11institutetext: Department of Astronomy and Astrophysics, Berlin Institute of
Technology, Hardenbergstr. 36, 10623 Berlin, Germany
11email<EMAIL_ADDRESS>22institutetext: Institute of Planetary
Research, German Aerospace Center, Rutherfordstr. 2, 12489 Berlin, Germany
33institutetext: Department of Earth Sciences, Free University of Berlin,
Malteserstr. 74-100, 12249 Berlin, Germany 44institutetext: Remote Sensing
Technology Institute, German Aerospace Center, 82234 Oberpfaffenhofen, Germany
55institutetext: Department of Planetary Sciences and Remote Sensing, Free
University of Berlin, Malteserstr. 74-100, 12249 Berlin, Germany

# Effect of mantle oxidation state and escape upon the evolution of Earth’s
magma ocean atmosphere

Nisha Katyal 11 Gianluigi Ortenzi 2233 John Lee Grenfell 22 Lena Noack 33
Frank Sohl 22 Mareike Godolt 1122 Antonio García Muñoz 11 Franz Schreier 44
Fabian Wunderlich 1122 Heike Rauer 112255

(Submitted on 2020/06/28 ; Accepted for publication on 2020/09/29)

###### Abstract

Context. The magma ocean period was a critical phase determining how Earth’s
atmosphere developed into habitability. However, there are major uncertainties
in the role of key processes such as outgassing from the planetary interior
and escape of species to space that play a major role in determining the
atmosphere of early Earth.

Aims. We investigate the effect of outgassing of various species and escape of
H2 for different mantle redox states upon the composition and evolution of the
atmosphere for the magma ocean period.

Methods. We included an important new atmosphere-interior coupling mechanism:
the redox evolution of the mantle, which strongly affects the outgassing of
species. We simulated the volatile outgassing and chemical speciation at the
surface for various redox states of the mantle by employing a C-H-O based
chemical speciation model combined with an interior outgassing model. We then
applied a line-by-line radiative transfer model to study the remote appearance
of the planet in terms of the infrared emission and transmission. Finally, we
used a parameterized diffusion-limited and XUV energy-driven atmospheric
escape model to calculate the loss of H2 to space.

Results. We have simulated the thermal emission and transmission spectra for
reduced and oxidized atmospheres present during the magma ocean period of
Earth. Reduced/thin atmospheres consisting of H2 in abundance emit more
radiation to space and have a larger effective height than oxidized/thick
atmospheres, which are abundant in H2O and CO2. We obtain that the outgassing
rates of H2 from the mantle into the atmosphere are a factor of ten times
higher than the rates of diffusion-limited escape to space. We estimate the
timescale of total mass loss of outgassed H2 via escape to be few tens of
million years, which is comparable to other studies.

Conclusions. Our work presents useful insight into the development of the
terrestrial atmosphere during the magma ocean period and provides input to
guide future studies that discuss exoplanetary interior compositions and their
possible links with atmospheric compositions that might be estimated from
observed infrared spectra by future missions.

###### Key Words.:

radiative transfer — methods: numerical — planets and satellites: terrestrial
planets — planets and satellites: atmospheres — infrared: planetary systems —
planets and satellites: interiors

##  1 Introduction

Understanding how the early atmosphere of Earth emerged into habitable
conditions is a central question not only for addressing our own origins, but
also for interpreting the fascinating new data in exoplanetary science (for
recent reviews, see, e.g., Jontof-Hutter 2019; Madhusudhan 2019). Magma oceans
(MOs) consisting of hot and molten silicates in the mantle exist during the
planetary accretion phase (Elkins-Tanton 2012) and facilitate the formation of
an atmosphere by outgassing of volatile species such as H2O, CO2, H2, CH4 ,
and CO . Thus, the MO period likely represents a key juncture on the pathway
to habitability because atmosphere-interior couplings (e.g., Elkins-Tanton
2008; Lebrun et al. 2013; Hamano et al. 2013; Schaefer et al. 2016)
established during this phase could have set the stage for subsequent
atmospheric evolution. The duration of the magma ocean phase is potentially
affected by the absence or presence of an atmosphere and the interior dynamics
of the mantle (e.g., Lebrun et al. 2013; Nikolaou et al. 2019) that affect the
thermal spectral evolution of the planet (e.g., Hamano et al. 2015; Katyal et
al. 2019). Other uncertainties such as the initial volatile inventory of the
mantle, for example, the amount of H2O stored in the mantle (Genda et al.
2017; Meech & Raymond 2019), and compositional constraints such as the mantle
oxygen fugacity (Hirschmann 2012; Schaefer & Fegley 2017) are also important
quantities that might critically affect the oxidation state of outgassed
volatiles and hence the subsequent climate and compositional evolution of the
atmosphere.

The budget and origin of volatiles in the mantle of Earth is one of the main
questions in geo-science (see, e.g., Meech & Raymond 2019). Recent studies
suggest quite diverse scenarios that range from a wet, late accretion to a dry
accretion to which water was added mostly during the “late veneer”, which is
proposed to have occurred roughly around 80-130 Myr after the isolation of
protoplanetary nebula at around 4.5 Gyr before present (Albarede et al. 2013).
Abundances of highly siderophile elements (HSE) and other volatile species
such as H2O suggest that volatile-rich material was added to Earth after its
core formed (Albarède 2009). Measurements of the abundances of water and
carbon in the bulk silicate Earth (BSE) show that the late veneer might indeed
have supplied 20% - 100% of the budget of the total hydrogen (H) and carbon
(C) in the BSE (Wang & Becker 2013). Some recent studies do not support the
late veneer as the main source of water on Earth (Dauphas 2017; Fischer-Gödde
& Kleine 2017; Hopp & Kleine 2018) and argue that water was likely added to
Earth already during the waning stages of accretion and thus was already
present during the late veneer. Interestingly, the reaction of terrestrial
water and Fe from the late veneer produced much hydrogen as a byproduct
(Zahnle et al. 2020). There is increasing isotopic evidence indicating that
the late veneer might also have supplied a significant amount of reducing
material onto the surface of Earth and into the atmosphere (Genda et al. 2017;
Greenwood et al. 2018; Zahnle et al. 2020).

The impact-degassing studies by Schaefer & Fegley (2007); Hashimoto et al.
(2007); Schaefer & Fegley (2010) and Fegley & Schaefer (2012) also in general
favored the outgassing of reduced gases upon impacts by certain types of
reduced meteoritic materials such as ordinary and enstatite chondrites.
Furthermore, changes in chondrites delivery rate are likely to affect the
redox state of the mantle and hence the atmospheric composition and amount, as
suggested recently by Schaefer & Fegley (2017). Their work also showed that
the variation in oxygen fugacities for mixtures of primitive meteoritic
material could lead to atmospheres ranging from highly oxidizing to highly
reducing ones (also see Hirschmann 2012).

The redox state of the minerals and melts is related to the oxygen fugacity
fO2 of the system that is equivalent to the partial pressure of the gas or the
availability of oxygen (Gaillard et al. 2015). This quantity is poorly
constrained for the early Earth (Frost & McCammon 2008). The current
understanding is that before the formation of the Earth core $\sim$4.56 Gyr
before present (Stevenson 1983), the lower fO2 of the upper mantle was
approximately three log units below the reducing iron-wüstite (IW) buffer,
that is, IW-3 and it evolved to a higher fugacity value that was approximately
equivalent to the current mantle of Earth, which resembles the quartz-
fayalite-magnetite (QFM) buffer, that is, IW+3.5 (Wood et al. 1990; O’Neill
1991). Direct geological evidence regarding the redox state of the mantle is
sparse, and it is not clear when the Earth may have been oxidized during the
Hadean (Trail et al. 2011). However, the geological record mainly suggests
that the upper mantle was initially reduced and became progressively oxidized
between 4.6 and 3.9 Gyr before present (Kasting et al. 1993; Delano 2001;
Schaefer & Fegley 2017), with a possible later further increase in redox state
by $\approx$ 1.3 at the end of the Archean (Aulbach & Stagno 2016).

Previous interior modeling studies investigating atmospheres at and around the
time of the magma ocean (e.g., Lebrun et al. 2013; Hier-Majumder & Hirschmann
2017; Nikolaou et al. 2019) have considered outgassing of volatiles such as
H2O and CO2, which means that they considered the mantle to be constantly
oxidized. These studies suggested outgassed pressures of H2O and CO2 ranging
from a few tens to hundreds of bar depending upon uncertainties, for instance,
in the initial volatile content, the timing of the MO, and internal properties
of the mantle. Recent modeling studies have now started to investigate the
effect of the redox state on the outgassing. Pahlevan et al. (2019)
investigated the effect of fO2 on the outgassing and atmosphere losses and
provided evidence for an early oxidation of silicate Earth using the D/H of
the oceans. Most recently, Ortenzi et al. (2020) calculated the expected
outgassing rates for the reduced and oxidized rocky planets and focused on
exploring the observational constraints for the atmospheric (and interior)
redox state of exoplanets.

Hydrodynamic escape of H2 during this time ($\sim$4.5 Gyr before present) was
likely not sufficient to desiccate the planet (Kasting et al. 1993), so that
conditions remained wet after the Moon-forming impact and crystallization of
the final MO (Lammer et al. 2018). Hydrogen would therefore have been a
sufficiently major component of an accretionary steam-based atmosphere during
the MO phase. However, hydrogen evolution around the time of the MO is poorly
understood because of the uncertainties in accretion, outgassing, ingassing,
and escape (see, e.g., Tian 2015). Several studies have considered the energy-
limited hydrodynamic escape of hydrogen from the early atmosphere (Archean) of
Earth (Tian et al. 2005; Kuramoto et al. 2013; Zahnle et al. 2019) and
calculated somewhat differing results based on including or excluding the
effect of H2 diffusive flux on the gas-density profile and processes such as
radiative cooling and thermal conduction that these escape models treat, for
example. Johnstone et al. (2019) have also discussed uncertainties in the
energy-limited mass-loss formula when applied to the early atmospheres. For
rocky planets around M dwarfs, large amounts of water could be lost depending
on the uncertainties in the stellar luminosity, for instance (e.g., Tian & Ida
2015; Schaefer et al. 2016).

We apply a coupled suite of interior and atmospheric models to investigate
outgassing and escape during the MO period and study the effect of varying the
mantle fugacity on the atmospheric evolution. We also calculate theoretical
atmospheric spectra that serve as a link for observations with early Earth-
sized planets. In Section 2 we present the volatile speciation model and the
atmospheric model. Section 3 presents the scenarios we adopted. Section 4
provides results of the effect of the redox state of the mantle on the
infrared emission and transmission spectra. The interplay between outgassing
and atmospheric loss of H2 is also presented in this section. In Section 5 we
discuss our findings. Finally, we provide a conclusion of our work in Section
6.

##  2 Methods and models

###  2.1 C-H-O based volatile speciation model

The outgassing and volatile chemical speciation were simulated following the
equilibrium and mass balance method (French 1966; Holloway 1981; Fegley 2013;
Gaillard & Scaillet 2014; Schaefer & Fegley 2017) that has recently been
presented in detail by Ortenzi et al. (2020). We calculated the outgassed
composition of the volatiles considering a broad range of temperatures,
pressures, and redox states. The four common petrological buffers used in the
literature and their mantle oxidation states are

$\ch{2Fe+SiO2+O2<=>Fe2SiO4},\hfill\rm{QIF}$ (1)

$\ch{2Fe+O2<=>2FeO},\hfill\rm{IW}$ (2)

$\ch{3Fe2SiO4+O2<=>2Fe3O4+3SiO2},\hfill\rm{QFM}$ (3)
$\ch{Ni+1/2O2<=>NiO}.\hfill\rm{NiNiO}$ (4)

Quartz-iron-fayalite (QIF) and IW buffers represent reduced conditions, and
quartz-fayalite-magnetite (QFM) and nickel-nickel-oxide (NiNiO) describe the
oxidizing redox states. The oxidation state (oxygen fugacity) for these
buffers was calculated following the parameterization from Holloway et al.
(1992),

$\log_{10}\rm f\ch{O2}=A-B/T+C(P-1)/T\;+Z,$ (5)

where the pressure (P) is in bars, the temperature (T) in is Kelvin, and the
parameters A, B and C are defined in Table 1. Z is a positive or negative
number denoting the deviation from the fugacity values with respect to the
buffers, as stated in the table. We mainly used the IW buffer and chose Z to
be -4 (highly reducing), 0 (reducing), and 4 (oxidizing) in order to
investigate the effect of a range of redox mantle states on outgassing.

