The Effect of Clouds as an Additional Opacity Source on the Inferred Metallicity of Giant Exoplanets

Abstract

Atmospheres regulate the planetary heat loss and therefore influence planetary thermal evolution. Uncertainty in a giant planet’s thermal state contributes to the uncertainty in the inferred abundance of heavy elements it contains. Within an analytic atmosphere model, we here investigate the influence that different cloud opacities and cloud depths can have on the metallicity of irradiated extrasolar gas giants, which is inferred from interior models. In this work, the link between inferred metallicity and assumed cloud properties is the thermal profile of atmosphere and interior. Therefore, we perform coupled atmosphere, interior, and evolution calculations. The atmosphere model includes clouds in a much simplified manner; it includes long-wave absorption but neglects shortwave scattering. Within that model, we show that optically thick, high clouds have negligible influence, whereas deep-seated, optically very thick clouds can lead to warmer deep tropospheres and therefore higher bulk heavy element mass estimates. For the young hot Jupiter WASP-10b, we find a possible enhancement in inferred metallicity of up to 10% due to possible silicate clouds at ∼0.3 bar. For WASP-39b, whose observationally derived metallicity is higher than predicted by cloudless models, we find an enhancement by at most 50%. However, further work on cloud properties and their self-consistent coupling to the atmospheric structure is needed in order to reduce uncertainties in the choice of model parameter values, in particular of cloud opacities.

1. Introduction

Metallicity and core mass of giant planets contain information on protostellar disks and on the process of planet formation. Therefore, planetary metallicity, or bulk heavy element mass fraction $Z_p$, is an important parameter. Core accretion formation models that reproduce the metallicity of the solar system giant planets [1] predict a rapid decrease of $Z_p$ with increasing planet mass $M_p$, still allowing for up to $14\times$ solar ($Z_p\sim 20\%$) for a Saturn-mass planet but for less than $3\times$ solar ($Z_P\sim 4.5\%$) for a 2 $M_{\mathrm{Jup}}$ planet.

Recently, Wakeford et al. (2018) [2] used transmission spectra to determine the metallicity in the atmosphere of the Saturn-mass planet WASP-39b. They retrieved a high value of ∼100–$200\times$ solar. This is not only higher than the prediction from core accretion formation but also higher than the upper limit of $55\times$ solar for the atmospheric metallicity as inferred from structure models for this planet [3]. Moreover, for some massive giant planets such as the 3 $M_{\mathrm{Jup}}$ planet WASP-10b [4], structure models predict a significant heavy element enrichment of $Z_p$ of 10% or more [5].

In this paper, we pursue the possibility of uncertainty in the planet’s inferred bulk metallicity due to an additional opacity source of limited vertical extent. We call it a cloud layer; however, we do not model any physical aspect of real clouds except the potential additional longwave opacity. Because of their optical properties, clouds in the atmosphere are known to modify the observable transmission spectrum [6] and the temperature structure of the atmosphere itself [7]. Clouds also influence the atmospheric scale height, which provides a direct link to the mean molecular weight of the atmosphere [8]. Since the latter depends on atmospheric metallicity, its value can be inferred from the observed transmission spectrum in combination with radiative transfer calculations, which yield the scale heights of the observed portion of the atmosphere. In this work, we follow a different approach—inferring the atmospheric metallicity from planetary structure models that are primarily constrained by the observed mass, radius, and age of the star as explained below. In gaseous planets, the radiative atmosphere transitions smoothly into the adiabatic deep interior. The pressure—temperature (P–T) conditions at this transition influence the internal temperatures and the possible intrinsic heat loss [9]. Higher temperatures at a given pressure level in a fluid planet lead to lower densities and to expansion if not compensated for by an increase in heavy element abundance, an effect that is still relevant for the ice giants Uranus and Neptune [10]. Therefore, atmospheric temperature profile and our inference of a planet’s metallicity are strongly coupled. We include a cloud layer into our coupled planetary atmosphere, interior, and evolution calculations by using the semi-analytic model of Heng et al. (2012) [11], which allows us to conveniently investigate the influence of assumed cloud opacity and assumed cloud pressure level on the atmospheric P–T profile. This model is applied to the two giant planets: WASP-10b and WASP-39b. Both planets may harbor clouds since their atmospheric P–T profiles intersect with a number of condensible species, as shown in Figure 1.

Candidates of cloud forming species for these planets are Na${}_2$S, MnS, Cr, and silicates. This study is not the first one to investigate the influence of cloudy and cloud-free atmospheres on the evolution of gaseous planets. Clouds have been considered in models for planets with hydrogen-dominated atmosphere before. For instance, Linder et al. (2018) [12] studied the influence on the spectra and thermal evolution of weakly irradiated exoplanets while Kurosaki et al. (2017) [13] studied the influence of water clouds on the cooling of the ice giant Uranus. For strongly irradiated hot Jupiters, Barman et al. (2001) [14] find a large heating effect in the upper atmosphere from reflection of stellar incident flux and absorption of dust grains at infrared wavelengths in comparison to clear atmospheres, with consequences on the emergent spectra, while Baraffe et al. (2003) [15] find a minor influence of dusty versus clear irradiated atmospheres for the luminosity evolution of hot Jupiters.

