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fractional uncertainty in the eclipse depth; on the Wien
tail, the fractional error on brightness temperature can
be smaller because the ﬂux is very sensitive to tempera-
ture.
By the same token, a secondary eclipse depth and
phase variation amplitude at a given wavelength can be
combined to obtain a measure of the planet’s night-side
brightness temperature at that waveband.
Since the albedo and recirculation eﬃciency of the
planet are not known ahead of time, it is not immedi-
atelyobviouswhich wavelengthsaresensitiveto reﬂected
light and which are dominated by thermal emission. For
each planet, we compute the expected blackbody peak if
the planet has no albedo and no recirculation of energy,
λε=0= 2898/Tε=0µm. Insofar as real planets will have
non-zero albedo and non-zero recirculation, the day side
should never reach Tε=0, and the actual spectral energy
distributionwillpeakatslightlylongerwavelengths. The
coolest planet in our sample, Gl 436b, would exhibit a
blackbody peak at λε=0= 3.1µm, while the hottest
planet we consider, WASP-12b, has λε=0= 0.9µm.
In practice this means that ground-based near-IR and
space-based mid-IR (e.g., Spitzer) observations are as-
sumed to measure thermal emission, while space-based
optical observations (MOST, CoRoT, Kepler) may be
contaminated by reﬂected starlight.
In Figure2, wedemonstratetwo alternativetechniques
to convert an array of brightness temperatures, Tb(λ),
into an estimate of a planet’s eﬀective temperature, Teﬀ.
The solid black line shows a model spectrum of ther-
mal emission from Fortney et al. (2008), with an ef-
fective temperature of Teﬀ= 1941 K shown with the
black dashed line. The expected blackbody peak of
the planet is marked with a vertical dotted line. The
red points are the expected brightness temperatures in
the J, H, and K sbands (crosses), as well as the IRAC
(asterisks) and MIPS (diamond) instruments on Spitzer
(Fazio et al. 2004; Rieke et al. 2004; Werner et al. 2004).
Since the majority of the observations of exoplanets have
been obtained with SpitzerIRAC, we focus on estimat-
ingTeﬀbasedonlyon brightness temperatures in thoseAlbedo and Heat Recirculation on Hot Exoplanets 5
Fig. 2.— The solid black line shows a model spectrum from
Fortney et al. (2008) including only thermal emission (ie: n o re-
ﬂected light). The planet’s eﬀective temperature is shown w ith the
black dashed line, while the expected wavelength of the blac kbody
peak of the planet is marked with a black dotted line. The red
points show the expected brightness temperatures in the J, H , and
Ksbands (crosses), as well as the IRAC (asterisks) and MIPS (di a-
mond) instruments on Spitzer. The linear interpolation technique
described in the text is shown with the red line.
four bandpasses.
Wien Displacement: The ﬁrst approach is to simply
adopt the brightness temperature of the bandpass clos-
est to the planet’s blackbody peak (the black dotted
line). If only the four IRAC channels are available, the
best one can do is the 3.6 µm measurement, yielding
Teﬀ= 1925 K. There is —however— some subtlety in
estimating the peak wavelength, as this is dependent on
knowing the planet’s temperature (and hence ABandε)
a priori.
Linear Interpolation: The linear interpolation tech-
nique, shown with the red line in Figure 2, obviates the
need for an estimate of the planet’s temperature. The
brightness temperature is assumed to be constant short-
ward of the shortest- λobservation, and longward of the
longest-λobservation. Between bandpasses, the bright-
ness temperature changes linearly with λ. As long as
the various brightness temperatures do not diﬀer grossly
from one another, this technique implicitly gives more
weight to observations near the hypothetical blackbody
peak. The bolometric ﬂux of this “model” spectrum is
then computed, and admits a single eﬀective tempera-
ture, which is Teﬀ= 1927 K for the current example.
Since we hope to apply our routine to planets with well
sampled blackbody peaks, we adopt the linear interpola-
tion technique, as it can make use of multiple brightness
temperature estimates near the peak.
Thetwotechniquesdescribedaboveproducesimilaref-
fective temperatures, though —unsurprisingly— neither
gives precisely the correct answer. But these system-
atic errors are comparable or smaller than the photo-
metric uncertainty in observations of individual bright-
ness temperatures (see Table 1). The best IR observa-
tions for the nearest, brightest planetary systems (e.g.,
HD 189733b and HD 209458b) lead to observational un-
certainties of approximately 50 K in brightness temper-
ature. For many planets, the uncertainty is 100–200 K.
By that metric, either the Wien displacement or the lin-
ear interpolation routines give adequate estimates of the
eﬀective temperature, with errors of 16 K and 14 K, re-spectively.
Wemakeamorequantitativeanalysisofthesystematic
uncertainties involved in the Linear Interpolation tem-
perature estimates as follows. We produce 8800 mock
data sets: 100 realizations for 11 models and data in
up to 8 wavebands (J, H, K, IRAC, MIPS; Since this nu-
mericalexperiment choosesrandom bands from the eight
available, the results should not be very diﬀerent if ad-
ditional wavebands are considered). We run our Linear
Interpolation technique on each of these and plot in Fig-
ure 3 the estimated day-side temperature normalized by

fractional uncertainty in the eclipse depth; on the Wien

tail, the fractional error on brightness temperature can

be smaller because the ﬂux is very sensitive to tempera-

ture.

