Omega\)**, as follows:**
\[\mathbf{AO_{4}}=\dfrac{R_{B}+R_{A}}{\sin{\Omega}}\,\mathbf{u}_{\rm AS}\,.\] (30)
**Except when body** \(B\) **falls completely in the penumbral cone, there will also be non–eclipsed pixels on the disk. When Eq.** 28 **holds, a pixel-by-pixel search is performed, in which the center of each pixel is checked for total or umbral (Sect.** B.1**), annular (Sect.** B.2**), or penumbral (Sect.** B.3**) eclipse conditions.**
### Total or umbral eclipses
**In the umbral zone (see Fig.** 14**), pixels experience a total stellar eclipse. If the umbral zonal is wide and the shadowed body** \(B\) **relatively small, such as in the Earth–moon system, all pixels on the disk of** \(B\) **can be simultaneously in the umbra, and factor** \(c_{i}=0\) **for all pixels. This is the case when**
\[\cos{\xi}>\cos{\Psi}\,,\] (31)
**with**
\[\sin{\Psi}=\dfrac{R_{S}-R_{A}}{\left\lVert\mathbf{r}_{\rm AS}\right\rVert}\,,\] (32)
**and**
\[\cos{\xi}=\frac{\left(\mathbf{r}_{\rm AB}-\mathbf{AO_{3}}\right)\cdot\mathbf{u }_{\rm AS}}{\left\lVert\mathbf{r}_{\rm AB}-\mathbf{AO_{3}}\right\rVert}\, \hskip 11.381102pt{\rm with}\hskip 11.381102pt\mathbf{AO_{3}}=-\dfrac{R_{A}-R_ {B}}{\sin{\Psi}}\,\mathbf{u}_{\rm AS}\,.\] (33)
**The disk of body** \(B\) **will only be partially inside the umbral shadow cone of body** \(A\) **when**
(34)
**and**
(35)
**Here**
\[\mathbf{O_{5}B}=\mathbf{r}_{\rm AB}-\mathbf{AO_{5}}\hskip 14.226378pt{\rm and} \hskip 14.226378pt\mathbf{O_{1}B}=\mathbf{r}_{\rm AB}-\mathbf{AO_{1}}\,,\] (36)
**with**
(37)
**If the disk of body** \(B\) **falls partially inside the umbral cone, the pixels where** \(\beta_{i}<\Psi\) **are inside the umbra, and** \(c_{i}=0\)**. Because** \(c_{i}\) **only applies to pixels on the illuminated part of the disk of** \(B\)**, this condition can be reformulated as**
\[\sin{\beta_{i}}<\sin{\Psi}\,.\] (38)
**Note that angle** \(\beta_{i}\) **of a pixel can be derived from**
(39)
**The pixels on the disk of** \(B\) **that are not in the umbral cone can be in the antumbral or annular eclipse zone, as described below.**
### Annular or antumbral eclipses
**Instead of crossing the umbral cone, body** \(B\) **can cross the antumbral cone, where eclipsing body** \(A\) **does not completely cover the stellar disk as seen from body** \(B\)**, thus yielding a so–called annular eclipse. Body** \(B\) **is in the antumbral shadow cone when**
\[\cos{\xi}<-\cos{\Psi}\,,\] (40)
**where** \(\xi\) **and** \(\Psi\) **follow from Eqs.** 33 **and** 32**. When Eq.** 40 **is satisfied, pixels on the disk of** \(B\) **are checked for their eclipsed status. A pixel is (partially) eclipsed if**
\[\cos{\beta_{i}}<-\cos{\Psi}\,.\] (41)
**For each pixel in the antumbral cone, factor** \(c_{i}\) **is given by the fraction of the stellar disk that is visible (see Fig.** 15**), i.e.**
\[c_{i}=\dfrac{\pi\alpha^{2}_{Si}-\pi\alpha^{2}_{Ai}}{\pi\alpha^{2}_{Si}}=1- \left(\dfrac{\alpha_{Ai}}{\alpha_{Si}}\right)^{2}\,,\] (42)
**where**
\[\alpha_{Si}=\arcsin{\dfrac{R_{Si}}{\left\lVert\mathbf{r}_{Si}\right\rVert}}\, \hskip 14.226378pt{\rm and}\hskip 14.226378pt\alpha_{Ai}=\arcsin{\dfrac{R_{Ai} }{\left\lVert\mathbf{r}_{Si}\right\rVert}}\,.\] (43)
**Here,** \(\mathbf{r}_{Si}\) **is the position vector of pixel** \(i\) **on body** \(B\) **with respect to the center of star** \(S\)**.**
<figure><img src="content_image/1807.10266/x15.png"><figcaption>Figure 15: The disks of star S and eclipsing body A seen from a pixel on bodyB in the antumbral zone. Angles αS and αA indicate the angular radii of thebodies.</figcaption></figure>
### Penumbral eclipses
**When Eq.** 28 **holds, all pixels that are not in the umbral or antumbral eclipse are examined for being in the penumbral shadow. Indeed, pixels with** \(\omega_{i}<\Omega\) **are within the penumbral cone, as can be seen in Fig.** 14**. This inequality can be rewritten as**
\[\cos{\omega_{i}}>\cos{\Psi}\,,\] (44)
**where** \(\cos{\Psi}\) **follows from Eq.** 32 **and** \(\cos\omega_{i}\) **is given by:**
(45)
**The magnitude of the eclipse at pixel** \(i\)**,** \(c_{i}\)**, can be calculated through the stellar and eclipsing body viewing angles, i.e. the angular diameter of the bodies,** \(2\alpha_{S}\) **and** \(2\alpha_{A}\)**, and the eclipsing body-to-star angular distance as seen from the shadowed body,** \(\delta\)**. Then, as** **follows from Fig.** 16**,** \(c_{i}\) **can be computed using**
\[c_{i}=\dfrac{\pi\alpha_{S}^{2}-A_{1}-A_{2}}{\pi\alpha_{S}^{2}}=1-\dfrac{A_{1}+ A_{2}}{\pi\alpha_{S}^{2}},\] (46)
**with**
\[A_{1}=\left\{{\begin{array}[]{*{20}{l}}\dfrac{\theta_{A}-\sin{\theta_{A}}}{2} \hskip 2.845276pt\alpha_{A}^{2}\hskip 28.452756pt\text{if}\hskip 5.690551pt \delta\geq l_{S}\\ \\ \pi-\dfrac{\theta_{A}-\sin{\theta_{A}}}{2}\hskip 2.845276pt\alpha_{A}^{2} \hskip 12.80374pt\text{if}\hskip 5.690551pt\delta<l_{S}\end{array}}\right.\,\] (47)
**and**
\[A_{2}=\left\{{\begin{array}[]{*{20}{l}}\dfrac{\theta_{S}-\sin{\theta_{S}}}{2} \hskip 2.845276pt\alpha_{S}^{2}\hskip 28.452756pt\text{if}\hskip 5.690551pt \delta\geq l_{A}\\ \\ \pi-\dfrac{\theta_{S}-\sin{\theta_{S}}}{2}\hskip 2.845276pt\alpha_{S}^{2} \hskip 12.80374pt\text{if}\hskip 5.690551pt\delta<l_{A}\end{array}}\right.\,\] (48)
<figure><img src="content_image/1807.10266/x16.png"><figcaption>Figure 16: Stellar (S) and eclipsing body (A) disks as seen from a pixel onthe shadowed body during a penumbral eclipse. The stellar shadowed area isdecomposed in two components A1 and A2. αS and αA stand for the angular radiusof the bodies, and δ is the angular separation between the bodies’ center. θSand θA are the central angles of the circular segments defined by the commoncord of the intersecting stellar and eclipsing body disks. The minimumdistance from the star/eclipsing body center to the common chord is defined aslS/lA.</figcaption></figure>
**Here,** \(l_{S}\) **and** \(l_{A}\) **are the distances from the centers of the stellar disk and eclipsing body** \(A\)**, respectively, to the line between the two points where the stellar and eclipsing body intersect. The angles** \(\theta_{S}\) **and** \(\theta_{A}\) **(see Fig.** 16**) follow from e.g. Heron’s formula:**
\[\theta_{S}=2\arcsin{\left(\dfrac{2}{\alpha_{S}\delta}\sqrt{S\left(S-\alpha_{S} \right)\left(S-\alpha_{A}\right)\left(S-\delta\right)}\right)}\,,\] (49)
**and**
\[\theta_{A}=2\arcsin{\left(\dfrac{2}{\alpha_{A}\delta}\sqrt{S\left(S-\alpha_{S} \right)\left(S-\alpha_{A}\right)\left(S-\delta\right)}\right)}\,,\] (50)
**with**
\[S=\dfrac{1}{2}\left(\alpha_{A}+\alpha_{S}+\delta\right)\,.\] (51)
**Pixels on body** \(B\) **that satisfy Eq.** 28 **but do not meet the conditions established by Eqs.** 31**,** 35**,** 41 **and** 44 **are not eclipsed and have** \(c_{i}=1.0\)**.**
## Appendix CNumber of pixels across the disk
**The results of our numerical simulations depend on the number of pixels that is used to compute the flux and polarization signals of the disk of the planet and moon. The number of pixels across the disk of the planet or the moon,** \(N^{\rm p}\) **and** \(N^{\rm m}\)**, respectively, determines the spatial resolution of the locally reflected Stokes vectors and hence the numerical error in the integration across the disk, in particular at large phase angles. However, the larger the number of pixels, the smaller the error but the longer the computational time. We have investigated the optimal number of pixels, expressed in the number of pixels across the equator of the planet and the moon,** \(N_{\rm eq}^{\rm p}\) **and** \(N_{\rm eq}^{\rm m}\)**, respectively, using a trade–off between the errors and the computational time, with time steps of 24 hours.**
**For the trade–off, we compute the flux** \(F\) **(Eq.** 53**) and degree of polarization** \(P\) **(Eq.** 53**) for consecutive values of** \(N_{\rm eq}\) **as functions of phase angle** \(\alpha\)**, both for the planet and the moon. In Fig.** 17**, we show the maximum and mean differences encountered across the whole phase angle range of the planet and the moon, together with the average disk integration time and the total phase curve computation time. The differences are defined as**
\[|\Delta F_{n}(\alpha)| = \dfrac{|F_{n}(\alpha)-F_{n-1}(\alpha)|}{F_{n-1}(\alpha=0\degr)},\] (52)
\[|\Delta P_{n}(\alpha)| = |P_{n}(\alpha)-P_{n-1}(\alpha)|,\] (53)
**with** \(n-1\) **and** \(n\) **two consecutive values of** \(N_{\rm eq}\)**.**
**Figure** 17 **shows that the flux and polarization differences decrease with increasing** \(N_{\rm eq}\)**, and that for the values of** \(N_{\rm eq}\) **considered, the computed flux and polarization curves have not yet completely converged. However, further increasing** \(N_{\rm eq}\) **increases the integration time across the planetary disk, as can be seen in Fig.** 17**c. In the simulations presented in this paper, we decided to use** \(N_{\rm eq}^{\rm p}=50\) **and** \(N_{\rm eq}^{\rm m}=14\)**, which yields an average disk–integration time of** \(\sim 0.8\) **seconds (thus an overall phase curve computation time of** \(\sim 2.4\) **minutes with a 24 h temporal resolution). These values for** \(N_{\rm eq}^{\rm p}\) **and** \(N_{\rm eq}^{\rm m}\) **produce smooth curves for individual transit and eclipse events for temporal resolutions as small as 1 minute.**
<figure><img src="content_image/1807.10266/x17.png"><figcaption>Figure 17: Analysis for the number of pixels along the equator Neq of theplanet (top) and moon (bottom). Shown are the maximum (solid line) and mean(dashed line) differences between results computed across the whole phaseangle range and for consecutive values of Neq values, for the reflected fluxF(α) relative to F(α=0∘) (a and d), and degree of polarization P (b and e,note that for the moon P=0). Also shown is the computational time (in minutes)for the computation of a full phase curve (with 24 hour temporal resolution)and the average disk integration (c and f). For Npeq (top), we used values of10, 20, 30, 40, 50, 60, and 70, and for Nmeq (bottom), 3, 5, 8, 11, 14, 16,and 19.</figcaption></figure>
|
1702.00161 | {
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] | # Multiple scroll wave chimera states
Volodymyr Maistrenko
1Scientific Center for Medical and Biotechnical Research, NAS of Ukraine,
Volodymyrska Str. 54, 01030 Kyiv, Ukraine 1 maistren@nas.gov.ua
Oleksandr Sudakov
1Scientific Center for Medical and Biotechnical Research, NAS of Ukraine,
Volodymyrska Str. 54, 01030 Kyiv, Ukraine 12Taras Shevchenko National University of Kyiv, Volodymyrska Str. 60, 01030 Kyiv, Ukraine 2
Oleksiy Osiv
1Scientific Center for Medical and Biotechnical Research, NAS of Ukraine,
Volodymyrska Str. 54, 01030 Kyiv, Ukraine 1
Yuri Maistrenko
1Scientific Center for Medical and Biotechnical Research, NAS of Ukraine,
Volodymyrska Str. 54, 01030 Kyiv, Ukraine 13Institute of Mathematics, NAS of Ukraine, Tereshchenkivska Str. 3, 01030 Kyiv, Ukraine3
###### Abstract
We report the appearance of three-dimensional (3D) multiheaded chimera states that display cascades of self-organized spatiotemporal patterns of coexisting coherence and incoherence. We demonstrate that the number of incoherent chimera domains can grow additively under appropriate variations of the system parameters generating thereby head-adding cascades of the scroll wave chimeras. The phenomenon is derived for the Kuramoto model of N identical phase oscillators placed in the unit 3D cube with periodic boundary conditions, parameters being the coupling radius \(r\) and phase lag \(\alpha\). To obtain the multiheaded chimeras, we perform the so-called ’cloning procedure’ as follows: choose a sample single-headed 3D chimera state, make appropriate scale transformation, and put some number of copies of them into the unit cube. After, start numerical simulations with slightly perturbed initial conditions and continue them for a sufficiently long time to confirm or reject the state existence and stability. It is found, by this a way, that multiple scroll wave chimeras including those with incoherent rolls, Hopf links and trefoil knots admit this sort of multiheaded regeneration. On the other hand, multiple 3D chimeras without spiral rotations, like coherent and incoherent balls, tubes, crosses, and layers appear to be unstable and are destroyed rather fast even for arbitrary small initial perturbations.
## 1 Introduction
Chimera states represent one of the most fascinating discoveries of modern nonlinear science at the border of the network and chaos theories. It has been found that networks of identical oscillators with non-local coupling can demonstrate robust co-existence of coherence and incoherence, such that a part of the network oscillators are synchronized but the others exhibit desynchronized and often chaotic behavior. First, the chimera phenomenon was described in 2002 for the _one-dimensional_ complex Ginzburg–Landau equation and its phase approximation, the Kuramoto model [1]. This paper stimulated the study [2] two years later. Both have opened a substantially new direction in the research of oscillatory networks. See recent review papers on the topic [3; 4].
At about the same time, this novel approach was extended to _two-dimensional_ networks of oscillators. Spiral waves with a randomized core were identified for a class of three-component reaction-diffusion systems in the plane and for the two-dimensional Ginzburg–Landau equation as well as for the corresponding non-locally coupled Kuramoto model [5; 6; 7]. They constitute a new class of chimera states, called the spiral wave chimeras [8]. This kind of spatio-temporal behavior is different from the oscillating 2D chimeras in which the coherent region only oscillates but does not spiral around the incoherence. The oscillating chimeras which are the natural counterparts of the 1D chimeras have been obtained in the form of stripes and spots, both coherent and incoherent [9; 10] and also twisted states [11]. An interesting observation is that chimeras of both classes, oscillating and spiraling, emerge in opposite corners of the system parameter space and thus cannot co-exist [9].
The first evidence of chimera states in _three-dimensions_ was reported two years ago in [12] for the Kuramoto model of coupled phase oscillators in three-dimensional (3D) grid topology
\[\dot{\varphi}_{ijk}=\omega+\frac{K}{N_{P}}\sum\limits_{(i^{\prime},j^{\prime}, k^{\prime})\in B_{P}(i,j,k)}\sin(\varphi_{i^{\prime}j^{\prime}k^{\prime}}- \varphi_{ijk}-\alpha),\] (1)
where \(\varphi_{ijk}\) are phase variables, and indexes \(i,j,k\) are periodic mod \(N\). The coupling is assumed long-ranged and isotropic: each oscillator \(\varphi_{ijk}\) is coupled with equal strength \(K\) to all its \(N_{P}\) nearest neighbors \(\varphi_{i^{\prime}j^{\prime}k^{\prime}}\) within a ball of radius \(P\), i.e., to those falling in the neighborhood
\[B_{P}(i,j,k):=\{(i^{\prime},j^{\prime},k^{\prime}){:}(i^{\prime}-i)^{2}+(j^{ \prime}-j)^{2}+(k^{\prime}-k)^{2}\leq P^{2}\},\]
where the distances are calculated taking into account the periodic boundary conditions of the network. The phase lag parameter \(\alpha\) is assumed to belong to the attractive coupling range from \(0\) to \(\pi/2\). The second control parameter, coupling radius \(r=P/N\) varies from \(1/N\) (local coupling) to \(0.5\) (close to global coupling). Without loss of generality, we put in Eq.(1) \(\omega=0\) and \(K=1\).
<figure><img src="content_image/1702.00161/Fig1.png"><figcaption>Figure 1: Parameter regions of 3D chimera states for Eq.(1). Regions for typeI oscillating and type II scroll wave chimeras appear in opposite corners ofthe parameter space. Snapshots of the states are shown in inserts. r=P/N.N=100.</figcaption></figure>
In [12], two principal families of 3D chimera states were obtained for Eq.(1): type I - oscillating chimeras, i.e., those without spiraling of the coherent region, and type II - spirally rotating chimeras, called _scroll wave chimeras_. Examples of the first class are coherent and incoherent balls, tubes, crosses, and layers in incoherent or coherent surrounding, respectively; the second class includes incoherent rolls of different modality and space disposition in a spiraling rotating coherent surrounding. As it is illustrated in Fig. 1, parameter regions for chimeras of both classes (type I) and (type II) do not intersect, while there is a huge multistability inside each of the classes.
Recently, two new kinds of the scroll wave chimeras, Hopf link and trefoil, with linked and knotted incoherent regions (”swelling” filaments) were detected in [13]. Our simulations confirm their existence in an ellipse-like parameter region which can be seen inside the type II chimera regions in Fig. 1 (delineated in black). Furthermore, there exist in Eq. (1) scroll wave chimeras in the form of chains with one and two links. Parameter regions for one- and two-link chain chimeras are shown in Fig. 1 too (delineated in red and brown). In the \(R^{3}\)-cube they look broken. However, they are indeed closed when considering them on the \(T^{3}\)-torus (which is topologically equivalent to the \(R^{3}\)-cube in the case of periodic boundary conditions). Note that all presented in Fig. 1 chimera patterns are obtained with randomly chosen initial conditions.
In the present paper, we study the appearance of multiheaded 3D chimera states built up on a base of the single- and low-headed states exhibited in Fig. 1. Similar to the 1D case [14], we design cascades of multiple 3D chimeras with an increasing number of incoherent regions and obtain parameter regions for their existence. To illuminate the multiheaded scroll wave appearances, we apply the so-called ”cloning procedure” as follows: glue a few copies of a chosen 3D chimera state, rescale them and stow them in the unit cube with periodic boundary conditions. Afterwards start simulation with the constructed multiheaded initial conditions perturbed slightly to prevent the symmetry capturing effect.
We show, with the use of massive numerical simulations, that the cloning procedure perfectly works for the type II, i.e. scroll wave chimeras including rolls, Hopf links, and trefoils. The calculations were performed, as a rule, up to \(t=10^{4}\) time units which corresponds to approximately 100 periods of the spiral rotations in the patterns.
On the other hand, it fails for for the type I oscillating chimeras, as well as chains. They disappear in the processes of simulation as soon as the symmetry imposed is violated. Cascades of the roll-type scroll wave chimeras with even numbers of incoherent rolls are constructed in Ch. 2. Hopf link and trefoil cascades are obtained in Ch.3. ”Hybrid chimeras” including different combinations of trefoil, Hopf links and parallel rolls are illustrated in Ch. 4. Examples of the multiheaded scroll wave dynamics are demonstrated by videos at http://chimera3d.biomed.kiev.ua/multiheaded.
Numerical simulations were performed on the base of Runge-Kutta solver DOPRI5 that was integrated into the software for large nonlinear dynamical networks [15], allowing for parallelized simulations with different sets of parameters and initial conditions. The simulations were performed at the computer cluster ”CHIMERA”, http://nll.biomed.kiev.ua/cluster, and the Ukrainian Grid Infrastructure providing distributed cluster resources and the parallel software [16].
## 2 Cascades of scroll wave chimeras with multiple incoherent rolls
<figure><img src="content_image/1702.00161/x1.png"><figcaption>Figure 2: Cascade of multiheaded scroll wave chimera states with parallelrolls. 3D screenshots and respective cross-sections are shown for: (a) - tworolls (α=0.8,r=0.165), (b) - four rolls (α=0.7,r=0.12), (c) - 6 rolls(α=0.7,r=0.09), (d) - 8 rolls (α=0.7,r=0.08), (e) - 10 rolls (α=0.64,r=0.07),(f) - 12 rolls (α=0.63,r=0.05), (g) - 14 rolls (α=0.64,r=0.056), (h) - 16rolls (α=0.6,r=0.06), N=100; (i) - 64 rolls (α=0.4,r=0.04). N=200. Coordinatesx=i/N,y=j/N,z=k/N.</figcaption></figure>
Scroll waves with parallel incoherent rolls represent one of the characteristic examples of the chimera states in three-dimensions. In [12], 2- and 4-rolled chimeras of this type were obtained, they exist in wide regions of the (\(\alpha,r\))-parameter space shown in Fig. 1. Due to the periodic boundary conditions they can be considered as scroll rings with incoherent cores (”swelling” filaments) on the \(T^{3}\)-torus. The microscopic dynamics inside the chimera rolls is chaotic while, in the large-scale, the rolls themselves are practically stationary, i.e. not moving in a significant manner. This can be seen in the supplementary videos of [12], also at http://chimera3d.biomed.kiev.ua/high-resolution (files fig5(a)hq-video.mkv, fig5(b)hq-video.mkv); as well as for 16, 64 parallel and 16 crossed rolls in http://chimera3d.biomed.kiev.ua/multiheaded/rolls/.
In Fig. 2, a cascade of scroll wave chimeras with pair-multiple incoherent parallel rolls is presented. The number of rolls increases additively from 2 [Fig.2(a)] to 16 [Fig.2(h)]. In addition, a 64-rolled chimera is shown in [Fig.2(i)]. The chimera rolls in Fig.2 are symmetrically located in the unit cube. However, the large-scale symmetry can be violated when other initial conditions are chosen. Note also that microscopic chaotic dynamics inside the rolls differ for different rolls of the same state, which is illustrated by the cross-sections shown below the 3D plots in Fig.2.
<figure><img src="content_image/1702.00161/Fig3.png"><figcaption>Figure 3: Parameter regions for h-headed parallel rolls chimera statesdelineated by the color lines: blue - h6, brown - h8, orange - h10 , green -h12, magenta - h14 rolls. r=P/N,N=100. Snapshots of respective chimera typesare shown in inserts.</figcaption></figure>
Regions for existence of the parallel rolled chimeras in the \((\alpha,r)\)-parameter plane are shown in Fig.3. They lie at intermediate phase shifts \(\alpha\) between \(0.1\) and \(0.95\), and at small coupling radius \(r<0.12\) including the minimal possible \(r=0.01\) when only 6 nearest neighbors are connected to each oscillator (\(N=100\)). Thus, the multiple rolled chimeras exist in the model (1) not only for non-local but also for the local coupling scheme. Based on this, we assume that such states should exist also in the limiting PDE case \(N\rightarrow\infty,r\to 0\). If so, the PDE obtained by this a way could be a rich source for multiple scroll waves (multiple scroll rings when written in the circular coordinates). Our simulations confirm that chimeras illustrated in Fig. 2 are robust 3D patterns as they survive for long integrating times of thousands of rotating periods. Further study in this direction would be interesting from both theoretical and practical points of view; е.g., in medicine as prospective models of spiral patterns formed on heart tissue during ventricular tachycardia and fibrillation (see [3; 12] and references therein).
To obtain the multi-rolled scroll wave chimeras we used the so-called ’cloning procedure’ as follows. First, take some number of 2- or 4-rolled parallel chimeras (previously obtained in [12]). Rescale them in an appropriate way and fill the unit cube with them. Afterwards, start calculations with these specially prepared initial conditions slightly perturb to prevent the symmetry capturing effect. Doing so, there is no guaranty that the resulting state will be of the form as assigned, i.e. with the initially chosen number of rolls. We often had to repeat the procedure trying different variants of the number and type of initial ’chimera clones’, as well as varying system parameters. Proceeding in such a way, after some number of trials the desired chimera pattern was usually obtained.
<figure><img src="content_image/1702.00161/Fig4.png"><figcaption>Figure 4: Parameter regions for h-headed parallel rolled chimera statesdelineated by the color lines: olive - h4 (N=50), red - h16 (N=100), blue -h64 (N=100), brown - h256 (N=200) rolls. r=P/N. Snapshots of respective 3Dchimera types are shown in inserts.</figcaption></figure>
<figure><img src="content_image/1702.00161/x10.png"><figcaption>Figure 5: Example of 1024-headed parallel scroll wave chimera state (a), andits cross-section at x=0.5 (b) with enlarged windows (c,d)(α=0.6,r=0.0075,N=400).</figcaption></figure>
Among other variants of the cloning procedure, there is one reliable approach always giving the chimera state we are looking for. This is in the case when just 8 identical parallel chimeras are taken as samples and placed, after rescaling, in the unit cube. Then, a 4-multiple rolled chimera is that is stable with respect to perturbations is obtained. We have never seen its destruction even at very long simulations. Therefore, we can successively repeat the multiple chimera regeneration as long as computer power allows to process it. As the system complexity grows exponentially, we were only able to produce \(4^{n}\)-rolled chimeras with \(n=1,2,3,4\) and \(5\). Our largest example is the 1024-rolled chimera calculated for \(N=400\), i.e. for \(N^{3}=64\) million oscillators up to 1000 time units, illustrated in Fig.5.
Stability regions for 4-, 16-, 64- and 256-headed parallel scroll wave chimera states are presented in Fig.4; the parameter point for the 1024-headed chimera in Fig.5 is also shown. As it can be observed, each next region in the cascade is twice thinner on the parameter \(r\) compared to the previous one. E.g., at \(\alpha=0.7\) the top border values of parameters \(r\) of the stability regions decreases approximately as 0.16, 0.08, 0.04, 0.02. We assume that with more computational power, the whole cascade can be obtained for the 4-multiple scroll wave chimeras with any \(2^{2(n+1)},n=1,2,3...\) number of heads. Note that this cloning procedure can be also applied successfully for 4 crossed rolls chimera.
## 3 Hopf link and trefoil chimera states
<figure><img src="content_image/1702.00161/Fig6abc.png"><figcaption>Figure 6: Cascade of multiple Hopf link chimera states: (a) - 1-headed(α=0.76,r=0.083,N=100), (b) - 2-headed (α=0.61,r=0.0215), (c) - 3-headed(α=0.61,r=0.027), (d) - 4-headed (α=0.68,r=0.03), (e) - 5-headed(α=0.68,r=0.018), (f) - 6-headed (α=0.7,r=0.017), (g) - 7-headed(α=0.74,r=0.02), (h) - 8-headed (α=0.84,r=0.03). N=200; (i) - 64-headed(α=0.72,r=0.01), N=400.</figcaption></figure>
Hopf link and trefoil chimera states represent 3D scroll waves with linked and knotted filaments. Due to the non-local coupling, the filaments are not singular (lines) as in standard scroll waves but ”swelled” proportionally to the radius of coupling. Moreover, they are filled by oscillators with unsynchronized, chaotic behavior. The Kuramoto model, Hopf link and trefoil chimeras were fist reported in [13], see also [17] where bifurcation transitions between this kind of 3D patterns are studied for a different model with local coupling.
### Cascades of multiple Hopf links and trefoils
<figure><img src="content_image/1702.00161/Fig7abc.png"><figcaption>Figure 7: Cascade of multiple trefoil chimeras states: (a) - 1-headed(α=0.68,r=0.07,N=100); (b) - 2-headed (α=0.6,r=0.023), (c) - 3-headed(α=0.68,r=0.02), (d) - 4-headed (α=0.585,r=0.02), (e) - 5-headed(α=0.672,r=0.02), (f) - 6-headed (α=0.72,r=0.02), (g) - 7-headed(α=0.78,r=0.02), (h) - 8-headed (α=0.72,r=0.02), N=200; (i) - 64-headed(α=0.72,r=0.01), N=400.</figcaption></figure>
In this chapter we design cascades of Hopf link and trefoil chimera states with additively growing the number of samples. Figs.6 and 7 illustrate the \(h\)-multiple cascades of Hopf links and trefoils, respectively, for \(h=1,2,3,...,8\) and \(64\). They are constructed by the cloning procedure using the the sampled initial conditions of identical Hopf links or trefoils, always with small perturbations to prevent the symmetry capturing. Constructed Hopf links and trefoils are not stationary patterns (in contrast to the parallel scroll waves in the previous Chapter), this is illustrated by videos at http://chimera3d.biomed.kiev.ua/multiheaded.
We find that multiple chimera states of this kind exist in wide enough regions of the (\(\alpha,r\))-parameter plane. The regions are heavily intersecting and do not shrink as \(h\) increases. This is illustrated by the bifurcation diagrams in Fig. 8 for \(N=200\). To our surprise, both parameter regions for single Hopf link and single trefoil coincide (delineated in black in Fig. 8). The same occurs for the respective multiple patterns. Indeed, 2-Hopf link and 2-trefoil states co-exist in the twice smaller region (delineated in blue). Moreover, the states of higher multiplicity \(h=3,4,...,8\), are all found in the same slightly smaller inclusive region (delineated in green). Therefore, parameter regions for the \(h\)-multiple states stabilize as \(h\) increases and, given our precision, they become indistinguishable beginning from \(h=3\).
<figure><img src="content_image/1702.00161/Fig8.png"><figcaption>Figure 8: Even cascades of Hopf link and trefoils: one-headed (delineated byblack line), 2-headed (delineated by blue line), 4-headed, 6-headed and8-headed (delineated by green line). r=P/N,N=200. Snapshots of the chimerastates are shown in inserts.</figcaption></figure>
Regions for Hopf link and trefoil chimera states are located in the intermediate range of the phase lag parameter \(\alpha\), approximately between 0.6 and 0.9, and for rather small values of the coupling radius \(r=P/N<0.08\). The lower boundary of the regions is given by the value \(r=0.012245...\) (\(N=200\)). This is the smallest value of \(r\), when the number of nearest neighbors \(\varphi_{i^{\prime}j^{\prime}k^{\prime}}\) coupled to each oscillator \(\varphi_{ijk}\) is \(N_{P_{min}}=81\). At smaller \(r<0.012245\), the number of coupled oscillators drops abruptly to 59. Our simulations confirm that both single and multiple Hopf links and trefoils do not survive with such low connectivity (\(P=59\) or smaller) and are fast transforming into some other state. We conclude that Hopf links and trefoils arise in the Kuramoto model (1) only with non-local, prolonged coupling. Local diffusive coupling is not enough to insure their stability, which is unlike to the roll-type chimeras surviving at local coupling, see Ch.2.
Multiple trefoil and Hopf link chimeras were obtained using the cloning procedure. It perfectly works, however, only when just 8 smaller copies of a state are laid into the unit cube. The other combinations, e.g., when looking for 2h, or 3h states, do not guarantee the desired result and require as a rule additional efforts. In many such cases the multi-compound structure appears to be unstable and is destroyed rather fast as simulations start. Then, we have to try again with different initial conditions and system parameters until the desired multi-headed Hopf link ot trefoil is eventually obtained.
### Chain chimeras
For the single-link chimera two more kind of linked scroll wave chimeras are given by single- and double-link chains, illustrated in Fig.9. In the \(R^{3}\)-cube the chains are broken, but they are indeed connected on the corresponding \(T^{3}\)-torus. Parameter regions for the chain chimeras are presented in Fig. 10. As it can be seen, similar to the Hopf links and trefoils, they both arise in Eq. (1) at the intermediate values of the phase lag parameter \(\alpha\) and for a rather small radius of coupling. For single-link chains the parameter \(\alpha\) should be approximately between 0.55 and 0.85, and \(r\) be smaller than 0.075.
<figure><img src="content_image/1702.00161/Fig9ab.png"><figcaption>Figure 9: Chain chimera states: (a) - single-linked chain(α=0.7,r=0.041,N=100), (b) - double-linked chain (α=0.61,r=0.033,N=200).</figcaption></figure>
The lower boundary of the single-link chain chimera is given by the coupling radius value \(r=0.00707...\) when each oscillator in the network is coupled to \(N_{P}=19\) its nearest-neighbors. At smaller \(r\) the number \(N_{P}\) of the couplers drops abruptly to \(7\) only. The state becomes unstable and is rapidly destroyed in the simulations. Parameter regions in Fig. 10 are obtained for Eq.(1) with \(N=200\). Interestingly, the same \(N_{P}=19\) lower bound on the number of coupled oscillators is also obtained for the chain chimera stability in Eq.(1) with \(N=100\).
<figure><img src="content_image/1702.00161/Fig10.png"><figcaption>Figure 10: Parameter regions for single-link (delineated by red line) anddouble-link (delineated by brown line) chain chimera states. Region for singleHopf link and trefoil are also shown (delineated by black line). r=P/N,N=200.</figcaption></figure>
Note that case of \(N_{P}=19\) couplers corresponds to the reliable numerical scheme for a PDE derived in [12], where 3D linked and knotted scroll waves have been obtained, however, only for a short time interval and their stability is not analyzed. Similarly, \(N_{P}=19\) in Eq.(1) can also be considered as local coupling. If so, we conclude that single-link chain chimera exist not only for the non-local but also for local coupling, and we expect that this can also be a robust pattern for the respective PDE in the limit \(N\rightarrow\infty,r\to 0\). This situation is different from the Hopf link and trefoil stability, which have the pure non-local coupling origination (Ch.3); on the other hand, is similar to the rolls chimeras (scroll rings on \(T^{3}\) ) , see Ch.2. The double-link chain chimera exists in a smaller parameter region, see Fig. 10, which is detached from the locally coupled case. Stability of the state begins with the non-locally coupling when \(N_{P}=27\) (\(r=0.01\) at \(N=200\)). In our simulations, we have also tried to ”clone” the chain chimeras with three and more links. However, they appear to be unstable and are rapidly destroyed in the presence of even very small perturbations.
### Large-scale dynamics and transformations
Our simulations show that multiple Hopf link and trefoil chimera states often change their structure with time in the following way. When starting with 8 practically identical patterns placed in the unit cube, we observe soon afterwards that they group into 4 visibly different pairs, where the pairwise objects are only slightly different. To our surprise, without or with only tiny perturbations (\(10^{-4}\)) the pairwise identity appears to be so strong that it practically is not affected by the asymmetry of the Runge-Kutta numerical algorithm, and only imposed asymmetry in the initial conditions can slowly destroy it. On the other hand, each such pair in the chimera state has its own shape, size and position in the 3D space, distinct from the others. This is illustrated by the video at http://chimera3d.biomed.kiev.ua/multiheaded/EvolutionT.
Let us follow the dynamics of a multiheaded state inside the stability region shown Fig.8. Take the 8-headed trefoil or Hopf links obtained from the single trefoil but with stronger perturbations of the initial conditions and start the simulations. Then, it is usually observed that in the some instants the samples collide and disappear or transform into other states. Eventually, as a rule, a hybrid state or a single-headed chimera is obtained.
Typical evolutions of a 8-headed trefoil and Hopf links chimera states with rather strong perturbations (\(0.1\)) of the identical initial conditions are demonstrated in the video at http://chimera3d.biomed.kiev.ua/multiheaded/EvolutionS . As one can observe there, the dynamics result finally in a single trefoil, Hopf link or other kind of chimera states. It depends on the chosen initial cloning chimera, parameter values \(\alpha\), \(r\) and perturbation value. Usually the trajectory evolutions were calculated up to \(t=10^{4}\) for \(N=200\).
To obtained the odd-headed Hopf links and trefoils chimera, an odd number of initial chimeras should be taken in the cloning procedure and placed in the unit cube. Sometimes, as we have often seen in the simulations, odd-headed chimeras arise from strong enough perturbations of the even-headed ones. In all cases considered, the probability to obtain a desired odd-headed chimera was rather small, however after some number of trials, we could eventually catch it.
Twice repeating the 8-cloning procedure gives birth to 64-headed Hopf links and trefoils. As it is illustrated in Fig.6(i) and Fig.7(i), each such pattern consist of 16 groups per 4 similar elements inside. Moreover, each 4 groups among the 16 are quite similar too. We expect that the head-adding sequence of the chimera states can be continued further, creating states with 512 and more objects. It requires, clearly, much more computational power to ensure the necessary accuracy of integrations. Indeed, the single Hopf link and trefoil states were obtained in our simulation of the \(N^{3}\)-dimensional network with \(N=100\); for the 8- and 64-headed states we had to take \(N=200\) and \(N=400\), respectively. In the latter case, a 64 million-dimensional nonlinear system should be integrated. The system complexity grows exponentially with further steps in the multiple chimera cascade.
### From trefoil to Hopf link
In our simulations, we often observed the situation when a trefoil chimera state transforms into a Hopf link (but not vise versa), see also [17; 13]. The transition starts in the moment when trefoil branches touch each other. Soon afterwards the state transforms into a Hopf links or breaks down completely and disappears.
<figure><img src="content_image/1702.00161/Fig11.png"><figcaption>Figure 11: Trefoil - Hopf link transformation: (a) - trefoil (t=900), (b) -transformation beginning (t=990), (c) - Hopf link (t=1000), α=0.632,r=0.04,N=100.</figcaption></figure>
There are three ways for the trefoil\(\rightarrow\)Hopf link transformation in Eq.(1). In the first case, when starting from random initial conditions, a trefoil is born, exists for some time interval, and then transforms into a Hopf link, see illustrative video at http://chimera3d.biomed.kiev.ua/multiheaded/Hopflink.
In the second case the trefoil\(\rightarrow\)Hopf link transformation occurs as a result of a strong enough perturbation of the initial conditions (generated originally a trefoil): http://chimera3d.biomed.kiev.ua/multiheaded/evolution. The third transformation scenario consists in the following: take a single trefoil inside its stability region, see Fig. 8, and move the parameter point to the boundary. When close to the boundary, it is often observed that trefoil can suddenly transform into a Hopf link.
To illustrate the latter scenario, fix parameters \(\alpha=0.6325,r=0.04\) close to the boundary of the trefoil stability region. Shift the parameter \(\alpha\) to \(0.632\) and start to simulations. At \(t=900\) the trefoil still exists (Fig. 11(a)) but soon after, at \(t=990\) two trefoil branches touch each other, and the transformation to a Hopf link begins ( in Fig. 11(b)). At \(t=1000\) (Fig. 11(c)) a Hopf link is created, and it exists for long further simulations.
## 4 Hybrid scroll wave chimeras
In the previous sections, different multiheaded scroll wave chimera states were reported for Eq.(1), each including similar incoherent elements such as rolls, Hopf links, trefoils, or chains only. Here, we demonstrate a possibility of hybrid-type organization for the multiheaded chimera states which can combine different of the above mentioned single-headed chimera types.