In Section 2, we list the relevant observed system parameters and describe our modeling approach for the atmosphere with a cloud layer, the interior, and the thermal evolution. Results for WASP-10b are presented in Section 3 and for WASP-39b in Section 4. In particular, we take the $Z_p$ value of Thorngren and Fortney (2019) [3] for WASP-39b as an input parameter for our models and see if the high predicted atmospheric metallicity of Wakeford et al. (2018) [2] can be reached just by including an additional opacity source which may mimic the effect of a cloud deck. We compare the obtained atmospheric models with self-consistent clear and cloudy models by Molliére et al. (2017) [17] in Section 5. A summary is given in Section 6.

2. Methods

2.1. Planet and Star Parameters

WASP-10b is a massive ($2.96M_{\mathrm{Jup}}$) and non-inflated ($T_{\mathrm{eq}}=950$ K) hot Jupiter. Its young age of $270\pm 80$Myr [18,19,20] makes it an interesting object to study planet formation and evolution. While early radius estimates predicted a rather large radius of ∼$1.27R_{\mathrm{Jup}}$ [19], subsequent careful analysis of the spots on the K5 dwarf host star suggested a $20\%$ smaller planet radius of $1.02R_{\mathrm{Jup}}$ [4], which we use in this study.

WASP-39b is a Saturn-mass planet ($0.28M_{\mathrm{Jup}}$) with a large radius ($1.27R_{\mathrm{Jup}}$) and therefore low density $\rho =0.141{\rho }_{\mathrm{Jup}}$ [21]. It is orbiting a late G-type star, which is smaller and, with an age of $9_{-4}^{+3}$Gyr, possibly older than the Sun. The observational parameters used here for WASP-10b and WASP-39b are listed in

Table 1. Stellar and planetary parameters.

	WASP-10b	WASP-39b ${}^5$
$M_P$	$2.{96}_{-0.17}^{+0.22}{M}_{\mathrm{Jup}}$${}^1$	$0.28\pm 0.03M_{\mathrm{Jup}}$
$R_P$	$1.{03}_{-0.03}^{+0.077}{R}_{\mathrm{Jup}}$${}^4$	$1.27\pm 0.04R_{\mathrm{Jup}}$
a	$0.{0369}_{-0.0014}^{+0.0012}$AU ${}^1$	$0.0486\pm 0.0005$AU
e	$0.013\pm 0.063$${}^3$	0
$T_{\mathrm{eq},\mathrm{A}=0}$	${950}_{-26}^{+30}$K ${}^4$	${1116}_{-32}^{+33}$K
P	3.09 d	4.05 d
$M_{\star }$	$0.75M_{\mathrm{Sun}}$${}^2$	$0.93\pm 0.03M_{\mathrm{Sun}}$
$R_{\star }$	$0.67R_{\mathrm{Sun}}$${}^4$	$0.895\pm 0.023R_{\mathrm{Sun}}$
$T_{\star }$	$4675\pm 100$K ${}^1$	$5400\pm 150$K
age ${\tau }_{\star }$	$270\pm 80$Myr${}^3$	$9_{-4}^{+3}$ Gyr

${}^1$ Ref. [19], ${}^2$ Ref. [18], ${}^3$ Ref. [20], ${}^4$ Ref. [4], ${}^5$ Ref. [21].

.

Since we are interested in the effect of clouds relative to cloudless atmospheres on the inferred planet metallicity, we compute here planet models for a variety of cloud parameters but do not account for the observational uncertainties in planet mass and radius. The only exception is thermal evolution calculations for WASP-39b, where we request its radius at present time to drop below the $1\sigma$ upper limit.

