11institutetext: (1) Université de Paris Cité, Institut de Physique du Globe
de Paris, CNRS F-75005 Paris, 1 rue Jussieu, 75005 Paris, France
11email<EMAIL_ADDRESS>
(2) Laboratoire AIM, CEA, CNRS, Univ. Paris-Sud, UVSQ, Université Paris-
Saclay, F-91191 Gif-sur-Yvette, France
(3) Université Paris-Saclay, UVSQ, CNRS, CEA, Maison de la Simulation, 91191,
Gif-sur-Yvette, France
(4) Institute of Geochemistry and Petrology, ETH Zürich, CH-8092, Zürich,
Switzerland
(5) CEED, PHAB, University of Oslo, Oslo, Norway

# The effect of a small amount of hydrogen in the atmosphere of ultrahot
magma-ocean planets: atmospheric composition and escape

Sébastien Charnoz 11 Aurélien Falco 1122 Pascal Tremblin 33 Paolo Sossi 44
Razvan Caracas 1155 Pierre-Olivier Lagage 22

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. Ultrahot (¿ 1500K) rocky exoplanets may be covered by a magma ocean,
resulting in the formation of a vapor rich in rocky components (e.g., Mg, Si,
Fe) with a low total pressure and high molecular mass. However, exoplanets may
have also captured a significant amount of hydrogen from the nebular gas
during their formation. Ultrahot rocky exoplanets around the Fulton gap ($\sim
1.8R_{\oplus}$) are sufficiently large to have retained some fraction of their
primordial hydrogen atmosphere.

Aims.  Here we investigate how small amounts of hydrogen (much smaller than
the mass of the planet) above a magma ocean may modify the atmospheric
chemistry and its tendency to thermally escape.

Methods. We use a chemical model of a magma ocean coupled to a gas equilibrium
code (that includes hydrogen) to compute the atmospheric composition at
thermodynamical equilibrium for various H contents and temperatures. An
energy-limited model is used to compute atmospheric escape and is scaled to
consider H-rich and H-poor atmospheres.

Results. The composition of the vapor above a magma ocean is drastically
modified by hydrogen, even for very modest amounts of H ($\ll 10^{-6}$
planetary mass). Hydrogen consumes much of the O2(g), which, in turn, promotes
the evaporation of metals and metal oxides (SiO, Mg, Na, K, Fe) from the magma
ocean. Vast amounts of $\mathrm{H_{2}O}$ are produced by the same process. At
high hydrogen pressures, new hydrogenated species such as $\mathrm{SiH_{4}}$
form in the atmosphere. In all cases, H, H2, and $\mathrm{H_{2}O}$ are the
dominant nonmetal-bearing volatile species. Sodium is the dominant atmospheric
metal-bearing species at T$<$ 2000K and low H content, whereas Fe is dominant
at high H content and low temperature, while SiO predominates at T¿3000 K. We
find that the atmospheric Mg/Fe, Mg/Si, and Na/Si ratios deviate from those in
the underlying planet and from the stellar composition. As such, their
determination may constrain the planet’s mantle composition and H content. As
the presence of hydrogen promotes the evaporation of silicate mantles, it is
conceivable that some high-density, irradiated exoplanets may have started
life as hydrogen-bearing planets and that part of their silicate mantle
evaporated (up to a few $10\%$ of Si, O, and Fe) and was subsequently lost
owing to the reducing role of H.

Conclusions. Even very small amounts of H can alter the atmospheric
composition and promote the evaporation to space of heavy species derived from
the molten silicate mantle of rocky planets. Through transit spectroscopy, the
measurement of certain elemental ratios, along with the detection of
atmospheric water or hydrogen, may help to determine the nature of a surface
magma ocean.

###### Key Words.:

exoplanets, magma oceans, hydrogen

##  1 Introduction

There are more than 350 ultrahot exoplanets (with equilibrium temperature of ¿
1500 K) discovered as of July 2022. About half of them have radii of ¡ 5 Earth
radii and have masses that are consistent with either pure rocky planets or
sub-Neptunes. At such temperatures and assuming a silicate-based mantle, their
surfaces should be partially or entirely covered by a magma ocean . Such
planets are of particular interest for future spectroscopic observations
because convection in the mantle should be sufficiently rapid (Solomatov,
2000) such that their atmospheres are readily replenished by chemical exchange
with the underlying molten surface. As such, any spectroscopic constraints on
the nature of their atmospheres may provide insights into the composition of
their molten mantles. However, the search for atmospheres on ultrahot rocky
worlds has so far brought mostly negative results. Spectroscopic observation
of LHS 3844b (with an equilibrium temperature $\sim$ 1000K) yielded results
that are compatible with the absence of an atmosphere (Kreidberg et al.,
2019). The detection of an atmosphere around 55 CNC e is debated. Although a
hotspot offset has been reported in the planet’s phase curve (Demory et al.,
2016) —which can be interpreted as the consequence of a thick and opaque
atmosphere—, detailed spectroscopic observations of 55 CNC e have not
unambiguously detected any gaseous species (Bourrier et al., 2018). Some hints
of the possible presence of HCN were reported in Tsiaras et al. (2016), but
this would seem unlikely on chemical equilibrium grounds given that hydrogen
itself has never been detected (see e.g., Ehrenreich et al. (2012)).

Recently, Zieba et al. (2022) reported the possible indirect detection of an
atmosphere around the ultrahot planet ($T_{eq}\sim\leavevmode\nobreak\ 2000$
K) K2-141b by comparing Spitzer to synthetic spectra. However, the resolution
of Spitzer spectra is too low to permit any clear identification of the
molecular species in the partial pressures expected above such a rocky planet.
Therefore, the detection of any sort of atmosphere around a lava planet
remains elusive, despite growing international research efforts towards this
goal. For the sake of completeness, we mention that a low-mean-density
atmosphere containing HCN and CH4 has been detected around the rocky world GJ
1132, but it is a ”warm” planet (T$\sim$ 530 K), and may not have a surface
magma ocean (Swain et al., 2021). Interestingly, ultrahot rocky planets tend
to have high average densities. For example, 55 Cnc e has an estimated density
of $\sim$ 5.9 g/cm3 (Crida et al., 2018; Mercier et al., 2022). When comparing
the density of exoplanets with their equilibrium temperatures (Fig. 1 bottom),
strongly irradiated rocky planets with equilibrium temperatures of Teq ¿ 1700
K have, on average, a much higher density than those planets with Teq below
1700 K.

More specifically, the population of terrestrial planets with densities ¡ 5
g/cm3 almost completely disappears at Teq larger than 1700 K (Fig. 1), and all
observed ultrahot rocky planets smaller than 3 Earth radii have densities
ranging from 5 to 11 g/cm3(with only one remarkable exception of a planet with
T${}_{eq}\sim\leavevmode\nobreak\ 2400$ K and $\rho\sim 3$ g/cm3). Otegi et
al. (2020) demonstrate that volatile-rich and rocky exoplanet populations may
be separated on the basis of their density, with a transition around 3 g/cm3.
Ultrahot mini-Neptunes and terrestrial planets may have a thin- to moderately
massive atmosphere (less than 50% of the planet’s mass) above a molten mantle
and an iron-rich core, as sketched out in Figure 2. The broad positive trend
of density versus equilibrium temperature might reflect a decreasing
mantle/core mass ratio with increasing surface temperature. This trend could
either result from efficient iron condensation (relative to silicates) in the
hottest region of the protoplanetary disk during the formation of first solids
(see e.g., Pignatale et al. (2017); Johansen & Dorn (2022)) or arise from
collisional (O’Neill & Palme, 2008; Denman et al., 2020) or evaporative (Young
et al., 2019; Charnoz et al., 2021; Jäggi et al., 2021) mass loss. However,
most terrestrial planets are thought to have moved inward during their
formation due to efficient migration, and so highly irradiated magma ocean
exoplanets could have formed further away at much lower temperatures and with
much higher hydrogen budgets than they have today.

In the present paper, we wish to investigate the possibility that the mantles
of rocky planets in a magma ocean state evaporate at high temperatures in the
presence of H2 gas. Depending on the efficiency of this process, it might lead
to a decrease in the mantle mass with increasing temperature, and thus an
increase in planet density, given that the iron-rich core remains isolated
from the evaporation occurring at the surface. Although the general treatment
is not new, and evaporation from a magma ocean has been studied under varying
circumstances (see e.g., Schaefer & Fegley, 2010; Lupu et al., 2014; Schaefer
et al., 2012; Ito et al., 2015; Ito & Ikoma, 2021; Charnoz et al., 2021; Jäggi
et al., 2021). It was found that the vapor above a magma ocean with a
composition like that of Earth’s mantle (also called bulk silicate Earth, or
BSE) has, on an anhydrous basis (i.e., neglecting H, C, Cl, S, N, and other
volatiles) very low pressure ($<<1$ bar in general) and is dominated by Na and
K for T ¡ 2500 K, and by SiO for T ¿ 3000 K (Schaefer & Fegley, 2010; Schaefer
et al., 2012; Ito et al., 2015; Ito & Ikoma, 2021). The escape of a mineral
atmosphere was studied in detail by Ito & Ikoma (2021), and was found to be
very inefficient. A mineral-based atmosphere released by a magma ocean is rich
in sodium, potassium, oxygen, iron, magnesium, and silicon (Schaefer & Fegley,
2010). Schaefer & Fegley (2004) show that the dominant metallic gas above a
silicate melt should be SiO for $T>2700$ K and Na for lower temperature. Ito &
Ikoma (2021) show that this mineral vapor dissociates to simple atomic and
ionic compounds in the upper atmosphere. Because Na and K are efficient
atmospheric coolants (Ito & Ikoma, 2021), most of the energy transferred to
the planet through irradiation by the star is re-emitted to space, making
thermal escape inefficient with resulting heating efficiency in the range of
$\sim 5x10^{-4}$ to $\sim 5x10^{-3}$, compared to 0.14 to 0.4 for hydrogen-
dominated atmosphere (Valencia et al., 2010). Ito & Ikoma (2021) therefore
concluded that atmospheric escape would have only a marginal influence on the
composition of ultrahot rocky planets covered by a magma ocean. Nonetheless,
Ito & Ikoma (2021) note that, at 3000 K, most of the Na and K content of the
planet’s mantle should escape during the lifetime of the star. However, as Na
and K may only represent a very small proportion of the mantles of such
planets (together they comprise ¡ 1 mol $\%$ in the BSE for example), even if
the entire planetary budget of Na and K were to escape, their loss would be
unlikely to significantly affect either the mass or the radius of the rocky
planet. For the atmospheric escape of the magma to have a significant impact
on the mass or radius of the planet, major components, such as O, Mg, or Si,
must be released in the vapor in very high quantities and then lost to space.

Here, we revisit the idea that atmospheric escape in the presence of hydrogen
can engender a significant loss of the mantle of a rocky planet. The
specificity of our approach lies in our consideration of the presence of a
hydrogen envelope above the magma ocean and in computation of the consequences
this has on the atmosphere’s speciation (which is modeled as a mixture between
the H captured from the protoplanetary disk and species evaporated from the
magma ocean) and atmospheric loss. To date, several studies have investigated
the dissolution of H2 or H2O in a silicate magma ocean, and the effect on the
planet’s mass and radius atmospheric loss (see e.g., Kite et al. (2020); Kite
& Schaefer (2021)), but neither the change in atmospheric speciation nor the
identities and partial pressures of the major rock-forming elements have been
investigated.

