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the actual model eﬀective temperature versus the num-
ber of wavebands used in the estimate. The temperature
estimates cluster near Test/Teﬀ= 1, indicating that the
technique is not signiﬁcantly biased. The scatter in es-
timates decreases as more wavebands are used, from a
standard deviation of 7.6% if only a single brightness
temperature is used, down to 2.4% if photometry is ac-
quired in eight bands. We incorporate this systematic
error into our analysis by adding it in quadrature to
the observational uncertainties described in the follow-
ing paragraph. This has the desirable eﬀect that planets
with fewer observations have a larger systematic uncer-
tainty on their eﬀective temperature.
Fig. 3.— The Linear Interpolation technique for estimating day-
side eﬀective as tested on a suite of eleven hot Jupiter spect ral
models provided by J.J. Fortney. The y-axis shows the estima ted
day-side eﬀective temperature normalized by the actual mod el ef-
fective temperature. The x-axis represents the number of br ight-
ness temperatures used in the estimate. Each color correspo nds to
one of the eleven models used in the comparison. The black err or
bars represent the standard deviation in the normalized tem pera-
ture estimates.
Inpractice,wewouldliketopropagatethephotometric
uncertainties to the estimate of Teﬀ. For the Wien Dis-
placement technique, this uncertainty propagates triv-
ially to the eﬀective temperature. For the linear inter-
polation technique, a Monte Carlo can be used to esti-
mate the uncertainty in Teﬀ: the input eclipse depths
are randomly shifted 1000 times in a manner consistent
with their photometric uncertainties —assuming Gaus-
sianerrors—andtheeﬀectivetemperatureisrecomputed
repeatedly. Thescatterintheresultingvaluesof Teﬀpro-
vides an estimate of the observational uncertainty in the
parameter, to which we add in quadrature the estimate
ofsystematicerrordescribedabove. The resultinguncer-
tainties are listed in Table 1. These uncertainties should6 Cowan & Agol
be compared to the uncertainties in Tε=0(also listed in
Table 1), which are computed using the uncertainty in
the star’s properties and the planet’s orbit.
There are two practical issues with the linear interpo-
lation temperature estimation technique. In some cases,
onlyupperlimitshavebeenobtained, thereforeonecould
setψ= 0, with the appropriate1-sigmauncertainty. But
this approach leads to huge uncertainties in Teﬀfor plan-
ets with a secondary eclipse upper-limit near their black-
body peak. Instead of “punishing” these planets, we opt
to not use upper-limits (though for completeness we in-
clude them in Table 1). Secondly, when multiple mea-
surements of an eclipse depth have been published for
a given waveband, we use the most recent observation,
indicated with a superscript “ e” in Table 1. In all cases
these observations either explicitly agree with their older
counterpart, or agree with the re-analyzed older data.
4.RESULTS
4.1.Looking for Reﬂected Light
For each planet, we use thermal observations (essen-
tially those in the J, H, K s, andSpitzerbands) to es-
timate the planet’s eﬀective day-side temperature, Td,
and —when phase variations are available— Tn. These
values are listed in Table 1. In ﬁve cases (CoRoT-
1b, CoRoT-2b, HAT-P-7b, HD 209458b, TrES-2b), sec-
ondary eclipses and/or phase variations have been ob-
tained at optical wavelengths. Such observations have
the potential to directly constrain the albedo of these
planets. One approach is to adopt the Tdfrom thermal
observations and calculate the expected contrast ratio at
optical wavelengths, under the assumption of blackbody
emission (see also Kipping & Bakos 2010). Insofar as
the observed eclipse depths are deeper than this calcu-
lated depth, one can invoke the contribution of reﬂected
light and compute a geometric albedo, Ag. If one treats
the planet as a uniform Lambert sphere, the geometric
albedo is related to the spherical albedo at that wave-
length byAλ=3
2Ag. These values are listed in Table 1.
But reﬂected light is not the only explanation for an
unexpectedly deep optical eclipse. Alternatively, the
emissivity of the planets may simply be greater at op-
tical wavelengths than at mid-IR wavelengths, in agree-
mentwith realisticspectralmodelsofhotJupiters, which
predict brightness temperatures greater than Teﬀon the
Wien tail (see, for example, the Fortney et al. model
showninFigure2, whichdoesnotincludereﬂectedlight).
Note that this increasein emissivityshould occurregard-
less of whether or not the planet has a stratosphere: by
deﬁnition, the depth at which the optical thermal emis-
sion is emitted is the depth at which incident starlight
is absorbed, which will necessarily be a hot layer —
assuming the incident stellar spectrum peaks in the op-
tical.
Determining the albedo directly (ie: by observing re-
ﬂected light) can be diﬃcult for short period planets,
because there is no way to distinguish between reﬂected
and re-radiated photons. The blackbody peaks of the
star and planet often diﬀer by less than a micron. There-
fore, unlike Solar System planets, these worlds do not
exhibit a minimum in their spectral energy distribution
between the reﬂected and thermal peaks. The hottest
—and therefore most ambiguous case— of the ﬁve tran-siting planets with optical constraints is HAT-P-7b. If
one takes the mid-IR eclipse depths at face value, the

the actual model eﬀective temperature versus the num-

ber of wavebands used in the estimate. The temperature

estimates cluster near Test/Teﬀ= 1, indicating that the

technique is not signiﬁcantly biased. The scatter in es-

timates decreases as more wavebands are used, from a

standard deviation of 7.6% if only a single brightness

temperature is used, down to 2.4% if photometry is ac-

quired in eight bands. We incorporate this systematic

error into our analysis by adding it in quadrature to

the observational uncertainties described in the follow-

ing paragraph. This has the desirable eﬀect that planets

with fewer observations have a larger systematic uncer-

tainty on their eﬀective temperature.