To obtain hybrid-type scroll wave chimeras we have tested different sample combinations in the cloning procedure. There is no guarantee that the process will be successful. In many trials the cloned hybrid-type chimeras get destroyed very fast, and the procedure has to be repeated starting from a different number organization of the initial ’chimera clones’, also varying the parameters and the magnitude of the perturbations.
Screenshots of characteristic hybrid-type scroll wave chimera states with additively growing number of samples are presented in Fig.12 such as: (a) - Hopf link and trefoil, (b) - Hopf link and 2 trefoils, (c) - 2 Hopf links and 2 trefoils, (d) - Hopf link and 4 trefoils, (e) - 2 Hopf links and 4 trefoils, (f) - 2 Hopf links and 5 trefoils, (g) - 2 Hopf links and 6 trefoils, (h) - 4 Hopf links and 4 rolls (\(\alpha=0.7,r=0.02\)), (i) - 4 Hopf links, 4 trefoils and 4 rolls.
<figure><img src="content_image/1702.00161/x13.png"><figcaption>Figure 12: Examples of multiheaded hybrid chimera states: (a) - one trefoiland one Hopf links (α=0.785,r=0.02), (b) - 2 trefoils and one Hopf links(α=0.755,r=0.02), (c) - 2 trefoils and 2 Hopf links (α=0.735,r=0.02), (d) - 4trefoils and one Hopf links (α=0.625,r=0.02), (e) - 4 trefoils and 2 Hopflinks (α=0.615,r=0.02), (f) - 5 trefoils and 2 Hopf links (α=0.68,r=0.02), (g)- 6 trefoils and 2 Hopf links (α=0.64,r=0.02), (h) - 4 Hopf links and 4 scrollwave rolls (α=0.7,r=0.02). (i) - 4 Hopf links, 4 trefoils and 4 scroll waverolls (α=0.76,r=0.02). N=200.</figcaption></figure>
These states exist for long times, up to \(t=10^{4}\) at least. They are not stationary objects in 3D and are usually characterized by the non-trivial temporal large-scale dynamics. In our simulations we have also observed some other hybrid patterns, also preserving for long-time simulations.
See videos at http://chimera3d.biomed.kiev.ua/multiheaded/hybrid, where more examples of the hybrid-type chimeras are shown.
To finalize, our last example is a 80-headed scroll wave chimera state including 32 Hopf links, 32 trefoils, and 16 rolls illustrated in Fig.13. The state is calculated for N=400 (i.e. 64 million oscillators), simulation time was \(t=1000\).
<figure><img src="content_image/1702.00161/Fig13.png"><figcaption>Figure 13: Example of 80-headed hybrid chimera states consisting of 32trefoils, 32 Hopf links, and 16 rolls (α=0.76,r=0.01). N=400.</figcaption></figure>
## 5 Conclusion
We have demonstrated a diversity of multiple scroll wave chimeras for the three-dimensional network of coupled Kuramoto phase oscillators with non-local coupling. Wide parameter regions are obtained for rolls, chains, Hopf links, and trefoil patterns. It follows, in particular, that rolled chimeras exist not only for non-local but also for local coupling schemes, beginning from only \(N_{P}=7\) nearest-neighbor couplers (as in the simplest diffusive coupling scheme). This fact is schematically indicated by the left corner inset in Fig. 14.
<figure><img src="content_image/1702.00161/Fig14.png"><figcaption>Figure 14: Number of oscillators NP coupled to each oscillator in the model(1) depending on the value of P=rN. Parallel and crossed rolled chimeras existbeginning from P=P1=1.0 (NP=7), single-link chains - from P=P2=1.414...(NP=19), double-link chains - from P=P3=2.0 (NP=27), trefoils and Hopf links -from P=P4=2.44948... (NP=81). The minimal values NP are found to be the samefor N=100,200, and 400.</figcaption></figure>
The next chimera state to appear when increasing the coupling radius \(r=N/P\) in Eq. (1) is the single-link chain, this occurs in the case of \(N_{P}=19\) couplers. At further increase of the coupling radius \(r\), first, the double-link chain stabilizes for \(N_{P}=27\) couplers (\(r=0.01\) at \(N=200\)). Then, only at \(N_{P}=81\) (\(r=0.02449\)) a variety in single and multiple Hopf links and trefoils become stable to exist further for this and larger values of \(r\), up to appr. 0.08.
An essential condition for the scroll wave chimeras appearance is the intermediate value of the phase shift \(\alpha\), appr. between 0.6 and 0.9. Therefore, Hopf link and trefoil chimera states exist in the Kuramoto model only with essentially non-local coupling and sufficiently large phase shift. On the other hand, multiple rolled chimeras which are actually the scroll rings in the circular coordinates are more stable objects. They grow not only for non-local but even for local, diffusive-type coupling schemes. Single-link chain chimeras are also preserved in the locally coupling case, but not the double-link ones which require some level of non-locality for the stabilization. We believe that the described fascinating scroll wave chimeras can be found in other, more realistic 3D networks displaying one of the inherent features of nature, that is due to non-local coupling.
## Acknowledgments
We thank B.Fiedler, E.Knobloch, P.Manneville and M.Hasler for illuminating discussions, and the Ukrainian Grid Infrastructure for providing the computing cluster resources and the parallel and distributed software.
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|
1401.4242 | {
"language": "en",
"source": "Arxiv",
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"content_image/1401.4242/x1.png",
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] | # Universal expression for adiabatic pumping in terms of nonequilibrium steady states
Naoko Nakagawa
College of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan
March 1, 2024
###### Abstract
We develop a unified treatment of pumping and nonequilibrium thermodynamics. We show that the pumping current generated through an adiabatic mechanical operation in equilibrium can be expressed in terms of the stationary distribution of the corresponding driven nonequilibrium system. We also show that the total transfer in pumping can be evaluated from the work imported to the driven counterpart. These findings lead us to a unified viewpoint for pumping and nonequilibrium thermodynamics.
pacs: 05.70.Ln, 05.40.-a, 05.60.Cd For centuries, heat pumping has been considered an important topic. The Carnot engine showed the direct relation between mechanical work and pumping of heat. Pumping induced by electric current known as the Peltier effect, was explained by the linear response theory as an example of the reciprocal relation. In molecular scales, the possibility of realizing heat pumps with thermal ratchets is suggested in [1; 2; 3; 4]
Apart from heat pumps, ion pumps or the directive transport of biomolecules are theoretically intensively studied. These are modeled using flashing ratchets [5; 6; 7; 8; 9] as stochastic pumps in molecular scales. The mechanism of pumping in flashing ratchets is related to geometric effects in the parameter space [10; 12; 11; 13]. The same property in heat pumps has also been discussed [14]. These studies suggest that the universal characteristics of pumps exist in various designs.
In this paper, we develop a unified viewpoint on pumping and nonequilibrium thermodynamics, from which one can derive the universal characteristics of pumps as well as examine efficient protocols for pumping. By pumping, we mean an equilibrium process in which the parameters of the system are varied through an external agent according to a fixed protocol in order to invoke the desired type of current through the system. For each setup of pumping, we introduce a corresponding “_driven counterpart_”, i.e., a nonequilibrium system in which the current flows spontaneously owing to an applied driving field. We then show that the pumping current is expressed in terms of the stationary probability distribution of the driven system and that it is well evaluated from the work imported to the driven counterpart operated using the same protocol.
## I Setup
We employ a classical system with a Hamiltonian \(H(\Gamma)\), where \(\Gamma=(\{\bm{x}\},\{\bm{p}\})\) denotes the system’s microstate. The Hamiltonian depends on a set of parameters \(\bm{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\). We assume a time-reversal symmetry for the Hamiltonian \(H(\Gamma)=H(\Gamma^{*})\), where \(\Gamma^{*}=(\{\bm{x}\},\{-\bm{p}\})\). We do not limit the number of the system’s degrees of freedom. It may be one or the Avogadro number. The system is not isolated but is in contact with an equilibrium environment (baths).
The time evolution of the system is governed by the deterministic dynamics according to the Hamiltonian \(H(\Gamma)\) and the stochastic Markovian dynamics owing to the external bath coupling. One operates the system mechanically by varying the parameters \(\bm{\alpha}\). The protocol for this operation is denoted as \(\hat{\bm{\alpha}}:=(\bm{\alpha}(t))_{t\in[0,\tau]}\). When discussing the time evolution of \(\Gamma\), we denote its value at time \(t\) by \(\Gamma(t)\) and its path in the whole time interval \([0,\tau]\) by \(\hat{\Gamma}=(\Gamma(t))_{t\in[0,\tau]}\).
In order to theoretically analyze pumping problems, we also study a system driven by a certain driving field \(\varepsilon\). We assume that the system reaches a unique nonequilibrium steady state (NESS) when we fix \(\varepsilon\) and \(\bm{\alpha}\) for a sufficiently long time. The transition probability associated with the path \(\hat{\Gamma}\) is denoted by \(\mathcal{T}_{\hat{\bm{\alpha}},\varepsilon}(\hat{\Gamma})\) in a protocol \(\hat{\bm{\alpha}}\) under the driving \(\varepsilon\). The probability distribution in the unique NESS is denoted by \(\rho_{\varepsilon}(\Gamma)\), with which we define
\[\psi^{\varepsilon}(\Gamma):=-\log\rho_{\varepsilon}(\Gamma).\] (1)
Note that \(\rho_{\varepsilon}(\Gamma)\) depends on \(\bm{\alpha}\) although we do not specify it for simplicity of notation. The canonical distribution \(\rho_{\mathrm{eq}}(\Gamma)\) corresponds to \(\rho_{0}(\Gamma)\), and we use \(\psi^{\mathrm{eq}}\) instead of \(\psi^{0}\).
For any function \(f(\hat{\Gamma})\) of a path, we define its average in the protocol \(\hat{\bm{\alpha}}\) as
\[\bigl{\langle}f\bigr{\rangle}_{\varepsilon}:=\int{\mathcal{D}}\hat{\Gamma}\rho _{\varepsilon}(\Gamma(0))\mathcal{T}_{\hat{\bm{\alpha}},\varepsilon}(\hat{ \Gamma})f(\hat{\Gamma}),\] (2)
where \(\int{\mathcal{D}}\hat{\Gamma}(\cdots)\) denotes the integral over all the possible paths \(\hat{\Gamma}\). For any function \(f(\Gamma)\) of a state, we define its average in the steady state as
\[\bigl{\langle}f\bigr{\rangle}_{\rho_{\varepsilon}}:=\int d\Gamma\rho_{ \varepsilon}(\Gamma)f(\Gamma).\] (3)
For equilibrium processes (\(\varepsilon=0\)), we use \(\langle f\rangle_{\mathrm{eq}}\) and \(\langle f\rangle_{\rho_{\mathrm{eq}}}\) instead of \(\langle f\rangle_{0}\) and \(\langle f\rangle_{\rho_{0}}\), respectively.
We assume that the current at time \(t\) is the function of \(\hat{\Gamma}\), \(J(\hat{\Gamma};t)\), i.e. it depends only on the system’s path but not on the system’s environment. Because the probability of the path depends on the environment and the applied protocol, the average \(\langle J\rangle_{\varepsilon}\) in turn depends on them. The total transfer in the whole time interval is given by
\[Q(\hat{\Gamma})=\int_{0}^{\tau}dtJ(\hat{\Gamma};t).\] (4)
In the context of a pump, \(Q(\hat{\Gamma})\) is the “_total pumping_” in a single execution of the protocol.
## II Pumping current and its conjugate driving
For a heat pump carrying energy from one place to the other, \(J(\hat{\Gamma};t)\) is the heat current between the two places and \(Q(\hat{\Gamma})\) is the total transferred heat.
It is crucial for us to observe that the mean heat current can be produced not only by the mechanical operation for pumping but also by imposing a difference in the temperatures at the two places. In the latter case, the mean current flows spontaneously along the natural direction, satisfying the second law of thermodynamics. The difference of the inverse temperatures is often called thermodynamic force corresponding to the heat current. In this paper, we call it the _conjugate driving_ corresponding to the heat current.
We refine the above situation as follows: In order to study the heat pumping in the system in contact with two separate isothermal heat baths indexed by \(k\) (\(k=1,2\)), for which the inverse temperature is denoted by \(\beta\), we also study its counterpart with the conjugate driving, i.e. the same system, for which the inverse temperatures \(\beta_{1}\), \(\beta_{2}\) of the baths are different. We choose \(\beta_{k}\) so as to satisfy \(\beta=(\beta_{1}+\beta_{2})/2\).
Letting \(J_{k}(\hat{\Gamma};t)\) be the heat current from the \(k\)th heat bath to the system at time \(t\) in the path \(\hat{\Gamma}\), the heat current from one heat bath to the other is formulated as
\[J(\hat{\Gamma};t)=\frac{J_{1}(\hat{\Gamma};t)-J_{2}(\hat{\Gamma};t)}{2}\] (5)
for both the pumping system and its driven counterpart. By computing the average, we have \(\langle J\rangle=\langle J_{1}\rangle=-\langle J_{2}\rangle\) under steady driving or any cyclic protocol. The conjugate driving, i.e. the thermodynamic force corresponding to the heat current is
\[\varepsilon=\beta_{2}-\beta_{1}.\] (6)
The entropy production owing to the heat current is \(\varepsilon J(\hat{\Gamma};t)=(\beta_{2}-\beta_{1})J(\hat{\Gamma};t)\).
For stochastic pumps represented using flashing ratchet models (see Fig. 1), we consider a particle in a potential with a periodic boundary condition in a certain coordinate \(x\). When applying a cyclic operation to the potential, the system may have a nonvanishing circulation in its microstates, and this may be observed as directed mean current \(\langle J\rangle_{\mathrm{eq}}\) of the particle, where
\[J(\hat{\Gamma};t)=\dot{x}(t).\] (7)
We notice that \(J(\hat{\Gamma};t)\) is determined by the system’s microstate and not by the operation.
The driven counterpart is the same system in which the particle is pulled by a constant nonconservative force \(f\) along the coordinate \(x\). The conjugate driving is
\[\varepsilon=\beta f,\] (8)
and the entropy production is \(\beta fJ(\hat{\Gamma};t)\), where \(J(\hat{\Gamma};t)\) for the driven system is the same as Eq. (7).
Even though we present our claims for a closed system setup in this paper, they can also be extended to include open systems with particle baths by modifying the setup, as discussed in Sec. 5 of [15]. For such open systems, we can consider particle pumping between two particle baths, where the particle current is defined parallel to Eq. (5). Here, the conjugate driving corresponds to \(\varepsilon=\beta(\mu_{2}-\mu_{1})\), where \(\mu_{k}\) is the chemical potential for the \(k\)th particle bath.
## III Main results
### The expression for total pumping
Our main result is the expression for the total pumping produced in equilibrium adiabatic operations, the derivation for which is given in the Appendix.
For the adiabatic protocol \(\hat{\bm{\alpha}}\) applied to an equilibrium system, the total pumping is
\[\bigl{\langle}Q\bigr{\rangle}_{\mathrm{eq}} = \int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot\Bigl{\langle}\left. \nabla_{\bm{\alpha}}\psi^{\mathrm{eq}}~{}\partial_{\varepsilon}\psi^{ \varepsilon}\right|_{\varepsilon=0}\Bigr{\rangle}_{\rho_{\mathrm{eq}}}\] (9)
\[= \int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot\Bigl{\langle}\nabla_{ \bm{\alpha}}\left.\partial_{\varepsilon}\psi^{\varepsilon}\right|_{\varepsilon =0}\Bigr{\rangle}_{\rho_{\mathrm{eq}}},\] (10)
where \(\int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdots\) is the line integral along the protocol \(\hat{\bm{\alpha}}\) in the parameter space of \(\bm{\alpha}\). It is remarkable that the total pumping is directly related to the steady probability distribution \(\rho_{\varepsilon}(\Gamma)\) for the driving counterpart. \(\rho_{\varepsilon}(\Gamma)\) depends on the type of the conjugate driving \(\varepsilon\), as does the equilibrium pumping.
The expression (9) indicates that the pumping is efficient when \(\nabla_{\bm{\alpha}}\psi^{\mathrm{eq}}(\Gamma)\) is parallel to \(\partial_{\varepsilon}\psi^{\varepsilon}(\Gamma)\) in the phase space of \(\Gamma\). In other words, it is efficient when the operation \(\hat{\bm{\alpha}}\) well mimics the nonequilibrium driving. It is worth noting that the kernels of Eqs. (9) and (10) correspond to the off-diagonal components of the Fisher information matrix because \(\langle\nabla_{\bm{\alpha}}\partial_{\varepsilon}\psi^{\varepsilon}|_{ \varepsilon=0}\rangle_{\rho_{\mathrm{eq}}}=\langle\nabla_{\bm{\alpha}}\partial _{\varepsilon}\psi^{\varepsilon}\rangle_{\rho_{\varepsilon}}\) for \(\varepsilon\to 0\).
In cyclic protocols \(\hat{\bm{\alpha}}_{\mathrm{cyc}}\), we can apply the Stokes’ theorem to \(\oint d{\bm{\alpha}}\cdot\rho_{\mathrm{eq}}(\Gamma)\nabla_{\bm{\alpha}}\psi^{\varepsilon}\) in the right-hand side of Eq. (10). Therefore,
\[\bigl{\langle}Q\bigr{\rangle}_{\mathrm{eq}}=\int_{S}dS~{}\bigl{\langle}{\cal J }\bigr{\rangle}_{\rho_{\mathrm{eq}}},\] (11)
where \(S\) is the region in the parameter space enclosed by the closed line of \(\hat{\bm{\alpha}}_{\mathrm{cyc}}\). We call \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\) the _pumping density_. When the number of parameters is two, i.e., \(\bm{\alpha}=(\alpha_{1},\alpha_{2})\), the pumping density is
\[{\cal J}(\Gamma)=\left.\partial_{\varepsilon}\left[\partial_{\alpha_{1}}\psi^{ \varepsilon}(\Gamma)\partial_{\alpha_{2}}\psi^{\mathrm{eq}}(\Gamma)-\partial_{ \alpha_{1}}\psi^{\mathrm{eq}}(\Gamma)\partial_{\alpha_{2}}\psi^{\varepsilon}( \Gamma)\right]\right|_{\varepsilon=0}.\] (12)
Various studies relating pumping to a geometric effect or the Berry phase [10; 12; 11; 13] report a result similar to Eq. (12), which is derived from the master equation or the cumulant generating function in cyclic operations in equilibrium. We emphasize that the key point of our formula (12) is the use of the probability distribution \(\rho_{\varepsilon}\) for the driven counterpart.
### Equilibrium pumping and work in a driven counterpart
To apply Eqs. (9), (10) or (12) to the pumping problem, we need to determine the probability distribution \(\rho_{\varepsilon}(\Gamma)\). Since \(\rho_{\varepsilon}(\Gamma)\) is not known in general, we show how \(\langle Q\rangle_{\mathrm{eq}}\) and \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\) can be approximately evaluated from an observable quantity.
We apply an adiabatic cyclic operation to both the equilibrium system and its driven counterpart. Then from Eq. (10) we get an approximate equality,
(13)
which relates the quantities of these distinct systems. The derivation of Eq. (13) is shown in the Appendix. Thus, we can evaluate “pumping in equilibrium” from the measurement of “work in a driven counterpart.” Here, the work is given by \(\langle W\rangle_{\varepsilon}=\int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot \langle\nabla_{\alpha}H\rangle_{\rho_{\varepsilon}}\). The relation (13) is consistent with the extended Clausius equality in [16; 17].
For general cyclic operations with a finite speed, the total pumping is related to work in the nonequilibrium counterpart as
(14)
where \(\langle\cdot\rangle^{\dagger}\) is the average along the reverse cyclic protocol \(\hat{\bm{\alpha}}^{\dagger}\), i.e. \(\hat{\bm{\alpha}}^{\dagger}=(\bm{\alpha}(\tau-t))_{t\in[0,\tau]}\). The relation (14) follows from an extended Jarzynski equality to NESS [18].
Relations (13) and (14) suggest a new approach to study pumping when \(Q\) is difficult to observe but \(W\) is measurable. Depending on the protocol, relation (13) or (14) may be useful. Note that the Jarzynski-like form (14) is more useful in mesoscopic pumps because the Jarzynski equality [19] is known to be efficient in mesoscopic systems.
We expect that the map of \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\) in the space of \(\bm{\alpha}\) can be a powerful tool to design an efficient protocol for pumping. Equation (11) indicates that \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\) is approximated by \(\langle Q\rangle_{\mathrm{eq}}/S\simeq-\beta\langle W\rangle_{\varepsilon}/( \varepsilon S)\), when we apply a cyclic protocol with a sufficiently small area \(S\) in the parameter space.
## IV Examples
### Numerical demonstration for stochastic pumping
<figure><img src="content_image/1401.4242/x1.png"><figcaption>Figure 1: (Color online) (a) Ratchet potentialV(x;α1,α2)={x4−(x+α1)2−5}{tanh(x+3+α2)−tanh(x−3)}/100 and the applied cyclicprotocol. The values of (α1,α2) are written in respective figures. Theparticle (solid red circle) evolves according to Eq. (15), where we take γ=1and kBT=0.3. For the respective change (green arrow), either α1 or α2 ischanged in a constant speed (∝τ−1). (b) Operation time τ vs total pumping⟨Q⟩eq observed in a cycle. We start from (α1,α2)=(1,0) after preparing itssteady state and continue the operation without stopping until we return to(1,0). After finishing the operation, we continue calculation until the systemreaches equilibrium. The dashed line corresponds to the estimate ⟨Q⟩eq=0.256explained in (c). (c) The map of pumping density ⟨J⟩ρeq. ρε(Γ) is calculatedin the counterpart driven by ε=2/3×10−2. ⟨Q⟩eq for the adiabatic limit of theprotocol in (a) is estimated as 0.256 from the integration of ⟨J⟩ρeq in Eq.(11). The lightest shading corresponds to 0≤⟨J⟩ρeq<0.15 and the black to⟨J⟩ρeq>1.5</figcaption></figure>
<figure><img src="content_image/1401.4242/x4.png"><figcaption>Figure 2: (Color online) (a) The reverse protocol applied to the system. Theparticle is always driven by f. (b) ε vs −log⟨e−βW⟩†ε (solid square) andβ⟨W⟩†ε (open triangle) for τ=1000. We start from the initial conditions in thesteady state under the conjugate driving and start the reverse of the cyclicprotocol in (a). We measure W up to the end of the change for α withoutcalculating the relaxation process after the change. The dashed lines areproportional to ε with a slope 0.256 whose value was estimated from thepumping density J(Γ) in Fig. 1 (c).</figcaption></figure>
We take a flashing ratchet model (see Fig. 1). The position of a particle evolves in a one-dimensional periodic potential \(V(x;\alpha_{1},\alpha_{2})\) according to the Langevin equation
\[\gamma\dot{x}=-\frac{\partial V}{\partial x}+\sqrt{2\gamma k_{\mathrm{B}}T}\xi (t),\] (15)
where \(\gamma\) is the friction constant, \(T\) is the temperature of the environment, and \(k_{\mathrm{B}}\) is the Boltzmann constant. The ratchet potential \(V(x;\alpha_{1},\alpha_{2})\) is operated externally by changing \((\alpha_{1},\alpha_{2})\) in the operation time \(\tau\) [Fig. 1(a)]. The total pumping \(\langle Q\rangle_{\mathrm{eq}}\) in this example corresponds to the mean shift of the particle.
As shown in Fig. 1(b), \(\langle Q\rangle_{\mathrm{eq}}\) converges to a certain value in larger values of \(\tau\), which will be the value for the adiabatic limit. Indeed, it is approximately equal to the expected total pumping in the adiabatic limit indicated by the dashed line, which is estimated from the calculation of the pumping density \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\). To determine \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\), we calculate \(\rho_{\varepsilon}(\Gamma)\) for the system under the conjugate driving \(\varepsilon=\beta f\),
\[\gamma\dot{x}=-\frac{\partial V}{\partial x}+f+\sqrt{2\gamma k_{\mathrm{B}}T} \xi(t)\] (16)
for a certain \(\bm{\alpha}\), from which we determine \({\cal J}(\Gamma)\) in Eq. (12) and take the average of \({\cal J}(\Gamma)\) by \(\rho_{\mathrm{eq}}(\Gamma)\). We repeat this procedure for various \(\bm{\alpha}\), and obtain the contour plot of \(\langle{\cal J}\rangle_{\rho_{\mathrm{eq}}}\) shown in Fig. 1(c).
Next, we apply the reverse cyclic protocol [Fig. 2(a)] to Eq. (16) and calculate the work \(W=\int_{\hat{\bm{\alpha}}^{\dagger}}d{\bm{\alpha}}\cdot\nabla_{\alpha}V\). From the ensemble of \(W\) for a slow operation, we calculate \(-\log\langle e^{-\beta W}\rangle^{\dagger}_{\varepsilon}\), which is proportional to \(\varepsilon\) as shown in Fig. 2(b). The slope in the figure (\(\tau=1000\)) is close to the total pumping \(\langle Q\rangle_{\mathrm{eq}}\) for the adiabatic limit. This coincidence corresponds to the convergence of \(\langle Q\rangle_{\mathrm{eq}}\) around \(\tau=1000\) [see Fig. 1(b)]. In Fig. 2(b), we supplementarily plot \(\beta\langle W\rangle^{\dagger}_{\varepsilon}\). Since \(-\log\langle e^{-\beta W}\rangle^{\dagger}_{\varepsilon}=\beta\langle W\rangle ^{\dagger}_{\varepsilon}+O(\tau^{-2})\), the line deviates from the origin at \(\varepsilon=0\) due to the finiteness of \(\tau\). However, the slope of \(\beta\langle W\rangle^{\dagger}_{\varepsilon}\) is also close to \(\langle Q\rangle_{\mathrm{eq}}\) for the adiabatic limit. When we take a smaller value of \(\tau\), the slope of \(\beta\langle W\rangle^{\dagger}_{\varepsilon}\) or \(-\log\langle e^{-\beta W}\rangle^{\dagger}_{\varepsilon}\) becomes less steep consistently with the decrease of \(\langle Q\rangle_{\mathrm{eq}}\). This result may suggest that the slope of \(\beta\langle W\rangle^{\dagger}_{\varepsilon}\) is an informative quantity for various pumping protocols with finite speed.
### Pumping densities in three state model
We here study a simpler example which can be solved exactly. We take a one-dimensional Markov jump model of three states (\(x=1,2\) and \(3\)) with a periodic boundary condition identifying \(x=3\) with \(x=0\). It acts as both a heat and a stochastic pump simultaneously.
In order to design the rate constants for the jump, we assume virtual energy barriers at every midpoint of the neighboring two states. We set the energies of the three states as \(v_{1}\), \(v_{2}\) and \(v_{3}\), and the energies of the barriers as \(u_{12}\), \(u_{23}\) and \(u_{31}\), respectively. Then, we express the jump rates \(R_{yx}\) from \(x\) to \(y\) as \(R_{yx}=e^{-\beta(u_{yx}-v_{x})}\). We assume the parameters for the operation as \(\bm{\alpha}=(v_{2},u_{23})\).
First, we show the pumping density when the system works as a heat pump. For this purpose, we assume the system is in contact with two heat baths: The one (say \(\beta_{1}\)) is in the region \(1\leq x<2.5\) and the other (say \(\beta_{2}\)) is in \(2.5\leq x<4(=1)\). The rate matrix for the conjugate driving \(\varepsilon=\beta_{2}-\beta_{1}\) is expressed as
\[R^{\varepsilon}=\left(\begin{array}[]{ccc}-\lambda_{1}^{\varepsilon}&R_{12}e^{ \frac{\varepsilon}{2}(u_{12}-v_{1})}&R_{13}e^{-\frac{\varepsilon}{2}(u_{13}-v_ {1})}\\ R_{21}e^{\frac{\varepsilon}{2}(u_{12}-v_{2})}&-\lambda_{2}^{\varepsilon}&R_{23 }e^{-\frac{\varepsilon}{2}(u_{23}-v_{2})}\\ R_{31}e^{-\frac{\varepsilon}{2}(u_{12}-v_{3})}&R_{32}e^{\frac{\varepsilon}{2}( u_{23}-v_{3})}&-\lambda_{3}^{\varepsilon}\end{array}\right),\] (17)
where \(\lambda_{x}^{\varepsilon}=\sum_{y\neq x}R_{xy}^{\varepsilon}\). We numerically calculate the probability distribution \(\rho_{\varepsilon}(x)\) for various values of \(\bm{\alpha}\). Figure 3(a) shows the pumping density resulting from the set of \(\rho_{0}(x)\) and \(\rho_{\varepsilon}(x)\).
<figure><img src="content_image/1401.4242/x6.png"><figcaption>Figure 3: Contour plot in a grey scale for pumping density |⟨J⟩ρeq|determined by ρeq(x) and ρε(x), where ρε(x) is calculated from the ratematrices. (a) Heat-pumping density calculated from the rate matrix (17). Theoperational parameters are v2 and u23 and the other parameters are fixed asv1=v3=0 and u12=u31=1. The lightest shading corresponds to |⟨J⟩ρeq|<0.011 andthe black to |⟨J⟩ρeq|>0.099. (b) Stochastic pumping density calculated fromthe rate matrix (18). The parameters are the same as in (a). The lightestshading corresponds to |⟨J⟩ρeq|<0.099 and the black to |⟨J⟩ρeq|>0.135.</figcaption></figure>
Second, we map the pumping density when the same system works as a stochastic pump. For the conjugate driving, we consider a uniform nonconservative force \(f\) in the direction of \(x\), i.e. \(\varepsilon=\beta f\). The rate constants \(R_{yx}^{\varepsilon}\) are
\[R^{\varepsilon}=\left(\begin{array}[]{ccc}-\lambda_{1}^{\varepsilon}&R_{12}e^{ -\frac{\varepsilon}{2}}&R_{13}e^{\frac{\varepsilon}{2}}\\ R_{21}e^{\frac{\varepsilon}{2}}&-\lambda_{2}^{\varepsilon}&R_{23}e^{-\frac{ \varepsilon}{2}}\\ R_{31}e^{-\frac{\varepsilon}{2}}&R_{32}e^{\frac{\varepsilon}{2}}&-\lambda_{3}^ {\varepsilon}\end{array}\right).\] (18)
The pumping density is shown in Fig. 3(b). These maps show that the system pumps both heat and particle simultaneously.
## V Discussions
We have developed a unified viewpoint on pumping and nonequilibrium thermodynamics by introducing a driven counterpart to pumping. With our unified viewpoint one can rederive various pumping results such as Eqs. (9), (10), (12), (13) and (14). From a theoretical point of view, the connection of total pumping \(\langle Q\rangle_{\mathrm{eq}}\) to the stationary distribution \(\rho_{\varepsilon}\) in the driven counterpart or to the Fisher information matrix (9) and (10) is most interesting. We expect that the accumulated knowledge on the Fisher information matrix provides a new viewpoint on pumping, while it remains as a future work.
From a point of applicability, we have related the total pumping \(\langle Q\rangle_{\mathrm{eq}}\) in equilibrium to the work \(\langle W\rangle_{\varepsilon}\) or \(-\log\langle e^{-\beta W}\rangle^{\dagger}_{\varepsilon}\) in the driven counterpart as shown in Eqs. (13) and (14). These relations are useful when \(Q\) is difficult to observe but \(W\) is measurable. As an example of application, we evaluate the pumping \(\langle Q\rangle_{\mathrm{eq}}\) from \(W\) in a mesoscopic pump. See Fig.2(b).
The work relation (13) accompanied by Eqs. (11) and (12) shows that meso- or macroscopic force in NESS is no longer a potential force due to the geometric effects of pumping. We need to use vector potential related to pumping in addition to the usual scalar potential. We comment that the geometric effect of excess heat reported in [20] has the same origin as the geometric effects of pumping in Eq. (12) and in [10; 12; 11; 13; 14]. This is because relation (13) is a version of an extended Clausius relation, which makes a connection between the excess heat and the entropy change [16].
AcknowledgementThe author is grateful to Hal Tasaki for stimulating discussions and a critical reading of the manuscript, and to Keiji Saito for suggestions and comments, especially on the relation of (9) to the Fisher information matrix. This work was supported by JSPS/MEXT KAKENHI Grants No. 23540435 and No. 25103002.
## VI Appendix
In this Appendix, we use fully specified notations: \(\rho^{\mathrm{st}}_{\bm{\alpha},\varepsilon}(\Gamma)\), \(\rho^{\mathrm{eq}}_{\bm{\alpha}}(\Gamma)\), \(\psi_{\bm{\alpha}}^{\varepsilon}(\Gamma)\) and \(\psi_{\bm{\alpha}}^{\mathrm{eq}}(\Gamma)\) instead of \(\rho_{\varepsilon}(\Gamma)\), \(\rho_{\mathrm{eq}}(\Gamma)\), \(\psi^{\varepsilon}(\Gamma)\) and \(\psi^{\mathrm{eq}}(\Gamma)\). The averages are \(\langle f\rangle^{\hat{\bm{\alpha}}}_{\varepsilon}\) and \(\bigl{\langle}f\bigr{\rangle}^{\bm{\alpha}}_{\rho_{\varepsilon}}\) instead of \(\langle f\rangle_{\varepsilon}\) and \(\bigl{\langle}f\bigr{\rangle}_{\rho_{\varepsilon}}\).
### Derivation of Eq. (9)
We reported in [15] that the probability distribution of NESS under the steady driving field \(\varepsilon\) has a linear response representation,
\[\rho^{\mathrm{st}}_{\bm{\alpha},\varepsilon}(\Gamma)=\rho^{ \mathrm{eq}}_{\bm{\alpha}}(\Gamma)~{}\exp{\left[-\varepsilon\bigl{\langle}Q \bigr{\rangle}^{(\bm{\alpha})}_{\Gamma^{*}\rightarrow\mathrm{eq}}\right]}+O( \varepsilon^{2}),\] (19)
where a conditioned expectation is defined as
\[\langle Q\rangle_{\Gamma\rightarrow\mathrm{eq}}^{(\bm{\alpha})}=\int{\mathcal{ D}}\hat{\Gamma}\,\delta(\Gamma(0)-\Gamma)\mathcal{T}_{(\bm{\alpha})}^{\mathrm{ eq}}[\hat{\Gamma}]\,Q(\hat{\Gamma}),\] (20)
with a fixed initial state \(\Gamma\). The notation \((\bm{\alpha})\) represents the protocol in which the parameters are kept constant at \(\bm{\alpha}\) and \(\mathcal{T}_{(\bm{\alpha})}^{\mathrm{eq}}=\mathcal{T}_{(\bm{\alpha}),0}\). The conditioned average \(\langle Q\rangle^{(\bm{\alpha})}_{\Gamma\rightarrow\mathrm{eq}}\) gives the total transfer observed in the relaxation process from the state \(\Gamma\). There is no transfer on average in equilibrium, i.e.,
(21)
We first concentrate on the protocol of an infinitesimal stepwise change from \(\bm{\alpha}\) to \(\bm{\alpha}^{\prime}=\bm{\alpha}+\mathit{\Delta}\bm{\alpha}\). Even though we do not observe any current before the stepwise change, we may observe it in the relaxation process after the stepwise change. Noting that \(\langle Q\rangle_{\Gamma\rightarrow\mathrm{eq}}^{(\bm{\alpha}^{\prime})}\) is the total transfer in the relaxation from the state \(\Gamma\), the total transfer after the stepwise change is
\[\bigl{\langle}Q\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\mathrm{eq}} =\] (22)
\[=\]
where we subtract Eq. (21) from the first line of Eq. (22) in order to obtain the expression in the second line. If \(\rho^{\mathrm{eq}}_{\bm{\alpha}^{\prime}}(\Gamma)=\rho^{\mathrm{eq}}_{\bm{ \alpha}}(\Gamma)+O(|\mathit{\Delta}\bm{\alpha}|)\) and \(\langle Q\rangle_{\Gamma\rightarrow\mathrm{eq}}^{(\bm{\alpha}^{\prime})}= \langle Q\rangle_{\Gamma\rightarrow\mathrm{eq}}^{(\bm{\alpha})}+O(|\mathit{ \Delta}\bm{\alpha}|)\), then
\[\bigl{\langle}Q\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\mathrm{eq}} = -{\mathit{\Delta}\bm{\alpha}}\cdot\int d\Gamma(\nabla_{\alpha} \rho^{\mathrm{eq}}_{\bm{\alpha}}(\Gamma))\bigl{\langle}Q\bigr{\rangle}_{\Gamma \rightarrow\mathrm{eq}}^{(\bm{\alpha})}\] (23)
with an error of \(O(|\mathit{\Delta}\bm{\alpha}|^{2})\).
As the next step, we refer to the representation (19), where \(\langle Q\rangle_{\Gamma\rightarrow\mathrm{eq}}^{(\bm{\alpha})}\) is related to \(\rho^{\mathrm{st}}_{\bm{\alpha},\varepsilon}\). Therefore, it is apparent that
(24)
Substituting Eq. (24) into Eq. (23), we have
\[\bigl{\langle}Q\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\mathrm{eq}} = -{\mathit{\Delta}\bm{\alpha}}\cdot\int d\Gamma(\nabla_{\alpha} \rho^{\mathrm{eq}}_{\bm{\alpha}}(\Gamma))\left.\partial_{\varepsilon}\psi_{\bm {\alpha}}^{\varepsilon}(\Gamma)\right|_{\varepsilon=0},\] (25)
\[= {\mathit{\Delta}\bm{\alpha}}\cdot\int d\Gamma\rho^{\mathrm{eq}}_{ \bm{\alpha}}(\Gamma)\left.\nabla_{\alpha}\psi_{\bm{\alpha}}^{\mathrm{eq}}( \Gamma)\partial_{\varepsilon}\psi_{\bm{\alpha}}^{\varepsilon}(\Gamma)\right|_{ \varepsilon=0},\]
where the negligible error term of \(O(|\mathit{\Delta}\bm{\alpha}|^{2})\) is ignored. We used \(\rho^{\mathrm{eq}}_{\bm{\alpha}}(\Gamma)=\rho^{\mathrm{eq}}_{\bm{\alpha}}( \Gamma^{*})\) to obtain the first line of Eq. (25).
Finally, we note that any adiabatic protocol is the accumulation of infinitesimal steps. We need to extend Eq. (25) to the line integral along the protocol \(\hat{\bm{\alpha}}\), as is expressed in Eq. (9).
In order to arrive at expression (10), we use an identity,
\[\int d\Gamma\rho(\Gamma)\frac{\partial^{2}\psi(\Gamma)}{\partial\alpha\partial \varepsilon}=\int d\Gamma\rho(\Gamma)\frac{\partial\psi(\Gamma)}{\partial \alpha}\frac{\partial\psi(\Gamma)}{\partial\varepsilon},\] (26)
which is derived from integration by parts and the conservation law \(\int\rho(\Gamma)d\Gamma=1\).