To our knowledge, the only other attempt to characterize the effect of H on
the composition of a hot rocky planet atmosphere was by Fegley et al. (2016),
who computed (based on the thermodynamic properties of hydroxide- and halide
gas species) the compositions of metal-bearing atmospheres above a BSE magma
for various partial pressures of H2O, but did not consider the case of a
captured H2 envelope above a magma ocean. Although the effect of H2 has been
examined in its influence on the escape of K from rocky planets (Erkaev et
al., 2022), these authors neglect H solubility in the magma ocean and do not
provide a self-consistent model of atmospheric speciation in equilibrium with
the magma. Lichtenberg et al. (2021); Dorn & Lichtenberg (2021) considered the
effect of dissolved $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$ on the structure of
an exoplanet, but did not compute the effect of H on the equilibrium chemistry
of the atmosphere. On the other hand, Sossi et al. (2020), Bower et al.
(2022), Gaillard et al. (2022), Sossi et al. (2023) determine the equilibrium
speciation of an atmosphere in equilibrium with magma oceans for a range of
H/C, fO2, and H masses, and argue that oxidized conditions promote the
dissolution of H2O in the magma ocean. However, these studies did not consider
the influence of H2O or H2 on the speciation of rock-forming elements in the
gas phase.

Because H2 is the dominant constituent of protoplanetary disks, it may be
captured during the planet formation process, even around growing rocky
planets (e.g., Olson & Sharp, 2019). The ubiquity of H may therefore play a
major role in influencing the chemistry of atmospheres degassing from rocky
planets. We concentrate our efforts on examining the effect of H on the
gaseous species present above silicate melts in the system Na-K-Mg-Al-Fe-Si-
O-H. We find that the evaporation of metal-bearing species from a BSE-like
magma ocean is strongly enhanced by the presence of hydrogen. It has been
shown that a H2-rich atmosphere can be lost rapidly for ultrashort period
planets, in less than 10 Myr (see e.g., Lopez (2017); Erkaev et al. (2022)).
Therefore, the effect of a captured H2 envelope on atmospheric chemistry and
its subsequent loss merits further investigation. In particular, we find that
even minor amounts of H (with respect to the planet’s mass) may have a strong
influence on the atmosphere composition.

In the first part of the present paper, we aim to quantify how the presence of
H modifies the vapor composition compared to a H-free case using a
thermodynamic model of vapor–magma ocean equilibrium. To illustrate the
physical processes at play, in section 2 we present a simplified model based
on two reactions that show how H modifies liquid–vapor equilibria in promoting
the vaporization of metal-bearing gases due to the decrease in oxygen fugacity
(= O2 partial pressure) it engenders, as well as the stabilization of
H-bearing metal molecules in the gas phase. This leads to extensive depletion
of Si and other metals from the silicate liquid. In section 3, we consider a
more realistic case of an infinite magma ocean with BSE composition in contact
with a hydrogen envelope. We compute the vapor composition as a function of
temperature and the amount of captured hydrogen. At the end of this section,
we also show that measurements of Na/Si, Mg/Si, and Mg/Fe and the detection of
H2O in the atmosphere of ultrahot rocky planets may be used as indicators of
the presence and nature of a magma ocean.

The second part of the paper explores how the presence of H promotes
atmospheric escape of heavy metallic species by increasing their concentration
in the atmosphere (SiO, O, O2, Fe, Mg, etc.). Should this process prevail over
long timescales, this may lead to considerable evaporation of mantle
components (relative to the Fe-Ni-rich core), and thus give rise to high-
density rocky planets. We use a simple energy-limited formalism modified to
take into account the inefficient loss of mineral species as reported in Ito
et al. (2015) and in Ito & Ikoma (2021) to quantify the extent of atmospheric
loss.

Figure 1:  Density of known exoplanets versus their equilibrium temperatures.
These data come from the online DACE database maintained by Observatory of
Geneva1 and are taken from the catalog of Otegi et al. (2020), which is an
exoplanet catalog based on reliable, robust, and —to the greatest possible
extent— accurate mass and radius measurements of transiting planets up to 120
earth masses (Otegi et al., 2020). (Top) Density vs. equilibrium temperature
for all planets. (Bottom) Same as above but limited to planets with radii of
smaller than 3 Earth radii. The dotted line shows a simple least-square fit to
the data.

1 https://dace.unige.ch/

Figure 2:  Schematic illustration of the internal structure of a rocky planet
as assumed in this paper, with an atmosphere containing a mixture of captured
hydrogen and gaseous species derived from a molten mantle that forms a global
magma ocean below the planet atmosphere. In the centre of the planet, an
iron/nickel core is present. The composition of the magma ocean is assumed to
be fixed, while the composition of the atmosphere is at thermodynamic
equilibrium with that of the magma ocean.

##  2 Toy model: A $\mathrm{SiO_{2}}$ magma ocean degassing in a
$\mathrm{H_{2}}$ atmosphere

We start with a simple chemical model to illustrate the effect of the presence
of a primordial hydrogen envelope at the surface of a degassing magma ocean.
For simplicity, we assume that the (infinite) magma ocean is made of
$\mathrm{SiO_{2}}$ only at temperature T and is surrounded by a
$\mathrm{H_{2}}$ atmosphere. To understand the basic ingredients, we reduce
the physics to two equations (1) the vaporization of $\mathrm{SiO}_{2(\ell)}$
(where the index $(\ell)$ stands for liquid and $(g)$ for gas) in
$\mathrm{SiO}_{(g)}$ and $\mathrm{O_{2}}$, and (2) the conversion of
$\mathrm{H_{2}}$ and $\mathrm{O_{2}}$ into $\mathrm{H_{2}O}$. These reactions
are

$\mathrm{SiO}_{2(\ell)}\Leftrightarrow\mathrm{SiO}_{(g)}+1/2\mathrm{O_{2}},$
(1)

$\mathrm{H_{2}O}\Leftrightarrow\mathrm{H_{2}}+1/2\mathrm{O_{2}}.$ (2)

At equilibrium between the liquid and the gas, the partial pressure of
$\mathrm{SiO}_{(g)}$ is given by the law of mass action at temperature $T$:

$P_{\mathrm{SiO}_{(g)}}P_{\mathrm{O_{2}}}^{1/2}=a[\mathrm{SiO}_{2(\ell)}]K_{1}(T),$
(3)

where $P_{\mathrm{SiO}_{(g)}}$ is the partial pressure of
$\mathrm{SiO}_{(g)}$, $a[\mathrm{SiO}_{2(\ell)}]$ is the activity of
$\mathrm{SiO_{2}}$ in the melt (here we consider an ideal and pure liquid so
that $a[\mathrm{SiO}_{2(\ell)}]=1$ here) and $P_{\mathrm{O_{2}}}$ is the
partial pressure of $\mathrm{O_{2}}$, that is the $f\mathrm{O_{2}}$ (we
consider an ideal gas, such that the fugacity, $f$ , is equal to the partial
pressure, $p$). Here, $K_{1}(T)$ is the equilibrium constant of this reaction
at temperature T. The equilibrium constant is then related to the Gibbs-free-
energy change of the evaporation reaction, $K_{1}(T)=e^{-\Delta G1/RT}$ where
$\Delta G1=\Delta G^{0}(\mathrm{SiO}_{(g)})+1/2\Delta
G^{0}(\mathrm{O_{2}})-\Delta G^{0}(\mathrm{SiO}_{2(\ell)})$ and $\Delta G^{0}$
are the Gibbs free energies of formation of the different species from the
elements. Similarly, for the $\mathrm{H_{2}O}$ formation equation (Eq. 2):

$P_{\mathrm{O_{2}}}^{1/2}P_{\mathrm{H_{2}}}=K_{2}(T)P_{\mathrm{H_{2}O}}.$ (4)

To close the system, we introduce the following limiting conditions. The first
is that all oxygen atoms that go in the atmosphere come from the dissociation
of $\mathrm{SiO_{2}}$, such that the atomic O:Si ratio in the atmosphere is
always equal to 2, which leads to the relation:

$P_{\mathrm{SiO}_{(g)}}+P_{\mathrm{H_{2}O}}+2P_{\mathrm{O_{2}}}=2P_{\mathrm{SiO}_{(g)}}.$
(5)

The above equation is established by considering that the total number of
oxygen moles in the system is
$N_{O}=N_{\mathrm{SiO}}+2N_{\mathrm{O_{2}}}+N_{\mathrm{H_{2}O}}$, and that
$N_{\mathrm{O}}=2N_{\mathrm{SiO}}$, if we assume that all oxygen comes from
the dissociation of $\mathrm{SiO}_{2(\ell)}$ . We therefore get the equality :
$2N_{\mathrm{SiO}}=N_{\mathrm{SiO}}+2N_{\mathrm{O_{2}}}+N_{\mathrm{H_{2}O}}$.
As partial pressures are proportional to the number of moles, one obtains
$2P_{\mathrm{SiO}}=P_{\mathrm{SiO}}+2P_{\mathrm{O_{2}}}+P_{\mathrm{H_{2}O}}$.
In addition, we assume that the total H content of the atmosphere is fixed and
is equal to $P^{0}_{H}$ , which is defined as:

$P^{0}_{H}=2P_{\mathrm{H_{2}O}}+2P_{\mathrm{H_{2}}},$ (6)

and is used as a proxy for the total quantity of H in the atmosphere in the
remainder of the paper. Equations 3 to 6 form a close set of equations with
unknowns
($P_{\mathrm{O_{2}}},P_{\mathrm{H_{2}O}},P_{\mathrm{SiO}_{(g)}},P_{\mathrm{H_{2}}}$)
and free parameters ($T,P_{\mathrm{H}}^{0}$). We solve this simple system
numerically.

Figure 3: Pressures of the different elements for the simple two-reaction
model with only $\mathrm{SiO}_{(g)}$, $\mathrm{H_{2}}$, $\mathrm{H_{2}O},$ and
$\mathrm{O_{2}}$. Here, the temperature is fixed to 3000 K. SiO solution 1 and
SiO solution 2 refer to asymptotic solutions with low and high hydrogen
content, respectively.  Figure 4: Pressures of the different elements for the
simple two-reaction model with only $\mathrm{SiO}_{(g)}$, $\mathrm{H_{2}}$,
$\mathrm{H_{2}O},$ and $\mathrm{O_{2}}$. Here, the hydrogen content is fixed
($P_{\mathrm{H}}^{0}=1$bar). SiO solution 1 and SiO solution 2 refer to
asymptotic solutions with low and high hydrogen content, respectively.

