###### 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 $\sim$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.

###### keywords:

extrasolar planets: hot Jupiters; atmospheres; clouds; individuals: WASP-10b,
WASP-39b

xx 1 5 Received: 16 October 2019; Accepted to Atmosphere: 26 October 2019;
Published: 30 October 2019 yes The Effect of Clouds as an Additional Opacity
Source on the Inferred Metallicity of Giant Exoplanets Anna Julia Poser 1,*,
Nadine Nettelmann 2 and Ronald Redmer 1Anna Julia Poser, Nadine Nettelmann and
Ronald Redmer Correspondence<EMAIL_ADDRESS>
##  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 Venturini et al. (2016) 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 $M_{\rm Jup}$ planet.

Recently, Wakeford et al. (2018) Wakeford et al. (2018) used transmission
spectra to determine the metallicity in the atmosphere of the Saturn-mass
planet WASP-39b. They retrieved a high value of $\sim$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 Thorngren and
Fortney (2019). Moreover, for some massive giant planets such as the 3 $M_{\rm
Jup}$ planet WASP-10b Maciejewski et al. (2011), structure models predict a
significant heavy element enrichment of $Z_{p}$ of 10% or more Thorngren et
al. (2016).

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 Morley et al. (2013) and
the temperature structure of the atmosphere itself Lines et al. (2018). Clouds
also influence the atmospheric scale height, which provides a direct link to
the mean molecular weight of the atmosphere Miller-Ricci et al. (2009). 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 Thorngren et al. (2019). 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 Podolak et al. (2019).
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) Heng et al. (2012), 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 Na2S, 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) Linder et al. (2019) studied the
influence on the spectra and thermal evolution of weakly irradiated exoplanets
while Kurosaki et al. (2017) Kurosaki and Ikoma (2017) studied the influence
of water clouds on the cooling of the ice giant Uranus. For strongly
irradiated hot Jupiters, Barman et al. (2001) Barman et al. (2001) 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) Baraffe et al. (2003) find a minor influence of
dusty versus clear irradiated atmospheres for the luminosity evolution of hot
Jupiters.

Figure 1: Condensation curves (grey dashed) of some species as labeled for
solar-metallicity atmospheres (taken from Ref. Kataria et al. (2016)) and
cloud-free $P$–$T$ profiles (solid) for WASP-10b (left) and WASP-39b (right).
Intersection points are possible cloud forming pressure levels. Orange $P$–$T$
profiles are our fits to the profiles for WASP-10b from J. Fortney (pers.
comm. 2012) as well as our fit to the global average profile of Wakeford et
al. (2018) for WASP-39b for different $T_{\rm int}$ values. Additionally, we
show the obtained clear profiles by Mollière et al. (2017) for their deduced
atmospheric enrichment in [Fe/H] (solid blue) and 10$\times$ smaller vs.
larger enrichment (dashed light blue vs. dashed dark blue).

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) Thorngren and Fortney (2019) for WASP-39b as an input
parameter for our models and see if the high predicted atmospheric metallicity
of Wakeford et al. (2018) Wakeford et al. (2018) 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) Mollière et al. (2017) in Section
5. A summary is given in Section 6.

##  2 Methods

###  2.1 Planet and Star Parameters

WASP-10b is a massive ($2.96\,M_{\text{Jup}}$) and non-inflated ($T_{\rm
eq}=950$ K) hot Jupiter. Its young age of $270\pm 80\,$Myr Johnson et al.
(2009); Christian et al. (2009); Maciejewski et al. (2010) makes it an
interesting object to study planet formation and evolution. While early radius
estimates predicted a rather large radius of $\sim$$1.27\,R_{\rm Jup}$
Christian et al. (2009), subsequent careful analysis of the spots on the K5
dwarf host star suggested a $20\%$ smaller planet radius of $1.02\,R_{\rm
Jup}$ Maciejewski et al. (2011), which we use in this study.

WASP-39b is a Saturn-mass planet ($0.28\,M_{\text{Jup}}$) with a large radius
($1.27\,R_{\text{Jup}}$) and therefore low density
$\rho=0.141\,\rho_{\text{Jup}}$ Faedi et al. (2011). It is orbiting a late
G-type star, which is smaller and, with an age of $9^{+3}_{-4}\,$Gyr, possibly
older than the Sun. The observational parameters used here for WASP-10b and
WASP-39b are listed in Table 1.

