text
stringlengths
0
44.4k
planet has a day-side effective temperature of ∼2000 K.
When combined with the Kepler observations, one com-
putesanalbedoofgreaterthan50%. Thelargeday-night
amplitude seen in the Kepler bandpass is then simply
due to the fact that the planet’s night-side reflects no
starlight, and the cool day-side can be attributed to high
ABand/orε. If, on the other hand, one takes the op-
tical flux to be entirely thermal in origin ( Aλ= 0), the
day-side effective temperature is ∼2800 K. This is very
close to that planet’s Tε=0, leaving very little power left
for the night-side, again explaining the large day-night
contrast observed by Kepler. The truth probably lies
somewhere between these two extremes, but in any case
this degeneracy will be neatly broken with Warm Spitzer
observations: the two scenarios outlined above will lead
to small and large thermal phase variations, respectively.
It is telling that the only optical measurement in Table 1
that is unanimously considered to constrain albedo —
and not thermal emission— is the MOST observations
of HD 209458b (Rowe et al. 2008), the coolest of the five
transiting planets with optical photometric constraints.
The bottom line is that extracting a constraint on re-
flected light from optical measurements of hot Jupiters is
best done with a detailed spectral model. But even when
reflectedlightcanbedirectlyconstrained,convertingthis
constraint on Aλinto a constraint on ABalso requires
detailedknowledgeofboththestarandtheplanet’sspec-
tral energy distributions, making for a model-dependent
exercise.
4.2.Populating the AB-εPlane
Setting aside optical eclipses and direct measurements
of albedo, we may use the rich near- and mid-IR data to
constrain the Bond albedo and redistribution efficiency
of short-period giant planets. We define a 20 ×20 grid in
ABandεand use Equations 4 & 5 to calculate the nor-
malized day-side and night-side effective temperatures,
Td/T0andTn/T0, at each grid point, ( i,j). For each
planet, we have an observational estimate of the day-side
effective temperature, and in three cases we also have an
estimate of the night-side effective temperature (as well
as associated uncertainties).
We first verifywhether ornot the observationsarecon-
sistent with a single ABandε. To evaluate this “null
hypothesis”, we compute the usual χ2=/summationtext24
i=1(model−
data)2/error2at each grid point. We use only the esti-
mates of day-side and (when available) night-side effec-
tive temperatures to calculate the χ2, giving us 27-2=25
degreesoffreedom. The“best-fit”has χ2= 132(reduced
χ2= 5.3), so the current observations strongly rule out
a single Bond albedo and redistribution efficiency for all
24 planets.
For 21 of the 24 planets considered here, we construct
a two-dimensional distribution function for each planet
as follows:
PDF(i,j) =1/radicalbig
2πσ2
de−(Td−Td(i,j))2/(2σd)2.(7)
This defines a swath through parameter space with the
same shape as the dotted line in Figure 1.
For the three remaining planets (HD 149026b,Albedo and Heat Recirculation on Hot Exoplanets 7
HD 189733b, HD 209458b), phase variation measure-
ments help break the degeneracy:
PDF(i,j) =1√
2πσ2
de−(Td−Td(i,j))2/(2σd)2
×1√
2πσ2ne−(Tn−Tn(i,j))2/(2σn)2.(8)
Fig. 4.— The global distribution function for short-period exo-
planets in the AB–εplane. The gray-scale shows the sum of the
normalized probability distribution function for the 24 pl anets in
our sample. The data mostly consist of infrared day-side flux es,
leading to the dominant degeneracy (see first the dotted line in
Figure 1).
We create a two-dimensional normalized probability
distribution function (PDF) for each planet, then add
these together to create the global PDF shown in Fig-
ure 4. This is a democratic way of representing the data,
since each planet’s distribution contributes equally.
In Figures 5 and 6 we show the distribution functions
for the albedo and circulation of the 24 planets in our
sample,obtainedbymarginalizingtheglobalPDFofFig-
ure 4 over either ABorε.
Fig. 5.— The solid black line shows the projection of the 2-
dimensional probability distribution function (the gray- scale of
Figure 4) projected onto the ε-axis. The dashed line shows the
ε-distribution if one requires that all planets have Bond alb edos
less than 0.1; under this assumption, we see hints of a bimoda l
distribution in heat circulation efficiency.Fig. 6.— The solid black line shows the projection of the 2-
dimensionalprobabilitydistributionfunction (the gray- scale ofFig-
ure 4) projected onto the AB-axis. The cumulative distribution
function (not shown) yields a 1 σupper limit of AB<0.35.
The solid line in Figure 5 shows no evidence of bi-
modality in heat redistribution efficiency, although there
is a wide range of behaviors. The dashed line in Figure 5
shows theε-distribution if one requires the albedo to be
low,AB<0.1. There are then many high-recirculation
planets, since advection is the only way to depress the
day-side temperature in the absence of albedo. Inter-
estingly, the dashed line doesshow tentative evidence of