1. Introduction {#s0005}
===============

The formation of Mars\' nebula-captured, catastrophically degassed and impact delivered protoatmosphere is directly connected to the planet\'s formation time scale, the nebula dissipation time, its orbital location and the planet\'s small mass compared to Earth and Venus. [@bib15; @bib16] investigated for the first time the hydrodynamic loss of oxygen from primitive atmospheres of Venus and Mars in detail. However, the pioneering studies of [@bib15; @bib16] are based on meanwhile outdated terrestrial planet formation models in which the time of the final accretion for terrestrial planets occurred $\geq 100\ {Myr}$ after the formation of the Sun ([@bib93]). Furthermore, in these pioneering studies by [@bib15; @bib16] the cooling phase of the magma ocean was expected to occur after $\sim 100\ {Myr}$, while more recent studies indicate that the solidification of magma oceans even with depths of up to $\sim 2000\ {km}$ is a fast process and mantle solidification of $\sim 98\%$ can be completed in $\leq 5\ {Myr}$ (e.g. [@bib25; @bib26; @bib63; @bib60; @bib37]). Moreover, it is also important to note that the assumption of several previous studies, that terrestrial planets, including early Mars finished their accretion late, resulted also in ages where the soft X-ray and extreme ultraviolet (XUV) flux of the young Sun was much lower compared to the high XUV flux values, which are now known from multi-wavelength observations of so-called young solar proxies (e.g., [@bib34; @bib78; @bib35; @bib20]). Because of the lack of accurate data, [@bib15; @bib16] applied an XUV enhancement factor as its highest value which was $\sim 25$ times higher than that of the present Sun.

In a recent review article on Mars\' origin [@bib12] argued that Mars\' small mass requires the terrestrial planets to have formed from a narrow annulus of material, rather than a disc extending to Jupiter. The truncation of the outer part of the disc was most likely related to migration of the gas giants, which kept the mass of Mars small. For the formation of the martian protoatmosphere this evidence from planet formation and latest dynamical models has important implications, because it would mean that Mars formed within a few million years and can be considered as a planetary embryo that never grew to a "real" more massive planet. Moreover, from the latest martian formation modeling scenarios most likely related to migration of the giants ([@bib91]), it is expected that most of the planet\'s building blocks consists of material that formed in a region just behind the ice line, so that the materials were more water-rich than the materials that were involved in the accretion of Venus and Earth.

[@bib12] suggests that the building blocks of early Mars could have consisted of $\sim 0.1–0.2\ {wt}.\%$ of H~2~O. The results presented in [@bib12] which are based on studies by [@bib91] agree in the amount of Mars\' initial water inventory with [@bib62] who applied also a dynamical model which yielded longer formation time scales. However, it should also be pointed out that model studies which consider different impact regimes than the before mentioned studies can also result in an early Mars which originated drier ([@bib42]). Although, it is obvious that our current knowledge of terrestrial planet formation and its related hydration is presently insufficient there is geomorphological evidence for water on early Mars, where $\sim 90\%$ was most likely outgassed and/or delivered during the first Gyr (e.g., [@bib16; @bib19; @bib8; @bib58]).

The main aim of the present study is to investigate in detail how long the before mentioned nebular captured and catastrophically outgassed protoatmospheres have been stable after Mars\' origin, and to understand how long the early planet\'s protoatmosphere survived against thermal atmospheric escape. In [Section 2](#s0010){ref-type="sec"} the formation of a nebula captured hydrogen envelope on early Mars and the expected catastrophically outgassed steam-type protoatmosphere based on materials which contain $\sim 0.1–0.2\ {wt}.\%$ H~2~O ([@bib12]) is described. In [Section 3](#s0025){ref-type="sec"} we discuss the early XUV radiation environment of the young Sun and the life time of the nebula gas which determines the age when the planet\'s protoatmosphere was exposed freely to the high solar XUV radiation field. In [Section 4](#s0030){ref-type="sec"} we study the upper atmosphere structure and the escape of the martian protoatmosphere by applying a time-dependent numerical algorithm, which is able to solve the system of 1-D fluid equations for mass, momentum, and energy conservation. Finally we describe the solar and atmospheric input parameters of the applied model and discuss the results.

2. Nebula-based and catastrophically outgassed protoatmospheres {#s0010}
===============================================================

For studying the potential habitability and atmosphere evolution of Mars, it is important to understand which sources and sinks contributed to the formation of the planet\'s initial atmosphere and water inventory. Furthermore, a detailed investigation on the escape-related evolution of the early martian protoatmosphere is important for understanding how long Mars may have had surface conditions that standing bodies of liquid water could have existed on the planet\'s surface. Generally four main processes are responsible for the formation of planetary atmospheres•capture of hydrogen and other gases (He, noble gases, etc.) from the solar nebular,•catastrophic outgassing of volatiles such as H~2~O, CO~2~, etc. and the formation of a steam atmosphere during and after the magma ocean solidification period,•impact delivery of volatiles by asteroids and comets,•degassing by volcanic processes during geological epochs.[Fig. 1](#f0005){ref-type="fig"} illustrates the expected atmosphere formation and loss scenarios for Mars during the planet\'s history. In the present work we focus on the origin and the evolution of the earliest martian protoatmosphere, consisting of hydrogen accumulated from the solar nebular and a catastrophically outgassed steam atmosphere after the planet finished its accretion and the magma ocean solidified.

2.1. Captured hydrogen envelope around early Mars {#s0015}
-------------------------------------------------

When proto-planets grow within the surrounding solar nebula by accretion of planetesimals, an extensive amount of gas will be attracted so that optically thick, dense hydrogen envelopes accumulate around a rocky core (e.g., [@bib66; @bib38; @bib95; @bib43; @bib44; @bib75]). The structure of such nebular-based hydrogen atmospheres was investigated decades ago by [@bib38] and [@bib68] for a wide range of planetary accretion rates, grain opacities, and gas disk densities. These pioneering studies obtained captured nebula gas around a Mars-mass body (i.e. $\sim 0.1M_{\oplus}$) of $8.4 \times 10^{24}\ g$ during nebular life times of $\sim 1–10\ {Myr}$, equivalent to the hydrogen content of $\sim 55$ Earth oceans ($1{EO}_{H} \approx 1.53 \times 10^{23}\ g$). More recent studies improved on these earlier results by adoption of realistic gas and dust opacities as well as a realistic equation of state leading to significantly lower atmosphere masses around bodies with masses that are $\sim 0.1M_{\oplus}$ ([@bib44]).

For the present investigation we computed a set of atmospheric models for Mars to obtain an estimate of the amount of gas collected from the protoplanetary disk into the planetary atmosphere. The hydrostatic structure equations have been solved by using the initial model integrator of the adaptive, implicit RHD-Code (TAPIR-code) the equation of state from [@bib80], gas opacities from [@bib29], and dust opacities by [@bib81]. Convective energy transport is included in TAPIR in the form of a turbulent convection model loosely based on the description by [@bib53].

For the conditions of the solar nebula at the position of the Mars orbit we assumed a gas density of $5 \times 10^{- 10}\ g\ {cm}^{- 3}$ and a temperature of 200 K. These values are in good agreement with restrains derived from the minimum-mass solar nebula ([@bib39]). The minimum-mass solar nebula (MSN) is a protoplanetary disk that contains the minimum amount of solid material which is necessary to build the planets of the Solar system.

The outer boundary conditions, i.e. nebula density and temperature, have been implemented at the Hill radius $r_{Hill}$ for all models as we consider $r_{Hill}$ to be a good approximation for the place where the essentially hydrostatic structure of the planetary atmosphere blends into the background disk structure. However, when calculating the captured atmospheric masses, i.e. the amount of gas in effect gravitationally bound to the planet, we used the minimum of $r_{Hill}$ and the Bondi radius $r_{Bondi}$, which turns out to be equal to the latter for all model runs by a margin of about a magnitude. The definition of the outer boundary condition seems to be, apart from the equation of state and nebular opacities, the main cause for the different captured atmospheric masses obtained by different authors. According to Ikoma (2012; private communication), the discrepancy between [@bib44] and [@bib38] is a case in point. In general, as also described by [@bib44], the atmospheres (and thus the atmospheric masses) of low-mass planets such as Mars are more dependent on outer boundary conditions than atmospheres of more massive Earth-like and super-Earth-type cores.