Table 1: Data collected from Holloway et al. (1992) to obtain the oxygen fugacity of the mantle buffers as listed.  Buffer | A | B | C
---|---|---|---
QIFaa$a$Quartz-iron-fayalite | 7.679 | 29673 | 0.05
IWbb$b$iron-wüstite | 6.899 | 27714 | 0.05
QFMcc$c$quartz-fayalite-magnetite | 8.555 | 24014 | 0.092
NiNiOdd$d$nickel-nickel-oxide | 8.951 | 24556 | 0.046

By simulating the oxidation state, that is, the oxygen fugacity fO2 of the
system, we are able to simulate the gas chemical speciation via the following
equilibria:

$\ch{CO+1/2O2<=>CO2},\;\rm and$ (6)

$\ch{2H2+O2<=>2H2O}.$ (7)

To calculate the ratio between the carbon species (Eq. 6), we considered the
equilibrium constant $K_{1}$ for the equilibrium as

$K_{1}=\exp\left(\frac{-\Delta_{r}G^{0}_{1}}{RT}\right)=\frac{X_{\ch{CO2}}}{X_{\ch{CO}}}\frac{1}{\rm
f\ch{O2}^{1/2}},$ (8)

where $R$ is the universal gas constant (8.314 J K-1 mol-1), $T$ is the
temperature of the outgassed material in Kelvin, and $\Delta_{r}G^{0}_{1}$ is
the Gibbs free energy of the reaction in Eq. (6), where

$\Delta_{r}G^{0}_{1}=\Delta_{f}G^{0}_{\ch{CO2}}-\Delta_{f}G^{0}_{\ch{CO}}.$
(9)

Substituting $\Delta_{r}G^{0}_{1}$ and the calculated fO2 in Eq. (6) gives us
all the necessary parameters needed to calculate the carbon species. For Eq.
(7), the Gibbs free energy of reaction $\Delta_{r}G^{0}_{2}$ is related only
to the Gibbs free energy of formation of water ($\Delta_{f}G^{0}_{\ch{H2O}}$),

$\Delta_{r}G^{0}_{2}=2\Delta_{f}G^{0}_{\ch{H2O}}.$ (10)

The values of $\Delta_{f}G^{0}$ for the different species were calculated
following the example of Fegley (2013) and compared to the literature (Chase
1999).

Similarly, the abundances of H2 and H2O Eq. (7) are related to the equilibrium
constant $K_{2}$ and the fugacity fO2 via

$\left(\frac{X_{\ch{H2O}}}{X_{\ch{H2}}}\right)^{2}=K_{2}\;\rm f\ch{O2},$ (11)

where $X_{\ch{H2O}}$ and ${X_{\ch{H2}}}$ are the mole fractions of H2O and H2
and are related to the partial pressure of each of the species. The rate of
outgassing of H2O can therefore be related to the rate of outgassing of H2 as

$r_{\ch{H2O}}=R_{1}\,\ r_{\ch{H2}},$ (12)

where $R_{1}=(K_{2}\;\rm f\ch{O2})^{0.5}$, $r_{\ch{H2}}$ , and $r_{\ch{H2O}}$
are the outgassing rates of H2 and H2O in units of m-2 s-1, respectively.

###  2.2 Coupled interior-atmospheric evolution

Figure 1: Schematic showing coupled interior-atmosphere exchange via
outgassing of reduced species such as H2 and CO (orange) for a reduced mantle
(e.g., IW buffer) and oxidized species such as H2O and CO2 (blue) for an
oxidized mantle (e.g., QFM buffer) as the magma ocean solidifies. The H2 in
the atmosphere is lost to space via escape.

A schematic for interior-atmosphere coupling is presented in Figure 1. We
started by taking the variation in surface pressure $p_{\rm s}$, surface
temperature $T_{\rm s}$ , and volatile abundances $f^{\rm init}_{\ch{H2O}}$
and $f^{\rm init}_{\ch{CO2}}$ from the interior model output of Nikolaou et
al. (2019), that is, a mantle temperature of 4000 K and an initial mantle
composition of $X_{\ch{H2O}}=0.05$ weight percent (wt%) (550 ppm) and
$X_{\ch{CO2}}=0.01$ wt% (130 ppm) as the input. We then applied the volatile
speciation model from Ortenzi et al. (2020) as described in Sect. 2.1 to
obtain the final outgassed molar abundance of the species as a function of
oxygen fugacity fO2 relative to the given mineral buffer (IW), as shown in
Figure 2. This figure shows that for a reduced mantle (between IW and IW-4)
and increasing input H/C from top to bottom, we obtain atmospheres that are
rich in CO, CO+H2 mixtures, and H2. On the other hand, for a more oxidized
mantle with typically higher fugacity values (between IW and IW+4) and
increasing input H/C from top to bottom (Fig. 2), we obtain an atmosphere that
is rich in CO2, H2O+CO2 , and H2O. The bottom panel of Fig. 2 corresponds to
an initially assumed 100% H2O which results in a almost pure H2 atmosphere for
a reduced mantle and a pure H2O atmosphere for an oxidized mantle.

Figure 2: Volatile chemical speciation in terms of outgassed mole fraction vs.
oxygen fugacity of the mantle. The x-axis indicates the oxygen fugacity range
in logarithmic units relative to the IW buffer. For reference (in log units),
buffers QIF = IW-1, QFM = IW+3.8, and NiNiO = IW+4.2 approximately (Wood et
al. 1990). From top to bottom, we show the BOM, active MO phase, EOM and a
steam-atmosphere phase that are characterized by different initial H/C between
hydrogen and carbon. The corresponding input values of the initial volume
mixing ratio of volatiles, surface pressures $P_{\rm s}$ , and temperatures
$T_{\rm s}$ (Nikolaou et al. 2019) for the speciation model are indicated at
the top of each panel.

The initial outgassed volatiles H2O and CO2 taken from Nikolaou et al. (2019)
with a mean molar mass $\mu_{\rm v}$ react with the melt and result in a
different composition of the species outgassed to the atmosphere as calculated
from the speciation model (Section 2.1). When the number of moles of the
species C-H-O is kept constant in the speciation model, a different atmosphere
with a new mean molar mass $\mu_{\rm atm}$ results, which has a different
pressure at the bottom (surface) of the atmosphere, which we now call $p_{\rm
boA}$. This pressure is calculated based on the volatile mass balance (Bower
et al. 2019),

$p_{\rm boA}=p_{s}\;\left(\frac{\mu_{\rm atm}}{\mu_{v}}\right),$ (13)

where $p_{s}$ is the initial (surface) pressure of the volatiles calculated as
a mono-gas atmosphere.

Starting from the mass of the outgassed species H2O given by $M=P\mbox{*}A/g$,
we obtained the following equation for the initial H2O outgassing rate from
the interior.

$R_{\ch{H2O}}^{\rm
init}=\frac{1}{A}\frac{dN}{dt}=\frac{N_{A}\;1000}{g\;m_{\ch{H2O}}}\left(\frac{dP}{dt}\right)\rm\,\
molecules\,\ m^{-2}s^{-1},$ (14)

where $dN$ is the number of H2O molecules outgassed in a time interval $dt$,
$N_{A}$ is Avogadro’s number (6.022 $\times 10^{23}$), $A$ is the area of the
Earth surface ($5.1\times 10^{14}\rm\,\ m^{2}$), m${}_{\ch{H2O}}$ is the
molecular weight (18 g/mol), $g=9.81$ m/s2 is the surface gravity, and $dp$ is
the difference in outgassed pressure at a time interval $dt$ taken from
coupled model result of Katyal et al. (2019) and Nikolaou et al. (2019).

To obtain the outgassing rate of H2, we used Eq. (12) and Eq. (14). To account
for the redox variation of the mantle and the volatile mass balance, we
assumed that the initial rate of H2O outgassing $R^{\rm init}_{\ch{H2O}}$ in
Eq. (14) is the total outgassing rate of combined (H2+H2O). Therefore, $R^{\rm
init}_{\ch{H2O}}=r_{\ch{H2}}+r_{\ch{H2O}}$. Inserting Eq. (12) here, we obtain
the rate of H2 outgassing as

$r_{\ch{H2}}=\frac{R^{\rm init}_{\ch{H2O}}}{1+R_{1}}\rm\,\ molecules\,\
m^{-2}s^{-1}.$ (15)

###  2.3 Convective lapse rate

The convection takes into account the heat transport by adiabatic expansion of
a mixture of a nonideal condensable gas H2O (because the critical point of
water at $T_{\rm c}=647$ K and $p_{\rm c}=220$ bar is close to the T-p range
considered in this study), denoted with subscript $c,$ and ideal,
noncondensable gases (mixture of CO2, CO, and H2) denoted with the subscript
$d$.

For a saturated water vapor atmosphere, the pressure exerted by the
condensable is given by the Clausius-Clapeyron equation,

$\frac{dp_{s}}{dT}=\frac{L_{c}(T)}{RT^{2}}p_{s},$ (16)

where $p_{s}$ is the saturation vapor pressure and $L_{c}(T)$ is the latent
heat of vaporization of the condensable, that is, H2O in this case. Inserting
this expression for $d\ln p_{c}/d\ln T$ into the formula given in
Pierrehumbert (2010) leads to the pseudoadiabatic slope as used in this study,
which is given by (see also Ding & Pierrehumbert 2016)

$\frac{d\ln P}{d\ln T}=\frac{p_{\rm s}}{P}\frac{L_{c}(T)}{R_{c}T}+\
\frac{p_{d}}{P}\frac{c_{pd}}{R_{d}}\frac{1+\left(\frac{c_{pc}}{c_{pd}}+\left(\frac{L}{R_{c}T}-1\right)\frac{L}{c_{pd}T}\right)r_{\rm
sat}}{1+\frac{L}{R_{d}T}r_{\rm sat}},$ (17)

where $c_{pd}$ and $c_{pc}$ are the specific heat capacities of the
noncondensable and condensable species, respectively. $r_{\rm sat}$ is the
saturation mass mixing ratio, given as (Catling & Kasting 2017)

$r_{\rm sat}=\epsilon f_{\rm sat}(\ch{H2O})=\epsilon\frac{p_{sat}}{P}.$ (18)

Here, $\epsilon$ is the ratio of mass of condensable and noncondensable
species, $P$ is the total pressure, and $f_{\rm sat}$ is the volume mixing
ratio of condensable at saturation. Eq. (17) reduces to Eq. (A13) of Kasting
(1988) at $r_{\rm sat}\to 0$. In this case, the atmosphere is dry and the
temperature profile is obtained by the dry adiabatic lapse rate given by (from
Eq. 17)

$\frac{d\ln p_{d}}{d\ln T}=\frac{c_{pd}(T)}{R_{d}},$ (19)

where $R_{d}$ is the gas constant for the noncondensable or the dry component
of the mixture and is obtained by Eq. (40) of Appendix A. When more than one
noncondensable species is present, the mean molecular weight of the mixture of
noncondensable is accounted for.

At temperatures that are high enough and for a condensable species reservoir
much larger than the atmospheric pressure exerted by the dry gases such that
$r_{\rm sat}\gg 1$, $P\to p_{\rm sat}$ and $p_{d}/P\to 0$, Eq. (17) reduces to
the Clausius-Clapeyron relation (Pierrehumbert 2010), which is valid for an
ideal gas equation. The temperature dependence of the latent heat of
vaporization $L_{c}(T)$ and saturation vapor pressure $p_{s}$ of water cannot
be neglected. For this, we used one of the most accurate formulations for
calculating the saturation vapor pressure of water known as the Tetens formula
(Tetens 1930). This formula provides a very good approximation for the
saturation vapor pressure of water with an error lower than $\sim 1$% (Huang
2018) in the temperature range 0-100 ∘C and is given as follows:

$\displaystyle p_{s}(T)=\begin{cases}p_{s}^{\rm
ref}\exp\left(\frac{17.625T}{T+238.3}\right),&\text{if }T\geq 0\\\ p_{s}^{\rm
ref}\exp\left(\frac{21.875T}{T+265.5}\right),&\text{otherwise}.\end{cases}$
(20)

Here $p_{s}^{ref}=610.78\,\rm Pa$ is the reference water vapor pressure at the
triple point, and $T$ is given in ∘C and equal to $T$\- 273 for $T$ in units
of Kelvin.

