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higher than the hemisphere-averaged temperature, ex-
|
cept in the ε= 1 limit. This is also why the quantity
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T4
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d+T4
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nis still a weak function of ε.
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Fig. 1.— Different kinds of idealized observations constrain the
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Bond albedo, ABand circulation efficiency, ε, differently. A mea-
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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
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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
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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
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the measurements complement each other: measuring
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two of the three quantities (Bond albedo, effective day-
|
side or night-side temperatures) uniquely determines the
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planet’s albedo and circulation efficiency. When obser-
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vations have some associated uncertainty, they define a
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swath through the AB–εplane.
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3.ANALYSIS
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3.1.Planetary & Stellar Data
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We begin by considering all the photometric obser-
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vations of short-period exoplanets published through
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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-
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tion5. This leaves us with 24 transiting exoplanets for
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which there are observations in at least one waveband
|
at superior conjunction, and in some cases in multiple
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wavebands and at multiple planetary phases.
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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.
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Knowing the stars’ Teff, loggand [Fe/H], we
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use the PHOENIX/NextGen stellar spectrum grids
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(Hauschildt et al. 1999) to determine their brightness
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temperatures at the observed bandpasses. At each wave-
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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.
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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
|
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