Understanding Trends Associated with Clouds in Irradiated Exoplanets

UNDERSTANDING TRENDS ASSOCIATED WITH CLOUDS IN IRRADIATED EXOPLANETS

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Citation Heng, Kevin, and Brice-Olivier Demory. “UNDERSTANDING TRENDS ASSOCIATED WITH CLOUDS IN IRRADIATED EXOPLANETS.” The Astrophysical Journal 777, no. 2 (October 18, 2013): 100. © 2013 American Astronomical Society.

As Published http://dx.doi.org/10.1088/0004-637x/777/2/100

Publisher Institute of Physics/American Astronomical Society

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Citable link http://hdl.handle.net/1721.1/93752

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Astrophysical Journal, 777:100 (11pp), 2013 November 10 doi:10.1088/0004-637X/777/2/100 C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

UNDERSTANDING TRENDS ASSOCIATED WITH CLOUDS IN IRRADIATED EXOPLANETS

Kevin Heng1 and Brice-Olivier Demory2 1 University of Bern, Center for Space and Habitability, Sidlerstrasse 5, CH-3012 Bern, Switzerland; [email protected] 2 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA; [email protected] Received 2013 May 13; accepted 2013 August 29; published 2013 October 18

ABSTRACT Unlike previously explored relationships between the properties of hot Jovian atmospheres, the geometric albedo and the incident stellar flux do not exhibit a clear correlation, as revealed by our re-analysis of Q0–Q14 Kepler data. If the albedo is primarily associated with the presence of clouds in these irradiated atmospheres, a holistic modeling approach needs to relate the following properties: the strength of stellar irradiation (and hence the strength and depth of atmospheric circulation), the geometric albedo (which controls both the fraction of starlight absorbed and the pressure level at which it is predominantly absorbed), and the properties of the embedded cloud particles (which determine the albedo). The anticipated diversity in cloud properties renders any correlation between the geometric albedo and the stellar flux weak and characterized by considerable scatter. In the limit of vertically uniform populations of scatterers and absorbers, we use an analytical model and scaling relations to relate the temperature–pressure profile of an irradiated atmosphere and the photon deposition layer and to estimate whether a cloud particle will be lofted by atmospheric circulation. We derive an analytical formula for computing the albedo spectrum in terms of the cloud properties, which we compare to the measured albedo spectrum of HD 189733b by Evans et al. Furthermore, we show that whether an optical phase curve is flat or sinusoidal depends on whether the particles are small or large as defined by the Knudsen number. This may be an explanation for why Kepler-7b exhibits evidence for the longitudinal variation in abundance of condensates, while Kepler-12b shows no evidence for the presence of condensates despite the incident stellar flux being similar for both exoplanets. We include an “observer’s cookbook” for deciphering various scenarios associated with the optical phase curve, the peak offset of the infrared phase curve, and the geometric albedo. Key word: planets and satellites: atmospheres Online-only material: color figures

1. INTRODUCTION tional definition of the “equilibrium temperature” and includes a dimensionless factor fdist that describes the efficiency of heat 1.1. Observational Motivation redistribution from the dayside to the nightside hemisphere: we have f = 2/3 and 1/2 for no and full redistribution, re- Built primarily to study the occurrence of Earth-like exoplan- dist spectively (Hansen 2008). The quantity Teq,0 is the equilibrium ets in our local cosmic neighborhood, the actual scientific scope temperature assuming full redistribution and a vanishing albedo. of the Kepler Space Telescope has expanded to include the study Strictly speaking, the geometric albedo is defined at a specific of hot Jupiters and their atmospheres. The detection of transits wavelength λ and at zero viewing angle. The spherical albedo and eclipses in the broadband optical channel of Kepler has en- (As) is the geometric albedo considered over all viewing an- abled the geometric albedos of about a dozen hot Jupiters to gles (Russell 1916; Marley et al. 1999; Sudarsky et al. 2000; be measured. In Figure 1, we show the geometric albedo (Ag) Seager 2010; Madhusudhan & Burrows 2012). For a Lamber- versus the incident stellar flux at the substellar point (F0)for tian sphere (isotropic scattering), we have Ag = 2As /3. When a sample of hot Jupiters, for which we have performed an im- integrated over all wavelengths, we obtain the Bond albedo proved analysis of Q0–Q14 data (see Section 2.1 for details). (A ). Since we are dealing with observations integrated over a Begining our discussion requires defining a few temperatures B broad optical bandpass, we assume AB = 3Ag/2; the considera- (Cowan & Agol 2011b), tion of more sophisticated scattering behavior will introduce an 1/2 order-of-unity correction factor to this relation. R For the range of T values listed in Table 1, we estimate T0 ≡ T , eq,0 a that the hot Jupiters examined in Figure 1 radiate mostly at 1/2 wavelengths of about 1–2 μm (using Wien’s law). Nevertheless, Rfdist 1/4 Teq ≡ T (1 − AB) , since the Kepler bandpass extends from λ1 ≈ 0.4 μmto a ≈ λ2 0.9 μm (Koch et al. 2010), it is instructive to estimate R 1/2 the fraction of thermal flux from the exoplanet radiated in this T ≡ T , (1) eq,0 2a bandpass, λ2 π Bλ(Teq)dλ λ1 with T denoting the stellar effective temperature, R the stellar fthermal = , (2) σ T 4 radius, and a the orbital semi-major axis of the exoplanet. The SB eq quantity T0 may be regarded as the “irradiation temperature” at by approximating the spectral energy distribution of a hot Jupiter = 4 = 4 the substellar point, such that F0 σSBT0 4σSBTeq,0 is the to be a blackbody function. In Figure 1,weplotfthermal as a incident stellar flux (or “stellar constant”), with σSB denoting function of F0 for both fdist = 2/3 and 1/2. It is apparent the Stefan–Boltzmann constant. The quantity Teq is our conven- that fthermal may not be small and generally increases with F0,

1 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory

Table 1 Geometric Albedos of Hot Jupiters from Kepler Photometry, Reﬁned Using Q0–Q14 Data

Object Name Ag Teq,0 (K) TrES-2b 0.015 ± 0.003 1444 ± 13 HAT-P-7b 0.225 ± 0.004 2139 ± 27 Kepler-5b 0.134 ± 0.021 1752 ± 17 Kepler-6b 0.091 ± 0.021 1451 ± 16 Kepler-7b 0.352 ± 0.023 1586 ± 13 Kepler-8b 0.051 ± 0.029 1638 ± 40 Kepler-12b 0.078 ± 0.019 1477 ± 26 Kepler-14b 0.012 ± 0.023 1573 ± 26 Kepler-15b 0.078 ± 0.044 1225 ± 31 Kepler-17b 0.106 ± 0.011 1655 ± 40 Kepler-41b 0.135 ± 0.014 1745 ± 43