Fig. 3.— The Linear Interpolation technique for estimating day-

side eﬀective as tested on a suite of eleven hot Jupiter spect ral

models provided by J.J. Fortney. The y-axis shows the estima ted

day-side eﬀective temperature normalized by the actual mod el ef-

fective temperature. The x-axis represents the number of br ight-

ness temperatures used in the estimate. Each color correspo nds to

one of the eleven models used in the comparison. The black err or

bars represent the standard deviation in the normalized tem pera-

ture estimates.

Inpractice,wewouldliketopropagatethephotometric

uncertainties to the estimate of Teﬀ. For the Wien Dis-

placement technique, this uncertainty propagates triv-

ially to the eﬀective temperature. For the linear inter-

polation technique, a Monte Carlo can be used to esti-

mate the uncertainty in Teﬀ: the input eclipse depths

are randomly shifted 1000 times in a manner consistent

with their photometric uncertainties —assuming Gaus-

sianerrors—andtheeﬀectivetemperatureisrecomputed

repeatedly. Thescatterintheresultingvaluesof Teﬀpro-

vides an estimate of the observational uncertainty in the

parameter, to which we add in quadrature the estimate

ofsystematicerrordescribedabove. The resultinguncer-

tainties are listed in Table 1. These uncertainties should6 Cowan & Agol

be compared to the uncertainties in Tε=0(also listed in

Table 1), which are computed using the uncertainty in

the star’s properties and the planet’s orbit.

There are two practical issues with the linear interpo-

lation temperature estimation technique. In some cases,

onlyupperlimitshavebeenobtained, thereforeonecould

setψ= 0, with the appropriate1-sigmauncertainty. But

this approach leads to huge uncertainties in Teﬀfor plan-

ets with a secondary eclipse upper-limit near their black-

body peak. Instead of “punishing” these planets, we opt

to not use upper-limits (though for completeness we in-

clude them in Table 1). Secondly, when multiple mea-

surements of an eclipse depth have been published for

a given waveband, we use the most recent observation,

indicated with a superscript “ e” in Table 1. In all cases

these observations either explicitly agree with their older

counterpart, or agree with the re-analyzed older data.

4.RESULTS

4.1.Looking for Reﬂected Light

For each planet, we use thermal observations (essen-

tially those in the J, H, K s, andSpitzerbands) to es-

timate the planet’s eﬀective day-side temperature, Td,

and —when phase variations are available— Tn. These

values are listed in Table 1. In ﬁve cases (CoRoT-

1b, CoRoT-2b, HAT-P-7b, HD 209458b, TrES-2b), sec-

ondary eclipses and/or phase variations have been ob-

tained at optical wavelengths. Such observations have

the potential to directly constrain the albedo of these

planets. One approach is to adopt the Tdfrom thermal

observations and calculate the expected contrast ratio at

optical wavelengths, under the assumption of blackbody

emission (see also Kipping & Bakos 2010). Insofar as

the observed eclipse depths are deeper than this calcu-

lated depth, one can invoke the contribution of reﬂected

light and compute a geometric albedo, Ag. If one treats

the planet as a uniform Lambert sphere, the geometric

albedo is related to the spherical albedo at that wave-

length byAλ=3

2Ag. These values are listed in Table 1.

But reﬂected light is not the only explanation for an

unexpectedly deep optical eclipse. Alternatively, the

emissivity of the planets may simply be greater at op-

tical wavelengths than at mid-IR wavelengths, in agree-

mentwith realisticspectralmodelsofhotJupiters, which

predict brightness temperatures greater than Teﬀon the

Wien tail (see, for example, the Fortney et al. model

showninFigure2, whichdoesnotincludereﬂectedlight).

Note that this increasein emissivityshould occurregard-

less of whether or not the planet has a stratosphere: by

deﬁnition, the depth at which the optical thermal emis-

sion is emitted is the depth at which incident starlight

is absorbed, which will necessarily be a hot layer —

assuming the incident stellar spectrum peaks in the op-

tical.

Determining the albedo directly (ie: by observing re-

ﬂected light) can be diﬃcult for short period planets,

because there is no way to distinguish between reﬂected

and re-radiated photons. The blackbody peaks of the

star and planet often diﬀer by less than a micron. There-

fore, unlike Solar System planets, these worlds do not

exhibit a minimum in their spectral energy distribution

between the reﬂected and thermal peaks. The hottest

—and therefore most ambiguous case— of the ﬁve tran-siting planets with optical constraints is HAT-P-7b. If

one takes the mid-IR eclipse depths at face value, the