In order to get some measure of the uncertainties involved in our modeling, we covered a small parameter space by varying the most important atmospheric parameters: the planetary luminosity $L_{pl}$ and the dust grain depletion factor *f*. [Table 1](#t0005){ref-type="table"} and [Fig. 2](#f0010){ref-type="fig"} summarize the results of our atmospheric calculations.

$L_{pl}$ is related to the rate of infalling planetesimals$$L_{pl} \simeq \mathit{GM}_{pl}{\overset{˙}{M}}_{acc}\left( {\frac{1}{r_{pl}} - \frac{1}{r_{Hill}}} \right),$$with *G* the Newton gravitational constant, planetary mass $M_{pl}$, planetary radius $r_{pl}$ and planetesimal accretion rate ${\overset{˙}{M}}_{acc}$. Taking into account that according to [@bib91] and [@bib12] Mars\' formation was completed before or soon after the nebular gas disappeared at $\sim 3–10\ {Myr}$, $M_{pl}/{\overset{˙}{M}}_{acc}$ should be several \~10^6^ years or larger. On the other hand, according to [@bib25] and [@bib37] the cooling time scale of a Mars-size planet could be well above 1 Myr and thus it seems plausible that during the nebula-gas accumulation phase the heat flux from the interior significantly adds to the planetary luminosity. The lower limit of $L_{pl}$ can be constrained by the radiogenic luminosity estimated to be $\sim 10^{20}\ {erg}\ s^{- 1}$ for Mars ([@bib90]).

For higher planetary luminosities $T_{s}$ almost reaches 1000 K and it is well likely that models with, e.g. other boundary conditions or different dust opacity data yield even higher surface temperatures. It is important to note that H~2~O can also be produced on a planet if $T_{s} > 1500\ K$. In such a case the planet\'s surface melts and atmospheric hydrogen can be oxidized by oxides such as wüstite, magnetite and fayalite, which are inside the planet to produce H~2~O on the planet ([@bib79; @bib44]). However, the model results which yield high surface temperatures are also those with only comparatively thin hydrogen envelopes, which is reasonable as high luminosities and temperatures tend to inflate a planetary atmosphere. Therefore, one may speculate that Mars atmospheres with $T_{s} > 1500\ K$ will be too thin to allow for efficient H~2~O production from a captured and oxidized hydrogen envelope.

Before we discuss the radiation environment of the young Sun during the first 100 Myr after Mars\' origin and before we model the escape of the nebula-based hydrogen envelope we investigate the possible range of catastrophically outgassed steam atmospheres.

2.2. Magma ocean and outgassing of a steam atmosphere on early Mars {#s0020}
-------------------------------------------------------------------

As discussed before the terrestrial planets are thought to have reached their final sizes by a series of giant accretionary impacts. These impacts were energetic enough to produce melting of some depth in the planet (e.g., [@bib87; @bib77; @bib60]). This hypothesis is supported by the discovery of ^142^Nd isotope anomalies in martian SNC meteoroids, which indicate that early Mars developed a magma ocean ([@bib36; @bib31; @bib22]). Therefore, the first major degassed volatile-rich atmospheres likely resulted from the solidification of these magma bodies, and their release into the growing atmosphere in excess of what can be held in crystallizing silicate minerals ([@bib1; @bib2; @bib4; @bib64; @bib99; @bib24; @bib22; @bib25; @bib26; @bib37; @bib60]). In these models the magma ocean is expected to solidify from the bottom upward, because the slope of the adiabat is steeper than the slope of the solidus and thus they first intersect at depth. Because the energy and size of late accretionary impacts on early Mars are unknown, we consider a 500 km-deep magma ocean and, as an end-member, a 2000 km-deep or whole mantle magma ocean.

H~2~O and CO~2~ will be integrated in solidifying minerals in small quantities, will be enriched in solution in magma ocean liquids as solidification proceeds, and will degas into a growing steam atmosphere. At pressures and temperatures of magma ocean crystallization no hydrous or carbonate minerals will crystallize ([@bib72; @bib96]). Details of the solidification process, the mineral considered, their H~2~O and carbon partitioning, and other methods can be found in [@bib25]. The quantity of water and carbon compounds available for degassing is dependent upon the bulk composition of the magma ocean. The terrestrial planets are likely to have been accreted from chondritic material and planetesimals built from chondrites.

[@bib6] recently demonstrated that Earth\'s water, and therefore likely Mars\' water, originated mainly from rocky meteoritic material. [@bib94] reports up to 20 wt% of H~2~O in primitive undifferentiated chondrites, and [@bib46] reports $\sim 3\ {wt}\%$ H~2~O in achondrites, though most are drier. Enstatite chondrites match the oxygen isotope composition of the Earth, but smaller fractions of the wide compositional range of other meteorite compositions (see also [@bib6] and [@bib23]; and references therein) though volatile-rich material from greater radii in the planetary disk may have been added later in planetary formation (e.g., [@bib76; @bib71]). Here we assume that water and carbon are added to the growing rocky planets from rocky chondritic material.

Though the original quantity of water and carbon added during giant impacts remains unconstrained, we model two possible starting compositions, according to [@bib12] one with 1000 ppm H~2~O, and one with 2000 ppm H~2~O, each with one-fifth the CO~2~ content. These initial compositions are conservatively supported by the data of [@bib46]. For simplicity the carbon is assumed to be degassed as CO~2~, though reducing conditions may have produced CO or even CH~4~.

[@bib25] showed that for a range of magma ocean bulk compositions with between $\sim 500\mspace{6mu}{and}\mspace{6mu} 5000\ {ppm}$ H~2~O, between $\sim 70\%$ and $\sim 99\%$ of the initial water and carbon is degassed into the planetary atmosphere. Magma ocean solidification is therefore the most significant degassing event in a planet\'s evolution; the remainder of the volatiles are stored in the interior, available for later degassing via volcanic processes (e.g., [@bib33]).

[Table 2](#t0010){ref-type="table"} shows the partial surface pressures of catastrophically outgassed steam atmospheres, depending on the assumed bulk magma ocean depths and the initial H~2~O and CO~2~ contents in the magma ocean in wt.% according to the model of [@bib25]. One can see that a global magma ocean with the depth of \~500 km can produce a steam atmosphere with total surface pressures of $\sim 60–130\ {bar}$. If the magma ocean contained the whole mantle, surface pressures between $\sim 150\mspace{6mu}{and}\mspace{6mu} 310\ {bar}$ could have been outgassed.

3. Radiation environment during Mars\' initial life time {#s0025}
========================================================

The efficiency of thermal atmospheric escape is related to the planet\'s temperature at the base of the thermosphere which is located near the mesopause--homopause location in combination with the amount of the XUV flux that is absorbed in the upper atmosphere. The predicted evolution of the Sun\'s bolometric luminosity relative to its present value and the related equilibrium temperature $T_{eq}$ at Mars is shown in [Fig. 3](#f0015){ref-type="fig"}. We have chosen two stellar evolution tracks ([@bib9; @bib86]) which predict the lowest and highest luminosities, respectively, between 1 and 10 Myr compared to other authors (cf. Fig. 14 of [@bib86]). From [@bib9], the track with parameters $M = 1M_{\odot}$, *Y*=0.282, *Z*=0.02 and mixing length parameter $\alpha = 1.9$ was adopted, the track from [@bib86] has $M = 1M_{\odot}$, *Y*=0.288, *Z*=0.02 and $\alpha = 1.68$. For planetary atmospheres that are in long-term radiative equilibrium the so-called planetary skin temperature is $T_{eff} \approx T_{eq}$. The lower panel of [Fig. 3](#f0015){ref-type="fig"} shows the corresponding evolution of the equilibrium temperature of Mars, which is about 200 K, 3--4 Myr after the Sun\'s origin. We assume a constant albedo over time and adopt a present-day value of $T_{eq} = 217\ K$. One should note that variations of the albedo due to the evolution of Mars\' early atmospheric composition and the Sun\'s spectral energy distribution could alter the predicted evolution of $T_{eq}$ as shown in [Fig. 3](#f0015){ref-type="fig"}. Thermal escape of the martian protoatmosphere was driven by the XUV emission of the young Sun. The evolution of this high-energy emission of a solar-type star can be roughly divided into two regimes, the saturation phase and the post-saturation evolution. During the saturation phase the stellar X-ray flux does not scale with the stellar rotation period and is saturated about 0.1% of the bolometric luminosity $L_{bol}$ ([@bib73; @bib45]). After the Sun settled on the main sequence and began to spin down from a possibly shorter period to about 2 days due to spin-down via angular momentum loss by the solar wind, the post-saturation phase began. During this phase, the XUV emission of the Sun was determined by its rotation period. A reconstruction of the XUV-evolution during this time period was attempted in the "Sun in Time" program ([@bib35] and references therein). By studying a sample of solar analogs of different ages [@bib78] found that the Sun\'s XUV flux enhancement factor $I_{XUV}$ at Earth\'s orbit in the wavelength range of 1--1200 Å can be calculated as$$I_{XUV} = {(t/4.56)}^{- 1.23}$$with the age *t* in Gyr. This relation was calibrated back to an age of 100 Myr corresponding to the youngest solar analog in their sample. However, deviations from this power law are possible during the first few hundred Myr because the stellar rotation periods, which determine the efficiency of the magnetic dynamo and, hence, the XUV emission during this phase, are not unique.