Figure 1. Geometric albedo vs. the incident stellar flux for a sample of contamination by thermal emission assuming full redistri- hot Jupiters studied with Kepler photometry. Error bars include possible = contamination by thermal emission. The curves labeled “fthermal” are estimations bution (fdist 1/2), the Ag values do not change much of the fraction of thermal flux radiated in the Kepler bandpass, for Ag = 0 (solid (and the Spearman rank coefficient remains, to the first = curve) and Ag 0.1, 0.2, 0.3, and 0.4 (dotted curves). The small and large open significant figure, unchanged). circles represent the geometric albedos corrected for contamination by thermal 2. When the correction is performed assuming no redistri- emission (see the text) assuming no and full redistribution, respectively. bution (f = 2/3), the trend flattens as expected (with (A color version of this figure is available in the online journal.) dist a Spearman rank coefficient of −0.1). Three of the data implying that the measured geometric albedo (Seager 2010), points are consistent with being zero. The key point is that 2 the correlation between Ag and F0 can only weaken, and not Fp,⊕ a strengthen, when heat redistribution is taken into account. Ag = , (3) F,⊕ Rp Our conclusion is that there exists no clear correlation may be contaminated by thermal emission “leaking” into between Ag and F0. Values of Ag ≈ 0.1 may be consistent the Kepler bandpass, causing Ag to be overestimated. The with Rayleigh scattering caused by hydrogen molecules alone quantities Fp,⊕ and F,⊕ are the fluxes from the star and the (Sudarsky et al. 2000), without the need for the presence of exoplanet, respectively, received at Earth, while the radius of clouds or condensates. The high geometric albedo associated ≈ the exoplanet is given by Rp. One may approximately correct with Kepler-7b (Ag 0.35) may require an explanation that for the contamination by thermal emission by considering the includes the effects of clouds or condensates (Demory et al. following equation: 2011, 2013). λ2 2 2 1.2. Theoretical Motivation ⊕ π Bλ(Teq)dλ Fp, λ1 Rp a Ag = − F,⊕ F0 R Rp Unlike other previously examined relationships between various properties of hot Jupiters (e.g., radius and heat redistribution λ2 2 π Bλ(Teq)dλ a versus T ; e.g., Cowan & Agol 2011b; Demory & Seager 2011; = − λ1 eq Ag,obs , (4) Laughlin et al. 2011; Perna et al. 2012), there is no clear trend F0 R of Ag with the incident stellar flux. One of the goals of the where Ag,obs is the measured value of the geometric albedo present study is to suggest that the absence of a clear trend is obtained by applying Equation (3). Since Teq depends on caused by a combination of opacity effects, possibly due to the Ag, the preceding expression is an implicit equation for the presence of condensates or clouds, and atmospheric circulation, geometric albedo, which may be solved to obtain the “de- the latter of which is often ignored in spectral analyses of hot contaminated” Ag, also shown in Figure 1. The small and large Jupiters. The study of clouds or hazes is emerging as a major open circles represent the corrected Ag values assuming no theme in the observations of hot Jupiters (e.g., Lecavelier des and full redistribution, respectively. Generally, decreasing the Etangs et al. 2008; Pont et al. 2008; Sing et al. 2011;Gibson efficiency of heat redistribution decreases the geometric albedo et al. 2012) and directly imaged exoplanets (e.g., Barman et al. obtained. A better approach is to allow for fdist to vary with 2011; Madhusudhan et al. 2011; Marley et al. 2012; Lee et al. F0, since the efficiency of heat redistribution worsens as F0 2013) and has long been an obstacle plaguing advances in the increases (Perna et al. 2012), but we do not attempt this as the understanding of brown dwarfs (e.g., Saumon & Marley 2008; functional form of fdist(F0) is not well known. For this reason, Artigau et al. 2009; Burrows et al. 2011; Helling et al. 2011; we do not specify the uncertainties associated with the corrected Buenzli et al. 2012). Ag values. On the theoretical front, several trends are now understood. Based on the results in Figure 1, there are two possible 1. The strength and depth of atmospheric circulation are interpretations: intimately tied to the intensity of stellar irradiation (Perna 1. Taken at face value (without performing a correction for et al. 2012). An “eddy diffusion coefficient” (Kzz)isoften contamination by thermal emission), the Ag versus F0 data used to mimic this behavior, but ultimately the relationship exhibit a weak correlation (Spearman rank coefficient of between atmospheric circulation and stellar flux can—and 0.6), although the Ag = 0.352 ± 0.023 measurement as- should—be calculated from first principles using global, sociated with Kepler-7b stands out. When we correct for three-dimensional (3D) simulations.