By the same token, a secondary eclipse depth and

phase variation amplitude at a given wavelength can be

combined to obtain a measure of the planet’s night-side

brightness temperature at that waveband.

Since the albedo and recirculation eﬃciency of the

planet are not known ahead of time, it is not immedi-

atelyobviouswhich wavelengthsaresensitiveto reﬂected

light and which are dominated by thermal emission. For

each planet, we compute the expected blackbody peak if

the planet has no albedo and no recirculation of energy,

λε=0= 2898/Tε=0µm. Insofar as real planets will have

non-zero albedo and non-zero recirculation, the day side

should never reach Tε=0, and the actual spectral energy

distributionwillpeakatslightlylongerwavelengths. The

coolest planet in our sample, Gl 436b, would exhibit a

blackbody peak at λε=0= 3.1µm, while the hottest

planet we consider, WASP-12b, has λε=0= 0.9µm.

In practice this means that ground-based near-IR and

space-based mid-IR (e.g., Spitzer) observations are as-

sumed to measure thermal emission, while space-based

optical observations (MOST, CoRoT, Kepler) may be

contaminated by reﬂected starlight.

In Figure2, wedemonstratetwo alternativetechniques

to convert an array of brightness temperatures, Tb(λ),

into an estimate of a planet’s eﬀective temperature, Teﬀ.

The solid black line shows a model spectrum of ther-

mal emission from Fortney et al. (2008), with an ef-

fective temperature of Teﬀ= 1941 K shown with the

black dashed line. The expected blackbody peak of

the planet is marked with a vertical dotted line. The

red points are the expected brightness temperatures in

the J, H, and K sbands (crosses), as well as the IRAC

(asterisks) and MIPS (diamond) instruments on Spitzer

(Fazio et al. 2004; Rieke et al. 2004; Werner et al. 2004).

Since the majority of the observations of exoplanets have

been obtained with SpitzerIRAC, we focus on estimat-

ingTeﬀbasedonlyon brightness temperatures in thoseAlbedo and Heat Recirculation on Hot Exoplanets 5

Fig. 2.— The solid black line shows a model spectrum from

Fortney et al. (2008) including only thermal emission (ie: n o re-

ﬂected light). The planet’s eﬀective temperature is shown w ith the

black dashed line, while the expected wavelength of the blac kbody

peak of the planet is marked with a black dotted line. The red

points show the expected brightness temperatures in the J, H , and

Ksbands (crosses), as well as the IRAC (asterisks) and MIPS (di a-

mond) instruments on Spitzer. The linear interpolation technique

described in the text is shown with the red line.

four bandpasses.

Wien Displacement: The ﬁrst approach is to simply

adopt the brightness temperature of the bandpass clos-

est to the planet’s blackbody peak (the black dotted

line). If only the four IRAC channels are available, the

best one can do is the 3.6 µm measurement, yielding

Teﬀ= 1925 K. There is —however— some subtlety in

estimating the peak wavelength, as this is dependent on

knowing the planet’s temperature (and hence ABandε)

a priori.

Linear Interpolation: The linear interpolation tech-

nique, shown with the red line in Figure 2, obviates the

need for an estimate of the planet’s temperature. The

brightness temperature is assumed to be constant short-

ward of the shortest- λobservation, and longward of the

longest-λobservation. Between bandpasses, the bright-

ness temperature changes linearly with λ. As long as

the various brightness temperatures do not diﬀer grossly

from one another, this technique implicitly gives more

weight to observations near the hypothetical blackbody

peak. The bolometric ﬂux of this “model” spectrum is

then computed, and admits a single eﬀective tempera-

ture, which is Teﬀ= 1927 K for the current example.

Since we hope to apply our routine to planets with well

sampled blackbody peaks, we adopt the linear interpola-

tion technique, as it can make use of multiple brightness

temperature estimates near the peak.

Thetwotechniquesdescribedaboveproducesimilaref-

fective temperatures, though —unsurprisingly— neither

gives precisely the correct answer. But these system-

atic errors are comparable or smaller than the photo-

metric uncertainty in observations of individual bright-

ness temperatures (see Table 1). The best IR observa-

tions for the nearest, brightest planetary systems (e.g.,

HD 189733b and HD 209458b) lead to observational un-

certainties of approximately 50 K in brightness temper-

ature. For many planets, the uncertainty is 100–200 K.

By that metric, either the Wien displacement or the lin-

ear interpolation routines give adequate estimates of the

eﬀective temperature, with errors of 16 K and 14 K, re-spectively.

Wemakeamorequantitativeanalysisofthesystematic

uncertainties involved in the Linear Interpolation tem-

perature estimates as follows. We produce 8800 mock

data sets: 100 realizations for 11 models and data in

up to 8 wavebands (J, H, K, IRAC, MIPS; Since this nu-

mericalexperiment choosesrandom bands from the eight

available, the results should not be very diﬀerent if ad-

ditional wavebands are considered). We run our Linear

Interpolation technique on each of these and plot in Fig-

ure 3 the estimated day-side temperature normalized by