Next the term $c_{pc}$ in Eq. (17) is the temperature-dependent specific heat
capacity of water adapted from Wagner & Pruß (2002) and applicable to high-
temperatures conditions given by (also see Katyal et al. 2019)

$\frac{c_{pc}(T)}{R_{c}}=1+n_{3}+\sum_{i=4}^{8}n_{i}\frac{(\gamma_{i}\tau)^{2}\exp(-\gamma_{i}\tau)}{[1-\exp(-\gamma_{i}\tau)]^{2}},$
(21)

where $\tau=T_{\text{c}}/T$ and $T_{\text{c}}$ is the critical temperature
(647 K). The values of the coefficients $n_{i}$ and $\gamma_{i}$ are taken
from Table 6.1 of Wagner & Pruß (2002). The total heat capacity of the dry
mixture $c_{pd}$ is calculated as a linear combination of individual heat
capacities of the noncondensable species weighted by their volume mixing
ratios. The total $c_{p}$ is thus the sum of specific heat capacity of moist
and dry component, weighted by the respective volume mixing ratios, and is
given by

$c_{p}(T)=\frac{\sum_{i}x_{di}c_{pdi}(T)+\sum_{i}x_{vi}c_{pci}(T)}{\sum_{i}x_{di}+\sum_{i}x_{ci}},$
(22)

which is valid for temperatures in the range 200-2000 K. The denominator in
this equation is equal to 1 for our case. The specific heat capacity of dry
gases such as CO2, CO, and H2 is derived from the Shomate equation (Parks &
Shomate 1940) in units of $\rm kJ/kg\;K$,

$c_{p}=A+BT+CT^{2}+DT^{3}+E/T^{2},$ (23)

where the coefficients $A,B,C,D,\text{and }E$ are taken from Chase (1998) and
$T=\rm Temperature/1000$ K for the temperature range valid between 298-6000 K.
For temperatures below 298 K, the specific heat capacities of CO, CO2 , and H2
are taken from analytical expressions based on a least-squares fit of data
from Lide (2000). The specific heat capacity as a function of temperature in
the range 200-3000 K for the four gases we considered is shown in Fig. 3.

Figure 3: Specific heat capacities as a function of temperature for volatile
species H2O, CO, and CO2 on the left x-axis and H2 on the right x-axis as
obtained from the Shomate equation described in the main text.

In order to obtain the temperature profiles, we followed the same procedure as
in Katyal et al. (2019) , that is, for surface temperatures $T_{s}>647$ K, we
used the dry adiabatic lapse rate (Eq. 19) by assuming a fixed surface
temperature and pressure at the bottom of the atmosphere (Eq. 13). The
temperature profile follows the moist adiabatic lapse rate (second term in Eq.
17) when the dry adiabatic lapse rate intersects with the moist adiabatic
lapse rate. The related work of Katyal et al. (2019) calculated temperature
profiles with only one gas species, that is, H2O, in the atmosphere. We have
additionally included CO2, CO, and H2 calculated from the volatile speciation
model described in Sect. 2.1. Our model accounts for the convective processes
in the atmosphere up to an altitude corresponding to the top of the atmosphere
pressure $p_{\rm ToA}=0.1$ Pa (similar to Pluriel et al. (2019)). The
atmospheric layers are vertically spaced in log pressure coordinates. When the
temperatures in the dry/moist adiabat fall below 200 K, the temperature
structure in our calculations follows an isothermal profile with a constant
tropospheric temperature of 200 K.

###  2.4 Radiative transfer code

The generic atmospheric radiation line-by-line infrared code (GARLIC; Schreier
et al. 2014) was used to compute thermal emission and transmission spectra
from 10 to 30,000 cm-1 with p,T and the composition of the atmosphere as the
input. A detailed description of the radiative transfer code is also available
in Katyal et al. (2019). For the verification and validation of GARLIC, see
Schreier et al. (2018a, b). The absorption coefficients were calculated line-
by-line (lbl) for H2O, CO2, H2, and CO from the HITRAN2016 database (Gordon et
al. 2017). Table $a$ shows the list of various molecules and sources for the
continuum available and relevant for this study.

Table 2: List of continua from the HITRAN CIA listaa$a$www.hitran.org/cia/ (Karman et al. 2019) and other sources.  Molecule | Continuum
---|---
H2 – H2 | Abel et al. (2011)
CO2 – CO2 | Baranov (2018)
H2O self | MT_CKD_2.5bb$b$http://rtweb.aer.com/continuum_frame.html (Mlawer et al. 2012)
H2O foreign | MT_CKD_2.5bb$b$http://rtweb.aer.com/continuum_frame.html (Mlawer et al. 2012)
CO2 foreign | MT_CKD_2.5bb$b$http://rtweb.aer.com/continuum_frame.html (Mlawer et al. 2012)
Table 3: List of measured reference Rayleigh cross sections $\sigma_{0,i}$ and central wavelengths $\lambda_{0,i}$ for the molecules.  Molecule |  $\sigma_{0,i}$ [cm2] |  $\lambda_{0,i}$ [$\mu$m] | Reference
---|---|---|---
CO2 | $12.4\cdot 10^{-27}$ | 0.53224 | Sneep & Ubachs (2005)
CO | $6.19\cdot 10^{-27}$ | 0.53224 | Sneep & Ubachs (2005)
H2 | $1.17\cdot 10^{-27}$ | 0.5145 | Shardanand & Rao (1977)

Rayleigh scattering by molecule H2O was obtained using the formalism described
in Murphy (1977) (also see Scheucher et al. 2020). For other molecules such as
CO2, CO, and H2, we have used the Rayleigh cross sections using the formalisms
of Sneep & Ubachs (2005) and Shardanand & Rao (1977) across the whole spectrum
in units of $\rm cm^{2}$/molecules expressed as

$\sigma_{\rm{Rayleigh},i}(\lambda)=\sigma_{0,i}\;\left(\frac{\lambda_{0,i}}{\lambda}\right)^{\alpha}.$
(24)

Here, $\alpha=4$, $\sigma_{0,i}$ and $\lambda_{0,i}$ are the reference cross
sections and wavelengths for the various molecules used in this study that are
listed in Table 3.

To mimic optically thick clouds in our simulations (Section 5.4), we assumed
clouds were present throughout the atmosphere (Kaltenegger & Traub 2009) with
the uppermost cloud deck coinciding with the lowermost $T=200$ K altitude. The
cloud-scattering cross section is expressed in a similar formalism as Eq.(24),
but with $\alpha=0$. This therefore acts as a scaling term with no wavelength
dependence. For reference, the terrestrial aerosol-scattering cross section is
about $10^{-27}\rm\,cm^{2}$ (Moran et al. 2018). That study (Moran et al.
2018) ruled out haze-scattering cross sections smaller than 9 $\times
10^{-25}$ cm2 to 3$\sigma$ and 3 $\times$ 10-23 cm2 to 1$\sigma$. Therefore we
assumed a cloud cross section $\sigma_{\rm
cloud}\equiv\sigma_{0_{i}}=6\times\rm 10^{-25}\rm\,cm^{2}$ , that is, $6\times
10^{2}$ times larger than the Earth-scattering cross section in our
simulations as presented in Section 5.4.

The aerosol optical depth integrated along the path is then given by

$\tau_{\rm aerosol}=\sigma_{\rm aerosol}\cdot N_{z},$ (25)

where $N_{z}$ is the column density of the molecular species in units of $\rm
molecules\,cm^{-2}$ (see also Kaltenegger & Traub 2009; Yan et al. 2015).

As a first step, we performed convective-radiative calculations without
coupled chemistry similar to previous modeling studies of the magma ocean
period. For future work, we plan to include the effect of chemical equilibrium
in the lower atmospheric layers by applying the self-consistent convection-
climate-photochemistry column model (Scheucher et al. 2020; Wunderlich et al.
2020) to the magma ocean period in order to study the effect upon transmission
and thermal emission spectra and atmospheric escape, for example.

###  2.5 Transmission spectra

The volatile chemical speciation model along with the atmospheric model
described in Sections 2.1-2.4 applicable to the early Earth magma ocean phase
can be used to characterize exoplanets in the magma ocean phase (see Figure 12
of Nikolaou et al. 2019), for instance, to detect the spectral features and
obtain the vertical extent of the atmosphere observationally. We used a well-
established diagnostic tool, transmission spectroscopy, to probe the molecular
composition of exoplanetary atmospheres. It provides the effective height of
the atmosphere as a function of wavelength (see, e.g., Benneke & Seager 2012).

In GARLIC, the theoretical transmission spectrum is expressed as the effective
height of the atmosphere given as follows (Kaltenegger & Traub 2009; Schreier
et al. 2018b; Katyal et al. 2019):

$h(\lambda)=\int_{0}^{\infty}A(z,\lambda)\;dz=\sum_{i}A_{i}(\lambda)\;\Delta
h_{i},$ (26)

where $A_{i}=1-\mathcal{T}_{i}$ is the absorption along the $i$th incident ray
with a transmission $\mathcal{T}_{i}$ that traverses through the
(exo)planetary atmosphere tangentially at a height $h_{i}$ and continues to
travel to the observer at Earth.

The atmospheric transit depth, $t_{\rm atm}$ is given by (Wunderlich et al.
2020)

$t_{\rm
atm}(\lambda)=\frac{(R_{p}+h(\lambda))^{2}}{R_{s}^{2}}-\frac{R_{p}^{2}}{R_{s}^{2}},$
(27)

where $R_{\rm p}$ is the planetary radius, and $R_{\rm s}$ is the stellar
radius.

###  2.6 Atmospheric escape

For the atmospheric escape of H2, we considered hydrodynamic fluxes of
escaping H2, possibly diffusion-limited at the homopause $\sim 100\;\rm km$,
in the presence of a static background atmosphere, similar to the approach of
Kuramoto et al. (2013); Zahnle & Catling (2017) and Zahnle et al. (2019). At
higher levels above the homopause, where the escaping H2 gas is accelerated as
a result of the absorption of stellar high-energy photons, hydrogen exists
mainly in atomic form. For the quantitative implementation of the escape
fluxes, we relied on Zahnle et al. (2019), who solved the hydrodynamic,
possibly diffusion-limited, problem in the atmosphere. In particular, we used
their Eq. 3, which parameterizes the transition from energy-limited escape to
diffusion-limited escape. Diffusion-limited escape becomes effective when the
atmosphere is more strongly irradiated and the escaping hydrogen cannot be
replenished from below the homopause.

The diffusion-limited flux $\phi_{\rm dl}$ generally provides an upper limit
to the escape of lighter species such as hydrogen in a background heavier and
stationary gas. The escape rate for H2 can be written as (Hunten 1973)

$\phi_{dl}=b_{aj}\;f_{\rm
tot}(\ch{H2})\left(\frac{1}{H_{a}}-\frac{1}{H_{\ch{H2}}}\right),$ (28)

where $b_{\rm aj}$ is the binary diffusion coefficient between the background
heavier gas $a$ such as CO2, CO and H2O and the escaping gas H2. Here, $f_{\rm
tot}\rm(H_{2})$ is the hydrogen volume mixing ratio (VMR) at the homopause.
$H_{a}$ and $H_{\ch{H2}}$ represent the unperturbed scale heights ($=kT/mg$)
of the heavier gas $a$ and the escaping gas H2 at the homopause, respectively.
We consider H2 as the dominant H-bearing species, that is, $f_{\rm tot}\equiv
f(\ch{H2})$. The binary diffusion coefficients $b_{aj}$ for H2 in CO and CO2
are roughly similar and taken to be $3\times 10^{21}\rm\,\ m^{-2}\,\ s^{-1}$
(Marrero & Mason 1972). We validated the escape rate obtained for the current
Earth using Eq. (28). The total hydrogen VMR was taken to be $f_{\rm
tot}\rm(H_{2})=7.15$ ppm with the background-dominant gas as N2. The diffusion
coefficient between the H2 and N2, $b_{aj}$ of $1.7\times 10^{21}\rm\,\
m^{-2}\,\ s^{-1}$ was taken from Hunten (1973). We verified that the
diffusion-limited escape of H2 is $\sim 1.5\times 10^{12}\,\ \rm\ch{H2}\,\ \rm
molecules\,\ \rm m^{-2}\,\ \rm s^{-1}$ , as also reported by Hunten & Donahue
(1976). In terms of the loss of H-atoms to space, this is equivalent to a
mass-loss rate of $\sim$3 kg/s.