2 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory 2. The geometric albedo of an irradiated atmosphere controls employ a Markov Chain Monte Carlo (MCMC) framework not only the amount of starlight penetrating the atmosphere, to characterize the posterior distribution of Ag.MCMCisa but also the depth of the penetration (Fortney et al. 2008; Bayesian inference method based on stochastic simulations Heng et al. 2012; Dobbs-Dixon et al. 2012). that samples the posterior probability distributions of adjusted 3. Whether a particle embedded in an atmospheric flow is held parameters for a given model. Our MCMC implementation aloft depends not only on its size and density, but also on (described in, e.g., Gillon et al. 2012) uses the Gibbs sampler the local temperature, pressure, and velocity field of the and the Metropolis–Hastings algorithm to estimate the posterior atmosphere (e.g., Spiegel et al. 2009). The abundance of distribution function of all jump parameters. Our nominal model the particles relative to the atmospheric gas, their sizes, and is based on a star and a transiting planet on a Keplerian orbit their composition in turn determine the geometric albedo, about their center of mass. which controls the heating of the atmosphere. Input data provided to the MCMC for each system consist To date, no model or simulation has succeeded in including of the Q0–Q14 Kepler photometry and the spectroscopic stellar all of these effects, which need to be studied in concert in order parameters (effective temperature T, metallicity [Fe/H], and to gain a global perspective on the atmospheres of irradiated log g) published in the references mentioned above. We correct exoplanets. Examining each factor in isolation is reasonable for the photometric dilution induced by neighbor stellar sources and necessary as an initial step, but the final word will come using a quarter-dependent dilution factor based on the dilution from studying their complex interplay. Conversely, investing values presented in the literature and on the contamination precision in some of these factors at the expense of others may values reported in the FITS files headers (Bryson et al. 2010). yield misleading results. We highlight some features of this We divide the total light curve into segments of duration of interplay using an analytical toy model and scaling relations about 24–48 hr and fit for each of them the smooth photometric in the hope that it will motivate more sophisticated, follow- variations due to stellar variability or instrumental systematic up studies. More succinctly, the goal of the present study is effects with a time-dependent quadratic polynomial. Baseline to relate the presence of condensates and their properties with model coefficients are determined at each step of the MCMC for the geometric albedo, the pressure of the atmospheric layer each light curve with the singular value decomposition method probed by the Kepler bandpass, and the strength of atmospheric (Press et al. 1992). The resulting coefficients are then used to circulation, which is tied to the stellar irradiation flux. correct the raw photometric light curves. To begin, Section 2 describes the details involved in construct- We assumed a quadratic law for the stellar limb darkening = = − ing Figure 1, as well as our theoretical method. Our results are (LD) and used c1 2u1 + u2 and c2 u1 2u2 as jump presented in Section 3. An example of comparative exoplane- parameters, where u1 and u2 are the quadratic coefficients. u1 and tology is described in Section 4 (Kepler-7b versus Kepler-12b), u2 were drawn from the theoretical tables of Claret & Bloemen along with an “observer’s cookbook” for diagnosing qualitative (2011) for the stellar parameters obtained from the references trends associated with the geometric albedo and phase curves. above. The MCMC has the following set of jump parameters: the 2. METHODOLOGY planet/star flux ratio, the impact parameter b, the transit duration 2.1. Data from first to fourth contact, the time of minimum light t0,the orbital period, the occultation depth,√ the two LD√ combinations Our goal is to perform detailed photometric analyses of all c1 and c2, and the two parameters e cos ω and e sin ω.The Kepler giant exoplanets confirmed to date in order to precisely latter two parameters allow us to fit for the occultation phase constrain their geometric albedos. In this section, we describe and width. A uniform prior distribution is assumed for all jump our methodology and results. parameters but c1 and c2, for which a normal prior distribution 2.1.1. Kepler Photometry is used, based on theoretical tables and the stellar parameters used as input data to the MCMC fit. This study is based on quarters Q0 through Q14 of Kepler We run two Markov chains of 100,000 steps each. The good data (see Batalha et al. 2013 for the Q0–Q8 data release) that are mixing and convergence of the chains are assessed using the available at the time of writing. In total, the data sets encompass Gelman–Rubin statistic criterion (Gelman & Rubin 1992). We about 1250 days of quasi-continuous photometric monitoring use the posterior distribution functions for the jump parameters 2 between 2009 May and 2012 October. We retrieved the Q0–Q14 ⊕ ⊕ 3 Fp, /F, , a/R, and (Rp/R) obtained from the MCMC FITS files from MAST and extracted the raw long-cadence to derive the geometric albedo posteriors for each planet. We photometry (Jenkins et al. 2010b) for each target. show the median of the posterior distribution function and its 2.1.2. Data Analysis and Derivation of the Geometric Albedo associated 1σ probability interval for Ag and Teq,0 in Table 2. As already mentioned in Section 1, the non-parametric We consider the following 11 giant exoplanets in this study: Spearman rank coefficient is 0.6 for the Ag versus F0 data shown TrES-2b (Barclay et al. 2012), HAT-P-7b (Pal et al. 2008; in Figure 1. Values of ±1 indicate a perfectly monotonically Christiansen et al. 2010), Kepler-8b (Jenkins et al. 2010a), increasing or decreasing trend, while those close to zero indicate Kepler-6b (Dunham et al. 2010), Kepler-5b (Koch et al. 2010), no correlation between the two quantities. Kepler-12b (Fortney et al. 2011), Kepler-7b (Latham et al. 2010), Kepler-14b (Buchhave et al. 2011), Kepler-15b (Endl 2.2. Models et al. 2011), Kepler-41b (Santerne et al. 2011; Quintana et al. 2013), and Kepler-17b (Desert´ et al. 2011). The purpose Any model constructed to describe clouds in exoplanetary of this analysis is to search for the planetary occultation atmospheres has to include both their radiative and dynamical (whose depth yields Fp,⊕/F,⊕)intheKepler bandpass and effects. Even when clouds contribute negligible mass to the derive the corresponding geometric albedo. To this end, we atmosphere, they introduce strong radiative forcing to it in the form of the reflection of incident starlight (cooling) and 3 http://archive.stsci.edu/kepler/ the retention of reprocessed starlight in the infrared (heating).

3 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory On Earth, these two significant effects almost cancel, but where g denotes the surface gravity of the exoplanet. In a purely not quite—it is this imperfect cancellation that is important reflecting atmosphere (AB = 1), there is no penetration of for determining details of terrestrial weather and the climate starlight (PD = 0). Note that this unique relationship between system. Getting these details correct to high precision remains the photon deposition layer and the Bond albedo only exists a formidable challenge (see, e.g., Pierrehumbert 2010). Clouds in the limit of uniform, vertical populations of scatterers and also interact with the atmospheric flow. To first order, we may absorbers in the 1D model atmosphere. When the assumption neglect the dynamical effects of clouds on the flow (unless the of spatial uniformity is relaxed, this unique relationship is dust-to-gas ratio is close to unity), but we may not neglect broken and it is now possible to specify cloud decks of varying the dynamical effects of the flow on the clouds. Atmospheric spatial and optical thicknesses, located at different altitudes, circulation sets the background state of velocity, temperature, that will produce the same Bond albedo. While being aware of pressure, and density—and all of its dependent quantities—that this possibility, we do not explore this complexity for several allows us to determine if a cloud particle will remain at a certain reasons: we are interested in elucidating trends rather than location in the atmosphere. making detailed predictions; to develop intuition, we confine To place the present study in context, we note that the 3D ourselves to analytical models as much as possible and it has simulations of Parmentier et al. (2013) examine the effects of the been previously shown that such a generalization breaks the atmospheric flow on embedded tracers that mimic the presence analytical nature of the model for the temperature–pressure of cloud particles, but these tracers do not feed back on the profile (Heng et al. 2012); and the atmospheric data associated flow in any way, either dynamically or radiatively. By contrast, with hot Jupiters are not (yet) of a high enough quality to warrant the 3D simulations of Dobbs-Dixon & Agol (2012) include such sophisticated investigations (unlike, for example, in the an extra opacity source to describe Rayleigh scattering, but do case of brown dwarfs). not consider the interaction between the flow and embedded particles. 2.2.2. Effects of Atmospheric Flow on Condensates It is practically impossible to include all of these effects In this sub-section, we use simple scaling relations to elu- in an analytical model in any rigorous manner. Instead, we cidate the relationship between vertical, atmospheric flow and subsume the absorptivity of the clouds into a general, infrared, its ability to loft condensates. The mean free path of molecules absorption opacity that also describes the atmospheric gas. within an atmosphere is L = m/ρσm, where m is the mean −15 2 The geometric albedo is a consequence of the scattering and molecular mass, ρ is the mass density, and σm ∼ 10 cm is absorption properties of the atmosphere, as we will discuss in the cross section for inter-molecular interactions. To describe Section 2.2.3. While being mindful of this fact, we prescribe the influence of the atmospheric flow on a particle embedded the geometric albedo as a free parameter when computing the within it, we use the analytical formulae previously described analytical temperature–pressure profile. To describe the action in Li & Wang (2003) and Spiegel et al. (2009) (and references of the atmospheric flow on the cloud particles, we use an therein). The terminal velocity of the (spherical) particle is analytical expression for the local terminal velocity of a given cloud particle and approximate the vertical velocity of the flow 2Cr2ρ g v = c c . (6) to be a fixed fraction of the local sound speed. Finally, we discuss t ρν the relationship between the properties of the cloud particles and the albedo. Its internal mass density is ρc, while its radius is rc.The kinematic viscosity of the atmospheric gas is ν ≈ Lcs , with 2.2.1. Radiative Forcing cs being the local sound speed. The quantity C is a correction We use analytical models of the temperature–pressure profiles factor accounting for the enhancement of the terminal velocity in rarefied media, of hot Jupiters in the present study from Heng et al. (2012), where the cloud-free models of Guillot (2010) were generalized 1.1 C = − to include a non-zero albedo. Whether the albedo is due to 1+Nk 1.256 + 0.4exp , (7) N Rayleigh scattering associated with molecules or scattering k associated with condensates or dust grains is unspecified. and depends on the Knudsen number, These models solve the one-dimensional (1D) radiative transfer L k T equation in the plane-parallel, two-stream approximation using N = = B the method of moments, while the incoming stellar radiation k r Pσ r c m c is approximated as a collimated beam. The free parameters −1 T P rc involved are the irradiation temperature (T0), the Bond albedo ∼ 10 , (8) 1500 K 0.1 bar 1 μm (AB), the absorption opacity in the infrared (κIR), and the absorption opacity in the optical (κO). We set collision-induced which in turn depends on the pressure level (P) and local absorption, associated with hydrogen molecules, to be the temperature (T) of the atmosphere considered. The Boltzmann dominant opacity source at pressures of 0.1 bar and greater constant is given by kB. ( = 51 for a bottom pressure of 10 bar). There is an option We define a quantity S ≡ vz/vt, which describes whether a to specify a purely absorbing cloud deck of intermediate width, particle is likely to be lofted by atmospheric circulation. We which we ignore for the purpose of simplicity and clarity. approximate the vertical component of the velocity to be Starlight incident upon an atmosphere is predominantly 1/2 absorbed at the following pressure level, known as the “photon γkBT deposition layer” (Heng et al. 2012), vz = Mzcs = Mz , (9) m − 0.63g 1 AB where γ = 7/5 is the adiabatic gas index. The vertical PD = , (5) κO 1+AB Mach number Mz is approximated to be constant, although