The XUV-evolution during the saturation phase was, as mentioned above, mainly determined by the evolution of $L_{bol}$. Due to the difficulty of observing stars in the EUV because of the strong absorption by the interstellar medium, much of what is known about the high-energy emission of very young stars is extrapolated from X-ray observations. Between the zero-age main sequence (ZAMS), which the Sun reached at an age of about 50 Myr according to stellar evolution models (e.g. [@bib9; @bib82]), and the end of the saturation phase, the solar XUV flux should have been approximately constant because of the more or less constant bolometric luminosity. For pre-main sequence (PMS) stars, the observed X-ray luminosities are in the order of a few $10^{30}\ {erg}\ s^{- 1}$ and show a large spread of more than an order of magnitude ([@bib74; @bib83]). These values are nevertheless consistent with the saturation level of main-sequence stars mentioned above because of the more luminous PMS-Sun and the observed evolution of the stellar X-ray emission during the first tens of Myr seems to be determined mainly by changes of $L_{bol}$ ([@bib74; @bib13]).

The estimated past evolution of the Sun\'s XUV flux, scaled to the orbit of Mars and normalized to the average present solar value of $2\ {erg}\ {cm}^{- 2}\ s^{- 1}$ (scaled from the present value at Earth of $4.64\ {erg}\ {cm}^{- 2}\ s^{- 1}$; [@bib78]), is shown in [Fig. 4](#f0020){ref-type="fig"}. The solid line indicates the post-saturation evolution after Eq. [(2)](#eq0010){ref-type="disp-formula"} and the symbols correspond to data from solar analog stars and the Sun. The dotted lines sketch a possible PMS-XUV evolution based on the evolution of $L_{bol}$ using theoretical evolutionary tracks for a solar mass star ([@bib9]) and assuming that the XUV emission consists mainly of X-rays, so that $L_{XUV}/L_{bol} \approx L_{X}/L_{bol} \approx 10^{- 3.2 \pm 0.3}$. The value of the saturation level is adopted from [@bib73] for stars of about one solar mass. The uncertainties of the Sun\'s XUV emission before the ZAMS are large because of the dependence of its activity level on the convection zone depth and the rotational history, which in turn depends on the disk locking history. Moreover, the contribution of EUV to the total XUV flux is observationally unconstrained because of strong absorption by the interstellar medium. Therefore we adopt a constant average XUV flux level of about 100 times the present value for our escape rate calculations.

The shaded area indicates the approximate formation time of Mars which occurred during the first few Myr ([@bib12]). The inner disk was still present after Mars formed, and the inner planets were still forming. An inner disk would have absorbed a significant fraction of the Sun\'s XUV radiation until it became optically thin so that the XUV flux actually received by Mars could have been lower than estimated in [Fig. 4](#f0020){ref-type="fig"}. Typically, inner disks disperse on timescales within a few Myr to 10 Myr (e.g. [@bib65; @bib69; @bib40]).

Thus, if one compares the latest views of Mars\' origin and age with that of the radiation history of the young Sun and the nebula dissipation time, Mars\' nebula-based and/or outgassed steam atmosphere as well as volatiles which were delivered by frequent impacts were exposed to an XUV flux which was $\sim 100$ times stronger compared to that of the present Sun during $\sim 95–100\ {Myr}$ after the planet\'s origin. In the following section we investigate how long early Mars could have kept these hydrogen-rich protoatmospheres against XUV-driven thermal atmospheric escape.

4. Thermal escape of Mars\' protoatmosphere {#s0030}
===========================================

At present Mars the CO~2~-rich thermosphere is in hydrostatic equilibrium, while a hydrogen-rich upper atmosphere of the protoatmosphere that is exposed to the high XUV flux of the young Sun will hydrodynamically expand and the bulk atmospheric particles can escape efficiently (e.g., [@bib92]; [@bib15; @bib16]; [@bib85; @bib57; @bib58; @bib59]). For this reason we apply a 1-D hydrodynamic upper atmosphere model to the martian protoatmosphere and calculate the XUV-heated hydrogen-dominated dynamically expanding upper atmosphere structure and the thermal hydrogen escape rates, including dissociated and dragged heavier atmospheric main species.

4.1. Energy absorption and model description {#s0035}
--------------------------------------------

The thermosphere is heated due to the absorption, excitation, dissociation and ionization of the gas by the incoming solar XUV radiation. By averaging the XUV volume heating rate over Mars\' dayside the volume heat production rate $q_{XUV}$ due to the absorption of the solar radiation can then be written as (e.g., [@bib27; @bib59])$$q\left( t,r \right) = \frac{\eta n\sigma_{a}}{2}\int_{0}^{\pi/2 + \arccos(1/r)}J\left( t,r,\Theta \right)\sin\ \Theta\ d\Theta,$$with the polar angle $\Theta$ and $J\left( t,r,\Theta \right) = J_{XUV}e^{- \tau(t,r,\Theta)}$, where$$\tau\left( t,r,\Theta \right) = \int_{r\ \cos\ \Theta}^{\infty}\sigma_{a}n\left( t,\sqrt{s^{2} + r^{2}\ \sin^{2}\Theta} \right)\ \mathit{ds},$$*q* is the volume heating rate depending on the radial distance, *n* the atmospheric number density which is a function of time and spherical radius *r*, *η* the heating efficiency which corresponds to the fraction of absorbed XUV radiation which is transformed into thermal energy. Depending on the availability of IR-cooling molecules such as $H_{3}^{+}$ or CO~2~ it is known from various studies that $\left. \eta \right.\sim 15–60\%$ ([@bib15; @bib16; @bib97; @bib55; @bib61; @bib52]). $\sigma_{a}$ is the absorption cross-section of hydrogen, and $J_{XUV}$ is the XUV energy flux of the young Sun outside the protoatmosphere.

For studying the XUV-exposed structure of the upper atmosphere we solve the system of the 1-D fluid equations for mass, momentum, and energy conservation in spherical coordinates by applying a non-stationary 1D hydrodynamic upper atmosphere model which is described in detail in [@bib27]$$\frac{\partial\rho r^{2}}{\partial t} + \frac{\partial\rho\mathit{vr}^{2}}{\partial r} = 0,$$$$\frac{\partial\rho\mathit{vr}^{2}}{\partial t} + \frac{\partial\left\lbrack {r^{2}\left( \rho v^{2} + P \right)} \right\rbrack}{\partial r} = \rho\mathit{gr}^{2} + 2\mathit{\Pr},$$$$\frac{\partial r^{2}\left\lbrack {\frac{\rho v^{2}}{2} + \frac{P}{\left( \gamma - 1 \right)}} \right\rbrack}{\partial t} + \frac{\partial\mathit{vr}^{2}\left\lbrack {\frac{\rho v^{2}}{2} + \frac{\gamma P}{\left( \gamma - 1 \right)}} \right\rbrack}{\partial r} = \rho\mathit{vr}^{2}g + q_{XUV}r^{2},$$with pressure$$P = \frac{\rho}{m_{H}}\mathit{kT},$$and gravitational acceleration$$g = - \nabla\Phi.$$We note that we neglect the conduction term in the equations because as shown later the energy flux related to thermal conductivity is less important under these extreme conditions compared to the energy flux of the hydrodynamic flow. Here, *ρ*, *v*, *P* and *T* are the mass density, radial velocity, pressure and temperature of the atmosphere, *r* is the radial distance from the center of the planet, $m_{H}$ is the mass of atomic hydrogen, *G* is Newton\'s gravitational constant, *γ* is the polytropic index or the ratio of the specific heats, and *k* is the Boltzmann constant.

