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date: 15 April 2021

# Steam Atmospheres and Magma Oceans on Planets

• Keiko HamanoKeiko HamanoEarth-Life Science Institute, Tokyo Institute of Technology

### Summary

A magma ocean is a global layer of partially or fully molten rocks. Significant melting of terrestrial planets likely occurs due to heat release during planetary accretion, such as decay heat of short-lived radionuclides, impact energy released by continuous planetesimal accretion, and energetic impacts among planetary-sized bodies (giant impacts). Over a magma ocean, all water, which is released upon impact or degassed from the interior, exists as superheated vapor, forming a water-dominated, steam atmosphere. A magma ocean extending to the surface is expected to interact with the overlying steam atmosphere through material and heat exchange.

Impact degassing of water starts when the size of a planetary body becomes larger than Earth’s moon or Mars. The degassed water could build up and form a steam atmosphere on protoplanets growing by planetesimal accretion. The atmosphere has a role in preventing accretion energy supplied by planetesimals from escaping, leading to the formation of a magma ocean. Once a magma ocean forms, part of the steam atmosphere would start to dissolve into the surface magma due to the high solubility of water into silicate melt. Theoretical studies indicated that as long as the magma ocean is present, a negative feedback loop can operate to regulate the amount of the steam atmosphere and to stabilize the surface temperature so that a radiative energy balance is achieved. Protoplanets can also accrete the surrounding $H2$-rich disk gas. Water could be produced by oxidation of $H2$ by ferrous iron in the magma. The atmosphere and water on protoplanets could be a mixture of outgassed and disk-gas components.

Planets formed by giant impact would experience a global melting on a short timescale. A steam atmosphere could grow by later outgassing from the interior. Its thermal blanketing and greenhouse effects are of great importance in controlling the cooling rate of the magma ocean. Due to the presence of a runaway greenhouse threshold, the crystallization timescale and water budget of terrestrial planets can depend on the orbital distance from the host star. The terrestrial planets in our solar system essentially have no direct record of their earliest history, whereas observations of young terrestrial exoplanets may provide us some insight into what early terrestrial planets and their atmosphere are like. Evolution of protoplanets in the framework of pebble accretion remains unexplored.

### Keywords

Silicate rocks, which primarily compose the crust and mantle of terrestrial planets, melt and form magma oceans at different times during planetary accretion. Planetary bodies that accreted early would have experienced melting due to intense heat generated from the decay of short-lived radionuclides, such as $26Al$, as evidenced by the presence of differentiated meteorites. High-velocity impacts are another potential heat source. Strong compression of solids by shock waves, followed by unloading due to the passage of rarefaction (release) waves, would have resulted in melting of the impactor and the target’s surface. Furthermore, giant collisions between planetary size bodies, which would have occurred at the final stage of planet formation, involve a vast amount of impact energy. Release of the enormous energy would have raised interior temperatures greatly, causing a global melting of the newly formed planet.

Magma oceans can be classified into several types. As rock temperature rises, melting starts at a temperature called the solidus and completes at a temperature called the liquidus. Rocks are partially melted at temperatures between the solidus and the liquidus. Magma oceans thus can be in a fully or partially molten state. The degree of melting strongly affects melt viscosity. On the basis of the rheological properties, Abe (1993a) categorized magma oceans as soft or hard. Magma oceans can also be transient or sustained, depending on the planetary heat budget (Abe, 1997). Furthermore, magma oceans can be grouped according to their radial and lateral length scales: deep and shallow (Abe, 1997) and global and local (Tonks & Melosh, 1993). In the latter classification, a local magma ocean is also called a magma pond. Elkins-Tanton (2012) used the terms surface and interior magma oceans, depending on whether the top of the magma ocean extends to the planetary surface. Furthermore, a molten layer that is located at the bottom of the mantle is called a basal magma ocean (Labrosse et al., 2007).

The term magma ocean as used in this article refers to a global molten layer extending to the surface. The degree of melting and planetary heat budget differ at different stages of planet formation. According to theoretical studies reviewed in this article, a magma ocean can be sustained in a partially molten state on protoplanets growing by planetesimal accretion, while a giant impact can form a deep and fully molten magma ocean, which thereafter crystallizes over time by emitting thermal radiation into space.

On the hot planetary surface, water would be kept in vapor form, forming “steam atmospheres.” Steam atmospheres are often mentioned in association with impact degassing from planetesimals, but in general the origin of the water does not matter. In a broad sense, the term is also used when the atmosphere contains gaseous species other than water vapor, as long as it is dominated by water vapor. Because water vapor is a good absorber of infrared radiation, the formation of a steam atmosphere can prevent heat from escaping and slow magma ocean crystallization. When heat is being released continuously, the surface temperature of a planetary body could increase as a result of the thermal blanketing of a steam atmosphere, leading to surface melting on a global scale.

A magma ocean extending to the surface is expected to interact with the overlying steam atmosphere through material and heat exchange. This article reviews the literature on the formation of steam atmospheres and their interplay with magma oceans. First the article addresses the formation of steam atmospheres and potential sources of water during accretion. The second part of the article describes the evolution of protoplanets growing by planetesimal accretion and the formation of a magma ocean by thermal blanketing effects of the atmosphere. The third part focuses on water budget during solidification of a deep magma ocean formed by giant impact, and the role of a steam atmosphere in the subsequent cooling of planets. Finally, topics relevant to terrestrial exoplanets are addressed.

### Formation of Steam Atmospheres During Accretion

Water can be delivered to protoplanets and planets as part of solids. In 1951, Rubey proposed that Earth’s atmosphere and water gradually built up on the airless surface via volcanic degassing after the birth of the Earth (Rubey, 1951). This view of continuous degassing prevailed through the 1970s, whereas it is now widely accepted that Earth and other terrestrial planets would have acquired their atmospheres throughout their accretion stages by a more efficient degassing mechanism involving impacts. Fanale (1971) and Arrhenius, Bibhas, and Alfvén (1974) proposed that rapid accretion, followed by core formation, could have caused melting of the surface or subsurface rocks, leading to early catastrophic degassing. Benlow and Meadows (1977) considered that impact vaporization could have released volatiles, which were bound in infalling dust particles.

