1001.0012.txt

arXiv:1001.0012v2  [astro-ph.EP]  20 Dec 2010Draft version May 20, 2018
Preprint typeset using L ATEX style emulateapj v. 8/13/10
THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPL ANETS
Nicolas B. Cowan1,2, Eric Agol2,
Draft version May 20, 2018
ABSTRACT
If both the day-side and night-side effective temperatures of a pla net can be measured, it is possible
to estimate its Bond albedo, 0 < AB<1, as well as its day–night heat redistribution efficiency,
0< ε <1. We attempt a statistical analysis of the albedo and redistribution efficiency for 24
transiting exoplanets that have at least one published secondary e clipse. For each planet, we show
how to calculate a sub-stellar equilibrium temperature, T0, and associated uncertainty. We then use
a simple model-independent technique to estimate a planet’s effective temperature from planet/star
flux ratios. We use thermal secondary eclipse measurements —tho se obtained at λ >0.8 micron—
to estimate day-side effective temperatures, Td, and thermal phase variations —when available— to
estimatenight-sideeffectivetemperature. Westronglyruleoutth e“nullhypothesis”ofasingle ABand
εforall 24planets. If wealloweachplanet to havedifferent paramete rs,we find that lowBond albedos
are favored ( AB<0.35 at 1σconfidence), which is an independent confirmation of the low albedos
inferred from non-detection of reflected light. Our sample exhibits a wide variety of redistribution
efficiencies. When normalized by T0, the day-side effective temperatures of the 24 planets describe
a uni-modal distribution. The two biggest outliers are GJ 436b (abno rmally hot) and HD 80606b
(abnormally cool), and these are the only eccentric planets in our sa mple. The dimensionless quantity
Td/T0exhibits no trend with the presence or absence of stratospheric in versions. There is also no
clear trend between Td/T0andT0. That said, the 6 planets with the greatest sub-stellar equilibrium
temperatures ( T >2400 K) have low ε, as opposed to the 18 cooler planets, which show a variety
of recirculation efficiencies. This hints that the very hottest trans iting giant planets are qualitatively
different from the merely hot Jupiters. We propose an explanation o f this trend based on how a
planet’s radiative and advective times scale with temperature: both timescales are expected to be
shorter for hotter planets, but the temperature-dependance of the radiative timescale is stronger,
leading to decreased heat recirculation efficiency.
Subject headings: methods: data analysis — (stars:) planetary systems —
1.INTRODUCTION
Short-period exoplanets are expected to have atmo-
spheric compositions and dynamics that differ signifi-
cantly from Solar System giant planets3. These planets
orbit∼100×closer to their host stars than Jupiter does
from the Sun. As a result, they receive ∼104×more flux
andexperiencetidalforces ∼106×strongerthanJupiter.
In contrast to Jupiter, which releases roughly as much
power in its interior as it receives from the Sun, short-
period exoplanets have power budgets dictated by the
flux they receive from their host stars. Roughly speak-
ing, the stellar flux incident on a planet does one of two
things: it is reflected back into space, or advected else-
where on the planet and re-radiated at different wave-
lengths. The physical parameters that describe these
processes are the planet’s Bond albedo and redistribu-
tion efficiency.
1.1.Albedo
1CIERA Fellow, Northwestern University, 2131 Tech Dr,
Evanston, IL 60208
email: n-cowan@northwestern.edu
2Astronomy Department, University of Washington, Box
351580, Seattle, WA 98195
3For our purposes a “short period” exoplanet is one where the
periastron distance is less than 0 .1 AU, regardless of its actual
period, and regardless of mass, which may range from Neptune -
sized to Brown Dwarf. They are all Class IV and V extrasolar
giant planets in the scheme of Sudarsky et al. (2003).Giant planets in the Solar System have albedos greater
than 50%because ofthe presenceofcondensedmolecules
(H2O, CH 4, NH3, etc.) in their atmospheres. Planets
with effective temperatures exceeding ∼400 K should be
cloud free, leading to albedos of 0.05–0.4 (Marley et al.
1999). If pressure-broadenedNa and K opacity is impor-
tant at optical wavelengths (as it is for brown dwarfs,
Burrows et al. 2000), then the Bond albedos of hot
Jupiters may be less than 10% (Sudarsky et al. 2000).
But the very hottest planets, the so-called class V extra-
solar giant planets ( Teff>1500 K), might have very high
albedosdue to a high silicate cloud layer(Sudarsky et al.
2000). For a planet whose albedo is dominated by
clouds (as opposed to Rayleigh scattering) the albedo
depends on the composition and size of cloud particles
(Seager et al. 2000).
Earlyattempts to observe reflected light from exoplan-
ets (Charbonneau et al. 1999; Collier Cameron et al.
2002a; Leigh et al. 2003a,b; Rodler et al. 2008, 2010) in-
dicated that they might not be as reflective as Solar Sys-
tem gas giants (for a review, see Langford et al. 2010).
Measurements of HD 209458b taken with the Cana-
dian MOST satellite revealed a very low albedo ( <8%,
Rowe et al.2008), andit hassincebeentakenforgranted
that all short-period planets have albedos on par with
that of charcoal.
From the standpoint of the planet’s climate, the im-
portant factor is not the albedo at any one wavelength,2 Cowan & Agol
Aλ, but rather the integrated albedo, weighted by the in-
cident stellar spectrum, known as the Bond albedo and
denoted in this paper as AB. The relation between Aλ
and the planet’s Bond albedo is not trivial. If the albedo
is dominated by gray clouds, then the albedo at a sin-
gle wavelength can indeed be extrapolated to obtain AB.
For non-grayreflectance spectra, however, it is critical to
measureAλat the peak emitting wavelength of the host
startoobtainagoodestimateofthe planet’senergybud-
get. For example, as pointed out in Marley et al. (1999),
planets with identical albedo spectra, Aλ, mayhaveradi-
cally different ABdepending on the spectraltype oftheir
host stars.