The energy-limited flux was obtained as the ratio of stellar XUV energy
incident upon the planet and the energy required to lift a given mass out of
the Earth potential well and into the space. The mass-loss rate of hydrogen
using the energy-limited formula (Watson et al. 1981) for hydrodynamic escape
(also see Koskinen et al. 2014; Hamano et al. 2015) is given by

$\dot{M}_{\rm el}=\frac{\pi\epsilon R_{p}R_{\rm XUV}^{2}F_{\rm
XUV}(t)}{GM_{p}}\;\rm g\,\ s^{-1},$ (29)

where $R_{\rm XUV}$ is the radial distance at which the XUV energy from the
star is deposited, and satisfies $R_{\rm XUV}$ ¿ $R_{p}$, and $\epsilon$ is
the heating efficiency ¡ 1. Upon assuming $R_{\rm XUV}\equiv R_{p}$ (Zhang
2020), this equation can be written as

$\dot{M}_{\rm el}=\frac{\pi\epsilon R_{p}^{3}F_{\rm XUV}(t)}{GM_{p}}\;\rm g\,\
s^{-1}.$ (30)

Here, $R_{\rm p}$ is the Earth radius (6.4 $\times$ 108 cm), $M$ is the mass
of Earth (5.4 $\times$ 1027 gm), $G$ is the gravitational constant (6.67
$\times$ 10-8 cm3 g${}^{-1}\rm s^{-2}$ ), $F_{\mathrm{XUV}}(t)$ is the time-
evolving flux from the host star at XUV wavelengths at 1 AU obtained by the
relation $F_{\mathrm{XUV}}(t)=5(4.5/t)^{1.24}$ in units of
$\rm\;erg\;cm^{-2}s^{-1}$, where $t$ is the age of the Sun in billion years
(Zahnle et al. 2019), and $\epsilon$ is taken to be 0.5 to be consistent with
Zahnle et al. (2019). Owing to the fast temporal evolution in the XUV output
of the young Sun (related to whether it was a fast, moderate, or slow
rotator), the $S$ value defined as $S(t)\equiv
F_{\mathrm{XUV}}/F_{\mathrm{XUV}\odot}$ can range between 10 and 100 during
the Hadean (Tu et al. 2015; Lammer et al. 2018). The lifetime of H2
atmospheres on early Earth is calculated as the ratio of mass-loss rate (Eq.
30) to the total mass of H2 in grams, $m_{\ch{H2}}$ , in the atmosphere as

$t\sim\left[\frac{G}{\epsilon\pi}\right]\left(\frac{M_{p}}{R_{p}^{3}}\right)\frac{m_{\ch{H2}}}{F_{\rm
XUV}}.$ (31)

Zahnle et al. (2019) have shown that the diffusion-limited escape applies to
conditions of higher levels of irradiation, whereas energy-limited escape may
be a better approximation for lower levels of irradiation. Building upon their
findings, we assumed that the flux is given by the expression

$\phi_{\ch{H2}}=\frac{1\times 10^{12}f_{\ch{H2}}S}{\sqrt{1+0.006S^{2}}}\,\ \rm
molecules\,\ cm^{-2}s^{-1},$ (32)

which is a good fit to their figure 5 if the VMR of H2 is lower than 0.2.

The hydrogen mass-loss $dM$ (in grams) due to escape at a particular time step
$dt$ is approximated as follows:

$dM=4\pi R_{p}^{2}\;\phi_{\ch{H2}}m_{\ch{H2}}\;dt.$ (33)

Here, $m_{\ch{H2}}$ is the mass of H2 in grams per molecule, and
$\phi_{\ch{H2}}$ is the escaping flux from Eq. (28) or Eq. (32).

The actual radius contribution in Eq. (30) and Eq. (33) comes from the sum of
the radius of the planet and height of the exobase, that is,
($R_{p}$+$H_{e}$). The height of the exobase, $H_{e}$ , is given by the
altitude level where the Lyman-edge opacity of hydrogen becomes 1. For the
scenario presented in Section 4.6, we obtain that ($R_{p}$+$H_{e})\sim R_{p}$
(calculation not shown here).

##  3 Volatile outgassing scenarios

We used three main scenarios in this work. Scenario 1 is for a surface
temperature of $3300\rm K$ and surface pressure of 76.7 bar, that is, at the
beginning of the magma ocean (BOM) period, as shown in Figure 2 (top) and
Table 4. Scenario 2 is similar to scenario 1, but for a surface temperature of
1650 K and pressure of 395 bar, that is, at the end of the magma ocean (EOM)
period, as shown in Figure 2 and Table 5. The conditions where the MO phase
ends are $T_{\rm s}=1650$ K at a solidification timescale of $\sim$ 1 Myr
according to the coupled interior-atmospheric model results of Katyal et al.
(2019) and Nikolaou et al. (2019). Previously, a coupled atmospheric-interior
model by Salvador et al. (2017), which was applied around the MO phase,
delivered P-T conditions with a similar H2O-CO2 composition as this study.

Scenario 3 is for representative cases taken from the study of Pluriel et al.
(2019), as shown in Table 6 referring to an H2O dominated and a CO2 dominated
atmosphere. Scenarios 1 and 2 cover a representative range of P, T, and mantle
oxidation state during the MO period. The buffers in the three scenarios were
chosen to represent the mantle redox state, ranging from reducing to
oxidizing.

Our choice of three scenarios (with varying initial volatile abundances of H2O
and CO2) and varying mantle fugacity, hence leading to different atmospheric
compositions, is motivated by previous works in the literature that also
explored the effect of composition by varying key species such as H2, CO, CO2,
H2O for magma ocean studies (Katyal et al. 2019; Nikolaou et al. 2019; Bower
et al. 2019), Venus-type planets with CO2 dominated atmosphere (Hamano et al.
2013; Pluriel et al. 2019), pure steam-dominated atmosphere (Katyal et al.
2019; Schaefer et al. 2016; Hamano et al. 2015), and the effect of adding H2
to the atmosphere of early Mars (Ramirez et al. 2014).

Table 4: Scenario 1 (BOM) for the three cases (1.1, 1.2, and 1.3). Columns 2 and 3 show the initial assumed mole fractions arising from H2O and CO2 outgassing, which is the input to the speciation model. The assumed buffer values for strongly reducing (IW-4), reducing (IW), and highly oxidizing (IW+4) have been used to study the effect of speciation under these conditions. The four columns on the right shows final outgassed species from the speciation model. For these scenarios, the surface p, T setting is fixed to be $T_{s}=3300$ K and $P_{\rm s}=76.7$ bar. The $p_{\rm boA}$ calculated from the new molecular weight of the atmosphere is also shown. Scenario 1.1 with IW+4 case resembles the p,T during the BOM as obtained by Nikolaou et al. (2019). The most dominant species in the atmosphere are marked in bold for each of the cases.  Scenario | Initial outgassing |  $P_{\rm s}$ (bar) | Buffer |  $p_{\rm boA}$ (bar) | Final outgassing
---|---|---|---|---|---
| $f^{\rm init}_{\rm\ch{H_{2}O}}$ | $f^{\rm init}_{\rm\ch{CO2}}$ |  |  |  | $f_{\rm\ch{CO2}}$ | $f_{\rm\ch{H2O}}$ | $f_{\rm\ch{H2}}$ | $f_{\rm CO}$
1.1 | 0.05 | 0.95 | 76.7 | IW-4 | 48 | 0.0018 | 0.0007 | 0.049 | 0.94
|  |  |  | IW | 53.2 | 0.15 | 0.029 | 0.02 | 0.79
|  |  |  | IW+4 | 75.3 | 0.90 | 0.049 | 0.000 | 0.046
1.2 | 0.75 | 0.25 | 76.7 | IW-4 | 27.1 | 0.000 | 0.01 | 0.74 | 0.25
|  |  |  | IW | 51 | 0.040 | 0.44 | 0.30 | 0.20
|  |  |  | IW+4 | 75.8 | 0.24 | 0.74 | 0.005 | 0.0122
1.3 | 1.0 | 0.00 | 76.7 | IW-4 | 9.5 | 0.0 | 0.014 | 0.98 | 0.00
|  |  |  | IW | 49.1 | 0.00 | 0.60 | 0.40 | 0.00
|  |  |  | IW+4 | 76.2 | 0.00 | 0.99 | 0.0069 | 0.00
Table 5: Same as Table 4, but for scenario 2 (EOM), investigating the effect of speciation under these conditions. The surface p,T setting is fixed at $T_{s}=1650$ K and $P_{s}=395$ bar. Scenario 2.2 with IW+4 resembles the p,T at the EOM, as obtained by Nikolaou et al. (2019). Scenario | Initial outgassing |  $P_{\rm s}$ (bar) | Buffer |  $p_{\rm boA}$ (bar) | Final outgassing
---|---|---|---|---|---
| $f^{\rm init}_{\rm\ch{H_{2}O}}$ | $f^{\rm init}_{\rm\ch{CO2}}$ |  |  |  | $f_{\rm\ch{CO2}}$ | $f_{\rm\ch{H2O}}$ | $f_{\rm\ch{H2}}$ | $f_{\rm CO}$
2.1 | 0.05 | 0.95 | 395 | IW-4 | 247.5 | 0.003 | 0.0005 | 0.049 | 0.94
|  |  |  | IW | 283.7 | 0.22 | 0.025 | 0.025 | 0.72
|  |  |  | IW+4 | 390.4 | 0.92 | 0.05 | 0.000 | 0.03
2.2 | 0.75 | 0.25 | 395 | IW-4 | 139 | 0.0007 | 0.007 | 0.74 | 0.25
|  |  |  | IW | 248 | 0.058 | 0.37 | 0.37 | 0.19
|  |  |  | IW+4 | 391 | 0.24 | 0.74 | 0.007 | 0.007
2.3 | 1.0 | 0.0 | 395 | IW-4 | 47.3 | 0.00 | 0.009 | 0.99 | 0.00
|  |  |  | IW | 219 | 0.00 | 0.50 | 0.50 | 0.00
|  |  |  | IW+4 | 391.5 | 0.00 | 0.99 | 0.0099 | 0.00
Table 6: Scenarios showing the output from the volatile speciation model applied to scenario 3.1, H2O-dominated atmosphere and scenario 3.2, CO2-dominated atmosphere at fixed $T_{s}=1500$ K for two different surface pressures. The atmospheric pressure $p_{boA}$ shown above is recalculated based on the new molecular weight of the outgassed species. The initial and final outgassed species are shown as mole fractions. The most dominant species in the atmosphere are marked in bold in each case. Scenario | Initial outgassing |  $P_{\rm s}$ (bar) | Buffer |  $p_{\rm boA}$ (bar) | Final outgassing | Reference
---|---|---|---|---|---|---
| $f^{\rm init}_{\rm\ch{H_{2}O}}$ | $f^{\rm init}_{\rm\ch{CO2}}$ |  |  |  | $f_{\rm\ch{CO2}}$ | $f_{\rm\ch{H2O}}$ | $f_{\rm\ch{H2}}$ | $f_{\rm CO}$ |
3.1 | 0.66 | 0.33 | 300 | None | 300 | 0.33 | 0.66 | - | - | Pluriel et al. (2019)
|  |  |  | IW | 191.8 | 0.083 | 0.311 | 0.35 | 0.24 |
|  |  |  | IW+4 | 297 | 0.32 | 0.66 | 0.006 | 0.007 |
3.2 | 0.02 | 0.98 | 510 | None | 510 | 0.98 | 0.02 | - | - | Pluriel et al. (2019)
|  |  |  | IW | 371.2 | 0.25 | 0.009 | 0.01 | 0.73 |
|  |  |  | IW+4 | 504.7 | 0.95 | 0.019 | 0.0 | 0.023 |

##  4 Results

###  4.1 Temperature profiles

Figures 4 and 5 show the pressure-temperature (p-T) and altitude-temperature
(z-T) profiles obtained using the convection lapse-rate formulation as
described in Section 2.3 for scenarios 1 and 2 as shown in Tables 4 and 5,
respectively. In Figure 4 scenario 1.1 (95% initial CO2, 5% initial H2O, top
left panel, p-T), the results suggest a dry adiabat in the lower unsaturated
troposphere that is related to the high surface temperature. The green curve
(oxidized buffer case) adiabat is steeper in this region than the blue and
orange curves because it is CO2-dominated, leading to a low heat capacity
(Figure 3) and accordingly to a steeper calculated lapse rate (equation 19).
This steep slope results in the green curve that intersects the saturated
vapor curve (dashed red line) at lower pressures than for the blue and orange
curves, which have lower CO2 but more CO and H2O (Table 5) owing to higher
heat capacities (Figure 3) and therefore shallower dry lapse rates (equation
19).

In Figure 4 for z-T (top right panel), the green curve features the lowest
atmospheric height (geometric thickness) due to its larger atmospheric
molecular weight and therefore smaller scale height than the other, lighter
atmospheres shown by the orange and blue curves. Above the dry adiabat regime
(e.g., above about 320 km for the orange line), the slope steepens
corresponding to the wet adiabat as temperatures are low enough to allow for
condensation. In the uppermost atmosphere, a fixed iso-profile temperature
($T=200$ K) (e.g., occurring above about 380 km for the orange line) is
imposed in our model for the radiative regime, thereby following other studies
(Lupu et al. 2014; Pluriel et al. 2019; Katyal et al. 2019).

In the middle and lower panels of Figure 4, the initial H2O (CO2) amounts are
increased (decreased) (see scenarios 1.2 and 1.3 in Table 4), which leads to
light, hydrogen-dominated atmospheres for the reduced buffer cases. Because H2
has a heat capacity between that of H2O and CO2 or CO (Figure 3), the green
slope in the middle and lower left panels is now closer to the blue and orange
slopes compared with the upper panel because the difference in heat capacities
is now smaller. In the middle and lower right panels, results suggest extended
scale heights for highly reduced atmospheres with dominant H2 (e.g., scenario
1.3, IW-4) because its molecular weight is lower than that of oxidized species
such as H2O or CO2 in the atmospheres (e.g., scenario 1.1 and 1.2, IW+4) and
subsequently the change in altitude range of the figure is notable.