For computational convenience we introduce normalized parameters$$\widetilde{P} = P/\left( n_{0}\mathit{kT}_{0} \right),\quad\widetilde{\rho} = \rho/\left( n_{0}m \right),\widetilde{v} = v/v_{0},\quad v_{0} = \sqrt{\mathit{kT}_{0}/m},\quad\widetilde{T} = T/T_{0},\widetilde{q} = \mathit{qr}_{0}/\left( \mathit{mn}_{0}v_{0}^{3} \right),\quad\widetilde{r} = r/r_{0},\widetilde{t} = \mathit{tv}_{0}/r_{0},\quad\beta = \mathit{GmM}_{pl}/\left( r_{0}\mathit{kT}_{0} \right).$$Here *r*~0~, *T*~0~, *n*~0~ and *v*~0~ are the radius, temperature, number density and thermal velocity at the lower boundary of the simulation domain. *β* is the so-called Jeans parameter ([@bib14]). For values of $\beta > 30$ the atmosphere can be considered as bound to the planet. For values which are lower classical Jeans escape happens. For *β* values that are \~2--3.5 the thermal escape can be very high ([@bib88]; [@bib89]) and for for values $\leq 1.5$ classical blow-off occurs and the atmosphere escapes uncontrolled. Using normalizations [(10)](#eq0050){ref-type="disp-formula"}, we obtain the normalized XUV flux distribution in the planetary atmosphere$$\widetilde{J}\left( \widetilde{r},\Theta \right) = J/J_{\mathit{XUV}0} = \exp\left\lbrack - \widetilde{\tau}\left( \widetilde{r},\Theta \right) \right\rbrack,$$where$$\widetilde{\tau}\left( \widetilde{r},\Theta \right) = \int_{\widetilde{r}\ \cos\ \Theta}^{\infty}a\widetilde{n}\left( \widetilde{t},\sqrt{s^{2} + {\widetilde{r}}^{2}\ \sin^{2}\Theta} \right)\ \mathit{ds},$$where $a = \sigma_{a}n_{0}r_{0}$ is obtained due to the normalization of Eq. [(4)](#eq0020){ref-type="disp-formula"}. The normalized heating rate is given by$$\widetilde{q}\left( \widetilde{r} \right) = A\widetilde{n}\int_{0}^{\pi/2 + \arccos(1/\widetilde{r})}\exp\left\lbrack - \widetilde{\tau}\left( \widetilde{r},\Theta \right) \right\rbrack\ \sin\ \Theta d\Theta,$$Integrating [(13)](#eq0065){ref-type="disp-formula"} over the whole domain we obtain the total energy absorption in the normalized units which is proportional to the incoming XUV flux.$$\int_{1}^{\infty}\widetilde{q}4\pi{\widetilde{r}}^{2}\ d\widetilde{r} = \pi\frac{J_{XUV}}{\mathit{mn}_{0}v_{0}^{3}}\ \frac{r_{{XUV}_{eff}}^{2}}{r_{0}^{2}},$$where $r_{{XUV}_{eff}}$ is the effective radius of the XUV energy absorption which is dependent on the density distribution. This effective radius can be determined from the following equation:$$r_{{XUV}_{eff}}^{2}/r_{0}^{2} = 1 + 2\int_{1}^{\infty}\left\lbrack 1 - \widetilde{J}\left( s,\pi/2 \right) \right\rbrack s\ \mathit{ds}.$$As shown by [@bib92] the effective radius can exceed the planetary radius quite substantially for a planetary body, which has a low gravity field and hence in low values of the *β* parameter when its atmosphere is exposed by high XUV fluxes. We get the appropriate coefficient$$A = \frac{\eta\sigma_{a}r_{0}J_{XUV}}{2\ \mathit{mv}_{0}^{3}}$$in formula [(13)](#eq0065){ref-type="disp-formula"} to satisfy Eq. [(14)](#eq0070){ref-type="disp-formula"} for a given value of $J_{XUV}$.

4.2. Boundary conditions at the lower thermosphere {#s0040}
--------------------------------------------------

The boundary conditions at the lower boundary of our simulation domain are the gas temperature *T*~0~, number density *n*~0~ and the corresponding thermal velocity *v*~0~ near the mesopause--homopause level *r*~0~, that is at present martian conditions located near the base of the thermosphere. The value of the number density *n*~0~ at the base of the thermosphere can never be arbitrarily increased or decreased as much as by an order of magnitude, even if the surface pressure on a planet varies during its life time by many orders of magnitude. The reason for this is that the value of *n*~0~ is strictly determined by the XUV absorption optical depth of the thermosphere. The temperature *T*~0~ at the base of the thermosphere $z_{0} = \left( r_{0} - r_{pl} \right)$ is determined only by the variation of the equilibrium or skin temperature of a planet, to which the base temperature *T*~0~ is usually quite close. In a hotter environment corresponding to the catastrophically outgassed steam atmosphere, which is for instance strongly heated by frequent impacts, *z*~0~ and the above estimated XUV effective radius $r_{XUV}$ simply rises to a higher altitude where the base pressure retains the same constant value as in a less dense atmosphere.

[@bib63] studied with a 1-D radiative--convective atmospheric model the coupling between magma oceans and outgassed steam atmospheres and found that for surface temperatures $T_{s} \geq 2350\ K$, the radiative temperature of a planetary atmosphere $T_{eff}$ can rise from $\sim 230\ K$ to $\sim 300–400\ K$, while $T_{eq}$ remains close to $\sim 200\ K$. However, such extreme surface temperatures are only be reached during the totally and partially molten stage of the magma ocean, which last only for $\approx 0.1\ {Myr}$ ([@bib60]). For this reason we assume in the following thermal escape calculations similar as in [Fig. 3](#f0015){ref-type="fig"} a temperature *T*~0~ of 200 K at the base of the thermosphere which corresponds to the equilibrium *T*~*eq*~, or skin temperature of Mars\' orbit. We point out that an uncertainty of $\pm 20\ K$ as shown in the evolutionary path of $T_{eq}$ in [Fig. 3](#f0015){ref-type="fig"} does not have a big influence in the modeled escape rates. We assume an atomic hydrogen density $n_{0} = 10^{13}\ {cm}^{- 3}$ at the lower boundary of the hydrogen-rich protoatmosphere (e.g., [@bib48; @bib84]). According to [@bib84], similar number density values can be expected also to H~2~O mixing ratios $\geq 50\%$ in a humid steam-like terrestrial planetary atmosphere.

The upper boundary of our simulation domain is chosen at $70r_{pl}$, but the results of our hydrodynamic model are considered as accurate only until the Knudsen number *Kn*, which is the ratio between the mean free path and the scale height, reaches 0.1 ([@bib47]). Because of the high XUV flux the whole bulk atmosphere reaches the martian escape velocity below or at this altitude level.