In the 1980s, a series of experimental works by Ahrens and colleagues demonstrated that impacting materials can degas at a velocity lower than the one required for melting or vaporization. They showed that a complete devolatilization of volatile-bearing minerals occurs when a shock pressure of several tens $GPa$ is attained (e.g., Lange & Ahrens, 1982a, 1982b, 1983, 1986; Lange, Lambert, & Ahrens, 1985; Tyburczy, Frisch, & Ahrens, 1986). For example, dehydration of serpentine ($H2Orelease$) occurs at a shock pressure of about $23GPa$ for a porous regolith (Lange & Ahrens, 1982a). Such a shock pressure can be attained by a vertical rock–rock collision with impact velocity exceeding $~3km/s$ (see O’Keefe & Ahrens, 1982), which is comparable to escape velocity from the moon. In general, shock pressure is expected to become high enough for impact degassing from rocky materials when the mass of a planetary body reaches that of the moon or Mars.

According to Stewart and Ahrens (2005), crystalline $H2O$ ice will melt completely on release from shock pressures above $2−4GPa$, and it will begin to vaporize at shock pressures above $7−9GPa$. If icy impactors are porous, melting and vaporization of $H2O$ ice will occur readily due to strong shock heating. Thus, degassing from ice-bearing planetesimals can occur on growing protoplanets as well. The composition of the impact-induced gas depends on the composition of solid impacting materials and the thermodynamic state after decompression. According to chemical equilibrium calculations, gas composition formed by outgassing of CI carbonaceous chondrites would mainly consist of $H2O$ and $CO2$, whereas gas composition in equilibrium with $H$ ordinary chondrites or enstatite chondrites would be more reducing, consisting of $H2$ and $CO$, with $H2O$ as the third component (Hashimoto, Abe, & Sugita, 2007; Schaefer & Fegley, 2007, 2010; Zahnle, Schaefer, & Fegley, 2010).

A gas molecule can escape from the gravity of a planetary body when it has sufficient thermal energy. Since the solidus temperature of mantle peridotites is typically higher than $1,400K$, it would be worthwhile to roughly estimate how large the planetary body should be to keep a steam atmosphere on the hot surface. One of the controlling parameters in thermal escape is the escape parameter $λ$, which is defined as the ratio of gravitational energy to thermal energy of a gas molecule as follows:

$Display mathematics$(1)

where $G$ is the gravitation constant, $kB$ the Boltzmann’s constant, $ME$ the Earth’s mass, $μgas$ the mass of a gas molecule in unified atomic mass units $(mu)$, $Ts$ the surface temperature, and $Mpl$, $Rpl$, and $ρpl$ are the mass, radius, and average density of the planetary body, respectively. The escape parameter can also be expressed as the ratio of the planetary radius to an atmospheric scale height $Ha$,

$Display mathematics$(2)

where

$Display mathematics$(3)

Here, we assume that atmospheric pressure, density, and temperature are related by the polytropic law:

$Display mathematics$(4)

where $γ$ is the polytropic exponent, which represents a dominant energy transfer process in the atmosphere, ranging from $1$ (isothermal) to the ratio of specific heat of a gas (adiabatic). In a polytropic atmosphere, the pressure approaches $0$ as altitude increases, when the following inequality holds:

$Display mathematics$(5)

Otherwise, atmospheric pressure approaches a finite value at infinity so that the atmosphere will expand in the absence of any inward pressure. In order to have the gas velocity be subsonic at the bottom of the atmosphere in spherical geometry, the following inequality also needs to hold:

$Display mathematics$(6)

(Parker, 1963). From the inequalities in Equations (5) and (6), a static atmosphere is possible when $λ>3$. This condition is satisfied when the planetary mass is larger than about the lunar mass. On the basis of the above discussions, water vapor can be released upon impact degassing and accumulate on a planetary body once its size exceeds the lunar or Mars mass, unless any other processes erode the impact-induced gas.

In order to produce steam atmospheres by impact degassing, building blocks of protoplanets and planets need to contain some water. Primitive meteorites in the solar system show a wide variety of water contents. Enstatite chondrites are almost dry, carbonaceous chondrites contain a high proportion of water of $5−10wt%$, and ordinary chondrites have a water content between enstatite chondrites and carbonaceous chondrites (see Alexander, 2017; Jarosewich, 1990; Kerridge, 1985). Dauphas and Morbidelli (2014) compiled the water content of the primitive meteorites and the presumed location of their parent bodies. Their result suggests that there seems to be a radial gradient in the water contents of the meteorites, which is often thought to reflect a temperature gradient in the protoplanetary disk. A simple extrapolation of this trend suggests that planetesimals formed in the terrestrial planet region would have been totally dry. If this is the case, no steam atmosphere would have formed on protoplanets that accreted the local planetesimals. On the other hand, there are several mechanisms proposed for the delivery of water to protoplanets. Computer simulations on interatomic potentials between $H2O$ molecules and forsterite showed that gaseous water could be directly absorbed into dust grains even in a hot disk (see Asaduzzaman, Muralidharan, & Ganguly, 2015; King et al., 2010; Muralidharan, Deymier, Stimpfl, de Leeuw, & Drake, 2008). Incorporation of the water-bearing dust grains into the local planetesimals might have allowed their water contents to deviate from the extremely low value extrapolated from the radial trend. The planetesimals in the terrestrial planet region might also acquire water by inward migration of a snow line in a protoplanetary disk. Although the position of a snow line and its time variation are controversial (Lodders, 2004), modeling studies of thermal structure in the disk demonstrated that the location of a snow line could move well inside 1 au as the disk accretion rate drops (see Bitsch, Johansen, Lambrechts, & Morbidelli, 2015; Hueso & Guillot, 2005; Oka, Nakamoto, & Ida, 2011). The planetesimals formed in the inner region may acquire a substantial amount of water by direct condensation of water vapor, or inward drifting of icy or phyllosilicate particles (Ciesla & Lauretta, 2005; Sato, Okuzumi, & Ida, 2016). Although the radial drift of hydrous particles might have been blocked by the formation of a proto-Jupiter in our solar system (Morbidelli et al., 2016), a rapidly growing giant planet would have also contributed to destabilization of the orbits of nearby hydrous planetesimals (see Wetherill, 1975). Raymond and Izidoro (2017) showed that gravitational scattering of hydrous planetesimals by a rapidly growing core(s), followed by gas drag to damp their eccentricities and inclinations, provides an efficient mechanism to deliver water to protoplanets in the terrestrial planet region. This radial mixing is a natural consequence of the formation of a giant planet at large orbital distance and does not necessarily require the migration of giant planets.

