THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPLANETS

Giant planets in the solar system have albedos greater than 50% because of the presence of condensed molecules (H₂O, CH₄, 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-broadened Na and K opacity is important 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 extrasolar giant planets (T_eff > 1500 K), might have very high albedos due 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).

Early attempts to observe reflected light from exoplanets (Charbonneau et al. 1999; Collier Cameron et al. 2002; Leigh et al. 2003a, 2003b; Rodler et al. 2008, 2010) indicated that they might not be as reflective as solar system gas giants (for a review, see Langford et al. 2010). Measurements of HD 209458b taken with the Canadian MOST satellite revealed a very low albedo (<8%; Rowe et al. 2008), and it has since been taken for granted that all short-period planets have albedos on par with that of charcoal.

From the standpoint of the planet's climate, the important factor is not the albedo at any one wavelength, A_λ, but rather the integrated albedo, weighted by the incident stellar spectrum, known as the Bond albedo and denoted in this paper as A_B. The relation between A_λ and the planet's Bond albedo is not trivial. If the albedo is dominated by gray clouds, then the albedo at a single wavelength can indeed be extrapolated to obtain A_B. For non-gray reflectance spectra, however, it is critical to measure A_λ at the peak emitting wavelength of the host star to obtain a good estimate of the planet's energy budget. For example, as pointed out in Marley et al. (1999), planets with identical albedo spectra, A_λ, may have radically different A_B depending on the spectral type of their host stars.

The first few measurements of hot Jupiter phase variations showed signs that these planets are not all cut from the same cloth. Harrington et al. (2006) and Knutson et al. (2007b) quoted very different phase function amplitudes for the υ Andromeda 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 wavelengths, and with very different observation schemes (in any case, subsequent re-analysis of the original data and newly acquired Spitzer observations of υ Andromeda 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 HD 209458b exhibit significantly different degrees of heat 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 temperature 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, simply put, is that optical absorbers high in the atmosphere of extremely hot Jupiters (equilibrium temperatures greater than ∼1700 K) would absorb incident photons where the radiative timescales are short, making it difficult for these planets to recirculate energy. The most robust detection of this temperature inversion is 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 efficiency—if any—is not clear.

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 combination 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 budget. In this work, we show how observations in different wavebands and for different planets can be meaningfully combined to estimate these planetary parameters.

In Section 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, A_B, and redistribution efficiency, ε, can be constrained by observations. In Section 3, we use published observations of 24 transiting planets to estimate day-side and—where appropriate—night-side effective temperatures. We construct a two-dimensional distribution function in A_B and ε in Section 4. We state our conclusions in Section 5.

Short-period planets have a power budget entirely dictated by the flux they receive from their host star, which dwarfs tidal heating or remnant heat of formation. Following Hansen (2008), we define the equilibrium temperature at the planet's sub-stellar point: T₀(t) = T_eff(R_*/r(t))^1/2, where T_eff and R_* are the star's effective temperature and radius, and r(t) is the planet–star distance (for a circular orbit r is simply equal to the semimajor axis, a). For shorthand, we define the geometrical factor a_* = a/R_*, which is directly constrained by transit light curves (Seager & Mallén-Ornelas 2003).

The incident flux on the planet is given by $F_{\rm inc} = \frac{1}{2}\sigma _{B}T_{0}^{4}$, and it is significant that this quantity has some associated uncertainty. For a planet on a circular orbit, the uncertainty in $T_{0}=T_{\rm eff}/\sqrt{a_{*}}$ is related—to first order—to the uncertainties in the host star's effective temperature, and the geometrical factor:

$$\begin{equation} \frac{\sigma _{T_{0}}^{2}}{T_{0}^{2}} = \frac{\sigma _{T_{\rm eff}}^{2}}{T_{\rm eff}^{2}} + \frac{\sigma _{a_{*}}^{2}}{4a_{*}^{2}}. \end{equation} \tag{ 1 }$$

For a planet with non-zero eccentricity, T₀ varies with time, but we are only concerned with its value at superior 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 r_sc = a(1 − e²)/(1 − esin ω), where e and ω are the planet's orbital eccentricity and argument of periastron, respectively.