The partial pressures of the various gas species as a function of the initial
H content and temperature are displayed in Figure 3. For low values of
$P_{\mathrm{H}}^{0}$ (¡ 1 bar), the atmosphere is dominated by
$\mathrm{O_{2}}$ and SiO(g) and corresponds to a pure rocky atmosphere in a
hydrogen-free environment. As $P_{\mathrm{H}}^{0}$ increases,
$\mathrm{H_{2}O}$ begins to form, and $P_{\mathrm{O_{2}}}$ drops sharply in
response. As the system becomes increasingly oxygen-poor, the quantity of
silicon in the atmosphere increases proportionally to $P_{O_{2}}^{-(1/2)}$, as
given by Eq. 3. In Figure 4, the hydrogen content of the atmosphere
$P_{\mathrm{H}}^{0}$ is set, while the temperature is varied. At low
temperatures, H2 predominates. However, as temperature increases, the
equilibrium (2) shifts to the left, and H2O becomes the prevailing H-bearing
gas species above $\sim$ 2700 K. Above $\sim$ 3000 K, the H2/H2O ratio remains
constant and SiO(g) and O2(g) increase in tandem according to the
$P_{O_{2}}^{-(1/2)}$ dependence.

We now consider two asymptotic cases. The partial pressure of
$\mathrm{SiO}_{(g)}$ in the H-poor and H-dominated regimes can be easily
computed as follows. If the pressure of hydrogen is extremely low, then
$\mathrm{H_{2}O}$ and $\mathrm{H_{2}}$ are mostly absent and the system
reduces to the single reaction written in Eq. 1.

Therefore, the relation between the pressures of $\mathrm{SiO}_{(g)}$ and
$\mathrm{O_{2}}$ is given by Eq. 3. We set $a[\mathrm{SiO}_{2(\ell)}]=1$ (pure
and ideal liquid). The dissociation of $\mathrm{SiO}_{2(\ell)}$ implies that
there are always two atoms of O for 1 atom of Si in the gas. Counting the
number of O and Si atoms in each species leads to the relation:

$2P_{\mathrm{O_{2}}_{(g)}}+P_{\mathrm{SiO}_{(g)}}=2P_{\mathrm{SiO}_{(g)}}.$
(7)

Combining Equations 3 and 7 and solving for $P_{\mathrm{SiO}_{(g)}}$ leads to
the simple solution:

$P_{\mathrm{SiO}_{(g)}}(T)=2^{1/3}K_{1}(T)^{2/3}.$ (8)

This solution is displayed in Figure 3 with the brown dashed line. In the
opposite case, in which hydrogen becomes a major component, $\mathrm{H_{2}}$
and $\mathrm{H_{2}O}$ control the system evolution. The system is therefore
controlled by the reaction

$\mathrm{SiO}_{2(\ell)}+\mathrm{H_{2}}\Leftrightarrow\mathrm{SiO}_{(g)}+\mathrm{H_{2}O}.$
(9)

The law of mass action gives the relationship between the partial pressures:

$P_{\mathrm{SiO}_{(g)}}P_{\mathrm{H_{2}O}}=K_{3}(T)P_{\mathrm{H_{2}}}a[\mathrm{SiO}_{2(\ell)}],$
(10)

where

$K3(T)=e^{-\frac{G^{0}(\mathrm{SiO}_{(g)})+G^{0}(\mathrm{H_{2}O})-G^{0}(\mathrm{H_{2}})-G^{0}(\mathrm{SiO}_{2(\ell)})}{RT}}=K_{1}(T)/K_{2}(T).$
(11)

As all oxygen in the vapor comes from the evaporation of $SiO2$, the
conservation of the oxygen atoms gives the relation
$P_{\mathrm{H_{2}O}}+P_{SiO}=2P_{SiO}$, which reduces to
$P_{\mathrm{H_{2}O}}=P_{SiO}$. The conservation of the hydrogen atoms gives
$2P_{\mathrm{H_{2}O}}+2P_{\mathrm{H_{2}}}=P^{0}_{H}$. We set
$a[\mathrm{SiO}_{2(\ell)}]=1$ as usual. Solving for $P_{\mathrm{SiO}_{(g)}}$
leads to a second-order polynomial whose only positive solution is

$P_{\mathrm{SiO}_{(g)}}(P_{\mathrm{H}}^{0},T)=\frac{-K_{3}+\left(K_{3}(T)^{2}+2K_{3}(T)P_{\mathrm{H}}^{0}\right)^{1/2}}{2}.$
(12)

This asymptotic solution is displayed as a purple dashed line in Figure 3, and
closely matches the SiO behavior for large hydrogen content. This also
explains why SiO and $\mathrm{H_{2}O}$ have the same values at high H
pressure. We see that at high hydrogen pressure, $\mathrm{H_{2}O}$ plays both
the role of O and H reservoir, while $\mathrm{O_{2}}$ is almost completely
absent. This clearly demonstrates that, as the pressure of H increases, the
content of Si in the atmosphere sharply increases also.

We conclude from this section that H plays a major in the composition of the
vapor: because of the formation of $\mathrm{H_{2}O}$, the O2 partial pressure
drops and, in turn, we observe an enhanced evaporation of SiO that scales with
$P_{\mathrm{H}}^{0^{1/2}}$. We now turn to the full model where this process
acts on all metals contained in the magma ocean.

##  3 Full model method calculation

Here, we describe a model of an infinite magma ocean (infinite = the liquid
magma composition is fixed) in equilibrium with a H-rich primordial
atmosphere. Our code is not unique; the MAGMA Fegley & Cameron (1987) and
VapoRock (Wolf et al., 2022) codes perform similar calculations. In
particular, the VapoRock code takes into account nonideal interactions via a
Regular solution model. For the vapor molecular composition, our approach uses
a Gibbs free-energy minimization code (we use the iconic CEA/NASA code
provided by Gordon & McBride (1996)). We modify the CEA code to couple it to a
vapor-liquid phase equilibrium model corresponding to liquid with a fixed
composition. The same approach may be used with any atmospheric code (e.g.,
Woitke et al., 2018). We assume that the magma ocean is made of a nonideal
mixture of liquid oxides, namely
$\mathrm{SiO}_{2(\ell)},\mathrm{Mg}\mathrm{O}_{(\ell)},\mathrm{Fe}\mathrm{O}_{(\ell)},\mathrm{Na}_{2}\mathrm{O}_{(\ell)}$,
and $\mathrm{K}_{2}\mathrm{O}_{(\ell)}$, with fixed molar abundances
$X^{l}_{i}$ (where $i$ stands for any of the liquid oxides). Each liquid oxide
may vaporize into gaseous species through the limited list of reactions
compiled in Table 1. The specificity of our approach compared to previous
works is that we assume there is a pre-existing H envelope whose total
pressure of monoatomic H is $P_{\mathrm{H}}^{0}$. For now, we assume that all
H stays in the vapor phase, whereas it is known that $\mathrm{H_{2}}$ and
$\mathrm{H_{2}O}$ may dissolve in the magma ocean (Hirschmann et al., 2012;
Sossi et al., 2023). This simplification is not critical here as
$P_{\mathrm{H}}^{0}$ may be interpreted as the quantity of H contained in the
atmosphere only.

###  3.1 The H monoatomic pressure: $P_{\mathrm{H}}^{0}$

The H envelope reacts with gaseous species derived from the magma ocean to
modify the atmospheric composition. Therefore, knowledge of the total amount
of H in the atmosphere is useful. In order to quantify the amount of H
independently of the atmosphere composition and planet mass, we introduce the
”hydrogen monoatomic pressure”, $P_{\mathrm{H}}^{0}$, which corresponds to the
partial pressure of H if all the H of the atmosphere is in pure monoatomic
form, and is computed as follows:

$P_{\mathrm{H}}^{0}=\sum_{i}\nu_{H}^{i}P_{i},$ (13)

where i is any molecule in the atmosphere, $P_{i}$ is the partial pressure of
molecule $i$, and $\nu_{H}^{i}$ is the stochiometric coefficient of H in
molecule $i$. For example, in an atmosphere in which all H is in
$\mathrm{H_{2}}$, $P_{\mathrm{H}}^{0}=2\times P_{\mathrm{H_{2}}}$ where
$P_{\mathrm{H_{2}}}$ is the partial pressure of $\mathrm{H_{2}}$.

###  3.2 Constitutive equations

For a given value of $P_{\mathrm{H}}^{0}$ and magma temperature, the
parameters we wish to determine are the molar fractions of each gaseous
element in the atmosphere, namely O, Si, Mg, Fe, Na, H, and K, which we refer
to collectively as $X^{g}_{i}$ (where $i$ stands for any of the previous
elements), as well as the total pressure ($P_{tot}$) of the atmosphere. For
any combination of $(X^{g}_{i},P_{tot}),$ the CEA/NASA code can provide the
molecular composition of the atmosphere at chemical equilibrium in the gas. To
couple the atmospheric model to the infinite magma ocean model, we must find
the only atmospheric composition that simultaneously satisfies (1) the
liquid/vapor equilibrium, (2) the mass conservation of O, and (3) the mass
conservation of H. The thermodynamic constraints are the following:

  * •

Liquid/gas equilibrium states that the partial pressure of metal-bearing
species evaporated from the melt (SiO, SiO2, Na, K, Fe, Mg) depends on the
atmospheric f$\mathrm{O_{2}}$. Therefore, the following relations must be
obeyed for the gas for each reaction $j$ reported in Table 1.

$P_{vapor_{j}}=\frac{K_{j}(T)a(liquid_{j})}{P_{\mathrm{O_{2}}}^{s_{j}}},$ (14)

where $vapor_{j}$ stands for the vapor species that bears the metal in
evaporation reaction $j$ (e.g., $Na_{(g)}$ in reaction $\\#3$), $liquid_{j}$
is the corresponding liquid oxide ($NaO_{0.5}$ for reaction $\\#3$), $s_{j}$
is the stochiometric coefficient (one-quarter in reaction $\\#3$), $K_{j}(T)$
is the equilibrium constant of vaporisation reaction $j$, and $a(liquid_{j})$
is the activity of liquid $\\#j$, that is
$a(liquid_{j})=X_{j}\times\Gamma_{j}$, where $X_{j}$ is the mole fraction of
$liquid_{j}$ in the magma ocean (assumed to be fixed) and $\Gamma_{j}$ is the
activity coefficient of $j$ in the liquid due to nonideal mixing effects.
Activity coefficients, $\Gamma_{j}$, of all melt oxide components are
interpolated using outputs of the VapoRock code (Wolf et al., 2022).

When the atmospheric composition verifies all the relations for 1¡j¡6 given by
Eq. 14, the gas is at thermodynamic equilibrium with the liquid.

  * •

We assume that all metals and O in the atmosphere come from the vaporization
of liquid oxides. This imposes a mass conservation relation between oxygen and
all evaporated metals, which is governed by the stoichiometric relations of
all reactions reported in Table 1. We simply get

$X^{g}_{\mathrm{O}}=2X^{g}_{\mathrm{Si}}+X^{g}_{\mathrm{Mg}}+\frac{1}{2}X^{g}_{\mathrm{Na}}+\frac{1}{2}X^{g}_{\mathrm{K}}+X^{g}_{\mathrm{Fe}},$
(15)

where $X^{g}_{i}$ is the number of moles of atom $i$ in the gas.

  * •

The initial total content of H (measured as a pressure) is fixed to a constant
value, $P_{\mathrm{H}}^{0}$, meaning that Equation 13 must also be verified.