### Derivation of Eq. (13)
We start from Eq. (10). Substituting Eq. (24) into Eq. (10), we have
\[\bigl{\langle}Q\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\mathrm{eq}} = \int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot\int d\Gamma\rho^{ \mathrm{eq}}_{\bm{\alpha}}(\Gamma)\nabla_{\alpha}\bigl{\langle}Q\bigr{\rangle} _{\Gamma^{*}\rightarrow\mathrm{eq}}^{(\bm{\alpha})}\] (27)
\[= \int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot\int d\Gamma\rho^{ \mathrm{eq}}_{\bm{\alpha}}(\Gamma)\bigl{\langle}Q\bigr{\rangle}_{\Gamma^{*} \rightarrow\mathrm{eq}}^{(\bm{\alpha})}\nabla_{\alpha}\psi_{\bm{\alpha}}^{ \mathrm{eq}}(\Gamma),\]
where we applied the integration by parts. As the expression (19) leads to
\[\rho^{\mathrm{eq}}_{\bm{\alpha}}(\Gamma)~{}\bigl{\langle}Q\bigr{\rangle}^{(\bm {\alpha})}_{\Gamma^{*}\rightarrow\mathrm{eq}}=-\frac{\rho^{\mathrm{st}}_{\bm{ \alpha},\varepsilon}(\Gamma)-\rho^{\mathrm{eq}}_{\bm{\alpha}}(\Gamma)}{ \varepsilon}+O(\varepsilon),\] (28)
Eq. (27) is transformed as
\[\bigl{\langle}Q\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\mathrm{eq}} = -\frac{1}{\varepsilon}\int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot \int d\Gamma[\rho^{\mathrm{st}}_{\bm{\alpha},\varepsilon}(\Gamma)-\rho^{ \mathrm{eq}}_{\bm{\alpha}}(\Gamma)]\nabla_{\alpha}\psi_{\bm{\alpha}}^{\mathrm{ eq}}(\Gamma)\] (29)
\[= \frac{\beta}{\varepsilon}\int_{\hat{\bm{\alpha}}}d{\bm{\alpha}} \cdot\int d\Gamma\rho^{\mathrm{st}}_{\bm{\alpha},\varepsilon}(\Gamma)\left( \nabla_{\alpha}F-\nabla_{\alpha}H(\Gamma)\right),\]
where \(F\) is the equilibrium free energy satisfying \(\langle\nabla_{\alpha}H\rangle^{\bm{\alpha}}_{\rho^{\mathrm{eq}}}=\nabla_{ \alpha}F\). Thus, we arrive at the final formula
\[\bigl{\langle}Q\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\mathrm{eq}}=-{\beta}\frac{ \bigl{\langle}W\bigr{\rangle}^{\hat{\bm{\alpha}}}_{\varepsilon}-\mathit{\Delta }F}{\varepsilon}+O(\varepsilon),\] (30)
where \(\mathit{\Delta}F=\int_{\hat{\bm{\alpha}}}d{\bm{\alpha}}\cdot\nabla_{\alpha}F\) and \(\langle W\rangle^{\hat{\bm{\alpha}}}_{\varepsilon}=\int_{\hat{\bm{\alpha}}}d{ \bm{\alpha}}\cdot\int d\Gamma\rho^{\mathrm{st}}_{\bm{\alpha},\varepsilon}( \Gamma)\nabla_{\alpha}H(\Gamma)\). Since \(\mathit{\Delta}F=0\) in cyclic protocols, we have Eq. (13) as a direct consequence of Eq. (30).
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] | # Sensitivity of IceCube-DeepCore to neutralino dark matter in the MSSM-25
[
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February 23, 2024
###### Abstract
We analyse the sensitivity of IceCube-DeepCore to annihilation of neutralino dark matter in the solar core, generated within a 25 parameter version of the minimally supersymmetric standard model (MSSM-25). We explore the 25-dimensional parameter space using scanning methods based on importance sampling and using DarkSUSY 5.0.6 to calculate observables. Our scans produced a database of 6.02 million parameter space points with neutralino dark matter consistent with the relic density implied by WMAP 7-year data, as well as with accelerator searches. We performed a model exclusion analysis upon these points using the expected capabilities of the IceCube-DeepCore Neutrino Telescope. We show that IceCube-DeepCore will be sensitive to a number of models that are not accessible to direct detection experiments such as SIMPLE, COUPP and XENON100, indirect detection using Fermi-LAT observations of dwarf spheroidal galaxies, nor to current LHC searches.
a]Hamish Silverwood, b]Pat Scott, c]Matthias Danninger, c,d]Christopher Savage,
c]Joakim Edsjö,
a]Jenni Adams, a]Anthony M Brown c]and Klas Hultqvist
[a]Department of Physics and Astronomy, University of Canterbury, Christchurch 8140, New Zealand [b]Department of Physics, McGill University, Montréal QC H2W2L8, Canada [c]Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden [d]Department of Physics & Astronomy, University of Utah, Salt Lake City, UT 84112, USA
## 1 Introduction
Much of the development of supersymmetry (SUSY) was motivated by problems within the Standard Model (SM), such as the fine tuning of the Higgs mass [1, 2] and the unification of gauge coupling constants at high energy [3, 4, 5, 6, 7]. When the lightest neutralino is the lightest supersymmetric particle (LSP) and \(R\)-parity is conserved, the lightest neutralino is a natural candidate for dark matter, providing a solution to one of the most important problems in astrophysics.
Neutralinos can accumulate in the centre of astrophysical bodies such as the Sun, where they would annihilate into a range of SM particles. Subsequent decays and interactions of these SM particles would yield neutrinos, which have the unique ability to escape the solar core and reach detectors on Earth [8, 9, 10].
One such detector is the IceCube Neutrino Telescope [11]. Previous analyses of data recorded by IceCube in the 22- and 40-string configurations have already provided limits on neutralino dark matter annihilation in the Sun, the neutralino-proton cross-section, and the resulting muon flux [12, 13]. Further papers have used projected capabilities of the full 86-string IceCube-DeepCore detector (IceCube-86) to analyse parameterisations of the MSSM in more detail, such as the four-parameter Constrained MSSM (CMSSM) [14, 15], the seven parameter MSSM-7 [16, 17], and the 19 parameter phenomenological MSSM (pMSSM) [18].
In this paper we present the results of an exploration of a 25 parameter version of the MSSM and the sensitivity of IceCube-86 to these models. This analysis involves two new elements compared to previous work: the parameter space has been enlarged to 25 parameters, reducing the artificial constrictions placed upon the MSSM; and the IceCube-86 model exclusion confidence level (CL) calculations now include point-by-point optimisation of the reconstructed cut angle around the Sun, which was not included in a similar analysis performed in [17].
The structure of this paper is as follows. In Section 2 we introduce the IceCube Neutrino Telescope and its salient characteristics. Then, in Section 3, we present the details of our analytical methods: the MSSM-25 parameterisation within which we worked (3.1), the method by which we explored this parameter space (3.2), the calculation of signal rates in IceCube (3.3), and finally the calculation of IceCube model exclusion CLs (3.4). In Section 4 the results of this exploration and analysis are presented and discussed.
## 2 The IceCube Neutrino Telescope
Completed in December 2010, the IceCube Neutrino Observatory consists of a cubic kilometre of extremely transparent natural ice deep below the South Pole, instrumented with 5160 digital optical modules (DOMs), along with 320 DOMs embedded in ice-filled tanks at the surface (known as the IceTop array). The in-ice DOMs are arranged along 86 strings lowered down vertical wells melted into the ice using a hot water drill. Seventy-eight of the strings are arranged in a hexagonal grid with a horizontal spacing of 125 m between strings, with DOMs separated vertically by 17 m and situated between 1450 m and 2450 m below the surface. The remaining eight strings are clustered in the centre with a horizontal spacing of about 72 m or less, and have 50 of their DOMs positioned with 7 m vertical spacing starting at the bottom. The remaining ten are arranged higher up the strings to act as a down-going muon veto. These eight strings and the 12 adjacent standard IceCube strings make up the DeepCore subarray. DeepCore provides increased sensitivity at lower energies and reduces the energy threshold to approximately 10 GeV [19]. The IceTop array is used to study cosmic ray flux and composition, and serves as a veto for high energy cosmic ray showers [11].
IceCube can reconstruct the type, energy and direction of the incoming neutrinos by observing the Čerenkov radiation emitted by leptons created in charged current interactions between the neutrinos and nuclei in the ice. Muons produced in such interactions are of particular interest, as their mean free paths can be up to 10km for the most energetic neutrinos. One can thus detect neutrinos that interact outside the volume of instrumented ice making up IceCube [11].
The sensitivity of IceCube to neutrinos of different energies is characterised by the effective area. For this study we use the effective area for muon neutrino detection by the 86-string IceCube configuration (IceCube-86), derived from simulation and presented in [20]. The sensitivity study presented in [20] is for the full 86 string detector with the initially proposed DeepCore geometry, that consisted of only six additional strings instead of the eight deployed. The study was performed as a full analysis in all details on simulated backgrounds, including realistic data processing and event selection. After the work presented in this paper was completed, preliminary results from the IceCube-79 detector were released [21]; final results were submitted just before this paper was accepted [22]. A comparison of the effective area from [20] and these more recent results suggests a moderate over-estimation of the effective area by the earlier sensitivity study: a factor of \(\sim\)3 at 100 GeV neutrino energy, dropping to \(\sim\)1.5 at 1 TeV. Given the order unity difference, the effective area from the IceCube-86 sensitivity analysis [20] will suffice for our purposes here.
## 3 Methods
### The MSSM-25 Parameterization
The full MSSM has 178 free parameters: 158 from the soft SUSY-breaking sector, one from the coupling of the two Higgs supermultiplets, and the remaining 19 from the Standard Model [23]. 54 of these parameters can be removed by suppressing the \(\mathbf{C}\)-terms of the soft SUSY-breaking Lagrangian, leaving 124 parameters. We direct the reader to [23] for more information. For phenomenological studies it is useful to reduce this number in order to make exploration of the parameter space computationally feasible. This reduction is performed via a series of assumptions that become progressively more severe as more parameters are eliminated. Previous analyses of neutralino dark matter signals in IceCube have used models with four, seven, nine and 19 parameters [16, 14, 15, 18, 17]. For this research we used a 25-parameter version of the MSSM.
The potential utility of increasing the number of parameters lies in reducing the artificial constrictions placed upon the MSSM and allowing different combinations of parameters that could perhaps yield dark matter candidates with previously unseen properties. The disadvantage of increasing the number of parameters is the increased computation time needed to sample the parameter space.
The parameters of the MSSM-25 are listed in the left column of Table 1. The Higgs sector is described by the ratio of Higgs vacuum expectation values (VEVs) \(\tan{\beta}\), the CP-odd Higgs mass \(m_{A}\) generated after electroweak symmetry breaking, and the \(\mu\) parameter from the MSSM superpotential. The gaugino sector of the soft SUSY-breaking Lagrangian \(\mathcal{L}_{\text{soft}}\) is parameterised by the three mass terms \(M_{1}\), \(M_{2}\), and \(M_{3}\).
The five Hermitian \(3\times 3\) sfermion mass-squared matrices in \(\mathcal{L}_{\text{soft}}\) are simplified by assuming the absence of flavour changing neutral currents (FCNCs), forcing all non-diagonal entries to zero. Each mass squared matrix therefore only has three real free parameters, and so the sfermion sector has a total of 15 parameters.
For the triple scalar couplings in \(\mathcal{L}_{\text{soft}}\) we again apply the prohibition on FCNCs to eliminate the off-diagonal entries. We also restrict the matrices to be Hermitian (and so the diagonal elements to be real) in order to eliminate any CP-violating phases [24]. We also make the assumption that the elements of the triple scalar coupling matrices are proportional to the corresponding elements of the Yukawa matrices \(\mathbf{y}_{\text{u}}\), \(\mathbf{y}_{\text{d}}\), and \(\mathbf{y}_{\text{e}}\). We can set the first and second entries of \(\mathbf{a}_{\text{u}}\) and \(\mathbf{a}_{\text{d}}\) to zero as the corresponding Yukawa couplings are negligibly small, but we retain one parameter, \(a_{\text{e}1/2}\), for the first and second entries of \(\mathbf{a}_{\text{e}}\), as it is relevant to the calculation of the anomalous magnetic moment of the muon [25]. We finally arrive at the following triple scalar coupling matrices:
\[\mathbf{a}_{\text{u}} =\left(\begin{array}[]{ccc}\phantom{ccc}0&0&0\\ 0&0&0\\ 0&0&a_{\text{u}3}Y_{u33}\end{array}\right)\hskip 14.226378pt\mathbf{a}_{\text{ d}} =\left(\begin{array}[]{ccc}\phantom{ccc}0&0&0\\ 0&0&0\\ 0&0&a_{\text{d}3}Y_{d33}\end{array}\right)\hskip 14.226378pt\mathbf{a}_{\text{ e}} =\left(\begin{array}[]{ccc}a_{\text{e}1/2}Y_{e11}&0&0\\ 0&a_{\text{e}1/2}Y_{e22}&0\\ 0&0&a_{\text{e3}}Y_{e33}\end{array}\right)\] (1)
### Parameter Scans
We performed scans of this parameter space using importance sampling with ADSCAN [26], an adaptive scanning programme based on the VEGAS algorithm [27]. The VEGAS algorithm works by performing an initial random scan of the parameter space and then focusing subsequent scans on areas that produce important values, as defined by an importance function \(G\). Our importance function was
\[G\left(\Omega_{\chi}h^{2}\right)=\text{exp}\left[-\frac{1}{2} \left(\frac{\Omega_{\chi}h^{2}-\Omega_{\text{WMAP-7}}h^{2}}{\sigma_{\Omega h^{ 2}}}\right)^{2}\right]\] (2)
where \(\Omega_{\text{WMAP-7}}h^{2}=0.1120\), the cold dark matter relic density derived from 7-year WMAP measurements, with \(1\sigma\) uncertainty of \(\pm 0.0056\)[28]. For the error in the denominator we chose \(\sigma_{\Omega h^{2}}=0.01\) to allow for some theoretical error above the \(1\sigma\) experimental error of \(\pm 0.0056\), as we did in the corresponding MSSM-7 ADSCAN parameter scans in [17]. The parameter \(\sigma_{\Omega h^{2}}\) determines the width of the Gaussian importance function used in the scanning process, with larger numbers generating a broader scan.
We ran the adaptive scanning programme multiple times with varying parameter limits, which are listed in Table 1. We rejected points in the parameter space that gave negative mass-squared terms, and where the lightest neutralino was not the LSP. We eliminated additional points by applying experimental limits on the accelerator observables produced by each point: the sparticle mass spectrum [29]; standard model and SUSY Higgs masses from LEP and CDF measurements [29]; the \(b\to s\gamma\) branching ratio using results from the Heavy Flavour Averaging Group [30]; the invisible \(Z^{0}\) boson decay width [29]; and the magnetic moment of the muon derived from data from the BaBar and KLOE experiments [31, 32, 33]. These limits are the defaults used by DarkSUSY.
As the primary focus of this paper is the impact of IceCube-DeepCore, our overall strategy was to deliberately lean towards a more inclusive scan, choosing to discard only points that disagreed with the most established and well-understood experimental limits. We made this choice so as to keep scans simple and easy to understand, as this helps to more clearly focus on the specific impacts of IceCube-DeepCore on the MSSM-25. In this way we avoid worrying about erroneously discarding some models due to e.g. the unknown model-dependence of existing LHC SUSY limits when translated into the MSSM-25, or the massive amount of computational resources required to obtain this knowledge.
From the database of points satisfying physicality, the demand for a neutralino LSP, and accelerator bounds, we selected those that gave a relic density within the \(2\sigma\) error bounds of the 7-year WMAP measurement, i.e. in the range \(0.1120-0.0112\leq\Omega_{\chi}h^{2}\leq 0.1120+0.0112\). We found a total of 6.02 million such points within our original database. ¹
[FOOTNOTE:1][ENDFOOTNOTE]
For calculating the values of all observables, we used the DarkSUSY 5.0.6 software package [34, 17], with default options except where otherwise specified. In particular, for the calculation of neutralino-nucleon scattering cross-sections this implies \(\Delta_{s}=-0.12\) for the strange quark content of the nucleus, and \(\Sigma_{\pi N}=45\) MeV, \(\sigma_{0}=35\) MeV for the hadronic matrix elements.
Parameter | Units | Limits
---|---|---
Gaugino mass terms M1, M2, and M3 | GeV | −16000 | 16000
Higgs parameter μ | GeV | −100000 | 100000
CP-odd Higgs mass mA | GeV | −8000 | 8000
Higgs VEV ratio tanβ | | 2 | 65
Sfermion mass parameters m~Q1, m~Q2, | GeV | 100 | 25000
m~Q3, m~¯u1, m~¯u2, m~¯u3, m~¯d1, m~¯d2, m~¯d3,
m~L1, m~L2, m~L3, m~¯e1, m~¯e2 , and m~¯e3
Triple scalar couplings au3, ad3, ae1/2, and ae3 | GeV | −50000 | 50000
| | |
Table 1: MSSM-25 Parameter Space Maximum Exploration Limits
### IceCube Signal Rates
We post-processed the 6.02 million points with suitable relic density using DarkSUSY 5.0.6, as described in [17], to find the IceCube signal rates that would be produced by each model.
We start by modelling the accumulation and annihilation of neutralinos in the Sun. The total population of neutralinos in the Sun \(N(t)\) is described by the equation [35]
\[\frac{\text{d}N(t)}{\text{d}t}=C_{c}-C_{a}N(t)^{2}.\] (3)
where \(C_{c}\) is the constant neutralino capture rate and \(C_{a}\) is defined as
(4)
The \(V_{j}\) terms are effective volumes, which take into account the quasi-thermal distribution of neutralinos in the Sun, and \(\langle\sigma_{a}v\rangle\) is the total self annihilation cross section multiplied by the relative velocity, taken in the limit of zero relative velocity.
The capture rate calculation depends on the halo density and velocity profile of dark matter, the neutralino mass and interaction cross-section, and the properties of the Sun; see e.g. [36] for a detailed description. We assume a standard dark matter halo model, with the Sun moving at \(v_{\odot}=220\) km s\({}^{-1}\) through a halo with local dark matter density \(\rho_{0}=0.3\) GeV cm\({}^{-3}\), and dark matter velocities following a Maxwell-Boltzmann distribution with dispersion \(\bar{v}=270\) km s\({}^{-1}\). We did not include the detailed effects of diffusion and planets upon the capture rate, as the simple free-space approximation [37] included in DarkSUSY turns out to be highly accurate [38]. The annihilation rate of neutralino pairs is given by
\[\Gamma_{a}(t)=\frac{1}{2}C_{a}N^{2}(t).\] (5)
Solving Equation 3 gives us the annihilation rate at a given time \(t\) as
\[\Gamma_{a}(t)=\frac{C_{c}}{2}\tanh^{2}\left(\frac{t}{\tau}\right)\] (6)
where
\[\tau=\frac{1}{\sqrt{C_{c}C_{a}}}\] (7)
is the capture-annihilation equilibrium time scale. Taking \(t\) as the age of the solar system (4.5 billion years) and approximating the structure of the Sun to be constant over its lifetime gives the current neutralino annihilation rate in the Sun.
From the annihilation rate \(\Gamma_{a}(t)\) and branching fractions, DarkSUSY calculates a predicted neutrino flux spectrum at the IceCube detector, using lookup tables previously computed with WimpSim [39]. WimpSim takes neutrino production in the Sun and propagation through the Sun, Earth and interplanetary space into account using a full three-flavour Monte Carlo simulation. Together with the effective area of IceCube-86 and the mean angular error obtained from [20], and some choice of analysis region around the Sun, the calculated flux yields an expected signal rate for any given model.
The other necessary ingredient for our model exclusion calculations is the number of background events. We generated these using a bootstrap Monte-Carlo re-simulation of the expected background rates away from the Sun, as derived in the sensitivity study [20]. The total expected number of background events over the whole sky was 15552 [40], consisting of atmospheric neutrino and muon backgrounds. Given only the total number of background events, we assumed all events to be uniformly distributed in declination. This is a good approximation for atmospheric neutrinos within the narrow declination band of the analysis, and a conservative choice for the down-going atmospheric muon component. This results in an overall robust background expectation. The single realisation of this expected background that we employed for this study contained 1452 simulated events within \(20^{\circ}\) of the solar centre.
### Likelihood Calculation
In this work we perform a model exclusion analysis similar to that of Section 4.3 in [17]. Compared to [17], we employ an expanded parameter space and optimise our angular event cut around the solar position. We also look in far more detail than [17] into the impacts of IceCube-86 on the model parameters and their corresponding derived observables. To quantify the exclusion CL we calculated the \(p\)-value for each model using the method outlined in Section 3.6 of [17], specifically Equation 3.28 and its contributors. The CL at which a model can be excluded is then \(1-p\).
We excluded all events with reconstructed angle outside some bin of width \(\phi\) centred on the Sun. We optimised the cut angle by repeating the \(p\)-value calculation for \(\phi=3^{\circ},6^{\circ},9^{\circ}\), and \(20^{\circ}\), and selecting the lowest \(p\)-value (i.e. highest exclusion CL) for each model. We optimised the cut angle because different models produce signals with different angular distributions. For instance, higher mass neutralinos tend to produce higher-energy neutrinos, which are more accurately reconstructed by IceCube and are therefore more densely clustered around the solar position. Using a small cut angle in this case reduces the number of background events included in the bin more than it reduces the number of signal events. A brief analysis of optimal cut angle is presented in Section 4.
Because we work entirely with simulated data in this paper, we can simply optimise the cut angle by choosing the value that produces the smallest \(p\)-value. Working with real data however, selecting the cut angle that gave the lowest \(p\)-value would constitute an _a posteriori_ analysis choice, which would make the statistical treatment less than rigorous. As we show later, the cut angle giving the lowest \(p\)-value is strongly correlated with the neutralino mass and somewhat dependent on the neutralino character, the quantities that define the expected neutrino energy spectrum (this is to be expected, because the angular distribution of muons is heavily dependent upon the neutrino energies). In an analysis of real data, one should therefore optimise the cut angle in advance using the expected neutrino energy spectrum for a given model, and an ensemble of background-only simulations like the one we employ here.
## 4 Results and Discussion
<figure><img src="content_image/1210.0844/x1.png"><figcaption>Figure 1: Spin-dependent (SD) neutralino-proton cross-section σSD,p (left),and spin-independent (SI) neutralino-proton cross-section σSI,p (right) asfunctions of lightest neutralino mass mχ, for points derived from explorationsof the MSSM-25 parameter space. In the left panel 90% CL spin-dependent WIMP-proton cross-section limits from SIMPLE [41] and COUPP [42] direct detectionexperiment are displayed as magenta and yellow lines respectively. In theright panel the 90% CL spin-independent WIMP-nucleon cross-section limit from225 live days of XENON100 direct detection experiment data is displayed as ayellow line [43]. Above mχ = 1 TeV the XENON100 limit is based on points fromthe XENON100 collaboration. Colour coding indicates predicted IceCube-86 modelexclusion CL. The areas of cyan and blue points show that IceCube-86 has theability to exclude models beyond the reach of current direct detectionexperiments such as SIMPLE, COUPP, and XENON100.</figcaption></figure>
<figure><img src="content_image/1210.0844/x2.png"><figcaption>Figure 2: Spin-dependent neutralino-proton cross-section σSD,p (left) andspin-independent neutralino-proton cross-section σSI,p (right) against muonflux in IceCube from neutralino annihilations, for points derived fromexplorations of the MSSM-25 parameter space. Colour coding indicates predictedIceCube-86 model exclusion CL.</figcaption></figure>
Figure 1 shows the results of the model exclusion analysis we performed on our set of 6.02 million points. Care must be taken in interpreting these plots for two reasons. First, we plot points with exclusion CL of \(3\sigma\) and above (cyan and blue) with larger dots than points with lower exclusion CL, as points with higher exclusion CL are less numerous but also more important to our discussions, and so need to be emphasized. Second, we plot points of higher exclusion CL on top of points with lower exclusion CL, and so the former can obscure the latter. An area of cyan or blue on Figure 1 means that IceCube has exclusion (or detection) capability of \(3\sigma\) (99.7% CL) or better, for a certain range of interaction cross-sections, neutralino masses, _and_ other MSSM-25 parameters. Nevertheless, one can see in the areas of blue and cyan points that IceCube has the ability to exclude models at better than 99% CL well beyond even the 90% CL reach of current direct detection experiments such as SIMPLE [41], COUPP [42] and XENON100 [44, 43].
In Figure 2 we see spin-dependent (SD) and spin-independent (SI) neutralino-proton cross-sections plotted against the predicted muon flux in IceCube-86 from solar neutralino annihilations. For these calculations we used a muon energy threshold of 1 GeV, and a maximum angular separation of 30\({}^{\circ}\) between the solar centre and muon arrival angle.
In general models which can be excluded at higher CL have higher muon fluxes – high muon flux leads to high signal, and high signal models are easier to rule out. This relationship, however, is not exact, as witnessed by the fact that the different regions of exclusion CL (i.e. different coloured points) are not cleanly banded. In Figure 2 one can see points with lower exclusion CL that nonetheless have higher muon fluxes than points with higher exclusion CL. This is a result of IceCube’s effective area increasing with energy, as seen in Figure 2 of [20]. A model can have a high muon flux but if it also has a comparatively low neutralino mass, then it will produce lower energy neutrinos and muons, which leads to a lower IceCube signal rate. Figure 4 throws this interplay into stark relief; models with a lower-mass neutralino will produce neutrinos with lower energy, and so compared to models with a higher neutralino mass a higher flux is necessary to compensate for the reduced effective area at lower energies.
<figure><img src="content_image/1210.0844/x3.png"><figcaption>Figure 3: Lightest neutralino mass mχ against muon flux in IceCube from solarneutralino annihilations, for points derived from explorations of the MSSM-25parameter space. Colour coding indicates predicted IceCube-86 model exclusionCL. The effect of the energy dependence of IceCube’s effective area can beseen in the distribution of high exclusion CL points: lower-mass neutralinoswill yield lower energy neutrinos/muons, and so a higher flux is necessary tocreate the same signal and thus exclusion CL in IceCube.</figcaption></figure>
Figure 4 compares spin-independent to spin-dependent neutralino-proton cross-sections. Note that many models have spin-dependent neutralino-proton cross-sections that are larger than their spin-independent neutralino-proton cross-sections. The majority of models with high exclusion CL have high spin-dependent cross-sections. The long vertical band of high exclusion CL at high spin-dependent cross-section are regions where spin-dependent scattering is the dominant capture mode of neutralinos in the sun. As one moves upwards along this band a ‘turn-off’ is reached at approximately \(\sigma_{SI,p}=10^{-44}\) cm\({}^{2}\); above this point, spin-independent nuclear scattering is a significant source of neutralino capture in the Sun, and even dominates the capture rates in many cases.
<figure><img src="content_image/1210.0844/x5.png"><figcaption>Figure 5: Gaugino fraction ratio Zg/(1−Zg) of the lightest neutralino againstlightest neutralino mass mχ (left) and spin-dependent neutralino-proton cross-section σSD,p (right), for points derived from explorations of the MSSM-25parameter space. Colour coding indicates predicted IceCube-86 model exclusionCL. Note that the largest spin-dependent cross-sections, and therefore thebest IceCube exclusion CL, occur where the lightest neutralino is a mixedgaugino-Higgsino.</figcaption></figure>
As shown in the left panel of Figure 5, IceCube has its strongest exclusion capability in the region where the neutralino is approximately equal parts gaugino and Higgsino. This corresponds to high spin-dependent neutralino-proton cross-section, as seen in the right panel of Figure 5. This agrees with the findings found in Section 4.3 of [17], where a similar analysis was performed using MSSM-7 models.
<figure><img src="content_image/1210.0844/x6.png"><figcaption>Figure 6: Gaugino fraction ratio Zg/(1−Zg) of the lightest neutralino againstlightest neutralino mass mχ, for points derived from explorations of theMSSM-25 parameter space. Colour coding indicates the optimal ϕcut angle foundfor each point.</figcaption></figure>
Of the 6.02 million points we analysed, 5.75 million had an optimal \(\phi_{\text{cut}}=3^{\circ}\), 32 had \(\phi_{\text{cut}}=6^{\circ}\), 156,493 had \(\phi_{\text{cut}}=9^{\circ}\), and 117,713 had \(\phi_{\text{cut}}=20^{\circ}\). Figure 7 shows the gaugino fraction ratio against the lightest neutralino mass, with colour coding indicating the optimal \(\phi_{\text{cut}}\) found for each of our 6.02 million points. An optimal \(\phi_{\text{cut}}\) of \(3^{\circ}\) is dominant at high neutralino mass \(m_{\chi}\), while larger optimal \(\phi_{\text{cut}}\) values become increasingly prevalent at lower neutralino masses. This confirms our earlier statement in Subsection 3.4: higher-mass neutralinos tend to produce higher energy neutrinos, which are more accurately reconstructed by IceCube and therefore more densely clustered around the solar position. In this case, a smaller cut angle reduces the number of background events more than it reduces the number of signal events. Conversely, lower-mass neutralinos will tend to produce lower energy neutrinos and so have a wider angular dispersion of reconstructed events; a wider cut angle is then required to maximise the number of signal events included in the analysis.
Figure 7 shows the self annihilation cross section for the lightest neutralino \(\langle\sigma v\rangle\) against its mass \(m_{\chi}\). Also shown are 95% CL limits on the neutralino self annihilation cross section derived from Fermi-LAT observations of dwarf spheroidal satellite galaxies (dSphs) of the Milky Way, assuming annihilation into either \(b\bar{b}\) or \(\tau^{+}\tau^{-}\) final states [45]. The vast majority of points lie below these exclusion lines, including all areas of blue and cyan, showing that IceCube-86 has exclusion capability at 99% CL or better beyond that of recent Fermi-LAT measurements.
<figure><img src="content_image/1210.0844/x8.png"><figcaption>Figure 8: Lightest squark mass (left) and lightest neutralino mass (right)against gluino mass for points derived from explorations of the MSSM-25parameter space. In the left panel the points to the bottom left of themagenta line are excluded by ATLAS at 95% CL, based on searches for directproduction of coloured sparticles and their decay into jets and missingtransverse energy (MET). The limits are derived by assuming that allsparticles are heavy (m=5 TeV) except for the gluino and first- and second-generation squarks (the masses of which are scanned over), and the lightestneutralino (which is taken as approximately massless) [46]. Note that ourMSSM-25 models do not rely on this assumption. In the right panel, points tothe bottom left of the magenta line are excluded by CMS at 95% CL. This lineis the combination of the best simplified model spectrum limits from the‘razor’ [47] and MT2 [48] analyses on gluino pair production and decay todiquarks + MET. These simplified analyses assume that each gluino in the pairdecays identically directly to q¯q~χ0, i.e. that there are no intermediatestates causing cascade decays of gluinos to neutralinos. This implies thatsquarks, charginos and the other neutralinos are either heavier than thegluino, or almost degenerate in mass with the lightest neutralino. Note thatour MSSM-25 models do not rely on any such assumptions. Colour codingindicates predicted IceCube-86 model exclusion CL.</figcaption></figure>
Figure 8 Left shows the lightest squark mass against the gluino mass for the points found during the MSSM-25 parameter space exploration. Also shown is the 95% CL exclusion limit derived from 4.71 fb\({}^{-1}\) of data at \(\sqrt{s}=7\) TeV from the ATLAS experiment at the Large Hadron Collider (LHC) [46]. This limit assumes a simplified set of SUSY models where all sparticles are given masses of 5 TeV except for the gluino, first- and second-generation squarks, and the lightest neutralino. The first- and second-generation squark masses are scanned over, and the lightest neutralino is taken to be approximately massless. The ATLAS analysis also assumes that the coloured sparticles are produced directly and have direct decays into jets and missing transverse energy. Points below and to the left of the ATLAS limit curve are excluded under these assumptions. The MSSM-25 models we have produced are not based upon any such assumptions. The large number of blue and cyan points above and to the right of this limit curve show that IceCube-86 is sensitive at better than 99% CL to many models beyond the 95% CL exclusion capability of ATLAS.
Figure 8 Right shows the lightest neutralino mass against the gluino mass for our database of MSSM-25 points. Also shown is the 95% CL exclusion limit derived from 4.7 fb\({}^{-1}\) of data taken at \(\sqrt{s}=7\) TeV by the CMS experiment at the LHC, with points below this line considered excluded [47, 48]. The line is the combination of the simplified model spectrum limits from the ‘razor’ [47] and \(M_{\text{T2}}\)[48] analyses on gluino pair production and decay to diquarks \(+\) missing transverse energy (MET). The former is based on limits from 4 \(b\)-tagged jets \(+\) jets \(+\) MET shown in Figure 6 Top Left of [47], and the latter from the \(\tilde{g}\to b\bar{b}\tilde{\chi}^{0}\) channel using the \(M_{\text{T2}}b\) analysis, shown in Figure 6 Top Right of [48]. Such simplified model spectra assume that the gluinos in the pair decay identically, and that their decays to neutralinos proceed directly via \(\tilde{g}\to q\bar{q}\tilde{\chi}^{0}\) only, without any cascade decays via intermediate states. This implies that other sparticles are either heavier than gluinos, uncoloured, or almost degenerate in mass with the LSP (such that their decay to neutralinos is kinematically suppressed or forbidden). In practice this makes such limits applicable to the MSSM-25 only in cases where each squark, neutralino and chargino is either heavier than the gluino, or very close to the LSP mass; our MSSM-25 models do not make this assumption, so the CMS limit must be interpreted with quite some care. Nonetheless, the large number of blue and cyan points above and to the right of this limit curve suggest that IceCube-86 has the capability to exclude many models at better than 99% CL that are beyond the current 95% CL reach of CMS.
<figure><img src="content_image/1210.0844/x9.png"><figcaption>Figure 9: Wino mass parameter M2 against bino mass parameter M1 for pointsderived from explorations of the MSSM-25 parameter space. Colour codingindicates predicted IceCube-86 model exclusion CL. This plot is similar to theplot generated by the M3 against M1 parameter combination.</figcaption></figure>
<figure><img src="content_image/1210.0844/x11.png"><figcaption>Figure 11: CP-odd Higgs mass mA (left) and ratio of Higgs VEVs tanβ (right)against Higgs parameter μ for points derived from explorations of the MSSM-25parameter space. Colour coding indicates predicted IceCube-86 model exclusionCL.</figcaption></figure>
<figure><img src="content_image/1210.0844/x12.png"><figcaption>Figure 12: Second generation L-type squark mass parameters M~Q2 against firstgeneration L-type squark mass parameter M~Q1 (left), and second generationL-type slepton mass parameter M~L2 against first generation L-type sleptonmass parameter M~L1 (right) for points derived from explorations of theMSSM-25 parameter space. Colour coding indicates predicted IceCube-86 modelexclusion CL. These plots are a representative sample of plots generated byother combinations of squark and slepton mass parameters respectively.</figcaption></figure>
Figures 10 to 12 show the exclusion CL across a selection of MSSM-25 parameters. In cases where combinations of parameters from the same category give similar plots, for instance squark mass parameters \(M_{\tilde{Q}_{1}}\) against \(M_{\tilde{Q}_{2}}\) and \(M_{\tilde{Q}_{1}}\) against \(M_{\tilde{Q}_{3}}\), we take one such plot as a representative sample. Several parameters display a strong influence on exclusion CL, as witnessed by concentrations of blue and cyan points. These concentrations can be found where 100 GeV \(\lesssim M_{1}\lesssim\) 200 GeV, (Figure 10), around \(\mu=200\) GeV (Figure 11 Left and Right), and in the region bounded roughly by 100 GeV \(\lesssim m_{A}\lesssim\) 500 GeV and 400 GeV \(\lesssim\mu\lesssim 1200\) GeV (Figure 11 Left). There is a clustering of high exclusion CL points at high values of the triple scalar couplings, as seen in Figure 10. The sfermion mass parameters (Figure 12) and \(\tan{\beta}\) (Figure 11 Right) show no influence over exclusion CL, as can be seen from the roughly even distribution of blue and cyan points over their entire ranges.
Here it is useful to recall our earlier note at the start of Section 4 regarding the interpretation of these plots. Points of high exclusion CL are plotted over those of lower exclusion CL, i.e. below the clusters of blue and cyan points there will be red and green points. Some parameter values produce large numbers of models that can be excluded at high CL by IceCube-86, but one cannot claim that IceCube-86 can exclude _all_ points with those parameter values at high CL. The only correct way to make a statement about which parts of a parameter space are preferred or excluded, and at what CL, is to perform a full statistical global fit, as described in [17].
The distribution of points across the parameter space shows some distinct edges at certain parameter values: at \(M_{2}=4000\) GeV in Figure 10, at \(m_{A}=4000\) GeV in the left panel of Figure 11, and the ‘box within a box’ features seen in Figures 10 and 12. These are artefacts of the scanning method and the parameter limits for various scans. For instance, we performed scans with limits on \(M_{2}\) of \(\pm 4000\) GeV and \(\pm 16000\) GeV, and so there are sharp cut-offs at these values. Similarly, we performed scans with limits on \(m_{A}\) of \(\pm 4000\) GeV and \(\pm 8000\) GeV, leading to similar features in Figure 11 Left. We performed a large number of scans with sfermion mass parameters limited to \(\pm 16000\) GeV, but only a few with limits expanded to \(\pm 25000\) GeV. Thus the region bounded by this first limit was much more heavily sampled, and so has a higher density of points. Similarly the majority of scans we performed had limits on triple scalar couplings \(a_{\text{u}3}\), \(a_{\text{d}3}\), \(a_{\text{e}1/2}\), and \(a_{\text{e}3}\) set to \(\pm 32000\) GeV, but only a few had these limits expanded to \(\pm 50000\) GeV.
<figure><img src="content_image/1210.0844/x13.png"><figcaption>Figure 13: Triple Scalar Couplings ad3 against au3 for points derived fromexplorations of the MSSM-25 parameter space and with CP-even Higgs mass in therange 124.6 - 126.8 GeV as indicated by recent ATLAS and CMS measurements [49,50]. Colour coding indicates predicted IceCube-86 model exclusion CL.Comparison to Figure 10 reveals the impact of the CP-even Higgs mass cut. Thisplot is a representative sample of plots generated by other combinations oftriple scalar couplings with CP-even Higgs mass cuts applied.</figcaption></figure>
Recent results from the ATLAS and CMS experiments indicate the existence of a particle compatible with the Standard Model Higgs boson and supersymmetric CP-even Higgs bosons within mass ranges of 125.2 - 126.8 GeV and 124.6 - 126.2 GeV respectively [49, 50]. To gauge the impact of this measurement upon our results we performed a cut on our database to extract all parameter space points producing a mass for at least one of the CP-even Higgs boson \(H^{0}\) or \(h^{0}\) within the combined ATLAS-CMS range of 124.6 - 126.8 GeV. Of the 6.02 million parameter space points in our database 1.22 million survived this cut. This cut does not alter the overall distribution of points across parameter space, except in the case of the triple scalar coupling \(a_{u3}\). Comparing the pre-cut distribution shown in Figure 10 to the post-cut distribution as shown in Figure 13 we can clearly see a large number of points with \(a_{u3}\) less than 2000 GeV are eliminated.
## 5 Conclusions
Our explorations of the MSSM-25 parameter space have yielded 6.02 million parameter space points that produce neutralino LSPs, satisfy constraints imposed by accelerator data, and produce a neutralino relic density within the \(2\sigma\) bounds of the 7-year WMAP measurement. Calculations of model exclusion CL for these points show that the 86-string configuration of IceCube will be able to rule out or detect at \(3\sigma\) (99.7% CL) or better a significant number of models that are beyond the 90% CL exclusion limits of current direct detection experiments such as SIMPLE, COUPP, and XENON100, and beyond the current 95% CL exclusion limits of Fermi-LAT, ATLAS, and CMS.
P.S. is supported by the Lorne Trottier Chair in Astrophysics, an Institute for Particle Physics Theory Fellowship and a Banting Fellowship, administered by the Natural Science and Engineering Research Council of Canada. C.S. is supported by the Swedish Research Council (VR) through the Oskar Klein Centre and the Department of Physics & Astronomy at the University of Utah. J.E. and K.H. are supported by VR through respective grants 621-2010-3301 and 621-2010-3705, and via the Oskar Klein Centre. J.A.A. and A.M.B. acknowledge support from the Marsden Fund Council from New Zealand Government funding, administered by the Royal Society of New Zealand.