2.2. Interior

To estimate the present structure of the planets, we connect the atmosphere to the interior and perform thermal evolution calculations. For the interior, we assume a three-layer structure of rocky core, an adiabatic, convective envelope, and a radiative atmosphere. Atmosphere and envelope consist of a mixture of hydrogen, helium and metals. Respective equations of state (EOS) are combined via the linear mixing rule. By heavy elements or metals, we denote all elements or molecules heavier than helium. $Z_{\mathrm{atm}}$ and $Z_{\mathrm{env}}$ are the heavy element mass fractions in the atmosphere and envelope, respectively, which we assume to be equal, $Z_{\mathrm{atm}}=Z_{\mathrm{env}}=Z$. This is an assumption, not ruling out other relations between atmospheric and envelope abundances [17,22]. The planetary bulk heavy element mass fraction is $Z_{\mathrm{P}}=Z_{\mathrm{env}}{M}_{\mathrm{env}}/M_{\mathrm{P}}+M_{\mathrm{core}}/M_{\mathrm{P}}$, and $M_{\mathrm{env}}$ and $M_{\mathrm{core}}$ are the masses of envelope and core. For the solar reference metallicity we use $Z_{\odot }=1.5\%$ [23]. For WASP-10b, we set $Z=Z_{\odot }$ and allow only the core mass to vary while, for WASP-39b, we allow also Z to vary. The helium to hydrogen mass fraction is set to the protosolar value of $Y=0.27$, where $Y=M_{\mathrm{He}}/(M_{\mathrm{He}}+M_{\mathrm{H}})$. For hydrogen and helium, we use the SCvH EOS [24]. Metals in the envelope are represented by that He-EOS scaled in density by a factor of four, or by the ice EOS presented in [25]. The rocky core obeys the pressure–density relation given in [25]. The density $\rho (P,T)$ is obtained from the linearly mixed EOS at the pressure P and temperate T by interpolation. We obtain the mixed EOS by adding heavy elements to the interior and the atmosphere via the linear mixing rule ${\rho }^{-1}(P,T)={\sum }_i{X}_i/{\rho }_i(P,T)$, where $X_i$ denotes the mass fraction of component i and $X_{\mathrm{H}}:=X$, $X_{\mathrm{He}}:=Y$, $X_{\mathrm{Z}}:=Z$ [26]. The density profile follows a pre-computed P–T profile along the adiabat of the envelope. Increasing the temperature at fixed pressure usually decreases the density. Lower densities in the mantle result in a larger core mass to conserve the given planet mass. This is why the P–T profile is so important. Otherwise, we rely on the usual structure equations for non-rotating, spherical giant planets as previously done in [26,27].

2.3. Atmosphere Model with Clouds

The atmosphere model yields the atmospheric P–T profile. We use the 1D, plane-parallel, analytical atmosphere model by Heng et al. (2012) [11] for hot Jupiters. It is based on the two-stream solution and dual band approximation, where the incoming and outgoing radiation fluxes are described by different frequency-averaged mean opacities. The incoming flux is represented by the short-wave opacity ${\kappa }_S$, equivalent to the opacity ${\kappa }_{\mathrm{vis}}$ for visual light used in [28], while the outgoing flux is described by the long-wave opacity ${\kappa }_L$ equivalent to ${\kappa }_{\mathrm{th}}$ in [28] for thermal radiation. Following Heng et al. [11], ${\kappa }_S$ is constant with respect to temperature and pressure while ${\kappa }_L$ may have a dependence on pressure. Indeed, gas opacities significantly depend on pressure because of pressure broadening or collision induced absorption. Cloud decks are included as an additional opacity source ${\kappa }_c\left(P\right)$ to the constant long-wave opacity ${\kappa }_{\mathrm{L},0}$ of the otherwise cloudless atmosphere,

$${\kappa }_L\left(P\right)={\kappa }_{\mathrm{L},0}+{\kappa }_c\left(P\right).$$

(1)

The analytic model atmosphere provides a relation between global mean temperature T and longwave optical depth $\mathrm{d}{\tau }_L={\kappa }_L\mathrm{d}m$, where m is column mass from top to bottom, as well as the parameter $\tau ={\kappa }_{L}m$. We call the latter here optical depth although this holds only if ${\kappa }_L=\mathrm{const}.$. The T–$\tau$ relation makes use of the Eddington coefficients ${\mathcal{E}}_1=1/3$ and ${\mathcal{E}}_2=1/2$ to close the set of equations for the moments of radiation transfer. It reads (cf. Equation (31) in [11])

$$T^4=\frac{T_{\mathrm{int}}^4}{4}\left(2+3{\int }_0^m{\kappa }_L\mathrm{d}{m}^{\prime }\right)+\frac{T_{\mathrm{eq}}^4}{2}\left[1+\frac{\gamma }{\sqrt{\xi }}{E}_2\left(\frac{{\kappa }_S{m}^{\prime }}{\sqrt{\xi }}\right)+3{\int }_0^m{\kappa }_L{E}_3\left(\frac{{\kappa }_S{m}^{\prime }}{\sqrt{\xi }}\right)\mathrm{d}{m}^{\prime }\right],$$

(2)

with $E_j\left(x\right)={\int }_1^{\infty }{y}^{-j}exp(-xy)\mathrm{d}y$ as the exponential integrals. Equation (2) depends on the cloud opacity through ${\kappa }_L$ and the opacity ratio $\gamma ={\kappa }_S/{\kappa }_L$. Furthermore, the global mean temperature T depends on the intrinsic heat flux $F_{\mathrm{int}}={\sigma }_B{T}_{\mathrm{int}}^4$, which is the outgoing flux from the planet at the bottom of the atmosphere, and on the zero-albedo irradiation flux ${\sigma }_B{T}_{\mathrm{eq},0}^4={\sigma }_B{T}_{\star }^4{(R_{\star }/2a)}^2$ where ${\sigma }_B$ is the Stephan–Boltzmann constant. Thus, ${(1-A)}^{1/4}{T}_{\mathrm{eq},0}$ is the globally averaged temperature a planet of albedo A would adopt if in radiation equilibrium with the incident flux. Since a scattering parameter $\xi <1$ would be inconsistent with a non-uniform opacity, here ${\kappa }_L\left(P\right)$, we set $\xi =1$ (no scattering) and take scattering into account only via the albedo in $T_{\mathrm{eq}}$, which we set to $A_B=0.3$ [29,30], while noting that other work suggests smaller values (e.g., [31]). More recently, the geometric albedo of several exoplanets has been derived from secondary eclipse data and found to be quite small, even less than 0.1 [32]. On the other hand, the Bond albedo value of Jupiter itself has recently been revised upward from its Voyager-data based value of 0.34 to the new Cassini-data based value of 0.5 [33]. To study the thermal evolution of irradiated giant planets as a function of uncertainty in albedo is left to future work. For a more consistent treatment of scattering in the presence of non-uniform absorption, see Ref. [34]. For the cloud-free (${\kappa }_L={\kappa }_{\mathrm{L},0}$) atmosphere without scattering ($\xi$=1), Equation (2) reduces to the global average temperature profile of Guillot (2010) [28]. The P–$\tau$ relation for constant gravity g and pressure-dependent longwave opacity reads