1.2.Redistribution Efficiency
The first few measurements of hot Jupiter phase vari-
ations showed signs that these planets are not all cut
from the same cloth. Harrington et al. (2006) and
Knutson et al. (2007a) quoted very different phase func-
tion amplitudes for the υAndromeda and HD 189733
systems. It was not clear whether the differences were
intrinsic to the planets, however, because the data
were taken with different instruments, at different wave-
lengths, and with very different observation schemes (in
any case, subsequent re-analysis of the original data and
newly aquired Spitzerobservations of υAndromeda b
paint a completely different picture of that system:
Crossfield et al. 2010).
The uniform study presented in Cowan et al. (2007),
on the other hand, showed that HD 179949b and
HD209458bexhibit significantlydifferentdegreesofheat
recirculation, confirming suspicions. But it was not clear
whether hot exoplanets were uni-modal or bi-modal in
redistribution: are HD 179949b and HD 209458b end-
members of a single distribution, or prototypes for two
fundamentally different sorts of exoplanets?
The presence or lack of a stratospheric tempera-
ture inversion (Hubeny et al. 2003; Fortney et al. 2006;
Burrows et al. 2007, 2008; Zahnle et al. 2009) on the
day-sides of exoplanets has been invoked to explain
a purported bi-modality in recirculation efficiency on
hot Jupiters (Fortney et al. 2008). The argument, sim-
ply put, is that optical absorbers high in the atmo-
sphere of extremely hot Jupiters (equilibrium temper-
atures greater than ∼1700 K) would absorb incident
photons where the radiative timescales are short, mak-
ingit difficult forthese planets torecirculateenergy. The
most robust detection of this temperature inversionis for
HD 209458b (Knutson et al. 2008), but this planet does
not exhibit a large day-night brightness contrast at 8 µm
(Cowan et al. 2007). So while temperature inversions
seem to exist in the majority of hot Jupiter atmospheres
(Knutson et al. 2010), their connection to circulation ef-
ficiency —if any— is not clear.
1.3.Outline of Paper
It has been suggested (e.g., Harrington et al. 2006;
Cowan et al. 2007) that observations of secondary
eclipses and phase variations each constrain a combina-
tion of a planet’s Bond albedo and circulation efficiency.
But observations —even phase variations— at a single
waveband do little to constrain a planet’s energy bud-
get. In this work we show how observations in differentwavebands and for different planets can be meaningfully
combined to estimate these planetary parameters.
In§2 we introduce a simple model to quantify the
day-side and night-side energy budget of a short-period
planet, and show how a planet’s Bond albedo, AB, and
redistribution efficiency, ε, can be constrained by ob-
servations. In §3 we use published observations of
24 transiting planets to estimate day-side and —where
appropriate—night-sideeffective temperatures. We con-
struct a two-dimensionaldistribution function in ABand
εin§4. We state our conclusions in §5.
2.PARAMETERIZING THE ENERGY BUDGET
2.1.Incident Flux
Short-period planets have a power budget entirely dic-
tated by the flux they receive from their host star,
which dwarfs tidal heating or remnant heat of forma-
tion. Following Hansen (2008), we define the equi-
librium temperature at the planet’s sub-stellar point:
T0(t) =Teff(R∗/r(t))1/2, whereTeffandR∗are the star’s
effective temperature and radius, and r(t) is the planet–
star distance (for a circular orbit ris simply equal to the
semi-major axis, a). For shorthand, we define the geo-
metrical factor a∗=a/R∗, which is directly constrained
by transit lightcurves (Seager & Mall´ en-Ornelas 2003).
The incident flux on the planet is given by Finc=
1
2σBT4
0, and it is significant that this quantity has some
associated uncertainty. For a planet on a circular orbit,
the uncertainty in T0=Teff/√a∗is related —to first
order— to the uncertainties in the host star’s effective
temperature, and the geometrical factor:
σ2
T0
T2
0=σ2
Teff
T2
eff+σ2
a∗
4a2∗. (1)
For a planet with non-zero eccentricity, T0varies with
time, but we are only concerned with its value at su-
perior conjunction: secondary eclipse occurs at superior
conjunction, when we are seeing the planet’s day-side.
At that point in the orbit, the planet–star distance is
rsc=a(1−e2)/(1−esinω), whereeandωare the
planet’s orbital eccentricity and argument of periastron,
respectively.
For planets with non-zero eccentricity, the uncertainty
inT0is given by
σ2
T0
T2
0=σ2
Teff
T2
eff+σ2
a∗
4a2∗+/parenleftBig
e2cos2ω
1−e2/parenrightBig
σ2
ecosω
+/parenleftBig
esinω
1−e2−1
2(1−esinω)/parenrightBig
σ2
esinω,(2)
whereσecosωandσesinωarethe observationaluncertain-
ties in the two components of the planet’s eccentricity4.
2.2.Emergent Flux
At secondary eclipse, and in the absence of albedo or
energy circulation, the equilibrium temperature of a re-
gion on the planet depends on the normalized projected
4This formulation is preferable to an error estimate based on σe
andσω, because the eccentricity and argument of periastron are
highlycorrelated inorbitalfits. Thatsaid, the uncertaint iesσecosω
andσesinωare often not included in the literature, in which case
we use a slightly different —and more conservative— formulat ion
of the error budget using σeandσω.Albedo and Heat Recirculation on Hot Exoplanets 3
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
higher than the hemisphere-averaged temperature, ex-
cept in the ε= 1 limit. This is also why the quantity
T4
d+T4
nis still a weak function of ε.
Fig. 1.— Different kinds of idealized observations constrain the
Bond albedo, ABand circulation efficiency, ε, differently. A mea-
surement of the secondary eclipse depth at optical waveleng ths is
a measure of albedo (solid line). A secondary eclipse depth a t
thermal wavelengths gives a joint constraint on albedo and r ecir-
culation (dotted line). A measurement of the night-side effe ctive
temperature from thermal phase variations yields a constra int (the
dashed line) nearly orthogonal to the day-side measurement .