After disk gas dissipates out, a large-scale radial mixing of rocky materials would occur. N-body simulations suggest that gravitational scattering of protoplanets and perturbations from giant planets implant hydrated planetesimals and protoplanets that originate in the outer disk into the terrestrial planet region (see Morbidelli et al., 2000; O’Brien, Morbidelli, & Levison, 2006; Raymond, Quinn, & Lunine, 2004, 2006, 2007). The dissipation of disk gas also triggers giant impacts among protoplanets. In the aftermath of giant impacts, rocky planets would undergo significant melting and vigorous convection would occur. As a result, water stored in the interior can outgas to the surface. According to numerical simulations, the abundance of water on finally formed planets is sensitive to an initial population of hydrous planetesimals and an assumed radial distribution of water content in building blocks. It can also vary widely due to the stochastic nature in the number of hydrated protoplanets accreted (Morbidelli et al., 2000).

The next parts of this article address the interplay between a steam atmosphere and a magma ocean on protoplanets and planets, with the assumption that water was delivered during accretion. If all building blocks of terrestrial planets were devoid of water (Albarède, 2009) or all the supplied water was instantaneously lost by some processes, no steam atmosphere would have formed or interacted with a magma ocean. In this case, all the water observed in the current terrestrial planets had to have been supplied by late accretion after the main stage of planet formation. A transient steam atmosphere might have formed through vaporization of (water) oceans by high-velocity impacts. According to Zahnle and Sleep (1997), asteroids with a diameter of several hundreds of kilometers would be enough to vaporize a substantial amount, or even all, of (water) oceans on Earth and Mars. Also, a thin silicate melt layer could have been formed on the surface by the rainout of an impact-induced rock vapor. Because the depth is estimated to be less than 10 m and the ground temperature likely drops to below the solidus in no more than 100 years (Segura, Zahnle, Toon, & McKay, 2013; Sleep & Zahnle, 1998; Zahnle & Sleep, 1997), this transient stage is not discussed here.

### Protoplanets Growing by Planetesimal Accretion

According to classical models of planet formation, planetary size bodies, protoplanets, first form in a protoplanetary disk through pairwise collisions between planetesimals. In a swarm of planetesimals, where numerous close encounters and gas drag set the relative velocity small, the largest few planetesimals grow rapidly at the expense of numerous smaller planetesimals. This is called runaway growth (see Wetherill & Stewart, 1989). The runaway growth phase ends as the planetesimals acquire velocity dispersion comparable to the escape velocity of the largest bodies through gravitational interactions. Afterward, the larger bodies with comparable masses grow at uniformly spaced orbits. This is called oligarchic growth (see Kokubo & Ida, 1998). The growth rate of the protoplanets becomes slower and slower and then ceases when they have accreted all planetesimals in their feeding zone. In our solar system, protoplanets are thought to have grown up to the lunar or Mars mass—which is large enough for impact degassing and retention of water vapor, as previously discussed—on a timescale of $106years$ through the runaway and oligarchic growth stages.

On protoplanets growing by continuous accretion of planetesimals, the impact energy is released at the surface. Assuming that the impact velocity of planetesimals $vimp$ is approximated by the escape velocity of a growing protoplanet, the accretion energy flux $E˙ac$ is given by

$Display mathematics$(7)

where $M˙pl$ represents the accretion mass flux of the protoplanet and $ΔM$ is the total mass of planetesimals accreting on the protoplanet on a timescale of $τac$. If protoplanets have no atmosphere, most of the released impact energy escapes from the surface into space by radiation. The resulting surface temperature $Tsnoatm$, which is determined by a balance of the accretion energy flux and the black-body radiation from the surface, is:

$Display mathematics$(8)

The surface temperature remains as low as $290K$, which is far below the solidus, with the accretion mass flux at which the planetesimals whose total mass is equivalent to the Mars mass accrete on a timescale of $3×106years$. Actually, some of the released energy would be buried in near-surface regions (see Kaula, 1979), being consumed to heat up the interior, so the surface temperature should become even smaller. Consequently, a global magma ocean seems unlikely to form without atmospheric thermal blanketing.

If protoplanets have their own atmosphere, it traps some of the thermal radiation and radiates it back to the surface, keeping part of the gravitational energy released by planetesimal impacts. The resulting surface temperature increases and can reach the solidus ($~1,400K$ for mantle peridotites), depending on the amount and composition of the atmosphere. In the case of a steam atmosphere, the surface temperature can rise rapidly when accretion energy flux exceeds the runaway greenhouse threshold. The runaway greenhouse threshold is the maximum radiation that a steam atmosphere can emit when water condensation occurs (Abe & Matsui, 1988; Kasting, 1988). The threshold fluxes obtained by recent radiative-convective models are $~280W/m2$ on Earth-sized planets and $~250W/m2$ on Mars-sized planets (Goldblatt et al., 2013; Kopparapu et al., 2014). From Equation (7), the typical accretion energy flux evaluates to $~420W/m2$, indicating that the surface of protoplanets can melt during planetesimal accretion due to strong thermal blanketing by a steam atmosphere.