$$P=m·g=(\tau /{\kappa }_L)g.$$

(3)

We use the cloud-free model to constrain the parameter $\gamma$. For WASP-10b, we fit to 1D, non-gray, atmospheric P–T profiles specifically calculated for this planet for different values of $T_{\mathrm{int}}$ [35] (see Figure 1). For WASP-39b, we fit ${\kappa }_S$ and ${\kappa }_{\mathrm{L},0}$ to the global averaged P–T profile from Ref. [16] for a $1\times$ solar composition metallicity. We find ${\kappa }_{\mathrm{L},0}=0.0136$ cm${}^2$/g, ${\kappa }_S=0.002$ cm${}^2$/g ($\gamma =0.147$) for WASP-10b and ${\kappa }_{\mathrm{L},0}=0.006$ cm${}^2$/g, ${\kappa }_S=0.00037$ cm${}^2$/g ($\gamma =0.062$) for WASP-39b. However, albeit using solar-composition models to fit our double-gray clear atmosphere, it is important to notice that an increase (or decrease) of atmospheric enrichment changes the position of the isotherm [36,37]. An increase in atmospheric metallicity leads to higher temperatures in the isothermal part of the atmosphere [38]. In the case of WASP-39b, where possibly the atmosphere is enriched by a factor of 100–200× solar value, the isotherm would be pushed to even hotter temperatures.

The results of these fits are shown in Figure 1. In Figure 2, we show the Rosseland mean opacities along the P–T profiles for the present planets using the fit formula of Valencia et al. [39] to the tabulated values of Freedman et al. [40]. We conclude that our obtained long-wave opacity values ${\kappa }_{\mathrm{L},0}\sim 0.01$ cgs are appropriate mean Rosseland mean opacities in the radiative atmospheres of both planets.

2.4. Cloud Model

The model of Heng et al. [11] for a purely absorbing cloud provides a simple toy model approach that reduces the complexity of the problem to few parameters while including the important greenhouse effect of clouds. The cloud opacity can be assumed to take the shape

$${\kappa }_c\left(P\right)={\kappa }_{c_0}exp\left[-{\Delta }_c{\left(1-P/P_c\right)}^2\right].$$

(4)

The cloud opacity depends on the normalization factor ${\kappa }_{{\mathrm{c}}_0}$, the location of the cloud deck $P_c$, and the cloud deck thickness parameter ${\Delta }_c$, where small ${\Delta }_c$ values yield vertically extended cloud decks while large ${\Delta }_c$ values lead to thin cloud decks. By construction, the cloud opacity adopts a Gaussian shape. The cloud optical depth ${\tau }_c$ adds to the longwave optical depth ${\tau }_L$. This is illustrated in Figure 3 and Figure 4 for WASP-10b and WASP-39b, respectively, for cloud parameters considered in this work. The cloud normalization opacity ${\kappa }_{c_0}$ was adjusted to reach optical depth values ${\tau }_{\mathrm{L}}$ as in [7,41].

In this cloud model, cloud decks lead to warming of the atmosphere above the cloud, although high above the cloud deck the effect may reverse and lead to cooling (not shown in Figure 3 and Figure 4). Nevertheless, the enhancement of opacity and optical depth in a limited region of the atmosphere leads to a strong heating of the deep atmosphere (left panel) and the typical isothermal region of temperature $T_{\mathrm{iso}}$, which is most clearly seen for $T_{\mathrm{int}}=0$, is shifted toward higher $T_{\mathrm{iso}}$ values. For $T_{\mathrm{int}}=0$, the isothermal region extends all the way down to the center of the planet. Of interest to this study is the question of how much the warming effect of the clouds affects the deep interior of planets of finite intrinsic heat fluxes ($T_{\mathrm{int}}>0$), and how much this warming effect affects our inferred heavy element abundances.

To address this question, we investigate six different possible cloud layers for WASP-10b and three for WASP-39b. They are selected based on the condensation curves of typical cloud species [16,42,43] shown in Figure 1. Possible cloud forming species and their approximate intersection pressures are listed in

Table 2. Cloud deck parameters considered in this work.