In Figure 1 we show how different kinds of observa-4 Cowan & Agol
tions constrain ABandε. For this example, we chose
constraints consistent with AB= 0.2 andε= 0.3. The
solid line is a locus of constant AB; the dotted line is
the locus of constant Td/T0; the dashed line is a lo-
cus of constant Tn/T0. From this figure it is clear that
the measurements complement each other: measuring
two of the three quantities (Bond albedo, effective day-
side or night-side temperatures) uniquely determines the
planet’s albedo and circulation efficiency. When obser-
vations have some associated uncertainty, they define a
swath through the AB–εplane.
3.ANALYSIS
3.1.Planetary & Stellar Data
We begin by considering all the photometric obser-
vations of short-period exoplanets published through
November 2010, summarized in Table 1. We have dis-
carded photometric observations of non-transiting plan-
ets because of their unknown radius and orbital inclina-
tion5. This leaves us with 24 transiting exoplanets for
which there are observations in at least one waveband
at superior conjunction, and in some cases in multiple
wavebands and at multiple planetary phases.
Stellar and planetary data are taken from the Ex-
oplanet Encyclopedia (exoplanet.eu), and references
therein. We repeated parts of the analysis with the
Exoplanet Data Explorer database (exoplanets.org) and
found identical results, within the uncertainties. When
the stellar data are not available, we have assumed typi-
cal parameters for the appropriate spectral class, and so-
lar metallicity. Insofar as we are only concerned with the
broadband brightnesses of the stars, our results should
not depend sensitively on the input stellar parameters.
Knowing the stars’ Teff, loggand [Fe/H], we
use the PHOENIX/NextGen stellar spectrum grids
(Hauschildt et al. 1999) to determine their brightness
temperatures at the observed bandpasses. At each wave-
band for which eclipse or phase observations have been
obtained, we determine the ratio of the stellar flux to the
blackbodyfluxatthatgridstar’s Teff. Wethenapplythis
factor to the Teffof the observed star.
It is worth noting that the choice of stellar model leads
to systematic uncertainties in the planetary brightness
that are of order the photometric uncertainties. For
example, Christiansen et al. (2010) use stellar models
for HAT-P-7 from Kurucz (2005), while we use those
of Hauschildt et al. (1999). The resulting 8 µm bright-
ness temperatures for HAT-P-7b differ by as much as
600 K, or slightly more than 1 σ. Our uniform use
of Hauschildt et al. (1999) models should alleviate this
problem, however.
3.2.From Flux Ratios to Effective Temperature
The planet’s albedo and recirculation efficiency gov-
ern its effective day-side and night-side temperatures, Td
andTn, respectively. Observationally, we can only mea-
sure the brightness temperature, ideally at a number of
different wavelengths: Tb(λ). If one knew, a priori, the
5For completeness, these are: τ-Bootis b, υ-Andromeda b,
51 Peg b, Gl 876d, HD 75289b, HD 179949b and HD 46375b
(Charbonneau et al. 1999; Collier Cameron et al. 2002b;
Leigh et al. 2003a,b; Harrington et al. 2006; Cowan et al. 200 7;
Seager & Deming 2009; Crossfield et al. 2010; Gaulme et al. 201 0)emergent spectrum of a planet, one could trivially con-
vert a single brightness temperature to an effective tem-
perature. Alternatively, if observations were obtained at
a number of wavelengths bracketing the planet’s black-
body peak, it would be possible to estimate the planet’s
bolometric flux and hence its effective temperature in a
model-independent way (e.g., Barman 2008).
We adopt the latter empirical approach of converting
observed flux ratios into brightness temperatures, then
using these to estimate the planet’s effective tempera-
ture. The secondary eclipse depth in some waveband di-
vided by the transit depth is a direct measureofthe ratio
of the planet’s day-side intensity to the star’s intensity
at that wavelength, ψ(λ). Knowing the star’s brightness
temperature at a given wavelength, it is possible to com-
pute the apparent brightness temperature of the planet’s
day side:
Tb(λ) =hc
λk/bracketleftbigg
log/parenleftbigg
1+ehc/λkT∗
b(λ)−1
ψ(λ)/parenrightbigg/bracketrightbigg−1
.(6)
On the Rayleigh-Jeans tail, the fractional uncertainty
in the brightness temperature is roughly equal to the
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
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
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
two separate peaks in ε: if short-period giant planets
have uniformly low albedos, then there appear to be two
modes of heat recirculation efficiency. We revisit this
idea below.
Figure 6 shows that planets in this sample are consis-
tent with a low Bond albedo. Note that this constraint
is based entirely on near- and mid-infrared observations,
and is thus independent from the claims of low albedo
based on searches for reflected light (Rowe et al. 2008,
and references therein). Furthermore, this is a constraint
on the Bond albedo, rather than the albedo in any lim-
ited wavelength range.
In Figure 7 we plot the dimensionless day-side effec-
tive temperature, Td/T0, against the maximum expected
day-side temperature, Tε=0. The cyan asterisks in Fig-
ure 7 show the four hot Jupiters without temperature
inversions, while most of the remaining planets have in-
versions (Knutson et al. 2010). The presence or absence
of an inversion does not appear to affect the efficiency of
day–night heat recirculation.
Planets should lie below the solid red line in Figure 7,
which denotes Tε=0= (2/3)1/4T0. Of the 24 planets in
our sample, only one (Gl 436b) has a day-side effective
temperature significantly above the Tε=0limit6. This
planet is by far the coolest in our sample, it is on an ec-
centric orbit, and observations indicate that it may have
a non-equilibrium atmosphere (Stevenson et al. 2010).