When the size of a protoplanet reaches the mass of the moon or Mars, impact-induced devolatilization of hydrous minerals will start to occur. Early studies focused on the formation of an impact-induced steam atmosphere on protoplanets growing in a disk-free environment, such as in vacuum (Abe & Matsui, 1985, 1986; Lange & Ahrens, 1982b; Matsui & Abe, 1986a, 1986b; Zahnle, Kasting, & Pollack, 1988). In these modeling studies, the growth of a steam atmosphere is slow and gradual below the decomposition temperature of hydrous minerals ($~1,000K$; Lange & Ahrens, 1982b), since some part of the released water is believed to recombine with surface rocks. When the surface temperature exceeds the dehydration temperature, the water supplied by water-bearing solids quickly builds up on the surface and forms a thick steam atmosphere. Due to the atmosphere’s thermal blanketing effects, the surface temperature also increases rapidly and then the surface rock starts to melt. Melting conditions of the rocky surface depend on the water content in planetesimals and the accretion energy released at the surface. Assuming that Earth accreted in about $5×107years$ at an accretion rate proposed by Safronov (1972), Abe and Matsui (1986) showed that, with water content of $0.1wt%$, the surface temperature reaches the solidus when the size of the protoplanet is about $0.4$ Earth’s radius (Figure 1).

Once a global, partially molten magma ocean has formed at the surface, the surface water starts to dissolve in the melt. Afterward, the amount of the steam atmosphere is likely controlled by mass balance with the magma ocean according to $H2O$ solubility in silicate melt. The resulting change in the thermal blanketing effects, in turn, affects the surface temperature and the mass of the silicate melt in the magma ocean. This interplay between a steam atmosphere and a magma ocean on protoplanets was first explored by a series of modeling studies by Abe and Matsui (Abe & Matsui, 1985, 1986; Matsui & Abe, 1986a, 1986b) and then was studied in more detail by Zahnle et al. (1988). They showed that, the interaction creates a negative feedback loop that regulates the amount of the steam atmosphere and stabilizes the surface temperature. If the amount of the steam atmosphere increases for some reason, this enhances the thermal blanketing effects of the steam atmosphere, leading to an increase in the surface temperature. As the degree of melting in the partially molten magma ocean increases, more steam dissolves into the magma ocean. Thus, the feedback loop brings the amount of the steam atmosphere back down, so that a radiative energy balance is achieved at the surface. This negative feedback loop can function even if the water content in planetesimals is as high as $1wt%$ (Abe & Matsui, 1986).

Zahnle et al. (1988) modeled various physical processes that were neglected in the previous studies by Matsui and Abe, such as impact stirring, convective heat transport with temperature-dependent viscosity, and self-consistent treatment of degassing/ingassing of water considering the evolution of the interior structure. They also considered two processes for the loss of water: hydrodynamic escape fuelled by extreme ultraviolet (EUV) radiation from the host star (see Kasting & Pollack, 1983; Watson, Donahue, & Walker, 1981) and impact erosion (Melosh & Vickery, 1989; Walker, 1986). They demonstrated that the negative feedback loop, which was first found by Matsui and Abe, can operate in a wide range of model parameters, with the exceptions of extremely dry planetesimals and/or too high EUV radiation. Loss of water can prevent or delay the accumulation of the steam atmosphere on the surface. If the EUV flux is too high, very little water remains on the protoplanet. In addition to supply of volatiles, planetesimal impacts are also instrumental in removing part of the preexisiting atmosphere. Zahnle et al. (1988) showed that the impact erosion by planetesimals becomes more efficient as the mass of the atmosphere increases up to some point so that there would be an equilibrium atmospheric pressure at which impact degassing balances with impact erosion. The estimated equilibrium pressure is approximately proportional to the water content in planetesimals. They reported that the water content of $0.1wt%$ is enough to melt the rocky surface during protoplanet accretion, considering both the loss and supply of water by planetesimal impacts.

One of the common features in the previous studies is that, as long as the surface is molten, the mass of the steam atmosphere is controlled to be almost constant, and the rest of the water is being stored in the mantle. As the accretion rate of planetesimals decreases toward the end of the growth stage of protoplanets, the water dissolved in the shallow interior is expelled into the surface in all the models above. The water released upon the surface at the end of planetesimal accretion is an amount comparable to the ocean mass on the present Earth (about $1.4×1021kg$; Abe & Matsui, 1985; Matsui & Abe, 1986a, 1986b; Zahnle et al., 1988).

The current models of planet formation have supported the theory that protoplanets would have undergone runaway accretion in a $H2$-rich protoplanetary disk, rather than in the disk-free environment assumed in the previous studies. Therefore, protoplanets embedded in the disk could have attracted the surrounding disk gas with gravity to form a dense $H2$-rich atmosphere. The thermal blanketing effects of the $H2$-rich proto-atmosphere could raise the surface temperature above the solidus (Hayashi, Nakazawa, & Mizuno, 1979; Lewis & Prinn, 1984). Based on the simple assumption that the proto-atmosphere has an adiabatic temperature profile, the surface temperature can reach the solidus when the protoplanet grows to the size of Mars (Lewis & Prinn, 1984). Hayashi et al. (1979) first explored a radiative-convective structure of captured proto-atmospheres with the opacity of gas and dust of solar composition. They showed that it is possible that the surface rock begins to melt when the mass of the protoplanet exceeds $~0.2$ Earth mass. Ikoma and Genda (2006) updated the opacity of gas and dust in the model by Hayashi et al. (1979) and investigated the sensitivity to the amount and size distribution of dust grains in the atmosphere. Their results suggest that the size of protoplanets needs to be larger than $~0.3$ Earth mass for the surface to melt in the case of the solar nebula.

The idea of global melting by thermal blanketing of a captured $H2$-rich atmosphere led Sasaki (1990) to propose the idea that the atmospheric $H2$ can react with ferrous iron in the surface magma and thus produce water. The additional $H2O$ in the atmosphere can enhance the surface temperature further due to both an opacity increase by $H2O$ molecules and a pressure increase due to growth of the mean molecular weight in the atmosphere. Once water was produced by this oxidation process and was incorporated into the interior, it could contribute to the water inventory on protoplanets, although Sasaki did not model any interactions between the atmosphere and the magma ocean. If a proto-atmosphere with a disk origin interacted with a magma ocean, part of the noble gases in the captured atmosphere might have similarly dissolved and remained in the interior. Modeling studies by Mizuno, Nakazawa, and Hayashi (1980, 1982) predicted that a considerable amount of neon with a solar isotopic composition might be trapped in the mantle if Earth was once surrounded by a solar-composition atmosphere. Currently available data on noble gases in the Earth’s mantle, however, show nonsolar isotopic compositions (Ozima & Podosek, 2001).