For planets with non-zero eccentricity, the uncertainty in T₀ is given by

$$\begin{eqnarray} \frac{\sigma _{T_{0}}^{2}}{T_{0}^{2}} &= & \frac{\sigma _{T_{\rm eff}}^{2}}{T_{\rm eff}^{2}} + \frac{\sigma _{a_{*}}^{2}}{4a_{*}^{2}}+\left(\frac{e^{2}\cos ^{2}\omega }{1-e^{2}}\right)\sigma _{e\cos \omega }^{2}\nonumber\\ && +\left(\frac{e\sin \omega }{1-e^{2}} - \frac{1}{2(1-e\sin \omega)}\right)\sigma _{e\sin \omega }^{2}, \end{eqnarray} \tag{ 2 }$$

where σ_ecos ω and σ_esin ω are the observational uncertainties in the two components of the planet's eccentricity.⁵

In the absence of albedo or energy circulation, every region on the planet is at its equilibrium temperature, which depends on the normalized projected distance, γ, from the center of the planetary disk as T(γ) = T₀(1 − γ²)^1/8. The thermal secondary eclipse depth in this limit is given by

$$\begin{eqnarray} \frac{F_{\rm day}}{F_{*}} &= & \left(\frac{R_{p}}{R_{*}} \right)^{2} \left(\frac{h c}{\lambda k T_{0}}\right)^{8}(e^{hc/\lambda k T_{\rm b}^{*}}-1)\cr & & \times \int _{0}^{(\lambda k T_{0}/h c)^{8}} \frac{dx}{\exp (x^{-1/8})-1}, \end{eqnarray} \tag{ 3 }$$

where T*_b is the brightness temperature of the star at wavelength λ.

In the no-circulation limit, then, the day-side emergent 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 temperature of the day side in the no-albedo, no-circulation limit: T_ε=0 = (2/3)^1/4T₀ (see also Burrows et al. 2008; Hansen 2008). Indeed, treating the planet's day side as a uniform hemisphere emitting at this temperature gives nearly the same wavelength dependence as the more complex Equation (3). The T_ε=0 temperatures for our sample of 24 transiting planets are shown in Table 1. These set the maximum 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 transported to the planet's night side, and/or to its poles. In this paper, we neglect the E–W asymmetry in the planet's temperature map due to zonal flows and hence ignore phase offsets in the thermal phase variations. Under this assumption, the day–night temperature contrast can more directly be extracted from the observed thermal phase variations.

In practice, many studies have adopted a single parameter to represent both zonal and meridional transport. It is instructive to consider the apparent day-side effective temperatures in various limits: uniform day-side temperature and T = 0 on the night side (this is often referred to as the planet's "equilibrium temperature"): T_equ = (1/2)^1/4T₀; in the case of perfect longitudinal transport but no latitudinal transport: T_long = (8/(3π²))^1/4T₀; and in the limit of uniform temperature everywhere on the planet: T_uni = (1/4)^1/4T₀.

Comparing the apparent day-side temperatures in the three limits of circulation above leads to the following simple parameterization of the day-side effective temperature in terms of the planetary albedo, A_B, and circulation efficiency, ε:

$$\begin{equation} T_{\rm d} = T_{0} (1-A_{B})^{1/4} \left(\frac{2}{3} - \frac{5}{12}\varepsilon \right)^{1/4}, \end{equation} \tag{ 4 }$$

where 0 < ε < 1. Note that ε is related to—but different from—the used in Cowan & Agol (2011). The former is merely a parameterization 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 = 0 case is equal to the ε = 0 case, while the → ∞ limit is equivalent to ≈ 0.95.

Our definition of ε is similar to the Burrows et al. (2006) definition of P_n and yields the same no-circulation limit. But our ε = 1 limit produces a lower day-side brightness than the P_n = 0.5 limit, because we assume that the planet's day side has a uniform temperature distribution in that limit (for a discussion of different redistribution parameterizations, see the appendix of Spiegel & Burrows 2010).

In reality, efficient longitudinal transport (read: fast zonal winds) may lead to more banding and therefore less efficient latitudinal transport. So one could argue that in the limit of perfect day–night temperature homogenization, both the day and night apparent temperatures should be T_d = (8/(3π²))^1/4T₀, in between the Burrows et al. value of T_d = (1/3)^1/4T₀ and that suggested by our parameterization, T_d = (1/4)^1/4T₀. At moderate day–night recirculation efficiencies, however, there is a good deal of latitudinal transport (I. Dobbs-Dixon 2010, private communication), so implicitly assuming a constant T ∝ cos^1/4 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 longitudinal transport is the dominant factor in determining day and night effective temperatures.