4 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory

Since we are interpreting vz to be the rms vertical velocity in Equations (9)–(11), min{1,S} may be interpreted as the relative abundance of embedded cloud particles summed horizontally over an entire atmospheric layer.

2.2.3. Relationship between Condensates and Albedo The scattering and absorption properties of cloud particles are mainly described by two quantities. The ﬁrst quantity is the “single scattering albedo”: ω0 = σscat/(σscat + σabs), where σscat and σabs are the scattering and absorption cross sections, respectively. The second quantity is the “asymmetry parameter” (g0), which is the mean cosine of the relative angle between the initial and scattered directions of a photon incident upon the particle. It is the sole parameter in the Henyey–Greenstein scattering phase function, which determines the spatial distribution of scattered light (Henyey & Greenstein 1941). Isotropic scattering occurs for g0 = 0. Particles described by g0 = 1pro- Figure 2. The rms vertical velocity (left panel) and Mach number (right panel) duce predominantly forward scattering, while those described as functions of pressure in the atmospheres of model hot Jupiters. See the text by g0 =−1 produce mostly backward scattering. Generally, ω0 for details of the simulations. and g0 depend on both the particle radius (rc) and the wavelength (A color version of this ﬁgure is available in the online journal.) considered (λ). (See also Madhusudhan & Burrows 2012.) we fully anticipate that it generally has the functional form Asymptotically, we expect that (e.g., Section 5.4 of Mz = Mz(P,T). It follows that Pierrehumbert 2010)

9M γk T g → 0,r/λ → 0, S ≈ z B . (10) 0 c (12) C 2 g0 → G0,rc/λ →∞, 2 ρcrc σmg

When S S0, atmospheric circulation keeps the particles lofted. where G0 > 0 is a constant. In other words, small particles When Smost + ω0F0 exp , cos ξ other published general circulation models of hot Jupiters, they 0 solve the primitive equations of meteorology, which assumes dF+ =−21 (1 − g0ω0) F− hydrostatic balance. Hydrostatic balance does not preclude dτ = − vz 0, since the vertical velocity is assumed to be sub-dominant − τ τ0 only in the vertical component of the momentum equation, but 21ω0g0 cos ξ0F0 exp . (14) cos ξ0 it does imply that any simulated value is probably a lower limit to the one obtained by a fully non-hydrostatic simulation. It is Here, τ denotes the optical depth measured from some reference −6 with these caveats that we adopt Mz = 10 at P ∼ 0.1 bar. depth in the model atmosphere, while τ0 is the optical depth

5 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory associated with the distance from this reference depth to the The preceding equation constitutes a solution of the radiative top of the model atmosphere. The blackbody flux is given by transfer equation. The blackbody efficiency of the particle is 4 πB = σSBT . The zenith angle ξ0 is the angle between the defined as πB incident stellar flux and the vertical axis; we allow for cos ξ0 = 1 fB ≡ . (20) in our derivation but later set it to be unity in our calculations. F0 cos ξ0 The dimensionless quantities 1 and 2 are closure relations In an isothermal atmosphere, we expect large particles to that depend on assumptions about the angular distribution behave like blackbodies (fB = 1) when they are observed of incoming versus outgoing radiation (Pierrehumbert 2010). at a wavelength λ 2πrc. By contrast, small particles are Physically, the terms involving F0 represent “direct beam” inefficient emitters of radiation (fB = 0) at a wavelength emission from the star, while those involving F± represent the λ 2πrc. diffuse emission. In the limit of ω0 = 0, we expect 3Ag/2 = fB, which Taking the derivative of the second equation in (14) and occurs only if 1 = 2. We adopt the “hemi-isotropic closure” eliminating F− using the first equation yields a second-order (1 = 2 = 1), which asserts that the flux is isotropic in each of ordinary differential equation for F+, the upward and downward hemispheres (Pierrehumbert 2010). We checked that Equation (19) yields 3Ag/2 = 1 when ω0 = 1, 2 = = d F+ 2 independent of g0 and fB. In the limit of g0 fB 0, we = K F+ − 812 (1 − ω0)(1 − g0ω0) πB dτ2 recover Equation (5.49) of Pierrehumbert (2010), − τ τ0 3Ag ω0 − 21ω0F0 [1+g0 (1 − ω0)] exp , (15) = √ √ . (21) cos ξ0 2 (1 + 2 1 − ω0 cos ξ0)(1 + 1 − ω0)