The high XUV flux of the young Sun will dissociate most H~2~ and H~2~O molecules in the thermosphere so that the upper part of the studied protoatmospheres should be mainly dominated by hydrogen atoms ([@bib48; @bib15; @bib97; @bib51; @bib58; @bib59]). As it was shown by [@bib63], during periods of magma ocean related hot surface temperatures the tropopause location in an overlaying steam atmosphere can move at an Earth or Venus-like planet from its present altitude of $\sim 30–40\ {km}$ up to higher altitudes of $\sim 300–550\ {km}$. Depending on the surface temperature and pressure of the steam atmosphere in such an environment the mesopause level would then also move to higher altitudes. By applying the model of [@bib63] to the outgassed steam atmospheres given in [Table 2](#t0010){ref-type="table"}, we obtain mesopause altitudes of $\sim 330–350\ {km}$, $\sim 450–465\ {km}$, $\sim 610–630\ {km}$ and $\sim 750–850\ {km}$ for surface temperatures of $\sim 1500\ K$, $\sim 2000\ K$, $\sim 2500\ K$ and $\sim 3000\ K$, respectively. This mesopause altitudes have been estimated by detailed modeling of the lowest 600 km of the steam atmospheres. The altitudes above 600 km are obtained from an extrapolation with a precision of $\sim 20\ {km}$ in the 600--700 km range and $\sim 50\ {km}$ above 700 km. The simulations used a grey approximation for the radiative transfer which can influence the profile by setting the mesospheric temperature and thus scale height to a slightly different value, but we do not expect that this uncertainty changes these altitudes by more than 20 km. As one can see, even in the most extreme case with a surface temperature of 3000 K, the mesopause altitude lies below 1000 km for a body with Earth\'s gravity. However, it will most likely be higher than 1000 km with a lower gravity such as Mars'. We plan to study the response to the mesopause location and its influence in the escape of outgassed steam atmospheres on Mars in detail in the near future.

However, for illustrating the importance and influence of the mesopause location in the escape efficiency we modeled also a case where we assumed that *z*~0~ is located at 1000 km above the planet\'s surface. That hydrogen-dominated gas envelopes with hot surface temperatures will have larger radii compared to planets with present time atmospheres is also addressed in [@bib67]. However, the planetary mass--radius relationship model results for small and low mass bodies remain highly uncertain.

5. Results {#s0045}
==========

5.1. Thermospheric profiles and escape rates {#s0050}
--------------------------------------------

By exposing the martian protoatmospheres with a 100 times higher XUV flux compared to today\'s solar value in martian orbit, we find that the convective thermal energy flux is less significant than the thermal energy flux related to the hydrodynamic flow. [Fig. 5](#f0025){ref-type="fig"} compares the thermal energy flux due to the hydrodynamic flow (curves at the top: dotted lines: $\eta = 15\%$; dashed-lines: $\eta = 40\%$) per steradian of the atmospheric particles with the convective thermal energy flux (curves at the bottom: dotted lines: $\eta = 15\%$; dashed-lines: $\eta = 40\%$), obtained by our hydrodynamic model. The two sudden decreases in the convective thermal energy flux curves can be explained, because this flux is proportional to the temperature gradient, and therefore it decreases in the vicinity of the temperature maximum and minimum. At first point we have a strong temperature maximum, and at the second point we have shallow temperature minimum. By comparing the two fluxes one can conclude that under such extreme conditions the influence of the thermal conduction on the atmospheric escape is expected to be rather small. Therefore we neglect the thermal conduction term in the energy equation.

[Fig. 6](#f0030){ref-type="fig"} shows examples of the XUV volume heating rate and the corresponding upper atmosphere structure of a hydrogen dominated upper atmosphere of early Mars for a heating efficiency *η* of 15 and 40% with $T_{0} = 200\ K$, and $n_{0} = 10^{13}\ {cm}^{- 3}$, which is exposed to a XUV flux which is 100 times higher at the planet\'s orbit compared to that of the present Sun and assumed mesopause locations at 100 km and 1000 km. Under these assumptions the bulk atmosphere reaches the escape velocity $v_{esc}$ at about 35*r*~0~ and 24*r*~0~ for heating efficiencies *η* of 15% and 40%, respectively. One can see from the volume heating rate $q_{XUV}$ and the connected temperature profile that the XUV deposition peak occurs above 1.5*r*~0~ for $z_{0} = 100\ {km}$ and at $\sim 2R_{0}$ if $z_{0} = 1000\ {km}$. This can also be seen in the temperature profiles, which decrease first due to adiabatic cooling until the high XUV flux of the young Sun balances the cooling process due to the XUV heating, resulting in the more or less constant temperature profile between $\sim 5\mspace{6mu}{and}\mspace{6mu} 35r_{pl}$ of $\sim 50–70\ K$. One can also see that for a heating efficiency *η* of 40% the adiabatic cooling is stronger at distances that are $\leq 2.0r_{0}$. The corresponding temperature drop is also larger for an *η* of 40% compared to that of 15%. For larger distances $r > 2r_{0}$, the energy absorption is larger and in the case of 40% efficiency, the additional heating exceeds the cooling. Therefore, the temperature decrease is less pronounced for large distances in the case of higher heating efficiencies compared to the lower value of $\eta = 15\%$.

[Table 3](#t0015){ref-type="table"} shows the thermal hydrogen atom escape rates and relevant atmospheric parameters at the critical distance where the bulk atmosphere reaches sonic speed for a lower and higher heating efficiency *η* of 15% and 40% and for z~0~ at 100 and 1000 km altitude. The temperature *T*~0~ and the number density *n*~0~ is assumed to be 200 K and 10^13^ cm^−3^ in all four cases. One can see from [Table 3](#t0015){ref-type="table"} that depending on *z*~0~ the thermal hydrogen escape rates can reach values between $\sim 2 \times 10^{32}$ and $\sim 10^{33}$ H atoms per second. The present time thermal hydrogen atom escape from Mars by the classical Jeans escape is about $\sim 1.5 \times 10^{26}\ s^{- 1}$ (e.g., [@bib54]), which indicates that the thermal escape of hydrogen from Mars\' protoatmosphere could have been up to $\sim 6–7$ orders of magnitude higher.

5.2. Escape of the nebula captured hydrogen envelope {#s0055}
----------------------------------------------------

By knowing the escape rate of hydrogen atoms we can now estimate the loss of the expected nebula-based hydrogen envelope from proto-Mars. If we use the most massive captured hydrogen envelope shown in [Table 1](#t0005){ref-type="table"} of $\sim 6.5 \times 10^{22}\ g$, corresponding to a luminosity of $\sim 2.4 \times 10^{22}\ {erg}\ s^{- 1}$ and a dust grain depletion factor *f* of 0.1, the envelope would be lost during $\sim 1.3–7.5\ {Myr}$. The escape time span depends on the heating efficiency and the location distance of the lower thermosphere. A more realistic captured atmosphere with a mass of $\sim 5 \times 10^{21}$ g would be lost in $\sim 0.1–0.5\ {Myr}$. From these escape estimates one can conclude that a captured nebular-based hydrogen envelope should have been lost very fast from the planet after the nebula dissipated. If the radius r~0~ in the nebula captured hydrogen envelope was at further distances compared to our assumed values, then the escape rates would be higher.

5.3. Escape of the catastrophically outgassed steam atmosphere {#s0060}
--------------------------------------------------------------

According to the outgassing of the magma ocean depth dependent steam atmospheres shown in [Table 2](#t0010){ref-type="table"}, even the deepest and most volatile-rich case completes solidification and degassing in $\leq 2 \times 10^{5}$ years. The heat loss from the small planetary body is fast enough to allow rapid solidification in a convecting magma ocean. Theoretical studies by [@bib25] showed that one can expect that the volatiles are likely to be released toward the end of solidification of the magma ocean in a "burst".

The applied magma ocean model of [@bib25] and the related results discussed in [Section 2.2](#s0020){ref-type="sec"} predict a surface temperature of $\geq 800\ K$ at the end of solidification, which lies above the condensation temperature for H~2~O of $\sim 645\ K$. If there is a solid-state mantle overturn, there will be a big temperature jump after $\sim 2–4\ {Myr}$, when the hot mantle cumulates rise up in Mars because of their buoyancy, and advect their great heat with them. According to [@bib12] Mars\' most likely finished its accretion or remained as a planetary embryo when the surrounding nebula was still present around the martian orbit location in its later stages. If this was the case the catastrophically outgassed volatiles could easily build up rapidly around the rocky embryo. As soon as this catastrophically outgassed steam atmosphere was released from the nebula, the efficient escape of the atmosphere which was driven by the high XUV flux of the young Sun began.