Notwithstanding the subtleties discussed above and noting that cooling tends to latitudinally homogenize night-side temperatures (Cowan & Agol 2011), we get a night-side temperature of

$$\begin{equation} T_{\rm n} = T_{0} (1-A_{B})^{1/4} \left(\frac{\varepsilon }{4} \right)^{1/4}. \end{equation} \tag{ 5 }$$

Note that T_d and T_n are the equator-weighted temperatures of their respective hemispheres (i.e., as seen by an edge-on viewer). As such, they will tend to be slightly higher than the hemisphere-averaged temperature, except in the ε = 1 limit. This is also why the quantity T⁴_d + T⁴_n is still a weak function of ε.

In Figure 1, we show how different kinds of observations constrain A_B and ε. For this example, we chose constraints consistent with A_B = 0.2 and ε = 0.3. The solid line is a locus of constant A_B; the dotted line is the locus of constant T_d/T₀; the dashed line is a locus of constant T_n/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 observations have some associated uncertainty, they define a swath through the A_B–ε plane.

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We begin by considering all the photometric observations of short-period exoplanets published through 2010 November, summarized in Table 1. We have discarded photometric observations of non-transiting planets because of their unknown radius and orbital inclination.⁶ 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 Exoplanet Encyclopedia (http://exoplanet.eu), and references therein. We repeated parts of the analysis with the Exoplanet Data Explorer database (http://exoplanets.org) and found identical results, within the uncertainties. When certain stellar data are not available, we have assumed typical parameters for the appropriate spectral class, and solar 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' T_eff, log g, and [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 waveband for which eclipse or phase observations have been obtained, we determine the ratio of the stellar flux to the blackbody flux at that grid star's T_eff. We then apply this factor to the T_eff of 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. (2010a) use stellar models for HAT-P-7 from Kurucz (2005), while we use those of Hauschildt et al. (1999). The resulting 8 μm brightness temperatures for HAT-P-7b differ by as much as 600 K, or slightly more than 1σ. Our uniform use of Hauschildt et al. (1999) models should alleviate this problem, however.

The planet's albedo and recirculation efficiency govern its effective day-side and night-side temperatures, T_d and T_n, respectively. Observationally, we can only measure the brightness temperature, ideally at a number of different wavelengths: T_b(λ). If one knew, a priori, the emergent spectrum of a planet, one could trivially convert a single brightness temperature to an effective temperature. Alternatively, if observations were obtained at a number of wavelengths bracketing the planet's blackbody 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 temperature. The secondary eclipse depth in some waveband divided by the transit depth is a direct measure of the 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 compute the apparent brightness temperature of the planet's day side:

$$\begin{equation} T_{\rm b}(\lambda) = \frac{h c}{\lambda k} \left[\log \left(1 + \frac{e^{h c/\lambda k T_{b}^{*}(\lambda)} - 1}{\psi (\lambda)}\right)\right]^{-1}. \end{equation} \tag{ 6 }$$

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 temperature.

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 immediately obvious which wavelengths are sensitive to 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, λ_ε=0 = 2898/T_ε=0 μm. Insofar as real planets will have non-zero albedo and non-zero recirculation, the day side should never reach T_ε=0, and the actual spectral energy distribution will peak at slightly longer wavelengths. The coolest planet in our sample, Gl 436b, would exhibit a blackbody peak at λ_ε=0 = 3.1 μm, while the hottest planet we consider, WASP-12b, has λ_ε=0 = 0.9 μm. In practice, this means that ground-based near-IR and space-based mid-IR (e.g., Spitzer) observations are assumed to measure thermal emission, while space-based optical observations (MOST, CoRoT, Kepler) may be contaminated by reflected starlight.

In Figure 2, we demonstrate two alternative techniques to convert an array of brightness temperatures, T_b(λ), into an estimate of a planet's effective temperature, T_eff. The solid black line shows a model spectrum of thermal emission from Fortney et al. (2008), with an effective temperature of T_eff = 1941 K shown with the black dashed line. The expected blackbody peak of the planet given this expected temperature is marked with a vertical dotted line. The red points are the brightness temperatures in the J, H, and K_s bands (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 Spitzer IRAC, we first focus on estimating T_eff based only on brightness temperatures in those four bandpasses.