Cameron and colleagues proposed that rainout of iron and silicate droplets in a gaseous giant could have created a molten rocky core (Cameron, Decampli, & Bodenheimer, 1982; DeCampli & Cameron, 1979; Slattery, 1978; Slattery, DeCampli, & Cameron, 1980). In this scenario, however, the formation of an Earth-sized rocky core requires accretion of solar materials with at least one Jupiter mass. Cameron and colleagues also argued that efficient loss of the massive disk-originated materials is problematic (DeCampli & Cameron, 1979).

Thus, the formation of a magma ocean on protoplanets has been investigated separately in the context of two distinct origins of planetary atmospheres: an outgassed $H2O$-rich atmosphere with a planetesimal origin and a $H2$-rich atmosphere with a disk origin. On the other hand, a modern view of the formation of the proto-atmosphere on protoplanets is that it could have been a mixture of outgassed and captured disk-gas components, as pointed out by Abe, Ohtani, Okuchi, Righter, and Drake (2000). The modern planetary atmospheres show depletion and distinct patterns in noble gas abundance compared with the solar abundance (see Pepin, 1991), while a low $D/Hratio$ recorded in deep mantle samples (Hallis et al., 2015) suggests that disk gas might have interacted with planetary reservoirs at some stage of planet formation. Recently, Saito and Kuramoto (2018) evaluated the thermal blanketing effects by considering a proto-atmosphere that has a two-layered structure: the upper layer, which has a chemical composition identical to the solar composition and connects to a protoplanetary disk at the top of the atmosphere, and the lower layer, which consists of an outgassed component. They showed that, in comparison to a pure disk-gas atmosphere, the presence of the outgassed component over the surface greatly enhances the thermal blanketing effects. They reported that the surface rock could melt even if the protoplanet were as small as half Mars’ radius, depending on the volatile content in the building blocks. Interactions between a magma ocean and a proto-atmosphere might have started earlier than considered in previous studies.

Mars may be a remnant protoplanet because of its small mass and rapid accretion suggested from Hf-W chronology (Dauphas & Pourmand, 2011). Although water abundance in the Martian mantle is controversial (see McCubbin et al., 2012; Wanke & Dreibus, 1994), early modeling studies predicted that protoplanets could have retained a significant fraction of water in the mantle after accretion (Abe & Matsui, 1986; Zahnle et al., 1988). This seems to support the idea that Mars had a wet early mantle. One of the potentially important processes omitted in the previous studies is chemical reaction with Fe metal. The melting of primitive rocks involves metal–silicate separation. Although metal–silicate partition coefficients of $H$ are still controversial (Clesi et al., 2018; Okuchi, 1997), $H$ (water) likely has a high affinity with metallic iron and preferentially enters a metallic phase. Also, chemical reaction of water with Fe metal, followed by $H2$ escape to the surface, may have caused oxidization in the interior (Sharp, McCubbin, & Shearer, 2013). Partitioning and chemical reaction of water with $Fe$ metal should be modeled in future work for further understanding of the water budget and the oxidation state of protoplanets. In the previous modeling studies on the interplay between a magma ocean and a steam atmosphere, radioactive heating by short-lived nuclide $26Al$ is also neglected. Considering the short formation timescale of Mars, the decay heat may have been so effective that it increased the temperature in a relatively cool undegassed silicate core later on (Pommier, Grove, & Charlier, 2012). Experimental studies showed that, at pressure $>3GPa$, water, which is released from coexisting hydrous minerals, can react with solid Fe below $1,000K$, forming iron hydroxides ($FeHx$; Iizuka-Oku et al., 2017). This also could have influenced the water stored in the mantle.

It remains challenging to theoretically predict a quantitative contribution of the two different origins of water on terrestrial planets. As described previously, the water content in building blocks of protoplanets remains unclear. Also, a production rate of water by redox reaction between disk gas and silicate melt, and its incorporation rate into the interior, would depend on many complex parameters, such as the mixing efficiency of the surface magma, the chemical composition and redox state of silicate melt, and the sedimentation rate of metallic iron $(Fe0)$, which is formed at the expense of $Fe2+$ in a magma ocean.

Wu et al. (2018) took another approach based on the D/H ratio in the bulk silicate Earth to quantify the relative contribution of the two different water reservoirs (Figure 2). Their model also includes a potential D-H fractionation on metal–silicate partitioning. They have concluded that water originating from chondritic materials accounts for the water inventory on the present Earth, with a minor contribution of water with a nebula origin (a few percent), although significant ingassing of solar nebula hydrogen could have occurred during protoplanet formation. In this study, however, the effect of H escape on D-H fractionation is neglected. Genda and Ikoma (2008) showed that the D/H ratio in the Earth’s ocean may have increased by a factor of 2–9 over the past $4.5Gyr$. If early Earth had a D/H ratio that was lower than the present value, the contribution of water with a nebula origin would become higher than that estimated by Wu et al. (2018).

### Planets Formed by Giant Impacts

After a protoplanetary disk dissipates out, dynamical interaction with the disk no longer damps eccentricities and inclinations of protoplanets, which are raised by mutual interaction among the protoplanets. This triggers impacts among them (giant impacts), which would last for about $108years$ (see Chambers, 2001). A giant impact deposits an enormous amount of energy in the planetary interior, which is sufficient to melt a substantial fraction of the protoplanet. Numerical studies based on hydrodynamic simulations suggest that early Earth’s mantle would have been largely or totally molten in the aftermath of the moon-forming impact (see Cameron & Benz, 1991; Lock et al., 2018; Nakajima & Stevenson, 2015). During this late-stage accretion, it would therefore be inevitable that terrestrial planets formed by giant impacts would start their lives in a deeply molten state, irrespective of the presence or absence of a planetary atmosphere. Instead, the thermal blanketing and greenhouse effects of a planetary atmosphere are of great importance to the subsequent cooling history.

