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planet has a day-side effective temperature of ∼2000 K.
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When combined with the Kepler observations, one com-
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putesanalbedoofgreaterthan50%. Thelargeday-night
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amplitude seen in the Kepler bandpass is then simply
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due to the fact that the planet’s night-side reflects no
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starlight, and the cool day-side can be attributed to high
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ABand/orε. If, on the other hand, one takes the op-
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tical flux to be entirely thermal in origin ( Aλ= 0), the
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day-side effective temperature is ∼2800 K. This is very
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close to that planet’s Tε=0, leaving very little power left
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for the night-side, again explaining the large day-night
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contrast observed by Kepler. The truth probably lies
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somewhere between these two extremes, but in any case
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this degeneracy will be neatly broken with Warm Spitzer
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observations: the two scenarios outlined above will lead
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to small and large thermal phase variations, respectively.
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It is telling that the only optical measurement in Table 1
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that is unanimously considered to constrain albedo —
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and not thermal emission— is the MOST observations
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of HD 209458b (Rowe et al. 2008), the coolest of the five
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transiting planets with optical photometric constraints.
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The bottom line is that extracting a constraint on re-
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flected light from optical measurements of hot Jupiters is
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best done with a detailed spectral model. But even when
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reflectedlightcanbedirectlyconstrained,convertingthis
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constraint on Aλinto a constraint on ABalso requires
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detailedknowledgeofboththestarandtheplanet’sspec-
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tral energy distributions, making for a model-dependent
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exercise.
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4.2.Populating the AB-εPlane
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Setting aside optical eclipses and direct measurements
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of albedo, we may use the rich near- and mid-IR data to
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constrain the Bond albedo and redistribution efficiency
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of short-period giant planets. We define a 20 ×20 grid in
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ABandεand use Equations 4 & 5 to calculate the nor-
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malized day-side and night-side effective temperatures,
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Td/T0andTn/T0, at each grid point, ( i,j). For each
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planet, we have an observational estimate of the day-side
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effective temperature, and in three cases we also have an
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estimate of the night-side effective temperature (as well
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as associated uncertainties).
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We first verifywhether ornot the observationsarecon-
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sistent with a single ABandε. To evaluate this “null
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hypothesis”, we compute the usual χ2=/summationtext24
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i=1(model−
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data)2/error2at each grid point. We use only the esti-
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mates of day-side and (when available) night-side effec-
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tive temperatures to calculate the χ2, giving us 27-2=25
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degreesoffreedom. The“best-fit”has χ2= 132(reduced
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χ2= 5.3), so the current observations strongly rule out
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a single Bond albedo and redistribution efficiency for all
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24 planets.
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For 21 of the 24 planets considered here, we construct
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a two-dimensional distribution function for each planet
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as follows:
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PDF(i,j) =1/radicalbig
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2πσ2
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de−(Td−Td(i,j))2/(2σd)2.(7)
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This defines a swath through parameter space with the
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same shape as the dotted line in Figure 1.
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For the three remaining planets (HD 149026b,Albedo and Heat Recirculation on Hot Exoplanets 7
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HD 189733b, HD 209458b), phase variation measure-
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ments help break the degeneracy:
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PDF(i,j) =1√
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2πσ2
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de−(Td−Td(i,j))2/(2σd)2
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×1√
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2πσ2ne−(Tn−Tn(i,j))2/(2σn)2.(8)
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Fig. 4.— The global distribution function for short-period exo-
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planets in the AB–εplane. The gray-scale shows the sum of the
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normalized probability distribution function for the 24 pl anets in
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our sample. The data mostly consist of infrared day-side flux es,
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leading to the dominant degeneracy (see first the dotted line in
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Figure 1).
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We create a two-dimensional normalized probability
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distribution function (PDF) for each planet, then add
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these together to create the global PDF shown in Fig-
|
ure 4. This is a democratic way of representing the data,
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since each planet’s distribution contributes equally.
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In Figures 5 and 6 we show the distribution functions
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for the albedo and circulation of the 24 planets in our
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sample,obtainedbymarginalizingtheglobalPDFofFig-
|
ure 4 over either ABorε.
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Fig. 5.— The solid black line shows the projection of the 2-
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dimensional probability distribution function (the gray- scale of
|
Figure 4) projected onto the ε-axis. The dashed line shows the
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ε-distribution if one requires that all planets have Bond alb edos
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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-
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dimensionalprobabilitydistributionfunction (the gray- scale ofFig-
|
ure 4) projected onto the AB-axis. The cumulative distribution
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function (not shown) yields a 1 σupper limit of AB<0.35.
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The solid line in Figure 5 shows no evidence of bi-
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modality in heat redistribution efficiency, although there
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is a wide range of behaviors. The dashed line in Figure 5
|
shows theε-distribution if one requires the albedo to be
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low,AB<0.1. There are then many high-recirculation
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planets, since advection is the only way to depress the
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day-side temperature in the absence of albedo. Inter-
|
estingly, the dashed line doesshow tentative evidence of
|
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