alx-ai/arxiv_papers · Datasets at Hugging Face

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distance,γ, from the center of the planetary disc as
T(γ) =T0(1−γ2)1/8. The thermal secondary eclipse
depth in this limit is given by:
Fday
F∗=/parenleftbiggRp
R∗/parenrightbigg2/parenleftbigghc
λkT0/parenrightbigg8/parenleftBig
ehc/λkT∗
b−1/parenrightBig
×/integraldisplay(λkT0/hc)8
0dx
exp(x−1/8)−1, (3)
whereT∗
bis the brightness temperature of the star at
wavelength λ.
In the no-circulation limit, then, the day-side emer-
gent spectrum is not exactly that of a blackbody, even
if each annulus has a blackbody spectrum. But these
diﬀerences are not important for the present study, since
we are concerned with bolometric ﬂux. By integrating
Equation 3 over λ, one obtains the eﬀective tempera-
tureoftheday-sideintheno-albedo,no-circulationlimit:
Tε=0= (2/3)1/4T0(see also Burrows et al. 2008; Hansen
2008). Indeed, treatingtheplanet’sday-sideasauniform
hemisphere emitting at this temperature gives nearly the
same wavelength dependence as the more complex Equa-
tion 3. The Tε=0temperatures for our sample of 24 tran-
siting planets are shown in Table 1. These set the max-
imum possible day-side eﬀective temperature we should
expect to measure.
The integrated day-side ﬂux in the general —non-zero
circulation— case is more subtle: heat may be trans-
ported to the planet’s night-side, and/or to its poles. In
this paper we neglect the E-W asymetry in the planet’s
temperature map due to zonal ﬂows and hence phase
oﬀsets in the thermal phase variations. Under this as-
sumption, the day-night temperature contrast can more
directly be extracted from the observed thermal phase
variations.
In practice, manystudies haveadopted asingle param-
eter to represent bothzonal and meridional transport. It
is instructive to consider the apparent day-side eﬀective
temperatures in variouslimits: uniform day-sidetemper-
ature andT= 0 on the night-side (this is often referred
to as the planet’s “equilibrium temperature”): Tequ=
(1/2)1/4T0; in the case of perfect longitudinal transport
but no latitudinal transport: Tlong= (8/(3π2))1/4T0;
and in the limit of a uniform temperature everywhere
on the planet: Tuni= (1/4)1/4T0.
Comparing the apparent day-side temperatures in the
three limits of circulation above leads to the following
simple parametrization of the day-side eﬀective temper-
ature in terms of the planetary albedo, AB, and circula-
tion eﬃciency, ε:
Td=T0(1−AB)1/4/parenleftbigg2
3−5
12ε/parenrightbigg1/4
,(4)
where 0< ε <1. Note that εis related to —but dif-
ferent from— the ǫused in (Cowan & Agol 2010). The
former is merely a parametrization of the observed disk-
integrated eﬀective temperature, while the latter, which
can take values from 0 to ∞, is a precisely deﬁned ratio
of radiative and advective timescales. The ǫ= 0 case is
precisely equal to the ε= 0 case, while the ǫ→ ∞limit
is equivalent to ǫ≈0.95.
Our deﬁnition of εis similar to the Burrows et al.(2006) deﬁnition of Pnandyieldsthe sameno-circulation
limit. But our ε= 1 limit produces a lower day-side
brightness than the Pn= 0.5 limit, because we as-
sume that the planet’s day-side has a uniform tempera-
ture distribution in that limit (for a discussion of diﬀer-
ent redistribution parameterizations, see the appendix of
Spiegel & Burrows 2010).
In reality, eﬃcient longitudinal transport (read: fast
zonalwinds) mayleadtomorebandingandthereforeless
eﬃcient latitudinal transport. So one could argue that
in the limit of perfect day-night temperature homoge-
nization, both the day and night apparent temperatures
should beTd= (8/(3π2))1/4T0, in between the Burrows
et al. value of Td= (1/3)1/4T0and that suggested by
our parameterization, Td= (1/4)1/4T0. At moderate
day-night recirculation eﬃciencies, however, there is a
good deal of latitudinal transport (I. Dobbs-Dixon, priv.
comm.), so implicitly assuming a constant T∝cos1/4
latitudinal dependence —as done by Burrows et al.— is
not founded, either. The bottom line is that any single-
parameter implementation of advection is incapable of
capturing the real complexities involved, but longitudi-
nal transport is the dominant factor in determining day
and night eﬀective temperatures.
Not withstanding the subtleties discussed above and
noting that cooling tends to latitudinaly homogenize
night-side temperatures (Cowan & Agol 2010), we get a
night-side temperature of:
Tn=T0(1−AB)1/4/parenleftBigε
4/parenrightBig1/4
. (5)
Note thatTdandTnare the equator-weighted tempera-
tures of their respective hemispheres (ie, as seen by an
edge-on viewer). As such, they will tend to be slightly

distance,γ, from the center of the planetary disc as

T(γ) =T0(1−γ2)1/8. The thermal secondary eclipse

depth in this limit is given by:

Fday

F∗=/parenleftbiggRp

R∗/parenrightbigg2/parenleftbigghc

λkT0/parenrightbigg8/parenleftBig

ehc/λkT∗

b−1/parenrightBig

×/integraldisplay(λkT0/hc)8

0dx

exp(x−1/8)−1, (3)

whereT∗

bis the brightness temperature of the star at

wavelength λ.