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Wien displacement. The first approach is to simply adopt the brightness temperature of the bandpass closest 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 T_eff = 1925 K. There is—however—some subtlety in estimating the peak wavelength, as this is dependent on knowing the planet's temperature (and hence A_B and ε) a priori.

Linear interpolation. The linear interpolation technique, shown with the red line in Figure 2, obviates the need for an a priori estimate of the planet's temperature. The brightness temperature is assumed to be constant shortward of the shortest-λ observation, and longward of the longest-λ observation. Between bandpasses, the brightness 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 temperature, which is T_eff = 1927 K for the current example. Since we hope to apply our routine to planets with well-sampled blackbody peaks, we adopt the linear interpolation technique, as it can make use of multiple brightness temperature estimates near the peak.

The two techniques described above produce similar effective temperatures, though—unsurprisingly—neither gives precisely the correct answer. But these systematic errors are comparable to or smaller than the photometric uncertainty in observations of individual brightness temperatures (see Table 1). The best IR observations for the nearest, brightest planetary systems (e.g., HD 189733b and HD 209458b) lead to observational uncertainties of approximately 50 K in brightness temperature. For many planets, the uncertainty is 100–200 K. By that metric, either the Wien displacement or the linear interpolation routines give adequate estimates of the effective temperature, with errors of 16 K and 14 K, respectively, in the above example.

We now make a more quantitative analysis of the systematic uncertainties involved in the linear interpolation temperature estimates. We produce 8800 mock data sets: 100 realizations for 11 spectral models and observations in up to eight wavebands (J, H, K, IRAC, MIPS; since this numerical experiment chooses random bands from the eight available, the results should not be very different if additional wavebands are considered). We run our linear interpolation technique on each of these and plot in Figure 3 the estimated day-side temperature normalized by the actual model effective temperature versus the number of wavebands used in the estimate. The temperature estimates cluster near T_est/T_eff = 1, indicating that the technique is not significantly biased. The scatter in estimates 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 acquired in eight bands. We incorporate this systematic error into our analysis by adding it in quadrature to the observational uncertainties described in the following paragraph. This has the desirable effect that planets with fewer observations have a larger systematic uncertainty on their effective temperature.

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In practice, we would also like to propagate the photometric uncertainties to the estimate of T_eff. For the Wien displacement technique, this uncertainty propagates trivially to the effective temperature. For the linear interpolation technique, a Monte Carlo approach can be used to estimate the uncertainty in T_eff: the input eclipse depths are randomly shifted 1000 times in a manner consistent with their photometric uncertainties—assuming Gaussian errors—and the effective temperature is recomputed repeatedly. The scatter in the resulting values of T_eff provides an estimate of the observational uncertainty in the parameter, to which we add in quadrature the estimate of systematic error described above. The resulting uncertainties are listed in Table 1. These uncertainties should be compared to the uncertainties in T_ε=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 interpolation temperature estimation technique. In some cases, only upper limits have been obtained, therefore one could set ψ = 0, with the appropriate 1σ uncertainty. But this approach leads to huge uncertainties in T_eff for planets with a secondary eclipse upper limit near their blackbody peak. Instead of "punishing" these planets, we opt to not use upper limits (though for completeness we include them in Table 1). Second, when multiple measurements 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.

For each planet, we use thermal observations (essentially those in the J, H, K_s, and Spitzer bands) to estimate the planet's effective day-side temperature, T_d, and—when phase variations are available—T_n. These values are listed in Table 1. In five cases (CoRoT-1b, CoRoT-2b, HAT-P-7b, HD 209458b, TrES-2b), secondary eclipses and/or phase variations have been obtained at optical wavelengths. Such observations have the potential to directly constrain the albedo of these planets. One approach is to adopt the T_d from thermal observations and calculate the expected contrast ratio at optical wavelengths, under the assumption of blackbody emission (see also Kipping & Bakos 2010a). Insofar as the observed eclipse depths are deeper than this calculated depth, one can invoke the contribution of reflected light and compute a geometric albedo, A_g. If one treats the planet as a uniform Lambert sphere, the geometric albedo is related to the spherical albedo at that wavelength by $A_{\lambda } = \frac{3}{2}A_{g}$. 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 optical wavelengths than at mid-IR wavelengths, in agreement with realistic spectral models of hot Jupiters, which predict brightness temperatures greater than T_eff on the Wien tail (see, for example, the Fortney et al. model shown in Figure 2, which does not include reflected light). Note that this increase in emissivity should occur regardless of whether or not the planet has a stratosphere: by definition, the depth at which the optical thermal emission 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 optical.

