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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