Figure 4: Left: Pressure-temperature profiles for the BOM scenarios 1.1
(upper), 1.2 (middle), and 1.3 (lower panel) for three different buffers IW-4,
IW, and IW+4. The dashed red line represents the saturation water vapor curve.
Pressures at the bottom of atmosphere values are taken from Table 4. Right:
Altitude-temperature profiles for scenarios 1.1 (upper), 1.2 (middle), and 1.3
(lower panel). We note the different altitude ranges for the panels on the
right. Figure 5:  Same as Fig. 4, but for p, T conditions and scenarios as
summarized in Table 5, representing the EOM.

Figure 5 illustrates the EOM scenarios (Table 5). Compared with the earlier
BOM case (Figure 4) with a surface temperature of 3300 K, the surface
temperature for the EOM scenario has cooled to 1650 K and the surface pressure
has increased from 76.7 bar (BOM) to 395 bar (EOM). The cooler temperatures
generally favor the formation of chemical species with weaker bond
dissociation energies, that is, H2 (4.52 eV) over H2O (5.1 eV) and CO2 (5.51
eV) over CO (11.1 eV) (where eV denotes electron Volt), which is consistent
with the changes produced by the chemical speciation model (compare Table 5
with Table 4). Larger changes of up to a few dozen percent in the species mole
fraction occur for scenario 2.2 compared with scenario 1.2, in which the
relative amount of C-H-O (0.75:0.25 for H2O:CO2) is the most similar. Figure 5
(EOM) shows modest changes in comparison to Figure 4 (BOM), which are
associated with the change in p,T and corresponding changes in speciation and
heat capacity, hence in the slope of the adiabats. The altitude decrease is
mainly associated with the decrease in temperature.

Figures 4 and 5 show that (1) the transition between dry and moist adiabat
(slope change) always appears to be located on the saturation curve $P_{\rm
sat}$ and it effectively occurs at lower temperature than the saturation
temperature (for similar pressure); and (2) for $f_{\ch{H2O}}$ $\geq$ 0.1 in
the atmosphere, the moist adiabat always coincides with the water vapor
saturation curve $P_{\rm sat}$.

Figure 6: Temperature profiles p-T for scenario 3.1 (upper panel) with an
initially H2O-dominated atmosphere and scenario 3.2 (lower panel) with an
initially CO2-dominated atmosphere (Table 6). The molar fraction of each
species is shown in the legend. ”No speciation” indicates that the values were
input directly by the user and were not calculated by the C-O-H speciation
model.

Figure 6 shows the temperature-pressure profiles obtained for scenarios 3.1
(upper panel) and 3.2 (lower panel) with H2O and CO2 dominated atmospheres of
Table 6, respectively. In the case of scenario 3.1 with its initially
H2O-dominated atmosphere (Fig. 6 upper panel), for the reducing buffer IW
case, the gradient of the dry adiabatic lapse rate is slightly reduced (orange
line) due to chemical speciation of H2 and CO as compared to the ”no
speciation” case (blue line). However, for an oxidizing buffer IW+4 (green
line), the dry adiabatic lapse rate is similar to the case for ”no speciation”
because the volume mixing ratio of outgassed species is similar to the initial
volatile outgassing. In Fig. 6, scenario 3.2 (lower panel), the chemical
speciation of the initially assumed CO2-dominated atmosphere (see Table 6),
leads to a CO-rich (IW buffer) and CO2-rich (IW+4 buffer) atmosphere. Thus,
the moist adiabatic lapse rate changes as compared to the upper panel results
and shows a deviation from pure water vapor saturation curve (dotted red
line). This is because of a low water vapor volume mixing ratio $f_{\rm
H_{2}O}=0.02$ and a higher CO2 volume mixing ratio $f_{\rm CO_{2}}=0.98,$
leading to an unsaturated troposphere. Moreover, the vertical range of the wet
adiabat regime (scenario 3.2) is smaller than in scenario 3.1 (upper panel),
and the radiative regime ($T=200\rm\,\ K$ isothermal profile) is evident at 40
Pa, for instance, for the green curve (lower panel) but occurs near the top of
the atmosphere, that is, around 0.1 Pa, for scenario 3.1. The differing
behavior in scenario 3.2 compared with scenario 3.1 arises mainly from the
different chemical speciation (see panel legends), which affects the heat
capacity and therefore the adiabatic gradients.

###  4.2 Thermal spectral emission

Using the temperature profiles as obtained in Fig. 4, Fig. 5 and composition
of various species for various scenarios of Table 4 and Table 5 as input to
the line-by-line (lbl) radiative transfer code GARLIC, we obtain the thermal
spectral emission from the atmosphere, also known as the outgoing longwave
radiation (OLR). Figure 7 shows the thermal emission spectra of scenario 1.1
(upper), 1.2 (middle), and 1.3 (lower) panel of Table 4 for the BOM. The
spectral features that arise due to the presence of reduced species such as CO
versus the oxidized species such as H2O and CO2 are easily distinguishable in
the three panels. In the top panel of Fig. 7 (scenario 1.1), a prominent CO
absorption feature is seen at 2.3 and 4.6 $\mu\rm m$ for the reduced buffer
case IW-4 (blue curve), whereas distinct CO2 features at 2.1, 4.3 and 15
$\mu\rm m$ are seen for the oxidized buffer case IW+4 (green curve). When the
input H2O outgassing value is increased (scenario 1.2), stronger H2O features
start to appear (orange and blue curves) along with the CO2 absorption
features for the oxidized buffer (green curve). Finally, when the input H2O
outgassing is increased to 100% (scenario 1.3), we obtain thermal spectra that
are dominated by water vapor with distinct water features between 1 to 2.6
$\mu\rm m$ and at 6.2 $\mu\rm m$ for the oxidized case (green curve). The
reduced atmosphere with IW-4 buffer produces an almost pure H2 atmosphere with
a similar spectrum as H2O but with a higher emitted flux.

Figure 8 shows the thermal emission spectra of scenario 2.1 (upper), scenario
2.2 (middle), and scenario 2.3 (lower) of Table 5 for the EOM. One notable
difference between Fig. 7 and Fig. 8 is a significant reduction in the emitted
flux especially at the smaller wavelengths, which is mainly related to the
lower surface temperature (i.e., $1650\rm\,\ K$) for the latter. The effect of
pressure at the bottom of the atmosphere on the thermal spectra is also seen
and is discussed in detail in Section 4.3. The effect of the large H2
continuum at 5-7 $\mu\rm m$ caused by self-collision of the H2 molecules could
reveal molecular H2 as a main constituent of an atmosphere not known a priori.

Figure 7: Comparison of thermal emission spectra for a reduced vs. an oxidized
mantle during the BOM presented for scenario 1.1 (upper panel), 1.2 (middle
panel), and 1.3 (lower panel) of Table 4. The output of the volatile
speciation model is shown in the legend. Values written above the legend show
the initial input values to the speciation model. Reference blackbody curves
for four different temperatures are plotted. Absorption features of key
chemical species are indicated. The spectra shown here have been binned to a
resolution of $\lambda/\Delta\lambda=1000$.  Figure 8: Same as Fig. 7, but for
scenarios of Table 5. The blackbody curves for three different surface
temperatures bracketing the emission spectra are plotted for reference.

Figure 9 compares the thermal emission spectra for reduced atmospheres
overlying a reducing buffer (IW-4) for the various scenarios 2.1, 2.2, and 2.3
of Table 5. In this figure, the prominent absorption features are seen for CO
at 2.3 and 4.6 $\rm\mu m$, while CO2 stands out at 2.1, 4.3, and 15 $\rm\mu
m$. A small amount of CO2 is always present even for this very reducing buffer
calculated by the mass-equilibrium method (Sect. 2.1). However, the strength
and width of CO2 absorption features are seen to decrease as the volume
fraction of initial CO2 decreases from 0.95 to 0.0 (see the zoomed 15 $\rm\mu
m$ feature in the right inset of Fig. 9). Another notable feature is visible
in scenario 2.3 with pure H2O (dissolved in the melt), which results in a H2
-dominated atmosphere and displays a large continuum absorption in the 5-7
$\rm\mu m$ region caused by collision-induced absorption.

Figure 10 is same as Fig. 9, but now for the oxidizing atmosphere cases (IW+4)
for scenarios 2.1, 2.2, and 2.3 of Table 5. A reduction in the emitted flux is
seen compared with the reduced atmosphere case (Fig. 9). This is because of
enhanced absorption by higher concentrations of greenhouse gases such as CO2
and H2O that are present in the oxidized atmosphere. Thus, as the initial H2O
volume mixing ratio (VMR) increases (from scenario 2.1 through 2.3), a further
reduction in the emitted flux is obtained. The most prominent absorption
features are seen for CO2 at 2.1, 4.3, and 15 $\rm\mu m$ and H2O at 1.0, 1.2,
1.4, 1.9, and 2.6 $\rm\mu m$, respectively. As a result of the thick overlying
steam atmosphere, the atmospheric window (8-10 $\mu\rm m$) becomes optically
thick and the 6.2 $\rm\mu\rm m$ H2O feature is not prominent (green curves in
scenarios 2.2 and 2.3). The inset of this figure shows the zoomed-in 4.3
$\rm\mu m$ CO2 feature. The CO feature at 4.6 $\rm\mu m$ is not seen here as
compared to the reduced case in Fig. 9. The inset on the far right shows a
zoomed-in view of the 15 $\rm\mu m$ CO2 absorption band as a function of CO2
VMR and indicates the change in band depths when the input CO2 VMR is reduced
from 0.95 to 0.25 (red to blue curve) and is then absent for the green curve
with no CO2. In summary, the main result is a diminishing of the CO band at
4.6 $\mu\rm m$ and the appearance of distinct H2O bands from $\sim$ 1 - 2.6
$\mu\rm m$. The H2O feature at 6.2 $\mu\rm m$ is less prominent because of the
thick steam atmosphere.

Figure 9: Thermal spectral emission for a reducing atmosphere overlying a
reduced mantle (IW-4) for scenarios 2.1, 2.2, and 2.3 of Table 5. The legend
shows the mole fraction of the outgassed species obtained from the speciation
model. Prominent features of absorbing species are seen at wavelengths marked
in the figure. Reference blackbody curves for three different surface
temperatures are plotted. The insets show the zoomed-in thermal spectra with
the most prominent absorption features of CO2, CO and H2. The spectra have
been binned to a resolution of $\lambda/\Delta\lambda=1000$. Figure 10: Same
as for Fig. 9, but for an oxidizing atmosphere overlying an oxidized mantle
(IW+4). Notable features are the emergence of prominent H2O features between
$\rm 1-2\;\mu m$ (scenario 2.3) and the disappearance of CO feature at $\rm
4.6\;\mu m$ as compared to Fig. 9 (scenario 2.1 and 2.2).

###  4.3 Dependence of OLR upon mantle fugacity

The thermal emission spectra were then averaged over the entire wavelength
range to obtain the outgoing longwave radiation (OLR) as shown in Figure 11
(right panels). The variation in pressure at the bottom of atmosphere
$p_{boA}$ (using Eq. 13) for three different buffers is shown in Figure 11
(left panels). The details of the various cases for which we investigated the
effect of speciation on the $p_{\rm boA}$ and subsequently on the OLR for the
BOM and EOM are shown in Tables 8 and 9 of Appendix B, respectively.

For the BOM (Fig. 11; top right panel) with $T_{\rm s}=3300\rm\;K$, the
highest OLR is obtained for the most reduced atmosphere (blue curve; IW-4
buffer) because the absorption bands are weaker and the atmospheric pressures
(top left panel) are lower than the IW buffer (orange curve), wherein a mix of
reduced and oxidized species are present in the atmosphere (see Table 8) that
contribute far more effectively to block radiation. It is interesting to note
that even for the most oxidizing atmosphere (green curve; IW+4 buffer), less
radiation is blocked by the atmosphere, which is mainly dominated by H2O ,
than for the IW buffer.

At lower temperatures (Fig. 11; bottom right panel) with $T_{\rm
s}=1650\rm\;K$ (EOM), the emitted radiation is far lower than in the high
temperature case ($T_{\rm s}=3300$ K) because for the latter, the radiation is
emitted in the infrared and visible wavelengths, which leads to a stronger
outgoing longwave radiation (OLR). Moreover, for the low temperature case
($T_{\rm s}=1650\rm\;K$), the OLR reaches a limit of $\sim$280 W/m2 (Nakajima
et al. 1992; Goldblatt et al. 2013; Marcq et al. 2017) for water-dominated
atmospheres (green curve; IW+4 buffer) at $\sim$1800 K (also see Katyal et al.
2019). The calculated OLR does not drop below the OLR limit because of the
impeding effect of absorption of radiation by H2O even if present in moderate
quantities. This is an important result that suggests that for a specific
surface temperature, atmospheres with lower surface pressures ($\sim$4-200
bar) can lead to strong outgoing emission of radiation and therefore to a more
effective cooling of the MO than for atmospheres with a high surface pressure
(¿ 200 bar). Studying the effect of pressure on OLR in this manner is
analogous to the effect of a threshold temperature as shown in Marcq et al.
(2017). We also refer to Figure 10 of Nikolaou et al. (2019), where the OLR
for pure steam atmospheres is plotted as a function of surface temperature and
surface pressures. According to their Figure 10 and Katyal et al. (2019),
higher OLR values are obtained for atmospheres with lower surface pressure
(for a similar surface temperature). This means that the outgoing radiation
for lower pressure cases may reach the OLR limit, but at temperatures lower
than $\sim 1800\rm\,\ K$ as previously reported (Kopparapu et al. 2013; Katyal
et al. 2019).