$\\#$ | Reaction | s | ref
---|---|---|---
1 | $\mathrm{SiO}_{2(\ell)}\Leftrightarrow\mathrm{SiO}_{(g)}+1/2\mathrm{O_{2}}$ | 1/2 | Chase(1998)
2 | $\mathrm{SiO}_{2(\ell)}\Leftrightarrow\mathrm{SiO}_{2(g)}$ | 0 | Chase(1998)
3 | $\mathrm{Na}\mathrm{O}_{1/2(\ell)}\Leftrightarrow\mathrm{Na}_{(g)}+1/4\mathrm{O_{2}}$ | 1/4 | Chase(1998)
4 | $\mathrm{K}\mathrm{O}_{1/2(\ell)}\Leftrightarrow\mathrm{K}_{(g)}+1/4\mathrm{O_{2}}$ | 1/4 | Chase(1998)
5 | $\mathrm{Mg}\mathrm{O}_{(\ell)}\Leftrightarrow\mathrm{Mg}_{(g)}+1/2\mathrm{O_{2}}$ | 1/2 | Chase(1998)
6 | $\mathrm{Fe}\mathrm{O}_{(\ell)}\Leftrightarrow\mathrm{Fe}_{(g)}+1/2\mathrm{O_{2}}$ | 1/2 | Chase(1998)

Table 1: Liquid-gas reactions

  * On the left-hand side are the liquid oxides and on the right the gas oxides. We note that for species Na2O and K2O, we have considered the oxide normalized by the number of metal atoms, as in Sossi et al. (2020) and Sossi et al. (2019). The mole fraction of $\mathrm{Na}\mathrm{O}_{1/2}$ in melt is of $\sqrt{\mathrm{Na}_{2}\mathrm{O}}$. The same applies to K2O. Values of all thermodynamic constants were taken from Chase (1998).

We note that we assume that Fe is only present as FeO in the melt, meaning
Fe2O3 is ignored, as is reasonable to expect in a magma ocean in equilibrium
with H2 gas.

The calculation is performed as follows:

  * •

Choose a magma composition ($X^{l}_{i}$), temperature, and a total hydrogen
content ($P_{\mathrm{H}}^{0}$) .

  * •

Determine the atomic composition of the atmosphere ($X^{g}_{i}$ for all atoms
i) and the total pressure $P_{tot}$. For all sets ($X^{g}_{i}$, $P_{tot}$),
the corresponding molecular composition of the atmosphere is computed using
the CEA/NASA chemical equilibrium code. The only solution to this problem is
the ($X^{g}_{i}$, $P_{tot}$) set for which the molecular composition of the
atmosphere simultaneously solves Equations 13 to 15.

  * •

The search is done using an iterative method, with a simplex minimization
algorithm. The calculation goes on until Equations 13 to 15 are solved up to a
relative accuracy of better than $10^{-3}$ for each partial pressure involved.

For the atoms considered in this paper (Si, Mg, K, Na, Fe, O, H), the
following gas species are included in the CEA/NASA thermolib table (and are
the same as those found in JANAF table): Fe, FeO, H, $\mathrm{H_{2}}$,
$\mathrm{H_{2}O}$, $\mathrm{H_{2}O_{2}}$, $\mathrm{HO_{2}}$, K,
$\mathrm{K_{2}}$, $\mathrm{K_{2}O}$, $\mathrm{K_{2}O_{2}}$,
$\mathrm{K_{2}O_{2}H_{2}}$, $\mathrm{KH}$, $\mathrm{KNa}$, $\mathrm{KO}$,
$\mathrm{KOH}$, $\mathrm{Mg}$, $\mathrm{Mg_{2}}$, $\mathrm{MgH}$,
$\mathrm{MgO}$, $\mathrm{MgOH}$, $\mathrm{Na}$,
$\mathrm{Na_{2}},\mathrm{Na_{2}O}$, $\mathrm{Na_{2}O_{2}}$,
$\mathrm{Na_{2}O_{2}H_{2}}$, $\mathrm{NaH}$, $\mathrm{NaO}$, $\mathrm{NaOH}$,
$\mathrm{O}$, $\mathrm{O_{2}}$, $\mathrm{O_{3}}$, $\mathrm{OH}$,
$\mathrm{Si}$, $\mathrm{Si_{2}}$, $\mathrm{Si_{3}}$, $\mathrm{SiH}$,
$\mathrm{SiH_{2}}$, $\mathrm{SiH_{3}}$, $\mathrm{SiH_{4}}$, $\mathrm{SiO}$,
$\mathrm{SiO_{2}}$.

###  3.3 Validation against other thermochemical codes

Appendix A displays the composition we find for the vapor in the absence of H.
This is a case that has been extensively investigated in the past (see e.g.,
Visscher & Fegley (2013); Ito et al. (2015)) and more recently in the VapoRock
(Wolf et al., 2022) and LavAtmos codes (van Buchem et al., 2022). Our
calculated partial pressures are in very good agreement with those calculated
by existing codes (compare our Appendix to Figure 4 of Wolf et al. (2022) and
van Buchem et al. (2022)).

###  3.4 Validation against molecular dynamics calculations

However, our calculations are limited to species that are included in the
thermodynamic tables, such as JANAF, which may not be complete. For the sake
of completeness, we compare our thermodynamic model against ab initio
molecular dynamics simulations. For this, we employ the VASP package based on
the planar augmented wavefunctions (Kresse & Hafner, 1993) of the density
functional theory. We consider pyrolite melt, with the bulk silicate Earth
composition (McDonough & Sun, 1995): 1/2Na2O.2CaO.3/2Al2O3.4FeO.30MgO.24SiO2.
This six-component composition was used previously to model the
crystallization (Caracas et al., 2019) and the structure of the magma ocean
(Solomatova & Caracas, 2019).

We performed simulations well inside the liquid–vapor dome at 3000 K and a
density of 0.4 g/cm3, at which neither liquid nor vapor are stable as a single
phase, but coexist as a mechanical mixture. The simulations are realized
within the generalized gradient approximation in the Perdew-Wang-Ernzernhof
formulation (Perdew et al., 1996), spin-polarized, and corrected for the van
der Waals interactions in the gas phase (Grimme et al., 2010).

We used five different initial configurations and ran simulations lasting
between 50 and 136 picoseconds with a time step of 2 femtoseconds. For each
simulation, we first build the entire interatomic connectivity matrix (Caracas
et al., 2021), which allows us to define the chemical species. At 3000 K, we
obtain a vapor whose main components are: SiO, FeO2, Na, O2, Mg, FeO, O, and
SiO2. The major difference with our thermodynamic calculation (and all other
published calculations; see Appendix A) comes from the presence of FeO2
observed in the ab initio simulation and not considered in any thermodynamic
calculation. Moreover, the FeO2 component is present at all temperatures in
the simulations. This may indicate that further work is needed to reconcile
thermochemical models and ab initio molecular dynamics calculations.

##  4 Results: Atmospheric composition for a magma ocean degassing in hydrogen

###  4.1 Composition above the magma ocean

Our calculations are performed between T = 1800 K and 3500 K, and for total
hydrogen pressures ranging from $10^{-6}$ to $10^{6}$ bar. At higher pressure,
hydrogen could be metallic (¿ 1 Mbar), which is beyond the scope of our paper,
as we focus on terrestrial to sub-Neptune planets, and ignore giant planets.
In addition, at such high pressures, the ideal-gas approximation on which the
present paper is based fails. Examination of the virial coefficients for
common gases indicate the assumption of ideality breaks down above $\sim$103
to 104 bar. Refractory species such as Al and Ca may also be of interest but
are not expected to comprise significant fractions of silicate atmospheres
(e.g. Fegley et al., 2016); neither have they been detected in exoplanetary
atmospheres. As such, we do not consider them in this work other than to
define the mole fractions of melt oxides in the BSE.

Table 2: Composition of the magma ocean corresponding to a BSE (Palme & O’Neill, 2014) $\\#$ Oxide | Molar fraction
---|---
$MgO$ | 0.468
$\mathrm{\mathrm{SiO}_{2}}$ | 0.388
$\mathrm{FeO}$ | 0.0579
$\mathrm{Na_{1}O_{1/2}}$ | 0.00578
$\mathrm{K_{1}O_{1/2}}$ | $3.42\times 10^{-4}$

Here, we present four fiducial calculations with magma oceans at 1800, 2000,
3000, and 3400 K and varying total H pressure. The total pressure of the
atmosphere above the magma ocean is plotted in Figure 5. Figure 6 provides the
conversion between H pressure and mass of hydrogen (with respect to the
planet’s mass). The composition of the atmospheres at equilibrium with the
magma oceans are plotted in Figs. 7 to 10. A rapid comparison of these figures
shows that the transition from a pure mineral atmosphere to a hydrogenated
mineral atmosphere occurs when $P_{H}^{O}$ becomes comparable to the mineral
vapor pressure (i.e., about $10^{-5}$, $10^{-4}$, $10^{-1}$, and 10 bars for
T=1800, 2000, 3000, 3400 K, respectively). These pressures can be converted to
a total mass of captured H using the following relation (assuming constant
surface acceleration at the planet’s surface:$g=GM_{p}/R_{p}^{2}$):

$M_{H}=\frac{4\pi
R_{p}^{4}}{GM_{p}}\sum_{i}\frac{P_{i}\nu^{i}_{H}\mu_{H}}{\mu_{i}},$ (16)

where $M_{p}$, is the planet mass, $i$ is any molecular species in the
atmosphere, $\nu^{i}_{H}$, $\mu_{H}$, and $\mu_{i}$ are the stochiometric
coefficient of H in molecule $i$, the molar mass of H, and the molar mass of
molecule $i,$ respectively. Assuming an Earth-like planet, the conversion
between $P_{H}^{0}$ and planet mass fraction is displayed in Figure 6.

We do see that for an Earth-like planet, amounts of H of
$P_{H}^{0}=10^{-5},10^{-4},10^{-1}$, and $10$ bars correspond to planet mass
fractions of $\sim 10^{-12},10^{-11},10^{-7}$, and $10^{-6}$, respectively.
This demonstrates that a negligible amount of H can efficiently hydrogenate a
mineral atmosphere. These mass fractions of H are several orders of magnitude
smaller than the initial amount of captured H during the formation of the
planet, which ranges from $10^{-3}$% to 100$\%$ of the planet’s mass (Owen et
al., 2020).

Considering the molecular composition of the atmosphere (Figs. 7, 8, 9, 10),
the same behavior is observed as in the simple two-reaction model (section 2).
As the H content increases (on the X axis), the partial pressures of gases
derived from evaporation of the magma ocean (those containing Si, Mg, Fe, and
K atoms) increase by several orders of magnitude relative to the H-absent case
(compare the left- and right-hand sides of Figs. 17 to 20), a key result of
the present paper. In Appendix B, we provide the atmospheric compositions
represented with partial pressures.

