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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

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