alx-ai/arxiv_papers · Datasets at Hugging Face

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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

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