In terms of molar fraction, for the T=1800 K case (Fig. 7), for low H contents
($P_{\mathrm{H}}^{0}<10^{-5}$bar) the atmosphere is dominated by
Na,$\mathrm{O_{2}}$ and K (in order of decreasing abundance), as in the
classic rocky atmosphere case (see e.g., Schaefer et al. (2012) ), and the
total pressure is about $10^{-5}$ bar. However, for cases in which
$P_{\mathrm{H}}^{0}>10^{-4}$ bar, the atmospheric composition becomes
increasingly dominated by $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$ as volatile
species, and the dominant metal-bearing species are now Fe and Na (in order of
decreasing abundance). Fe is efficiently released and increases linearly with
$P_{H}^{0}$. For $P_{H}^{0}$=1 bar, NA is 100 times more abundant that in the
H free case(see partial pressure plot in Appendix 17), and Fr is 1000 times
more abundant than in the H-free case . Interestingly, for
$P_{\mathrm{H}}^{0}>10^{3}$ bar, a new gas species becomes dominant (just
after $\mathrm{H_{2}}$): $\mathrm{SiH_{4}}$ (”Silane”, a toxic gas with a
strong repulsive odor) and $\mathrm{SiH_{3}}$ (”Silanide”; 100-1000 times less
abundant).

At 2000 K, the behavior is qualitatively the same as for 1800 K. While Na is
the most abundant metallic species in the absence of H for T=1800 and 2000 K,
it is always overtaken by Fe and SiO for H content larger than $10$ bar.

For T = 3000 K (Fig. 9), the same qualitative behavior is observed but more
species are involved. At this temperature, a pure rocky atmosphere (i.e., low
H) is dominated by SiO, Na, O2,O, Fe,SiO2, FeO, Mg, and MgO, with a total
pressure of about 0.1 bar. When $P_{\mathrm{H}}^{0}\sim 0.1$ bar, the dominant
volatile species become $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$, and the
abundance of $\mathrm{O_{2}}$ decreases. For $1<P_{\mathrm{H}}^{0}<10^{4}$
bar, the dominant heavy species are SiO, Fe, Mg, and Na. For
$P_{\mathrm{H}}^{0}>10^{4}$ bar, SiO decreases sharply as silicon is converted
into $\mathrm{SiH_{4}}$ and $\mathrm{SiH_{3}}$. In turn, the
$P_{\mathrm{O_{2}}}$ increases again for $P_{\mathrm{H}}^{0}>10^{4}$ bar
because of the liberated oxygen in this process (see Fig. 19).

Finally, for T = 3400 K (Fig. 10), at low H content, the pure rocky atmosphere
has a total pressure of about 3 bar, and is dominated by SiO, O2, O,
$\mathrm{SiO}_{2(g)}$ , Na, Fe, and $\mathrm{Mg}\mathrm{O}_{(g)}$.

For $P_{\mathrm{H}}^{0}>100$ bar, the atmosphere becomes dominated by
$\mathrm{H_{2}}$, $\mathrm{H_{2}O}$, and SiO while the abundance of
$\mathrm{O_{2}}$ decreases. For $P_{\mathrm{H}}^{0}>10^{4}$ bar, SiH4 and SiH2
become the most abundant heavy molecules. The shift of this transition to
higher initial H contents is mostly due to the temperature dependence of
reaction 2, in which the pH2/pH2O ratio decreases with increasing temperature,
such that the formation of the reduced species silane and silanide are delayed
to higher total H contents in hotter atmospheres.

From the above considerations, the following conclusions can be drawn:

  * •

Adding hydrogen to the evaporated atmosphere above the magma ocean increases
the partial pressure of all evaporated species owing to the decrease in
$f\mathrm{O_{2}}$. This therefore promotes efficient evaporation of the magma
ocean as more material goes into the atmosphere, and will also favor
spectroscopic detection as the column density of evaporated species will be
higher and the scale heights larger.

  * •

The vapor becomes strongly hydrogenated (where $\mathrm{H_{2}}$ and
$\mathrm{H_{2}O}$ become the dominant species and where the atmosphere
chemistry deviates strongly for pure mineral atmosphere) when
$P_{\mathrm{H}}^{0}>P_{RA}$ where $P_{RA}$ is the pressure of the rocky
atmosphere in the absence of H. Converted into a mass of atmospheric H, this
represents from $10^{-12}$ to $10^{-6}$ times the planet mass only.

  * •

The appearance of $\mathrm{SiH_{4}}$ and $\mathrm{SiH_{3}}$ occurs for
$P_{\mathrm{H}}^{0}>10^{3}$ bar in every case (representing $10^{-3}$ times
the planet mass of hydrogen for a Earth like planet) owing to association
reactions; for example, SiO + 2H2 = SiH4 \+ H2O, in which there are three
moles of gas in the reactants compared with only two in the products. At these
high H pressures, $\mathrm{SiH_{4}}$ and $\mathrm{SiH_{3}}$ become the most
abundant heavy molecules (after $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$).

Figure 5: Total pressure of the gas at equilibrium with the magma ocean for
different temperatures. Figure 6: Total pressure of monoatomic hydrogen
($P_{H}^{0}$) versus planet mass fraction, assuming the planet has the mass
and radius of the Earth. Figure 7: Molar abundances of most abundant species
versus total monoatomic hydrogen pressure $P_{\mathrm{H}}^{0}$ for an
atmosphere at equilibrium with a magma ocean at T=1800K. The same plot but
with partial pressures is provided in Appendix B.  Figure 8: Molar abundances
of most abundant species versus total monoatomic hydrogen pressure
$P_{\mathrm{H}}^{0}$ for an atmosphere at equilibrium with a magma ocean at
T=2000K. The same plot but with partial pressures is provided in appendix B.
Figure 9: Molar abundances of most abundant species versus total monoatomic
hydrogen pressure $P_{\mathrm{H}}^{0}$ for an atmosphere at equilibrium with a
magma ocean at T=3000 K. The same plot but with partial pressures is provided
in appendix B. Figure 10: Molar abundances of most abundant species versus
total monoatomic hydrogen pressure ($P_{\mathrm{H}}^{0}$) for an atmosphere at
equilibrium with a magma ocean at T=3400K. The same plot but with partial
pressures is provided in appendix B.

###  4.2 Signature of a hydrogenated magma ocean?

It is interesting to speculate as to how we could detect the presence of a
magma ocean beneath a hydrogen-rich atmosphere. In the absence of hydrogen,
the results presented above, and in previous studies (Schaefer et al., 2012;
Schaefer & Fegley, 2010; Lupu et al., 2014; Schaefer et al., 2012; Ito et al.,
2015; Ito & Ikoma, 2021; Jäggi et al., 2021; Charnoz et al., 2021), show that
Na should be released by a magma ocean for T ¿ 1500 K and should remain the
dominant species up to 2500 K. At higher temperatures (¿ 2500 K), SiO should
be the most abundant mineral species in the vapor.

When H is present —even in small quantities—, $\mathrm{H_{2}O}$, Fe, Na, and
SiO become dominant in the atmosphere. This suggests that the detection of
$\mathrm{H_{2}O}$, Si, Na, or Fe for a rocky planet with an equilibrium
temperature of $T_{eq}>2000K$ may point to the presence of a magma ocean below
the atmosphere and the presence of a small amount of H. As of today, heavy
species, such as Si or Mg, have never been detected in the atmosphere of a
rocky or sub-Neptune planet, whereas they have been detected in the
atmospheres of giant exoplanets (Zieba et al., 2022). In order to provide a
framework for their detection in the future, below we report some atomic
ratios that may be of interest for future observations designed to unveil the
presence of a magma ocean.

The Mg/Si ratio in the atmosphere above the magma ocean is found to deviate
strongly from that of the planet’s mantle (Fig.11). For $P_{H}^{0}$ $<$ 1000
bar, the atmospheric Mg/si ratio is almost constant at about $10\%$ of the
mantle value. For higher hydrogen content, the Mg/Si ratio decreases —by
several orders of magnitude— below the values of both the star and the mantle.
As Mg and Si are two major elements that condense at relatively high
temperatures ($\sim$ 1350 K at 10-4 bar) from the nebular gas, the
fractionation of these elements by condensation or evaporation during
planetary accretion should be limited, at least for stars with solar-like C/O
ratios (Larimer & Bartholomay, 1979). Therefore, assuming that the Mg/Si ratio
of the planetary mantle is equal to that of the host star, their detection on
an ultrahot rocky planet may provide insights into the magma ocean
composition. A caveat associated with this conclusion is that Si remains
lithophilic in the planetary mantle, that is, it does not enter the core in
appreciable amounts, as is the case on Mercury for example (e.g., Malavergne
et al., 2010).

The $Mg/Fe$ ratio of the atmosphere is of interest because our models show
that it is independent of H content (Fig. 12). This is because pMg and pFe are
proportional to aMgO and aFeO in the liquid phase, and inversely proportional
to fO${}_{2}^{1/2}$ (cf. Eq. 14). Measuring the $Mg/Fe$ ratio in a molten
exoplanet atmosphere may provide insights into the Mg/Fe mantle composition
independently of the H content of the atmosphere.

On the other hand, the Na/Si ratio of the atmosphere is strongly dependent on
the H content of the planet (Fig. 13). If this ratio can be measured (or
deduced based on SiO partial pressure, see Wolf et al. (2022)), it may
provided constraints as to the H content of the atmosphere. Such a conclusion
is dependent upon Na still being present in the magma ocean today despite its
tendency to evaporate and escape over the lifetime of the planet (see
following section, and Erkaev et al. (2022)).

Figure 11: Molar Mg/Si ratio in the vapor above a magma ocean as a function of
surface temperature. Colors display the amount of hydrogen in the planet
(Log($P_{\mathrm{H}}^{0}$) in bars). The blue dashed line shows the solar
value of Mg/Si and the black dashed line shows the BSE value.  Figure 12:
Molar Mg/Fe ratio in the vapor above a magma ocean as a function of surface
temperature. Colors display the amount of hydrogen in the planet
(Log($P_{\mathrm{H}}^{0}$) in bars). The blue dashed line shows the solar
value of Mg/Fe and the black dashed line shows the BSE value.  Figure 13:
Molar Na/Si ratio in the vapor above a magma ocean as a function of surface
temperature. Colors display the amount of hydrogen in the planet
(Log($P_{\mathrm{H}}^{0}$) in bars). The blue dashed line shows the solar
value of Na/Si and the black dashed line shows the BSE value.

##  5 Atmospheric escape

Now that we are able to compute the atmosphere composition above the magma
ocean, we wish to compute the efficiency of escape. This is a very complex
problem, and has mostly been investigated for hydrogen-rich planets. The
treatment of the full problem using an MHD-hydrocode including multiple
species and complex chemistry (as we do here) is well beyond the scope of the
present paper. Still, to obtain an approximate estimate of the escape
efficiency of such an atmosphere, we developed a model inspired by the popular
energy-limited model, which we present below.

###  5.1 Method of calculation

We try to quantify the efficiency of atmospheric escape from planets with a
long-lived magma ocean embedded in a primordial hydrogen envelope. We focus on
planets very close to their star so that stellar heating is strong enough to
maintain a magma ocean during the lifetime of the star. While it is well
established that hot-Jupiter planets are massive enough to retain their
atmospheres (see e.g., Valencia et al. (2010)), Earth-sized planets should
have lost their primordial H envelope in less than 10 Myr ( e.g., Valencia et
al. (2010); Lopez (2017); Erkaev et al. (2022)). What happens in the
intermediate regime of super-Earths/sub-Neptunes is comparatively poorly
understood. Of particular importance is the observation of a bimodal radius
distribution of super-Earths, with a well-identified valley in the radius
range 1.5-2 $R_{\oplus}$ (see e.g., Fulton et al. (2017); Gupta & Schlichting
(2019)). It has been proposed that this ”valley” marks the transition from the
smaller rocky planets (”super-Earths”) to planets containing a few per cent of
the planet’s total mass in a H/He- rich envelope (”sub-Neptunes”; see e.g.,
Owen & Wu (2013); Lopez & Fortney (2014); Ginzburg et al. (2016); Ginzburg &
Sari (2017)). This transition may be due to rapid photoevaporation of highly
irradiated H-rich atmosphere, which may not survive for planets with radii of
$<1.5-2R_{\oplus}$. In this section, we do not investigate the origin of this
transition; rather, we study the consequence of the presence of a small amount
of hydrogen on the efficiency of magma evaporation and escape of the
atmosphere produced in such a scenario.