Table 1: Stellar and planetary parameters.

| WASP-10b |  WASP-39b 5
---|---|---
$M_{P}$ |  $2.96^{+0.22}_{-0.17}\,M_{\text{Jup}}$ 1 | $0.28\pm 0.03\,M_{\text{Jup}}$
$R_{P}$ |  $1.03^{+0.077}_{-0.03}\,R_{\text{Jup}}$ 4 | $1.27\pm 0.04\,R_{\text{Jup}}$
a |  $0.0369^{+0.0012}_{-0.0014}\,$AU 1 |  $0.0486\pm 0.0005\,$AU
e |  $0.013\pm 0.063$ 3 | 0
$T_{\text{eq,A=0}}$ |  $950^{+30}_{-26}\,$K 4 |  $1116^{+33}_{-32}\,$K
P | 3.09 d | 4.05 d
$M_{\star}$ |  $0.75\,M_{\text{Sun}}$ 2 | $0.93\pm 0.03\,M_{\text{Sun}}$
$R_{\star}$ |  $0.67\,R_{\text{Sun}}$ 4 | $0.895\pm 0.023\,R_{\text{Sun}}$
$T_{\star}$ |  $4675\pm 100\>$K 1 |  $5400\pm 150\>$K
age $\tau_{\star}$ |  $270\pm 80\,$Myr3 |  $9^{+3}_{-4}$ Gyr

1 Ref. Christian et al. (2009), 2 Ref. Johnson et al. (2009), 3 Ref.
Maciejewski et al. (2010), 4 Ref. Maciejewski et al. (2011), 5 Ref. Faedi et
al. (2011).
---

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. $Z_{\text{atm}}$ and
$Z_{\rm{env}}$ are the heavy element mass fractions in the atmosphere and
envelope, respectively, which we assume to be equal,
$Z_{\rm{atm}}=Z_{\rm{env}}=Z$. This is an assumption, not ruling out other
relations between atmospheric and envelope abundances Mollière et al. (2017);
Mordasini et al. (2016). The planetary bulk heavy element mass fraction is
$Z_{\rm P}=Z_{\rm{env}}M_{\text{env}}/M_{\rm P}+M_{\text{core}}/M_{\rm P}$,
and $M_{\text{env}}$ and $M_{\text{core}}$ are the masses of envelope and
core. For the solar reference metallicity we use $Z_{\odot}=1.5\%$ Lodders
(2003). 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 $Y=M_{\rm
He}/(M_{\rm He}+M_{\rm H})$. For hydrogen and helium, we use the SCvH EOS
Saumon and Chabrier (1995). 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
Hubbard and Marley (1989). The rocky core obeys the pressure–density relation
given in Hubbard and Marley (1989). 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 $X_{\rm H}:=X$, $X_{\rm He}:=Y$, $X_{\rm{Z}}:=Z$
Nettelmann et al. (2011). 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
Nettelmann et al. (2011); Fortney and Nettelmann (2010).

###  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) Heng et al.
(2012) 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
$\kappa_{\rm vis}$ for visual light used in Guillot (2010), while the outgoing
flux is described by the long-wave opacity $\kappa_{L}$ equivalent to
$\kappa_{\rm th}$ in Guillot (2010) for thermal radiation. Following Heng et
al. (2012), $\kappa_{S}$ 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
$\kappa_{c}(P)$ to the constant long-wave opacity $\kappa_{\text{L,0}}$ of the
otherwise cloudless atmosphere,

$\kappa_{L}(P)=\kappa_{\text{L,0}}+\kappa_{c}(P)~{}\textrm{.}$ (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 $\kappa_{L}=\rm{const.}$. 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 Heng et
al. (2012))

$T^{4}=\frac{T_{\text{int}}^{4}}{4}\left(2+3\int_{0}^{m}\kappa_{L}\mathrm{d}m^{\prime}\right)+\frac{T_{\text{eq}}^{4}}{2}\left[1+\frac{\gamma}{\sqrt{\xi}}E_{2}\left(\frac{\kappa_{S}m^{\prime}}{\sqrt{\xi}}\right)+3\int_{0}^{m}\kappa_{L}E_{3}\left(\frac{\kappa_{S}m^{\prime}}{\sqrt{\xi}}\right)\mathrm{d}m^{\prime}\right]~{}\textrm{,}$
(2)