2 where K ≡ 412(1 − g0ω0)(1 − ω0). The homogeneous 3. RESULTS solution for F+ takes the form 3.1. Albedo Spectra: The Degeneracy between Condensate Size and Relative Abundance to Sodium F+,h = f± exp [±K (τ − τ0)]. (16) The top panel of Figure 3 shows Ag as a function of ω0 for For a deep atmosphere, we expect the homogeneous solution to different values of g0 and fB. As the particle becomes more not diverge when τ →−∞, which compels us to set f− = 0. forward scattering (g0 > 0), the geometric albedo generally = The particular solution takes the form decreases (see also Sudarsky et al. 2000). When ω0 0 and 1, = 3Ag/2 fB and 1, respectively, as expected. Generally, larger τ − τ0 particles (higher fB values) correspond to higher geometric F+,p = fp,1 exp + fp,2. (17) albedos. Next, we compute albedo spectra by considering the cos ξ 0 limiting case where Rayleigh scattering is entirely due to the presence of small condensates (2πr /λ 1) and absorption The constants f and f are determined by substituting c p,1 p,2 is due to the sodium D doublet (Sudarsky et al. 2000). (The Equation (17) into Equation (15) and matching the coefﬁcients potassium doublet absorbs at somewhat longer wavelengths: found on both sides of the equation. The full solution (F +F ) +,h +,p about 0.77 μm.) Details of the scattering and absorption cross then becomes sections used are described in the Appendix. This simple model contains three parameters: the particle radius (r ), the τ − τ0 c F+ = f+ exp [K (τ − τ0)] + fp,1 exp + fp,2. (18) scattering refractive index of the condensates (n ), and the ξ r cos 0 relative abundance of sodium atoms to the condensates by number (fNa). The single scattering albedo becomes All that remains is to determine the constant f+, which may be accomplished by substituting the full solution into the second σscat ω0 = (22) equation in (14), which yields an expression for F− and hence σscat + fNaσNa F − F− = 2F↓. Applying the boundary condition of F↓ = 0 + and is somewhat insensitive to nr. We choose nr = 1.6 to mimic when τ = τ produces an expression for f . 0 + the presence of silicates such as enstatite. We set g0 = 0. In the Returning to our expression for F−, we again apply the middle and bottom panels of Figure 3, we show calculations condition τ = τ to obtain F↑ = 3/2A F cos ξ , which yields 0 g 0 0 of Ag using Equations (19) and (22)forrc ∼ 0.1 μm and the somewhat unwieldy formula for the geometric albedo, ∼10 nm and compare them to the measurement of the albedo spectrum of the hot Jupiter HD 189733b by Evans et al. 2[g ω + ω (1 − g ω )] A = 0 0 0 0 0 (2013). These assumed particle radii are consistent with those g − − 2 2 3(1 g0ω0)(1 K cos ξ0) inferred by Lecavelier des Etangs et al. (2008). Generally, K 16 f (1 − ω )(1 − g ω ) the computed geometric albedo is a degenerate function of rc + 1 2 B 0 0 0 2 (1 − g ω )+K 3K2 and fNa. However, since the Rayleigh cross section scales as 1 0 0 σ ∝ r6, a small change in the particle radius needs to be − scat c − 41(1 g0ω0) g0ω0 compensated by a large change in the relative abundance of 3[21(1 − g0ω0)+K] 1 − g0ω0 sodium atoms to condensates. This property may prove to be useful for constraining r in future observations of exoplanetary − 2K c 3[2 (1 − g ω )+K] atmospheres. We do not use Equation (19) to compute Ag for 1 0 0 use in the analytical temperature–pressure proﬁles, but rather [1 + 2 (1 − g ω ) cos ξ ][g ω + ω (1 − g ω )] × 1 0 0 0 0 0 0 0 0 . specify it as a free parameter while being mindful that the 2 2 (1 − g0ω0)(1 − K cos ξ0) geometric albedo is an emergent property of the scattering and (19) absorption properties of the atmosphere.

6 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory

Figure 4. Examples of temperature–pressure proﬁles adopting some of the parameters of Kepler-7b. The dots indicate the locations of the photon deposition layers, where starlight is predominantly absorbed. The Kepler bandpass probes the photon deposition layer. (A color version of this ﬁgure is available in the online journal.)

Figure 5. “Lofting parameter” S as a function of Teq,0 (see the text for deﬁnition) and the geometric albedo of the hot Jupiter for three different cloud particle radii (rc = 1, 10, and 100 μm). (A color version of this ﬁgure is available in the online journal.)

In the limit of vertically uniform populations of scatterers and absorbers, starlight is absorbed and reflected mostly at the Figure 3. Calculations of the geometric albedo. Top panel: Ag as a function photon deposition layer (P ). It is also the pressure level the of the single scattering albedo (ω ) for different values of the asymmetry D 0 Kepler optical bandpass is probing. Thus, the presence of a parameter (g0) and the blackbody efficiency of the particle (fB). Middle panel: Ag as a function of wavelength (λ) for Rayleigh scattering by condensates and non-zero albedo has two effects. The first, obvious one is to absorption by the sodium D doublet assuming particle radii of rc ∼ 0.1 μm. diminish the amount of heat deposited in an atmosphere. The ∼ Bottom panel: same as for the middle panel, but for rc 10 nm. second, less obvious effect is to alter the location at which most (A color version of this figure is available in the online journal.) of the starlight is being deposited. More reflecting atmospheres tend to have their starlight deposited higher in altitude. Both properties are reflected in Figure 4. Consequently, the effect of 3.2. The Effects of Geometric Albedo on Thermal Structure a non-zero albedo is to produce a temperature inversion if Ag is of a high enough value. Figure 4 shows some examples of temperature–pressure profiles. We have adopted some of the parameters of Kepler-7b: = = 3.3. The Relationship between Thermal Structure, Teq 1586 K and log g 2.62 (Demory et al. 2011). We pick Condensate Size, and Condensate Abundance 2 −1 κIR = 0.004 cm g such that the infrared photosphere lies at ∼0.1 bar. Since the opacity sources in the optical range of Next, we wish to examine the lofting properties of the wavelengths remain poorly known for hot Jupiters in general, atmosphere at P = PD. We evaluate the temperature at the 2 −1 we adopt κO = 0.003 cm g as an illustration. We consider photon deposition layer using our analytical T–P profiles: = ≡ Ag = 0, 0.17, and 0.34 to demonstrate the effects of varying the T TD where TD T (PD). In Figure 5,weshowS as a function albedo, corresponding to PD ≈ 0.09, 0.04, and 0.03 bar. of Teq,0 for various values of Ag. We show three sets of curves