	Cloud Species	${\mathit{P}}_{\mathit{c}}$ [bar]	${\mathbf{\Delta }}_{\mathit{c}}$ [-]	${\mathit{\kappa }}_{{\mathit{c}}_0}$ [m${}^2$/kg]
WASP-10b	KCl/ZnS	0.01	1	0.01
	KCl/ZnS	0.01	1	0.1
	Na${}_2$S	0.3	1	0.01
	Na${}_2$S	0.3	1	0.1
	MnS	10	10	0.01
	MnS	10	10	0.1
WASP-39b	Na${}_2$S	0.001	1	0.2
	MnS	0.3	1	0.1
	MgSiO${}_3$/Cr	30	10	0.01

. We take those pressures as the cloud deck mean location $P_c$ in Equation (4). We consider optically thick (${\kappa }_{c_0}\sim 10{\kappa }_{\mathrm{L},0}$, ${\tau }_c>1$, see Figure 3 and Figure 4) and optically very thick (${\kappa }_{c_0}\sim 100{\kappa }_{\mathrm{L},0}$, ${\tau }_c>10$) cloud decks. However, for simplicity, we label them optically thin and optically thick, respectively. The vertical extension is set to ${\Delta }_c=1$ where possible in order to allow for a non-zero (${\Delta }_c$ sufficiently large) but not tremendously too strong (${\Delta }_c$ sufficiently small) effect. In the real planet, several cloud decks may be present simultaneously and they may be patchy, while, in this model, only one permanent cloud deck is considered and assumed to be uniform.

2.5. Atmosphere-Interior Connection

The transition to the adiabatic interior is made where the local numeric temperature gradient ${\nabla }_{\mathrm{T},\mathrm{local}}$ is larger than the adiabatic gradient ${\nabla }_{\mathrm{ad}}$ taken from the EOS table. Further, we see a convective region forming in most cloudy models above the cloud deck. As the starting point for the adiabatic interior we take the lower intersection of ${\nabla }_{\mathrm{T},\mathrm{local}}$ with ${\nabla }_{\mathrm{ad}}$. Generally, the boundary moves to lower pressures with increasing $T_{\mathrm{int}}$ and $T_{\mathrm{eq}}$ [9].

2.6. Planetary Evolution

To determine the present $T_{\mathrm{int}}$ value of a planet, we perform thermal evolution calculations. The planets are assumed to be of the same age as the parent star within an uncertainty of a few Myr. Further, we assume an orbital location constant in time. Of course, the planets once migrated to their present location, but this is thought to have happened on a comparably short timescale during the first 10 Myr [44]. Integrating the energy balance equation over time, we obtain the evolution of luminosity L and radius $R_P\left(\mathrm{t}\right)$

$$\begin{array}{c}\hfill L_{\mathrm{eff}}-L_{\mathrm{eq}}=L_{\mathrm{int}}=L_{\sec }+L_{\mathrm{radio}}+L_{\mathrm{extra}}\end{array}$$

(5)

with $L_{\mathrm{eq}}=4\pi R_P^2{\sigma }_B{T}_{\mathrm{eq}}^4$ being the absorbed and re-emitted flux. The heat loss from the interior $L_{\mathrm{int}}=4\pi R_P^2{\sigma }_B{T}_{\mathrm{int}}^4$ contains three further components. $L_{\sec }=-4\pi R_P^2{\int }_0^M\mathrm{d}mT\left(m\right)\frac{\mathrm{d}s}{\mathrm{d}t}$ accounts for cooling and contraction of the planet, $L_{\mathrm{radio}}$ stands for radiogenic heating, but is of minor importance for H/He-dominated gas giants, and $L_{\mathrm{extra}}=ϵ4\pi R_P^2{\sigma }_B{T}_{\mathrm{eq}}^4$ denotes an extra energy that may be needed to inflate the planet. In Ref. [45], the statistically most likely values of $ϵ$ as a function of irradiation flux are determined for a sample of planets that exclude planets with $M_P<0.5M_{\mathrm{Jup}}$. Here, we need the extra heating term in order to reach the large age of the 0.28 $M_{\mathrm{Jup}}$ planet WASP-39b. Depending on the distribution of heavy elements in the envelope vs. core, we find $ϵ$ = 2.75–4.00% compared to the majority of hot Jupiters where $ϵ$ = 1–3% [45]. In Figure 5, we show the radius evolution of WASP-39b with and without extra heating. For the young WASP-10b, we do not need extra heating to explain its measured radius.

3. Results for WASP-10b

In Figure 6, we show atmospheric P–T profiles for WASP-10b for finite $T_{\mathrm{int}}$ values for a cloud deck at $0.3$bar and two different cloud opacities ${\kappa }_c=0.01$ and 0.1 m${}^2$/kg.

As shown in Figure 3 for $T_{\mathrm{int}}=0$K, clouds can shift the temperature in the isothermal region significantly toward higher values. This is also the case for finite $T_{\mathrm{int}}$ values. Figure 6 also shows that clouds shift the onset of the adiabatic interior to deeper regions. Both effects become more pronounced with increasing cloud opacity. However, we find that the interior adiabat follows the adiabat of the cloud-free case of same $T_{\mathrm{int}}$ value.