Here, we consider the case of an atmosphere in thermodynamic equilibrium with
a magma ocean of constant temperature at its base as described in the previous
section. Models of strongly irradiated planets show that escape is always
efficient for terrestrial-mass planets (Ito et al., 2015; Benedikt et al.,
2020; Ito & Ikoma, 2021) and that the particle velocities in the atmospheres
of these planets far exceed those necessary for hydrostatic equilibrium.
Although this conclusion is unlikely to hold for a cold atmosphere (which may
indeed be hydrostatic), this assertion may be justified for ultrahot planets
(T ¿ 1800 K) with irradiated atmospheres, depending on the average molar mass
of the atmosphere. For heavy species (Si, Mg), Ito & Ikoma (2021) find strong
atmospheric escape (with mass flux of about 0.4 $M_{\oplus}/Gyr$) in coupled
hydrodynamical and radiative simulations, for an Earth-size planet at 0.02 AU
devoid of hydrogen. These authors conclude that the atmosphere is never at
hydrostatic equilibrium and is in ”blow-off” regime, with a Parker-wind-type
transonic velocity profile. Konatham et al. (2020) argue that significant to
catastrophic escape occurs if the atomic thermal velocities reach $>10\%$ of a
planet’s escape velocities. This means that an Earth-like planet at a
temperature of 3000 K may lose all molecules up to 100 atomic mass units,
whereas a planet of 2 Earth masses may lose atoms up to gasses with molar
weight up to $\sim 25$ atomic mass units. Similarly, Johnstone et al. (2019)
find that even Ar (40 AMU) can escape from a strongly irradiated Earth-like
planet (but these authors do not include heavier species). In the latter
study, the H/N/O/C/He/Ar atomic ratios in the upper escaping atmosphere are
found to vary between 0.75 and 2.5 relative to the same ratios in the lower
atmosphere. As long as the atmosphere is in the hydrodynamic escape regime
(blow-off), it seems that no strong atomic fractionation occurs during the
escape, within a factor of a few (Johnstone et al., 2019).

In all the studies mentioned above, the gravitational stratification of
species is not found because diffusive timescales are always longer than
advective timescales. This is why gravitational stratification is ignored
here.

We note that cooling via vibrational bands of $\mathrm{H_{2}O}$ (Yoshida et
al., 2022) and atomic lines for N2 and O2 (Nakayama et al., 2022) has been
found to be efficient enough to cool down the atmosphere and prevent escape
from habitable planets under very strong XUV irradiation (up to about $1000$
times the present Earth value, which is relevant for very young stars).
However, in the case we consider here, that is, ultrahot and ultrashort-period
exoplanets, the XUV flux may be $>10^{4}$ times the value for Earth today
owing to the very short stellar distance (0.01-0.02 au). In such extreme
cases, Yoshida et al. (2022) do find hydrodynamical escape.

A full treatment of the computation of atmospheric escape is beyond the scope
of the present paper. Here, we limit our analysis to simple estimates based on
the popular ”energy limited” approximation, following the method of Salz et
al. (2015), which provides a useful scaling law for energy-limited escape;
this law is calibrated in 1D numerical simulations for irradiated, hydrogen-
dominated planets. The mass flux (Kg/s) escaping from the atmosphere is
approximated by:

$\dot{M}=\frac{-3\beta^{2}\eta F_{XUV}}{4KG\rho_{p}},$ (17)

where $\beta=R_{XUV}/R_{p}$ is the ratio of the radius of absorption of the
radiation from an XUV star to the physical radius of the planet ($R_{p}$), and
$\eta$ is a parameter standing for the ”heating efficiency”, which is the
fraction of input radiative energy converted into thermal expansion of the
atmosphere. $F_{XUV}$ is the incident stellar flux in the XUV band (in
$J/s/m^{2}$) and K is a parameter accounting for the effect of stellar tides
(which facilitate escape). Here, we use K=1 for simplicity.

Numerous physical mechanisms are subsumed into $\eta$ and its value is highly
dependent on the stellar spectrum, the gravity of the planet, and the
atmospheric composition (see e.g., Valencia et al. (2010); Salz et al. (2016);
Ito & Ikoma (2021)). Salz et al. (2016) find that $\beta$¡2 and depends
sensitively on the planet’s gravitational potential well and the flux of the
star. The following parametrization is proposed (Salz et al., 2016):

$\begin{split}Log_{10}(\beta)=Max[0.,-0.185Log_{10}\left(\frac{\Phi_{G}}{erg/g}\right)+\\\
0.021Log_{10}\left(\frac{F_{XUV}}{erg/cm^{2}/s}\right)+2.42],\end{split}$ (18)

where $\Phi_{G}=GM_{p}/R_{p}$. Numerical simulations by Salz et al. (2016),
which are valid for a H2-dominated atmosphere, show that for planets with a
deep gravitational well (i.e., large $\Phi$), escape is difficult and $\eta$
may be $\ll\leavevmode\nobreak\ 1$ (giant planets), whereas for planets with
shallow gravitational potential (i.e., small $\Phi$), escape is relatively
efficient (Earth- and super-Earth-mass planets). Following Salz et al. (2016)
and noting $\nu=Log_{10}\left(\frac{\Phi_{G}}{erg/g}\right)$ $\eta_{H}$ for a
hydrogen-dominated atmosphere:

$Log_{10}(\eta_{H})=\cases{-}0.5-0.44(\nu-12.)&\text{ for $12.<\nu<13.11$ }\\\
-0.98-7.29(\nu-13.11)\text{for $13.11<\nu<13.6.$ }{}$ (19)

The above fit, which is valid for H2-dominated atmospheres, may not be valid
for pure mineral atmospheres. Ito & Ikoma (2021) find that, in pure mineral
atmospheres, atmospheric cooling is much more efficient than in H2-dominated
atmospheres, resulting in low escape efficiency with $2\times
10^{-4}<\eta<4\times 10^{-3}$, which is much lower than $0.1<\eta<0.5$ for
H-dominated atmospheres. To take this compositional effect into account, we
replace $\eta$ in Eq. 17 by:

$\eta_{eff}=\eta_{H}\times\left(\frac{1.-e^{-x}}{1.-e^{-1}}\right)+5\times
10^{-4},$ (20)

where $x$ is the atomic fraction of H in the atmosphere. Although not optimal,
the above prescription ensures that H2-dominated atmospheres escape at the
efficiency given by Salz et al. (2016), whereas mineral atmospheres escape
with an efficiency in the (lower) range of the values reported by Ito & Ikoma
(2021).

Finally, concerning the time-dependent XUV flux at the location of the planet,
we adopt the fit to XUV flux data reported for young stars by Valencia et al.
(2010):

$F_{XUV}(J/m^{2}/s)=\cases{2}9.7\times
10^{-3}(0.1)^{-1.23}\left(\frac{1}{a^{2}}\right)\text{ for t <
0.1\leavevmode\nobreak\ Gyr}\\\ \par 29.7\times
10^{-3}\left(\frac{t}{1\leavevmode\nobreak\
gyrs}\right)^{-1.23}\left(\frac{1}{a^{2}}\right)\par\text{ for t >
0.1\leavevmode\nobreak\ Gyr.}{}$ (21)

The calculation is performed as follows. We start with a planet mass $M_{p}$
and radius $R_{p}$ with surface acceleration $g$, surface temperature $T,$ and
mass of H in the atmosphere $M_{H}$, corresponding to an initial pressure of
monoatomic H: $P_{\mathrm{H}}^{0}=M_{H}*g/(4\pi{R_{p}}^{2})$. For a given $T$
and $P_{\mathrm{H}}^{0}$ , the atmospheric molecular composition above the
magma ocean is computed as described in section 3. We compute the total mass
loss according to Eq. 17. We assume the star is 5 Gyr old. Following the two
preceding arguments regarding inefficient gravitational stratification in the
blow-off regime, the total mass flux of each atom is $\dot{M}\times X_{i}$,
where $X_{i}$ is the atomic mass fraction of atom $i$ in the escaping gas.

We do not consider here the dissolution of $\mathrm{H_{2}}$ or
$\mathrm{H_{2}O}$ in the magma ocean as this will be the subject of a future
study. We note that the dissolution of $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$
into the magma ocean may lead to the constitution of a reservoir of H and O,
which may lengthen the duration of H escape (Hier-Majumder & Hirschmann, 2017;
Bower et al., 2022). Consequently, the timescales of H escape reported in the
following section should be considered as lower bounds. In addition, the
substantial solubility of H as $\mathrm{H_{2}O}$ in the magma ocean (Sossi et
al., 2023) indicates that it would be largely ($\geq$ 95 % of its total
budget) dissolved should the entire mantle of an Earth-mass planet remain
molten. However, as the magma ocean cools, the solubility of $\mathrm{H_{2}O}$
exceeds the mass fraction of available liquid in which to dissolve, leading to
late-stage release of significant amounts of O and H (as H2O) into the
atmosphere when the atmospheric pressure is sufficiently low (Solomatova &
Caracas, 2021). This may serve to increase the fO2 relative to that initially
imposed by the H2-rich nebular gas, and thus modify atmospheric chemistry. We
warn the reader that these effects are not considered here.

###  5.2 Results

Figure 14 displays the mass of hydrogen as a function of time for an Earth-
like planet ($M_{p}=1\leavevmode\nobreak\ M_{\oplus}$,
$R_{p}=1\leavevmode\nobreak\ R_{\oplus}$) located at 0.02 au, starting with
different hydrogen content from $10^{-6}$ to $10^{-1}$ times the planet mass.
The top panel shows that most H is lost within about 100 Kyr to 10 Myr,
depending on $M_{H}^{0}/M_{p}$. The bottom panel displays the lost mass of Na
(a proxy for the moderately volatile elements) divided by its mass in the
planet’s mantle magma ocean (in red; we assume to first order that the magma
ocean is $50\%$ of the planet mass). For $M_{H}^{0}/M_{p}=10^{-6}$ (red solid
line), Na is lost at a roughly constant rate as long as hydrogen is present
and the star is ¡ 10 Myr old when its XUV flux is intense. When hydrogen has
disappeared, the efficiency of Na escape increases. This result may be
surprising, but can be reconciled with the fact that, for high H contents, the
Na mole fraction is very low; that is, the atmosphere is dominated by SiO or
Fe (see previous section). In contrast, when the abundance of H drops to zero,
the mole fraction of the atmosphere comprised of H increases by three orders
of magnitude, meaning the escape rate of Na increases by a proportionate
factor. Over gigayear timescales, the loss of Na is dominated by loss during
the last 1 billion years and we find, at the end, that about $30\%$ of the
mantle Na content is lost.