In the no-circulation limit, then, the day-side emer-

gent spectrum is not exactly that of a blackbody, even

if each annulus has a blackbody spectrum. But these

diﬀerences are not important for the present study, since

we are concerned with bolometric ﬂux. By integrating

Equation 3 over λ, one obtains the eﬀective tempera-

tureoftheday-sideintheno-albedo,no-circulationlimit:

Tε=0= (2/3)1/4T0(see also Burrows et al. 2008; Hansen

2008). Indeed, treatingtheplanet’sday-sideasauniform

hemisphere emitting at this temperature gives nearly the

same wavelength dependence as the more complex Equa-

tion 3. The Tε=0temperatures for our sample of 24 tran-

siting planets are shown in Table 1. These set the max-

imum possible day-side eﬀective temperature we should

expect to measure.

The integrated day-side ﬂux in the general —non-zero

circulation— case is more subtle: heat may be trans-

ported to the planet’s night-side, and/or to its poles. In

this paper we neglect the E-W asymetry in the planet’s

temperature map due to zonal ﬂows and hence phase

oﬀsets in the thermal phase variations. Under this as-

sumption, the day-night temperature contrast can more

directly be extracted from the observed thermal phase

variations.

In practice, manystudies haveadopted asingle param-

eter to represent bothzonal and meridional transport. It

is instructive to consider the apparent day-side eﬀective

temperatures in variouslimits: uniform day-sidetemper-

ature andT= 0 on the night-side (this is often referred

to as the planet’s “equilibrium temperature”): Tequ=

(1/2)1/4T0; in the case of perfect longitudinal transport

but no latitudinal transport: Tlong= (8/(3π2))1/4T0;

and in the limit of a uniform temperature everywhere

on the planet: Tuni= (1/4)1/4T0.

Comparing the apparent day-side temperatures in the

three limits of circulation above leads to the following

simple parametrization of the day-side eﬀective temper-

ature in terms of the planetary albedo, AB, and circula-

tion eﬃciency, ε:

Td=T0(1−AB)1/4/parenleftbigg2

3−5

12ε/parenrightbigg1/4

,(4)

where 0< ε <1. Note that εis related to —but dif-

ferent from— the ǫused in (Cowan & Agol 2010). The

former is merely a parametrization of the observed disk-

integrated eﬀective temperature, while the latter, which

can take values from 0 to ∞, is a precisely deﬁned ratio

of radiative and advective timescales. The ǫ= 0 case is

precisely equal to the ε= 0 case, while the ǫ→ ∞limit

is equivalent to ǫ≈0.95.

Our deﬁnition of εis similar to the Burrows et al.(2006) deﬁnition of Pnandyieldsthe sameno-circulation

limit. But our ε= 1 limit produces a lower day-side

brightness than the Pn= 0.5 limit, because we as-

sume that the planet’s day-side has a uniform tempera-

ture distribution in that limit (for a discussion of diﬀer-

ent redistribution parameterizations, see the appendix of

Spiegel & Burrows 2010).

In reality, eﬃcient longitudinal transport (read: fast

zonalwinds) mayleadtomorebandingandthereforeless

eﬃcient latitudinal transport. So one could argue that

in the limit of perfect day-night temperature homoge-

nization, both the day and night apparent temperatures

should beTd= (8/(3π2))1/4T0, in between the Burrows

et al. value of Td= (1/3)1/4T0and that suggested by

our parameterization, Td= (1/4)1/4T0. At moderate

day-night recirculation eﬃciencies, however, there is a

good deal of latitudinal transport (I. Dobbs-Dixon, priv.

comm.), so implicitly assuming a constant T∝cos1/4

latitudinal dependence —as done by Burrows et al.— is

not founded, either. The bottom line is that any single-

parameter implementation of advection is incapable of

capturing the real complexities involved, but longitudi-

nal transport is the dominant factor in determining day

and night eﬀective temperatures.

Not withstanding the subtleties discussed above and

noting that cooling tends to latitudinaly homogenize

night-side temperatures (Cowan & Agol 2010), we get a

night-side temperature of:

Tn=T0(1−AB)1/4/parenleftBigε

4/parenrightBig1/4

. (5)

Note thatTdandTnare the equator-weighted tempera-

tures of their respective hemispheres (ie, as seen by an

edge-on viewer). As such, they will tend to be slightly