7 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory for rc = 1, 10, and 100 μm. For rc = 1 μm, we obtain S 1 configurations of TD that will produce a sinusoidal functional for Teq,0 = 1000–3000 K, implying that micron-sized particles form for S with a peak that is offset from the secondary eclipse. should be readily lofted in hot Jovian atmospheres, consistent The first and simplest configuration is a shifted Heaviside func- with the results from the 3D simulations of Parmentier et al. tion, i.e., TD has two values, one in each hemisphere, but it is (2013). Micron-sized particles have Nk 1 and S ∝ PD. translated in longitude by some amount. In this case, the cor- Since we do not allow the optical opacity (κO) to depend on responding brightness temperature in the Kepler bandpass as a the intensity of incident, stellar irradiation, S is a flat function function of orbital phase will be a sinusoidal function that has of Teq,0. However, since PD depends on Ag, the pressure level a peak offset (Cowan & Agol 2008). The second configuration probed is lower (higher altitude) and the corresponding value of is for TD ∝ sin(φ ± φ0), with φ being the longitude of the S is lower for larger values of Ag.Forrc = 100 μm(Nk 1), exoplanet and φ0 being a constant offset, which also produces S now has a dependence on Teq,0 as we have S ∝ TD. However, a sinusoidal function for S. In the right circumstances (i.e., since S 1 for all values of the equilibrium temperature rc ∼ 10 μm), the variation in temperature and pressure may examined, we do not expect particles with rc = 100 μmto cause S to possess values ranging from 0.1 to 1. Such a variation be lofted by atmospheric circulation. Particles with rc = 10 μm in S produces a longitudinal variation in the abundance of lofted have S ∼ 0.1–1 and are expected to be partially lofted as a cloud particles, which in turn produces a longitudinal varia- population, again consistent with the results of Parmentier et al. tion in the associated albedo and the flux of reflected starlight. (2013). The observations of Kepler-7b (Demory et al. 2013) are consis- If small cloud particles or grains (Nk 1) are robustly tent with such a scenario. By contrast, if small cloud particles formed in hot Jovian atmospheres, then they should be om- (Nk 1) are embedded in the atmosphere of Kepler-12b, we nipresent due to the ease at which atmospheric circulation will expect S ∝ PD, which produces a flat phase curve for a given keep them aloft. The variations in their scattering and absorp- albedo value if the opacity in the optical range of wavelengths tion properties (see Section 2.2.3), which are determined by is roughly constant with longitude. In other words, when small their sizes and compositions, are expected to produce a scatter grains are present, we expect their abundance to be zonally uni- in the values of the measured geometric albedos. form; if g, κO, and Ag are constant, then PD and hence S are constant across longitude. Small particles are consistent with a 4. DISCUSSION lower albedo if rc λ/2π. However, there is an important detail in the optical phase 4.1. Comparative Exoplanetology: curve of Kepler-7b that is difficult to reconcile with our simple Kepler-7b versus Kepler-12b explanation. The peak of the optical phase curve peaks after secondary eclipse, implying that the corresponding brightness The bounty of exoplanets discovered by the Kepler mission map peaks westward of the substellar point (Demory et al. 2013). has emphasized the importance of comparative exoplanetol- Irradiated atmospheres in the hot Jupiter regime are expected to ogy. Within our sample of hot Jupiters examined in Figure 1, possess temperature maps that peak eastward of the substellar Kepler-7b (Latham et al. 2010; Demory et al. 2011) and point (Showman & Polvani 2011), a theoretical expectation that Kepler-12b (Fortney et al. 2011) provide for an intriguing com- is corroborated by 3D simulations of atmospheric circulation. parison. They receive similar degrees of stellar heating (T ≈ eq,0 That this property is independent of whether Newtonian cooling 1500 K), and thus we expect the strength of atmospheric circula- (e.g., Showman & Guillot 2002; Heng et al. 2011b), dual- tion to be comparable in both atmospheres. They possess compa- band radiative transfer (Heng et al. 2011a; Rauscher & Menou rable surface gravities (log g ≈ 2.6) and radii (R ≈ 1.6–1.7R ). J 2012; Perna et al. 2012), or multi-wavelength radiative transfer They orbit somewhat quiescent, Sun-like stars of comparable (Showman et al. 2009) is utilized suggests that it is a robust metallicity ([Fe/H] ≈ 0.1). Yet their measured geometric albe- outcome of the hot Jupiter regime. If an infrared phase curve of dos are non-negligibly different: A ≈ 0.35 (for Kepler-7b) g Kepler-7b is measured with this property (eastward shift), then versus 0.08 (for Kepler-12b). For Kepler-7b, there is evidence it is a “smoking gun” for the presence of condensates—reflected for the presence of condensates at the atmospheric layer probed light and thermal emission are not tracing each other, implying by the Kepler bandpass (Demory et al. 2011, 2013). Further- that the spatial distribution of the condensates is being modified more, the optical phase curve of Kepler-7b exhibits a sinusoidal by atmospheric dynamics. functional form with a peak that is offset from the secondary eclipse (Demory et al. 2013). The infrared secondary eclipses 4.2. Caveats and Future Work of Kepler-7b, as detected by the Spitzer Space Telescope at 3.6 and 4.5 μm, imply brightness temperatures that are markedly Several caveats and unexplored aspects provide opportunities lower than that in the optical (Demory et al. 2013). By contrast, for future work. We have not used an eddy diffusion coefficient the phase curve of Kepler-12b exhibits no sinusoidal variations (Kzz) to mimic atmospheric circulation and instead approxi- (at the sensitivity level of Kepler) with a period similar to the mated the vertical velocity as a fixed fraction of the local sound orbital motion of the exoplanet, while registering brightness speed. Although such an approach includes the effect of varying temperatures of about 1400–1600 K in the infrared (Fortney the stellar irradiation, since it is tied to the temperature–pressure et al. 2011). profile, it fails to capture the dependence of the depth of atmo- We apply the lessons learnt from the analytical models pre- spheric circulation on the incident stellar flux (Fortney et al. sented in this study. All else being equal, we expect the photon 2008; Dobbs-Dixon et al. 2012; Perna et al. 2012). The vigor deposition layer to reside at a higher altitude (lower pressure) of atmospheric circulation in irradiated exoplanetary atmo- for Kepler-7b due to its higher albedo. If large cloud parti- spheres means that if clouds form, they will be well mixed from cles (Nk 1) are embedded in the atmosphere of Kepler-7b, ∼1 mbar all the way down to ∼10 bar, depending on F0 and then we expect S ∝ TD to describe the lofting property of its Ag. Performing 3D simulations of atmospheric circulation with photon deposition layer. Large particles generally produce a a diversity of cloud configurations will inform 1D models on higher geometric albedo if rc λ/2π. There are two possible how to “paint” clouds onto their T-P profiles.