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|
1706.06227 | {
"language": "en",
"source": "Arxiv",
"date_download": "2024-12-03T00:00:00"
} | {
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"content_image/1706.06227/x1.png"
] | # Fluctuation-enhanced electric conductivity in electrolyte solutions
Jean-Philippe Péraud\({}^{1}\), Andy Nonaka\({}^{1}\), John B. Bell\({}^{1}\), Aleksandar Donev\({}^{2}\), and Alejandro L. Garcia\({}^{3}\)
\({}^{1}\) Computational Research Division, Lawrence Berkeley National Laboratory
1 Cyclotron Road, Berkeley, CA 94720
\({}^{2}\) Courant Institute of Mathematical Sciences, New York University
251 Mercer Street, New York, NY 10012
\({}^{3}\) Department of Physics and Astronomy, San Jose State University
1 Washington Square, San Jose, CA 95192
February 26, 2024
###### Abstract
In this letter we analyze the effects of an externally applied electric field on thermal fluctuations for a fluid containing charged species. We show in particular that the fluctuating Poisson-Nernst-Planck equations for charged multispecies diffusion coupled with the fluctuating fluid momentum equation, result in enhanced charge transport. Although this transport is advective in nature, it can macroscopically be represented as electrodiffusion with renormalized electric conductivity. We calculate the renormalized electric conductivity by deriving and integrating the structure factor coefficients of the fluctuating quantities and show that the renormalized electric conductivity and diffusion coefficients are consistent although they originate from different noise terms. In addition, the fluctuating hydrodynamics approach recovers the electrophoretic and relaxation corrections obtained by Debye-Huckel-Onsager theory, and provides a quantitative theory that predicts a non-zero cross-diffusion Maxwell-Stefan coefficient that agrees well with experimental measurements. Finally, we show that strong applied electric fields result in anisotropically enhanced velocity fluctuations and reduced fluctuations of salt concentrations.
fluctuating hydrodynamics, computational fluid dynamics, Navier-Stokes equations, low Mach number methods, multicomponent diffusion, electrohydrodynamics, Nernst-Planck equations pacs: 05.40.-a, 47.11.-j, 47.10.ad, 47.11.St, 47.55.pd, 47.65.-d Introduction –The interaction between ionic species and an externally imposed electric field is at the core of many electrokinetic problems and applications [1] such as electrophoresis. Studying these types of problems usually involves solving the Poisson-Nersnt-Planck equation, which assumes that the solution is ideal with no cross-diffusion between the different ions. In a recent publication [2], we presented a numerical scheme based on fluctuating hydrodynamics for simulating electrokinetic problems at mesoscopic scales where thermal fluctuations are non-negligible. In this approach, the generalized Poisson-Nernst-Planck equation is combined with the fluctuating Landau-Lifshitz Navier-Stokes equations yielding a set of stochastic partial differential equations that can be solved either analytically or numerically. In this letter we use theoretical calculations to show that, in dilute electrolyte solutions, under an applied electric field there exists a coupling phenomenon between the fluctuations of local net charges and fluid velocity. This coupling results in an effective enhancement of the electric conductivity, which we call “fluctuation-induced electroconvection”. We highlight both the similarities and the differences between this result and the enhancement of mass diffusion [3; 4; 5] associated with giant fluctuations [6; 7]. Furthermore, we show that in the presence of an electric current there exists a coupling between the fluctuations of ion density and charge density that results in a reduction of the electric conductivity. We show that the renormalized conductivity is consistent with Onsager’s reciprocal relations provided that there exists a Maxwell-Stefan (MS) cross-diffusion coefficient between the cation and anion, as, indeed, is measured in experiments. Lastly, we show that the coupling produces an anisotropic enhancement of the momentum fluctuations of the fluid that goes as the square of the magnitude of the applied electric field.
Problem description –We model a homogeneous solution composed of a neutral solvent fluid (e.g., water) and two ionic solute species of opposite charge. We assume that a uniform electric field \(\bm{E}_{\text{ext}}\) is externally applied. Both for simplicity and for the sake of focusing on the coupling between charge fluctuations and the applied electric field, we assign the same physical parameters to the anion and the cation (i.e., equal “bare” diffusivity \(D_{0}\) in the solvent, absolute charge per mass \(z\), and molecular mass \(m\)). Generalizations to different ions is straightforward. We denote the mass fraction of the cation and anion by \(w_{+}\) and \(w_{-}\), respectively, which are both \(w_{0}\) in the homogeneous system. The density \(\rho\), the kinematic and dynamic viscosity \(\nu\) and \(\eta\) (\(\eta=\rho\nu\)), and the dielectric permittivity \(\epsilon\) are all assumed constant. We assume that the system remains isothermal at temperature \(T\) and neglect both viscous and ohmic heating.
The theoretical system we consider is infinite in all directions. The fluid is subjected to fluctuations in species mass flux and stress tensor consistent with the fluctuation-dissipation theorem [7]. We use a low Mach approximation [8] and neglect density fluctuations, i.e., \(\bm{\nabla}\cdot\bm{v}=0\), where \(\bm{v}(\mathbf{x},t)\) refers to the velocity vector with components \((v_{x},v_{y},v_{z})\). Assuming the electrolytes are dilute, the equations describing the mass fractions are:
\[\partial_{t}w_{\pm}+\mathbf{v}\cdot\nabla w_{\pm}=D_{0}\nabla^{2}w_{\pm}\mp \frac{D_{0}mz}{k_{B}T}\nabla\cdot\left(\bm{E}w_{\pm}\right)+\nabla\cdot\left( \sqrt{2D_{0}m{\rho}^{-1}w_{\pm}}\bm{Z}_{\pm}\right)\] (1)
where \(k_{B}\) is Boltzmann’s constant. We used Nernst-Einstein relation, which states that the electric mobility is given by \(D_{0}mz/(k_{B}T)\). Symbols \(\bm{Z}_{+}\) and \(\bm{Z}_{-}\) refer to two independant Gaussian white noise vector fields, and, assuming that the dielectric coefficient is constant, the total electric field is the solution to \(\epsilon\nabla\cdot\bm{E}=z\rho(w_{+}-w_{-})\equiv q_{f}\). The velocity field follows the fluctuating Navier-Stokes equation
\[\partial_{t}\bm{v}+\nabla\cdot\left(\bm{v}\bm{v}^{T}\right)=\nu\nabla^{2}\bm{v }+{\rho}^{-1}{\nabla p}+{\rho}^{-1}{q_{f}\bm{E}}+\sqrt{\nu{\rho}^{-1}k_{B}T}\; \nabla\cdot\left(\bm{\mathcal{W}}+\bm{\mathcal{W}^{T}}\right)\] (2)
where \(p\) is the pressure and \(\bm{\mathcal{W}}(\bm{r},t)\) is a white noise tensor field; superscript \(T\) denotes transpose. Physically, the fluctuation-induced electroconvection we are studying is due to mass fraction fluctuations, given by Eq. (1), which result in enhanced velocity fluctuations through the term \(q_{f}\bm{E}\) in Eq. (2). The calculations we carry out next closely resemble previous linearized or “one-loop renormalization” calculations of fluctuation-enhanced diffusivity in non-ionic binary mixtures [3; 4; 5].
Structure factors –We now use linearized fluctuating hydrodynamics to compute the spectrum of the steady-state concentration and velocity fluctuations. We define the fluctuations \(\delta w_{\pm}=w_{\pm}-w_{0}\) but use the sum \({\delta n}=\delta w_{+}+\delta w_{-}\) and difference \({\delta c}=\delta w_{+}-\delta w_{-}\), which are more suited to describe the problem than the individual mass fractions. Linearizing Eq. (1) yields:
\[\begin{split}\partial_{t}{\delta n}&=D_{0}\nabla^{2} {\delta n}-\frac{D_{0}mz}{k_{B}T}\bm{E}_{\text{ext}}\cdot\nabla{\delta c}+ \sqrt{2D_{0}m{\rho}^{-1}w_{0}}\nabla\cdot(\bm{Z}_{+}+\bm{Z}_{-})\\ \partial_{t}{\delta c}&=D_{0}\nabla^{2}{\delta c}- \frac{D_{0}}{\lambda^{2}}{\delta c}-\frac{D_{0}mz}{k_{B}T}\bm{E}_{\text{ext}} \cdot\nabla{\delta n}+\sqrt{2D_{0}m{\rho}^{-1}w_{0}}\nabla\cdot(\bm{Z}_{+}-\bm {Z}_{-})\end{split}\] (3)
where the Debye length \(\lambda\) is defined by \(\lambda^{2}=\epsilon k_{B}T/(2\rho mw_{0}z^{2})\). We also linearize (2) and, as in [7], we apply a double curl operator in order to eliminate the pressure term. We obtain, in Fourier space:
\[\begin{split}-\bm{k}\times\bm{k}\times&\partial_{t} \hat{\bm{v}}=-\nu k^{4}\hat{\bm{v}}-z\bm{k}\times\bm{k}\times\left[\bm{E}_{ \text{ext}}\hat{{\delta c}}\right]\\ &-i\sqrt{\nu{\rho}^{-1}k_{B}T}\bm{k}\times\bm{k}\times\left[\bm{k }\cdot(\hat{\mathcal{W}}+\hat{\mathcal{W}}^{T})\right]\end{split}\] (4)
We take \(\bm{E}_{\text{ext}}=E_{\text{ext}}\bm{e}_{x}\), where \(\bm{e}_{x}\) is a unit vector in the \(x\) direction, and let \(\theta\) denote the angle between \(\bm{E}_{\text{ext}}\) and the wavevector \(\bm{k}\). In that case, the \(x-\)component of (4) becomes:
\[\partial_{t}\hat{v_{x}}=-\nu k^{2}\hat{v_{x}}+zE_{\text{ext}}\sin^{2}(\theta) \hat{{\delta c}}+ik\sin(\theta)\sqrt{2\nu{\rho}^{-1}k_{B}T}\hat{\mathcal{V}}\] (5)
and where \(\hat{\mathcal{V}}(\bm{k},t)\) is a scalar white-noise process.
Taking the Fourier transform of (3) and combining it with (5), we obtain that the vector \(\hat{\mathcal{U}}(\bm{k},t)=(\hat{{\delta n}}(\bm{k},t),\hat{{\delta c}}(\bm{k },t),\hat{v}_{x}(\bm{k},t))\) is described by the Ornstein-Uhlenbeck process \(\partial_{t}\hat{\mathcal{U}}=\bm{M}\hat{\mathcal{U}}+\bm{N}\hat{\mathcal{Z}}\) with
\[\bm{M}=\left(\begin{array}[]{c c c}-D_{0}k^{2}&-ik\cos(\theta)\frac{E_{\text{ ext}}D_{0}mz}{k_{B}T}&0\\ -ik\cos(\theta)\frac{E_{\text{ext}}D_{0}mz}{k_{B}T}&-D_{0}\left(k^{2}+\lambda^ {-2}\right)&0\\ 0&zE_{\text{ext}}\sin^{2}(\theta)&-\nu k^{2}\end{array}\right)\] (6)
where \(\hat{\mathcal{Z}}(\bm{k},t)\) is a vector of three uncorrelated white noise processes. The variance matrix is diagonal,
\[\bm{N}\bm{N}^{*}=k^{2}\rho^{-1}\text{Diag}\left\{4mD_{0}w_{0},4mD_{0}w_{0},2 \nu k_{B}T\sin^{2}(\theta)\right\}.\]
The steady-state spectrum of the fluctuations, i.e., the matrix of static structure factors \(\bm{S}(\bm{k})=\langle\hat{\mathcal{U}}\hat{\mathcal{U}}^{*}\rangle\), where \(\langle\cdot\rangle\) denotes the steady-state average, is given by the solution of the linear system \(\bm{M}\bm{S}+\bm{S}\bm{M}^{*}=-\bm{N}\bm{N}^{*}\)[9].
The complete expression for \(\bm{S}(\bm{k})\) is quite involved. Here we focus on the linear response to the applied field. For sufficiently weak electric fields, there are only two correlations that are altered by the electric field to linear order in \(E_{\text{ext}}\):
\[S_{\hat{{\delta c}},\hat{v}_{x}} =\frac{2mw_{0}}{\rho D_{0}}\frac{zk^{2}\lambda^{4}\sin^{2}(\theta )}{\left[1+(\text{Sc}+1)k^{2}\lambda^{2}\right]\left[1+\lambda^{2}k^{2}\right] }E_{\text{ext}}\] (7)
\[S_{\hat{{\delta c}},\hat{{\delta n}}} =i\frac{2mw_{0}}{\rho k_{B}T}\frac{mzk\lambda^{2}\cos(\theta)}{(1 +k^{2}\lambda^{2})(1+2k^{2}\lambda^{2})}E_{\text{ext}}\] (8)
The auto-correlations \(S_{\hat{{\delta n}},\hat{{\delta n}}}\), \(S_{\hat{{\delta c}},\hat{{\delta c}}}\) and \(S_{\hat{v}_{x},\hat{v}_{x}}\) are, to leading order, quadratic in \(E_{\text{ext}}\).
Enhancement of electric conductivity –The electroconvective coupling results in a net charge flux. From (1), we may write the average charge flux as:
\[\langle\bm{F}_{q}\rangle=\underbrace{\frac{2\rho D_{0}mz^{2}}{k_{B}T}\langle \bm{E}\rangle w_{0}}_{\bm{F}_{q,0}}+\underbrace{\rho z\langle\bm{v}{\delta c} \rangle}_{\bm{F}_{q,\text{adv}}}+\underbrace{\frac{\rho D_{0}mz^{2}}{k_{B}T} \langle\delta\bm{E}{\delta n}\rangle}_{\bm{F}_{q,\text{relx}}},\] (9)
where \(\delta\bm{E}\equiv\bm{E}-\langle\bm{E}\rangle\). Here \(\bm{F}_{q,\text{0}}=C_{0}\bm{E}_{\text{ext}}\) where \(C_{0}\equiv 2m\rho w_{0}z^{2}D_{0}/(k_{B}T)\) is the electric conductivity resulting from the Nernst-Einstein relation. On the other hand the two other terms modify the charge flux because the correlations \(\langle\bm{v}{\delta c}\rangle\) and \(\langle\delta\bm{E}{\delta n}\rangle\) are non-zero as we show below. This additional charge flux is proportional to the electric field in the linearized regime and can be related to an enhanced electric conductivity.
We first examine the advective charge flux \(\bm{F}_{\text{adv}}\), which is intuitively the most direct consequence of the coupling and results from the correlation between the velocity and the charge density fluctuations. It is also the most important quantitatively. We can physically interpret \(\lim_{k\rightarrow\infty}S_{\hat{{\delta c}},\hat{v}_{x}}=0\) as charge fluctuations with small wavelength diffusing away before the Lorentz force can advectively accelerate the charged regions. The component of the advective flux parallel to \(\bm{E}_{\text{ext}}\) can be expressed as an integral of Fourier components over all wavevectors,
\[\bm{F}_{q,\text{adv}}\cdot\bm{e}_{x} =C_{\text{adv}}E_{\text{ext}}=\frac{\rho z}{8\pi^{3}}\int_{k<k_{c }}S_{\hat{{\delta c}},\hat{v}_{x}}d\bm{k}\] (10)
\[=\frac{\rho z}{4\pi^{2}}\int_{k=0}^{k_{c}}\int_{\theta=0}^{\pi}S_ {\hat{{\delta c}},\hat{v}_{x}}k^{2}\sin(\theta)d\theta dk\] (11)
where, as done in prior work on renormalization of diffusion coefficients [5], we define a cutoff \(k_{c}={\pi}/{a}\), where \(a\) is a molecular scale. This is necessary since the integrand is not integrable because it converges towards a non-zero quantity for large wavenumbers. This “ultraviolet divergence” is actually a consequence of a breakdown of the validity of the hydrodynamic equations at molecular scale. Performing the integral in (11) using (7), and using the fact that the Schmidt number in liquids is large, \(\text{Sc}\gg 1\), we obtain the approximation
\[C_{\text{adv}} \approx \frac{2mw_{0}z^{2}}{3\pi D_{0}a\text{Sc}}\left[1-\frac{a}{\pi \lambda}\arctan\left(\frac{\pi\lambda}{a}\right)\right]\] (12)
\[\approx \frac{2m\rho w_{0}z^{2}}{k_{B}T}\left[\frac{k_{B}T}{3\pi a\eta}- \frac{k_{B}T}{6\pi\lambda\eta}\right]\equiv C_{\text{enh}}+C_{\text{ep}},\] (13)
where in (13) we expand to leading order in \(a/\lambda\) since \(\lambda\gg a\) for dilute solutions. We note that \(C_{\text{ep}}\) is known as the electrophoretic term, derived within the Debye-Huckel-Onsager (DHO) theory by rather different means [10; 11]. We note that the term in bracket in (13) can be interpreted as a difference of Stokes-Einstein coefficients for a sphere of radius \(a/2\) and a sphere of radius \(\lambda\). This corresponds to the classical physical picture that the Stokes friction on an ion needs to be adjusted because an ion must drag its ionic atmosphere with it [12] (equivalently, the ion experiences fluid drag relative to ionic cloud [10]).
The flux \(\bm{F}_{\text{relx}}\) is derived here by using the fact that \(\epsilon\langle\bm{E}{\delta n}\rangle=\rho z\langle\nabla\left[\nabla^{-2}{ \delta c}\right]{\delta n}\rangle\) and going to Fourier space:
\[\bm{F}_{q,\text{relx}}=\frac{\rho^{2}D_{0}mz^{3}}{8\pi^{3}\epsilon k_{B}T}\int _{\bm{k}}i\frac{\bm{k}}{k^{2}}S_{\hat{{\delta c}},\hat{{\delta n}}}d\bm{k} \equiv C_{\text{relx}}\bm{E}_{\text{ext}}\] (14)
which, after using Eq. (8), becomes:
\[C_{\text{relx}}=-\frac{D_{0}\rho m^{3}z^{4}w_{0}\sqrt{2}}{12\lambda\pi\epsilon k _{B}^{2}T^{2}(1+\sqrt{2})}.\] (15)
Physically, this is due to the anisotropic counter-ionic cloud surrounding a given ion and known in the DHO theory as the relaxation term [10; 11].
Both \(C_{\text{ep}}\) and \(C_{\text{relx}}\) go as \(w_{0}^{1/2}\) and vanish in the limit of infinitely dilute solutions where \(\lambda\gg a\). Expressions (13) and (15) show that the deterministic linear response that is obtained by ensemble-averaging the equations is not the “bare” response expressed by the conductivity \(C_{0}\), but is instead enhanced, or _renormalized_ by the enhanced conductivity \(C_{\text{enh}}\) due to fluctuation-induced charge transport. Macroscopically, this suggests that the quantity that is experimentally accessible is the renormalized or “dressed” \(C=C_{0}+C_{\text{enh}}+C_{\text{ep}}+C_{\text{relx}}\), and that particular care should be taken when setting the simulation parameters of a fluctuating hydrodynamics solver, so that this enhancement effect is not double-counted [8].
Renormalized transport coefficients –The renormalization of the electric conductivity is connected to the renormalization of the diffusion coefficient that results from giant fluctuations [3; 4; 5]. In [5], a calculation very similar to the one performed above is carried out for the renormalization of the diffusion coefficient in a non-ionic mixture, and it is found that diffusion is renormalized by ¹\(D_{\text{enh}}=k_{B}T/(3\pi a\eta)\). While this result was derived for non-ionic solutions, it can easily be generalized since analyzing the giant fluctuations in the linear regime requires imposing electroneutrality of the steady state. Consequently, the macroscopic gradients of the species charge densities must be equal, which in our case reduces to \(\nabla w_{+}=\nabla w_{-}=\nabla w_{0}\). With this condition, the approach developed in [5] shows that the renormalized mass flux for \({\delta n}\) is the same as that of non-ionic solutions. Qualitatively, the renormalization of the diffusion coefficient is not affected by the presence of charges because the thermal velocity fluctuations advect both the ion and the counterion together, thus maintaining electroneutrality. As with non-ionic mixtures, the renormalized diffusion coefficient is \(D\equiv D_{0}+D_{\text{enh}}\). On the other hand, the electric conductivity \(C=C_{0}+C_{\text{enh}}+C_{\text{ep}}+C_{\text{relx}}\) is renormalized to:
[FOOTNOTE:1][ENDFOOTNOTE]
\[C \approx \frac{2mw_{0}z^{2}\rho}{k_{B}T}\left(D_{0}+\frac{k_{B}T}{3\pi a \eta}-\frac{k_{B}T}{6\pi\lambda\eta}-\frac{D_{0}z^{2}m^{2}}{12(2+\sqrt{2})\pi \lambda\epsilon k_{B}T}\right)\]
\[\approx \frac{2mw_{0}z^{2}\rho}{k_{B}T}\left(D-\frac{A}{\lambda}\right)=C _{\text{PNP}}-C_{0}\frac{A}{D_{0}\lambda}.\]
where \(C_{\text{PNP}}=(2mw_{0}z^{2}\rho/k_{B}T)D\) is the electric conductivity obtained from the Poisson-Nernst-Planck equations with the renormalized diffusivity \(D\) and where \(A\) is a coefficient independent of the concentration of electrolytes.
For infinitely dilute solutions (\(\lambda\rightarrow\infty\)), the renormalizations of the electric conductivity and the diffusivity are consistent with the Poisson-Nernst-Planck equation, i.e., \(C=C_{\text{PNP}}\), which amounts to assuming that Fick’s diffusion matrix is diagonal. This is a manifestation of the overall consistency of fluctuating hydrodynamics, even though the two enhancement phenomena stem from distinct noise terms ²; it is worth noting that in the fully nonlinear diffusion model studied in [13] the only noise term is the stochastic stress and all diffusion arises by advection by thermal velocity fluctuations.
[FOOTNOTE:2][ENDFOOTNOTE]
For finite \(\lambda\), the renormalized diffusion coefficient \(D_{0}+D_{\text{enh}}\) and the renormalized electric conductivity (Renormalized transport coefficients –) do not satisfy the Nernst-Einstein relation so the renormalized Poisson-Nernst-Planck equation must be corrected to leading order in \(a/\lambda\) to be consistent with Onsager’s reciprocal relations. Specifically, the renormalized Fick’s diffusion matrix must include off-diagonal coefficients; to satisfy both renormalized coefficients, the mass fluxes \(\bm{F}_{+}\) and \(\bm{F}_{-}\) of the two ionic species must be expressed as:
\[\left(\begin{array}[]{c}\bm{F}_{+}\\ \bm{F}_{-}\end{array}\right)=-\rho\underbrace{\left(\begin{array}[]{c c}D- \frac{A}{2\lambda}&\frac{A}{2\lambda}\\ \frac{A}{2\lambda}&D-\frac{A}{2\lambda}\end{array}\right)}_{\bm{D}}\cdot\nabla \left(\begin{array}[]{c}w_{+}+\frac{mz}{k_{B}T}w_{+}\phi\\ w_{-}-\frac{mz}{k_{B}T}w_{-}\phi\end{array}\right)\] (17)
where \(\phi\) is the electric potential (\(\bm{E}=-\nabla\phi\)).
In order to give a more physical interpretation to the cross-diffusion coefficient, we link the renormalized Fickian diffusion matrix to a renormalized Maxwell-Stefan (MS) diffusion matrix [14]. The MS diffusion coefficients can be physically interpreted as inverse friction coefficients between _pairs_ of distinct species. For a very dilute solution, it has been assumed when writing Eqs. (1) that the (bare) MS cross-diffusion coefficient between the two ionic species, \(\mathfrak{D}^{(+,-)}_{0}\), is 0, and that the (bare) cross-diffusion coefficient between the solvent and either ion is identical, i.e. \(\mathfrak{D}^{(s,+)}_{0}=\mathfrak{D}^{(s,-)}_{0}=D_{0}\). However, this is inconsistent with the renormalized Fickian diffusion matrix \(\bm{D}\) with nonzero off-diagonal coefficients. Introducing the renormalized MS diffusion coefficients \(\mathfrak{D}^{(+,-)}\) and \(\mathfrak{D}^{(s,+)}=\mathfrak{D}^{(s,-)}\) and writing the friction matrix as the inverse of the Fickian diffusion matrix, we obtain, to first order in \(w_{0}\), \(\mathfrak{D}^{(s,+)}=D\), and the cross-diffusion coefficient:
\[\mathfrak{D}^{(+,-)}\approx 12\pi D^{2}\left[\frac{k_{B}T}{\eta}+\frac{D_{0}z^ {2}m^{2}}{2(2+\sqrt{2})\epsilon k_{B}T}\right]^{-1}\frac{M}{m}\lambda w_{0}\] (18)
where \(M\) denotes the molecular mass of the solvent.
Using the complete formulas for the electrophoretic (\(C_{\text{ep}}\)) and relaxation (\(C_{\text{relx}}\)) terms from DHO theory [10], one can easily generalize Eq. (18) to unequal ions. With parameters of water (molecular mass \(M=3\times 10^{-26}\) kg, \(\eta=1.05\times 10^{-3}\) kg/ms), we find \(\mathfrak{D}^{(+,-)}\approx 0.9\times 10^{-10}\sqrt{c}\) for salt solutions (\(D_{\text{Na}}\approx 1.3\times 10^{-9}\) m\({}^{2}\)/s and \(D_{\text{Cl}}\approx 2.0\times 10^{-9}\) m\({}^{2}\)/s) where \(c\) is in mol/L and where the result is in m\({}^{2}\)/s, in very good agreement (within 10% difference) with published experimental measurements [12; 15; 16].
Enhancement of velocity fluctuations –The fluctuation-induced electroconvection derived in this paper is associated with a corollary phenomenon, namely, the enhancement of velocity fluctuations in the direction of the electric field, as shown by the expression of \(S_{\hat{v}_{x},\hat{v}_{x}}\), written below in the case where \(\bm{k}\) and \(\bm{E}_{\text{ext}}\) are orthogonal (\(\theta=\pi/2\)):
\[S_{\hat{v}_{x},\hat{v}_{x}}^{\perp}=\frac{k_{B}T}{\rho}+\frac{2mw_{0}}{\rho\nu D _{0}}\frac{z^{2}E_{\text{ext}}^{2}\lambda^{4}}{\left[1+(\text{Sc}+1)k^{2} \lambda^{2}\right]\left[1+\lambda^{2}k^{2}\right]}\] (19)
In Figure 1 we show a comparison between the theoretical structure factor of velocity fluctuations (when the wavevector and the electric field are orthogonal) and the same quantity obtained with the code developed in [2]. The main finding here is that, provided the field is strong enough, the amplitude of the low wavenumber fluctuations is noticeably enhanced. As in the phenomenon of giant fluctuations [6; 7], this results in large scale patterns, with the key difference that these patterns are found in the \(x-\)component (i.e. colinear to the applied electric field) of the velocity instead of the mass fractions.
<figure><img src="content_image/1706.06227/x1.png"><figcaption>Figure 1: Structure factor of velocity fluctuations parallel to the appliedfield versus wavenumber. The computational system used to verify thetheoretical calculations is a cubic domain of dimension L, with periodicboundary conditions in all directions, with: T=300 K, ϵ=6.91×10−19s2⋅C2⋅cm−3⋅g−1, ν=1.05×10−2 cm2⋅s−1, D0=10−5 cm2⋅s−1, z=103 C⋅g−1, w0=10−5,m=3×10−23 g, ρ=1.0 g⋅ cm−3, E=106 V ⋅ cm−1=1013g ⋅ cm⋅s−2⋅C−1. Since a singlecomputation can not cover the wide range of wavenumbers shown here, thecomputational results combine three different systems, of sizes 20 microns, 2microns and 200 nm. The number of computational cells is indicated in thelegend. The theoretical calculation is corrected to account for the discreteLaplacian effect LowMachElectrolytes .</figcaption></figure>
For small wavenumbers (large length scales), the structure factor for wavevectors orthogonal to \(\bm{E}_{\text{ext}}\) (\(\theta=\pi/2\)) converges toward
\[S_{\hat{v}_{x},\hat{v}_{x}}^{\perp}(k\to 0)=\frac{k_{B}T}{\rho}\left[1 +\frac{\epsilon E_{\text{ext}}^{2}\lambda^{2}}{\rho\nu D_{0}}\right].\] (20)
The effect of the electric field on the velocity fluctuations is significant when \(E_{\text{ext}}\geq\lambda^{-1}\sqrt{\epsilon^{-1}\rho\nu D_{0}}\). Using the Maxwell approximation \(\nu\approx v_{\text{th}}\lambda_{\text{th}}\) where \(v_{\text{th}}\) and \(\lambda_{\text{th}}\) refer respectively to the molecular speed and length (e.g., sound speed and mean free path), we can write it as
\[\frac{\epsilon E_{\text{ext}}^{2}/2}{\rho v_{\text{th}}^{2}/2}\gtrsim\frac{1}{ \text{Sc}}\frac{\lambda_{\text{th}}^{2}}{\lambda^{2}},\] (21)
where the left-hand side is the ratio of the electric and the thermal energy densities. This condition may seem constraining, but in dilute liquid solutions the right hand side is much smaller than 1. In fact, with the parameters chosen for Figure 1, the condition on the electric field is \(E_{\text{ext}}\gtrsim\) 6 kV/cm which is the higher end of the electric fields applied in electrophoresis experiments [17].
We note that, on the other hand, the fluctuations of \({\delta n}\) are reduced anisotropically by the electric field,
\[S_{\hat{{\delta n}},\hat{{\delta n}}}(k\to 0) =2mw_{0}\rho^{-1}\left[1+\left(\frac{mzE_{\text{ext}}\cos(\theta) \lambda}{k_{B}T}\right)^{2}\right]^{-1}\] (22)
\[=2mw_{0}\rho^{-1}\left[1+\frac{m\epsilon E_{\text{ext}}^{2}\cos^{ 2}(\theta)}{\rho n_{0}k_{B}T}\right]^{-1},\] (23)
where \(n_{0}\) is the total ion mass fraction. Note that the second term in the brackets is the ratio of the typical magnitude of the Maxwell stress tensor and the osmotic pressure of the ions. The reduction of the ion number density fluctuations is significant when \(mz\lambda E_{\text{ext}}\gtrsim k_{B}T\), or, equivalently, when the energy lost (or gained) by an ion crossing a distance \(\lambda\) in the direction of the field is larger than the thermal energy \(k_{B}T\). Using parameters for sodium at concentration \(w_{0}=10^{-5}\) gives \(E_{\text{ext}}\gtrsim\) 20 kV/cm.
Concluding remarks –In summary, using a fluctuating hydrodynamics formulation, we show that there exists a coupling between the fluctuations in charge density and fluid velocity that is proportional to the applied electric field. This coupling leads to an effective enhancement, or renormalization, of the measured electric conductivity of an ionic mixture. This enhancement is comparable to the enhancement of the diffusion coefficients that results from giant fluctuations, in that the enhancement coefficients match in the limit of infinite dilution. For finite dilution, the renormalization of mass diffusivity and electric conductivity are different. This shows that, although we started from a diagonal Fickian diffusivity matrix, renormalizing the fluctuating Poisson-Nernst-Planck equations yields an off-diagonal Fickian diffusion term, itself linked with a non-zero renormalized cross-diffusion Maxwell-Stefan coefficient between the two counterions, in good agreement with experimental coefficients reported in the literature. In fact, in our prior work [2] we demonstrated that results from Debye-Huckel theory, including the non-analytic Debye-Huckel correction to the internal energy, can be obtained from a fluctuating hydrodynamics theory of dilute electrolyte solutions. The present work further demonstrates that fluctuating hydrodynamics provides a generalizable and systematic approach to derive corrective transport coefficients such as the electrophoretic and the relaxation term. Finally, for large electric fields, the applied field can significantly amplify the velocity fluctuations and suppress fluctuations of salt concentration. We expect this phenomenon to be observable experimentally and by molecular dynamics simulations.
The theory developed here can readily be extended in a number of important directions. Firstly, the assumption of dynamically-identical ions can be removed so that a more direct comparison with experimental measurements for different salts can be performed, including polyvalent salts. It is also important to consider solutions with one ion and two counterions, such as for example solutions of NaCl and KCl in water. Such extensions would reveal whether the surprising experimental observation of negative Maxwell-Stefan diffusion coefficients[18; 19] between co-ions [20] can be explained by fluctuating hydrodynamics and renormalization. Here we only considered strong electrolytes but the generalization to weak electrolytes is possible by using FHD for reactive fluids [21]. Lastly, we started here with fluctuating hydrodynamics equations based on the PNP equations, i.e., we assumed an ideal solution with no cross-diffusion, so our starting equations had only one mobility coefficient per ion, instead of one Maxwell-Stefan coefficient per pair of ions. The renormalized equations, on the other hand, have cross-diffusion and also a non-ideal Debye-Huckel contribution to the free energy density. This suggests that a more proper theory should start from the more complete equations, allowing for a nonzero _bare_ MS cross-coefficient \(\mathfrak{D}^{(+,-)}_{0}\). As explained in [22] for non-electrolytes, bare diffusion coefficients can be given a microscopic interpretation in terms of Green-Kubo expressions and can therefore, in principle, be measured in molecular dynamics simulations, and the renormalization due to thermal fluctuations computed numerically using a numerical fluctuating hydrodynamics solver [2]. Carrying out such an ambitious program for electrolyte solutions is a worthy challenge for the future.
## Acknowledgements
We thank Burkhard Duenweg and Mike Cates for illuminating discussions about linear response theory and renormalization. This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program under Award Number DE-SC0008271 and contract DE-AC02-05CH11231. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
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|
1708.05286 | {
"language": "en",
"source": "Arxiv",
"date_download": "2024-12-03T00:00:00"
} | {
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} | [] | # Simple Open Stance Classification for Rumour Analysis
Ahmet Akera,b
Leon Derczynskia
Kalina Bontchevaa
Department of Computer Science, University of Sheffielda
Department of Information Engineering, University of Duisburg-Essenb
a.aker@is.inf.uni-due.de, leon.derczynski@sheffield.ac.uk
K.Bontcheva@sheffield.ac.uk
###### Abstract
Stance classification determines the attitude, or stance, in a (typically short) text. The task has powerful applications, such as the detection of fake news or the automatic extraction of attitudes toward entities or events in the media. This paper describes a surprisingly simple and efficient classification approach to open stance classification in Twitter, for rumour and veracity classification. The approach profits from a novel set of automatically identifiable problem-specific features, which significantly boost classifier accuracy and achieve above state-of-the-art results on recent benchmark datasets. This calls into question the value of using complex sophisticated models for stance classification without first doing informed feature extraction.
## 1 Introduction
Stance detection is the problem of classifying the attitude taken by an author in a short piece of text. Typical stances include showing support, denying, commenting on or querying an existing claim or fact. Knowing the stance that authors hold in response to claims, e.g. in online commentary, gives useful insights. It can reveal rumours and fake news claims as the discourse around them is monitored Procter et al. (2013). Stance reflects how certain authors are of a claim’s veracity Biber (2006), which enables the effective detection of potential false rumours Lukasik et al. (2015). Stance also reveals how online populations react to business and political news.
This paper addresses the general-purpose, or _open_ stance classification task. This is distinct from _target-specific_ stance classification, as in augenstein-EtAl:2016:EMNLP2016 and mohammad2016semeval, which focus on stances towards known, pre-determined targets. In the latter task, the target has already been extracted, from e.g. conversational cues. Target-specific stance classification is suited to situations where the target is already known, such as analyses of a specific product or political actor. In contrast, the open stance classification task is appropriate in emerging news or novel contexts, such as working with online media or streaming news analysis.
Open stance classification is often applied in rumour resolution. Since attitudes in discourse around a claim are indicative not only of the controversiality of the claim, but also can act as a proxy for its veracity, it is reasonable to consider the application of open stance detection for rumour analysis. Indeed, many approaches to rumour and fake news analysis rely on this signal Derczynski et al. (2017)¹. In veracity analysis, the claim is already known, and the goal is to gather observations and analyse crowd reaction in order to resolve the claim. Instead of being concerned with specific targets, we apply non-targeted – _open_ – stance analysis to messages replying to a claim, where the target may vary but the high-level rumour topic rumour remains the same.
[FOOTNOTE:1][ENDFOOTNOTE]
Our simple approach to open stance classification implements common features used in stance classification reported by related work (e.g. bag-of-words, named entities, user activity information, URL presence). We extend this with problem-specific features (which we refer to as the AF features) designed to capture how users react to tweets and express confidence in them. Our results show adding these features gives significantly higher performance on benchmark datasets, compared to recent state-of-the-art systems. We make our classifier freely available on the PHEME software repository.²
[FOOTNOTE:2][ENDFOOTNOTE]
The outline of the paper is as follows. First we describe related work (Section 2) and then introduce our method along with the classification techniques used and features extracted (Section 3). Next, Section 4 describes our experimental set-ups, followed by results in Section 5. We report on feature analysis in Section 6, prior to concluding the paper (Section 7).
## 2 Related Work
The first study that tackles automatic stance classification is that of Qazvinian et al. Qazvinian et al. (2011). With a dataset containing 10K tweets and using a Bayesian classifier and three types of features categorised as “content”, “network” and “Twitter specific memes”, the authors achieved an accuracy of 93.5%. Similar to them, Hamidian and Diab Hamidian and Diab (2015) perform rumour stance classification by applying supervised machine learning using the dataset created by Qazvinian et al. Qazvinian et al. (2011). However, instead of Bayesian classifiers, the authors use J48 decision tree implemented within the Weka platform Hall et al. (2009). The features from Qazvinian et al. Qazvinian et al. (2011) are adopted and extended with time-related information and the hastags themselves, instead of the content of the hashtag as used by Qazvinian et al. Qazvinian et al. (2011). In addition to the feature categories introduced above, Hamidian and Diab Hamidian and Diab (2015) introduce another feature category, namely “pragramatic”. The pragmatic features include named entity, event, sentiment and emoticons. The evaluation of the performance is casted as either 1-step problem containing a 6 class classification task (not rumour, 4 classes of stance and not determined by the annotator) or 2-step problem containing first a 3 class classification task (non-rumour, rumour, and not determined), followed by a 4 class classification task (stance classification). The two step approach achieves better performance with 82.9% F-1 measure, compared to 74% with the 1-step approach. The authors also report that the best performing features were the content based features and the worst performing ones – the network and Twitter specific features. In their most recent paper, Hamidian and Diab Hamidian and Diab (2016) introduce the Tweet Latent Vector (TLV) approach that is obtained by applying the Semantic Textual Similarity model proposed by Guo and Diab Guo and Diab (2012). The authors compare the TLV approach to their own earlier system, as well as to the original features of Qazvinian et al. Qazvinian et al. (2011) and show that the TLV approach outperforms both baselines.
Liu et al. Liu et al. (2015) use a rule-based method and show that it outperforms the approach reported by Qazvinian et al. Qazvinian et al. (2011). Zeng et al. Zeng et al. (2016) enrich the feature sets investigated by earlier studies by features derived from the Linguistic Inquiry and Word Count (LIWC) dictionaries Tausczik and Pennebaker (2010). Lukasik et al. Lukasik et al. (2016) investigate Gaussian Processes as rumour stance classifier. For the first time the authors also use Brown Clusters to extract the features for each tweet. Unlike researchers above, Lukasik et al. evalute on the rumour data released by Zubiaga et al. Zubiaga et al. (35), where they report an accuracy of 67.7%. This result is achieved when the classifier is trained on \(n-1\) rumours and tested on the n\({}^{th}\) rumour. However, the authors achieve substantially better results when a small proportion of the in-domain data (data from the n\({}^{th}\) rumour) is also included in the training (68.6% accuracy). Performance scores differ substantially from those in the studies described above, given that Lukasik et al. Lukasik et al. (2016) tackled classification of stance in new rumours that differ from those in the training set.