There is no reason, on the other hand, that planets
shouldn’t lie below the red dotted line in Figure 7:
all it would take is non-zero Bond albedo. That said,
only 3 of the 24 planets we consider are in this region,
6This is driven by the abnormally high 3.6 micron brightness
temperature; including the 4.5 micron eclipse upper limit d oes not
significantly change our estimate of this planet’s effective temper-
ature.8 Cowan & Agol
Fig. 7.— The dimensionless day-side effective temperature,
Td/T0, plotted against the maximum expected day-side temper-
ature,Tε=0. The red lines correspond to three fiducial limits of
recirculation, assuming AB= 0: no recirculation (solid), uniform
day-hemisphere (dashed), and uniform planet (dotted). The gray
points indicate the default values (using only observation s with
λ >0.8 micron) for the four planets whose optical eclipse depths
may be probing thermal emission rather than just reflected li ght
(from left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-P-7b ).
For these planets we have here elected to include optical mea sure-
ments in our estimate of the day-side bolometric flux and effec tive
temperature, shown in black. The cyan asterisks denote thos e hot
Jupiters known notto have a stratospheric inversion according
to (Knutson et al. 2010). They are, from left to right: TrES-1 b,
HD 189733b, TrES-3b, WASP-4b. The two red x’s denote the ec-
centric planets in our sample, which are also the two worst ou tliers.
with the greatest outlier being HD 80606b, a planet on
an extremely eccentric orbit with superior conjunction
nearly coinciding with periastron. As such, it is likely
that much of the energy absorbed by the planet at that
point in its orbit performs mechanical work (speeding up
winds, puffingupthe planet, etc. SeealsoCowan & Agol
2010) rather than merely warming the gas. Gl 436b and
HD 80606b are denoted by red x’s in Figure 7.
The gray points in Figure 7 indicate the default val-
ues (using only observationswith λ>0.8 micron) for the
four planets whose optical eclipse depths may be probing
thermal emission rather than just reflected light (from
left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-
P-7b). For these planets we have here elected to use
all available flux ratios (including optical observations
potentially contaminated by reflected light) to estimate
the day-side bolometric flux and effective temperature,
shown as black points in Figure 7.
If one takes these day-side effective temperature es-
timates at face value, it appears that the planets with
Tε=0<2400 K exhibit a wide-variety of redistribution
efficiencies and/or Bond albedos, but are consistent with
AB= 0. It is worth noting that many of the best char-
acterized planets in this region have Td/T0≈0.75, and
this accounts for the sharp peak in the dotted line of Fig-
ure 5 atε= 0.75. The hottest 6 planets, on the other
hand, have uniformly high Td/T0, indicating that they
have both low Bond albedo andlow redistribution effi-
ciency. These planets must not have the high-altitude,
reflective silicate clouds hypothesized in Sudarsky et al.
(2000). But this conclusion is dependent on how one
interprets the Keplerobservations of HAT-P-7b: if the
large optical flux ratio is due to reflected light, then this
planet is cooler than we think, and even the hottest tran-siting planets exhibit a variety of behaviors.
5.SUMMARY & CONCLUSIONS
We have described how to estimate a planet’s incident
power budget ( T0), where the uncertainties are driven by
the uncertainties in the host star’s effective temperature
and size, as well as the planet’s orbit. We then described
a model-independent technique to estimate the effective
temperature of a planet based on planet/star flux ra-
tiosobtained at variouswavelengths. When the observed
day-side and night-side effective temperatures are com-
pared, one can constrain a combination of the planet’s
Bond albedo, AB, and its recirculation efficiency, ε. We
applied this analysis on 24 known transiting planets with
measured infrared eclipse depths.
Our principal results are:
1. Essentially all of the planets are consistent with low
Bond albedo.
2. We firmly rule out the “null hypothesis”, whereby all
transiting planets can be fit by a single ABandε. It
is not immediately clear whether this stems from differ-
ences in Bond albedo, recirculation efficiency, or both.
3. In the few cases where it is possible to unambiguously
infer an albedo based on optical eclipse depths, they are
extremely low, implying correspondingly low Bond albe-
dos (<10%). If one adopts such low albedos for all
the planets in our sample, the discrepancies in day-side
effective temperature must be due to differences in recir-
culation efficiency.
4. These differences in recirculation efficiency do not
appear to be correlated with the presence or absence of
a stratospheric inversion.
5. Planets cooler than Tε=0= 2400 K exhibit a wide va-
riety of circulation efficiencies that do not appear to be
correlated with equilibrium temperature. Alternatively,
theseplanetsmayhavedifferent (but generallylow)albe-
dos. Planets hotter than Tε=0= 2400 K have uniformly
low redistribution efficiencies and albedos.
The apparent decrease in advective efficiency with
increasing planetary temperature remains unexplained.
One hypothesis, mentioned earlier, is that TiO and VO
would provide additional optical opacity in atmospheres
hotter than T∼1700 K, leading to temperature in-
versions and reduced heat recirculation on these plan-
ets (Fortney et al. 2008). But if our sample shows any
sharp change it behavior it occurs near 2400 K, rather
than 1700K. One couldinvokeanotheroptical absorber,
but in any case the lack of correlation —pointed out in
thisworkandelsewhere—betweenthepresenceofatem-
perature inversionand the efficiency of heat recirculation
makes this explanation suspect. Another possible expla-
nation for the observed trend is that the hottest planets
have the most ionized atmospheres and may suffer the
most severe magnetic drag (Perna et al. 2010).
The simplest explanation for this trend is simply that
the radiative time is a steeper function of temperature
than the advective time: advective efficiency is given
roughly by the ratio of the radiative and advective times
(eg: Cowan & Agol 2010). In the limit of Newtonian
cooling, the radiative time scales as τrad∝T−3. If one
assumes the wind speed to be of order the local sound
speed, then the advective time scales as τadv∝T−0.5.
One might therefore naively expect the advective effi-
ciency to scale as T−2.5. Such an explanation would notAlbedo and Heat Recirculation on Hot Exoplanets 9
explain the apparent sharp transition seen at 2400 K,
however.