The role of H in influencing the Si content of the planet’s mantle is more
marked. The blue solid line in Fig. 14 shows Si evolution for
$M_{H}^{0}/M_{p}=10^{-6}$. After 5 Gyr, the planet devoid of H has lost only
$\sim$ $1\%$ of its Si. Conversely, for planets starting with hydrogen
contents of $10\%$ of the mass of the entire planet, about $10\%$ of the total
budget of Si is lost after 5 Gyr. This strong increase in Si loss with H
content reflects the fact that when H is abundant, SiO becomes the dominant
metal-bearing gas species. In addition, the presence of H increases the
efficiency of atmospheric heating ($\eta\leavevmode\nobreak\
>\leavevmode\nobreak\ 0.1$, see section 5.1) and decreases the mean molar mass
of the atmosphere. We highlight the fact that, due to the simplicity of the
models, the above values may not be understood as exact solutions, but rather
as order-of-magnitude estimates of the extent of escape. Indeed, the values
could be somewhat underestimated owing to the conservative approach that was
taken in estimating the quantities used in section 5.1, though it should be
mentioned that the composition of the magma ocean is assumed to remain fixed
rather than evolve, as would be expected for a finite mass undergoing Rayleigh
(fractional) vaporisation.

We also explored different planet masses and radii and find that the main
factors limiting the amount of metals and metal-bearing oxides and hydrides
able to escape is the lifetime of the H envelope (controlled by the planet’s
density) and the surface temperature. For a more complete view of the effect
of escape on each atmospheric constituent, Fig. 15 shows the lost mass
fraction of every atom (O, Si, Mg, Fe, Na, K) for all combinations of
temperature and initial mass fraction of hydrogen and for an Earth-like planet
located at 0.02 au. The color of the plot indicates the mass fraction of each
atom that is lost (with respect to the content in the mantle). White denotes
that all atoms in the mantle are lost, purple indicates a loss of about
10$\%$, and so on. The most volatile elements, Na and K, are completely lost
for $T_{surf}<3500$ K, which is in line with the results of Ito & Ikoma
(2021). For $T_{surf}>3500$ K, the efficiency of Na and K loss is decreased
because their molar fraction in the atmosphere is supplanted by other species
(Si, Fe, Mg), thus diminishing their net loss in comparison to other species.
Indeed the mass flux (eq. 17) only depends on the F(XUV) flux and the planet
mass and radius. In addition, the heating efficiency is low (¡ 0.001), such
that when temperature is ¿ 3200 K and H content is low, the species that
escape are mostly those that predominate in the atmosphere, that is, mostly
Si, O, and Mg. If we now turn to Si, we see that more than 10$\%$ of the
planet’s mantle Si is lost for $T_{surf}>2600$ K and for a H mass fraction of
$M_{H}/M_{p}>0.1$. The same result applies for oxygen. As oxygen is the most
abundant species by mass (about $50\%$ of the mass of Earth’s mantle), this
implies significant removal of the planet’s mantle and an increase in average
density (an effect that is not taken into account here). Concerning Fe, it is
substantially lost (¿10$\%$) for temperatures ¿ 2600K and
$M_{H}^{0}/M_{p}>0.01$ due to its high abundance in the vapor combined with
its low abundance (6.3 mol %) in the mantle.

Magnesium shows an intermediate behavior between those of Si and Na, with most
efficient loss occurring for $T_{surf}>3200$ K and $M_{H}^{0}/M_{p}>10^{-2}$.
Up to $8\%$ of Mg can be lost from the mantle in such extreme conditions. The
less efficient loss of Mg stems from its lower volatility than Si and Fe for
the BSE composition considered here.

Figure 14: Escape from a 1 $M_{\oplus}$ and $R_{\oplus}$ planet located at
0.02 au from its star. Top: Mass of H vs time for different initial mass
fractions of hydrogen. Bottom: Mass fraction of lost Si(red) and Na(blue) vs.
time. The solid line shows these relations for an initial H mass fraction of
$10^{-6}$ times the planet mass, and the dashed line is for an initial H mass
fraction of 0.1 times the planet mass.  Figure 15: Each figure shows a color
plot of the mass fraction of an atom that is lost for temperature T (X axis)
and initial hydrogen mass (Y axis) of the planet. The planet’s solid core here
is 1 $M_{\oplus}$ and 1 $R_{\oplus}$.

Therefore, in conclusion, we do see that for extreme surface temperatures (¿
2600 K) and initial H mass content of ¿ 1$\%,$ a significant fraction of the
mantle’s content ($\sim$ a few 10$\%$) could be evaporated during the planet’s
evolution. We note that the results of this section must be taken with care
because of the simplicity of the model and the numerous simplifications we had
to introduce. The energy-limited escape is known to overestimate the escape,
and the escape efficiency is not well known for atmosphere heavier than solar
composition. In addition, vibrational and atomic radiative cooling lines
during the escape (see e.g., Yoshida et al., 2022; Nakayama et al., 2022) may
act against escape. We therefore conclude that our simple model suggests that
escape of heavy molecules may be made easier by the presence of H, but further
investigation of this process is required to confirm this finding.

##  6 Conclusions

We have studied the composition and the escape of an atmosphere above a magma
ocean planet in the presence of hydrogen. In the present study, a special
effort was dedicated to computing the atmospheric composition at
thermodynamical equilibrium with the magma ocean, taking into account the
effect of H on the atmospheric chemistry, and its retro-action on the
liquid–atmosphere equilibrium. Our main results are summarized below:

  * •

Hydrogen present in the atmosphere has a dramatic effect on the vapor
composition. For T=2000 K and 3000 K, the atmosphere composition is strongly
modified for atmospheric hydrogen contents of about $10^{-8}$ and $10^{-6}$
times the planet mass, respectively (assuming an Earth mass planet) and
becomes dominated by $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$.

  * •

Hydrogen lowers the O2 partial pressure (=oxygen fugacity $fO_{2}$) to make
H2O, which happens in proportion to the mass of H added. As a consequence, the
addition of hydrogen promotes efficient degassing of metals Na, K, Fe, Mg, and
Si. These elements all evaporate according to reactions,
$M^{x+s}O_{(x+s)/2}(l)=M^{x}O_{x/2}(g)+s/4O_{2}(g)$ in which the
stoichiometric coefficient, s, is positive. Therefore, a small amount of
hydrogen (comparable to the total pressure of the evaporated elements, and
nearly negligible compared to the planet’s mass) greatly enhances the
extraction of these elements from the magma to the vapor phase.

  * •

The atomic abundances of heavy (i.e., rock-forming) species in the atmosphere
may diverge significantly from the mantle value, and are controlled by the
magma–atmosphere equilibrium.The Mg/Si ratio in the atmosphere is constant and
independent of H content for T¿3000 K and may be directly related to the
composition of the planet’s mantle. Conversely, the Na/Si ratio is very
sensitive to H content (owing to the different dependence of the partial
pressures of their major gas species on $fO_{2}$, and may be useful in
constraining the presence of H). Finally, the Mg/Fe ratio is independent of H
and may be linked to the mantle composition. This suggests that spectroscopic
observations of magma ocean planets could provide a means to constrain the
mantle composition, even in the presence of hydrogen.

  * •

For Earth-sized planets of ¡ 0.02 AU and with a temperature above the magma
ocean, that is ¿ 2000 K, most H is lost in ¡ 10 Myr, regardless of its
speciation.

  * •

For Earth-sized planets of ¡ 0.02 AU and with a temperature above the magma
ocean, that is ¿ 3000 K, we find significant loss of Fe, Si, and O (¿ 10$\%$
of their budget in the planet’s mantle). In such extreme conditions, this may
lead to a decrease in the mass of the mantle, and in turn to an increase in
the mean density of the planet (due to core/mantle ratio decreases). This may
potentially explain the high densities of strongly irradiated rocky planets.

In conclusion, we show that the presence of relatively small fractions of H in
the atmosphere may enhance the evaporation of metals from ultrahot rocky
planets covered by a magma ocean, and, at very extreme temperatures, may lead
to significant loss of heavy atoms such as Si, O, or Fe. Whereas no heavy
species have yet been detected in rocky planetary atmospheres, we identify
ratios of rock-forming elements in the vapor phase that are potentially able
to reveal the presence and nature of a magma ocean, and subsequently probe the
composition of the planet mantle. In contrast to some recent studies
(Zilinskas et al., 2022), we do find that the presence of H strongly increases
the partial pressure of major degassed species (see Figures 17 to 20). This
will significantly impact the resulting atmospheric spectra and (P,T)
structure of the exoplanet. This will be the subject of a future study.

In the future, we will study the effect of dissolution of $\mathrm{H_{2}}$ and
$\mathrm{H_{2}O}$ on the atmospheric composition of magma ocean exoplanets. We
anticipate that the effect of dissolution of $\mathrm{H_{2}O}$ will be to
oxidize the atmosphere, thus lowering the reducing effect of H, which may lead
to less efficient release of metallic species (boosted by H2); it will also
create an additional reservoir of H, probably resulting in an extended escape
time of H. This will be studied in a future paper.

On a more technical note, we also emphasize that improvement of ab initio
calculations and comparison with tabulated thermochemical data will be useful
in the future, as ab initio calculations reveal the formation of a great
diversity of species (sometimes not available in common data tables such as
JANAF or CEA-NASA) and at higher temperatures than are typically accessible
experimentally.

###### Acknowledgements.

SC, AF, PT acknowledge financial support by LabEx UnivEarthS (ANR-10-LABX-0023
and ANR-18-IDEX-0001) and by the CNES (Centre National d’Études Spatiales). PT
would also like to acknowledge and thank the ERC for funding this work under
the Horizon 2020 program project ATMO (ID: 757858). RC acknowledges support
from the European Research Council under EU Horizon 2020 research and
innovation program (grant agreement 681818 – IMPACT to RC), the Research
Council of Norway with project number 223272 and project HIDDEN 325567, and
access to supercomputing facilities via eDARI stl2816 grants, PRACE RA4947
grant, Uninet2 NN9697K grant. PS thanks the Swiss National Science Foundation
(SNSF) via an Ambizione Fellowship (180025), an Eccellenza Professorship
(203668) and the Swiss State Secretariat for Education, Research and
Innovation (SERI) under contract number MB22.00033, a SERI-funded ERC Starting
Grant ’2ATMO’. We thank the anonymous reviewer for their insightful comments
that improved the quality of the paper.