Subsequent work has also tackled stance classification for new, unseen rumours. Zubiaga et al. Zubiaga et al. (34) moved away from the classification of tweets in isolation, focusing instead on Twitter ’conversations’ Tolmie et al. (2015) initiated by rumours, as part of the Pheme project Derczynski and Bontcheva (2014). They looked at tree-structured conversations initiated by a rumour and followed by tweets responding to it by supporting, denying, querying or commenting on the rumour.
Rumour stance classification for tree structured conversations has also been studied in the RumourEval shared task at SemEval 2017 Derczynski et al. (2017). Subtask A there consisted of stance classification of individual tweets discussing a rumour within a conversational thread as one of _support_, _deny_, _query_, or _comment_. Eight participanting teams submitted results to this task. Most of the systems viewed this task as a 4-way single tweet classification task, with the exception of the best performing system by Kochkina et al. Kochkina et al. (2017), as well as the systems by Wang et al. Wang et al. (2017) and Singh et al. Singh et al. (2017). The winning system addressed the task as a sequential classification problem, where the stance of each tweet takes into consideration the features and labels of the preceding tweets. The system by Singh et al. Singh et al. (2017) takes as input pairs of source and reply tweets, whereas Wang et al. Wang et al. (2017) addressed class imbalance by decomposing the problem into a two step classification task – first distinguishing between comments and non-comments and then classifying non-comment tweets as one of support, deny or query. Half of the systems employed ensemble classifiers, where classification was obtained through majority voting Wang et al. (2017); García Lozano et al. (2017); Bahuleyan and Vechtomova (2017); Srivastava et al. (2017). In some cases the ensembles were hybrid, consisting both of machine learning classifiers and manually created rules, with differential weighting of classifiers for different class labels Wang et al. (2017); García Lozano et al. (2017); Srivastava et al. (2017). Three systems used deep learning, with Kochkina et al. Kochkina et al. (2017) employing LSTMs for sequential classification, Chen et al. Chen et al. (2017) using convolutional neural networks (CNN) for obtaining the representation of each tweet, assigned a probability for a class by a softmax classifier and García Lozano et al. García Lozano et al. (2017) using CNN as one of the classifiers in their hybrid conglomeration. The remaining two systems by Enayet and El-Beltagy Enayet and El-Beltagy (2017) and Singh et al. Singh et al. (2017) used support vector machines with a linear and polynomial kernel respectively.
## 3 Method
### Data
In our experiments we used two different data sets: RumourEval dataset Derczynski et al. (2017) and the PHEME dataset Zubiaga et al. (35). In the PHEME dataset the authors identify rumours associated with events, collect conversations sparked by those rumours in the form of replies and annotate each of the tweets in the conversations for stance. These data consist of tweets from 5 different events: Ottawa shooting, Ferguson riots, Germanwings crash, Charlie Hebdo and Sydney siege. Each dataset has a different number of rumours where each rumour contains tweets marked with stance annotations: “supporting”, “questioning”, “denying” or “commenting”. A summary of the data is given in Table 1.
Dataset | Rumours | S | D | Q | C
---|---|---|---|---|---
Ottawa shooting | 58 | 161 | 76 | 64 | 481
Ferguson riots | 46 | 192 | 83 | 94 | 685
Charlie Hebdo | 74 | 236 | 56 | 51 | 710
Sydney siege | 71 | 89 | 4 | 99 | 713
Table 1: PHEME Data: Counts of tweets with supporting (S), denying (D),
questioning (Q) and commenting (C) labels in each event collection.
The RumourEval dataset is derived from the PHEME dataset, however, for the purpose of the RumourEval shared Task A the data has a given split into training and testing. This provides an established basis for evaluation. The training data draws from stories in 2014–2016, from the earlier PHEME dataset. The evaluation split covers two new stories, both from 2016: first, the disappearance of Marina Joyce, a British Youtube personality, who was rumoured to have been abducted in July 2016. There was significant speculation in social media, and the case was brought to a concrete resolution as the police investigated and posted an open public response. The second story was that Hillary Clinton had pneumonia during mid-September 2016. The prevalence and spread of this story could be tracked easily, and it emerged in a short space of time, though among background noise of speculative, unsubstantiated claims about her and her opponent’s health. More details about this dataset can be obtained from the SemEval website³.
[FOOTNOTE:3][ENDFOOTNOTE]
In keeping with prior work Zeng et al. (2016); Lukasik et al. (2016); Zubiaga et al. (34), our experiments assume that incoming tweets already belong to a particular rumour, e.g. a user is tracking tweets related to a certain rumour. For each new tweet, features are extracted into a feature vector, which is then used to assign each tweet its stance towards the rumour.
### Classifiers
We experiment with three different, well known machine learning classifiers: (1) a decision tree, J48; (2) Random Forests Breiman (2001); and (3) an Instance Based classifier (K-NN). For the Random Forest we use 50 trees (_-I 50_). Pruning is enabled for J48. Finally we run the Instance Based classifier with _-I -K 10_ settings.
### Features
Prior work on stance classification investigated various features which can be categorized into linguistic, message-based, and topic-based categories Mendoza et al. (2010); Qazvinian et al. (2011); Hamidian and Diab (2015); Liu et al. (2015); Zeng et al. (2016); Lukasik et al. (2016); Zubiaga et al. (34). The following list summarizes the features adopted in this work.
* **BOW (Bag of words):** For this feature we first create a dictionary from all the tweets in the out-of-domain dataset. Next each tweet is assigned the words in the dictionary as features. For words occurring in the tweet the feature values are set to the number of times they occur in the tweet. For all other words “0” is used.
* **Brown Cluster:** Brown clustering is a hard hierarchical clustering method and we use it to cluster words in hierarchies. It clusters words based on maximising the probability of the words under the bigram language model, where words are generated based on their clusters Liang (2005). In previous work, it has been shown that Brown clusters yield better performance than directly using the BOW features Lukasik et al. (2015). Brown clusters are obtained from a bigger tweet corpus that entails assignments of words to brown cluster ids. We used 1000 clusters, i.e. there are 1000 cluster ids. All 1000 ids are used as features however only, ids that cover words in the tweet are assigned a feature value “1”. All other cluster id feature values are set to “0”.
* **POS tag:** The BOW feature captures the actual words and is domain dependent. To create a feature that is not domain dependent, we added Part of Speech (POS) tags as additional feature. Similar to the BOW feature we created a dictionary of POS tags from the entire corpus (excluding the health data) and used this dictionary to label each tweet with it -- binary, i.e. whether a POS tag is present.⁴ However, instead of using just single POS tags, we created sequences containing bi-gram, tri-gram and 4-gram POS tags. Feature values are the frequencies of POS tag sequences occurring in the tweet. [FOOTNOTE:4][ENDFOOTNOTE]
* **Sentiment:** This is another domain-independent feature. Sentiment analysis reveals the sentimental polarity of the tweet such as whether it is positive or negative. We used the Stanford sentimentSocher et al. (2013) tool to create this feature. The tool returns a range from 0 to 4 with 0 indicating “very negative” and 4 “very positive”. First, we used this as a categorical feature but turning it to a numeric feature gave us better performance. Thus each tweet is assigned a sentiment feature whose value varies from 0 to 4.
* **NE:** Named entity (NE) is also domain independent. We check for each tweet whether it contains _Person, Organization, Date, Location_ and _Money_ tags and for each tag present, “1” is added, or a “0” otherwise.
* **Reply:** This is a binary feature, which is assigned “1” if the tweet is a reply to a previous one, or a “0” otherwise. Tweet reply information is extracted from the tweet metadata. Again this feature is domain independent.
* **Emoticon:** We created a dictionary of emoticons using Wikipedia⁵. In Wikipedia those emoticons are grouped by categories, which we use as a feature. If any emoticon from a category occurs in the tweet, we assign for that category feature the value “1” – otherwise “0”. Again similar to the previous features this feature is domain independent. [FOOTNOTE:5][ENDFOOTNOTE]
* **URL:** This is again domain independent. We assign the tweet “1” if it contains any URL, or “0” otherwise.
* **Mood:** Mood detection analyses textual content using different view points or angles. Mood detection is performed using the tool from Celli et al. (2016), which analyses tweets from five different angles: amused, disappointed, indignant, satisfied and worried. For each of this angles it returns a value from -1 to +1. We use the different angles as the mood features and the returned values as the feature value.
* **Originality score**: This is the count of tweets the user has produced, i.e. “statuses count” in the Twitter API.
* **isUserVerified(0-1)**: Whether the user is verified or not.
* **NumberOfFollowers**: Number of followers the user has.
* **Role score**: This is the ratio between the number of followers and followees (i.e. NumberOfFollowers/NumberOfFollowees).
* **Engagement score**: the number of tweets divided by the number of days the user has been active (number of days since the user account creation till today).
* **Favourites score**: The “favourites count” divided by the number of days the user has been active.
* **HasGeoEnabled(0-1)**: User has enabled geo-location or not.
* **HasDescription(0-1)**: User has description or not.
* **LenghtOfDescription in words**: The number of words in the user description.
* **averageNegation**: We determine using the Stanford parser Chen and Manning (2014) the dependency parse tree of the tweet, count the number of negation relation (“neg”) that appears between two terms and divide this by the number of total relations.
* **hasNegation(0-1)**: Tweet has negation relationship or not.
* **hasSlangOrCurseWord(0-1)**: A dictionary of key words⁶ is used to determine the presence of slang or curse words in the tweet. [FOOTNOTE:6][ENDFOOTNOTE]
* **hasGoogleBadWord(0-1)**: Same as above but the dictionary of slang words is obtained from Google.⁷ [FOOTNOTE:7][ENDFOOTNOTE]
* **hasAcronyms(0-1)**: The tweet is checked for presence of acronyms using a acronym dictionary.⁸ [FOOTNOTE:8][ENDFOOTNOTE]
* **averageWordLength**: Average length of words (sum of word character counts divided by number of words in each tweet).
* **hasQuestionMark(0-1)**: The tweet has “?” or not.
* **hasExclamationMark(0-1)**: The tweet has “!” or not.
* **hasDotDotDot(0-1)**: Whether the tweet has “…” or not.
* **numberOfQuestionMark**: Count of “?” in the tweet.
* **NumberOfExclamationMark**: Count of “!” in the tweet.
* **numberOfDotDotDot**: Count of “…” in the tweet.
* **Binary regular expressions applied on each tweet**: .*(rumor?—debunk?).*, .*is (that—this—it) true.*, etc. In total there are 10 features covering regular expressions.
This work extends the features above, with new additional problem-specific features (**AF features**). AF features score the level of confidence in a tweet. We compute scores for surprise (_surpriseScore (SS)_), doubt (_doubtScore (DS)_), certainty (_noDoubtScore (NDS)_) and support (_supportScore (SPS)_) towards rumourous tweets. For each of these features a list of typical words is collected. We use this list to compute a cumulative vector using word2Vec Mikolov et al. (2013). For each word in the list, we obtain its word2Vec representation, add them together and finally divide the resulting vector by the number of words to obtain the cumulative vector. Similarly a cumulative vector is computed for the words in the tweet excluding acronyms, named entities and URLs. We use cosine to compute the angle between those two cumulative vectors to determine each of the scores. Our word embeddings comprise the vectors published by baroni2014don. The full list of tweet confidence AF features is as follows:
* **surpriseScore (SS)**: cosine between embedding of tweet content and the list of surprise words, e.g. “surprise”, “wonder”, etc.
* **doubtScore (DS)**: cosine between embedding of tweet content and the list of doubt words, e.g. “doubt”, “uncertain”, etc.
* **noDoubtScore (NDS)**: cosine between embedding of tweet content and the list of certainty words, e.g.“surely”, “sure”, etc.
* **supportScore (SPS)**: cosine between embedding tweet content and the list of support words, e.g. “support”, “confirm”, etc. Furthermore, the following two AF features are included:
* **initialTweetSim (ITS)** captures tweets that tend to support rumours. Every rumour is initiated by a tweet. We compute the cosine similarity based on word2Vec of the tweet being classified to the first tweet in the rumour thread. If the tweet is just a simple re-retweet of the initial tweet, this is taken as an evidence that the tweet is supportive of that tweet.
* **isQuestion (IQ)** indicates whether a tweet starts with an interrogative. The feature is binary and aims to capture questioning tweets.
## 4 Experimental Setup
Classifier | All features | w.o. AF
---|---|---
Decision tree | 74.16 | 72.25
Random Forest | 79.02 | 76.54
IBk | 75.59 | 73.02
Baseline-Turing | 78.4 | –
Table 2: Accuracy scores of different stance classifiers for the RumourEval
dataset. The baseline is the best performing system in the SemEval evaluation
Turing.
classifier | Ottawa shooting | Ferguson riots | Charlie Hebdo | Sydney siege | macro mean
---|---|---|---|---|---
IBk | 70.31* | 72.35 | 78.33 (ref) | 75.44 | 74.10
Decision tree | 76.28 (ref) | 75.20 (ref) | 78.21 | 80.01 (ref) | 77.42
Random Forest | 69.39* | 69.16 | 74.57 | 74.49 | 71.90
Baseline - GP | 62.28 | 64.31 | 70.66 | 65.04 | 65.57
Baseline - HP | 67.77 | 68.44 | 72.93 | 68.59 | 69.43
Table 3: Accuracy scores for different stance classifiers on the PHEME
dataset. * indicates a significant difference to (“ref”) scores for each
column of the table respectively as indicated by the paired t-test with
p<0.001.
classifier | Ottawa shooting | Ferguson riots | Charlie Hebdo | Sydney siege | macro mean
---|---|---|---|---|---
IBk / AF | 69.26 | 69.54 | 77.09 | 73.28 | 72.29
J48 / AF | 75.62 | 74.85 | 77.05 | 79.21 | 76.68
Random Forest / AF | 67.87 | 68.31 | 75.40 | 72.57 | 71.03
Table 4: Accuracy scores of different stance classifiers on the PHEME dataset
with AF features removed.
Features | Accuracy
---|---
All features | 79.02
All without AF | 76.54
All without ITS | 78.55
All without SS | 77.59
All without SPS | 78.16
All without DS | 78.36
All without NDS | 77.59
All without IQ | 78.64
Table 5: Contribution of each AF feature. Accuracy scores are for the Random
Forest classifier on RumourEval data set with each feature removed in turn.
### Baselines
On the RumourEval dataset we run different classifiers (see Section 3.2). We compare the performance of these classifiers against the best-performing system from the RumourEval challenge, namely **Turing** Kochkina et al. (2017).
We also run all the classifiers from the RumourEval dataset on the PHEME dataset. The results are compared against the following baseline systems reported on the PHEME dataset:
* **Gaussian Processes (GP)** reported by lukasik2015classifying.
* **Hawkes Processes (HP)** reported by lukasik2016hawkes. HPs make use of both temporal and textual information of tweets.
### Training-Testing Settings
We have two different settings. In the first setting we use the SemEval training data to train the models and apply on the testing data. In the second setting we perform training and testing on the PHEME dataset. For the PHEME dataset, we follow the leave one out (LOO) strategy taken by Lukasik et al. Lukasik et al. (2016) to construct the training and testing data. In LOO n-1 rumours (all tweets within these rumours) are used for training and the resulting model is tested on the n\({}^{th}\) rumour. Finally, results are macro-averaged.
## 5 Results
As shown in Table 2 (column two of the table) the best performing learner on the RumourEval dataset is the Random Forest classifier. It achieves the accuracy of _79.02_, higher than any participating system in the RumourEval Task A.⁹
[FOOTNOTE:9][ENDFOOTNOTE]
The results on the PHEME dataset are shown in Table 3. Overall the best performing classifier is the J48 decision tree learner. The difference in accuracy scores between the classfiers is tested for significance using paired t-test (p\(<\)0.001). J48 is only significantly better than IBk and J48 for the _Ottawa shooting_ event type. In the remaining event types, J48 performs better, but not significantly better than IBk and Random Forest.
All classifiers J48, IBk and Random Forest, however, outperform the **GP** and **HP** baselines on all event types ¹⁰.
[FOOTNOTE:10][ENDFOOTNOTE]
What these results demonstrate is that simpler classifiers, such as J48 and Random Forest can outperform significantly more sophisticated machine learning methods (GPs and HPs in this case, and LSTMs in the RumourEval case), thanks to the additional knowledge captured in the rich feature set. In contrast, for example, the GP and HP models relied primarily on BOW and Brown clustering features.
## 6 Feature Analysis
The results described in Section 5 are based on features reported by related work, enhanced by us with AF features (Section 3.3). We repeat the experiments with AF features removed from the feature set, in order to quantify the extent of their contribution.
For the RumourEval dataset the results are shown in column 3 of Table 2. The omission of the AF features leads to a performance decrease for all classifiers. The accuracy scores also fall below that of the SemEval winner **Turing** – the state-of-the-art system on the RumourEval dataset.
The results on the PHEME dataset are shown in Table 4. The exclusion of the AF features leads to an overall drop in performance when compared to the same classifiers in Table 3. However, these differences are not significant and the classifiers with AF features removed still perform at least as well as the GP and HP baselines (for the event type _Ferguson riots_), or outperform the baselines (for all other event types).
Table 5 shows the accuracy scores of the Random Forest stance classifier, the best performing system on the RumourEval dataset when each AF feature is removed in turn. The results indicate that each AF feature contributes to the accuracy boost in stance classification. The highest accuracy loss results from removing the surprise (SS) and certainty (NDS) scores and the least – when the _isQuestion_ (IQ) feature is removed. None of the AF feature removals cause a significant drop in accuracy. However, the loss is significant (\(p<0.0001\)) when all AF features are removed.
Both RumourEval and PHEME dataset evaluations show that the AF features play an important role in terms of achieving higher accuracy for tweet-based stance classification. They also show the importance of task or problem-specific feature engineering and point out that it is possible with some feature engineering effort to outperform state-of-the-art techniques that are typically considered more powerful and sophisticated than traditional learning methods.
## 7 Conclusion
This paper tackled the problem of stance classification of tweets towards rumours. In our approach we use a simple classification approach, combining common features reported by related studies with our novel AF features, to boost overall accuracy. Our results show that this approach leads to significantly better results on both RumourEval and PHEME datasets compared to current state-of-the-art systems. Furthermore, our results show that the omission of the AF features proposed in this work leads to significantly lower performance. Adding AF to the feature set causes our approach to outperform the best performing system on the RumourEval dataset. These results show the importance of task- or problem-oriented feature engineering.
The proposed features are content based and work on text level. In our future work we plan to investigate features that are able to capture communication behaviours between users. We also plan to apply stance information as a feature in rumour veracity classification.
## Acknowledgments
This work was partially supported by the European Union funded COMRADES (grant agreement No. 687847) and PHEME projects (grant agreement No. 611223), as well as an EPSRC career acceleration fellowship (EP/I004327/1).
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by
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February 22, 2024
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A mathematical model from a previous work was re-fitted and analyzed for experimental data regarding the cellular immune response to the lymphocytic choriomeningitis virus. Specifically, the \(CD8^{+}\) T cell response to six MHC class I-restricted epitopes (GP* and NP*) and \(CD4^{+}\) T cell responses to two MHC class II-restricted epitopesDe Boer, Homann, and Perelson (2003). In this work, we use calibration through log likelihood maximization to investigate if different parameters can produce a more accurate fit of the model presented previously in the paper titled _Different Dynamics of CD\(4^{+}\) and CD\(8^{+}\) T Cell Responses During and After Acute Lymphocytic Choriomeningitis Virus InfectionDe Boer, Homann, and Perelson (2003)_.
## I Historical Background
The leading cause of death among people under the age of 45 is trauma. A main cause of death after trauma is internal bleeding of the abdominal organs. Of those abdominal organs, the spleen is one of the most often injured due to blunt traumavan der Vlies _et al._ (2011), so much so that it is affected in 32% of patients that have traumatic abdominal injuriesvan der Vlies _et al._ (2012). Every year, about 39,000 patients are admitted in hospitals throughout the United States of America for the treatment of blunt splenic trauma (BSI). Of those admitted patients, 28,000 will undergo nonoperative management careRequarth (2010).
It was once believed that the spleen could not heal spontaneously and would also rupture later during patient recovery. These ideas started to be re-examined in the 1970s when data about post-operative infections regarding laparotomies and removing the spleen was publishedvan der Vlies _et al._ (2011). Due to this new knowledge, more doctors are now performing procedures such as splenic artery embolization (SAE) which preserve the spleen and also hold a high success ratevan der Vlies _et al._ (2012).
Like the appendix, the spleen’s function in the human body was unknown to many. In 1919, two individuals (Morris and Bullock) showed that dogs with their spleen that were infected with rat plague bacillus had a higher survival rate as opposed to those that were infected without their spleen. Due to limitations in diagnostic medicine at the time, splenectomy was a main standard of care. Until about 65 years ago, no one even questioned the idea of splenectomy as a treatment en masse when there were a plethora of overwhelming polstsplenectomy infection (OPSI) cases. Things started to change roughly fifty years ago when care shifted to splenorraphies and then again about a decade later when clinicians started expectant management of patients became mainstream. Contrary to older ideas regarding the spleen’s function, we now know the spleen assists with immune function. It can help filter specific antigens and microorganisms and act in cell regenerationKaseje _et al._ (2008).
## II Introduction
The lymphocytic choriomeningitis (LCM) virus is a rodent disease that an be passed in many different ways from rodent to rodent, human to human, and rodent to human. It can stay dormant in mice for as long as 35 yearsLehmann-Grube (1971). A member of the arenaviridae family of viruses, lymphocytic choriomeningitis virus (LCMV) is a human pathogen that can infect a large amount of the human populationBonthius (2012).
## III Materials and Methods
In the previous study by De Boer et al.De Boer, Homann, and Perelson (2003), data was collected by first injecting six to eight week old mice, male and female, with LCMV. At different time intervals, spleen cell samples were extracted and measured from an average of three to four mice per data point. The number of specific T cells per spleen was measured. A set of linear differential equations was used to describe the data. Due to the linear nature of the model, a solution to the differential equations can be obtained. De Boer, et al. estimated the parameters of the model using the DNLS1 subroutine from the Common Los Alamos Software Library and based on the Levenberg-Marquardt algorithmMoré (1978).
We employ a different, more efficient strategy to arrive at the same destination: parameter optimization through log likelihood maximization.
<figure><img src="content_image/1705.02380/Fig1_All.png"><figcaption>Figure 1: Cubic spline interpolation of specific CD4+ T cells per spleen (tworesponses GP61 and NP309) with respect to days after LCMV infectionDe Boer,Homann, and Perelson (2003).</figcaption></figure>
<figure><img src="content_image/1705.02380/Fig2_All.png"><figcaption>Figure 2: Cubic spline interpolation of specific CD4+ T cells per spleen (tworesponses GP61 and NP309) with respect to days after LCMV infectionDe Boer,Homann, and Perelson (2003).</figcaption></figure>
<figure><img src="content_image/1705.02380/Fig4_All.png"><figcaption>Figure 3: Cubic spline interpolation of specific CD8+ T cells per spleen (sixresponses GP33, GP118, NP205, NP396, GP276, GP92) with respect to days afterLCMV infectionDe Boer, Homann, and Perelson (2003).</figcaption></figure>
In order to calculate the model results with the necessary parameters, we first calculate the values of the population of activated cells, A and the memory cells, M. The death rate of memory cells and activated cells is given by \(\delta_{M}\) and \(\delta_{A}\), respectively.
There are three phases in the model: rapid expansion, rapid contraction, and slower contraction of the T cell population. The three phases for modeling the amount of T cells per spleen correspond to the following time intervals: t \(<\) T, t between T and T+\(\Delta\), and T\(>\)T+\(\Delta\).
The population of A cells in the initial phase of rapid cellular growth is given by the following differential equation:
\[\frac{dA}{dt}=\rho A,\]
the parameter \(\rho\) is the net expansion rate.
The population of A and M cells during the contraction phase is given by the following equations:
\[\frac{dA}{dt}=-(r+\alpha+\delta_{A})A\]
\[\frac{dM}{dt}=rA-\delta_{M}M\]
The parameter \(\alpha\) is the cell death rate, also known as rapid apoptosis and \(r\) is the contraction phase rate. At time T, A reaches its peak value and is calculated by \(A(T)=A(0)\exp[\rho T]\).
We then calculate the value of A at T + \(\Delta\). The \(\Delta\) term is the duration of the phase of rapid contraction of T cells population. That phase is followed by a phase of slower contraction of the population. Now, \(A(T+\Delta)=A(T)\exp[-\delta(\Delta)]\). Next, we calculate the value of M at T+\(\Delta\),
\[M(T+\Delta)=\frac{\exp[-\delta_{M}(T+\Delta)](rA(T)(1-\exp[-(\delta_{A}-\delta _{M})(T+\Delta)]))}{\delta-\delta_{M}}\]
For \(t<T\), \(A=A(0)\exp[\rho t]\) and M=0.
For \(t>T\), \(A=A(T)\exp[-\delta(t-T)]\) and
\[M=\frac{\exp[-\delta_{M}(t)]rA(T)(1-\exp[-(\delta-\delta_{M})t])}{\delta- \delta_{M}},\]
where \(\delta=r+\delta_{A}+\alpha\).
For \(T<t<T+\Delta\), \(A=A(T+\Delta)\exp[-\delta^{\prime}(t-T-\Delta)]\) and
\[M(t)=\frac{\exp[-\delta_{M}t](rA(T+\Delta)(1-\exp[-(\delta^{\prime}-\delta_{M} )t]))+M(T+\Delta)(\delta^{\prime}-\delta_{M})}{\delta^{\prime}-\delta_{M}},\]
and \(\delta^{\prime}=r+\delta_{A}\).
Data was collected from citation #De Boer, Homann, and Perelson (2003) using the PlotDigitizer application. A Macbook Pro running Anaconda Python 3.6 with the Jupyter Notebooks was used for all calculations.
### Sensitivity Analysis
Sobol method was used first to see which parameters have the most influence on the fluctuations of the results. The results of these calculations were inconclusive and we were not able to ascertain which parameter had the greatest effect. We suspect that all parameters have a similar influence on the results of the differential equations.
## IV Discussion
Our sensitivity analysis on the parameters of the model did not show that any particular parameter is significantly more important than the others.
It was found that the optimized parameters calculated in this work, using a simple log likelihood maximization method, are noticeably different from those calculated in the paperDe Boer, Homann, and Perelson (2003), using the Levenberg-Marquard algorithm.
A few factors that influenced the results are the following. The data was acquired directly from the graphs in the published paperDe Boer, Homann, and Perelson (2003) and not provided via a database from the corresponding author due to time constraints of the project. The use of this method is subject to errors that are difficult to quantify due to human-machine interaction. Another influential factor is that our study used only the data corresponding to the first 70 days, while the original article used the full data set up to 921 days. Despite these issues, in most cases the newly estimated parameters fall within the 95% combined confidence intervals reported in the paper.
Parameter | Log Likelihood Maximization Method | 95% Combined CIDe Boer, Homann, and Perelson (2003) | Units
---|---|---|---
ρ | 1.82 | 0.98-2.21 | d−1
δA | 0.01182 | 0.01-0.47 | d−1
δM | 0.00017 | 0.0006-0.003 | d−1
r | 0.00094 | 0.001-0.022 | d−1
α | 0.16750 | 0.12-0.82 | d−1
T | 7.72 | 8-9.1 | Days
Δ | 10.4 | 2.2-9.8 | Days
A(0) | 19.3 | 0.4-518.8 | Cells
Table 1: Results for implementation of log-likelihood maximization, compared
to De Boer et al.’s results from the paperDe Boer, Homann, and Perelson
(2003).
## V Conclusion
We can conclude that, in cases where a quick and simple verification is in order, “data thieving” from published graphs and the log likelihood maximization method can be useful tools. In this case, we were able to obtain results consistent with previously publishedDe Boer, Homann, and Perelson (2003) work using the calibration likelihood estimation parameter optimization.
List of Figures
* 1 Cubic spline interpolation of specific \(CD4^{+}\) T cells per spleen (two responses GP61 and NP309) with respect to days after LCMV infectionDe Boer, Homann, and Perelson (2003).
* 2 Cubic spline interpolation of specific \(CD4^{+}\) T cells per spleen (two responses GP61 and NP309) with respect to days after LCMV infectionDe Boer, Homann, and Perelson (2003).
* 3 Cubic spline interpolation of specific \(CD8^{+}\) T cells per spleen (six responses GP33, GP118, NP205, NP396, GP276, GP92) with respect to days after LCMV infectionDe Boer, Homann, and Perelson (2003).
List of Tables
* 1 Results for implementation of log-likelihood maximization, compared to De Boer et al.’s results from the paperDe Boer, Homann, and Perelson (2003).
## References
* Bonthius (2012)Bonthius, D. J., in _Seminars in pediatric neurology_, Vol. 19 (Elsevier, 2012) pp. 89–95.
* De Boer, Homann, and Perelson (2003)De Boer, R. J., Homann, D., and Perelson, A. S., The Journal of Immunology **171**, 3928 (2003).
* Kaseje _et al._ (2008)Kaseje, N., Agarwal, S., Burch, M., Glantz, A., Emhoff, T., Burke, P., and Hirsch, E., The American Journal of Surgery **196**, 213 (2008).
* Lehmann-Grube (1971)Lehmann-Grube, F., in _Lymphocytic Choriomeningitis Virus_ (Springer, 1971) pp. 1–173.
* Moré (1978)Moré, J. J., in _Numerical analysis_ (Springer, 1978) pp. 105–116.
* Requarth (2010)Requarth, J. A., Journal of Trauma and Acute Care Surgery **69**, 1423 (2010).
* van der Vlies _et al._ (2012)van der Vlies, C. H., Hoekstra, J., Ponsen, K. J., Reekers, J. A., van Delden, O. M., and Goslings, J. C., Cardiovascular and interventional radiology **35**, 76 (2012).
* van der Vlies _et al._ (2011)van der Vlies, C. H., Olthof, D. C., Gaakeer, M., Ponsen, K. J., van Delden, O. M., and Goslings, J. C., International Journal of Emergency Medicine **4**, 47 (2011).
|
1512.02411 | {
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] | # Identifying the source of super-high energetic electrons in the presence of pre-plasma in laser-matter interaction at relativistic intensities
D. Wu
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, China
S. I. Krasheninnikov
University of California-San Diego, La Jolla, California, 92093, USA
S. X. Luan
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, China
W. Yu
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, Shanghai 201800, China
February 29, 2024
###### Abstract
The generation of super-high energetic electrons influenced by pre-plasma at relativistic intensity laser-matter interaction is studied in a one-dimensional slab approximation with particle-in-cell simulations. Different pre-plasma scale-lengths of \(1\ \mu\text{m}\), \(5\ \mu\text{m}\), \(10\ \mu\text{m}\) and \(15\ \mu\text{m}\) are considered, showing an increase in both particle number and cut-off kinetic energy of electrons with the increase of pre-plasma scale-length, and the cut-off kinetic energy greatly exceeding the corresponding laser ponderomotive energy. A two-stage electron acceleration model is proposed to explain the underlying physics. The first stage is attributed to the synergetic acceleration by longitudinal electric field and laser pulse, with its efficiency depending on the pre-plasma scale-length. These electrons pre-accelerated in the first stage could build up an intense electrostatic potential barrier with its maximal value several times as large of the initial electron kinetic energy. Part of energetic electrons could be further accelerated by the reflection off the electrostatic potential barrier, with their finial kinetic energies significantly higher than the values pre-accelerated in the first stage.
pacs: 52.38.Kd, 41.75.Jv, 52.35.Mw, 52.59.-f
## I Introduction
The influence of pre-plasma in laser-matter interaction at relativistic intensities has attracted great attention from both experimental and theoretical investigations, because of its significant effects on a number of applications, such as laser driven ion acceleration[1; 2; 3; 4; 5; 6; 7], fast ignition[8; 9; 10; 11] and bright x/\(\gamma\) ray sources[12; 13], etc. The pre-plasma produced by the intrinsic laser pre-pulse (usually with ns duration) can be as high as \(10\ \mu\text{m}\) for the energetic main pulses of energies tens of kJ with a typical contrast ratio \(10^{-5}\). In the fast ignition related experiments with relatively long pulses (with tens of ps duration), high intensity and high power laser, even the contrast ratio can be as high as \(10^{-8}\), considerable pre-plasma can still build up in front of a solid density target. The pre-formed plasma always exists in laser-matter interaction at relativistic intensities, thus the laser pre-plasma interaction is inevitable.
The fast electron generation due to relativistic intensity laser-matter interaction influenced by preformed plasma has been addressed in a number of experimental and theoretical studies[10; 14; 15; 16; 17; 18; 19; 20], suggesting that the presence of pre-plasma can significantly affect the fast electron distributions. Both experiments and numerical simulations have reported an increase of fast electron generation efficiency with the increase of pre-plasma scale-length. The recent particle-in-cell simulations[18] have observed super-high energetic electrons with the cut-off kinetic energy as high as \(100\ \text{MeV}\) at laser of intensity \(10^{20}\ \text{W}/\text{cm}^{2}\) and pre-plasma of scale length \(10\ \mu\text{m}\). However the underlying physics, i) _the increase in the generation efficiency of energetic electrons with the increase of pre-plasma scale-length_, and ii) _the acceleration mechanism of super-high energetic electrons with kinetic energy greatly exceeding the ponderomotive energy_, is still unclear, which is the aim of this work.
In order to simulate laser-matter interaction with pico-second duration in the presence of large scale pre-plasma, we choose to use one-dimensional (1-D) particle-in-cell (PIC) simulations[4], because it is computationally cheap. Although multi-dimensional effects, such as laser filamentation and self-focusing[21; 22], might play roles in these processes, they are neglected in the present work. We focus on the role of pre-plasma in energetic electron beam generation by using systematic particle-in-cell simulations, where the laser is of intensity \(10^{20}\ \text{W}/\text{cm}^{2}\), and different pre-plasma scale-lengths, \(1\ \mu\text{m}\), \(5\ \mu\text{m}\), \(10\ \mu\text{m}\) and \(15\ \mu\text{m}\), are considered. The questions, i) “why the generation efficiency of energetic electrons is increasing with the increase of pre-plasma scale-length”, and ii) “what is underlying acceleration mechanism of super-high energetic electrons with kinetic energy greatly exceeding the ponderomotive energy”, are answered. A two-stage acceleration model is proposed to identify the source of super-high energetic electrons. The first stage is the synergetic acceleration by longitudinal electric field and laser pulses, with its efficiency depending on the pre-plasma scale-length. The second stage is related to the intense electrostatic potential building in front of the target and the accompanying electron reflection by the intense electrostatic potential barrier.
This paper is arranged as follows: The details of numerical modelling and simulation results are demonstrated in Sec. II. The two-stage acceleration model by analysing the simulation results is proposed in Sec. III to explain the impacts of pre-plasma and identify the sources of energetic electrons. In Sec. IV, the acceleration model is further addressed analytically and numerically. The conclusions are given in Sec. V.
<figure><img src="content_image/1512.02411/x1.png"><figcaption>Figure 1: (color online) Schematic of simulation set-up. (a) A linearlypolarized laser enters into the simulation box from left boundary andpropagates in z-direction. The laser is of intensity 1020 W/cm2, where thelaser wavelength is 1 μm and pre-plasma scale-length is 10 μm. The simulationbox is 400 μm, and the simulation time is 400T0, i.e. 1.3 ps. To analyse theelectron energy distributions in detail, we place two diagnostic planes atz=100 μm and z=300 μm (shown by the thick black lines), which could time-integrally record the energy distributions of electrons passing through. (b)In order to ensure the accuracy of numerical simulation, we record thetemporal variation of laser energy flux (blue line), ∫(E×B)zdSdt, at the leftsimulation boundary (i.e. z=0), the electromagnetic energy (black line),∫(1/2)(E2+B2)dV, in the simulation box and the particle kinetic energy (redline), ∑pmp(γp−1), in the simulation box.</figcaption></figure>
## II Numerical simulation results
The simulations are performed with 1-D PIC code. In order to simulate laser-matter interactions with large scale pre-plasmas, the weighted particle technology is applied in the numerical simulations, which is proven to be more efficient than uniform weighted particles in large density gradients calculations[23]. In addition, a 4-th order particle cloud and 4-th order FDTD method are applied in our simulations, because these feature makes it suitable for simulating laser solid-density-plasma interactions at relativistic intensities[23]. The laser is of intensity \(10^{20}\ \text{W}/\text{cm}^{2}\) or normalized amplitude \(a=8.54\) (with laser wavelength \(1\ \mu\text{m}\)), entering the simulation box from the left boundary. The initial plasma density profile is taken as \(n_{e}=n_{\text{solid}}/(1+\exp[-2(z-z_{0})/L_{p}])\), where \(n_{\text{solid}}=50n_{c}\) is the solid plasma density and \(L_{p}\) is the pre-plasma scale-length. As the electron recoiling due to the artificial electrostatic field on the right boundary could interrupt the physics we are studying, to avoid this boundary effect, we choose a large simulation box with the size of \(400\ \mu\text{m}\), which is divided into \(40000\) cells, with each cell containing \(1000\) electrons and \(1000\) ions. In our simulations, the region \(0<z<100\ \mu\text{m}\) is left as vacuum, \(L_{p}\) varies from \(L_{p}=1\ \mu\text{m}\), \(5\ \mu\text{m}\), \(10\ \mu\text{m}\) to \(15\ \mu\text{m}\), \(z_{0}\) is fixed as \(180\ \mu\text{m}\) and the minimum plasmas density is set as \(0.001n_{c}\) for all simulation cases. In order to analyse the electron energy distributions in detail, we have placed two diagnostic planes to temporally record the electrons passing through. As shown in Fig. 1 (a), the first diagnostic plane is located at \(z=100\ \mu\text{m}\) to record the electron going through in -z-direction, and the other one is located at \(z=300\ \mu\text{m}\), recording the electron passing through in z-direction.
<figure><img src="content_image/1512.02411/x2.png"><figcaption>Figure 2: (color online) The laser is of intensity 1020 W/cm2, and laserwavelength is 1 μm. (a) Electron energy spectra recorded at z=300 μm at thefinial time of simulations. Black line records the energy spectra for pre-plasma of scale-length 1 μm, red line is the case for pre-plasma of scale-length 5 μm, green line is of scale-length 10 μm and blue line is of 15 μm.(b) The black (red) crosses show the spectra of electrons (passing through in-z-direction) recorded by the diagnostic plane located at z=100 μm and theblack (red) triangles show the energy spectra of electrons (in z-direction)recorded by the diagnostic plane located at z=300 μm with pre-plasma of scale-length 10 μm (5 μm).</figcaption></figure>
To ensure the accuracy of the simulation, as we have done previously[24], we record the energy history of laser flux energy entering the simulation box (\(E_{l}\), blue line), electromagnetic field energy in the simulation box (\(E_{em}\), black line), and particle kinetic energy in the simulation box (\(E_{k}\), red line), which is shown in Fig. 1 (b). It is clearly demonstrated that, at \(t=180T_{0}\), part of laser flux energy is starting to convert to plasmas. The energy conservation condition \(E_{l}=E_{em}+E_{k}\) is always satisfied in the whole simulation. The total simulation time is set to be \(400T_{0}\), to avoid the electron recoiling effect.