A giant impact would cause most of the rocky parts to become devolatilized and, at the same time, strip part of the preexisting atmosphere from the protoplanet. The atmosphere in the vicinity of the impact site would be driven off by propagation of a strong shock wave and expansion of impact-induced rock-vapor plumes (Melosh & Vickery, 1989). The strong shock wave also would propagate from the impact site through the planetary interior and accelerate the ground radially upward. The induced ground motion then pushes the overlying atmosphere up and drives its escape (Ahrens, 1993). According to 1D hydrodynamic simulations, a blow-off of the entire atmosphere requires that the ground velocity exceed the escape velocity of the planet (Genda & Abe, 2003; Schlichting, Sari, & Yalinewich, 2015). This means that a substantial escape of atmosphere is unlikely to occur as long as the impact velocity of the giant impact is comparable to the planetary escape velocity, because the energy of shock wave quickly dissipates with distance, so that the ground velocity is expected to be smaller than the impact velocity itself on most of the planetary surface (Schlichting et al., 2015). Recent 3D simulations have shown that a planetary atmosphere would suffer less erosion with grazing impacts (Kegerreis, Eke, Massey, & Teodoro, 2020).

Genda and Abe (2005) also showed that if (water) ocean is present on protoplanets before the impact, its expansion can expel the overlying atmosphere effectively, whereas the water in the ocean can still remain after the impact. Gases that are much less soluble in water should become more depleted. Consequently, protoplanets can retain a substantial fraction of water through multiple giant impacts. This suggests that water endowment of protoplanets, which is established during the growth stage by planetesimal accretion, is therefore one of the key factors for the formation of a steam atmosphere after giant collisions. As noted above, large-scale radial mixing due to gravitational interaction of protoplanets and perturbations from giant planets is also an important mechanism for delivering water to terrestrial planets.

The crystallization and thermal structures of deep magma oceans have been extensively investigated since the 1980s (see Abe, 1993b, 1997; Agee & Walker, 1988; Miller, Stolper, & Ahrens, 1991; Solomatov & Stevenson, 1993a, 1993b; Tonks & Melosh, 1990), with a primary focus on melt–solid separation. Although the importance of thermal blanketing by planetary atmosphere was already recognized in these early studies, linking cooling of a deep magma ocean with the growth of an outgassed atmosphere was first modeled by Elkins-Tanton (2008, 2011) for $H2O−CO2$ atmospheres, and it has now been explored by several groups targeting terrestrial planets in the solar system and beyond (Hamano, Abe, & Genda, 2013; Hamano, Kawahara, Abe, Onishi, & Hashimoto, 2015; Katyal et al., 2019; Lebrun et al., 2013; Salvador et al., 2017; Schaefer, Wordsworth, Berta-Thompson, & Sasselov, 2016).

#### Water Partitioning

A high degree of melting suggests that magma has a liquidlike low viscosity $(~0.1Pas)$.

A hot deep magma ocean with a planetary scale $(~106m)$ would be turbulently convective. Water, which may still remain on the surface after the impact, would be dissolved in the surface magma, which is rapidly transported down into the deep interior and is homogenized by vigorous convection. Thus, an ingassing process could have occurred at the very early stage, depending on the pre-impact distribution of water between the surface and the interior.

Following an energetic giant impact, part of the mantle of the impacted protoplanet is strongly heated so that a photospheric surface temperature from the newly formed planet could exceed $16,000K$ (see Cameron & Benz, 1991). The post-impact planet would be surrounded by a hot silicate-vapor atmosphere. The atmosphere would be made optically thick by opacities of $SiO$ and other rock-forming components (Fegley & Schaefer, 2014), and would emit high radiation from hot silicate clouds into space (Zahnle et al., 2007). As the silicate vapor cools and rains out, volatile gaseous species like $H2O$ are expected to become increasingly dominant in the atmosphere, although the transition of thermal and chemical structure from a silicate-dominated to a steam-dominated atmosphere has not been modeled yet. Most studies on interplay between a steam atmosphere and a magma ocean have started calculations from surface temperatures less than $3,500K$, at which the estimated silicate vapor pressure is about 2 bars or less (Fegley & Schaefer, 2014). Hot steam atmospheres can dissolve rocky elements like Si, Mg, and Fe from magma. Fegley et al. (2016) have shown that the rock-forming elements can vaporize fractionally in steam atmosphere. For example, Si could partition in a steam atmosphere more preferentially than Mg and Fe, depending on the temperature and partial pressure of water vapor. Fegley and colleagues proposed that the fractional vaporization of rocky elements, followed by escape of the steam atmosphere, could change the composition of rocky planets.

Assuming that a steam atmosphere is in solubility equilibrium with water in the whole magma ocean, mass balance calculations suggest that the majority of water would be partitioned in the magma ocean until the very end of the solidification stage. This is because of the high solubility of water into silicate melt and the vast mass of the magma ocean relative to water. Experimental and modeling studies demonstrated that water solubility in various silicate melts can have a negative temperature dependence (see Moore, Vennemann, & Carmichael, 1995; Spera, 1974; Yamashita, 1999), which was not taken into account in the previous mass balance calculations. The retrograde temperature dependence may offset the dissolution of water into the hot magma ocean (Fegley & Schaefer, 2014). The total pressure may also affect the fugacity of water in melts and thus its solubility, although this effect has not been well quantified.

Upon crystallization, water behaves as an incompatible component. At high pressures and temperatures relevant to a deep magma ocean, no hydrous phase will crystallize (Ohtani, Litasov, Hosoya, Kubo, & Kondo, 2004). Water can go into common mantle minerals but in small quantities. An incorporation rate of water by solid–melt partitioning $dmH2Opart/dt$ can be modeled as:

$Display mathematics$(9)

where $dmsol/dt$ is the solidification rate of a magma ocean, $wmelt$ is the water content in silicate melt, and $ci$, $DH2Oi$, and $wisat$ are the proportion, the solid–melt partition coefficient of water, and the saturation limit in water content for a cumulate mineral $i$. Partition coefficients between nominally anhydrous minerals and melts are typically much less than $1:0.02$ for pyroxene, $0.002$ for olivine, and $0.0001$ for Mg-perovskite (see Elkins-Tanton, 2008, and references therein). These minerals also have a saturation limit for water content to be incorporated into crystallographic defects, which is typically $1,000−2,000ppm$. Water in excess of this limit will be exsolved from cumulates. Therefore, water would preferentially accumulate in the residual liquid during magma ocean crystallization.