According to [@bib60], who studied the thermal evolution of an early martian magma ocean in interaction with a catastrophically outgassed $\sim 43\ {bar}$ H~2~O and $\sim 14\ {bar}$ CO~2~ steam atmosphere, water vapor would start to condense into liquid H~2~O after $\sim 0.1\ {Myr}$. On the other hand, such a fast cooling of the steam atmosphere contradicts the isotopic analysis of martian SNC meteorites by [@bib22], where analyzed data can be best explained by a progressive crystallization of a magma ocean with a duration of up to $\sim 100\ {Myr}$. Therefore, [@bib60] suggest that frequent impacts of large planetesimals and small embryos, which have been not included in their study, could have kept the surface during longer times warmer. This suggestion is quite logical because one can also expect that during the first 100 Myr after the origin of the Solar System the young planets have been frequently hit by large impactors (e.g., [@bib3; @bib4; @bib32; @bib58]), which may have enhanced the input energy flux above the value which is defined by the solar flux alone. In such a case one will obtain a hotter surface that prevent atmospheric H~2~O vapor from condensing (e.g., [@bib38; @bib32; @bib56; @bib57; @bib60]).

One should also note that for the surface temperatures of $\sim 500\ K$, which are expected during the "Mush" stage ([@bib60]), according to [@bib49] one can also expect water vapor mixing ratios at the mesopause level near to 1. For this reason H~2~O will continue to escape effectively, even if there are periods of liquid water on the planet\'s surface. However, the mesopause level will be closer to the planet\'s surface and the escape rates will be reduced and may have values which correspond to case CI in [Table 3](#t0015){ref-type="table"}.

In the outgassed steam atmosphere, the H~2~O molecules in the upper atmosphere will be dissociated by the high XUV flux of the young Sun and by frequently occurring impacts in the lower thermosphere (e.g., [@bib16; @bib56; @bib57]). [@bib85] showed that for XUV fluxes which are $> 10$ times that of today\'s Sun, CO~2~ and/or CH~4~ molecules in the martian upper atmosphere will also be destroyed, so that C atoms can escape similar to O atoms with escape flux values which are $\geq 10^{11}\ {cm}^{- 2}\ s^{- 1}$. From this study one can expect that for an XUV flux which is $\sim 100$ times stronger than the present solar value most CO~2~ and/or CH~4~ molecules will be dissociated as soon as they are exposed to the high XUV radiation. Therefore, one can assume that O and C atoms should also populate the lower hydrogen dominated thermosphere so that they can be dragged by the dynamically outward flowing hydrogen atom flux ([@bib98; @bib100; @bib15; @bib16; @bib41; @bib56; @bib58]).

As initial amount and composition of the outgassed atmosphere we adopt the four cases presented in [Table 2](#t0010){ref-type="table"}. With the given partial surface pressures of H~2~O and CO~2~ and assuming that all molecules are dissociated under the high XUV flux of the young Sun, we calculate the initial inventories of atomic H, O, and C. For all four cases atomic hydrogen is the most abundant species ($N_{H}/N = 0.61$), followed by oxygen ($N_{O}/N = 0.36$), whereas C is just a minor constituent ($N_{C}/N = 0.03$). Hydrogen is assumed to escape at rates given in [Table 3](#t0015){ref-type="table"}. The fractionation factors $x_{i} = L_{i}/\left( L_{H}f_{i} \right)$ for an escaping atmosphere composed of two major (here H, O) and several minor species (here only one, namely C) are given by Eqs. (19) and (35) of [@bib100], where $f_{i} = n_{i}/n_{H} = N_{i}/N_{H}$ is the mixing ratio with respect to H and $L_{i}$ are the escape fluxes of the heavy species *i* given in s^−1^. Using the definition of *x* the escape fluxes of O and C can then by written as$$L_{O} = L_{H}f_{O}x_{O} = L_{H}f_{O}\left( {1 - \frac{\mu_{O} - 1}{\mu_{O}\Phi_{O}}\ \frac{1}{1 + f_{O}}} \right)$$$$L_{C} = L_{H}f_{C}\frac{1 - \frac{\mu_{C} - 1}{\mu_{C}\Phi_{C}} + \frac{b_{HC}}{b_{OC}}f_{O}x_{O} + \frac{b_{HC}}{b_{HO}}f_{O}\left( 1 - x_{O} \right)}{1 + \frac{b_{HC}}{b_{OC}}f_{O}}$$with $\mu_{i} = m_{i}/m_{H}$, the binary diffusion parameters *b*, and the parameter$$\Phi_{i} = \frac{L_{H}\mathit{kT}}{3\pi\mathit{GMm}_{i}b_{Hi}}$$which represents approximately the ratio of drag to gravity (drag dominates if $\Phi_{i} > \left( \mu_{i - 1} \right)/\mu_{i}$). The factor $3\pi$ stems from our adopted solid angle over which we assume that escape takes place and which is therefore included in the values of $L_{H}$. The binary diffusion parameter of O in H $b_{HO} = 4.8 \times 10^{17}T^{0.75}\ {cm}^{- 1}\ s^{- 1}$ was taken from Table 1 of [@bib98]. $b_{HC}$ was assumed to be equal to $b_{HO}$, and $b_{OC}$ is roughly estimated as $2 \times 10^{17}T^{0.75}\ {cm}^{- 1}\ s^{- 1}$. However, we note that changing these parameters, as well as the adopted temperature, does not affect the results if the hydrogen escape rate is large.

Eqs. [(17) and (18)](#eq0085 eq0090){ref-type="disp-formula"} were derived under the assumption that the flow is isothermal and subsonic ([@bib100]), which is actually not valid during the phase of saturated solar XUV emission studied here. However, they showed that these simpler analytic approximations become comparable to the non-isothermal transonic solutions if $x_{i}\operatorname{âª¢}1/\mu_{i}$ and $\Phi_{i}$ is large. These conditions are both fulfilled here because the masses of O and C are much larger than H and hydrogen escapes very efficiently (hence, $\Phi_{i}\operatorname{âª¢}$). It was also assumed that the mixing ratios *f*~*i*~ are approximately constant with height. Expressions for *x*~*i*~ without this constraint include terms with an exponential function that goes to zero for large $\Phi_{i}$ ([@bib98; @bib100]) and would therefore vanish for the cases studied here.

[Figs. 7--10](#f0035 f0040 f0045 f0050){ref-type="fig"} show the temporal evolution of the partial surface pressures of H, O, and C normalized to the initial total surface pressure for the four cases of the outgassed atmospheres given in [Table 2](#t0010){ref-type="table"}. These results have been obtained by adopting the modeled hydrogen loss rates shown in the cases CI, CII, CIII and CIV in [Table 3](#t0015){ref-type="table"} corresponding to 100 times the present solar XUV flux and a lower boundary temperature *T*~0~ of 200 K but low and high heating efficiencies *η* of 15% and 40%, and *z*~0~ at 100 km and 1000 km altitude. This temperature is also used for evaluating $\Phi_{i}$, but choosing a different value does not affect the results because the large $L_{H}$ dominates. The initial hydrogen inventory evolves with a constant escape rate, because the timescale for total hydrogen loss occurs during a time frame between $\sim 0.4$ and 12 Myr, well below the time it takes the Sun to drop out of its saturation phase. The evolution of the inventories, and hence partial surface pressures, of O and C are found numerically by integration of Eqs. [(17) and (18)](#eq0085 eq0090){ref-type="disp-formula"}.

From these figures one can see that the timescale for complete loss of H, O, and C, for a $\sim 50\ {bar}$ H~2~O and $\sim 10\ {bar}$ CO~2~ atmosphere for low *η* and z~0~ occurs in less than 2.5 Myr. If the base of the thermosphere would expand from 100 km to 1000 km and $\eta = 40\%$, such a steam atmosphere would be lost after $\sim 0.4\ {Myr}$. Depending on the initial volatile content and assumed heating efficiencies and *z*~0~, steam atmospheres with $\sim 260\ {bar}$ H~2~O and $\sim 55\ {bar}$ CO~2~ would be lost from early Mars between $\sim 2.1$ and 12 Myr. We also note that a magnetosphere would not protect the escape of the bulk atmosphere under these conditions because most of the atoms escape as neutrals until they become ionized due to the interaction with the early solar wind and plasma environment at large planetary distances ([@bib50; @bib57]).