with $E_{j}(x)=\int^{\infty}_{1}y^{-j}\exp(-xy)\>\mathrm{d}y$ 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 $F_{\rm int}=\sigma_{B}\,T_{\rm int}^{4}$, which is the outgoing flux
from the planet at the bottom of the atmosphere, and on the zero-albedo
irradiation flux
$\sigma_{B}T_{\text{eq,\,0}}^{4}=\sigma_{B}T_{\star}^{4}(R_{\star}/2a)^{2}$
where $\sigma_{B}$ is the Stephan–Boltzmann constant. Thus, $(1-A)^{1/4}T_{\rm
eq,0}$ 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
$\kappa_{L}(P)$, we set $\xi=1$ (no scattering) and take scattering into
account only via the albedo in $T_{\text{eq}}$, which we set to $A_{B}=0.3$
Marley et al. (1999); Gelino et al. (1999), while noting that other work
suggests smaller values (e.g., Sudarsky et al. (2000)). 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 Madhusudhan et al.
(2014). 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 Li et al. (2018). 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. Heng et al. (2014). For the
cloud-free ($\kappa_{L}=\kappa_{\text{L,\,0}}$) atmosphere without scattering
($\xi$=1), Equation (2) reduces to the global average temperature profile of
Guillot (2010) Guillot (2010). The $P$–$\tau$ relation for constant gravity
$g$ and pressure-dependent longwave opacity reads

$P=m\cdot g=(\tau/\kappa_{L})\>g~{}\textrm{.}$ (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 $T_{\rm int}$ Fortney et al. (2007)
(see Figure 1). For WASP-39b, we fit $\kappa_{S}$ and $\kappa_{\rm L,\,0}$ to
the global averaged $P$–$T$ profile from Ref. Kataria et al. (2016) for a
$1\times$ solar composition metallicity. We find $\kappa_{\text{L,0}}=0.0136$
cm2/g, $\kappa_{S}=0.002$ cm2/g ($\gamma=0.147$) for WASP-10b and
$\kappa_{\text{L,0}}=0.006$ cm2/g, $\kappa_{S}=0.00037$ cm2/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
Fortney et al. (2006); Mollière et al. (2015). An increase in atmospheric
metallicity leads to higher temperatures in the isothermal part of the
atmosphere Wakeford et al. (2017). In the case of WASP-39b, where possibly the
atmosphere is enriched by a factor of 100–200$\times$ 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. (2013) to the tabulated values of
Freedman et al. (2008). We conclude that our obtained long-wave opacity values
$\kappa_{\text{L,0}}\sim 0.01$ cgs are appropriate mean Rosseland mean
opacities in the radiative atmospheres of both planets.

Figure 2: Fit of Valencia et al. (2013) to the Rosseland mean opacities
$\kappa_{\rm R}$ of Freedman et al. (2008) along isobars (dashed) and
$\kappa_{R}$ along the present $P$–$T$ profiles for WASP-10b (solid orange)
and WASP-39b (solid blue). The transition from the atmosphere to the interior
is marked by grey circles.

###  2.4 Cloud Model

The model of Heng et al. (2012) 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

$\kappa_{c}(P)=\kappa_{c_{0}}\exp\left[-\Delta_{c}\left(1-P/P_{c}\right)^{2}\right]\quad.$
(4)

The cloud opacity depends on the normalization factor $\kappa_{\rm{c_{0}}}$,
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 Figures 3 and 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 $\tau_{\rm L}$ as
in Lines et al. (2018); Marley (2000). 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 Figures 3 and
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 $T_{\rm iso}$, which
is most clearly seen for $T_{\rm int}=0$, is shifted toward higher $T_{\rm
iso}$ values. For $T_{\rm int}=0$, 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 ($T_{\rm int}>0$), and how much this warming
effect affects our inferred heavy element abundances.

Figure 3:  Influence of two cloud decks (optically thin, blue; optically
thick, violet, cloud-free, orange) on the $P$–$T$ relation of WASP-10b for
$T_{\rm int}=0\>$K (left) and $P$–$\tau$-relation (right) for optical depth
$\tau_{L}$ (dashed) and cloud optical depth $\tau_{c}$ (solid), which
contributes to $\tau_{L}$ (cf. Equation (49) in Heng et al. (2012)). The cloud
decks are located at 0.3 bar and are of vertical extension $\Delta_{c}=1$.
Figure 4:  Similar to Figure 3 but for WASP-39b and three possible cloud
locations at 0.001 (green), 0.3 (purple) and 30 bar (orange) as well as the
cloud-free atmosphere (blue, only left panel). For the cloud deck at 30 bar,
we show the resulting $P$–$T$ profiles for $T_{\rm int}=300,400,500$ K (from
darker to lighter orange) as well as the isotherm (dash-dotted, orange).