The fast electron energy spectra obtained for different pre-plasma scale-length (\(L_{p}=1\ \mu\text{m}\), \(5\ \mu\text{m}\), \(10\ \mu\text{m}\) and \(15\ \mu\text{m}\)) while keeping the laser of intensity \(10^{20}\ \text{W}/\text{cm}^{2}\) fixed, are analysed in Fig. 2. The dependence of electron energy distributions on the pre-plasma scale-length is plotted in Fig. 2 (a), which records the energy spectra of electrons passing through the diagnostic plane located at \(z=300\ \mu\text{m}\). There is a clear relation between cut-off kinetic energy and pre-plasma scale-length, the larger the scale-length the higher the cut-off kinetic energy, which is in agreement with earlier published works[18]. We have also found that the cut-off electron kinetic energy greatly exceeds the corresponding laser ponderomotive energy, which is \(3.8\ \text{MeV}\) at intensity \(10^{20}\ \text{W}/\text{cm}^{2}\). For pre-plasma of scale-length \(1\ \mu\text{m}\), \(5\ \mu\text{m}\), \(10\ \mu\text{m}\) and \(15\ \mu\text{m}\), the corresponding cut-off energies are \(28\ \text{MeV}\), \(44\ \text{MeV}\), \(92\ \text{MeV}\) and exceeding \(150\ \text{MeV}\), respectively. In Fig. 2 (b), we pick up two cases with pre-plasma of scale-length \(5\ \mu\text{m}\) and \(10\ \mu\text{m}\), and include the energy spectra of electrons recorded by the diagnostic plane located at \(z=100\ \mu\text{m}\). By comparing the two energy spectra recorded by two different diagnostic planes, we can find that the cut-off energy recorded at \(z=300\ \mu\text{m}\) is about three times of the value recorded at \(z=100\ \mu\text{m}\). For pre-plasma of scale-length \(5\ \mu\text{m}\), the cut-off energy recorded at \(z=100\ \mu\text{m}\) is \(15\ \text{MeV}\), while that recorded at \(z=300\ \mu\text{m}\) is \(44\ \text{MeV}\). In addition, for scale-length \(10\ \mu\text{m}\), the cut-off energy recorded at \(z=100\ \mu\text{m}\) is \(32\ \text{MeV}\), while that recorded at \(z=300\ \mu\text{m}\) is \(92\ \text{MeV}\).
<figure><img src="content_image/1512.02411/x3.png"><figcaption>Figure 3: (color online) The laser is of intensity 1020 W/cm2, and laserwavelength is 1 μm. Comparisons of z-pz phase-space plots with different pre-plasma scale-length, (a) 1 μm, (b) 5 μm, (c) 10 μm and (d) 15 μm,respectively. The red lines covered on the phase-space plots are theelectrostatic potential curves (∫zEzdz), normalized by −eϕ/mec2. Note that thephase-space mixing region and the maximal value of the electrostatic potentialbarrier increase with the increase of the pre-plasma scale-length. The firststage is due to the synergetic acceleration by the longitudinal electric fieldEz and the ponderomotive force of the reflected laser pulse, and the secondstage is attributed to the intense electrostatic potential building and theaccompanying reflection of the energetic electrons off the potential barrier.</figcaption></figure>
## III Explaining of simulation results
We have found that the cut-off kinetic energy of electrons increases with the increase of the pre-plasma scale-length. In the meanwhile, we have also noticed that the cut-off electron kinetic energy recorded by diagnostic plane located at \(z=300\ \mu\text{m}\) is three times more or less that recorded at \(z=100\ \mu\text{m}\). The aim of this work is to uncover the mysteries, i) the increase in the generation efficiency of energetic electrons with the increase of pre-plasma scale-length and ii) the source of super-high energetic electrons with energy greatly exceeding the corresponding laser ponderomotive energy. In order to understand the underlying physics of the observed phoneme, we now refer to analysing the \(z\)-\(p_{z}\) phase-space dynamics. Fig. 3 describes the phase-space patterns of laser pre-plasma interactions with laser of intensity \(10^{20}\ \text{W}/\text{cm}^{2}\) and pre-plasma of scale-length \(L_{p}=1\ \mu\text{m}\) [Fig. 3 (a)], \(5\ \mu\text{m}\) [Fig. 3 (b)], \(10\ \mu\text{m}\) [Fig. 3 (c)] and \(15\ \mu\text{m}\) [Fig. 3 (d)], respectively. The phase-space density \(D(z,p_{z})\) gives a value proportional to the number of electrons found between \(z\) and \(z+dz\) having longitudinal momentum ranged between \(p_{z}\) and \(p_{z}+dp_{z}\). The normalized electrostatic potential, \(-e\phi/m_{e}c^{2}\), due to the longitudinal charge separation field \(E_{z}\), is shown in red curves covered on phase plots. The electron longitudinal momentum \(p_{z}\) is in the dimensionless units of \(\gamma\beta\) and \(z\) is in the units of laser wavelength, which is \(1\ \mu\text{m}\).
In the very earlier stage of the laser propagation in under-dense preformed plasma, part of electrons are swept away in the forward direction by the laser ponderomotive force, leaving behind immobile ions. The electric field \(E_{z}\) due to charge separation within the under-dense plasma region tries to pull the electrons in the backward direction. When the laser arrives at the critical density surface and is reflected back, the ponderomotive force of the reflected laser pulse can further accelerate the electrons in the backward direction. Actually, the first stage acceleration is due to the synergetic effects by this longitudinal charge separation field \(E_{z}\) and the ponderomotive force of the reflected laser pulse. From Woodward-Lawson theorem[25], we know that a single electron in vacuum, oscillating coherently with a propagating plane laser pulse would gain zero cycle averaged energy since the electron energy gain in one half cycle is exactly equal to the energy loss in the next half cycle. However, when there exists an external electric field[18; 26; 27; 28], even though this field is very week, the Woodward-Lawson theorem can be broken and the electron can obtain none zero energy from the synergetic effects by the external electric field and the laser pulse.
When the incident laser arrives at the critical density surface and is reflected back, a strong delta-like charge separation field or the step-like electrostatic potential, as shown in Fig. 3, is build up therein, which is strong enough to drive electrons to very high velocity within very short time and short length. Imagine we are standing on the frame of a backward propagating electron, we will find that the incident laser pulse is oscillating very fast, and its only contribution to the motions of the electron is to increase its mass by a factor \(\gamma=(1+a^{2}/2)^{1/2}\) in an average way (Appendix A), however the reflected laser pulse is oscillating so slow that this electron can be captured and continually be accelerated backward by its ponderomotive force. Actually the first stage acceleration strongly depends on the pre-plasma scale-length. As clearly demonstrated in Fig. 3 (c) [\(10\ \mu\text{m}\)], the first stage acceleration is stronger than that in Fig. 3 (a)[\(1\ \mu\text{m}\)] and (b) [\(5\ \mu\text{m}\)], but not as efficient as that in Fig. 3 (d) [\(15\ \mu\text{m}\)]. According to Woodward-Lawson theorem[25], a single electron can not gain none zero cycle-averaged energy from one plane wave. However, in our case, there exists an external electric field \(E_{z}\) due to the charge separation field in the under-dense pre-plasma region. Actually, as we shall analyse in the next section, the pre-plasma scale-length determines the space extension of \(E_{z}\), which eventually determines the maximum possible electron energy gain in this synergetic acceleration process.
The energetic electrons pre-accelerated in the first stage continuously propagate backward and expand freely, building up an intense electrostatic potential barrier therein, as shown by the red curves in Fig. 3. Actually the peak value of the electrostatic potential barrier is three times as large of the kinetic energy of these electrons pre-accelerated. However at present, the claiming “three times” only have statistical meanings. As we know, for an electron with kinetic energy \(E_{k\text{in}}\) initially located at position with zero electrostatic potential energy, it is impossible to arrive at the position with potential energy \(U_{p}>E_{k\text{in}}\) without any external forces. However, for the continuously emitting electron beams or separated multi electron bunches, we find that part of electrons can arrive at positions where the potential energies are several times as large of their initial kinetic energies. When these electrons are reflected back to their original positions, the obtained kinetic energies of the returned electrons will increase to \(E_{k\text{f}\max}=N\times E_{k\text{in}}\). Although it seems impossible, this process conserves the total energies of the system, and \(\sum n_{\text{in}}E_{k\text{in}}=\sum n_{\text{f}}E_{k\text{f}}\) is always satisfied, with \(E_{k\text{f}}\) having \(E_{k\text{f}\min}<E_{k\text{in}}<E_{k\text{f}\max}\). In the next section, solid interpretations are presented, including analytical analysis and electrostatic numerical simulations, for the building process of electrostatic potential and the accompanying electron kinetic enhancement by the reflection off this potential barrier.
## IV Two-stage acceleration model
_The synergetic acceleration by longitudinal electric field and laser ponderomotive force–_ We consider the relativistic electron dynamics in the presence of two counter-propagating plane laser waves with vector potential \(a_{+}\) and \(a_{-}\) and longitudinal field \(E_{z}\). \(a_{+}\) means the propagating of laser pulse is with the same propagation direction of electron in the presence of electric field \(E_{z}\). Considering the electron propagates with high velocity along the z-direction, the only contribution of the incident wave \(a_{-}\) is to increase the electron mass in an averaged way. The z-momentum and energy equation, in normalized units, can be written as
\[\frac{d(\gamma v_{z})}{dt}=\frac{-1}{2\gamma}\frac{\partial a_{+}^{2}}{ \partial z}+E_{z},\] (1)
\[\frac{d\gamma}{dt}=\frac{1}{2\gamma}\frac{\partial a_{+}^{2}}{\partial t}+E_{z }v_{z},\] (2)
where \(v_{z}\) is the electron velocity component along z-direction and the relativistic factor \(\gamma\) defined as \(\gamma=\gamma_{A}\gamma_{z}\) with \(\gamma_{A}=(1+a^{2}/2+a_{+}^{2})^{1/2}\), \(a^{2}/2\) is the average mass increase due to the incident laser wave of the form \(a_{-}=a\sin(t+x)\), and \(\gamma_{z}=1/(1-v_{z}^{2})^{1/2}\).
<figure><img src="content_image/1512.02411/x4.png"><figcaption>Figure 4: (color online) Schematic demonstration of the electron dynamics inthe presence of two counter-propagating laser waves and longitudinal electricfield Ez. Note that the a+ means the propagating of laser pulse is with thesame propagation direction of electron in the presence of electric field Ez,where r2 is the reflection rate compared with the incident laser a−.</figcaption></figure>
<figure><img src="content_image/1512.02411/x5.png"><figcaption>Figure 5: (color online) Factor η as function of σE and R.</figcaption></figure>
For reflecting plane wave of the form \(a_{+}=a_{+}\sin(t-z)\), from Eqs. (1) and (2), we find
\[\frac{d}{dt}\gamma_{A}\gamma_{z}(1-v_{z})=-E_{z}(1-v_{z}).\] (3)
For constant electric field, Eq. (3) can be integrated and we have
\[\gamma_{A}\gamma_{z}(1-vz)=\sigma_{\tau_{0}}-E_{z}(t-t_{0}-z-z_{0}),\] (4)
where \(t_{0}\) is the time at which the electron crosses \(z=z_{0}\) and \(\sigma_{\tau_{0}}=\gamma_{A}\gamma_{z}(1-v_{z})|_{t=t_{0},z=z_{0}}\). Note for the highly relativistic case, we have \(\sigma_{\tau_{0}}\sim(1/2)(\gamma_{A}/\gamma_{z})\ll 1\).
The trajectory of the electron \(z\) can be found by introducing a local time \(\tau=t-z\), in which \(d\tau/d\tau=dt/d\tau-dz/d\tau\) and \(dt/d\tau=(dz/dt)(dt/d\tau)/v_{z}\), as
\[\frac{dz}{d\tau}=\frac{v_{z}}{1-v_{z}}.\] (5)
Using \(v_{z}\) from Eq. (4), \(dz/d\tau\) can be found to be
\[\frac{dz}{d\tau}=\frac{1}{2}[f^{2}(\tau)-1],\] (6)
where \(f(\tau)=\gamma_{A}(\tau+\tau_{0})/(\sigma_{\tau_{0}}-E_{z}\tau)\).
The change in the electron energy only due to the contribution of laser waves, \(\sigma\varepsilon(\tau)\) is given by \(\Delta\varepsilon(\tau)=\gamma_{A}(\tau+\tau_{0})\gamma_{z}(\tau+\tau_{0})- \gamma_{A}(\tau_{0})\gamma_{z}(\tau_{0})-E_{z}[z(\tau+\tau_{0})-z(\tau_{0})]\). Following the Eq. (4) and making use of the inequality (\(\sigma_{\tau_{0}}\ll 1\), \(\sigma_{\tau+\tau_{0}}\ll 1\) and \(E_{z}\tau\ll 1\)), \(\Delta\varepsilon(\tau)\) can then be rewritten as
\[\Delta\varepsilon(\tau)=\frac{1}{2}\int_{0}^{\tau}{\frac{d\gamma_{A}^{2}(\tau+ \tau_{0})/d\tau}{\sigma_{\tau_{0}}-E_{z}\tau}}d\tau\] (7)
Through Eqs. (6) and (7), we can find the maximal-possible energy gain within the limited longitudinal scale length \(L\) and the maximal in-phase time \(\tau=\pi/2\),
\[L=\frac{1}{2E_{z}^{2}}[\frac{\gamma_{A}^{2}(\pi/2+\tau_{0})}{\sigma_{E}-\pi/2} -\frac{\gamma_{A}^{2}(\tau_{0})}{\sigma_{E}}-a_{+}^{2}f(\sigma_{E})]-\frac{\pi }{4},\]
\[\Delta\varepsilon(\pi/2)=\frac{a_{+}^{2}}{2E_{z}}f(\sigma_{E}),\] (8)
where we define \(\sigma_{E}=\sigma_{\tau_{0}}/E_{z}\geq\pi/2\), and
\[f(\sigma_{E})=\int_{0}^{\pi/2}\frac{\sin{(2x)}}{\sigma_{E}-x}dx.\] (9)
As \(\tau_{0}\) is just an arbitrary initial local time, for simplicity we set \(\tau_{0}=0\) in the following expressions. Assuming \(a\gg 1\), \(L\gg 1\) and \(a_{+}^{2}=Ra^{2}\), where \(R\) is the reflection rate, based on Eq. (IV) we can obtain,
\[E_{z}=\frac{a}{L^{1/2}}\sqrt{[\frac{R\sigma_{E}+\pi/4}{2\sigma_{E}(\sigma_{E}- \pi/2)}-\frac{R}{2}f(\sigma_{E})]}.\] (10)
Combining Eq. (8) and Eq. (10), the maximal-possible electron kinetic energy gain within the limited longitudinal length \(L\) from the laser of incident amplitude \(a\) and reflection rate \(R\) can be expressed as,
\[\Delta\varepsilon=\eta aL^{1/2}=\frac{R}{2}\frac{f(\sigma_{E})}{g(\sigma_{E})} aL^{1/2},\] (11)
with \(g^{2}(\sigma_{E})=(R\sigma_{E}+\pi/4)/[2\sigma_{E}(\sigma_{E}-\pi/2)]-{Rf( \sigma_{E})}/{2}\).
In Eq. (11), the coefficient \(\eta\) is the function of \(R\) and \(\sigma_{E}\). From Fig. 5, for the typical reflection rate \(R=0.9\), \(\alpha\) almost saturates at \(0.5\) for a large range of \(\sigma_{E}\). Finally, we give a scaling law which describes the maximal-possible electron energy gain for the synergetic acceleration process, where the laser intensity \(I\) is normalized by \(1.37\times 10^{18}\ \text{W}/\text{cm}^{2}\) and the longitudinal length \(L\sim\beta L_{p}\) is normalized by \(\mu\text{m}\),
\[\varepsilon\ [\text{MeV}]=0.64\times\beta^{1/2}\times I^{1/2}\times L_{p}^{1/2}.\] (12)
In Eq. (12), we assume that the longitudinal length is on the order of pre-plasma scale-length with \(L\sim\beta L_{p}\). Here we give an estimated value of \(\beta\) as \(2.5\), by comparing the actual longitudinal acceleration extension \(L\sim 25\ \mu\text{m}\) and pre-plasma scale-length \(L_{p}=10\ \mu\text{m}\) in Fig. 3 (c) and the actual longitudinal acceleration extension \(L\sim 40\ \mu\text{m}\) and pre-plasma scale-length \(L_{p}=15\ \mu\text{m}\) in Fig. 3 (d). According to the scaling law of Eq. (12), we can see that the first stage acceleration, or the synergetic acceleration by longitudinal electric field \(E_{z}\) and the ponderomotive force of the reflected laser, depends on both the incident laser intensity and the pre-plasma scale-length.
_Electrostatic potential building and the accompanying electron reflection–_ To get the insights on both i) the possibility of the formation of the electrostatic potential barrier with the maximal value significantly larger than electron kinetic energy, and ii) the role of the potential barrier in electron acceleration, let us consider 1-D model problem. Assume that at \(t=0\) we have a bunch of electrons with density \(n_{b}\) occupied region \(0<z<z_{b}\) (\(z_{b}\ll\lambda_{De}\)) with momentum \(p_{0}>0\) and a bunch of immobile ions, located at \(z<0\) such that total electron and ion charges compensate each other. We consider dynamics of electron bunch expansion assuming that the electrons, which come back to their original positions, do not move any-more. Since we are considering the 1-D geometry, then the electric field acting on electron is solely depends on its original position at \(t=0\) and does not vary in time. Therefore, for the electron having \(z(t=0)=z_{0}<z_{b}\) we have the following equation of motion,
\[\frac{d}{dt}\frac{p}{\sqrt{1-p^{2}}}=-E_{z}(z_{0}),\] (13)
where \(E_{z}(z_{0})\) is the original electric field which is normalized by \(e/m_{e}c\). From Eq. (13) we find the time dependence of the position \(z(t,z_{0})\) of the electron initially located at \(z_{0}\) as
\[z(t,z_{0})=z_{0}+\int_{0}^{t}\frac{p_{0}-E_{z}(z_{0})t^{{}^{\prime}}}{\sqrt{1+ [p_{0}-E_{z}(z_{0})t^{{}^{\prime}}]^{2}}}dt^{{}^{\prime}}=z_{0}-\frac{1}{E_{z} (z_{0})}\{\sqrt{1+[p_{0}-E_{z}(z_{0})t]^{2}}-{\sqrt{1+p_{0}^{2}}}\},\] (14)
where \(p_{0}=p(t=0)\). From Eqs. (13) and (14) one can easily see that within the setting of the problem the electrons coming back to its original position have \(p=-p_{0}\) and, therefore, acquire the original energy.
The original increase of the normalized electrostatic potential within the electron bunch, \(\delta\phi_{0}\), can be easily found from Poisson equation,
\[\delta\phi_{0}=\frac{1}{2}(\frac{\omega_{pe}z_{b}}{c})^{2},\] (15)
where \(\omega_{pe}^{2}=4\pi e^{2}n_{b}/m\). Now we will analyse time variation of the electrostatic potential at relatively large time \(t>p_{0}/E_{z}(z_{0})\), when the majority of electrons already came back to their original positions. Estimating the magnitude of \(E_{z}(z_{0})\) from the Poisson equation, we can re-write this inequality as,
\[t>\tau_{b}=\frac{p_{0}c}{\omega_{pe}^{2}z_{b}}.\] (16)
Then the difference of the normalized electrostatic potential, \(\Delta\phi(t)\), between the head of expanding electron bunch, \(z_{h}(t)=z(t,z_{b})\), and the coordinate \(z_{r}(t)\) with \(z_{r}=z(t,z_{r})\) of electrons returning to its original position at time \(t\), can be written as follows,
\[\Delta\phi(t)=\int_{z_{r}(t)}^{z_{h}(t)}E_{z}(z)dz,\]
or,
\[\Delta\phi(t)=-\int_{0}^{E_{z}[z_{r}(t)]}E_{z}(z_{0})\frac{dz(t,z_{0})}{dE_{z} (z_{0})}dE_{z}(z_{0})\\ =-\frac{1}{2}(\frac{\omega_{pe}}{c})^{2}[z_{b}^{2}-z_{r}^{2}(t)]+\int_{0}^{2p_ {0}}\frac{\sqrt{1+p_{0}^{2}}-\sqrt{1+(p_{0}-\xi)^{2}}}{\xi}d\xi.\] (17)
Since we are considering the time \(t\gg\tau_{b}\) where \(z_{r}(t)\to z_{b}\), we find the following asymptotic expression, \(\Delta\phi_{\infty}=\Delta\phi(t\rightarrow\infty)\),
\[\Delta\phi_{\infty}=\int_{0}^{2p_{0}}[\sqrt{1+p_{0}^{2}}-\sqrt{1+(p_{0}-\xi)^{ 2}}]\frac{d\xi}{\xi}.\] (18)
From Eq. (18) we derive \(\Delta\phi_{\infty}\sim p_{0}^{2}\) for \(p_{0}\ll 1\) and \(\Delta\phi_{\infty}\sim 2\ln(2)p_{0}^{2}\) for \(p_{0}\gg 1\). In other words, for non-relativistic case \(\Delta\phi_{\infty}\) is twice of the initial electron kinetic energy \(E_{k\text{in}}\), while for a super-relativistic case \(\Delta\phi_{\infty}\sim 2\ln(2)E_{k\text{in}}\sim 1.4E_{k\text{in}}\).
As we mentioned before, electrons, being finally reflected back by potential, will come to their original positions and obtain their original kinetic energy. So that in the process of launching just one electron bunch, there is no possibility of increasing electron energy. However, situation changes drastically, when we launch a few electron bunches separated by a dwell time \(\tau_{\text{dw}}\). To get an insight in electron acceleration mechanism, consider the case of two bunches. The first bunch, launched at \(t=0\) will expand as it was discussed before. At time \(t=\tau_{\text{dw}}>\tau_{b}\), the second bunch starts launching. At that time, the first bunch has formed the “potential barrier” between the head of the first bunch and the launch point, with \(\phi_{\text{bar}}=\Delta\phi_{\infty}\). However, almost all electrons of the first bunch already came back to their original position and the electric field within the “potential barrier” becomes very small, with \(E\sim\Delta\phi_{\infty}/p_{0}t\ll E_{z}(z_{0})\). As a result, second bunch also expands virtually into vacuum and at the time \(t=2\tau_{\text{dw}}\), the cumulative contribution of the first and second bunches will create the “potential barrier” with \(\phi_{\text{bar}}=2\times\Delta\phi_{\infty}\). In addition, relatively small amount of electrons at the head of the first bunch turning back after the expansion of the second bunch will finally acquire not only their initial kinetic energy but also potential energy created by the second bunch. As a result, their total kinetic energy when they are reaching the launching location will increase in two times. Their additional energy come in expense of electron energy from the second bunch, which are de-accelerated somewhat, while passing through the second bunch electrons.
We can consider the injection of many identical electron bunches with the dwell time between them such that the previous bunches do not impact the dynamics of latter ones. One can easily find that the amount of such bunches is limited by \(N_{b}\sim\ln(t/\tau_{b})\). Therefore, maximum kinetic energy, acquired by the returned electrons of the very first bunch, after being accelerated by electric field of all bunches can be estimated as \(E_{k\max}\sim N_{b}\times E_{k\text{in}}\), which, nonetheless, can be significantly larger than \(E_{k\text{in}}\).
We can consider also continuous injection of electrons into a half-space taking the time-dependent distribution function of launching electrons \(f(t,v)\). Considering the non-relativistic case we take,
\[f(t,v)=\frac{n_{0}\delta(v-v_{0})}{1-\alpha t\omega_{pe}(n_{0})},\] (19)
where \(\alpha\ll 1\). This temporal evolution of electron launch, limited by \(\alpha t\omega_{pe}(n_{0})\), resembles the rate of bunch launches. The final energy by electric field of all bunches can be estimated as \(N_{b}\times E_{k\text{in}}\), which, nonetheless, can be significantly larger than \(E_{k\text{in}}\).
<figure><img src="content_image/1512.02411/x6.png"><figcaption>Figure 6: (color online) Simulation parameters, tb=L0/v0, tp=1/ωpe andτb=2τ2p/τb: L0=0.2, v0=0.7 and ωpe=0.5, corresponding to tb=0.28571, tp=2.0and τb=28.0. (a) (d) and (c) are the z-vz phase, electric field (black line)and potential (red line) profile at t=0.5, t=28 and t=80 respectively. (d) isthe maximal electric field (black line) and potential (red line) evolutionwith time.</figcaption></figure>
In order to confirm the above theoretical analysis, we also run a serious of 1-D electrostatic PIC simulations, which is solved by an energy conserving method (Appendix B). The electrostatic PIC simulations solve the following equations,
\[\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial z}+\frac{eE}{m_{e}} \frac{\partial f}{\partial v}=0,\] (20)
\[\frac{\partial E}{\partial z}=4\pi e\int{fdv},\] (21)
\[f(t=0)=n_{e}\delta(v-v_{0}),\] (22)
with \(\omega_{pe0}=4\pi n_{0}e^{2}/m_{e}\), \(v=\bar{v}[c]\), \(t=\bar{t}[1/\omega_{pe0}]\), \(z=\bar{z}[c/\omega_{pe0}]\), \(E_{z}=\bar{E_{z}}[c\omega_{pe0}m_{e}/e]\), \(\phi=\bar{\phi}[m_{e}c^{2}/e]\), \(\omega_{pe}=\bar{\omega_{pe}}[\omega_{pe0}]\), \(n_{e}=\bar{\omega_{pe}}^{2}[n_{0}]\) and \(f=\bar{f}[n_{0}/c]\). We define a reference density \(n_{0}\), corresponding to an reference plasma frequency \(\omega_{pe0}\). \(1/\omega_{pe0}\) define the time scale in simulation, \(c/\omega_{pe0}\) define the length scale and \(c\) is speed of light. We can change the plasma density in simulation by adjusting \(\bar{\omega_{pe}}\). If \(\bar{\omega_{pe}}=1\), the plasma density used in simulation is exactly \(n_{0}\), if \(\bar{\omega_{pe}}=0.5\), the corresponding plasma density in simulation is \(0.5\times 0.5\times n_{0}\).
Fig. 6 shows the simulation results, in which an electron bunch of velocity \(v_{0}=0.7\), thickness \(L_{0}=0.2\) and plasma frequency \(\omega_{pe}=0.5\) is emitted from the surface \(z=0\). Fig. 6 (a), (b) and (c) show the time-snap of \(z\)-\(v_{z}\) phase-space, electric field and potential profile at \(t=0.5\), \(t=28\) and \(t=80\), which clearly demonstrates that at \(t=28\), the electrons in the rear start returning to the emitting point at \(z=0\), well consistent with the theoretical analysis, \(\tau_{d}=(2/\omega^{2}_{pe})(v_{0}/L_{0})=28\). In our simulation, we include a numerical friction mechanism to stopping electrons when re-entering into the emitting point. Fig. 6 (d) shows the maximal electric field and potential evolution with time, and we find that the maximal potential almost keeps constant even when the back edge of the bunch returns to the emitting point, which is also consistent with theoretical prediction. As expected by theoretical analysis, the maximal electric field decrease with time as \(\tau_{b}/t\) when \(t>\tau_{b}\). The kinetic energy of the returned electron is exactly equal to the initial value, having \(v=-v_{0}\), which is, nonetheless, consistent with the theoretical prediction.
<figure><img src="content_image/1512.02411/x7.png"><figcaption>Figure 7: (color online) Simulation parameters, tb=L0/v0, tp=1/ωpe andτb=2τ2p/τb: L0=0.2, v0=0.7 and ωpe=0.5, corresponding to tb=0.28571, tp=2.0and τb=28.0. (a)-(b) corresponds to two bunches cases, with the second bunchemitted at t=200. (c)-(d) corresponds to three bunches cases, with the thirdbunch emitted at t=300. (e)-(f) corresponds to four bunches cases, with thefourth bunch emitted at t=350. (a) (c) and (e) are the z-vz phase-space,electric field (black line) and potential (red line) profile at t=400 for two,three and four bunches cases respectively. (b) (d) and (f) are thecorresponding maximal electric field (black line) and potential (red line)evolution with time.</figcaption></figure>
Let us consider the situation of emitting multi bunches. Fig. 7 (a) and (b) show the two bunches cases with the dwell time \(\tau_{\text{dw}}=280\) greatly larger than \(\tau_{d}=28\). We noticed that the maximal potential energy can be further increased by the emission of the second bunch, finally reaching four times as large of original kinetic energy. The velocity of the returned electron can be as high as \(v=-0.99\) compared with the initial value \(v_{0}=0.7\), confirming the theoretical prediction that the kinetic energy of returned electron is increased by twice. Fig. 7 (c) (d) (e) and (f) are cases of three (\(\tau_{\text{dw}1}=200\) and \(\tau_{\text{dw}2}=100\)) and four (\(\tau_{\text{dw}1}=200\), \(\tau_{\text{dw}2}=100\) and \(\tau_{\text{dw}3}=50\)) bunches, the maximal potential and the returned electron kinetic can be further increased as expected. Limited to the computational ability of our simulation, if the dwell time is long enough, the finial maximal potential energy will be close to the theoretically predicted value \(E_{k\max}\sim\ln(t/\tau_{b})E_{k\text{in}}\).
<figure><img src="content_image/1512.02411/x8.png"><figcaption>Figure 8: (color online) Simulation parameters: electron beam with constantvelocity v0=0.4 and density profile ω2peexp(tωpe), where ωpe=0.125. (a) is thez-vz phase-space and potential profile plotted at t=40. (b) is the maximalpotential evolution with time. (c) is the velocity spectra of the returnedelectrons collected at the emitting point.</figcaption></figure>
As shown in Fig. 3, the emission of electrons is a continuous process. Here in Fig. 8, we show the simulation results of continuous emission of electron beam with constant velocity \(v_{0}=0.4\) and density profile \(\omega_{pe}^{2}\exp{(t\omega_{pe})}\), where \(\omega_{pe}=0.125\). The simulation results, as shown in Fig. 8 (b), indicate that the maximal potential energy is more than three times as large of the initial kinetic energy at \(t=40\) and is still increasing gradually with time. Please note the oscillation of maximal potential energy, with its oscillation frequency increasing with time. These oscillations come from the plasma intrinsic oscillations, with its frequency determined by the density of the emitting electron beam. With the increase of density, the maximal potential energy and oscillation frequency are also increasing with time. Fig. 8 (c) records the velocity spectra of the returned electrons collected at the emission point, indicating that the returned electrons actually span a large velocity range, from \(-0.3\) to \(-0.7\). This spanning of the velocity range is also consistent with the theoretical prediction, with some of the electron having velocity higher than the initial value \(0.4\), and some of the electron having velocity smaller than \(0.4\). The cut-off kinetic energy of the returned electrons can be about three times as large of the initial value.
## V Conclusions
The generation of super-high energetic electrons influenced by pre-plasma in relativistic intensity laser matter interaction is studied in a one-dimensional slab approximation with particle-in-cell simulations. Different pre-plasma scale-lengths of \(1\ \mu\text{m}\), \(5\ \mu\text{m}\), \(10\ \mu\text{m}\) and \(15\ \mu\text{m}\) are considered, which shows an increase in both particle number and cut-off energy of energetic electrons with the increase of the pre-plasma scale-length, and the obtained cut-off energy of electrons greatly exceeding the corresponding laser ponderomotive energy. The two questions, i) “why the generation efficiency of energetic electrons is increasing with the increase of pre-plasma scale-length”, and ii) “what is underlying acceleration mechanism of super-high energetic electrons with kinetic energy greatly exceeding the ponderomotive energy”, are answered in this work.
A two-stage electron acceleration model is proposed to explain the underlying physics in detail. The first stage is attributed to the synergetic acceleration by the longitudinal charge separation electric field \(E_{z}\) and the ponderomotive force of the reflected laser pulse. The efficiency of the first stage acceleration depends on the pre-plasma scale-length. The maximal possible energy gain during the first stage acceleration is analysed, and a scaling law is obtained by solving the relativistic electron motions in the presence of two counter-propagating plane laser waves and the external electric field due to the charge separation within limited space extension on the order of pre-plasma scale-length. The maximal-possible energy gain in the first stage is estimated to be \(\varepsilon\ [\text{MeV}]=0.64\times\beta^{1/2}\times I^{1/2}\times L_{p}^{1/2}\), where \(I\) is laser intensity normalized by \(1.37\times 10^{18}\ \text{W}/\text{cm}^{2}\) and \(L_{p}\) is pre-plasma scale-length normalized by \(\mu\text{m}\). The scaling law indicates that with the increase of pre-plasma scale-length and incident laser intensity, the maximal-possible electron energy is also increasing, which agrees well with the simulation results.
The energetic electrons pre-accelerated in the first stage could build up an intense electrostatic potential barrier with the potential energy several times as large of electron kinetic energy. Part of energetic electrons could be reflected by this potential, obtaining finial kinetic energies several times as large of the initial values. The potential building and the accompanying electron kinetic enhancement process by this potential barrier are analysed theoretically and confirmed by electrostatic PIC simulations, where the theoretical prediction and electrostatic PIC simulations are in good agreement.
The multidimensional effects of laser propagation through under-dense plasmas are neglected in the present studies. We plan to address the multi-dimension effects in future studies.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China (11304331, 11174303, 61221064), the National Basic Research Program of China (2013CBA01504, 2011CB808104) and USDOE Grant DENA0001858 at UCSD.
## Appendix A Confirmation of the reduced model
We have studied the motion of a single electron in the field of \(a_{+}\), \(a_{-}\) and \(E_{z}\) by numerically solving the \(1\)D-\(3\)V electron equation of motion with the standard Boris algorithm. Fig. 9 (a) shows the motion of a single electron in the fields of only \(a_{+}\) and \(E_{z}\). It indicates that when the Woodward-Lawson theorem is broken, electron will be continuously accelerated forward and the final kinetic energy is increasing with the increase of the acceleration length. Fig. 9 (b) shows when there exists two counter-propagating laser pulses, i.e. \(a_{+}\) and \(a_{-}\), the dynamics of the electron initially at rest is quite complicated, resulting in stochastic-like motions.
However, when electron with a initial large momentum \(p_{z}\) enters the fields of two counter-propagating laser waves and longitudinal electric field, the influence of the incident laser \(a_{-}\) can be simplified. The only contribution of the incident laser wave \(a_{-}\) is to increase the electron mass in an averaged way. In Fig. 9 (c), black line shows the full dynamics of the electron under \(a_{+}\), \(a_{-}\) and \(E\), and the red line shows the dynamics of electron only under \(a_{+}\) and \(E\) but replacing \(\gamma_{a}=(1+a_{+}^{2}+a_{-}^{2})^{1/2}\) to \(\gamma_{a}=(1+a_{+}^{2}+a^{2}/2)^{1/2}\). The results of full dynamics and reduced model are well fitted, confirming our assumption.
<figure><img src="content_image/1512.02411/x9.png"><figcaption>Figure 9: (color online) Parameters: (a) a+(t−z)=2.0, a−(t+z)=0.0, Ez=−0.02,pz(t=0,z=0)=0.0, (b) a+(t−z)=2.0, a−(t+z)=2.0, Ez=−0.02, pz(t=0,z=0)=0.0 and(c) a+(t−z)=5.0, a−(t+z)=5.0, Ez=−0.5, pz(t=0,z=0)=10.0. (a) and (b) blackline represents the evolution of γaγz−γa(t=0,z=0)γx(t=0,z=0)−Ezz vs z, redline represents sin(t−z)2 vs z and blue line represents sin[2(t−z)] vs z. In(c) black line represents the evolution of γaγz−γa(t=0,z=0)γx(t=0,z=0)−Ezz vsz from full simulation, and red line represents the evolution ofγaγz−γa(t=0,z=0)γx(t=0,z=0)−Ezz vs z from the reduced simulation.</figcaption></figure>
## Appendix B Simulation method of electrostatic PIC
A new PIC method, which conserves energy exactly, is used. The equations of motion of particles and the Maxwell’s equations are differenced implicitly in time by the mid-point rule and solved concurrently by a Jacobian-free Newton Krylov (JFNK) solver. The particle average velocities and the electrostatic field are calculated self-consistently by the JFNK solver to preserve the total energy of the system.
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1911.06546 | {
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] | # Idealised simulations of the deep atmosphere of hot jupiters:
deep, hot, adiabats as a robust solution to the radius inflation problem F. Sainsbury-Martinez
1Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ, Université Paris-Saclay 1 felix.sainsbury@cea.fr
P. Wang
2Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France 21Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ, Université Paris-Saclay 1
S. Fromang
4Laboratoire AIM, CEA/DSM-CNRS-Université Paris 7, Irfu/Departement d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette, France 4
P. Tremblin
1Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ, Université Paris-Saclay 1
T. Dubos
5Laboratoire de Météorologie Dynamique (LMD/IPSL), Sorbonne Université, Centre National de la Recherche Scientifique, École Polytechnique, École Normale Supérieure, Paris 5
Y. Meurdesoif
6Laboratoire des Sciences du Climat et de l’Environnement/Institut Pierre-Simon Laplace, Université Paris-Saclay, CEA Paris-Saclay 6
A. Spiga
5Laboratoire de Météorologie Dynamique (LMD/IPSL), Sorbonne Université, Centre National de la Recherche Scientifique, École Polytechnique, École Normale Supérieure, Paris 57Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France. 7
J. Leconte
7Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France. 7
I. Baraffe
8Astrophysics Group, University of Exeter, Exeter, Devon 83Ecole Normale Superieure de Lyon, CRAL, UMR CNRS 5574 3
G. Chabrier
3Ecole Normale Superieure de Lyon, CRAL, UMR CNRS 5574 38Astrophysics Group, University of Exeter, Exeter, Devon 8
N. Mayne
8Astrophysics Group, University of Exeter, Exeter, Devon 8
B. Drummond
8Astrophysics Group, University of Exeter, Exeter, Devon 8
F. Debras\({}^{,}\)
9IRAP, Université de Toulouse, CNRS, UPS, Toulouse, France 9
Received 02 August 2019; accepted 14 Nov 2019
Key Words.:**Planets and satellites: interiors - Planets and satellites: atmospheres - Planets and satellites: fundamental parameters - Planets: HD209458b - Hydrodynamics**
###### Abstract
Context:The anomalously large radii of hot Jupiters has long been a mystery. However, by combining both theoretical arguments and 2D models, a recent study has suggested that the vertical advection of potential temperature leads to an adiabatic temperature profile in the deep atmosphere hotter than the profile obtained with standard 1D models.
Aims:In order to confirm the viability of that scenario, we extend this investigation to three dimensional, time-dependent, models.
Methods:We use a 3D General Circulation Model (GCM), DYNAMICO to perform a series of calculations designed to explore the formation and structure of the driving atmospheric circulations, and detail how it responds to changes in both the upper and deep atmospheric forcing.
Results:In agreement with the previous, 2D, study, we find that a hot adiabat is the natural outcome of the long-term evolution of the deep atmosphere. Integration times of order \(1500\) years are needed for that adiabat to emerge from an isothermal atmosphere, explaining why it has not been found in previous hot Jupiter studies. Models initialised from a hotter deep atmosphere tend to evolve faster toward the same final state. We also find that the deep adiabat is stable against low-levels of deep heating and cooling, as long as the Newtonian cooling time-scale is longer than \(\sim 3000\) years at \(200\) bar.