As discussed above, the time scale for cooling of the steam atmosphere to temperature-pressure values, that water can condense and build lakes or even oceans is very important and the influence of energy deposition on planetary surfaces by frequent impacts of large planetsimals or small embryos have to be studied in coupled magma ocean-protoatmosphere models in the future. Therefore, it is also possible that all of our studied steam atmosphere scenarios presented in [Table 2](#t0010){ref-type="table"} may have been lost within a few Myr, before the atmospheres cooled to temperatures that big lakes or oceans could have formed.

However, for outgassed steam atmospheres with surface pressures $\operatorname{âª¢}50\ {bar}$, the timescale for total escape compared to the steam atmosphere cooling timescale could be larger, so that large lakes or water oceans could have been formed sporadically. In such scenarios water condensed and could have been present on the planet\'s surface for short time until the high thermal escape rates and impactors evaporated it again ([@bib32]). During this time and also during later stages a fraction of condensed, or via later impacts delivered H~2~O, may have been again incorporated by hydrothermal alteration processes such as serpentinization, so that remaining parts of it could be stored even today in subsurface serpentine ([@bib19]).

We point out that a detailed photochemical study, which includes processes such as dissociation, ionization, etc. of the outgassed CO~2~ molecules is beyond the scope of the present study. Our expectation that no dense CO~2~ atmosphere has build up on early Mars during the first 100 Myr is also supported by a study of [@bib85], who showed that the thermal escape of C atoms was so efficient even during the early Noachian, $> 4.1\ {Gyr}$ ago, that a CO~2~-dominated martian atmosphere could not have been maintained, and Mars most likely has begun its origin colder. In agreement with [@bib58] by the mid to late Noachian, as one can see from [Fig. 4](#f0020){ref-type="fig"}, the solar XUV flux would have become much weaker allowing the build up of a secondary CO~2~ atmosphere by volcanic outgassing ([@bib33]) and/or impact delivered volatiles.

Our results are also in agreement with the conclusions of [@bib11], which are based on the so far not detected carbonates, that no major surface sink of CO~2~ is present and the initial CO~2~, if it was more abundant, should have been lost from Mars very early other than being stored in surface reservoirs after having been dissolved in liquid water at the surface. However, it should be noted that the accumulation of a secondary outgassed CO~2~ atmosphere and volatiles, which could have been delivered by later impacts is highly dependent on less efficient atmospheric escape processes after the strong early hydrodynamic loss during the XUV-saturation phase of the young Sun as well as by the efficiency of carbonate precipitation, and serpentinization during the Hesperian and Amazonian epochs (e.g., [@bib17; @bib18; @bib58; @bib70]) .

Our result that Mars lost most likely the majority of its initial H~2~O inventory very early is in support of the hypothesis presented by [@bib5] that the planet could not develop an efficient plate tectonic regime due to the rapid removal of water by hydrodynamic escape. These authors suggest that the resulting low abundance of the remaining water in the martian mantle combined with weaker gravity than on Earth acted against the bending and foundering of lithospheric plates and the planet instead took the dynamic route of developing a thick stagnant lid. Because of the low size and gravity of Mars not enough water could be incorporated into the Martian mantle before it was lost to space so that plate tectonics never began.

6. Conclusions {#s0065}
==============

The production and loss of the earliest martian atmosphere which consisted of captured nebula gas (H, He, etc.) and outgassed and impact delivered volatiles (e.g. H~2~O, CO~2~, and CH~4~) have been studied. By using the latest knowledge of the origin of Mars summarized in [@bib12], we estimated the protoatmosphere masses and partial pressures and applied a 1-D hydrodynamic upper atmosphere model to the extreme XUV conditions of the young Sun. Depending on the amount of the outgassed volatiles, as well as the assumed heating efficiency and altitude location of the lower thermosphere, our results indicate that early Mars lost its nebular captured hydrogen envelope and catastrophically outgassed steam atmosphere most likely within $\sim 0.4–12\ {Myr}$ by hydrodynamic escape of atomic hydrogen. The main reasons for the fast escape of even a steam atmosphere with an amount of $\sim 70\%$ of an Earth ocean and $\sim 50\ {bar}$ CO~2~ within $< 12\ {Myr}$ are Mars\' low gravity and the $\sim 100$ times higher XUV flux of the young Sun, which lasted $\sim 100\ {Myr}$ after the Solar Systems origin. The efficient escape of atomic hydrogen, drags heavier atoms within the escaping bulk atmosphere so that they can also be lost to space. Our results support the hypotheses of [@bib85] that early Mars could not build up a dense CO~2~ atmosphere during the early Noachian. The results are also in agreement with the hypothesis presented in [@bib58] that after the planet lost its protoatmosphere the atmospheric escape rates were most likely balanced with the volatiles, which have been outgassed by volcanic activity and delivered by impacts until the activity of the young Sun decreased, so that the atmospheric sources could dominate over the losses $\sim 4.2–3.8\ {Gyr}$ ago.

P. Odert, H. Lammer, K. G. Kislyakova and Yu. N. Kulikov acknowledge support from the Helmholtz Alliance project "Planetary Evolution and Life". E. Dorfi, M. Güdel, K. G. Kislyakova, H. Lammer, A. Stökl and E. A. Dorfi acknowledge the Austrian Science Fund (FWF) for supporting this study via the FWF NFN project S116 "Pathways to Habitability: From Disks to Active Stars, Planets and Life", and the related FWF NFN subprojects, S 116 02-N1 "Hydrodynamics in Young Star-Disk Systems", S116 604-N16 "Radiation & Wind Evolution from T Tauri Phase to ZAMS and Beyond", and S11607-N16 "Particle/Radiative Interactions with Upper Atmospheres of Planetary Bodies Under Extreme Stellar Conditions". M. Leitzinger and P. Odert also acknowledge the support from the FWF project P22950-N16. N. V. Erkaev acknowledges support by the RFBR Grant no 12-05-00152-a. Finally, H. Lammer thanks M. Ikoma from the Department of Earth and Planetary Science, of the University of Tokyo, Japan, for discussions related to the accumulation of nebular-based hydrogen envelopes around Mars-mass bodies. Finally the authors thank an anonymous referee for the interesting and important suggestions and recommendations that helped to improve the results of our study.

![Illustration of Mars\' origin and protoatmosphere formation and evolution. The dotted lines correspond to the accumulation during the growth and escape of nebula-based hydrogen from proto-Mars. The onset of escape corresponds to the nebula dissipation time around $\sim 3–10\ {Myr}$, which is also the expected time period when Mars finished its accretion ([@bib12]). The short dashed lines illustrate the catastrophically outgassed volatiles and their expected escape after the planet\'s magma ocean solidified. Later on when the solar activity decreased a secondary CO~2~ atmosphere could have build up by volcanic activity ([@bib33; @bib58]) and the late heavy bombardment may also have delivered volatiles to Mars $\sim 3.8\ {Gyr}$ ago.](gr1){#f0005}

![Nebular-captured hydrogen envelopes for a Mars size and mass object at 1.5 AU, in units of Earth ocean equivalent amounts of hydrogen ($1{EO}_{H} = 1.53 \times 10^{23}\ g$) as a function of luminosity for three different dust grain depletion factors *f*=0.001 (dashed-dotted line), *f*=0.01 (dashed line), and *f*=0.1 (dotted line).](gr2){#f0010}

![Evolution of the Sun\'s bolometric luminosity relative to its present value (upper panel) and the equilibrium temperature of Mars (lower panel). The solid line corresponds to an evolution track of [@bib86] and the dotted line to [@bib9], both for a star of solar mass and metallicity. The present-day values in both panels are indicated by diamonds. For the evolution of the $T_{eq}$, a constant albedo was assumed.](gr3){#f0015}