Figure 11: Left: Pressure at the bottom of the atmosphere $p_{\rm boA}$ for
various cases referring to different initial volatile abundances and three
different mantle buffers: reduced (IW-4), IW, and oxidized (IW+4). The details
of speciation for various cases on the x-axis are shown in Table 8 ($T_{\rm
s}=3300$ K) and Table 9 ($T_{\rm s}=1650$ K) in the Appendix B. The horizontal
dashed black line indicates the total surface pressure of the initial
outgassed volatile for the various cases. Right: Outgoing longwave radiation
(OLR) for various cases corresponding to the atmospheric surface pressure
shown in the left panels and for three different buffers.

In Figure 11, cases 1 and 2 are mainly dominated by CO2 and/or CO (see Tables
8 and 9), which show only few absorption bands and thus display the strongest
OLR. Cases 3 and 4, which are composed of a mixture H2O+CO2+CO+H2 (see Tables
8 and 9), result in a minimum OLR where the spectra shows absorption lines of
almost all the species present. Cases 5 and 6 are dominated by absorption
through H2O and/or H2, which leads to a stronger OLR than the mixed-species
cases 3 and 4, where absorption of all species strongly reduces the OLR for
the high-temperature cases. For the low-temperature cases (lower right panel),
the effect of the absorption by H2O dominates because the H2O band and
continuum absorption cover the entire infrared wavelength regime.

###  4.4 Early H2 atmosphere scenario

While hydrogen is prone to escape to space via strong incoming extreme UV
(EUV) during the Hadean (more details in Section 4.6), it is also an important
gas in enhancing the greenhouse effect in the atmosphere. The efficiency of
heating the atmosphere, however, depends upon the surface partial pressure of
hydrogen.

We explored a case similar to scenario 2.3 of Table 4 and shown previously in
Figure 9 with a pure H2 atmosphere, but with a varying hydrogen surface
pressure $p_{\ch{H2}}$ and a fixed surface temperature $T_{\rm s}=2800\,\rm
K$. We obtained the thermal emission of radiation by varying the surface
pressure $p_{\ch{H2}}$ in the range 0.6-185 bar, as shown in Figure 16 (upper
panel) of Appendix C. We observe a large number of undulations around 2
$\mu\rm m$ in the obtained thermal spectra, especially at higher surface
pressures. These undulations are caused by the effects of collision-induced
absorption (CIA) by H2. The lower panel of Fig. 16 shows a decrease in the OLR
with the (surface) partial pressure of H2 attributed to the higher absorption
of radiation at higher pressures, as also shown in Fig. 11 (left panels). We
therefore conclude that several bars of H2 in the atmosphere could
significantly affect the climate in the cases studied, leading to additional
greenhouse warming (also see Ramirez et al. 2014). A similar trend in the OLR
versus $p_{\ch{H2}}$ as shown here was suggested by Pierrehumbert & Gaidos
(2011).

###  4.5 Transmission spectra

We obtained the transmission spectra for the BOM and the EOM using the lbl
radiative transfer code GARLIC and the temperature profiles (Sect. 4.1) and
concentration as an input. The resultant spectra are shown as the wavelength-
dependent effective height of the atmosphere (see Eq. 26 in Figure 12). The
spectra were binned to a resolution of $R=\lambda/\Delta\lambda=1000$ over the
wavelength range shown.

Figure 12: Transmission spectra calculated by GARLIC shown as atmospheric
effective height vs. wavelength for the BOM scenarios (left panels) and EOM
scenarios (right panels) for three different redox buffers IW-4, IW, and IW+4.
The mole fraction of final outgassed species in the atmosphere for various
scenarios is the same as provided in Tables 4 and 5. The effective height of a
reduced atmosphere (with low molecular weight) for the case of reduced mantle
IW-4 (blue curve; scenarios 1.2, 1.3, 2.2, and 2.3) is lower than an
atmosphere with larger molecular weight (blue curve; scenarios 1.1 and 2.1)
and oxidizing atmospheres (green curve; all scenarios). For more details, see
the text. The spectra have been binned to a resolution of
$\lambda/\Delta\lambda=1000$.

As shown in Fig. 12, the effective height of the atmosphere increases to one
order of magnitude for the reduced atmospheres (IW-4) from scenarios 1.1 to
1.3 (and 2.1 to 2.3) for both BOM (left panel) and EOM (right panel)
scenarios. This is related to the increasing atmospheric scale height for the
lighter H2 -dominated atmospheres. Similarly, for the oxidized atmospheres
(IW+4), the effective height increases from a dense and thick CO2 -dominated
atmosphere (scenarios 1.1 and 2.1) to lighter H2O -dominated atmospheres
(scenarios 1.2, 1.3, 2.2, and 2.3). In the visible-optical range, the slope of
the so-called Rayleigh-scattering extinction feature is somewhat steeper for
the reducing atmosphere (blue line) because of the different scale heights.
For scenarios 1.3 and 2.3, H2O absorption features are seen ubiquitously from
roughly 0.9 to 2.6 $\rm\mu m$. For a high CO2 initial volatile outgassing
(scenario 1.1 and 2.1), CO absorption features are seen at 2.3, 4.6 $\rm\mu m$
for IW-4 buffer, and CO2 absorption features are seen as shown for 2.1, 4.3,
and 15 $\rm\mu m$ for the IW+4 buffer. For the EOM scenarios (right panels),
the cooler surface temperature (1650 K) leads to a decrease in scale height
($H=kT/mg$) compared with the BOM (left panels) scenarios with higher surface
temperature (3300 K).

A decrease in the depth of the absorption bands for CO (IW-4) and CO2 (IW+4)
is seen at 4.3 and $\rm 15\;\mu\rm m$, respectively, which is related to a
decrease in the input CO2 from scenarios 1.1 to 1.2 at $T_{\rm s}=1650\;\rm K$
(right panel). A similar trend is seen for scenarios at $T_{\rm s}=1650\;\rm
K$ (right panel). For the oxidizing atmospheres (IW+4) in scenarios 1.3 and
2.3, the most prominent H2O features between $\rm 1-2\;\mu\rm m$ are seen, but
the 6.2 $\rm\mu m$ H2O feature is suppressed. This is likely related to the
overlying optically thick atmosphere of H2O (blanketing effect) with a very
high surface pressure ($\sim 395\rm bar$), wherein the radiation will be
dominated by the emission from the top of the atmosphere. This hinders
chemical species in the lower atmosphere from being detected in the upper
regions sampled by transmission spectroscopy. It is therefore difficult to
probe dense atmospheres with high optical depths.

###  4.6 Atmospheric evolution and escape over the MO period

Table 7 shows the range of surface partial pressure of H2 for a highly reduced
(IW-4) and an oxidized mantle (IW+4) over a period of MO solidification $\sim
1\;\rm Myr$. For the reducing-buffer case (IW-4), over the period of MO, the
surface pressure of atmospheric hydrogen increases from 2.3 to 102 bar, while
for the oxidized case, it increases from 0.02 to 2.7 bar. To investigate the
evolution of H2 along with other species in the atmosphere, we split the MO
period into ten time steps where the change in species abundance from scenario
1.1 (BOM) to scenario 2.2 (EOM) in Table 7 is set to change with a linear time
step of 0.1 Myr (only shown for H2 in the table). The resulting changes in
species abundances across the MO can be attributed to outgassing and chemical
speciation. As a first step, the input data for the various scenarios were
taken from Nikolaou et al. (2019); Katyal et al. (2019), but in a future
study, we plan to perform time-dependent, coupled interior-atmosphere
evolutionary scenarios over the magma ocean phase applying a newly developed
climate-chemistry atmospheric model (Scheucher et al. 2020; Wunderlich et al.
2020).

Table 7: Hydrogen abundance in the atmosphere from the beginning (scenario 1.1) to the end (scenario 2.2) of the magma ocean for a reduced (IW-4) and oxidized (IW-4) mantle.  Buffer | t = 1 yr (scenario 1.1) | t = 1 Myr (scenario 2.2)
---|---|---
|  $p_{boA}$(bar) |  $p_{\ch{H2}}$ (bar) | $f_{\ch{H2}}$ |  $p_{boA}$(bar) |  $p_{\ch{H2}}$ (bar) | $f_{\ch{H2}}$
IW-4 | 48 | 2.3 | 0.05 | 139 | 102.8 | 0.74
IW+4 | 75.3 | 0.02 | 0.0003 | 391 | 2.7 | 0.007

We now consider the evolution of atmospheric species over the MO period. The
mole fraction $f$ of the resultant outgassed species $i$ is translated into
the mass fraction via $w_{i}=f_{i}\;M_{i}/\bar{M}$, where $M_{i}$ is the
molecular weight of the species $i$ and $\bar{M}$ is the mean molar mass of
the atmosphere. The atmospheric mass of the individual species is obtained by
multiplying the calculated mass fraction $w_{i}$ with the total mass of the
atmosphere $M=P\mbox{*}A/g$.

Figure 13 shows the resulting atmospheric mass evolution of a reducing
atmosphere lying above a reduced mantle, that is, IW-4 (upper panel) and an
oxidizing atmosphere lying above an oxidized mantle, that is, IW+4 (lower
panel) during the MO period. The circles show the mass of the H2 escaping via
diffusion-limited escape according to Eq. (33) as described in Section 2.6,
where the color of the circles indicates the total mass-loss rate (g/s). The
mass-loss rate obtained from energy-limited escape formalism as in Eq. (30) is
obtained to be $5.2\times 10^{7}\;\rm g/s$ for $S=10$ and $5.2\times
10^{8}\;\rm g/s$ for $S=100$. As described in Sect. 2.6, ($R_{p}+H_{e})\sim
R_{p}$ for this scenario (see Table 7). The mass loss of H2 (in grams) is then
calculated and shown as the yellow shaded region within these $S$ ranges in
Fig. 13 (both upper and lower panels). In general, the atmospheric H2 mass
loss (in grams) is seen to increase for both the escape processes because the
H2 abundance increases throughout the MO evolution timescale. The residual H2
in the atmosphere after accounting for escape is seen to overlie the total H2
content of the atmosphere, implying that a significant amount of H2 remains in
the atmosphere. We also show a shaded region obtained from the parameterized
Eq.(32) of Zahnle et al. (2019), where the lower and upper boundaries of the
shaded region indicate $S=10$ and $S=100$, respectively, and $f_{\ch{H2}}$
values are taken over the evolution timescale (see Table 7).

For the oxidized-mantle (IW+4) case (Fig. 13 lower panel), the H2 abundance is
very low (see Table 7), causing this to be the bottleneck for H2 to diffuse
through heavy CO2 or H2O up to the homopause. Moreover, the plentiful amount
of XUV energy available here (with a high loss rate) can in fact lead to the
complete removal of hydrogen. For the oxidizing atmosphere lying above an
oxidized mantle, that is, IW+4, escape of H2 is therefore not energy limited
but diffusion limited. Finally, we conclude that for both the redox states of
the mantle, the outgassing of H2 into the atmosphere dominates the escape of
H2 (Kuramoto et al. 2013; Ramirez et al. 2014).

Figure 13: Evolution/escape of atmospheric species in terms of the atmospheric
mass (left axis) for the duration of the magma ocean (x-axis) for a reduced
mantle case (upper panel) and an oxidized mantle case (lower panel). The
colored solid lines refer to atmospheric masses integrated over the whole
planet that change due to outgassing and chemical speciation (see text). The
dotted brown line shows the residual mass of H2 in the atmosphere after
accounting for outgassing and DL escape. The shaded region in cyan shows the
mass loss of H2 in the range for $S=10$ to $S=100$ (XUV relative to the modern
value) using the Zahnle et al. (2019) best-fit formalism (see text). The
shaded region in yellow shows the energy-limited loss of H2 for the lower
range $S=10$ to upper range $S=100$. The dotted blue horizontal line shows the
hydrogen content of one (modern) Earth ocean (M${}_{\rm HO}=1.6\times
10^{23}\rm\,\ g$). The filled circles indicate the escaping mass of H2
obtained using DL. The colors indicate the respective mass-loss rates (right
axis).

As an illustration, Figure 14 compares the escape rates for scenario 2.2 (EOM)
calculated for H2 with our estimated H2 outgassing rate obtained using
Equation (14) as described in Section 2.1.