Solidifying cumulates may retain some water by trapping water-enriched interstitial liquids. When the mass fraction of trapped melts is defined as $ξ$, the incorporation rate of water can be approximately expressed as follows by considering both solid–melt partitioning of water and trapping of interstitial liquids:

$Display mathematics$(10)

where the first term on the righthand side represents the water that partitions into minerals, and the second term shows the contribution of water that is contained in trapped melts. The actual value of trapped melt fraction $ξ$ in the solidifying magma ocean is not well constrained. A geochemical model of the lunar magma ocean indicates that a trapped melt fraction of a few percent could explain observed characteristics of high-$Ti$ mare basalt (Snyder, Taylor, & Neal, 1992). Also, chemical analysis of abyssal peridotites sampled from the southwest Indian ridge suggests that efficient melt separation would start at the degree of melting less than $1%$ (Johnson & Dick, 1992). Modeling studies on magma-ocean-atmosphere coupling often assume a constant fraction of trapped melts of $0−3%$ during crystallization (see Elkins-Tanton, 2008; Hamano et al., 2013). This assumption gives rise to a relatively small amount of water storage in the mantle in the results—around $20%$ of the total water inventory at most.

Solubility of water in silicate melt is a strong function of water partial pressure. As the magma rises and the pressure decreases, the solubility of water decreases, and, when the water content in the magma exceeds the solubility, water will start to degas. Release of water to the surface can occur through two processes: nucleation and growth of bubbles, and diffusion of water molecules across a thermal boundary layer. The first process would be dominant if bubbles can grow fast and large enough to separate from the magma flow. Once bubbles nucleate, small bubbles seem to rapidly coalesce into larger ones in a low-viscosity melt flow, which ascends with a decreasing velocity as it approaches the surface (Massol et al., 2016). The nucleation rate depends on many parameters, such as the degree of supersaturation, the presence of crystals to assist nucleation (e.g., $Fe−Tioxides$), and the surface tension for liquids (for a review, see Shea, 2017).

When bubble nucleation requires a high degree of supersaturation, the second process, diffusion of water molecules, may become dominant. The degassing rate depends on the gradient of water content across a thermal boundary layer, as well as a diffusion coefficient. As long as the cumulate fraction in the surface magma is sufficiently low for the magma viscosity to be as low as liquid, the thermal boundary layer is typically as thin as the order of cm, with a high heat flux of $104$ to $106W/m2$ (see Solomatov, 2007). It allows water to degas efficiently even with a molecular diffusion coefficient, which typically ranges from $10−11$ to $10−8m2/s$ for basaltic melt with water content of $1,000ppm$ (Zhang, Xu, Zhu, & Wang, 2007). The effective diffusion coefficient should be much larger than the molar diffusion coefficient due to mixing by eddies. On the basis of these estimates, water vapor in the atmosphere is assumed to be in solubility equilibrium with a magma ocean in the modeling studies (Elkins-Tanton, 2008; Hamano et al., 2013; Lebrun et al., 2013; Salvador et al., 2017). A common feature of these studies is that the early mantle retains only a modest fraction of water and the remaining majority water is exsolved from the magma ocean, forming a massive atmosphere.

In a highly viscous partially molten layer with a high cumulate fraction, melt separation would be an important process to transport volatiles to the surface. The efficiency of melt separation would depend on the rate of crystallization and the rate of liquid percolation or matrix compaction. According to Lebrun et al. (2013), the melt percolation velocity is higher than the propagation velocity of a complete solidification front (Figure 3) in the shallow viscous magma ocean. So, they assumed that the atmosphere is in solubility equilibrium with the liquid in the magma ocean at the late stage of solidification as well. On the other hand, Hier-Majumder and Hirschmann (2017) recently combined a small-scale parameterization on melt extraction with thermal evolution of a magma ocean and indicated that it is possible that magma ocean crystallization proceeds faster than compaction in a partially molten layer. As a result, the larger fraction of melt might be trapped in the solidifying cumulates. In their model, more than half of the water could be left in the mantle in some cases.

The depth at which solid crystals first form in a magma ocean depends on the relation between an adiabat in the magma ocean and the liquidus curve of mantle silicate. If the liquidus is steeper than the adiabat through the entirety of the magma ocean, solidification would start from its bottom and a crystallization front would proceed upward (see Andrault et al., 2011). In the bottom-up crystallization scenario, all water in the magma ocean can be involved in partitioning with the atmosphere, as long as vigorous convection homogenizes the magma ocean and rapidly transports water to the surface. By contrast, if the adiabat has a steeper slope than the liquidus, solidification would start at middle depths and the solidification front would advance upward and downward (see Stixrude et al., 2009). In the case of the middle-out solidification, the magma ocean could be separated into upper and lower parts. Water present in the lower part may be isolated from the upper convective part and may be stored deep in the mantle. The actual evolution of the magma ocean structure would also depend on the mode of crystallization (fractional vs. batch), which affects the density structure (Caracas, Hirose, Nomura, & Ballmer, 2019). If an unstable density profile is produced as a result of fractional crystallization, the dense water-enriched residue liquid may overturn and bring the water into the deep mantle (Tikoo & Elkins-Tanton, 2017).

#### Heat Budget

A magma ocean loses its heat through vigorous convection, which is driven by a temperature difference between the surface and the interior. The convective heat flux is given as:

$Display mathematics$(11)

and

$Display mathematics$(12)

where $Ra$ is the Rayleigh number, $Tm$ and $Lm$ the potential temperature and the depth of the magma ocean, $g$ is the planetary gravity, and $α$, $κ$, $k$, and $ν$ represent, respectively, the thermal expansion, the thermal diffusivity, the thermal conductivity, and the kinematic viscosity in the magma ocean (Siggia, 1994; Solomatov, 2007). The convective heat flux depends on the surface temperature, whereas the surface temperature can vary as a result of the combined effects of any heat sources, including the convective heat flux itself, and the thermal blanketing effects of the planetary atmosphere.