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 Lodders and Fegley (2006); Morley
et al. (2015); Kataria et al. (2016) shown in Figure 1. Possible cloud forming
species and their approximate intersection pressures are listed in Table 2. We
take those pressures as the cloud deck mean location $P_{c}$ in Equation (4).
We consider optically thick ($\kappa_{c_{0}}\sim 10\>\kappa_{\text{L,0}}$,
$\tau_{c}>1$, see Figures 3 and 4) and optically very thick
($\kappa_{c_{0}}\sim 100\>\kappa_{\text{L,0}}$, $\tau_{c}>10$) 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.

Table 2: Cloud deck parameters considered in this work. | Cloud Species |  $P_{c}$ [bar] |  $\Delta_{c}$ [-] |  $\kappa_{c_{0}}$ [m2/kg]
---|---|---|---|---
| KCl/ZnS | 0.01 | 1 | 0.01
| KCl/ZnS | 0.01 | 1 | 0.1
| Na2S | 0.3 | 1 | 0.01
WASP-10b | Na2S | 0.3 | 1 | 0.1
| MnS | 10 | 10 | 0.01
| MnS | 10 | 10 | 0.1
| Na2S | 0.001 | 1 | 0.2
WASP-39b | MnS | 0.3 | 1 | 0.1
| MgSiO3/Cr | 30 | 10 | 0.01

###  2.5 Atmosphere-Interior Connection

The transition to the adiabatic interior is made where the local numeric
temperature gradient $\nabla_{\rm T,local}$ is larger than the adiabatic
gradient $\nabla_{\rm ad}$ 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
$\nabla_{\rm T,local}$ with $\nabla_{\rm ad}$. Generally, the boundary moves
to lower pressures with increasing $T_{\rm int}$ and $T_{\rm eq}$ Thorngren et
al. (2019).

###  2.6 Planetary Evolution

To determine the present $T_{\rm int}$ 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 Alexander and Armitage (2009). Integrating
the energy balance equation over time, we obtain the evolution of luminosity
$L$ and radius $R_{P}(\rm t)$

$\displaystyle
L_{\text{eff}}-L_{\text{eq}}=L_{\text{int}}=L_{\text{sec}}+L_{\text{radio}}+L_{\text{extra}}$
(5)

with $L_{\text{eq}}=4\pi R_{P}^{2}\sigma_{B}T_{\text{eq}}^{4}$ being the
absorbed and re-emitted flux. The heat loss from the interior
$L_{\text{int}}=4\pi R_{P}^{2}\sigma_{B}T_{\text{int}}^{4}$ contains three
further components. $L_{\text{sec}}=-4\pi
R_{P}^{2}\int_{0}^{M}\mathrm{d}mT(m)\,\frac{\mathrm{d}s}{\mathrm{d}t}$
accounts for cooling and contraction of the planet, $L_{\text{radio}}$ stands
for radiogenic heating, but is of minor importance for H/He-dominated gas
giants, and $L_{\text{extra}}=\epsilon 4\pi
R_{P}^{2}\sigma_{B}T_{\text{eq}}^{4}$ denotes an extra energy that may be
needed to inflate the planet. In Ref. Thorngren and Fortney (2018), the
statistically most likely values of $\epsilon$ as a function of irradiation
flux are determined for a sample of planets that exclude planets with
$M_{P}<0.5\>M_{\rm Jup}$. Here, we need the extra heating term in order to
reach the large age of the 0.28 $M_{\rm Jup}$ planet WASP-39b. Depending on
the distribution of heavy elements in the envelope vs. core, we find
$\epsilon$ = 2.75–4.00% compared to the majority of hot Jupiters where
$\epsilon$ = 1–3% Thorngren and Fortney (2018). 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.

Figure 5: Radius evolution for WASP-39b with (bluish) and without (purple)
extra heating. The planet stays hot and inflated for several Gyrs as the
additional heating $\epsilon$ prevents further contraction.

##  3 Results for WASP-10b

In Figure 6, we show atmospheric $P$–$T$ profiles for WASP-10b for finite
$T_{\rm int}$ values for a cloud deck at $0.3\>$bar and two different cloud
opacities $\kappa_{c}=0.01$ and 0.1 m2/kg.