The combination of optical observations of secondary
eclipses and thermal observations of phase variations is
the best way to constrain planetary albedo and circu-
lation. The optical observations should be taken near
the star’s blackbody peak, both to maximize signal-to-
noise, and to avoidcontaminationfrom the planet’s ther-
mal emission, but this separationmay not be possible for
the hottest transiting planets. The thermal observations,
likewise, should be near the planet’s blackbody peak to
better constrain its bolometric flux. Note that this wave-
length is shortwardof the ideal contrastratio, which typ-
ically falls on the planet’s Rayleigh-Jeans tail. Further-
more, the thermal phase observations should span a full
planetaryorbit: thelightcurveminimumisthemostsen-
sitive measure of ε, and should occur nearly half an orbit
apart from the light curve maximum, despite skewed di-
urnal heatingpatterns (Cowan & Agol 2008, 2010). This
means that observing campaigns that only cover a little
more than half an orbit (transit →eclipse) are probably
underestimating the real peak-trough phase amplitude.A possible improvement to this study would be to per-
form a uniform data reduction for all the Spitzerexo-
planet observations of hot Jupiters. These data make up
the majority of the constraints presented in our study
and most are publicly available. And while the pub-
lished observations were analyzed in disparate ways, a
consensus approach to correcting detector systematics is
beginning to emerge.
N.B.C. acknowledges useful discussions of aspects of
this work with T. Robinson, M.S. Marley, J.J. Fort-
ney, T.S. Barman and D.S. Spiegel. Thanks to our
referee B.M.S. Hansen for insightful feedback, and to
E.D. Feigelson for suggestions about statistical methods.
N.B.C. was supported by the Natural Sciences and Engi-
neering Research Council of Canada. E.A. is supported
by a National Science Foundation Career Grant. Sup-
port for this work was provided by NASA through an
award issued by JPL/Caltech. This research has made
use of the Exoplanet Orbit Database and the Exoplanet
Data Explorer at exoplanets.org.
REFERENCES
Agol, E., Cowan, N. B., Knutson, H. A., Deming, D., Steffen,
J. H., Henry, G. W., & Charbonneau, D. 2010, ApJ, 721, 1861
Alonso, R. et al. 2009a, A&A, 506, 353
Alonso, R., Deeg, H. J., Kabath, P., & Rabus, M. 2010, AJ, 139,
1481
Alonso, R., Guillot, T., Mazeh, T., Aigrain, S., Alapini, A. ,
Barge, P., Hatzes, A., & Pont, F. 2009b, A&A, 501, L23
Anderson, D. R. et al. 2010, A&A, 513, L3+
Barman, T. S. 2008, ApJ, 676, L61
Barnes, J. R., Barman, T. S., Prato, L., Segransan, D., Jones ,
H. R. A., Leigh, C. J., Collier Cameron, A., & Pinfield, D. J.
2007, MNRAS, 382, 473
Beerer, I. M., et al. 2010, arXiv:1011.4066
Borucki, W. J. et al. 2009, Science, 325, 709
Burrows, A., Budaj, J., & Hubeny, I. 2008, ApJ, 678, 1436
Burrows, A., Hubeny, I., Budaj, J., Knutson, H. A., &
Charbonneau, D. 2007, ApJ, 668, L171
Burrows, A., Marley, M. S., & Sharp, C. M. 2000, ApJ, 531, 438
Burrows, A., Sudarsky, D., & Hubeny, I. 2006, ApJ, 650, 1140
Campo, C. J. et al. 2010, ArXiv e-prints
Charbonneau, D. et al. 2005, ApJ, 626, 523
Charbonneau, D., Knutson, H. A., Barman, T., Allen, L. E.,
Mayor, M., Megeath, S. T., Queloz, D., & Udry, S. 2008, ApJ,
686, 1341
Charbonneau, D., Noyes, R. W., Korzennik, S. G., Nisenson, P .,
Jha, S., Vogt, S. S., & Kibrick, R. I. 1999, ApJ, 522, L145
Christiansen, J. L. et al. 2010, ApJ, 710, 97
Christiansen, J. L., et al. 2010b, arXiv:1011.2229
Collier Cameron, A., Horne, K., Penny, A., & Leigh, C. 2002a,
MNRAS, 330, 187
—. 2002b, MNRAS, 330, 187
Cowan, N. B., Agol, E., & Charbonneau, D. 2007, MNRAS, 379,
641
Cowan, N. B., & Agol, E. 2008, ApJ, 678, L129
Cowan, N. B., & Agol, E. 2010, arXiv:1011.0428
Croll, B., Albert, L., Lafreniere, D., Jayawardhana, R., &
Fortney, J. J. 2010a, ApJ, 717, 1084
Croll, B., Jayawardhana, R., Fortney, J. J., Lafreni` ere, D ., &
Albert, L. 2010b, ApJ, 718, 920
Croll, B., Lafreniere, D., Albert, L., Jayawardhana, R., Fo rtney,
J. J., & Murray, N. 2010c, arXiv:1009.0071
Crossfield, I. J. M., Hansen, B. M. S., Harrington, J., Cho,
J. Y.-K., Deming, D., Menou, K., & Seager, S. 2010, ApJ, 723,
1436
Deming, D., Harrington, J., Laughlin, G., Seager, S., Navar ro,
S. B., Bowman, W. C., & Horning, K. 2007, ApJ, 667, L199Deming, D., Harrington, J., Seager, S., & Richardson, L. J. 2 006,
ApJ, 644, 560
Deming, D., Seager, S., Richardson, L. J., & Harrington, J. 2 005,
Nature, 434, 740
Deming, D., et al. 2010, arXiv:1011.1019
Demory, B.-O. et al. 2007, A&A, 475, 1125
Fazio, G. G. et al. 2004, ApJS, 154, 10
Fortney, J. J., Cooper, C. S., Showman, A. P., Marley, M. S., &
Freedman, R. S. 2006, ApJ, 652, 746
Fortney, J. J., Lodders, K., Marley, M. S., & Freedman, R. S.