The optically thin cloud (blue) in Figure 6 shifts $T_{\mathrm{iso}}$ by about 400 K from ∼1400 K to ∼1800 K. Under these conditions, the initially assumed Na${}_2$S molecules would no longer condense while silicate clouds (Mg${}_2$SiO${}_3$ and Mg${}_2$SiO${}_4$) might form in present WASP-10b. In this sense, we consider the optically thin cloud at 0.3 bar a more likely option for WASP-10b. On the other hand, the assumption of the optically thick cloud at 0.3 bar (purple) clearly shifts $T_{\mathrm{iso}}$ far beyond any temperature regime where heavy elements might condense out. Similar reasoning applies to the four other cloud cases considered for WASP-10b. Our optically thick clouds cause too strong heating, evaporating any clouds, while in the atmosphere heated by the optically thin clouds condensible species could still condense out. This is the picture that emerges if using condensation curves for solar-metallicity atmospheres. Despite the apparent inconsistencies with the optically thick clouds, we keep them in the loop. This allows us to place an upper limit on the quantitative influence of assumed long-wave absorbers on the inferred metallicity.

We proceed with the case of the cloud at $0.3$ bar and show the radius evolution in Figure 7.

Cooling times in agreement with the known age of the system can easily be obtained for all considered cloud models. Lower assumed $T_{\mathrm{int}}$ values for the present planet lead to longer cooling times. We find that optically thick clouds with their strong heating effect slow down the heat loss from the interior, leading to higher $T_{\mathrm{int}}$ values. They also slow down the contraction of the planet. To obtain a radius for the present planet in agreement with the known age and radius, the planet with optically very thick clouds must harbor a larger amount of heavy elements. That leads to the link between $M_{\mathrm{core}}$ in representation of planetary bulk heavy element mass $Z_P$ and $T_{\mathrm{int}}\left(t_0\right)$ shown in Figure 8. Thick lines in Figure 8 show the models matching $R_P$, $M_P$ and the error range of the age of the system.

The higher is $T_{\mathrm{iso}}$, rising with the optical thickness and $P_c$, the larger is the core mass. The more likely option of the optically thin cloud deck at $0.3$ bar (solid dark blue) leads to a 10% higher core mass compared to the cloud-free model (orange). For optically thin clouds high in the atmosphere, the heating effect on the atmosphere is lower and the influence on inferred metallicity is negligible (the blue-dashed curve in Figure 8 coincides with the orange curve). For the optically thin clouds deep in the atmosphere the heating effect is strong and therefore we had to make the cloud more tenuous by increasing ${\Delta }_c$ instead. The maximum enhancement in inferred heavy element abundance is about 10% and well represented by the medium-height cloud at $0.3$ bar. For optically thick clouds, which are not likely options, we obtain a maximum increase in inferred heavy element content of up to 100%.

Thorngren and Fortney (2019) find for WASP-10b $Z_{\mathrm{P}}=0.12\pm 0.02$, using $M_P=3.15M_{\mathrm{Jup}}$ and $R_P=1.08R_{\mathrm{Jup}}$, in agreement with our results for the cloud-free model, where we obtain $Z_{\mathrm{P}}=0.13$ at $T_{\mathrm{int}}=400$ K.

4. Results for WASP-39b

4.1. Cloud Height

In Figure 9, we show the atmospheric P–T profiles for WASP-39b with and without cloud decks. We find that the high cloud deck at 0.001 bar would heat the upper atmosphere so much that only silicates could condense out at such low pressures.

The analysis in Ref. [2] indicates the presence of clouds on only one side of the limb while a clear sky on the other. In their 3D global circulation models, a high-metallicity atmosphere was clearly required to explain the spectra while optically thick, uniform clouds would not much influence the fit. Thus, the observations do not well constrain the presence of clouds, in particular in the deep atmosphere below ∼0.1 bar or deeper. We proceed with the cloud deck at 30 bar. According to Figure 1 and Figure 9, this cloud deck could be a more likely solution for the 10–30 bar region while the heating of the deeper troposphere for the deep-seated cloud at 30 bar is very strong. At 3000–4000 K, condensible species will not condense out. On the other hand, a uniform silicate cloud layer at 30 bar may impose a compositional gradient, which itself may inhibit convection unless the super-adiabaticity becomes sufficiently strong. As a result, the temperature gradient needed to transport the internal heat outward must be larger than in the adiabatic case without cloud. For the solar system giant planets, this effect may amount up to several 100 Kelvins [46]. Therefore, we consider the deep, optically thin cloud at 30 bar a possible option for WASP-39b. We caution that a number of further effects may lead to a more complex picture than drawn here. Condensation of heavier species decreases the mean molecular weight of the surrounding medium and condensates may decouple from the gas phase, affecting the density difference between vertically moving parcels and the background state and thus the possible stability. Moreover, since the Rosseland mean opacity depends on metallicity [40], redistribution of condensible species by condensation also influences the radiative gradient of the background state. Leconte et al. (2017) [46] also found that possible stability requires a sufficiently high mixing ratio of condensible species. Whether sufficient conditions for stability are satisfied in the atmospheres of the hot Jupiters remains to be investigated.