As shown in Figure 3 for $T_{\rm int}=0\>$K, clouds can shift the temperature
in the isothermal region significantly toward higher values. This is also the
case for finite $T_{\rm int}$ 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 $T_{\rm
int}$ value.

Figure 6: Atmospheric $P$–$T$ profiles of WASP-10b for finite $T_{\text{int}}$
values of 300, 400, 500, and 600 K (from bottom to top) and three cloud deck
scenarios: optically thin at 0.3 bar (blue), optically thick at 0.3 bar
(purple), and cloud-free (orange). Circles mark the transition between
atmosphere and adiabatic interior.

The optically thin cloud (blue) in Figure 6 shifts $T_{\rm iso}$ by about 400
K from $\sim$1400 K to $\sim$1800 K. Under these conditions, the initially
assumed Na2S molecules would no longer condense while silicate clouds (Mg2SiO3
and Mg2SiO4) 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 $T_{\rm iso}$ 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.

Figure 7: Radius evolution of WASP-10b for the cloud-free atmosphere (orange)
as well as for optical thick (purple) and optical thin (blue) clouds decks at
$P_{c}=0.3$ bar. Solid lines yield a cooling time of 270 Myr while dashed
lines within the $1\sigma$ uncertainty of the age, e.g., light dashed lines
then describe models reaching the lower limit of the age constraint of 190
Myr.

Cooling times in agreement with the known age of the system can easily be
obtained for all considered cloud models. Lower assumed $T_{\text{int}}$
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 $T_{\text{int}}$ 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 $M_{\text{core}}$ in representation of planetary bulk heavy
element mass $Z_{P}$ and $T_{\text{int}}(t_{0})$ 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 $T_{\rm iso}$, 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 $Z_{\rm P}=0.12\pm 0.02$, using
$M_{P}=3.15\,M_{\rm Jup}$ and $R_{P}=1.08\,R_{\rm Jup}$, in agreement with our
results for the cloud-free model, where we obtain $Z_{\rm P}=0.13$ at $T_{\rm
int}=400$ K.

Figure 8: $M_{\text{core}}$-$T_{\text{int}}$ relation for WASP-10b assuming
cloud-free atmosphere (orange) and six different cloud decks in the
atmosphere. Thin lines indicate the results obtained by $M_{P}$, $R_{P}$,
thick bars highlight the solutions that also satisfy the age constraint.

##  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.

Figure 9: Atmospheric $P$–$T$ profiles of WASP-39b for finite $T_{\text{int}}$
values of 400, 500, and 600 K for three decks located at 0.001 (green), 0.3
(purple), and 30 (orange) bar as well as the cloud-free atmosphere (blue).
Circles mark the transition between atmosphere and adiabatic interior.

The analysis in Ref. Wakeford et al. (2018) 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 $\sim$0.1 bar or deeper.
We proceed with the cloud deck at 30 bar. According to Figures 1 and 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 Leconte et al. (2017). 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 Freedman et al.
(2008), redistribution of condensible species by condensation also influences
the radiative gradient of the background state. Leconte et al. (2017) Leconte
et al. (2017) 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 Sing
et al. (2016) and VLT FORS2 Nikolov et al. (2016) and in the infrared obtained
with _Spitzer_ IRAC Sing et al. (2016) 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
$\sim$$151^{+48}_{-46}\times$ solar abundances, though their free-chemistry
model yielded a lower metallicity of $\sim$$117^{+14}_{-30}\times$ solar
abundances.

The high metallicity of 100–200$\times$ solar corresponds to a heavy element
mass fraction $Z_{\rm env}\sim 0.25$–0.75 (see Table 3). Cloud-free structure
model of WASP-39b yield a maximum $Z_{p}$ value of 0.25 Thorngren and Fortney
(2019), where $Z_{p}\geq Z_{\rm env}$ 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. Thorngren and Fortney (2019) for cloud-free models. Because this
planet seems to be inflated (see Section 2.6), we account for extra heating
$\epsilon>0$. For $Z_{\text{env}}=0.05$, we find $\epsilon\approx 2.75\%$,
whereas for $Z_{\text{env}}=0.2$ we find $\epsilon\approx 3.90\%$. These
$\epsilon$ 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 $T_{\rm int}$, we also use the same range of $T_{\rm int}$
values as found for the cloud-free case, so that no additional evolution
calculations are necessary. Figure 10 shows the results on $Z_{\rm env}$ and
$Z_{p}$. Even for the extreme case of the optically thick cloud, high
atmospheric metallicities of $Z_{\rm env}\sim 0.5$ 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. (2018). 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 $Z_{\rm env}$ 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
(Becker et al., 2014), 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.