Figure 14: Hydrogen escape and outgassing rates as a function of redox state
of the mantle for scenario 2.2 (EOM). The solid red line shows the H2
outgassing rate at the end of magma ocean. The closed red circles show the
diffusion-limited escape of H2 for the EOM. The shaded region in yellow shows
the loss rate due to energy-limited escape within the range S=10 (lower
boundary) to S=100 (upper boundary) i.e. XUV relative to the modern value. The
vertical dotted line marks the region separating a high H2 VMR to low H2 VMR
and illustrates the most efficient escape process occurring. The present Earth
H2 degassing rate with an uncertainty range (Ramirez et al. 2014) is marked
for reference.

The outgassing rates obtained by us during the magma ocean period in the
Hadean (3.8-4.0 Gyr before present) for a variable redox state of the mantle
are higher than in the Archean (2-4.0 Ga), that is, $9\times 10^{14}\rm\,\
molecules\,\ m^{-2}s^{-1}$ (Tian et al. 2005; Kuramoto et al. 2013). First,
one of our central results is that outgassing (red solid line) proceeds faster
than diffusion-limited escape (closed red circles) by a factor of x10 and
energy-limited escape depending upon the uncertainty in the XUV radiation as
evident from the $S$ values displayed in the figure. Second, for an atmosphere
with a high hydrogen abundance, that is, VMR(H2) ¿ 0.2 (-4 ¡ IW $\leq$ 0.5),
the energy-limited escape is more effective, whereas for an atmosphere with
lower hydrogen abundances, that is, $\rm VMR(\ch{H2})<0.2\;(0.5<IW<4)$, the
diffusion-limited escape is more effective, as shown in Fig. 14.

##  5 Discussion

We have studied the effect of the redox state of the mantle and volatile
outgassing of major outgassed species H2O and CO2 from the interior and their
interaction with the melt resulting in outgassing of newer species at the
surface. The effect on the atmospheric profiles, thermal infrared emission,
transmission, and escape of H2 is discussed below.

###  5.1 Mantle redox and atmospheric temperature

Our study has suggested an interesting coupling between interior outgassing,
volatile speciation due to the mantle redox state, atmospheric composition,
and atmospheric temperature that affects atmospheric extent, mixing, and the
location of the dry and moist adiabat. During the BOM, more oxidized
atmospheres switch from dry to wet adiabat at lower pressures (Fig. 4, green
lines) than reduced atmospheres due to differences in their mean molecular
weight and heat capacities, which leads to a lower scale height. The pressure
at which the moist adiabat is reached affects the onset of saturation, thereby
affecting the microphysical processes such as condensation and ultimately
ocean formation. Detailed microphysical processes are not included in our
study and need further investigation with more detailed models. On the other
hand, the more reduced atmospheres have larger scale heights, and the switch
to the moist adiabat lies at higher altitudes (Fig. 4, blue curve).
Precipitation initiated at high altitudes has to sediment over a greater
distance to rain out, which could favor evaporation in the lower, warmer
atmospheric layers, and which might therefore be less efficient at condensing
to form the oceans. For the EOM case (Figure 5), although the atmosphere cools
(from 3300K to 1650K at the surface) and the surface pressure increases by a
factor of 5, the temperature structures for the EOM are similar to those for
the BOM (Figure 4). This is because the pressure increase (which increases
atmospheric thickness) is offset by the cooling effect (which decreases
atmospheric thickness). In summary, the above processes likely affected the
evolution of climate on the early Earth by affecting key processes such as
cooling to space and greenhouse heating, but also the onset of condensation
and accordingly, the formation of terrestrial ocean.

###  5.2 Thermal emission and transmission spectra

Our results suggest that the thermal emission of radiation is affected by key
parameters such as surface temperature, surface pressure, mantle redox state,
and initial volatile content of the two main species H2O and CO2 that are
outgassed during the magma ocean phase. The most prominent absorption features
of oxidized versus reduced species in the atmosphere for the three different
mantle oxidation states IW-4, IW, and IW+4 are shown in Figs. 7 and 8. For the
reducing mantle cases, the volatile speciation model would result in a mixture
of both oxidized and reduced species in the atmosphere depending upon the
initial volatile contents $f^{\rm init}_{\ch{H2O}}$ and $f^{\rm
init}_{\ch{CO2}}$ (see Tables 4, 5, 8, and 9). As shown in Fig. 11 (right
panels), the OLR for the most reduced atmosphere case overlying a reduced
mantle (IW-4) consists of H2, CO or CO+H2 (depending upon the case) and
results in the highest OLR because these molecules have only a few absorption
bands. On the other hand, we obtain lower OLR for other moderately reduced
buffer, that is, IW, resulting in a mixed atmosphere of CO2+CO+H2+H2O with the
largest number of absorption bands covering the entire spectral range.
Finally, the lowest OLR is obtained for an oxidizing atmosphere overlying an
oxidizing buffer (IW+4), resulting in CO2, H2O or H2O+CO2 atmospheres
(depending upon the case).

As a result, a planet with a reduced mantle buffer (IW-4) and low water
content (cases 1 and 2) is expected to cool faster than a planet with a high
water content in the atmosphere (cases 4, 5, and 6). For a moderately reduced
mantle buffer (IW), the planet will cool down slowly as the OLR is the lowest
for it. For a planet with an oxidized mantle buffer (IW+4), the cooling
timescales will be slower than those of the IW-4, but faster than the IW case.

The initial outgassed water content along with the surface temperature and the
redox state of the mantle are important factors driving planetary cooling.
Figure 11 shows for lower temperature ($T_{\rm s}=1650$ K) and oxidized mantle
(IW+4 buffer) that the OLR limit $\sim$280 W m-2 (Goldblatt et al. 2013; Marcq
et al. 2017) occurs only for H2O -rich atmospheres (cases 5 and 6). However,
this is not the case for a high temperature ($T_{\rm s}=3300$ K) and an
oxidized mantle (IW+4) because at such high temperatures, the emission occurs
in visible wavelengths along with infrared, and H2O is not able to block the
radiation in the visible as effectively as in infrared. Radiation is thus not
able to reach the OLR limit.

Oxidized atmospheres containing heavier species such as CO2 result in a dense
atmosphere with small effective heights for transmission spectra (Fig. 12,
scenarios 1.1 and 2.1) as compared to a reduced and lighter atmosphere with
H2, resulting in extended atmospheres with a large effective scale height, as
shown in scenarios 1.3 and 2.3 of Fig. 12. The slope of the Rayleigh-
scattering feature in the ultraviolet and visible wavelength range could
provide information on the bulk atmospheric composition. However, when
aerosols, thick hazes, or clouds are considered, this slope changes
considerably because of the change in the wavelength dependence of the
scattering (see Eq. 25), which changes the atmospheric optical depth and
effective height of the atmosphere (also see Wunderlich et al. 2020).

###  5.3 Outgassing and escape: loss timescales

Our results suggest very different regimes in terms of outgassing of various
species and escape of H2 depending on the mantle redox state during the MO
period. For the scenarios that consider the reducing and oxidizing state of
the mantle, the outgassing rate of H2 is greater than the escape rate, and
there is enough interior outgassing of H2 to form an atmosphere. A similar
result was suggested by Ramirez et al. (2014) for early Earth and Mars.

Because the VMR of H2 is high (e.g., in a reducing atmosphere), the escape of
H2 most likely occurs by energy-driven, that is, energy-limited, escape. On
the other hand, if less hydrogen is present (e.g., in an oxidizing
atmosphere), the diffusion of hydrogen through a heavy gas to reach the upper
atmospheric regions and escape is limited. Therefore, the atmosphere would
probably enter a DL configuration. Thus, depending upon the abundance of H2 in
the atmosphere, our results suggest a switch from energy-limited (EL) escape
(high VMR) of H2 to DL escape (low VMR), thus complying with Zahnle et al.
(2019).

For a reduced early atmosphere with 102 bar of H2 by the end of MO, we
estimate the timescale of H2 atmosphere removal to be $\sim$10 Myr using both
DL and EL ($S=10$) escape. On the other hand, because the H2 mass-loss rate
and the surface partial pressure of surface hydrogen are lower (2.7 bar) for
an oxidized atmosphere, the mass-loss timescale is estimated to be $\sim$16
Myr assuming DL escape. Similar mass-loss timescales ($\sim$106-107 yr) for
surface pressures of 10 to 100 bar are obtained by hydrodynamic escape
modeling of Pahlevan et al. (2019). The results for total hydrogen mass-loss
obtained via two different parameterized escape models (this work) and
hydrodynamic model (Pahlevan et al. 2019) are therefore approximately similar.
Moreover, as we showed in Fig. 13, the mass loss obtained using the energy-
limited (hydrodynamic) approach following Zahnle et al. (2019) (cyan shaded
region) approximates the diffusion-limited mass loss at higher levels of
irradiation, $S=100$ (also see Lammer et al. 2018). Furthermore, it is worth
stating that several works (e.g., Lammer et al. 2011) have noted the potential
importance of EUV in driving energy-limited escape on early Earth and
speculated that this quantity is not known to within a factor 100 or more
times the modern-day value depending upon whether the early Sun was a fast or
slow rotator.

###  5.4 Clouds

Clouds could be rather common features in extrasolar atmospheres, and they
could have a potentially strong effect upon the atmospheric spectra, climate,
and photochemistry. Spectral features can be significantly reduced depending
upon the extent (layer location and thickness) and the microphysical
properties (size, shape, number density distribution, and refractive index) of
the clouds occurring over a wavelength range that is sensitive to the cloud
diameter, for example. The general effect of clouds upon atmospheric spectra
and retrieval by the James Webb Space Telescope (JWST) was discussed by Mai &
Line (2019) and Fauchez et al. (2019).

In magma ocean worlds, thick water cloud layers could form (see, e.g., Lebrun
et al. 2013; Marcq et al. 2017) as the outgassed steam atmospheres cool. These
cloud layers could help prevent the loss of planetary infrared radiation to
space, and if thick water clouds were to extend over the region sampled by
transmission spectroscopy, for instance, they might significantly reduce the
strength of the spectral features detected.

In Fig. 15 we show the atmospheric transit depth calculated using Eq. (27) for
planet Earth orbiting an M-type star (0.4 solar radii) in the top panel and a
G-type star (bottom panel) for two different redox states of the mantle, IW+4
and IW. A clear increase in transit depth is visible for the IW buffer with
50% H2O and 0.5% H2 because the molecular mass of the atmosphere is lower than
in the pure H2O case (IW+4 buffer). Results also suggest a substantial
increase in the transit depth when observing via an M-type star because its
stellar radius is smaller, thus favoring the possibility of detecting
atmospheric species. We also compared the cloud-free scenarios with the cloudy
simulations. With a cloud deck (see Section 2.4), the spectral features are
weakened (depending upon the cloud cross section), as shown in Figure 15. The
results of the comparison between cloud-free and cloudy scenarios ($\alpha=0$
and enhanced scattering cross section as compared to Rayleigh) agree well with
Turbet et al. (2019). The transit depths in the optically thick cloudy
scenarios can be considered as maximum values (because thick clouds
effectively block shortwave radiation) compared with the smaller transit
depths of the clear atmosphere cases without clouds. A sensitivity study by
Moran et al. (2018) that involved increasing the haze-scattering cross section
also suggested a weakening in spectral features when a global layer of
Rayleigh-scattering haze was added to the TRAPPIST-1 d, e, and f planets. A
more involved sensitivity study including the geometry of the cloud, such as
its cross section (Kitzmann et al. 2011a, b), requires the use of detailed
cloud microphysics (Zsom et al. 2012) that is applicable for Earth-like
exoplanets. We did not explore this here and aim to investigate it in future
related studies.

Figure 15: Atmospheric transit depth of a cloud-free planetary atmosphere
around an M star (top) and G star (bottom) at the EOM phase for an atmosphere
simulated from IW+4 mantle buffer (red) and IW mantle buffer (blue) is
compared with the transit depth for cloudy scenarios ($\alpha=0$ and enhanced
cross-section as compared to the Rayleigh scattering).

###  5.5 Application to exoplanets

Several hundred rocky exoplanets classified as super-Earths with mass ranging
between $M=1-8M_{\rm E}$ and radius $R=1-1.8R_{\rm E}$ are being discovered by
various current space missions such as TESS111Transiting Exoplanet Survey
Satellite and will be discovered by future missions such as PLATO222PLAnetary
Transits and Oscillations of stars, JWST333James Webb Space Telescope, and
CHEOPS444CHaracterising ExOPlanets Satellite. Atmospheric characterization of
these planets via spectroscopic analysis can benefit greatly from interior
geochemical models that consider outgassing during the magma ocean phase
(Bower et al. 2019) and volcanic outgassing after the magma ocean phase
(Ortenzi et al. 2020). Ortenzi et al. (2020) showed that the redox state of
the mantle affects the atmospheric thickness and overall evolution, thus
constraining the interior chemical state of rocky exoplanets. Puffy outgassed
atmospheres for hot, molten Earth-like planets often lead to an effect known
as “radius inflation” that depends upon the initial volatile inventory of H2O
and CO2 (Bower et al. 2019) and on the irradiation that they receive from the
star (Turbet et al. 2019).