2008, ApJ, 678, 1419
Fressin, F., Knutson, H. A., Charbonneau, D., O’Donovan, F. T.,
Burrows, A., Deming, D., Mandushev, G., & Spiegel, D. 2010,
ApJ, 711, 374
Gaulme, P., et al. 2010, A&A, 518, L153
Gillon, M. et al. 2009, A&A, 506, 359
—. 2010, A&A, 511, A260000+
Hansen, B. M. S. 2008, ApJS, 179, 484
Harrington, J., Hansen, B. M., Luszcz, S. H., Seager, S., Dem ing,
D., Menou, K., Cho, J. Y.-K., & Richardson, L. J. 2006,
Science, 314, 623
Hauschildt, P. H., Allard, F., & Baron, E. 1999, ApJ, 512, 377
Hubeny, I., Burrows, A., & Sudarsky, D. 2003, ApJ, 594, 1011
Kipping, D. M., & Bakos, G. ´A. 2010, arXiv:1004.3538
Kipping, D. M., & Bakos, G. ´A. 2010b, arXiv:1006.5680
Knutson, H. A., Charbonneau, D., Allen, L. E., Burrows, A., &
Megeath, S. T. 2008, ApJ, 673, 526
Knutson, H. A. et al. 2007a, Nature, 447, 183
Knutson, H. A., Charbonneau, D., Burrows, A., O’Donovan,
F. T., & Mandushev, G. 2009a, ApJ, 691, 866
Knutson, H. A., Charbonneau, D., Cowan, N. B., Fortney, J. J. ,
Showman, A. P., Agol, E., & Henry, G. W. 2009b, ApJ, 703,
769
Knutson, H. A. et al. 2009c, ApJ, 690, 822
Knutson, H. A., Charbonneau, D., Deming, D., & Richardson,
L. J. 2007b, PASP, 119, 616
Knutson, H. A., Howard, A. W., & Isaacson, H. 2010, ApJ, 720,
1569
Kurucz, R. L. 2005, Memorie della Societa Astronomica Itali ana
Supplement, 8, 14
Langford, S. V., Wyithe, J. S. B., Turner, E. L., Jenkins, E. B .,
Narita, N., Liu, X., Suto, Y., & Yamada, T. 2010,
arXiv:1006.5492
Laughlin, G., Deming, D., Langton, J., Kasen, D., Vogt, S.,
Butler, P., Rivera, E., & Meschiari, S. 2009, Nature, 457, 56 2
Leigh, C., Cameron, A. C., Horne, K., Penny, A., & James, D.
2003a, MNRAS, 344, 127110 Cowan & Agol
Leigh, C., Collier Cameron, A., Udry, S., Donati, J.-F., Hor ne,
K., James, D., & Penny, A. 2003b, MNRAS, 346, L16
L´ opez-Morales, M., Coughlin, J. L., Sing, D. K., Burrows, A .,
Apai, D., Rogers, J. C., Spiegel, D. S., & Adams, E. R. 2010,
ApJ, 716, L36
Machalek, P., Greene, T., McCullough, P. R., Burrows, A., Bu rke,
C. J., Hora, J. L., Johns-Krull, C. M., & Deming, D. L. 2010,
ApJ, 711, 111
Machalek, P., McCullough, P. R., Burke, C. J., Valenti, J. A. ,
Burrows, A., & Hora, J. L. 2008, ApJ, 684, 1427
Machalek, P., McCullough, P. R., Burrows, A., Burke, C. J.,
Hora, J. L., & Johns-Krull, C. M. 2009, ApJ, 701, 514
Marley, M. S., Gelino, C., Stephens, D., Lunine, J. I., &
Freedman, R. 1999, ApJ, 513, 879
Nymeyer, S., et al. 2010, arXiv:1005.1017
O’Donovan, F. T., Charbonneau, D., Harrington, J.,
Madhusudhan, N., Seager, S., Deming, D., & Knutson, H. A.
2010, ApJ, 710, 1551
Perna, R., Menou, K., & Rauscher, E. 2010, ApJ, 719, 1421
Richardson, L. J., Deming, D., & Seager, S. 2003, ApJ, 597, 58 1
Rieke, G. H. et al. 2004, ApJS, 154, 25
Rodler, F., K¨ urster, M., & Henning, T. 2008, A&A, 485, 859
—. 2010, A&A, 514, A23+Rogers, J. C., Apai, D., L´ opez-Morales, M., Sing, D. K., &
Burrows, A. 2009, ApJ, 707, 1707
Rowe, J. F. et al. 2008, ApJ, 689, 1345
Seager, S., & Deming, D. 2009, ApJ, 703, 1884
Seager, S., Whitney, B. A., & Sasselov, D. D. 2000, ApJ, 540, 5 04
Seager, S., & Mall´ en-Ornelas, G. 2003, ApJ, 585, 1038
Sing, D. K., & L´ opez-Morales, M. 2009, A&A, 493, L31
Snellen, I. A. G., & Covino, E. 2007, MNRAS, 375, 307
Snellen, I. A. G., de Mooij, E. J. W., & Albrecht, S. 2009, Natu re,
459, 543
Snellen, I. A. G., de Mooij, E. J. W., & Burrows, A. 2010, A&A,
513, A76+
Spiegel, D. S., & Burrows, A. 2010, ApJ, 722, 871
Stevenson, K. B. et al. 2010, Nature, 464, 1161
Sudarsky, D., Burrows, A., & Hubeny, I. 2003, ApJ, 588, 1121
Sudarsky, D., Burrows, A., & Pinto, P. 2000, ApJ, 538, 885
Todorov, K., Deming, D., Harrington, J., Stevenson, K. B.,
Bowman, W. C., Nymeyer, S., Fortney, J. J., & Bakos, G. A.