Figure 10: Range of envelope metallicity $Z_{\rm env}$ due to different
atmosphere models without clouds (blue), with optically thick cloud deck at
$0.3\>$bar (purple), and optically thin at $30\>$bar (favored case, orange).
Circles/diamonds indicate the maximum $Z_{\rm env}$ value for a fully-mixed
planet. The solid and dashed colored lines indicate models with different
$T_{\rm int}$/$\epsilon$-values obtained for the cloud-free models. The arrows
indicate the increase of $Z_{\rm env}$ when clouds are switched on.  Table 3:
Constraints on atmospheric metallicity of WASP-39b.

| WASP39-b |  |
---|---|---|---
Wakeford et al. (iso. eq.) | [M/H]=$151^{+48}_{-46}\,\times$ solar | $Z_{\rm env}=0.514^{+0.25}_{-0.24}$ |
Wakeford et al. (free-chem.) | [M/H]=$117^{+14}_{-30}\,\times$ solar | $Z_{\rm env}=0.45^{+0.09}_{-0.17}$ |
Thorngren and Fortney | Z:H${}_{\text{P}}=40.51\pm 8.3\,\times$ solar |  $Z_{P}=0.22\pm 0.03$ (=$Z_{\rm env}$, fully mixed) |
This work, cloud-free |  | $Z_{\rm env,max}=0.2$ |
This work, $0.3\>$bar cloud deck |  | $Z_{\rm env,max}=0.47$ |
This work, $30\>$bar cloud deck |  | $Z_{\rm env,max}=0.31$ |

Notes. For conversion of [M/H] to Z, we use Equation (3) in Thorngren and
Fortney (2019) with water as heavy element. The ratio Z:HP is the atmosphere
abundance for a fully mixed planet, as derived derived from interior models in
Thorngren and Fortney (2019), Equation (3).
---

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 Chabrier and
Baraffe (2007) and in Saturn itself Leconte and Chabrier (2013).

##  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.,
Malik et al. (2019); Allard et al. (2001); Hauschildt and Baron (1999)), we
here compare the ad-hoc approach of Heng et al. (2012) Heng et al. (2012) to
the self-consistent atmosphere models with clouds of Mollière et al. (2017)
Mollière et al. (2017), who used the petitCODE Mollière et al. (2015). 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) Ackerman and Marley (2001) cloud model for
clouds composed of MgAl2O4, Mg2SiO4, Fe, KCl and Na2S. 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 $T_{\rm int}=400$ 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 Mollière et
al. (2017)). 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 Na2S 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 Na2S 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 $\sim$1 kbar,
while for WASP-39b our favored 30 bar cloud model is supported to $\sim$100
bar and thus may overestimate the here obtained influence on the $Z_{\rm env}$
of WASP-39b.

Figure 11: Atmospheric temperature structures for WASP-10b (left) and WASP-39b
(right). Our result for the clear atmosphere is shown in orange for $T_{\rm
int}=0$ K as well some of our cloudy solutions in dark blue. The bunch of
profiles in red (cold models) and light blue (hot models) are from Mollière et
al. (2017) for different cloud model parameter, see text. The condensation
curves assume solar elemental abundance.

##  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) Heng et al. (2012). Our major findings are as
follows:

  1. (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.

  2. (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.

  3. (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.

  4. (IV)

For WASP-39b we find a maximum atmospheric metallicity of $Z_{\rm env}=0.31$
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 Thorngren and Fortney (2018), $\epsilon$ > 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.

A.J.P. developed large parts of the computer code used to perform the model
computations, obtained the results presented here, and prepared the figures.
A.J.P. and N.N. wrote the paper, and R.R. approved it. N.N. and R.R. designed
the project. All authors discussed and contributed to the methodology.

A.J.P. and R.R. thank the DFG for support within the SPP 1992.

###### Acknowledgements.

We thank the referees for helping to improve this manuscript. We thank R.
Neuhäuser and G. Maciejewski for discussions on the observational parameters
of WASP-10b. N.N. thanks the Swiss PlanetS Program for hospitality in March
2018. We thank P. Mollière for providing us with data from petitCODE for the
atmospheres of WASP-10b and WASP-39b, and thank J. Fortney and L. Scheibe for
discussions. The authors declare no conflict of interest. References

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