8 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory Another important interplay we have not explored concerns the atmosphere. With most of the starlight being deposited atmospheric chemistry. Specifically, the strength and spectral at lower pressures, the infrared photosphere also lies at distribution of the incident stellar flux modify the absorption lower pressures. With the atmosphere being more radiative (κO) and scattering (Ag) properties of both the condensates and (or less advective) at higher altitudes, a small phase offset in the gas in the irradiated atmosphere. To understand the lofting the infrared is expected (Showman & Guillot 2002;Cowan behavior of small cloud particles as a function of the incident & Agol 2011a; Perna et al. 2012; Heng 2012). Conversely, stellar flux requires this interplay to be elucidated. a larger infrared phase offset is expected for a low albedo In recent years, retrieval models, originally developed for (Fortney et al. 2008). the modestly irradiated planets/moons of the solar system, have 4. Low albedo, small infrared phase offset. For highly irradi- been employed to infer the temperature–pressure profiles and at- ated hot Jupiters, the intense stellar flux trumps any effect mospheric chemistry/composition of hot Jupiters (Madsudhan due to opacity (e.g., albedo), and small infrared phase off- & Seager 2009; Lee et al. 2012; Line et al. 2012). While being sets are generally expected (Perna et al. 2012). Conversely, an important development in the study of exoplanetary atmo- hot Jupiters with lower levels of incident irradiation are spheres, these published works have so far been “blue sky” and expected to possess large infrared phase offsets. omitted the effects of clouds. Benneke & Seager (2012) specify As an example, we consider the prototypical case of the the albedo as a free parameter in their analysis, but only include hot Jupiter HD 189733b, for which large peak offsets have one of its effects as suggested by their use of the Guillot (2010) been measured in the infrared phase curves (Knutson et al. model: the diminution of the incident stellar flux, but not the 2007, 2009). This is consistent with the low intensity of stellar modification of its vertical absorption profile. They also allow irradiation impinging upon its atmosphere (T ≈ 1200 K). for the cloud-top pressure to be a fitting parameter. To illus- eq,0 Transit observations in the ultraviolet and optical range of trate the importance of including a non-zero albedo, consider wavelengths reveal a spectral slope, punctuated by sodium lines, the limiting case of A = 1, in which case we expect the photon g consistent with Rayleigh scattering by condensates present at deposition layer to reside at the top of the atmosphere and the the day–night terminators (Lecavelier des Etangs et al. 2008; infrared photosphere to be undefined (if interior heat from the Pont et al. 2008, 2013; Sing et al. 2011; Gibson et al. 2012). irradiated exoplanet is negligible). When the geometric albedo That the infrared peak offsets are not small suggests that the is just below unity, we expect the infrared photosphere(s) to be albedo associated with the condensates is small, or even close located just below the top of the atmosphere. When A = 0, g to zero, at and near the peak of the stellar spectrum. Based the contribution functions are correctly computed by the pub- on a tentative comparison of several brightness temperature lished retrieval models. Since we expect physical quantities to points with an analytical temperature–pressure profile, Heng vary continuously, the contribution functions, at various infrared et al. (2012) estimate that the Bond albedo of HD 189733b is wavelengths, should shift to lower pressures as A increases, an g about 0.1. The albedo spectrum of HD 189733b suggests a low effect that needs to be included in the retrieval models. Future to vanishing geometric albedo in the Kepler bandpass (Evans work that includes a simple model of clouds, with a small num- et al. 2013). Optical phase curves of HD 189733b will further ber of free parameters, in the retrieval technique will set some constrain the properties of the condensates, including if they are empirical constraints on the cloud properties, although some small or large (as already described by the preceding scenarios). degeneracy is anticipated. Case studies that contradict these scenarios (e.g., Crossfield Although 3D simulations that treat only the dynamical or et al. 2010) hint at the possibility of missing physics or chem- radiative interaction between the atmospheric flow and the cloud istry and will inspire novel ways of thinking about irradiated particles are insufficient for a full understanding of the effects of exoplanetary atmospheres. Some of the degeneracies described clouds in irradiated exoplanetary atmospheres, neither is a 1D may be broken by examining the colors of these atmospheres. analytical model that employs simple, approximate treatments for these effects, as presented in this study. Nevertheless, both approaches drive us toward constructing falsifiable models that K.H. acknowledges financial and/or logistical support from are realistic yet simple enough to be confronted by observations the University of Bern, the Swiss-based MERAC Foundation, and feasibly included in future 3D simulations. and the University of Zurich.¨ We are grateful to Bruce Draine, Jaemin Lee, Michael Gillon, and Nikku Madhusudhan for useful 4.3. An “Observer’s Cookbook” conversations. K.H. benefited from discussions conducted at the Exeter–Oxford exoplanet workshop in 2013 April and at the To provide an executive summary of the observational rele- PPVI conference in 2013 July. We thank the anonymous referee vance of our study, we highlight a few scenarios and suggest for constructive comments that improved the quality and clarity possible (and possibly non-unique) interpretations. When the of the manuscript. peak amplitude of the phase curve is on the order of the occultation depth, then we term it to be “sinusoidal”; if not, we term APPENDIX it to be “flat.” When the angular offset of the peak of the phase curve from secondary eclipse is ∼1◦ or close to zero, we term MODELING SODIUM ABSORPTION it to be “small.” Phase offsets of ∼10◦ are “large.” AND RAYLEIGH SCATTERING 1. High albedo, sinusoidal optical phase curve. Large cloud The quantum mechanical properties of the sodium D doublet particles or dust grains (∼10 μm). As discussed, a possible are well known (e.g., Draine 2011). The absorption cross section example is Kepler-7b. is πe2 2. Low albedo, flat optical phase curve. Small cloud particles σNa = fluφν , (A1) (1 μm). As discussed, a possible example is Kepler-12b. mec 3. High albedo, small infrared phase offset. A high albedo where e is the elementary unit of electric charge, me is the mass implies that the photon deposition layer resides higher in of the electron, c is the speed of light, flu is the oscillator strength,

9 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory and φν is the line proﬁle function, described by a Lorentz proﬁle, REFERENCES

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10 The Astrophysical Journal, 777:100 (11pp), 2013 November 10 Heng & Demory

Pont, F., Sing, D. K., Gibson, N. P., et al. 2013, MNRAS, 432, 2917 Saumon, D., & Marley, M. S. 2008, ApJ, 689, 1327 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numer- Seager, S. 2010, Exoplanet Atmospheres (Princeton, NJ: Princeton Univ. Press) ical Recipes in FORTRAN: The Art of Scientiﬁc Computing (Cambridge: Showman, A. P., Fortney, J. J., Lian, Y., et al. 2009, ApJ, 699, 564 Cambridge Univ. Press) Showman, A. P., & Guillot, T. 2002, A&A, 385, 166 Quintana, E. V., Rowe, J. F., Barclay, T., et al. 2013, ApJ, 767, 137 Showman, A. P., & Polvani, L. M. 2011, ApJ, 738, 71 Rauscher, E., & Menou, K. 2012, ApJ, 750, 96 Sing, D. K., Pont, F., Aigrain, S., et al. 2011, MNRAS, 416, 1443 Russell, H. N. 1916, ApJ, 43, 173 Spiegel, D. S., Silverio, K., & Burrows, A. 2009, ApJ, 699, 1487 Santerne, A., Bonomo, A. S., Hebrard,´ G., et al. 2011, A&A, 536, A70 Sudarsky, D., Burrows, A., & Pinto, P. 2000, ApJ, 538, 885