![Evolution of the Sun\'s XUV emission normalized to the present value and scaled to the present martian orbit at 1.52 AU. The solid line indicates the evolution during the post-saturation phase ([@bib78]) with data of solar analogs (black dots) and the Sun indicated. The dotted lines indicate the approximate evolution of the saturated XUV emission estimated by $10^{- 3.2 \pm 0.3}L_{bol}$ ([@bib73]), with the bolometric luminosity taken from stellar evolution tracks of a solar mass star ([@bib9]). The shaded area indicates the expected formation time of Mars ([@bib12]). The dashed line shows our adopted average XUV value during the Sun\'s saturation phase.](gr4){#f0020}

![Comparison of the thermal energy flux per steradian of the hydrodynamical flow (upper dashed line: $\eta = 40\%$; dotted line: $\eta = 15\%$) with the thermal energy flux related only to the thermal conductivity (lower dashed: $\eta = 40\%$; dotted line: $\eta = 15\%$).](gr5){#f0025}

![Example of the XUV volume heating production rate (top left), temperature profile (top right), density profile (bottom left) and the velocity profile (bottom right) for a hydrogen-rich martian upper atmosphere with *z*~0~=100 km by assuming a heating efficiency of 15% (dotted lines) and 40% (dashed lines) and a temperature *T*~0~ at the base of the thermosphere of 200 K as a function of distance in planetary radii for a hydrogen-dominated upper atmosphere at Mars, that is exposed to a 100 times higher XUV flux compared to today\'s solar value. The solid line shown in the velocity profiles corresponds to the escape velocity $v_{esc}$ as a function of distance. The dashed--dotted lines ($\eta = 15\%$) and the dashed--dotted--dotted--dotted lines ($\eta = 40\%$) correspond to similar profiles but with $z_{0} = 1000\ {km}$. The hydrogen atoms reach the escape velocity below the theoretical exobase level at a location of $\sim 35r_{0}$ for $\eta = 15\%$ and at \~24 $r_{pl}$ for $\eta = 40\%$.](gr6){#f0030}

![Temporal evolution of the partial surface pressures $P_{surf}$ of H, O, and C normalized to the total initial surface pressure $P_{total}$ for the four compositions of outgassed atmospheres described in [Table 2](#t0010){ref-type="table"}. The hydrogen inventory evolves assuming a constant escape rate and parameters according to CI in [Table 3](#t0015){ref-type="table"} valid for 100 XUV. Both O and C are dragged along with the escaping H.](gr7){#f0035}

![Temporal evolution of the partial surface pressures $P_{surf}$ of H, O, and C normalized to the total initial surface pressure $P_{total}$ for the four compositions of outgassed atmospheres described in [Table 2](#t0010){ref-type="table"}. The hydrogen inventory evolves assuming a constant escape rate and parameters according to CII in [Table 3](#t0015){ref-type="table"} valid for 100 XUV. Both O and C are dragged along with the escaping H.](gr8){#f0040}

![Temporal evolution of the partial surface pressures $P_{surf}$ of H, O, and C normalized to the total initial surface pressure $P_{total}$ for the four compositions of outgassed atmospheres described in [Table 2](#t0010){ref-type="table"}. The hydrogen inventory evolves assuming a constant escape rate and parameters according to CIII in [Table 3](#t0015){ref-type="table"} valid for 100 XUV. Both O and C are dragged along with the escaping H.](gr9){#f0045}

![Temporal evolution of the partial surface pressures $P_{surf}$ of H, O, and C normalized to the total initial surface pressure $P_{total}$ for the four compositions of outgassed atmospheres described in [Table 2](#t0010){ref-type="table"}. The hydrogen inventory evolves assuming a constant escape rate and parameters according to CIV in [Table 3](#t0015){ref-type="table"} valid for 100 XUV. Both O and C are dragged along with the escaping H.](gr10){#f0050}

###### 

Integral parameters for Mars model atmospheres models with dust depletion factors *f* of 0.1, 0.01, and 0.001 and for accretion rates ${\overset{˙}{M}}_{acc}$ between $1 \times 10^{- 6}$ and $1 \times 10^{- 9}$ Earth masses per year. *L* is the luminosity resulting from the accretion of planetesimals; $M_{atm}$ is the atmospheric mass up to the Bondi radius; and surface pressure and temperature on the surface are denoted as $P_{s}$ and $T_{s}$, respectively.

  ${\overset{˙}{M}}_{acc}$ ($M_{Mars}/{yr}$)   $f_{dust}$   *L* (erg/s)             $M_{atm}$ (g)           $P_{s}$ (bar)   $T_{s}$ (K)
  -------------------------------------------- ------------ ----------------------- ----------------------- --------------- -------------
  $9.35 \times 10^{- 9}$                       0.001        $2.39 \times 10^{22}$   $6.58 \times 10^{22}$   7.81            600
  $9.35 \times 10^{- 9}$                       0.01         $2.39 \times 10^{22}$   $3.21 \times 10^{22}$   3.38            639
  $9.35 \times 10^{- 9}$                       0.1          $2.39 \times 10^{22}$   $1.00 \times 10^{22}$   8.08            690
                                                                                                                            
  $9.35 \times 10^{- 8}$                       0.001        $2.38 \times 10^{23}$   $2.66 \times 10^{22}$   2.58            693
  $9.35 \times 10^{- 8}$                       0.01         $2.38 \times 10^{23}$   $9.76 \times 10^{21}$   0.763           724
  $9.35 \times 10^{- 8}$                       0.1          $2.38 \times 10^{23}$   $2.86 \times 10^{21}$   0.155           754
                                                                                                                            
  $9.35 \times 10^{- 7}$                       0.001        $2.38 \times 10^{24}$   $8.81 \times 10^{21}$   0.628           795
  $9.35 \times 10^{- 7}$                       0.01         $2.38 \times 10^{24}$   $2.84 \times 10^{21}$   0.151           784
  $9.35 \times 10^{- 7}$                       0.1          $2.38 \times 10^{24}$   $5.25 \times 10^{20}$   0.028           841
                                                                                                                            
  $9.35 \times 10^{- 6}$                       0.001        $2.38 \times 10^{25}$   $2.70 \times 10^{21}$   0.132           885
  $9.35 \times 10^{- 6}$                       0.01         $2.38 \times 10^{25}$   $5.14 \times 10^{20}$   0.028           862
  $9.35 \times 10^{- 6}$                       0.1          $2.38 \times 10^{25}$   $3.21 \times 10^{19}$   0.005           960

###### 

Modelled atmospheric partial surface pressures $P_{H_{2}O}$ and $P_{{CO}_{2}}$ in units of bar of catastrophically outgassed steam atmospheres dependent on initial H~2~O and CO~2~ contents in wt% inside a magma ocean with a minimum depth of 500 km and a maximum depth of 2000 km.

  Bulk magma ocean   Initial H~2~O (wt.%)   Initial CO~2~ (wt.%)   $P_{H_{2}O}$ (bar)   $P_{{CO}_{2}}$ (bar)
  ------------------ ---------------------- ---------------------- -------------------- ----------------------
  500 km deep        0.1                    0.02                   52                   11
                     0.2                    0.04                   108                  22
                                                                                        
  2000 km deep       0.1                    0.02                   122                  26
                     0.2                    0.04                   257                  54

###### 

Modeled atmospheric parameters and thermal hydrogen atom escape rates $L_{th}$ corresponding to a 100 times higher XUV flux compared to today\'s Sun at the critical distance $r_{c} \leq r_{exo}$, where the dynamically outward flowing hydrogen dominated bulk atmosphere reaches (sonic speed) above the planetary surface and two heating efficiencies *η* of 15% and 40%.

  CASES   *η* (%)   z~0~ (km)   $r_{{XUV}_{eff}}$ (r~0~)   $r_{c}$ (r~0~)   $n_{c}$ (cm^-3^)      $T_{c}$ (K)   $L_{th}$ (s^−1^)
  ------- --------- ----------- -------------------------- ---------------- --------------------- ------------- ------------------
  CI      15        100         3.4                        32.5             $2 \times 10^{6}$     40            1.8×10^32^
  CII     15        1000        4.5                        30               $6.2 \times 10^{6}$   36            7.0×10^32^
  CIII    40        100         3.2                        21               $6.5 \times 10^{6}$   60            3.0×10^32^
  CIV     40        1000        4.2                        20               $1.7 \times 10^{7}$   50            1.0×10^33^