4.2. Metallicity

WASP-39b is an interesting planet because of its observationally determined atmospheric water abundance. Recently, Wakeford et al. (2018) completed the existing transmission spectrum data in the optical obtained with HST STIS [47] and VLT FORS2 [48] and in the infrared obtained with Spitzer IRAC [47] by adding spectral data in the near infrared using the HST WFC3 camera. The clearly detected water absorption features allowed them to retrieve the atmospheric metallicity, temperature, and cloudiness of the observationally accessible part of the atmosphere amongst other parameters. Combined likelihood analysis of their isothermal equilibrium model yielded a high-metallicity atmosphere of ∼${151}_{-46}^{+48}\times$ solar abundances, though their free-chemistry model yielded a lower metallicity of ∼${117}_{-30}^{+14}\times$ solar abundances.

The high metallicity of 100–200× solar corresponds to a heavy element mass fraction $Z_{\mathrm{env}}\sim 0.25$–0.75 (see

Table 3. Constraints on atmospheric metallicity of WASP-39b.

	WASP39-b
Wakeford et al. (iso. eq.)	[M/H]=${151}_{-46}^{+48}\times$ solar	$Z_{\mathrm{env}}=0.{514}_{-0.24}^{+0.25}$
Wakeford et al. (free-chem.)	[M/H]=${117}_{-30}^{+14}\times$ solar	$Z_{\mathrm{env}}=0.{45}_{-0.17}^{+0.09}$
Thorngren and Fortney	Z:H${}_{\mathrm{P}}=40.51\pm 8.3\times$ solar	$Z_P=0.22\pm 0.03$ (=$Z_{\mathrm{env}}$, fully mixed)
This work, cloud-free		$Z_{\mathrm{env},\max }=0.2$
This work, $0.3$bar cloud deck		$Z_{\mathrm{env},\max }=0.47$
This work, $30$bar cloud deck		$Z_{\mathrm{env},\max }=0.31$

Notes. For conversion of [M/H] to Z, we use Equation (3) in [3] with water as heavy element. The ratio Z:H${}_P$ is the atmosphere abundance for a fully mixed planet, as derived derived from interior models in [3], Equation (3).

). Cloud-free structure model of WASP-39b yield a maximum $Z_p$ value of 0.25 [3], where $Z_p\ge Z_{\mathrm{env}}$ due to the possible presence of a core.

First, we require our cloud-free models to have $Z_P=0.22\pm 0.03$ as found in Ref. [3] for cloud-free models. Because this planet seems to be inflated (see Section 2.6), we account for extra heating $ϵ>0$. For $Z_{\mathrm{env}}=0.05$, we find $ϵ\approx 2.75\%$, whereas for $Z_{\mathrm{env}}=0.2$ we find $ϵ\approx 3.90\%$. These $ϵ$ values are then used also for the models with clouds. From our experience with the models for WASP-10b, where optically thin clouds have a minor effect on $T_{\mathrm{int}}$, we also use the same range of $T_{\mathrm{int}}$ values as found for the cloud-free case, so that no additional evolution calculations are necessary. Figure 10 shows the results on $Z_{\mathrm{env}}$ and $Z_p$. Even for the extreme case of the optically thick cloud, high atmospheric metallicities of $Z_{\mathrm{env}}\sim 0.5$ as observationally derived can barely be reached. Interestingly, however, for the optically thin deep cloud the enhancement in inferred metallicity amounts to about 50%, which allows us to obtain solutions just within the observational uncertainty of Wakeford et al. [2]. A summary of the metallicities for WASP-39b is given in Table 3.

With our favored cloud model for WASP-39b, the optically thin deep cloud, we obtain a maximum envelope metallicity of 0.3, which is 50% higher than our value in the cloud-free case. However, the maximum $Z_{\mathrm{env}}$ value still falls short of the observed value. Our results therefore confirm the conclusion of Thorngren and Fortney (2019) of additional sources of uncertainty relevant to WASP-39b. One source of uncertainty is the EOS. While the H/He-EOS was found to induce an uncertainty of a few percent only for massive hot Jupiters and brown dwarfs, wherein matter is largely degenerate [49], this effect might be stronger for warm, lower-mass planets where temperature effect on the P–$\rho$ relation can be stronger. The composition of heavy elements matters as well. Icy cores typically have a 50% higher mass than rocky cores if otherwise the same modeling procedure is applied.

Since WASP-39b is likely to be heavy element-rich, it could also be that the heavy elements are not homogeneously distributed but that their abundance increases with depth. Even slight compositional gradients can suppress convection and delay cooling. This may be the case in exoplanets [50] and in Saturn itself [51].