2010, ApJ, 708, 498
Werner, M. W. et al. 2004, ApJS, 154, 1
Wheatley, P. J., et al. 2010, arXiv:1004.0836
Zahnle, K., Marley, M. S., Freedman, R. S., Lodders, K., &
Fortney, J. J. 2009, ApJ, 701, L20Albedo and Heat Recirculation on Hot Exoplanets 11
TABLE 1
Secondary Eclipses & Phase Variations of Exoplanets
Planet Tε=0[K]aλ[µm]bEclipse DepthcTbright[K] Phase AmplitudecDerived Quantitiesd
CoRoT-1b12424(84) 0.60(0.42) 1 .6(6)×10−42726(141) Td=2674(144) K
0.71(0.25) 1 .26(33)×10−42409(75) 1 .0(3)×10−4Aλ<0.1
2.10(0.02) 2 .8(5)×10−32741(125) Td(A= 0)=2515(84) K
2.15(0.32) 3 .36(42)×10−32490(157)
3.6(0.75) 4 .15(42)×10−32098(116)
4.5(1.0) 4 .82(42)×10−32084(106)
CoRoT-2b21964(42) 0.60(0.42) 6(2) ×10−52315(85) Td=1864(233) K
0.71(0.25) 1 .02(20)×10−42215(49) Aλ= 0.16(7)
1.65(0.25) <1.7×10−3(3σ) Td(A= 0)=2010(144) K
2.15(0.32) 1 .6(9)×10−31914(292)
3.6(0.75) 3 .55(20)×10−31798(40)
4.5(1.0)e4.75(19)×10−31791(33)
4.5(1.0) 5 .10(42)×10−3
8.0(2.9) 4 .1(1.1)×10−3
8.0(2.9)e4.09(80)×10−31318(143)
Gl 436b3934(41) 3.6(0.75) 4 .1(3)×10−41145(23) Td=1082(38) K
4.5(1.0) <1.0×10−4(3σ)
5.8(1.4) 3 .3(1.4)×10−4797(106)
8.0(2.9)e4.52(27)×10−4737(17)
8.0(2.9) 5 .7(8)×10−4
8.0(2.9) 5 .4(7)×10−4
16(5) 1 .40(27)×10−3963(126)
24(9) 1 .75(41)×10−31016(182)
HAT-P-1b41666(38) 3.6(0.75) 8 .0(8)×10−41420(47) Td=1439(59) K
4.5(1.0) 1 .35(22)×10−31507(100)
5.8(1.4) 2 .03(31)×10−31626(128)
8.0(2.9) 2 .38(40)×10−31564(151)
HAT-P-7b52943(95) 0.65(0.4) 1 .30(11)×10−43037(35) 1 .22(16)×10−4Td=2086(156) K
3.6(0.75) 9 .8(1.7)×10−42063(152) Aλ= 0.58(5)
4.5(1.0) 1 .59(22)×10−32378(179) Td(A= 0)=2830(86) K
5.8(1.4) 2 .45(31)×10−32851(235)
8.0(2.9) 2 .25(52)×10−32512(403)
HD 80606b61799(50) 8.0(2.9) 1 .36(18)×10−31137(73) Td=1137(113) K
HD 149026b71871(17) 8.0(2.9)e3.7(0.8)×10−4976(276) 2 .3(7)×10−4Td=1571(231) K
8.0(2.9) 8 .4(1.1)×10−4Tn=976(286) K
HD 189733b81537(16) 2.15(32) <4.0×10−4(1σ) Td=1605(52) K
3.6(0.75) 2 .56(14)×10−31639(34) Tn=1107(132) K
4.5(1.0) 2 .14(20)×10−31318(45)
5.8(1.4) 3 .10(34)×10−31368(69)
8.0(2.9) 3 .381(55)×10−3
8.0(2.9) 3 .91(22)×10−31.2(2)×10−3
8.0(2.9)e3.440(36)×10−31259(7) 1 .2(4)×10−3
16(5) 5 .51(30)×10−31338(52)
24(9) 5 .98(38)×10−3
24(9)e5.36(27)×10−31202(46) 1 .3(3)×10−3
HD 209458b91754(15) 0.5(0.3) 7(9) ×10−62368(156) Td=1486(53) K
2.15(0.32) <3×10−4(1σ) Aλ= 0.09(7)
3.6(0.75) 9 .4(9)×10−41446(45) Td(A= 0)=2031(128) K
4.5(1.0) 2 .13(15)×10−31757(57) Tn=1476(304) K
5.8(1.4) 3 .01(43)×10−31890(149)
8.0(2.9) 2 .40(26)×10−31480(94) <1.5×10−3(2σ)
24(9) 2 .60(44)×10−31131(143)
OGLE-TR-56b102874(84) 0.90(0.15) 3 .63(91)×10−42696(116) Td=2696(236) K
OGLE-TR-113b111716(33) 2.15(0.32) 1 .7(5)×10−31918(164) Td=1918(219) K
TrES-1b121464(16) 3.6(0.75) <1.5×10−3(1σ) Td=998(67) K
4.5(1.0) 6 .6(1.3)×10−4972(56)
8.0(2.9) 2 .25(36)×10−31152(94)
TrES-2b131917(21) 0.65(0.4) 1 .14(78)×10−52020(132) Td=1623(76) K
2.15(0.32) 6 .2(1.2)×10−41655(80) Aλ= 0.06(3)
3.6(0.75) 1 .27(21)×10−31490(84) Td(A= 0) = 1751(80) K
4.5(1.0) 2 .30(24)×10−31652(74)
5.8(1.4) 1 .99(54)×10−31373(177)
8.0(2.9) 3 .59(60)×10−31659(163)
TrES-3b142093(32) 0.7(0.3) <6.2×10−4(1σ) Td=1761(66) K
1.25(0.16) <5.1×10−4(3σ)
2.15(0.32) 2 .41(43)×10−3
2.15(0.32)e1.33(17)×10−31770(58)
3.6(0.75) 3 .46(35)×10−31818(73)12 Cowan & Agol
TABLE 1
Secondary Eclipses & Phase Variations of Exoplanets
4.5(1.0) 3 .72(54)×10−31649(107)
5.8(1.4) 4 .49(97)×10−31621(173)
8.0(2.9) 4 .75(46)×10−31480(82)
TrES-4b152250(37) 3.6(0.75) 1 .37(11)×10−31889(63) Td=1891(81) K
4.5(1.0) 1 .48(16)×10−31727(83)
5.8(1.4) 2 .61(59)×10−32112(283)
8.0(2.9) 3 .18(44)×10−32168(197)
WASP-1b162347(35) 3.6(0.75) 1 .17(16)×10−31678(87) Td=1719(89) K
4.5(1.0) 2 .12(21)×10−31923(91)
5.8(1.4) 2 .82(60)×10−32042(253)
8.0(2.9) 4 .70(46)×10−32587(176)