Determining the albedo directly (i.e., by observing reflected 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. Therefore, 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 transiting 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 computes an albedo of greater than 50%. The large day–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 A_B and/or ε. If, on the other hand, one takes the optical flux to be entirely thermal in origin (A_λ = 0), the day-side effective temperature is ∼2800 K. This is very close to that planet's T_ε=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 reflected light from optical measurements of hot Jupiters is best done with a detailed spectral model. But even when reflected light can be directly constrained, converting this constraint on A_λ into a constraint on A_B also requires detailed knowledge of both the star and the planet's spectral energy distributions, making for a model-dependent exercise.

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 A_B and ε and use Equations (4) and (5) to calculate the normalized day-side and night-side effective temperatures, T_d/T₀ and T_n/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 verify whether or not the observations are consistent with a single A_B and ε. To evaluate this "null hypothesis," we compute the usual χ² = ∑²⁴_i=1(model − data)²/error² at each grid point. We use only the estimates of day-side and (when available) night-side effective temperatures to calculate the χ², giving us 27 − 2 = 25 degrees of freedom. 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:

$$\begin{equation} \mbox{PDF}(i,j) = \frac{1}{\sqrt{2\pi \sigma _{\rm d}^{2}}}e^{-(T_{\rm d} - T_{\rm d}(i,j))^{2}/(2\sigma _{\rm d})^{2}}. \end{equation} \tag{ 7 }$$

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, HD 189733b, HD 209458b), thermal phase variation measurements help break the degeneracy:

$$\begin{eqnarray} \mbox{PDF}(i,j) &= & \frac{1}{\sqrt{2\pi \sigma _{\rm d}^{2}}}e^{-(T_{\rm d} - T_{\rm d}(i,j))^{2}/(2\sigma _{\rm d})^{2}}\cr && \times \frac{1}{\sqrt{2\pi \sigma _{\rm n}^{2}}}e^{-(T_{\rm n} - T_{\rm n}(i,j))^{2}/(2\sigma _{\rm n})^{2}}. \end{eqnarray} \tag{ 8 }$$

We create a two-dimensional normalized probability distribution function (PDF) for each planet, then add these together to create the global PDF shown in Figure 4. This is a democratic way of representing the data, since each planet's distribution contributes equally.

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In Figures 5 and 6, we show the distribution functions for the albedo and circulation of the 24 planets in our sample, obtained by marginalizing the global PDF of Figure 4 over either A_B or ε, respectively.

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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 the ε-distribution if one requires the albedo to be low, A_B < 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. Interestingly, the dashed line does show 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 consistent 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 limited wavelength range.

In Figure 7 we plot the dimensionless day-side effective temperature, T_d/T₀, against the maximum expected day-side temperature, T_ε=0. Planets should lie below the solid red line, which denotes T_ε=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 T_ε=0 limit.⁷ This planet is by far the coolest in our sample, it is on an eccentric 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 should not 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, 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, puffing up the planet, etc., see also Cowan & Agol 2011) rather than merely warming the gas. Gl 436b and HD 80606b are denoted by red "×"s in Figure 7.

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The cyan asterisks in Figure 7 show the four hot Jupiters without temperature inversions, while most of the remaining planets have inversions (Knutson et al. 2010). The presence or absence of an inversion does not appear to affect the efficiency of day–night heat recirculation.

The gray points in Figure 7 indicate the default values (using only observations with λ>0.8 μm) 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 estimates at face value, it appears that the planets with T_ε=0 < 2400 K exhibit a wide-variety of redistribution efficiencies and/or Bond albedos, but are consistent with A_B = 0. It is worth noting that many of the best characterized planets in this region have T_d/T₀ ≈ 0.75, and this accounts for the sharp peak in the dotted line of Figure 5 at ε = 0.75. The hottest six planets, on the other hand, have uniformly high T_d/T₀, indicating that they have both low Bond albedo and low redistribution efficiency. 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 Kepler observations 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 transiting planets exhibit a variety of behaviors.