5. Comparison to Self-Consistent Cloud Models

Clouds will not only be important at infrared wavelengths, but they can also contribute to absorption and scattering of irradiation at short wavelengths. This is neglected in the cloud model we use. Out of the codes capable of calculating the structure of self-luminous and/or irradiated planets (e.g., [52,53,54]), we here compare the ad-hoc approach of Heng et al. (2012) [11] to the self-consistent atmosphere models with clouds of Mollière et al. (2017) [17], who used the petitCODE [37]. Within this code, models with clouds and different metallicities have been calculated specifically for WASP-10b and WASP-39b. That code calculates radiative-convective equilibrium atmospheric structures and spectra of extrasolar planets self-consistently, assuming chemical equilibrium. The radiative transfer model implements absorption, emission and scattering. It implements the Ackerman and Marley (2001) [55] cloud model for clouds composed of MgAl${}_2$O${}_4$, Mg${}_2$SiO${}_4$, Fe, KCl and Na${}_2$S. Particle opacities are calculated using Mie theory (assumption of spherical, homogeneous grains) or the distribution of hollow spheres (approximating irregularly shaped dust aggregates). For both planets, WASP-10b and WASP-39b, we plot the clear and cloudy solutions of Mollière et al. (2017) in comparison to our clear and cloudy atmosphere models for $T_{\mathrm{int}}=400$ K in Figure 11. Mollière et al. (2017) used cloud models which differ in the assumptions of the grain shape, the standard settling parameter $f_{sed}$ from the Ackerman and Marley model, the maximum cloud mass fraction, the width of the cloud particle size distribution as well the inclusion of iron clouds (see Table 2 in [17]). The different model assumptions result in different atmospheric structures. For temperate giant planets, such as WASP-10b and WASP-39b, they investigated cold cloud models as well, where only Na${}_2$S and KCl are considered as possible cloud species (Figure 11, red) as for this temperature regime higher temperature condensates may not mix up from their deep cloud deck locations.

For both planets, the cloudy atmospheric structures from Mollière et al. (2017) lead to both cooler and hotter isotherms. Their favored cold cloud models, only using Na${}_2$S and KCl as cloud species, lead to cooler isotherm for all different cloudy model parameters compared to the clear atmosphere in orange. In contrast, in this work, the fit parameter of the double-gray atmosphere and the added cloud opacity lead to a warmer atmosphere beneath the cloud deck for all of our assumed cloud decks in the atmosphere. For WASP-10b, there is only one model (dashed light blue) that yields a hotter isotherm, whereas for WASP-39b there are three cloudy models that yield a hotter isotherm compared to the cloud-free case. This comparison suggest that our favored 0.3 bar cloud model for WASP-10b may be supported by the hot cloud model of Mollière et al. (2017), down to significant depths of ∼1 kbar, while for WASP-39b our favored 30 bar cloud model is supported to ∼100 bar and thus may overestimate the here obtained influence on the $Z_{\mathrm{env}}$ of WASP-39b.

6. Conclusions

We performed coupled atmosphere, interior, and evolution calculations for the two giant planets WASP-10b and WASP-39b and investigated the effect of additional absorbers, which we call cloud decks, of the inferred metallicity. We assumed cloud optical thicknesses of ${\tau }_c\approx$ 1–10 (named optically thin) and ${\tau }_c\approx$ 1–100 (named optically thick), as well as different cloud heights in the atmosphere corresponding to certain condensible species. The clouds decks are purely absorbing and based on the model of Heng et al. (2012) [11]. Our major findings are as follows:

(I)

Through their additional infrared opacity, these cloud decks tend to warm the atmosphere beneath. This leads to a more or less pronounced enhancement in inferred heavy element abundance.

(II)

For the optically thicker cloud decks, the heating is too strong so that condensible species would no longer condense out. This puts an upper limit on the enhancement in metallicity of 100% on both planets.

(III)

For optically thin clouds, the heating of the atmosphere can be sufficiently small so that condensible species can condense out. In this case, we find an increase of the core mass of up to 10% for WASP-10b.

(IV)

For WASP-39b we find a maximum atmospheric metallicity of $Z_{\mathrm{env}}=0.31$ if we assume a deep cloud at 30 bars in the troposphere that in addition would lead to inhibited convection. Even in this favored case, the possible envelope metallicity is still near the lower limit of the observationally inferred value. Further effects that lead to a heating of the planet are clearly required. Since the heating efficiency is empirically not yet constrained, as it is for hot Jupiters [45], $ϵ$ > 3% is not excluded for such planets. Such high values would help to bring the observationally inferred Z in agreement with interior-model inferred Z.

Due to the complexity of modeling clouds in a realistic manner, we applied a simple cloud model that is a crude representation of real cloud decks. The predictive power of that model stands and falls with the assumed cloud opacity, cloud height, and cloud thickness, which are poorly known parameters. Another important point is how to couple this cloud model to the atmospheric structure self-consistently. Nevertheless, this work suggests that deep cloud decks below the pressure level accessible to transmission spectra observations may influence the temperature structure in the atmosphere and the inferred metallicity to some extent. A more sophisticated approach is desired.