WASP-2b171661(69) 3.6(0.75) 8 .3(3.5)×10−41264(164) Td=1280(121) K
4.5(1.0) 1 .69(17)×10−31380(53)
5.8(1.4) 1 .92(77)×10−31299(232)
8.0(2.9) 2 .85(59)×10−31372(154)
WASP-4b182163(60) 3.6(0.75) 3 .19(31)×10−32156(97) Td=2146(140) K
4.5(1.0) 3 .43(27)×10−31971(75)
WASP-12b193213(119) 0.9(0.15) 8 .2(1.5)×10−43002(104) Td=2939(98) K
1.25(0.16) 1 .31(28)×10−32894(149)
1.65(0.25) 1 .76(18)×10−32823(88)
2.15(0.32) 3 .09(13)×10−33018(51)
3.6(0.75) 3 .79(13)×10−32704(49)
4.5(1.0) 3 .82(19)×10−32486(68)
5.8(1.4) 6 .29(52)×10−33167(179)
8.0(2.9) 6 .36(67)×10−32996(229)
WASP-18b203070(50) 3.6(0.75) 3 .1(2)×10−33000(107) Td=2998(138) K
4.5(1.0) 3 .8(3)×10−33128(150)
5.8(1.4) 4 .1(2)×10−33095(103)
8.0(2.9) 4 .3(3)×10−32991(153)
WASP-19b212581(49) 1.65(0.25) 2 .59(45)×10−32677(135) Td=2677(244) K
XO-1b221526(24) 3.6(0.75) 8 .6(7)×10−41300(32) Td=1306(47) K
4.5(1.0) 1 .22(9)×10−31265(34)
5.8(1.4) 2 .61(31)×10−31546(89)
8.0(2.9) 2 .10(29)×10−31211(87)
XO-2231685(33) 3.6(0.75) 8 .1(1.7)×10−41447(102) Td=1431(98) K
4.5(1.0) 9 .8(2.0)×10−41341(105)
5.8(1.4) 1 .67(36)×10−31497(155)
8.0(2.9) 1 .33(49)×10−31179(219)
XO-3241982(82) 3.6(0.75) 1 .01(4)×10−31875(30) Td=1871(63) K
4.5(1.0) 1 .43(6)×10−31965(40)
5.8(1.4) 1 .34(49)×10−31716(330)
8.0(2.9) 1 .50(36)×10−31625(236)
aThe planet’s expected day-side effective temperature in the absence of reflection or recirculation ( AB= 0,ε= 0). The 1 σuncertainty is shown
in parenthese.
bThe bandwidth is shown in parenthese.
cEclipse depths and phase amplitudes are unitless, since the y are measured relative to stellar flux.
dTdandTndenote the day-side and night-side effective temperatures o f the planet, as estimated from thermal secondary eclipse de pths and
thermal phase variations, respectively. The estimated 1 σuncertainties are shown in parentheses. The default day-si de temperature is computed
using only observations at λ >0.8µm. Eclipse measurements at shorter wavelengths may then be u sed to estimate the planet’s albedo at those
wavelengths, Aλ. Note that this is a spherical albedo; the geometric albedo i s given by Ag=2
3Aλ. If —on the other hand— AB= 0 is assumed,
then all the day-side flux is thermal, regardless of waveband , yielding the second Tdestimate.
eWhen multiple measurements of an eclipse depth have been pub lished in a given waveband, we use the most recent observatio n. In all cases
these observations are either explicitly agree with their o lder counterpart, or agree with the re-analyzed older data.
1Snellen et al. (2009); Alonso et al. (2009b); Gillon et al. (2 009); Rogers et al. (2009); Deming et al. (2010),2Alonso et al. (2009a); Snellen et al.
(2010); Gillon et al. (2010); Alonso et al. (2010); Deming et al. (2010),3Deming et al. (2007); Demory et al. (2007); Stevenson et al. ( 2010);
Knutson et al. in prep.,4Todorov et al. (2010),5Borucki et al. (2009); Christiansen et al. (2010),6Laughlin et al. (2009),7Knutson et al.
(2009b),8Deming et al. (2006); Knutson et al. (2007a); Barnes et al. (2 007); Charbonneau et al. (2008); Knutson et al. (2009c); Ago l et al.
(2010),9Richardson et al. (2003); Deming et al. (2005); Cowan et al. ( 2007); Rowe et al. (2008); Knutson et al. (2008),10Sing & L´ opez-Morales
(2009),11Snellen & Covino (2007),12Charbonneau et al. (2005); Knutson et al. (2007b),13O’Donovan et al. (2010); Croll et al. (2010a);
Kipping & Bakos (2010b),14Fressin et al. (2010); Croll et al. (2010b); Christiansen et al. (2010b),15Knutson et al. (2009a),16,17Wheatley et al.
(2010),18Beerer et al. (2010),19L´ opez-Morales et al. (2010); Campo et al. (2010); Croll et a l. (2010c),20Nymeyer et al. (2010),21Anderson et al.
(2010),22Machalek et al. (2008),23Machalek et al. (2009),24Machalek et al. (2010)