Furthermore, as shown in Fig. 15, the redox state of the mantle and initial
inventory of volatiles can induce strong variations in the transit depths for
planets orbiting G and M stars. This would translate into changes in the
planetary radius to be observed by missions such as TESS and PLATO (Rauer et
al. 2014). However, the detectability of molecular spectral features may be
severely affected by the presence of clouds (see e.g. Fauchez et al. 2019).

Kaltenegger et al. (2007) modeled the observable spectra of Earth-like planets
about two billion years ago to a present-day atmosphere. They reported a
number of constraints, such as the resolution of the telescopes for detecting
the molecular species in the atmosphere. It is therefore important to
characterize the atmosphere of Earth through its geological history and that
our study present useful insights on this. Recent interior-atmosphere modeling
studies have shown that pure steam-based atmospheres of highly irradiated
exoplanets orbiting M dwarfs such as GJ 1214 b (Schaefer et al. 2016) and LHS
3844 b are subject to strong XUV of the star and could lose the outgassed
atmosphere and become a bare rock (Kreidberg et al. 2019). See also Kite &
Barnett (2020) for a detailed study of atmospheric loss and subsequent revival
relevant for exoplanets. In all these models, the water is dissociated into
hydrogen and oxygen, which escape to space, and some remaining oxygen is
dissolved into the magma ocean or remains in the atmosphere (e.g., Schaefer et
al. 2016).

We have provided important testable predictions to inform future space
missions such as the JWST, Ariel555Atmospheric Remote-sensing Infrared
Exoplanet Large Survey (Tinetti et al. 2018; Edwards et al. 2019),
LIFE666Large Interferometer for Exoplanets (Quanz et al. 2019), and
ELT777Extremely Large Telescope (Maiolino et al. 2013) that will characterize
planetary atmospheres in terms of detecting species such as CO, CO2 , and H2O
in spectral observations of Earth-sized planets.

Planets in close-in orbits around cooler stars are favored targets in
exoplanetary science because of their improved contrast ratios, higher transit
frequency, and increased geometric transit probability Scalo et al. (2007);
Shields et al. (2016), for example. Furthermore, the detectability of
atmospheric spectral signals from MO exoplanets (Ito et al. 2015) and thick
atmospheres with H2O and CO2 with a high sensitivity and large spectral
coverage of the JWST (NIRSpec, MIRI) instruments (Ducrot et al. 2020) would
bring new possibilities to constrain such atmospheres.

##  6 Conclusions

The main conclusions of this study are listed below.

  1. 1.

The redox state of the mantle can affect not only atmospheric composition, but
also the vertical temperature structure and therefore the mixing and the
hydrological cycle. These interior-atmospheric couplings could be important
for steam-condensation timescales and accordingly, for the formation of the
Earth oceans.

  2. 2.

Reduced atmospheres emit thermal radiation more strongly, which leads to
faster cooling than in oxidized atmospheres because the latter feature strong
absorbers such as H2O.

  3. 3.

Reduced atmospheres (CO and H2) have much larger spectral features in
transmission than oxidized atmospheres (H2O and CO2) because the scale heights
are larger.

  4. 4.

Thick optical clouds and hazes can mute or weaken the spectral signatures of
molecules in the transmission spectra and enhance the transit depth because of
the cloud cover. It therefore becomes difficult to probe cloudy atmospheres.
On the other hand, clouds absorb much of the radiation and reduce the outgoing
longwave radiation by 1-2 orders of magnitude (Marcq et al. 2017), leading to
longer magma ocean cooling timescales.

  5. 5.

The interplay between outgassing and escape suggests that outgassing of
hydrogen proceeds faster than the atmospheric escape of H2 and that enough
outgassed hydrogen is available to form an atmosphere over the magma ocean
duration of $\sim$1 Myr. The timescale for total H2 mass loss, however, is
estimated to be within some dozen million years. Furthermore, a more
sophisticated hydrodynamical code to study the loss of H2 and the heavy
atmospheric species H2O and CO2 is desirable and planned for future studies.

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## Acknowledgements

We thank the anonymous reviewer and the editor for their valuable comments
that greatly enhanced the quality of the manuscript. N.K. and G.O. acknowledge
funding from the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 263649064 – TRR 170. This is TRR 170 Publication No.
117. M.G. acknowledges funding from DFG - Project-ID GO 2610/1-1. F.S.
acknowledges DFG project SCHR 1125/3-1.

##  Appendix A Gas constant for a mixture

When a mixture of two gases $a$ and $b$, with partial pressure $p_{a}$ and
$p_{b}$ , exists at the same volume $V$ and temperature $T$, the mixture
behaves like a perfect gas, and according to Dalton’s law of partial
pressures,

$(p_{a}+p_{b})V=(m_{a}R_{a}+m_{b}R_{b})T.$ (34)

Because the mixture behaves like a perfect gas, for $m=nM$ moles of a gas, the
total pressure is given by

$PV=mR_{\rm mix}T,$ (35)

where $R_{\rm mix}$ is the gas constant for the mixture. From Eqs. (34) and
(35), $mR_{\rm mix}=m_{a}R_{a}+m_{b}R_{b}$. Hence,

$R_{\rm mix}=\frac{m_{a}R_{a}+m_{b}R_{b}}{m}.$ (36)

In terms of the molar volume fraction,

$R_{\rm mix}=\frac{n_{a}M_{a}R_{a}+n_{b}M_{b}R_{b}}{\sum_{i}n_{i}M_{i}}.$ (37)

Because $MR=\bar{R}$, where $\bar{R}$ is the universal gas constant (= 8.31415
J kg-1 K-1), this equation can be written as

$R_{\rm mix}=\frac{n_{a}\bar{R}+n_{b}\bar{R}}{n_{a}M_{a}+n_{b}M_{b}}.$ (38)

For a single-component gas $a$, $R=\bar{R}/M_{a}$. For a two-component gas
mixture, Eq.(38) can be written as

$R_{\rm mix}=\bar{R}\left(\frac{n_{a}+n_{b}}{\mu}\right),$ (39)

where $\mu=n_{a}M_{a}+n_{b}M_{b}$ is the mean molecular weight of the mixture.
For an ith-component mixture of gases,

$R_{\rm mix}=\bar{R}\left(\frac{\sum n_{i}}{\sum n_{i}M_{i}}\right).$ (40)

##  Appendix B Volatile speciation model output

Table 8: Cases for BOM used to produce Figure 11. Cases 1, 4, and 6 are similar to scenarios 1.1, 1.2, and 1.3, respectively, which are also presented in Table 4. Columns 2 and 3 show the initial assumed mole fractions arising from H2O and CO2 outgassing, which is the input to the speciation model. The assumed buffer values for strongly reducing (IW-4), reducing (IW), and highly oxidizing (IW+4) have been used to study the effect of speciation under these conditions. The four columns on the right show final outgassed species from the speciation model. For these scenarios, the surface p, T setting is fixed to be $T_{\rm s}=3300$ K and $P_{\rm s}=76.7$ bar. The $p_{\rm boA}$ calculated from the new molecular weight of the atmosphere is also shown. The most dominant species in the atmosphere are marked in bold for each of the cases.  Case | Initial outgassing |  $P_{\rm s}$ (bar) | Buffer |  $p_{\rm boA}$ (bar) | Final outgassing
---|---|---|---|---|---
| $f^{\rm init}_{\rm\ch{H_{2}O}}$ | $f^{\rm init}_{\rm\ch{CO2}}$ |  |  |  | $f_{\rm\ch{CO2}}$ | $f_{\rm\ch{H2O}}$ | $f_{\rm\ch{H2}}$ | $f_{\rm CO}$
1 | 0.05 | 0.95 | 76.7 | IW-4 | 48 | 0.0018 | 0.0007 | 0.049 | 0.94
|  |  |  | IW | 53.2 | 0.15 | 0.029 | 0.02 | 0.79
|  |  |  | IW+4 | 75.3 | 0.90 | 0.049 | 0.000 | 0.046
2 | 0.25 | 0.75 | 76.7 | IW-4 | 44.14 | 0.0014 | 0.036 | 0.24 | 0.75
|  |  |  | IW | 52.83 | 0.12 | 0.15 | 0.10 | 0.62
|  |  |  | IW+4 | 75.43 | 0.71 | 0.245 | 0.0016 | 0.036
3 | 0.5 | 0.5 | 76.7 | IW-4 | 37.43 | 0.00 | 0.007 | 0.49 | 0.50
|  |  |  | IW | 52.12 | 0.08 | 0.30 | 0.20 | 0.42
|  |  |  | IW+4 | 75.6 | 0.47 | 0.29 | 0.003 | 0.024
4 | 0.75 | 0.25 | 76.7 | IW-4 | 27.1 | 0.000 | 0.01 | 0.74 | 0.25
|  |  |  | IW | 51 | 0.040 | 0.44 | 0.30 | 0.20
|  |  |  | IW+4 | 75.8 | 0.24 | 0.74 | 0.005 | 0.0122
5 | 0.95 | 0.25 | 76.7 | IW-4 | 14.0 | 0.0 | 0.013 | 0.94 | 0.0049
|  |  |  | IW | 49.6 | 0.008 | 0.56 | 0.38 | 0.042
|  |  |  | IW+4 | 76.1 | 0.047 | 0.94 | 0.0063 | 0.002
6 | 1.0 | 0.00 | 76.7 | IW-4 | 9.5 | 0.0 | 0.014 | 0.98 | 0.00
|  |  |  | IW | 49.1 | 0.00 | 0.60 | 0.40 | 0.00
|  |  |  | IW+4 | 76.2 | 0.00 | 0.99 | 0.0069 | 0.00
Table 9: Same as for Table 8, but for EOM cases used to produce Figure 11. Cases 1, 4, and 6 are similar to scenarios 2.1, 2.2, and 2.3, respectively, which are also presented in Table 5. The surface p,T setting is fixed at $T_{\rm s}=1650$ K and $P_{\rm s}=395$ bar.  Case | Initial outgassing |  $P_{\rm s}$ (bar) | Buffer |  $p_{\rm boA}$ (bar) | Final outgassing
---|---|---|---|---|---
| $f^{\rm init}_{\rm\ch{H_{2}O}}$ | $f^{\rm init}_{\rm\ch{CO2}}$ |  |  |  | $f_{\rm\ch{CO2}}$ | $f_{\rm\ch{H2O}}$ | $f_{\rm\ch{H2}}$ | $f_{\rm CO}$
1 | 0.05 | 0.95 | 395 | IW-4 | 247.5 | 0.003 | 0.0005 | 0.049 | 0.94
|  |  |  | IW | 283.7 | 0.22 | 0.025 | 0.025 | 0.72
|  |  |  | IW+4 | 390.4 | 0.92 | 0.05 | 0.000 | 0.03
2 | 0.25 | 0.95 | 395 | IW-4 | 227.2 | 0.002 | 0.002 | 0.25 | 0.74
|  |  |  | IW | 277.2 | 0.176 | 0.124 | 0.125 | 0.57
|  |  |  | IW+4 | 390.6 | 0.72 | 0.24 | 0.002 | 0.023
3 | 0.5 | 0.5 | 395 | IW-4 | 192.4 | 0.0015 | 0.005 | 0.49 | 0.49
|  |  |  | IW | 266 | 0.11 | 0.25 | 0.25 | 0.38
|  |  |  | IW+4 | 390.7 | 0.48 | 0.49 | 0.005 | 0.0157
4 | 0.75 | 0.25 | 395 | IW-4 | 139 | 0.0007 | 0.007 | 0.74 | 0.25
|  |  |  | IW | 248.9 | 0.058 | 0.37 | 0.37 | 0.19
|  |  |  | +4 | 391 | 0.24 | 0.74 | 0.007 | 0.007
5 | 0.95 | 0.25 | 395 | IW-4 | 70.66 | 0.00 | 0.009 | 0.94 | 0.05
|  |  |  | IW | 226.7 | 0.011 | 0.47 | 0.47 | 0.038
|  |  |  | IW+4 | 391.3 | 0.048 | 0.94 | 0.0094 | 0.001
6 | 1.0 | 0.0 | 395 | IW-4 | 47.3 | 0.00 | 0.009 | 0.99 | 0.00
|  |  |  | IW | 219 | 0.00 | 0.50 | 0.50 | 0.00
|  |  |  | IW+4 | 391.5 | 0.00 | 0.99 | 0.0099 | 0.00

##  Appendix C OLR of H2 atmospheres

Figure 16: _T_ op: Thermal emission spectra for an H2-dominated atmosphere
overlying a highly reduced mantle (IW-4) for a varying surface partial
pressure of hydrogen as shown in the legend. Bottom: OLR plotted against
p${}_{\ch{H2}}$ for a fixed surface temperature of $T_{\rm s}=2800$ K showing
a decrease in OLR with increase in H2 surface partial pressure.