Conclusions:We conclude that the steady-state vertical advection of potential temperature by deep atmospheric circulations constitutes a robust mechanism to explain hot Jupiter inflated radii. We suggest that future studies of hot Jupiters are evolved for a longer time than currently done, and, when possible, include models initialised with a hot deep adiabat. We stress that this mechanism stems from the advection of entropy by irradiation induced mass flows and does not require (finely tuned) dissipative process, in contrast with most previously suggested scenarios.
## 1Introduction
**The anomalously large radii of highly irradiated Jupiter-like exoplanets, known as hot Jupiters, remains one of the key unresolved issues in our understanding of extrasolar planetary atmospheres. The observed correlation between the stellar irradiation of a hot Jupiter and its observed inflation** **(for examples, see** Demory and Seager****2011; Laughlin et al.****2011; Lopez and Fortney****2016; Sestovic et al.****2018**)** **suggests that it is linked to the amount of energy deposited in the upper atmosphere. Several mechanisms have been suggested as possible explanations (see** Baraffe et al.2009, 2014; Fortney and Nettelmann 2010**, for a review). These solutions include tidal heating and physical (i.e. not for stabilisation reasons) dissipation (**Leconte et al.2010; Arras and Socrates 2010; Lee 2019**), ohmic dissipation of electrical energy (**Batygin and Stevenson 2010; Perna et al.2010; Batygin et al.2011; Rauscher and Menou 2012**), deep deposition of kinetic energy (**Guillot and Showman 2002**), enhanced opacities which inhibit cooling** **(**Burrows et al.****2007**)** **or ongoing layered convection that reduces the efficiency of heat transport** **(**Chabrier and Baraffe****2007**)****. At present time, however, there is no consensus across the community on a given scenario because the majority of these solutions require finely tuned physical environments which either deposit additional energy deep within the atmosphere or affect the efficiency of vertical heat transport.**
**Recently,** Tremblin et al.**(**2017**)****, hereafter** [2]**, suggested a mechanism that naturally arises from first physical principles. Their argument goes as follows: consider the equation for the evolution of the potential temperature** \(\Theta\)**, which is equivalent to entropy in this case:**
\[\frac{D\Theta}{Dt}=\frac{\Theta H}{Tc_{p}}\,,\] (1)
**where** \(D/Dt\) **stands for the Lagrangian derivative in spherical coordinates,** \(H\) **is the local heating or cooling rate,** \(c_{p}\) **is the heat capacity at constant pressure, and** \(\Theta\) **is defined as a function of the temperature** \(T\) **and pressure,** \(P\)**:**
\[\Theta=T\left(\frac{P_{0}}{P}\right)^{\frac{\gamma-1}{\gamma}}\,,\] (2)
**where** \(P_{0}\) **is a reference pressure and** \(\gamma\)**=**\(C_{p}/C_{v}\) **is the adiabatic index. In a steady state,** **Equation 1** **reduces to**
\[\bm{u}\cdot\nabla\Theta=\frac{\Theta H}{Tc_{p}}\,,\] (3)
**where** \(\bm{u}\) **is the velocity. In the deep atmosphere, radiative heating and cooling both tend to zero (i.e.** \(H\to 0\)**) because of large atmospheric opacities. In this case (with** \(H\to 0\)**), and if the winds do not vanish (i.e.** \(\left|\bm{u}\right|\neq 0\)**, see** **subsection 3.2****), the potential temperature** \(\Theta\) **must remain constant for** **Equation 3** **to be valid. In other words, the temperature-pressure profile must be adiabatic and satisfy the scaling:**
\[P\propto T^{\frac{\gamma}{\gamma-1}}\,.\] (4)
**We emphasise that this adiabatic solution is an equilibrium that does not require any physical dissipation. There is an internal energy transfer to the deep atmosphere, through an enthalpy flux, but there is no dissipation from kinetic, magnetic, or radiative energy reservoirs to the internal energy reservoir. Dissipative processes** \(D_{\mathrm{dis}}\) **would act as a source term with** \(\bm{u}\cdot\nabla\Theta\propto D_{\mathrm{dis}}\) **and would drive the profile away from the adiabat.**
**Physically, as discussed by** [2]**, this constant potential temperature profile in the deep atmosphere is driven by the vertical advection of potential temperature from the outer and highly irradiated atmosphere to the deep atmosphere by large scale dynamical motions where it is almost completely homogenised by the residual global circulations (which themselves can be linked to the conservation of mass and momentum, and the large mass/momentum flux the super-rotating jet drives in the outer atmosphere). The key point is that it causes the temperature-pressure profile to converge to an adiabat at lower pressures than those at which the atmosphere becomes unstable to convection. As a result, the outer atmosphere connects to a hotter internal adiabat than would be obtained through a standard, ’radiative-convective’ single column model. This potentially leads to a larger radius compared with the predictions born out of these 1D models.**
**Whilst** [2] **was able to confirm this hypothesis through the use of a 2D stationary circulation model, there are still a number of limitations to their work. Maybe most importantly, the models they used only considered the formation of the deep adiabat within a 2D equatorial slice. The steady-state temperature-pressure profiles at other latitudes remains unknown, as well as the nature of the global circulations at these high pressures in the equilibrated state. Strong ansatzes were also made about the nature of the meridional (i.e. vertical and latitudinal) wind at the equator, with their models prescribing the ratio of latitudinal to vertical mass fluxes, that could potentially affect the proposed scenario. The purpose of this paper is to reduce and constrain these assumptions and limitations and to demonstrate the viability of a deep adiabat at equilibrium. This is done by means of a series of idealised 3D GCM calculations designed such as to allow us to fully explore the structure of the deep atmospheric circulations in equilibrated hot Jupiter atmospheres, as well as investigate the time-evolution of the deep adiabat. As we demonstrate in this work, the adiabatic profile predicted by** [2] **naturally emerges from such calculations and appears to be robust against changes in the deep atmosphere radiative properties. This is the core result of this work.**
**The structure of the work is as follows. Our simulations properties are described in** **section 2****, where we introduce the GCM DYNAMICO, used throughout this study. We then demonstrate that, when using DYNAMICO, not only are we are able to recover standard features observed in previous short-timescale studies of hot Jupiter atmospheres (****subsection 3.1****), but also that, when the simulations are extended to long-enough time-scales, an adiabatic profile develops within the deep atmosphere (****subsection 3.2****). We then explore the robustness of our results by presenting a series of sensitivity tests, including changes in the outer and deep atmosphere thermal forcing (****subsection 3.3****). Finally, in** **section 4****, we provide concluding remarks, including suggestions for future computational studies of hot Jupiter atmospheres and a discussion about implications for the evolution of highly irradiated gas giants.**
## 2Method
**DYNAMICO is a highly computationally efficient GCM that solves the primitive equation of meteorology (see** Vallis 2006 **for a review and** Dubos and Voitus 2014 **for a more detailed discussion of the approach taken in DYNAMICO) on a sphere** **(**Dubos et al.****2015**)****. It is being developed as the next state--of--the art dynamical core for Earth and planetary climate studies at the Laboratoire de Météorologie Dynamique and is publicly available**¹**. It has recently been used to model the atmosphere of Saturn at high resolution** **(**Spiga et al.****2020**)****. Here, we present some specificities of DYNAMICO (section** 2.1**) as well as the required modifications we implemented to model hot Jupiter atmospheres (section** 2.2**).**
[FOOTNOTE:1][ENDFOOTNOTE]
### DYNAMICOs numerical scheme
Quantity (units) | Description | Value
---|---|---
dt (seconds) | Time-step | 120
Nz | Number of Pressure Levels | 33
d | Number of Sub-divisions | 20
N(∘) | Angular Resolution | 3.5
Ptop (bar) | Pressure at Top | 7×10−3
Pbottom (bar) | Pressure at Bottom | 200
g (m.s−2) | Gravity | 8.0
RHJ (m) | HJ Radius | 108
Ω (s−1) | HJ Angular Rotation Rate | 2.1×10−5
cp (J.kg−1.K−1) | Specific Heat | 13226.5
R (J.kg−1.K−1) | Ideal Gas Constant | 3779.0
Tinit(K) | Initial Temperature | 1800
Table 1: Parameters for Low Resolution Simulations
**Briefly, DYNAMICO takes an energy-conserving Hamiltonian approach to solving the primitive equations. This has been shown to be suitable for modelling hot Jupiter atmospheres** **(**Showman et al.****2008; Rauscher and Menou****2012**)****, although this may not be valid in other planetary atmospheres** **(**Mayne et al.****2019**)****. Rather than the traditional latitude-longitude horizontal grid (which presents numerical issues near the poles due to singularities in the coordinate system - see the review of** Williamson 2007 **for more details), DYNAMICO uses a staggered horizontal-icosahedral grid (see** Thuburn et al.2014 **for a discussion of the relative numerical accuracy for this type of grids) for which the number of horizontal cells** \(N\) **is defined by the number of subdivisions** \(d\) **of each edge of the main spherical icosahedral**²**:**
[FOOTNOTE:2][ENDFOOTNOTE]
\[N=10d^{2}+2.\] (5)
**As for the vertical grid, DYNAMICO uses a pressure coordinate system whose levels can be defined by the user at runtime. Finally, the boundaries of our simulations are closed and stress-free with zero energy transfer (i.e. the only means on energy injection and removal are the Newtonian cooling profile and the horizontal, numerical, dissipation). Note that, unlike some other GCM models of hot Jupiters (e.g.** Schneider and Liu 2009; Liu and Showman 2013; Showman et al.2019**), we do not include an additional frictional (i.e. Rayleigh) drag scheme at the bottom of our simulation domain, instead relying on the hyperviscosity and impermeable bottom boundary to stabilise the system.**
**As a consequence of the finite difference scheme used in DYNAMICO, artificial numerical dissipation must be introduced in order to stabilise the system against the accumulation of grid-scale numerical noise. This numerical dissipation takes the form of a horizontal hyper-diffusion filter with a fixed hyperviscosity and a dissipation time-scale at the grid scale, labelled** \(\tau_{dissip}\)**, which serves to adjust the strength of the filtering (the longer the dissipation time, the weaker the dissipation). Technically DYNAMICO includes three dissipation timescales, each of which either diffuses scalar, vorticity, or divergence independently. However, for our models, we set all three timescales to the same value. It is important to point out that the hyperviscosity is not a direct equivalent of the physical viscosity of the planetary atmosphere, but can be viewed as a form of increased artificial dissipation that both enhances the stability of the code, and accounts for motions, flows, and turbulences which are unresolved at typical grid scale resolutions. This is known as the large eddy approximation and has long been standard practice in the stellar (e.g.** Miesch 2005**) and planetary (e.g** Cullen and Brown 2009**) atmospheric modelling communities. Because it acts at the grid cell level, the strength of the dissipation is resolution dependent at a fixed** \(\tau_{dissip}\) **(this can be seen in our results in** **Figure 7****).**
**In a series of benchmark cases,** Heng et al.**(**2011**)** **(hereafter** [1]**) have shown that both spectral and finite-difference based dynamical cores which implement horizontal hyper-diffusion filters can produce differences of the order of tens of percent in the temperature and velocity fields when varying the dissipation strength. We also found such a similar sensitivity in our models: for example, the maximum super-rotating jet speed varies between** \(3000\,\mathrm{ms^{-1}}\) **and** \(4500\,\mathrm{ms^{-1}}\) **as the dissipation strength is varied. The dissipation strength must thus be carefully calibrated. In the absence of significant constraints on hot Jupiter zonal wind velocities, this was done empirically by minimising unwanted small-scale numerical noise as well as replicating published benchmark results (An alternative, which is especially useful in scenarios where direct or indirect data comparisons are unavailable, is to plot the spectral decomposition of the energy profile and adjust the diffusion such that the energy accumulation on the smallest scales is insignificant). We found that setting** \(\tau_{dissip}=2500\,\mathrm{s}\) **in our low resolution runs leads to benchmark cases in good agreement with the results of, for example** Mayne et al.**(**3**)****, whilst also exhibiting minimal small-scale numerical noise. This is in reasonable agreement with other studies, with our models including a hyper-diffusion of the same order of magnitude as, for example,** [1]**. Note that, due to differences in the dynamics between those of Saturn and that observed in hot Jupiters, and in particular due to the presence of the strong super-rotating jet, we must use a significantly stronger dissipation to counter grid-scale noise than that used in previous atmospheric studies calculated using DYNAMICO** **(**Spiga et al.****2020**)****.**
### Newtonian cooling
**In our simulations of hot Jupiter atmospheres using DYNAMICO, we do not directly model either the incident thermal radiation on the day-side, or the thermal emission on the night-side, of the exoplanet. This would be prohibitively computationally expensive for the long simulations we perform in the present work. Instead we use a simple thermal relaxation scheme to model those effects, with a spatially varying equilibrium temperature profile** \(T_{eq}\) **and a relaxation time-scale** \(\tau\) **that increases with pressure throughout the outer atmosphere. Specifically, this is done by adding a source term to the temperature evolution equation that takes the form:**
\[\frac{\partial T\left(P,\theta,\phi\right)}{\partial t}=-\frac{T\left(P,\theta ,\phi\right)-T_{eq}\left(P,\theta,\phi\right)}{\tau\left(P\right)}\,.\] (6)
**This method, known as Newtonian Cooling has long been applied within the 3D GCM exoplanetary community (i.e.** Showman and Guillot **(**2002**)****,** Showman et al.**(**2008**)****,** Rauscher and Menou 2010**,** Showman and Polvani **(**2011**)****,** Mayne et al.42**,** Guerlet et al.2014 **or** Mayne et al.3**), although it is gradually being replaced by coupling with simplified, but more computationally expensive, radiative transfer schemes (e.g.** Showman et al.2009**,** Rauscher and Menou 2012 **or** Amundsen et al.2016**) due to its limitations (e.g. it is difficult to use to probe individual emission or absorption features, such as non-equilibrium atmospheric chemistry or stellar activity).**
**The forcing temperature and cooling time-scale we use within our models have their basis in the profiles** Iro et al.**(**2005**)** **calculated via a series of 1D radiative transfer models. These models were then parametrised by** [1]**, who created simplified day-side and night-side profiles. The parametrisation used here is based upon this work, albeit modified in the deep atmosphere since this is the focus of our analysis. As a result, it somewhat resembles a parametrised version of the cooling profile considered by** Liu and Showman **(**2013**)****.**
**Specifically,** \(T_{eq}\) **is calculated from the pressure dependent night-side profile (**\(T_{night}\left(P\right)\)**) according to the following relation:**
(7)
**where** \(\Delta T\) **is the pressure dependent day-side/night-side temperature difference,**
\[\Delta T(P)=\left\{\begin{array}[]{ll}\Delta T_{0}&\textrm{if }P<P_{low}\\ \Delta T_{0}\log(P/P_{low})&\textrm{if }P_{low}<P<P_{high}\\ 0&\textrm{if }P>P_{high}\end{array}\right.\,,\] (8)
**in which we used** \(\Delta T_{0}=600\) **K,** \(P_{low}=0.01\) **bar and** \(P_{high}=10\) **bar. The night-side temperature profile** \(T_{night}\) **is parametrised as a series of linear interpolations in** \(\log(P)\) **space between the points**
\[\left(\frac{T}{1\textrm{K}},\frac{P}{1\textrm{ bar}}\right)=(800,10^{-6}) \textrm{, }(1100,1)\textrm{ \& }(1800,10)\,.\] (9)
**For** \(P>10\) **bar, we set** \(T_{eq}=T_{night}=T_{day}=1800\)**K.**
**Likewise, at pressures smaller than** \(10\) **bar,** \(\tau\) **is linearly interpolated, in** \(\log(P)\) **space, between the points**
\[\left(\log\left(\frac{\tau}{1\textrm{sec}}\right),\frac{P}{1\textrm{ bar}}\right)=(2.5,10^{-6})\textrm{, }(5,1)\textrm{, }(7.5,10)\textrm{ \& }( \log(\tau_{220}),220)\,.\] (10)
**For** \(P>10\) **bar, we consider a series of models that lie between two extremes: at one extreme we set** \(\log(\tau_{220})\) **(which we define as the decimal logarithm of the cooling time-scale** \(\tau\) **at the bottom of our model atmospheres: i.e. at** \(P=220\) **bar) to infinity, which implies that the deep atmosphere is radiatively inert, with no heating or cooling. As for the other extreme, this involves setting** \(\log(\tau_{220})=7.5\)**, which implies that radiative effects do not diminish below** \(10\) **bar. In** **section 3** **we explore results at the first extreme, with no deep radiative dynamics. Then, in** **subsubsection 3.3.3****, we explore the sensitivity of our results to varying this prescription.**
## 3Results
Model | Description
---|---
A | The base low resolution model, in which the deep atmosphere is isothermally initialised
B | Like model A, but with the deep atmosphere adiabatically initialised
C | Mid Resolution version of model A (d=30)
D | High Resolution version of model A (d=40)
E→I | Highly evolved versions of model A, which have reached a deep adiabat and then had deep isothermal Newtonian cooling introduced at various strengths: For E log(τ220)=7.5, F log(τ220)=11, G log(τ220)=15, H log(τ220)=20, and I log(τ220)=22.5
J & K | Highly evolved versions of model A which have reached a deep adiabat, and then had their outer atmospheric Newtonian cooling modified to reflect a different surface temperature: 1200K in model J and 2200K in model K
Table 2: Models discussed in this work
**The default parameters used with our models are outlined in** **Table 1****, with the resultant models, as well as the simulation specific parameters, detailed in** **Table 2****.**
**In** **subsection 3.2****, we use the results of models** _A_ **and** _B_ **to demonstrate the validity of the work of** [2] **in the time-dependent, three-dimensional, regime. We next explore the robustness and sensitivity of our results to numerical and external effects in** **subsection 3.3****. Note that, throughout this paper, all times are either given in seconds or in Earth years - specifically one Earth year is exactly 365 days.**
### Validation of the hot jupiter model
<figure><img src="content_image/1911.06546/latP_T_1800K_t_2_liney.png"><figcaption>(a) Temperature Contours</figcaption></figure>
**We start by exploring the early evolution of model** _A_**, testing how well it agrees with the benchmark calculations of** [1]**. The model is run for an initial period of** \(30\) **years in order to reach an evolved state before we take averages over the next five** **years** **of data. Note that this model was also used to calibrate the horizontal dissipation (**\(\tau_{dissip}\)**). In** **Figure 1****, we show zonally and temporally-averaged plots of the zonal wind and the temperature as a function of both latitude and pressure.**
**We find that the temperature (left panel) is qualitatively similar to that reported by both** [1] **and** Mayne et al.**(**3**)****. The temperature range we find (**\(\sim\!\!750\textrm{K}\rightarrow\sim\!\!2150\textrm{K}\)**) matches their results (**\(\sim 700\to 2000$\mathrm{K}$\)**) to within a 10% margin of uncertainty. This is satisfactory given the differences between the various set-ups and numerical implementations of the GCMs, as well as the variations that occur when adjusting the length of the temporal averaging window.**
**The zonal wind displays a prominent, eastward, super-rotating equatorial jet that extends from the top of the atmosphere down to approximately** **10 bar** **(Note that, as we continue to run this model for more time, the vertical extent of the jet increases, eventually reaching significantly deeper that 100 bar after 1700 years). It exhibits a peak wind velocity of** \(\approx 3500$\mathrm{m}\,\mathrm{s}^{-1}$\)**, depending upon the averaging window considered, in good agreement with the work of both** [1] **and** Mayne et al.**(**3**)** **who found peak jet speeds on the order of** \(3500\to 4000$\mathrm{m}\,\mathrm{s}^{-1}$\)**. In the upper atmosphere, it is balanced by counter-rotating (westward) flows at extratropical and polar latitudes. The zonal wind is also directed westwards at all latitudes below** \(\sim\!\!\textrm{50 bar}\)**, with this wind also contributing to the flows balancing the large mass and momentum transport of the super-rotating jet.**
**The differences we find between our models and the reference models are not unexpected. As discussed by** [1]**, the jet speed and temperature profile are indeed highly sensitive not only to the numerical scheme adopted by the GCM (i.e. spectral vs finite difference - see their Figure 12) but also to the form and magnitude of horizontal dissipation and Newtonian cooling used. In our models, unlike** [1]**, we explicitly set our deep (**\(P>10\textrm{bar}\)**) cooling to zero, which may explain the enhanced deep temperatures observed in our models, most likely an early manifestation of the deep adiabat we expect to eventually develop.**
**As noted by other works (e.g.** Menou and Rauscher 2009; Rauscher and Menou 2010; Mayne et al.3**), it takes a long time for the the deep atmosphere to reach equilibrium, and the above simulation is by no means an exception: the eastward equatorial jet extends deeper and deeper as time increases, with no sign of stopping by the end of the simulated duration. This long time-scale evolution is explored in detail in the following section.**
### The formation of a deep adiabat
<figure><img src="content_image/1911.06546/x1.png"><figcaption>(a) Isothermal Initialisation</figcaption></figure>
<figure><img src="content_image/1911.06546/x3.png"><figcaption>Figure 3: Time evolution of the equatorially averaged T–P profile withinmodel A covering the >1500 (Earth) years of simulation time required for it toreach equilibrium. The light grey dashed line shows the initial temperatureprofile for P>10 bar, whilst the dark grey line shows the forcing profile forP<10 bar. The time evolution is represented by the intensity of the lines,with the least evolved (and thus lowest visual intensity) snapshot starting att≈30 years followed by later snapshots at increments of approximately 60 years</figcaption></figure>
<figure><img src="content_image/1911.06546/wind_temp_p32_T_1800K_snap.png"><figcaption>(a) Snapshot of the zonal wind and temperature profile at P=0.72 mbar</figcaption></figure>
**As discussed by** [2]**, and in** **section 1****, an adiabatic profile in the deep atmosphere (i.e.** \(P>\sim\!\!1\rightarrow\sim\!\!\textrm{10 bar}\)**) should be a good representation of the steady state atmosphere. In order to confirm that this is the case, we performed a series of calculations with a radiatively inert deep atmosphere (i.e. no deep heating or cooling, as required by the theory of** [2]**).**
**We explore this using two models,** _A_ **and** _B_**, which only differ in both their initial condition and their duration. In model** _A_**, the atmosphere, including the deep atmosphere, is initially isothermal with** \(T\)**=**\(1800\)**K and is evolved for more than** \(1500\) **Earth years in order to reach a steady state in its** \(T\)**–**\(P\) **profile (as shown in** **(a)****). As a consequence of the long time-scales required for the model to reach equilibrium,** **and the computational cost of such an endeavour, model** _A_ **(and** _B_**) is run at a relatively low resolution**³**. We will investigate the sensitivity of our results to spatial resolution in** **subsubsection 3.3.1****. As for model** _B_**, it is identical to model** _A_ **except in the deep atmosphere, where it is initialised with an adiabatic** \(T\)**–**\(P\) **profile for** \(P\)**¿**\(10\) **bar. As a result of this model being initialised close to the expected equilibrium solution, model** _B_ **was then run for only** \(100\) **years in order to confirm the stability of the steady-state. In both cases, we find that the simulation time considered is long enough such that the thermodynamic structure of the atmosphere has not changed for multiple advective turnover times** \(t_{adv}\sim 2\pi R_{HJ}/u_{\phi}\)**.**
**Figure 2** **shows that both models have evolved to the same steady state: an outer atmosphere whose** \(T-P\) **profile is dictated by the Newtonian cooling profile, and a deep adiabat which is slightly hotter** \(\left(\sim\!\!1900\textrm{K}\right)\) **than the cooling profile at** \(P=\textrm{10 bar}\)****\(\left(1800\textrm{K}\right)\)**. This is reinforced by the latitudinal and longitudinal temperature profile throughout the simulation domain. In** **Figure 4** **we plot the zonal wind and temperature profile at three different heights (pressures). Here we can see that, in the outer atmosphere (panels** _a_ **and** _b_**) the profile is dominated by the newtonian cooling, with horizontal advection (and the resulting offset hotspot) starting to become significant as we move towards middle pressures. As for the deep atmosphere (snapshot in panel** _c_ **and time average in panel** _d_**), here we start to see evidence of both the heating and near-homogenisation of the deep atmosphere. Note that we refer to the atmosphere as nearly homogenised because the temperature fluctuations at, for example,** \(P=10\) **bar are less than** \(1\%\) **of the mean temperature.**
**Importantly, this convergence to as deep adiabat not only occurs in the absence of vertical convective mixing (an effect which is absent from our models, which contain no convective driving), but also at a significantly lower pressure** \(\left(P=\textrm{10 bar}\right)\) **than the pressure (**\(\sim\)**40 bar for HD209458b -** Chabrier et al.2004**) at which we would expect the atmosphere to become unstable to convection (and so, in the traditional sense, prone to an adiabatic profile).**
**Therefore, the characteristic entropy profile of the planet is warmer than the entropy profiles calculated from standard 1D irradiated models. We will discuss the implications of this result for the evolution of highly irradiated gas giant in** **section 4****.**
**In model** _A_**, the steady state described above is very slow to emerge from an initially isothermal atmosphere. This is illustrated in** **Figure 3** **which shows the time evolution of the** \(T\)**-**\(P\) **profile. It takes more than 500 years of simulation time to stop exhibiting a temperature inversion in the deep atmosphere, let alone the** \(>\)**1500 years required to reach the same steady state as model** _B_**.**
[FOOTNOTE:3][ENDFOOTNOTE]
**As will be further discussed in** **section 4****, this slow evolution of the deep adiabat is probably one of the main reason why this result has not been reported by prior studies of hot Jupiter atmospheres.**
**The mechanism advocated by** [2] **relies on the existence of vertical and latitudinal motions that efficiently redistribute potential temperature. In order to determine their spatial structure, we plot in** **Figure 5** **the zonally and temporally averaged meridional mass-flux stream function and zonal wind velocity for model** _A_**.**
**Starting with the zonal wind profile (grey lines) we can see evidence for a super-rotating jet that extends deep into the atmosphere, with balancing counter flows at the poles and near the bottom of the simulation domain. In the deep atmosphere, this jet has evolved with the deep adiabat, extending towards higher pressures as the developing adiabat (almost) homogenises (and hence barotropises) the atmosphere. This barotropisation on long timescales seems similar to the drag-free simulation started from a barotropic zonal wind in** Liu and Showman **(**2013**)******
**The meridional mass-flux stream function is defined according to**
\[\Psi=\frac{2\pi R_{HJ}}{g\cos{\theta}}\int_{P_{top}}^{P}u_{\phi}\,dP.\] (11)
**We find that the meridional (latitudinal and vertical) circulation profile is dominated by four vertically aligned cells extending from the bottom of our simulation atmospheres to well within the thermally and radiatively active region located in the upper atmosphere. These circulation cells lead to the formation of a strong, deep, down-flow at the equator (which can be linked to the high equatorial temperatures in the upper atmosphere), weaker, upper atmosphere, downflows near the poles, and a mass conserving pair of upflows at mid latitudes (**\(\theta=20^{\circ}\to 30^{\circ}\)**). The meridional circulation not only leads to the vertical transport of potential temperature (as high potential temperature fluid parcels from the outer atmosphere are mixed with their ‘cooler’ deep atmosphere counterparts), but also to the almost complete latitudinal homogenisation of the deep atmosphere (with only small temperature variations remaining). In a fully radiative model, these circulations would also mix the outer atmosphere, leading to the equilibrium temperature profiles we instead impose via Newtonian cooling (see, for example,** Drummond et al.2018, 2018 **for more details about the 3D mixing in radiative atmospheres).**
**Note that the vertical extent the zonal wind, and the structure of the lowest cells in the mass-flux stream function, appear to be affected by the bottom boundary, suggesting that they extend deeper into the atmosphere. Whilst this is interesting and important, it should not affect the final state our P-T profiles reach, but does suggest that models of hot-Jupiters should be run to higher pressures to fully capture the irradiation driven deep flow dynamics.**
****
**The primary driver of the latitudinal homogenisation are fluctuations in the meridional circulation profile, which are visible within individual profile snapshots, but are averaged out when we take a temporal average. This includes contributions from spatially small-scale velocity fluctuations at the interface of the large-scale meridional cells. Evidence for these effects can be seen in snapshots of the zonal and meridional flows, in an RMS analysis of the zonal velocity, and of course in deep temperature profile that these advective motions drive. The first reveals complex dynamics, such as zonally-asymmetric and temporally variable flows, that are hidden when looking at the temporal average, but which mask the net flows when looking at a snapshot of the circulation. The second reveals spatial and temporal fluctuations on the order of** \(5\to 10$\mathrm{m}\,\mathrm{s}^{-1}$\) **in the deep atmosphere. Finally the third (as plotted in panels** _c_ **and** _d_ **of** **Figure 4****, which show snapshots or the time average of the zonal wind and temperature profile, respectively) reveals small scale temperature and wind fluctuations, which are likely associated with the deep atmosphere mixing, that are lost when looking at the average, steady, state.**
**However, a more detailed analysis of the dynamics of this homogenisation, as well as the exact nature of the driving flows and dynamics, is beyond the scope of this paper. Although interesting in its own right, the mechanism by which the circulation is set up in the deep atmosphere of our isothermally initialised simulations might not be relevant to the actual physical mechanism happening in hot Jupiters with hot, deep, atmospheres.**
<figure><img src="content_image/1911.06546/streamfunction_lat_pressure_T_1800K_alt_neo2.png"><figcaption>Figure 5: Zonally and temporally-averaged (over a period of ≈30 years) stream-function for model A. Clockwise circulations on the meridional plane are shownin red and anticlockwise circulations are shown in blue. Additionally thezonally and temporally averaged zonal wind is plotted in black (solid =eastward, dashed = westward).</figcaption></figure>
<figure><img src="content_image/1911.06546/x4.png"><figcaption>(a) Longitudinal Variations</figcaption></figure>
**As a consequence of both the meridional circulations described above, and the zonal flows that form as a response to the strong day-side/night-side temperature differential, the deep atmosphere** \(T\)**–**\(P\) **profile is independent of both longitude (****(a)****) and latitude (****(b)****). Only in the upper atmosphere (**\(P<10\) **bar) do the temperature profiles start to deviate from one another, reflecting the zonally and latitudinally varying Newtonian forcing. Taken together, the two panels of** **Figure 6** **confirm that the latitudinal and vertical steady-state circulation, the super-rotating eastward jet, and any zonally-asymmetric flows act to advect potential temperature throughout the deep atmosphere, leading at depth to the formation of a hot adiabat without the need for any convective motions.**
### Robustness of the results
**Having confirmed that a deep adiabatic temperature profile connecting with the outer atmospheric temperature profile at** \(P=\textrm{10 bar}\) **is a good representation of the steady state within our hot Jupiter model atmospheres, we now explore the robustness of this result.**
#### 3.3.1Sensitivity To Changes In The Horizontal Resolution
<figure><img src="content_image/1911.06546/x6.png"><figcaption>Figure 7: Equatorially averaged T-P profile snapshots for three initiallyisothermal (see grey dashed line in the deep atmosphere) models run with thesame dissipation time (tdissip=2500s), vertical resolution, and Newtoniancooling profile (dark grey), but different horizontal resolutions (Models A(yellow), C (light green), and D (orange)).</figcaption></figure>
**We start our exploration of the robustness of our results by confirming that the eventual convergence of the deep atmosphere on to a deep adiabat appears resolution independent.**
**Figure 7** **shows the** \(T\)**–**\(P\) **profiles obtained for three models at the same time (**\(t\approx 1800\) **years) but with different resolutions (our ‘base’ resolution model,** _A_**, a ‘mid-res’ model,** _C_**, and a ‘high-res’ model,** _D_**). The mid resolution model (**_C_**) has almost reached the exact same equilibrium adiabatic profile as the low resolution case (**_A_**): comparing this with the time-evolution of model** _A_ **(****Figure 3****) confirms that they are both on the path to the same equilibrium state, and that a significant amount of computational time would be required to reach it. This becomes even clearer when we look at a high resolution model (**_D_**). Here we find that, despite the long time-scale of the computation, the deep atmosphere still exhibits a temperature inversion, suggesting, in comparison to** **Figure 3****, that the model has a long way to go until it reaches the same, deep adiabat, equilibrium.**
**In general, we have found the better the resolution the more slowly the atmosphere temperature profile evolves** **towards the adiabatic steady state solution. This stems most likely from the fact that horizontal numerical dissipation, on a fixed dissipation time-scale, decreases with increasing resolution. Note that we kept the horizontal dissipation timescale constant due to both the computational expense of the parameter study required to set the correct dissipation at each resolution, and the numerical dissipation independence of the steady-state in the deep atmosphere.**
**Evidence for the impact of the small-scale flows on this slow evolution can be seen in the temporal and spatial RMS profiles of the zonal flows, which reveal that, as we increase the resolution by a factor 2, the magnitude of the small-scale velocity fluctuations decreases by roughly the same factor. These results are in agreement with the effect of changing the numerical dissipation timescale (**\(\tau_{\textrm{dissip}}\)**) at a fixed resolution, where longer timescales also slow down the circulation, thereby increasing the time required to reach a steady** \(T\)**-**\(P\) **profile in the deep atmosphere (not shown). Despite these numerical limitations, it remains clear that the, the presence, and strength, of any numerical dissipation does not affect the steady state solutions of the simulation, which remains as an adiabatic P-T profile in the deep atmosphere.**
#### 3.3.2Sensitivity to changes in the upper atmosphere forcing function
<figure><img src="content_image/1911.06546/x7.png"><figcaption>Figure 8: Equatorially averaged T–P profiles for three models: A (green), J(yellow) and K (orange). The orange (K) and yellow (J) models have had theirouter atmosphere cooling modified such that Teq=2200 K or 1200 K,respectively. The solid lines represent the equilibrium T–P profiles whilstthe dashed lines represent the T–P profiles 200 years after the outeratmospheres forcing was adjusted (shown in dark grey for each model). Notethat, after 200 years of ‘modified’ evolution, only the 2200K model has notreached equilibrium.</figcaption></figure>
**We next explore how the deep adiabat responds to changes in the outer atmosphere irradiation and thermal emission (via the imposed Newtonian cooling). The aim is not only to test the robustness of the deep adiabat, but also to explore the response of the adiabat to changes in the atmospheric state. As part of this study, the two scenarios we consider were initialised using the evolved adiabatic profile obtained in model A, but with a modified outer atmosphere cooling profile such that** \(T_{night}=1200\)**K (model** _J_**) or** \(T_{night}=2200\)**K (model** _K_**).** **Figure 8** **shows the equilibrium** \(T\)**–**\(P\) **profiles (solid lines) as well as snapshots of the** \(T\)**–**\(P\) **profiles after only** \(200\) **years of ‘modified’ evolution (dashed lines). It also includes a plot of model** _A_ **to aid comparison.**
**Model** _J_ **evolves in less than 200 years towards a new steady state profile that corresponds to the modified cooling profile. The deep adiabat reconnects with the outer atmospheric profile at** \(P=\textrm{10 bar}\) **and** \(\sim\!\textrm{1250K}\) **(in agreement with the relative offset found in our** \(1800\textrm{K}\) **models,** _A_ **and** _B_**). The meridional mass circulation (not shown) displays evidence for the same qualitative flows driving the vertical advection of potential temperature as models** _A_ **and** _B_**. However it also shows signs that it is still evolving, suggesting that the steady state meridional circulation takes longer to establish than the vertical temperature profile.**
**In model** _K_**, we find that,** \(200\) **years after modifying the outer atmospheres cooling profile, the deep atmosphere has not yet reached a steady state. In fact it takes approximately** \(1000\) **years of evolution for it to reach equilibrium, which we show as a solid line in** **Figure 8****. This confirms that, model** _K_**, although slow to evolve relative to the cooling case (model** _J_**), does eventually settle onto a deep, equilibrium, adiabat. Additionally, this adjustment occurs significantly faster than the equivalent evolution of a deep adiabat from an isothermal start.**
**Based on the results of this section, we conclude that it is faster for the deep atmosphere to cool than to warm when it evolves toward its adiabatic temperature profile. In order to understand this time-scale ordering, we have to note that the only way for the simulation to inject or extract energy is through the fast Newtonian forcing of the upper atmosphere and also that the thermal heat content of the deep atmosphere is significantly larger than that of the outer layers. The deep (**\({}_{d}\)**) and upper (**\({}_{u}\)**) atmospheres are connected by the advection of potential temperature that we will rewrite in a conservative form as an enthalpy flux:** \(\rho c_{p}Tu\) **and we simplify the process to two steps between the two reservoirs (assuming they have similar volumes): injection/extraction by enthalpy flux and Newtonian forcing in the upper atmosphere.**
* **In the case of cooling, the deep atmosphere contains too much energy and needs to evacuate it. It will setup a circulation to evacuate this extra-energy to the upper layers with an enthalpy flux that would lead to an upper energy content set by** \(\rho_{u}c_{v}T_{u}\sim\rho_{u}c_{v}T_{u,init}+\rho_{d}c_{v}(T_{d,init}-T_{d,eq})\) **if we ignore first Newtonian cooling.** \(T_{u}\) **would then be very large essentially because of the density difference between the upper and lower atmosphere. The Newtonian forcing term proportional to** \(-(T_{u}-T_{u,eq})/\tau\) **is then very large and can efficiently remove the energy from the system.**
* **In the case of heating, the deep atmosphere does not contain enough energy and needs an injection from the upper layers. This injection is coming from the Newtonian forcing and can at first only inject** \(\rho_{u}c_{v}(T_{u,eq}-T_{u,init})\) **in the system. The enthalpy flux will then lead to an energy content in the deep atmosphere given by** \(\rho_{d}c_{v}T_{d}\sim\rho_{d}c_{v}T_{d,init}+\rho_{u}c_{v}(T_{u,eq}-T_{u,init})\) **if we assume that all the extra-energy is pumped by the deep atmosphere. Because of the density difference and the limited variations in the temperature caused by the forcing, the temperature change in the deep atmosphere is small and will require more injection from the upper layers to reach equilibrium. However, even in the most favourable scenario in which all the extra energy is transferred, the Newtonian forcing cannot exceed** \(-(T_{u,init}-T_{u,eq})/\tau\) **which explains why it will take a much longer time to heat the deep atmosphere than to cool it.**
#### 3.3.3Sensitivity to the addition of newtonian cooling to the deep atmosphere
<figure><img src="content_image/1911.06546/x8.png"><figcaption>Figure 9: Newtonian cooling relaxation time-scale profiles used in the modelsshown in Figure 10. Note that a smaller value of τ means more rapid forcingtowards the imposed cooling profile (which in all cases is isothermal in thedeep atmosphere, where P>10 bar), and that the relaxation profiles areidentical for P<10 bar (grey line).</figcaption></figure>
<figure><img src="content_image/1911.06546/x9.png"><figcaption>Figure 10: Snapshots of the T–P profile for five, initially adiabaticsimulations (coloured lines - based on model B, and with the same outeratmosphere cooling profile (dark grey)) which are then forced to a deepisothermal profile (grey dashed line) with varying log(τ220) (Equation 10).</figcaption></figure>