## References

  * Benedikt et al. (2020) Benedikt, M. R., Scherf, M., Lammer, H., et al. 2020, Icarus, 347, 113772
  * Bourrier et al. (2018) Bourrier, V., Dumusque, X., Dorn, C., et al. 2018, A&A, 619, A1
  * Bower et al. (2022) Bower, D. J., Hakim, K., Sossi, P. A., & Sanan, P. 2022, The Planetary Science Journal, 3, 93
  * Caracas et al. (2019) Caracas, R., Hirose, K., Nomura, R., & Ballmer, M. D. 2019, Earth and Planetary Science Letters, 516, 202
  * Caracas et al. (2021) Caracas, R., Kobsch, A., Solomatova, N. V., et al. 2021, Journal of Visualized Experiments
  * Charnoz et al. (2021) Charnoz, S., Sossi, P. A., Lee, Y.-N., et al. 2021, Icarus, 364, 114451
  * Chase (1998) Chase, M. 1998, NIST-JANAF Thermochemical Tables, 4th Edition (American Institute of Physics, -1)
  * Crida et al. (2018) Crida, A., Ligi, R., Dorn, C., & Lebreton, Y. 2018, ApJ, 860, 122
  * Demory et al. (2016) Demory, B.-O., Gillon, M., de Wit, J., et al. 2016, Nature, 532, 207
  * Denman et al. (2020) Denman, T. R., Leinhardt, Z. M., Carter, P. J., & Mordasini, C. 2020, Monthly Notices of the Royal Astronomical Society, 496, 1166
  * Dorn & Lichtenberg (2021) Dorn, C. & Lichtenberg, T. 2021, ApJ, 922, L4
  * Ehrenreich et al. (2012) Ehrenreich, D., Bourrier, V., Bonfils, X., et al. 2012, A&A, 547, A18
  * Erkaev et al. (2022) Erkaev, N., Scherf, M., Herbort, O., et al. 2022, Monthly Notices of the Royal Astronomical Society, 518, 3703
  * Fegley & Cameron (1987) Fegley, B. & Cameron, A. G. W. 1987, Earth and Planetary Science Letters, 82, 207
  * Fegley et al. (2016) Fegley, Bruce, J., Jacobson, N. S., Williams, K. B., et al. 2016, ApJ, 824, 103
  * Fulton et al. (2017) Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109
  * Gaillard et al. (2022) Gaillard, F., Bernadou, F., Roskosz, M., et al. 2022, Earth and Planetary Science Letters, 577, 117255
  * Ginzburg & Sari (2017) Ginzburg, S. & Sari, R. 2017, MNRAS, 469, 278
  * Ginzburg et al. (2016) Ginzburg, S., Schlichting, H. E., & Sari, R. 2016, ApJ, 825, 29
  * Gordon & McBride (1996) Gordon, S. & McBride, B. J. 1996, NASA reference publication 1311
  * Grimme et al. (2010) Grimme, S., Antony, J., Ehrlich, S., & Krieg, H. 2010, The Journal of Chemical Physics, 132, 154104
  * Gupta & Schlichting (2019) Gupta, A. & Schlichting, H. E. 2019, MNRAS, 487, 24
  * Hier-Majumder & Hirschmann (2017) Hier-Majumder, S. & Hirschmann, M. M. 2017, Geochemistry, Geophysics, Geosystems, 18, 3078
  * Hirschmann et al. (2012) Hirschmann, M. M., Withers, A. C., Ardia, P., & Foley, N. T. 2012, Earth and Planetary Science Letters, 345, 38
  * Ito & Ikoma (2021) Ito, Y. & Ikoma, M. 2021, MNRAS, 502, 750
  * Ito et al. (2015) Ito, Y., Ikoma, M., Kawahara, H., et al. 2015, ApJ, 801, 144
  * Jäggi et al. (2021) Jäggi, N., Gamborino, D., Bower, D. J., et al. 2021, The planetary science journal, 2, 230
  * Johansen & Dorn (2022) Johansen, A. & Dorn, C. 2022, A&A, 662, A19
  * Johnstone et al. (2019) Johnstone, C. P., Khodachenko, M. L., Lüftinger, T., et al. 2019, A&A, 624, L10
  * Kite et al. (2020) Kite, E. S., Fegley, Bruce, J., Schaefer, L., & Ford, E. B. 2020, ApJ, 891, 111
  * Kite & Schaefer (2021) Kite, E. S. & Schaefer, L. 2021, ApJ, 909, L22
  * Konatham et al. (2020) Konatham, S., Martin-Torres, J., & Zorzano, M.-P. 2020, Proceedings of the Royal Society of London Series A, 476, 20200148
  * Kreidberg et al. (2019) Kreidberg, L., Koll, D. D. B., Morley, C., et al. 2019, Nature, 573, 87
  * Kresse & Hafner (1993) Kresse, G. & Hafner, J. 1993, Physical review. B, Condensed matter, 47, 558
  * Larimer & Bartholomay (1979) Larimer, J. W. & Bartholomay, M. 1979, Geochimica et Cosmochimica Acta, 43, 1455
  * Lichtenberg et al. (2021) Lichtenberg, T., Bower, D. J., Hammond, M., et al. 2021, Journal of Geophysical Research (Planets), 126, e06711
  * Lopez (2017) Lopez, E. D. 2017, MNRAS, 472, 245
  * Lopez & Fortney (2014) Lopez, E. D. & Fortney, J. J. 2014, ApJ, 792, 1
  * Lupu et al. (2014) Lupu, R. E., Zahnle, K., Marley, M. S., et al. 2014, ApJ, 784, 27
  * Malavergne et al. (2010) Malavergne, V., Toplis, M. J., Berthet, S., & Jones, J. 2010, Icarus, 206, 199
  * McDonough & Sun (1995) McDonough, W. & Sun, S.-s. 1995, Chemical Geology, 120, 223
  * Mercier et al. (2022) Mercier, S. J., Dang, L., Gass, A., Cowan, N. B., & Bell, T. J. 2022, accepted in Astron. J., arXiv:2209.02090
  * Nakayama et al. (2022) Nakayama, A., Ikoma, M., & Terada, N. 2022, The Astrophysical Journal, 937, 72
  * Olson & Sharp (2019) Olson, P. L. & Sharp, Z. D. 2019, Physics of the Earth and Planetary Interiors, 294, 106294
  * O’Neill & Palme (2008) O’Neill, H. S. C. & Palme, H. 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366, 4205
  * Otegi et al. (2020) Otegi, J. F., Bouchy, F., & Helled, R. 2020, A&A, 634, A43
  * Owen et al. (2020) Owen, J. E., Shaikhislamov, I. F., Lammer, H., Fossati, L., & Khodachenko, M. L. 2020, Space Sci. Rev., 216, 129
  * Owen & Wu (2013) Owen, J. E. & Wu, Y. 2013, ApJ, 775, 105
  * Palme & O’Neill (2014) Palme, H. & O’Neill, H. 2014, in Treatise on Geochemistry (Second Edition), second edition edn., ed. H. D. Holland & K. K. Turekian (Oxford: Elsevier), 1–39
  * Perdew et al. (1996) Perdew, J. P., Burke, K., & Ernzerhof, M. 1996, Physical Review Letters, 77, 3865
  * Pignatale et al. (2017) Pignatale, F. C., Gonzalez, J. F., Cuello, N., Bourdon, B., & Fitoussi, C. 2017, MNRAS, 469, 237
  * Salz et al. (2016) Salz, M., Schneider, P. C., Czesla, S., & Schmitt, J. H. M. M. 2016, A&A, 585, L2
  * Schaefer & Fegley (2004) Schaefer, L. & Fegley, B. 2004, Icarus, 169, 216
  * Schaefer & Fegley (2010) Schaefer, L. & Fegley, B. 2010, Icarus, 208, 438
  * Schaefer et al. (2012) Schaefer, L., Lodders, K., & Fegley, B. 2012, ApJ, 755, 41
  * Solomatov (2000) Solomatov, V. 2000, Origin of the Earth and Moon, 1, 323
  * Solomatova & Caracas (2019) Solomatova, N. V. & Caracas, R. 2019, Journal of Geophysical Research: Solid Earth, 124, 11232
  * Solomatova & Caracas (2021) Solomatova, N. V. & Caracas, R. 2021, Science Advances, 7, eabj0406
  * Sossi et al. (2020) Sossi, P. A., Burnham, A. D., Badro, J., et al. 2020, Science advances, 6, eabd1387
  * Sossi et al. (2019) Sossi, P. A., Klemme, S., O’Neill, H. S. C., Berndt, J., & Moynier, F. 2019, Geochimica et Cosmochimica Acta, 260, 204
  * Sossi et al. (2020) Sossi, P. A., Moynier, F., Treilles, R., et al. 2020, Geochim. Cosmochim. Acta., 288, 316
  * Sossi et al. (2023) Sossi, P. A., Tollan, P. M., Badro, J., & Bower, D. J. 2023, Earth and Planetary Science Letters, 601, 117894
  * Swain et al. (2021) Swain, M. R., Estrela, R., Roudier, G. M., et al. 2021, AJ, 161, 213
  * Tsiaras et al. (2016) Tsiaras, A., Rocchetto, M., Waldmann, I. P., et al. 2016, ApJ, 820, 99
  * Valencia et al. (2010) Valencia, D., Ikoma, M., Guillot, T., & Nettelmann, N. 2010, A&A, 516, A20
  * van Buchem et al. (2022) van Buchem, C., Miguel, Y., Zilinskas, M., & van Westrenen, W. 2022, arXiv e-prints, Submitted, arXiv:2210.10463
  * Visscher & Fegley (2013) Visscher, C. & Fegley, Bruce, J. 2013, ApJ, 767, L12
  * Woitke et al. (2018) Woitke, P., Helling, C., Hunter, G. H., et al. 2018, A&A, 614, A1
  * Wolf et al. (2022) Wolf, A. S., Jäggi, N., Sossi, P. A., & Bower, D. J. 2022, arXiv e-prints, arXiv:2208.09582
  * Yoshida et al. (2022) Yoshida, T., Terada, N., Ikoma, M., & Kuramoto, K. 2022, ApJ, 934, 137
  * Young et al. (2019) Young, E., Shahar, A., Nimmo, F., et al. 2019, Icarus, 323, 1
  * Zieba et al. (2022) Zieba, S., Zilinskas, M., Kreidberg, L., et al. 2022, A&A, 664, A79
  * Zilinskas et al. (2022) Zilinskas, M., van Buchem, C. P. A., Miguel, Y., et al. 2022, A&A, 661, A126

##  Appendix A Composition of vapor in the absence of H

Figure 16 shows the vapor composition at equilibrium with a liquid with BSE
composition, computed with the procedure described in (Charnoz et al. 2021);
it compares very well with composition reported in Ito et al. (2015) Visscher
& Fegley (2013) van Buchem et al. (2022).

Figure 16: Molar fractions in the vapor at equilibrium with a magma with BSE
composition, and without hydrogen.

##  Appendix B Partial pressures in the hydrogenated atmosphere

Figure 17: Molar abundances of most abundant species versus total monoatomic
hydrogen pressure $P_{\mathrm{H}}^{0}$ for an atmosphere at equilibrium with a
magma ocean at T=1800 K. Figure 18: Molar abundances of most abundant species
versus total monoatomic hydrogen pressure $P_{\mathrm{H}}^{0}$ for an
atmosphere at equilibrium with a magma ocean at T=2000 K. Figure 19: Molar
abundances of most abundant species versus total monoatomic hydrogen pressure
$P_{\mathrm{H}}^{0}$ for an atmosphere at equilibrium with a magma ocean at
T=3000 K. Figure 20: Molar abundances of most abundant species versus total
monoatomic hydrogen pressure $P_{\mathrm{H}}^{0}$ for an atmosphere at
equilibrium with a magma ocean at T=3400 K.