ERRATUM: “UNDERSTANDING TRENDS ASSOCIATED WITH CLOUDS IN IRRADIATED EXOPLANETS” (2013, ApJ, 777, 100)

Online-only material: color ﬁgures

We report on a technical error that affects some of the numbers reported in our study, but do not alter its general conclusions. A confusion between division and multiplication in a computer code used to post-process our three-dimensional (3D) simulations led to an under-estimation of the vertical velocity by several orders of magnitude. We note that this error does not affect previous, published work. Correcting this error implies that Figures 2, 4, and 5 of the published version of this article have to be amended. The previous claim that the vertical Mach number has an order-of-magnitude value of 10−6 is now revised to 10−3. These changes only affect the part of our study that deals with the lofting properties of condensates, but do not affect the parts dealing with the measured geometric albedos of hot Jupiters and interpreting the reﬂection spectrum of HD 189733b. For the latter, we correct a second error associated with the Rayleigh scattering cross section for particles, which amends Figure 3 and Equations (A5) and (A7) of the published article. All of the salient and major conclusions of our study, as stated in our original abstract, remain unchanged. In 3D simulations of atmospheric circulation that solve the primitive equations of meteorology, it is common to specify the pressure (P) as the vertical coordinate (rather than the spatial coordinate z). A direct output of such 3D simulations, including ours, is the quantity DP ω ≡ . (1) Dt To compute the vertical velocity in more familiar physical units, we invoke the ideal gas law and use

− ∂P 1 RT DP v = ω =− , (2) z ∂z Pg Dt where R denotes the specific gas constant, T the temperature, and g the surface gravity. Since R and g are input parameters, P is obtained from the vertical grid and T and ω are direct outputs of the simulation; vz may be straightforwardly computed. Unfortunately, an error was made in the post-processing computer code used to evaluate vz, ∂P Pg DP v = ω =− , (3) z ∂z RT Dt due to a confusion between division and multiplication. This error does not affect previous, published work as ω, and not vz,was the quantity being explicitly simulated/computed. It substantially revises Figure 2 in the published version of this article, which is corrected here. Again, rms values are presented. First, the order-of-magnitude values of vz,rms are now significantly larger, allowing us to achieve consistency with other published results of atmospheric circulation (e.g., Rauscher & Menou 2012). Second, both vz,rms and the vertical Mach number (Mz,rms) display greater sensitivity to the stellar irradiation flux and the absence or presence of a temperature inversion. Third, they are somewhat insensitive, across pressure, between ∼10 μbar and ∼0.1 bar. Roughly speaking, −3 −6 one may adopt Mz,rms ∼ 10 within this pressure range. This is the value we now adopt, rather than the Mz,rms ∼ 10 value assumed in the published article. The only equation affected by this error is Equation (11) of the published version of this article, which now reads 2.7M γP S ≈ z ρcrcg − M γ P ρ r g 1 ∼ 1 z c c . (4) 10−3 7/5 0.1 mbar 3gcm−3 1 μm 103 cm s−2 Essentially, we expect micron-sized grains to be lofted up to ∼0.1 mbar by atmospheric circulation. In addition to Figure 2, Figures 4 and 5 of the published article are now revised and presented in this erratum. For the 2 −1 temperature–pressure profile presented in Section 3.2 and Figure 4 of the published article, we now adopt κIR = 4cm g 2 −1 and κO = 3cm g and place the bottom boundary of the model at 1 bar (instead of 100 bar). These changes allow us to examine the profile at higher altitudes (lower pressures). We see that the qualitative effect of changing the geometric albedo remains invariant. The temperature values are largely similar. The lofting parameter S is then computed using these three temperature–pressure profiles.

1 The Astrophysical Journal, 785:80 (4pp), 2014 April 10 Erratum: 2013,ApJ,777, 100

Figure 2. The rms vertical velocity (left panel) and Mach number (right panel) as functions of pressure in the atmospheres of model hot Jupiters. (A color version of this ﬁgure is available in the online journal.)

Figure 3. Calculations of the geometric albedo. (A color version of this ﬁgure is available in the online journal.)

2 The Astrophysical Journal, 785:80 (4pp), 2014 April 10 Erratum: 2013,ApJ,777, 100

Figure 5. “Lofting parameter” S as a function of Teq,0 and the geometric albedo of the hot Jupiter for three different cloud particle radii (rc = 1, 10, and 100 μm). (A color version of this ﬁgure is available in the online journal.)

Figure 5 shows that the qualitative conclusion is unchanged: micron-sized particles are easily lofted, rc = 10 μm particles are partially lofted as a population, while we do not expect rc = 100 μm particles to be lofted by atmospheric circulation. In Equation (A5) of the published article, we correct an error associated with the Rayleigh scattering cross section for particles, 27π 5 n2 − 1 2 σ = r r6λ−4, (5) scat 2 c 3 nr +2 where the extra factor of 26 derives from the fact that it is the diameter and not the radius that should be used in the formula. Also, we correct a coding error that originates from the conversion between μm and cm for the physical units of λ. These changes imply that the middle and bottom panels of Figure 3 of the published article have to be amended; a corrected version is shown in this erratum. Only a minute amount of condensates, relative to sodium atoms by number, needs to be present to produce the measured geometric albedo spectrum of HD 189733b. Furthermore, Equation (A7) is revised, 1/3 1 (n − 1) n2 +2 r > r r f 1/6. (6) c 2 − gas 2πn nr 1

= × 17 −3 = = 1/6 1/6 Using n 5 10 cm , nr 1.0001, and nr 1.6, we obtain 0.5fgas nm for the critical particle radius (instead of 0.9fgas nm), an order of magnitude larger than the Bohr radius. A technical error associated with the post-processing simulation output led to the vertical velocity of the atmospheric circulation to be underestimated, which we rectify in this erratum. We provide corrected versions of Figures 2, 4, and 5 and Equation (11). Some of the reported numbers associated with the part of the study dealing with the lofting of condensates have changed, but the overarching conclusions remain unchanged. The other parts of the study, which deal with the measured geometric albedos of hot Jupiters and the interpretation of the reﬂection spectrum of HD 189733b are unaffected. For the latter, we correct an error associated with the Rayleigh

3 The Astrophysical Journal, 785:80 (4pp), 2014 April 10 Erratum: 2013,ApJ,777, 100 scattering cross section, which amends Figure 3 and Equations (A5) and (A7) of the published version. The major conclusions of the study, as described in the published version of the abstract, remain unchanged.

We thank Vivien Parmentier and Tom Evans for bringing these issues to our attention.

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