<figure><img src="content_image/1911.06546/streamfunction_lat_pressure_T_1900K_alt_15_neo.png"><figcaption>(a) log(τ220)=15</figcaption></figure>
**It is unlikely that the atmosphere will suddenly turn thermally inert at pressures greater than** \(10\) **bar. Rather, we expect the thermal time-scale will gradually increase with increasing pressure. In this section, we examine the sensitivity of the deep atmospheric flows, circulations, and thermal structure to varying levels of Newtonian cooling. Additionally we are motivated to quantify the maximum amount of Newtonian cooling under which the deep atmosphere is still able to maintain a deep adiabat.**
**To explore this, we consider five models each with different cooling time-scales at the bottom of the atmosphere (i.e. five different values of** \(\log(\tau_{220})\)**). From this, we can then linearly interpolate the relaxation time-scale in** \(\log(P)\) **space between** \(10\) **and** \(220\) **bar. The resultant profiles are plotted in** **Figure 9****, and can be split into three distinct groups: 1) The relaxation profile with** \(\log(\tau_{\textrm{220}})=7.5\) **(model** _E_**) represents a case with rapid Newtonian cooling that does not decrease with increasing pressure; 2) The case** \(\log(\tau_{\textrm{220}})=11\) **(model** _F_**) is a simple linear continuation of the relaxation profile we use between** \(P=\textrm{1 bar and 10 bar}\)**. It is the simplest possible extrapolation of the upper atmosphere thermal time-scale profile, and likely represents the strongest realistic forcing in the deep atmosphere; 3) The remaining relaxation profiles,** \(\log(\tau_{\textrm{220}})=\textrm{15, 20, 22.5}\) **(models** _G, H_ **and** _I_**), represent heating and/or cooling processes that get progressively slower in the deep atmosphere, in accordance with expectations born out from 1D atmospheric models of hot Jupiter atmospheres (see, for example,** Iro et al.2005**).**
**The results we obtained are summarised by the** \(T\)**–**\(P\) **profiles we plot in** **Figure 10****. For low levels of heating and cooling in the deep atmosphere (models** _G, H_ **and** _I_**), the results are almost indistinguishable from models** _A_ **and** _B_**, with only a decrease in the outer atmosphere connection temperature of a few Kelvin in model** _G_**. We find a more significant reduction in the temperature of the** \(T\)**–**\(P\) **when we investigate model** _F_**, in which we set** \(\log(\tau_{\textrm{220}})=11\)**. In particular, there is a deepening of the connection point between the outer atmosphere and the deep adiabat, which only becomes apparent for** \(P>20\) **bar in this model. This result suggest that model** _F_ **falls near the pivot point between models in which the deep atmosphere is adiabatic and those that relax toward the imposed temperature profile. This is confirmed by model** _E_**, in which** \(\tau_{\textrm{220}}=7.5\)**, where we find that the deep adiabat has been rapidly destroyed (in** \(<\textrm{30 years}\)**), such that the deep** \(T\)**–**\(P\) **profile corresponds to the imposed cooling profile throughout the atmosphere. This occurs because the Newtonian time-scale has become smaller than the advective time-scale, which means that the imposed temperature profiles dominates over any dynamical effects.**
**Before closing this section, let us briefly comment on the meridional circulation profiles obtained in those models that converge onto a similar deep adiabatic temperature profile (models** _G, H_ **and** _I_**). For all of them, we recover the same qualitative structure we found for model** _A_**, characterised by meridional cells of alternating direction that extend from the deep atmosphere to the outer regions. The finer details of the circulations, however, differ from the ones seen in model** _A_**. This is illustrated in** **Figure 11** **which displays the meridional circulation and zonal flow profiles for models** _G_ **(****(a)****) and** _H_ **(****(b)****). As the Newtonian cooling becomes faster in the deep atmosphere, the number of meridional cells increases (see also** **Figure 5****), to the point that, in model** _E_**, no deep meridional circulation cells exists and the deep circulation profile is essentially unstructured.** **Despite these differences in the shape of the meridional circulation, the steady state profiles obtained in these simulations in the deep atmosphere is again an adiabatic PT profile provided the Newtonian cooling is not (unphysically) strong.**
## 4Conclusion and discussion
<figure><img src="content_image/1911.06546/x10.png"><figcaption>Figure 12: Evolution of the sub-stellar point (i.e. day-side) Temperature-Pressure profile in a simulation (detailed in Amundsen et al. 2016) calculatedusing the Met Office GCM, the Unified Model, (Mayne et al. 2014a) andincluding a robust two-stream radiation scheme (Amundsen et al. 2014). Here weshow snapshots of the T-P profile at 0.25 (purple), 2.5 (green), and 25(orange) Earth years, along with two example adiabats (grey dotted and dashedlines) designed to show how the deep atmosphere gets warmer and connects tosteadily warmer adiabats as the simulation progresses. Note that thisprogression is, at the end of the simulated time, ongoing towards a deep, hot,adiabat, albeit at an increasingly slow rate.</figcaption></figure>
### Conclusions of the simulation results
**By carrying out a series of 3D GCM simulations of irradiated atmospheres, we have shown in the present paper that:**
* **If the deep atmosphere is initialised on an adiabatic PT profile, it remains, as a steady state, on this profile,**
* **If the deep atmosphere is initialised on a too hot state, it rapidly cools down to the same steady state adiabatic profile,**
* **If the deep atmosphere is initialised on a too cold state, it slowly evolves towards the steady state adiabatic profile.**
**Furthermore, in all the above cases, the deep adiabat forms at lower pressures that those at which we would expect, from 1D models, the atmosphere to be convectively unstable. We have also shown that this steady-state adiabatic profile is stable to changes in the deep Newtonian cooling and is independent of the details of the flow structures, provided that the velocities are not completely negligible. The hot adiabatic deep atmosphere is the natural final outcome of the simulations, for various resolutions, even though the time-scale to reach steady-state is longer at higher resolution when starting from a too cold initial state.**
**When the simulations are initialised on a too cold profile, the time-scale to reach the steady state is of the order of** \(t\sim\textrm{1000 years}\)**, explaining why the** **formation of a deep adiabat has not been seen in previous GCM studies: this time-scale far outstrips the time taken for the outer atmosphere to reach an equilibrium state (**\(t<\sim\!\textrm{$1$ year}\) **for** \(P<\textrm{1 bar}\)**). As a result, the vast majority of published GCM models only contain a** _partially evolved deep atmosphere_**, the structure of which is directly comparable to the early outputs of our isothermally initialised calculation. Examples of this early evolution of the deep atmosphere towards a deep adiabat (as seen in the early outputs plotted in** **Figure 3****) include Figure 6 of** Rauscher and Menou **(**2010**)** **(where the deep temperature profile shows signs of heating from its initial isothermal state, albeit only on the irradiated side of the planet), Figure 7 of** Amundsen et al.**(**2016**)** **(where we see a clear shift from their initially isothermal deep atmosphere towards a deep adiabat), and Figure 8 of** Kataria et al.**(**2015**)** **(where we again see a temperature inversion and a push towards a deep adiabat for Wasp-43b). It is tempting to think that if these simulations were run longer, they would evolve to a similar, deep adiabatic structure (with a corresponding increase in the exoplanetary radius). In order to investigate this possibility, we have extended the model of** Amundsen et al.2016**, run with the Unified Model of the Met Office (which includes a robust two-stream radiation scheme that replaces the Newtonian Cooling in our models), for a total of** \(\approx 25\) **Earth years. The results are shown in** **Figure 12****, where we plot the pressure-temperature profile at three different times, along with examples of the approximate deep adiabat that best matches each snapshot. We see that the deep atmosphere rapidly converges towards a deep adiabat with further vertical advection of potential temperature warming up this adiabat as the** **simulation goes on. Since this process keeps going on during the simulation, the result not only reinforces our conclusions but suggests that our primary Newtonian cooling profile represents a reasonable approximation of the incident irradiation and radiative loss.**
**The results obtained in the present simulations suggest that future hot Jupiter atmosphere studies should be initialised with a hot, deep, adiabat starting at the bottom of the surface irradiation zone (**\(P\sim\!\!\textrm{10 bar}\) **for HD209458b). Furthermore, in a situation where the equilibrium profile in the deep atmosphere is uncertain, we suggest that this profile should be initialised with a hotter adiabat than expected rather than a cooler one. The simulation should then be run long enough for the deep atmosphere to reach equilibrium. This is in agreement with the results of** Amundsen et al.**(**2016**)****, who also suggested that future GCM models should be initialised with hotter profiles than currently considered. For instance, recent 3D simulations of HD209458b have been initialised with a hotter interior T-P profile (for example, one of their models is initialised with an isotherm that is** \(800\) **K hotter than typically used in GCM studies, thus bringing the deep atmosphere closer towards its deep adiabat equilibrium temperature), and show important differences, on the time-scales considered, between the internal dynamics obtained with this set-up, and the ones obtained with a cooler, more standard, deep atmospheric profile (see,** Lines et al.2018, 2018, 2019**). Using aforementioned more correct atmosphere initial profiles should not only bring these models towards a more physical hot Jupiter parameter regime (with then a correct inflated radius), but also provide a wealth of information on how the deep adiabat responds to changes in parameter and computational regime.**
### Evolution of highly irradiated gas giants
**The results obtained in the present GCM simulations have strong implications for our understanding of the evolution of highly irradiated gas giants. As just mentioned, we first emphasise that simulations initialised from a too cold state are not relevant for the evolution of inflated hot Jupiters (although it could be of some interest for re-inflation, but this is beyond the scope of this paper). Indeed, inflated hot Jupiters are primarily in a hot initial state and, as far as the evolution is concerned, only the steady state of the atmosphere matters. The shorter timescales needed to reach this steady state are irrelevant for the evolution (with a typical Kelvin Helmholtz timescale of** \(\sim 1\textrm{Myr}\)**).**
**As shown in the present simulations, provided they are run long enough, hot Jupiter atmospheres converge at depth, i.e. in the optically thick domain, to a hot adiabatic steady-state profile without the need to invoke any dissipation mechanism such as ohmic, or kinetic energy, dissipation. These 3D dynamical calculations thus confirm the 2D steady-state calculations of** [2]**. Importantly enough, the transition to an adiabatic atmospheric profile occurs at lower pressures than the ones at which the medium is expected to become convectively unstable (thus adiabatic according to the Schwarzchild criterion). This means that the planet lies on a hotter internal entropy profile than suggested by 1D irradiation models, yielding a larger radius. The mechanism of potential temperature advection in the atmosphere of irradiated planets thus provides a robust solution to the radius inflation problem.**
**As mentioned previously, almost all scenarios suggested so far to resolve the anomalously inflated planet problem rely on the (uncomfortable) necessity to introduce finely tuned parameters. This is true, in particular, for all the different dissipation mechanisms, whether they involve kinetic energy, or ohmic and tidal dissipation. This is in stark contrast with the present mechanism, in which** _entropy_ **(potential temperature) is advected from the top to the bottom of the atmosphere. High entropy fluid parcels are moved from the upper to the deep atmosphere and toward high latitude while low entropy fluid parcels come from the deep atmosphere and are deposited in the upper atmosphere. This gradually changes the entropy profile until a steady state situation is obtained. Although an enthalpy (and mass and momentum) flux is associated with this process, down to the bottom of the atmosphere (characterised by some specific heat reservoir), this does not require a dissipative process (from kinetic, magnetic or radiative energy reservoirs into the internal energy reservoir).**
**In order to characterise this deep heating flux, and confirm that our hot, deep, adiabat would not be unstable due to high temperature radiative losses, we also explored the vertical enthalpy flux in our model and compared it to the radiative flux, as calculated for a deep adiabat using ATMO (**[2]**). This analysis reveals that the vertical enthalpy flux dominates the radiative flux at all** \(P>1\) **bar: For example, averaging over a pressure surface at** \(P=10\) **bar, we find a net vertical enthalpy flux (**\(\rho c_{p}Tu_{z}\)**) of** \(-1.04\times 10^{8}\mathrm{ergs^{-1}cm^{-2}}\) **compared to a outgoing radiative flux of** \(7.68\times 10^{6}\mathrm{ergs^{-1}cm^{-2}}\)**, suggesting that any deep radiative losses are well compensated by energy (enthalpy) transport from the highly irradiated outer atmosphere. This result is reinforced by UM calculation we show in** **Figure 12****, which intrinsically includes this deep radiative loss and show no evidence of cooling due to deep radiative effects.**
**This (lack of a requirement for additional dissipative processes) is of prime importance when trying to understand the evolution of irradiated planets. Whereas dissipative processes imply an extra energy source in the evolution (**\(\int_{M}{\dot{\epsilon}}dm\)**, where** \({\dot{\epsilon}}\) **is the energy dissipation rate, to be finely tuned), to slow down the planet’s contraction, there is no need for such a term in the present process. Indeed, as an** _isolated_ **substellar object (i.e. without nuclear energy source) cools down, its gravitational potential energy is converted into radiation at the surface, with a flux** \(\sigma T_{\mathrm{eff}}^{4}\)**. Let us now suppose that the same object is immersed into an isotropic medium characterised by a pressure** \(P=\)**220 bars and a temperature** \(T\sim 4000\) **K, typical conditions in the deep atmosphere of 51Peg-B like hot Jupiters.** **Once the object’s original inner adiabat (after its birth) has cooled down to 4000 K at 220 bars, the thermal gradient between the external and internal media will be null, which essentially reduces the local convective flux and the local optically thick radiative flux to zero. Thus the cooling flux will be reduced to almost zero. At this point, the core cannot significantly cool any more and is simply in thermal equilibrium with the surrounding medium. Both the contraction and the cooling flux are essentially insignificant:** \(dR/dt\approx 0,\sigma T_{\mathrm{int}}^{4}\approx 0\)**, in which we define** \(\sigma T_{\mathrm{int}}^{4}\) **as the radiative and convective cooling flux at the interior-atmosphere boundary. Indeed, convection will become inefficient in transporting energy and any remaining radiative loss in the optically thick deep core will be compensated by downward energy transport from the hot outer atmosphere. For a highly irradiated gas giant, the irradiation flux is not isotropic, but the combination of irradiation and atmospheric circulation will lead to a similar situation, with a deep atmosphere adiabatic profile of** \(\sim\)**4000 K at 220 bars for all latitudes and longitudes. Therefore, the planet’s interior does not significantly cool any more and we also have** \(\sigma T_{\mathrm{int}}^{4}\approx 0\)**. The evolution of the planet is stopped (**\(dS/dt\approx 0\)**), or let say its cooling time is now prohibitively long, and the planet lies on a constant adiabat determined by the equilibrium between the inner and atmospheric ones at the interior-atmosphere boundary. The situation will last as long as the planet-star characteristics will remain the same, illustrating the robustness of the potential temperature advection mechanism to explain the anomalous inflation of these bodies.**
**Therefore, the irradiation induced advection of potential temperature appears to be the most natural and robust processes to resolve the radius-inflation puzzle. Note that, this does not exclude other processes (e.g. dissipative ones) from operating within hot Jupiter atmospheres, but they are unlikely to be the dominant mechanisms responsible for the radius inflation.**
###### Acknowledgements.
**FSM and PT would like to acknowledge and thank the ERC for funding this work under the Horizon 2020 program project ATMO (ID: 757858). NJM is part funded by a Leverhulme Trust Research Project Grant and partly supported by a Science and Technology Facilities Council Consolidated Grant (ST/R000395/1). JL acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 679030/WHIPLASH). GC was supported by the Programme National de Planétologie (PNP) of CNRS-INSU cofunded by CNES. IB thanks the European Research Council (ERC) for funding under the H2020 research and innovation programme (grant agreement 787361 COBOM). FD thanks the European Research Council (ERC) for funding under the H2020 research and innovation programme (grant agreement 740651 NewWorlds). BD acknowledges support from a Science and Technology Facilities Council Consolidated Grant (ST/R000395/1).**
**The authors also wish to thank Idris, CNRS, and Mdls for access to the supercomputer Poincare, without which the long time-scale calculations featured in this work would not have been possible. The calculations for Appendix A were performed using Met Office software. Additionally, the calculations in Appendix A used the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment is funded by BIS National E-Infrastructure capital grant ST/K000373/1 and STFC DiRAC Operations grant ST/K0003259/1. DiRAC is part of the National E-Infrastructure.**
**Finally the authors would like to thank the Adam Showman for their careful and insightful review of the original manuscript.**
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|
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] | # Ultraviolet energy dependence of particle production sources in relativistic heavy-ion collisions
Georg Wolschin
g.wolschin@thphys.uni-heidelberg.de
Institut für Theoretische Physik der Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany, EU
February 28, 2024
###### Abstract
The energy dependence of particle production sources in relativistic heavy-ion collisions is investigated from RHIC to LHC energies. Whereas charged-hadron production in the fragmentation sources follows a \(\ln(s_{NN}/s_{0})\) law, particle production in the mid-rapidity gluon-gluon source exhibits a much stronger dependence \(\propto\ln^{3}(s_{NN}/s_{0})\), and becomes dominant between RHIC and LHC energies. The production of particles with pseudorapidities beyond the beam rapidity is also discussed.
pacs: 25.75.-q,24.10.Jv,24.60.-k †
[FOOTNOTE:†][ENDFOOTNOTE]
## I Introduction
The investigation of charged-hadron production in relativistic heavy-ion collisions has generated a vast amount of energy- and centrality-dependent data at energies reached at both, the Relativistic Heavy-Ion Collider RHIC Alver _et al._ (2011), and the Large Hadron Collider LHC Cole (2014). It has been shown Wolschin (2013) within the framework of a nonequilibrium-statistical relativistic diffusion model (RDM) Wolschin (1999, 2004) that the energy-dependent multiplicity of produced charged hadrons is well understood quantitatively based on a mid-rapidity low-\(x\) gluonic source and the two fragmentation sources. This applies not only to AuAu collisions at RHIC Wolschin _et al._ (6) and PbPb at LHC Röhrscheid and Wolschin (2012), but also to asymmetric systems such as dAu at RHIC Wolschin _et al._ (8) and \(p\)Pb at LHC Wolschin (2013).
The relativistic diffusion model is in scope and character located between the (equilibrium) statistical model for multiple hadron production that was proposed by Fermi Fermi (1950) and Hagedorn Hagedorn (1968), and much more detailed numerical models that aim at a microscopic description of the collision, such as the Color Glass Condensate (CGC, see Gelis _et al._ (2010)) for the initial state, hydrodynamics for the main part of the time evolution (e.g. Koide _et al._ (2007); Luzum and Romatschke (2008); Alver _et al._ (2010); Heinz and Snellings (2013)), and codes like URQMD for the final state Bass _et al._ (2013).
The statistical hadronization (or thermal) model has been further developed and compared to a large amount of data by many authors such as Braun-Munzinger et al. or Becattini et al. Manninen and Becattini (2008); Braun-Munzinger _et al._ (1995); Andronic _et al._ (2006), and it has consistently – with only few exceptions – provided good descriptions of particle production yields, in particular, at mid rapidity. As a consequence of its ambition to account for particle production with few parameters (temperature, chemical potential, characteristic volume) in an equilibrium setting with collective expansion, the thermal model does, however, not describe effects such as the plateau occurring in rapidity distributions \(dN/dy\) of produced particles at higher (RHIC and above) energies, the corresponding dip in pseudo rapidity \(dN/d\eta\), and other outstanding features such as limiting fragmentation at RHIC and LHC energies.
To account for such non-equilibrium effects and model the collision in full detail requires in current scenarios to match the CGC initial state smoothly to viscous hydrodynamics when the coupling constant becomes too strong in the course of the time development for perturbative QCD techniques to be applicable Gale _et al._ (2013), and finally use Cooper-Frye freeze out Cooper and Frye (1974) or another code that accounts for the final-state interactions Bass _et al._ (2013).
However, since even the most sophisticated codes that purport to describe the full time evolution will contain a certain amount of arbitrariness and can not fully replace the experiment, it appears indicated to permit phenomenological models such as the RDM that include non-equilibrium effects to some extent, reproduce substantial features of the data and have some predictive power, but do not claim to fully account for every detail of the collision and of the ensuing particle production.
The nonequilibrium-statistical relativistic diffusion model is – in its linear approximation Wolschin (2004) – based on an analytically solvable transport equation with three sources. It does not only consider particle production from a central source as the thermal model does, but also from the fragmentation sources. The latter evolve in time and eventually tend to merge with the central source towards an overall thermal equilibrium distribution, but since the interaction time is extremely short at RHIC and LHC energies, this equilibrium state is not reached, and in particular the rapidity and pseudorapidity distributions show characteristic nonequilibrium features.
In this work I present an investigation of the energy dependence of the charged-hadron production sources within the relativistic diffusion model in symmetric systems, AuAu at RHIC c.m. energies per nucleon pair of 19.6, 62.4, 130 and 200 GeV, and PbPb at LHC energies of 2.76 and 5.52 TeV. The gluon-dominated source, in addition to the fragmentation sources related to the valence part of the nucleons, had been implemented earlier into the RDM Wolschin (2004); Wolschin _et al._ (6). A related model with a gluonic source at mid rapidity had also been proposed by Bialas and Czyz Bialas and Czyz (2005).
In Wolschin (2013) it has been found that the fragmentation sources for produced charged hadrons – which are clearly visible in net-proton rapidity distributions where the gluonic source cancels out Mehtar-Tani and Wolschin (2009) – have the expected logarithmic dependence on \(\sqrt{s_{NN}}\), whereas the particle content in the mid rapidity gluon-gluon induced source that rises strongly with energy is close to a power law. This result has since been corroborated through other independent investigations of charged-particle and transverse energy production Sahoo _et al._ (2014); Sahoo and Mishra (2014) such that a renewed and more precise consideration in particular of the central source is indicated.
The fragmentation sources are responsible for most of the yield in the regions close to the beam rapidities. Here limiting fragmentation scaling Alver _et al._ (2011) is valid not only at RHIC, but also at LHC energies Röhrscheid and Wolschin (2012). This is in contrast to earlier predictions of the thermal model Cleymans _et al._ (2008) which find a violation of extended longitudinal scaling at LHC energies, providing another indication that equilibrium statistical concepts are invalid in the fragmentation region. Here the yields in pseudorapidity also extend beyond the value of the beam rapidity, and in the final paragraph of this note the origin of this effect is discussed.
## II Hadron production sources
For a detailed phenomenological investigation of the charged-hadron particle content in the three particle-production sources, the nonequilibrium-statistical relativistic diffusion model Wolschin (1999, 2004, 2013) is used. The fragmentation sources \(R_{1,2}(y,t=\tau_{int})\) with charged-particle content \(N_{ch}^{qg,1}\) (projectile-like), \(N_{ch}^{gq,2}\) (target-like) and the midrapidity low-\(x\) gluon-gluon source \(R_{gg}(y,t=\tau_{int})\) with charged-particle content \(N_{ch}^{gg}\) are added incoherently to generate the total pserudorapidity density distribution as
\[\frac{dN^{tot}_{ch}(y,t=\tau_{int})}{dy}=N_{ch}^{qg,1}R_{1}(y, \tau_{int})\] (1)
\[\qquad\qquad+N_{ch}^{gq,2}R_{2}(y,\tau_{int})+N_{ch}^{gg}R_{gg}(y ,\tau_{int})\]
with the rapidity \(y=0.5\cdot\ln((E+p)/(E-p))\), and the interaction time \(\tau_{int}\). The latter corresponds to the total integration time of the underlying partial differential equation, which is a linear partial differential equation of the Fokker-Planck type, as described in Wolschin (2013).
Converting the rapidity distribution \(dN/dy\) for produced charged hadrons to the corresponding pseudorapidity distribution \(dN/d\eta\) (\(\eta\) = - ln(tan(\(\theta/2\))) ) with the proper Jacobian transformation \(dy/d\eta\) and minimizing the analytical solutions of the transport equation with respect to available pseudorapidity data then yields the particle content of the sources as functions of \(\sqrt{s_{NN}}\)Wolschin (2013). The corresponding RDM-parameters for central collisions have been published in Tab. 1 of Wolschin (2013).
<figure><img src="content_image/1501.03026/x1.png"><figcaption>Figure 1: (Color online) The RDM pseudorapidity distribution functions forcharged hadrons in central AuAu (RHIC) and PbPb (LHC) collisions at c.m.energies of 19.6 GeV, 130 GeV, 200 GeV, 2.76 TeV and 5.02 TeV shown here areoptimized in χ2−fits with respect to the PHOBOS Back _et al._ (2003); Alver_et al._ (2011) (bottom) and ALICE Guilbaud _et al._ (2013) (top) data, withparameters from Wolschin (2013). The upper distribution function is anextrapolation to the LHC design energy of 5.52 TeV. At the lowest energy, onlythe fragmentation sources contribute (dash-dotted curves).</figcaption></figure>
<figure><img src="content_image/1501.03026/x2.png"><figcaption>Figure 2: (Color online) The RDM pseudorapidity distribution functions forcharged hadrons in central 200 GeV AuAu (top frame) and 2.76 TeV PbPbcollisions are adjusted through χ2−minimizations to the PHOBOS Alver _et al._(2011) (see also Wolschin _et al._ (2006a)) and ALICE Guilbaud _et al._(2013) data, see Wolschin (2013). The underlying particle production sourcesare shown: dash-dotted curves are the fragmentation sources, dashed curves themid rapidity gluon-gluon sources, and dotted curves the central sourceswithout the effect of the Jacobian transformation from rapidity topseudorapidity. The particle content in the gluon-gluon source rises stronglywith increasing c.m. energy, and constitutes the largest source at LHCenergies.</figcaption></figure>
<figure><img src="content_image/1501.03026/x3.png"><figcaption>Figure 3: (Color online) Number of produced charged hadrons as function of thec.m. energy √sNN from RDM-fits of the available data for central heavy-ioncollisions at 0.019, 0.062, 0.13, 0.2 TeV (RHIC, AuAu), 2.76 TeV (LHC, PbPb),plus extrapolation to 5.52 TeV. Circles are the total numbers, following apower law ∝s0.23NN. Triangles are particles from the fragmentation sources∝log(sNN/s0). Squares are hadrons produced from the midrapidity source, with adependence ∝log3(sNN/s0). A power law ∝s0.44NN Wolschin (2013) is also shown(short-dashed curve), but fails to fit the extrapolated 5.52 TeV yield. Thegluon-gluon source (dashed) becomes the main source of particle productionbetween RHIC and LHC energies.</figcaption></figure>
<figure><img src="content_image/1501.03026/x4.png"><figcaption>Figure 4: (Color online) The total charged-hadron production in central AuAuand PbPb collision in the energy region 19.6 GeV to 5.52 TeV is following apower law Ntot∝(sNN/s0)0.23 (solid line), whereas the particle content in thefragmentation sources is Nqg∝ln(sNN/s0), dash-dotted curve. The particlecontent in the mid-rapidity source obeys Ngg∝ln3(sNN/s0), dashed curve, nottoo far from a power law (short-dashed line) only in the intermediate energyrange 0.1–2.76 TeV. The energy dependence of the mid rapidity yield is shownas a dotted line, with PHOBOS data Alver _et al._ (2011) at RHIC energies,and ALICE data Aamodt _et al._ (2011) at 2.76 TeV.</figcaption></figure>
Results of this approach are summarized in Fig. 1, where the charged-hadron pseudorapidity distributions are shown from low RHIC energies of 19.6 GeV, via 130 GeV, 200 GeV, to 2.76 TeV, plus a prediction at 5.52 TeV. It is noted that the midrapidity source is found to be absent at 19.6 GeV and appears only at the higher energies, rising in particle content with \(\sqrt{s_{NN}}\). The individual sources are displayed in Fig. 2 at 200 GeV and 2.76 TeV, where the effect of the Jacobian transformation from rapidity \(y\) to pseudorapidity \(\eta\) is also shown. The central gluon-gluon source is seen to become dominant as the energy is increased from RHIC to LHC.
The corresponding particle contents of the sources are displayed in Fig. 3, which resembles the analogous figure in Wolschin (2013), but differs in a decisive detail. The total particle content is found to follow a power law,
\[N_{ch}^{tot}=1.1\cdot 10^{4}(s_{NN}/s_{0})^{0.23}\] (2)
with \(s_{0}=1~{}\)TeV\({}^{2}\), whereas the particle content in the two fragmentation sources is as expected a logarithmic function of the energy
\[N_{ch}^{qg}=695\cdot\ln(s_{NN}/s_{0})\] (3)
with \(s_{0}=100~{}\)GeV\({}^{2}\). The midrapidity gluon-gluon source is approximated by the thin dashed line following a power law as was already proposed in Wolschin (2013)
\[N_{ch}^{gg}\simeq 4\cdot 10^{3}(s_{NN}/s_{0})^{0.44}\] (4)
with \(s_{0}=1~{}\)TeV\({}^{2}\). However, when considering also the yield predicted within the relativistic diffusion model (RDM) for the LHC design energy of 5.52 TeV, the power law fails to fit the expected yield, whereas a cubic log dependence agrees with the prediction,
\[N_{ch}^{gg}=7.5\cdot\ln^{3}(s_{NN}/s_{0})\] (5)
where \(s_{0}=\)169 GeV\({}^{2}\).
It remains to be seen whether the data actually follow the model prediction. In the upcoming PbPb run at the LHC in 2015, the c.m. energy is scheduled to be 5.125 TeV, corresponding to 13 TeV \(pp\). The total charged-hadron yield predicted by Eq. (2) at this energy is \(N_{ch}^{tot}=23,327\), with the central source contributing \(N_{ch}^{gg}=12,811\) charged hadrons according to Eq. (5). The RQM-value for the total charged-hadron production at the lower LHC energy of 2.76 TeV is \(N_{ch}^{tot}=17,327\) according to Tab. 1 of Wolschin (2013); the power law Eq. (2) yields 17,546. The ALICE collaboration meanwhile quotes an extrapolated value of \(17,146\pm 722\)Abbas _et al._ (2013).
When examining the RDM results for the particle content of the sources more closely also in the low-energy region where RHIC data are available, it turns out that the power law Eq. (4) is an acceptable approximation to \(N_{ch}^{gg}\) only between about 100 GeV and 2.76 TeV.
This becomes particularly obvious in Fig. 4, where the same plot is shown using a double-logarithmic scale, following a suggestion by Trainor Trainor (31). Here power laws appear as straight lines – such as the one for the total charged-hadron production, or also for the midrapidity yield
\[\frac{dN_{ch}^{tot}}{d\eta}|_{\eta\simeq 0}=1.15\cdot 10^{3}(s_{NN}/s_{0})^{0. 165}\] (6)
with \(s_{0}=1~{}\)TeV\({}^{2}\) (dotted line, and data points from Phobos Alver _et al._ (2011) and ALICE Aamodt _et al._ (2011)).
The cubic-log dependence of the gluon-gluon source (dashed) is seen to fit the points extracted from the RDM-analyses Wolschin (2013) of PHOBOS and ALICE data rather precisely at the available energies, and it agrees with the RDM-prediction at the LHC design energy of 5.52 TeV.
As required by the RDM analysis of the 19.6 GeV AuAu data, the gluon-gluon contribution becomes unimportant below 20 GeV – whereas a power law would still predict a yield of about 100 charged hadrons in this energy region. Although a hybrid function with a log-dependence at RHIC energies that turns into a power law at LHC energies may appear as a reasonable compromise Sahoo and Mishra (2014); Sahoo _et al._ (2014), it can not compete with the ln\({}^{3}\)-dependence for the central source regarding the precision of the fit to the RDM-results.
## III Energy dependence of the mid-rapidity source
The origin of the cubic-log dependence of the total charged-hadron yield on \(s_{NN}\) (or \(\sqrt{s_{NN}}\)) in the mid-rapidity gluon-gluon source can be traced schematically neglecting for the moment the precise value of the proportionality factor appearing in Eq. (5). The width of the gluon-gluon distribution is expected to scale roughly with the beam rapidity,
\[\sigma\propto y_{\text{beam}}=\ln{(\sqrt{s_{NN}}/m_{p})}=0.5\ln{({s_{NN}}/m_{p }^{2})}\] (7)
where \(m_{p}\) is the proton mass. With respect to the midrapidity value, the STAR collaboration observed in 2004-2006 that dijet production which generates the hard component of the spectrum is at midrapidity proportional to the square of the soft-component density, that is associated with low-\(x\) gluons Adams _et al._ (2006); Trainor (33).
Since the yield of low-\(x\) gluons is proportional to the logarithm of the c.m. energy, the density at midrapidity that arises from dijet production is proportional to \(\ln^{2}s\). Hence, the integrated yield in the gluon-gluon source can be estimated as
\[N_{ch}^{gg}\simeq\int_{-y_{\text{beam}}}^{y_{\text{beam}}}\frac{dN}{d\eta}|_{ gg}d\eta\propto\ln^{3}(s_{NN}/s_{0})\\\] (8)
in agreement with the above result of the phenomenological RDM-analysis.
On the theoretical side, the \(gg\to gg\) scattering amplitude has been evaluated in the presence of a classical color field e.g. by Cheung and Chiu Cheung and Chiu (2011). They find that the classical color field modifies the \(gg\to gg\) elastic scattering amplitude, and suppresses it when the longitudinal momentum fraction \(x\) of the incident gluon is small. The rise of the cross section with energy in the central distribution – that is driven by the growth of the gluon density at small \(x\) – is therefore suppressed by the quantum-classical interaction from the dense medium Cheung and Chiu (2011). The predicted cross section has a ln\({}^{2}s\) asymyptotic behavior that satisfies the Froissart bound Froissart (1961), and the integral over rapidity becomes proportional to \(\ln^{3}s\).
It is interesting to compare the results of the present analysis with the rapidity distributions from the hydrodynamic approach of Landau and Belen’kji Landau (1953); Belen’kji and Landau (1955), and applications to particle production by Carruthers and Duong-van Carruthers and Duong-van (1972, 1973), as well as Steinberg Steinberg (2006). There the width (FWHM) \(\Gamma=\sqrt{8\ln 2}\cdot\sigma\) of a gaussian pseudorapidity distribution for produced charged particles is obtained from the variance Carruthers and Duong-van (1973)
\[\sigma^{2}_{\text{Landau}}=\ln\gamma=\ln{(\sqrt{s_{NN}}/2m_{p})}\] (9)
with the Lorentz-factor \(\gamma=1/\sqrt{(}1-\beta^{2}),\beta=p/E.\) It turns out that this expression is in reasonable agreement Steinberg (2006) with data from AGS and SPS where the stopping fraction is sizeable. Deviations start to become visible at RHIC – where nuclear transparency Bjorken (1983) with well-separated fragmentation sources is already obvious – and, in particular, at LHC where the measured width of the \(dN/d\eta\)-distributions for charged hadrons is substantially broader than predicted by Eq. (9), as shown by the ALICE collaboration Abbas _et al._ (2013). It has therefore been concluded ’…that Landau hydrodynamics does not explain the expansion dynamics at LHC energies’ Sahoo _et al._ (2014).
Whereas this is certainly true for the overall pseudorapidity distribution of charged particles, Landau’s approach may still be viable for a proper description of the mid-rapidity source which accounts for particles generated from low-\(x\) gluons. Indeed for 2.76 TeV PbPb, the RDM-analysis yields a width in rapidity \(y\) of \(\Gamma_{gg}=6.24\)Wolschin (2013), compared to a Landau result of \(\Gamma_{gg}^{\text{Landau}}=6.36\) in \(\eta\) according to Eq. (9). At RHIC energies, the Landau result is, however, larger than the RDM result for the mid-rapidity source, and the results at the higher LHC energies of 5.125 TeV and 5.519 TeV PbPb remain to be seen.
## IV Yields beyond the beam rapidity
Already in the investigation of AuAu collisions at RHIC energies Alver _et al._ (2011) it has been observed that pseudorapidity yields of produced charged particles extend significantly beyond the value of the beam rapidity. This is particularly obvious in PHOBOS AuAu results at 130 GeV where \(dN/d\eta\) data have been taken beyond \(y_{\text{beam}}\)Back _et al._ (2001). The RDM solutions for 200 GeV AuAu and 2.76 TeV PbPb also clearly indicate expected yields beyond \(y_{\text{beam}}\) at these higher energies, see Fig. 5.
Obviously it is not excluded that this can partly be due to a real physical effect, with a few charged particles produced at larger rapidities than that of the beam value. However, the bulk of the large charged-particle pseudorapidity density in the region at and beyond the beam rapidity – which amounts to more than 100 charged particles – is likely due to the transformation from rapidity to pseudorapidity.
Reconsider the expressions for rapidity \(y\), longitudinal velocity \(\beta_{||}\), and pseudorapidity \(\eta\)
\[y=\frac{1}{2}\ln{\frac{1+\beta_{||}}{1-\beta_{||}}}\] (10)
\[\beta_{||}=\frac{\exp{(2y)}-1}{\exp{(2y)}+1}\] (11)
\[\eta=-\ln{(\tan(\theta/2)).}\] (12)
The transformation between \(\eta\) and \(y\) is
\[y=\frac{1}{2}\ln{\frac{\sqrt{(m/p_{T})^{2}+\cosh^{2}y}+\sinh\eta}{\sqrt{(m/p_{ T})^{2}+\cosh^{2}y}-\sinh\eta}}.\] (13)
Here \(m\) is the mass of the particle species considered. The relative particle abundances in central (0-5%) PbPb collisions at 2.76 TeV are 83% pions, 13% kaons, and 4% protons, with the pion fraction increasing to 84% for more peripheral (50-60%) collisions. Hence, I use an accordingly averaged effective mass for \(m\) as described in detail in Röhrscheid and Wolschin (2012).
Since only the ratio \(m/p_{T}\) enters the Jacobian, one can also fix the mass at the pion mass \(m=m_{\pi}\), and calculate the corresponding effective transverse momentum from \(<p_{T,\text{eff}}>=m_{\pi}J_{y=0}/\sqrt{1-J_{y=0}^{2}}\) with the experimentally determined Jacobian \(J_{y=0}\) at \(90^{0}\), see Röhrscheid and Wolschin (2012) for 2.76 TeV PbPb. The values of \(m/p_{T}\) used in the calculations shown in Fig. 5 are \(m/p_{T}\) = 0.466; 0.349; 0.585 for \(\sqrt{s_{NN}}\) = 0.13; 0.2; 2.76 TeV, respectively.
The above expression for the transformation from \(y\) to \(\eta\) has the limits \(y\rightarrow\eta-\ln(m/p_{T})\) for \(m<<p_{T}\), and \(y\rightarrow\eta\) for \(p_{T}<<m\). Since most of the produced charged hadrons at a LHC energy of 2.76 TeV are pions, the limit \(y\approx\eta\) at small transverse momenta – very forward angles – is reached for charged hadrons at larger values of \(\eta\) than for protons (net protons determine the value of the beam rapidity). Hence, the \(dN/d\eta\) distribution for charged hadrons which are mostly pions can easily extend beyond \(y_{\text{beam}}\).
<figure><img src="content_image/1501.03026/x5.png"><figcaption>Figure 5: (Color online) Produced charged particles in central AuAu collisionsat √sNN=130 and 200 GeV (RHIC/ PHOBOS data Alver _et al._ (2011), bottom, andin 2.76 TeV PbPb Guilbaud _et al._ (2013), top, in comparison with the RDMsolutions. The values of the beam rapidities are indicated as arrows(ybeam=4.932,5.362,7.987). The pseudorapidity yields extend beyond ybeam,which is particularly evident in case of the 130 GeV PHOBOS data.</figcaption></figure>
## V Conclusion
The analysis of the energy dependence of charged-hadron pseudorapidity distributions in AuAu collisions at RHIC energies, and PbPb collisions at LHC energies in the phenomenological nonequilibrium-statistical relativistic diffusion model reveals the expected ln(\(s\))-dependence for the total particle content of the two fragmentation sources, but a ln\({}^{3}s\)-dependence for the total charged hadron content of the gluon-gluon source. Modifying the conclusion of an initial investigation Wolschin (2013), it is only in a limited energy region of about 100 GeV to 2.76 TeV that this dependence may be approximated by a power law.
###### Acknowledgements.
I thank Tom Trainor for pointing out that a \(\ln^{3}(s_{NN})\)-dependence of the gluon-gluon source gives a better representation of the RDM-results for particle production, than a power law. Discussions with Jean-Paul Blaizot, Larry McLerran and Dmitri Melikhov during their stays at the Institute for Theoretical Physics in Heidelberg are gratefully acknowledged.
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