One of the potential heat sources is heat production by radioactive decay in rocks. Lebrun et al. (2013) included radioactive heating by $238U,235U,232Th,40K$, and $26Al$ in their model by considering their chondritic abundance. They showed that if a giant impact occurred later than $5Myr$ after the formation of the oldest $CAI$, the short-lived radionuclide $26Al$ would have become mostly extinct, so that its decay would have had almost no effect on the magma ocean crystallization. They also showed that the other long-lived radionuclides would have had little impact on the crystallization timescale of a fully and partially molten magma ocean, but they could have delayed crystallization of a shallow partially molten layer which remained underneath solid crusts.

In addition to global melting, a giant impact can produce a large moon very close to the newly formed planet. N-body simulations predict that the moon would have formed just beyond the Roche limit (the gravitational field) of Earth in the aftermath of the moon-forming impact (see Kokubo, Ida, & Makino, 2000). Tidal interaction with the nearby moon could have generated heat in the magma ocean and thus delayed its crystallization (Sears, 1992). Zahnle, Lupu, Dobrovolskis, and Sleep (2015) coupled an orbital evolution of the moon with the thermal evolution of early Earth to address the effects of tidal heating. Their model calculations showed that tidal heating can sustain the terrestrial magma ocean over $2−10Myr$ and enhance the subsequent geothermal heat flow in comparison to the moonless case. They also found a negative feedback loop between viscosity-dependent tidal heating and temperature-dependent viscosity in the magma ocean. The energy dissipation inside the magma ocean depends on the viscosity and is most effective in materials that are just beginning to solidify. If the internal temperature increases as a result of tidal dissipation, the viscosity becomes lower, which in turn results in decreasing tidal dissipation. Zahnle et al. (2015) have shown that this feedback can operate in the presence of a thick atmosphere and, meanwhile, the moon’s orbital evolution slows.

In the absence of disk gas and dust, stellar radiation reaches terrestrial planets and provides another important heat source. Although the stellar radiation would be much smaller than an intense radiation emitted from the hot molten surface itself (typically $>106W/m2$), it can affect the cooling rate of a magma ocean under the presence of a thick steam atmosphere.

Elkins-Tanton (2008) modeled the thermal and chemical evolution of a magma ocean on Earth and Mars with the growth of an outgassed atmosphere consisting of $H2O$ and $CO2$. She showed that $70−95%$ of water would have finally degassed to the surface, forming a substantial atmosphere during solidification. The high outgassed fraction suggests that the initial $H2O$ content of less than $0.1wt%$ results in the formation of a massive atmosphere with hundreds or thousands of bars in the modeling study. Taking into account the thermal blanketing effects, Elkins-Tanton also showed that crystallization of a magma ocean would have ended within $5Myr$ on Earth in the range of water contents considered in this study. This duration is significantly longer than that of the case with no atmosphere, which is estimated to be on the order of thousands of years (see Solomatov, 2007). The obtained crystallization timescale is still short enough to reconcile with the geochemical constraint that liquid water would have been present at the surface at $4.2Ga$ or earlier (Valley, Peck, King, & Wilde, 2002; Wilde, Valley, Peck, & Graham, 2001).

Considering that the atmosphere is in radiative equilibrium, Elkins-Tanton (2008) evaluated the heat flux from the magma ocean $F$ as:

$Display mathematics$(13)

where $σ$ is the Stephan-Boltzmann constant, $ε$ is the emissivity that increases with the optical depth of the atmosphere, and $T⋆$ is the equilibrium temperature, which is determined such that the blackbody radiation with $T⋆$ is in balance with the incoming net stellar radiation. This expression implies that the heat flux is always positive, so that the magma ocean continues to lose its heat, until the surface temperature is equal to the equilibrium temperature, which is of the order of $100K$ in the terrestrial planet region of the solar system. On the other hand, numerical studies with radiative convective models of steam atmospheres had demonstrated that, if a steam atmosphere is in a runaway greenhouse state, a surface temperature higher than the solidus of rocks can be maintained solely by stellar radiation (Abe & Matsui, 1988; Kasting, 1988).

Figure 4 illustrates the planetary flux from the top of a steam atmosphere as a function of surface temperature. If the atmosphere is assumed to be in a thermal equilibrium state, the planetary flux should be in balance with the total heat flux on the planet—that is, the sum of the net stellar radiation and the convective heat flux from the magma ocean. As the surface temperature decreases and the atmospheric pressure increases, the thermal blanketing becomes more effective. If the total heat flux exceeds $280W/m2$(the runaway greenhouse threshold) and at least a few tens of bars of a steam atmosphere are present, the surface temperature exceeds the solidus. By contrast, if the total heat flux is smaller than it, the surface temperature should be lower than a few hundred degrees Kelvin, regardless of the mass of the steam atmosphere.

The presence of the threshold flux led Hamano et al. (2013) and Lebrun et al. (2013) to couple a radiative convective model of steam atmospheres with thermal evolution of a magma ocean. The runaway greenhouse threshold defines the innermost edge of the habitable zone (HZ) as a distance at which the net stellar radiation is equal to the threshold flux. From this definition, a planet in the HZ should receive net stellar radiation smaller than the threshold flux. As mentioned above, only when the convective heat flux from the interior is high enough does the steam atmosphere stay in a runaway greenhouse state, keeping the surface in a molten state. The previous studies showed that the crystallization age of the terrestrial magma ocean would have been around $1Myr$ (Lebrun et al., 2013), or at most $4Myr$ (Hamano et al., 2013). These crystallization timescales are comparable to those obtained by Elkins-Tanton (2008), with an amount of water comparable to the total inventory of the present Earth. Hamano et al. (2013) also predicted that Earth-size planets formed outside the innermost edge of the HZ would have crystallized on a timescale of several million years, even if the initial water endowment was as large as 10 ocean masses of water. As crystallization proceeds, the viscosity in the magma ocean will jump up when the cumulate fraction reaches $~60%$ (Abe, 1997). The viscosity transition would drastically reduce the efficiency of heat transfer by convection in the magma ocean. When the convective total heat flux becomes small enough for the steam atmosphere to get out of a runaway greenhouse state, the surface temperature would quickly drop, and the steam atmosphere would collapse into water oceans, leaving a shallow partially molten layer below the solid surface (Lebrun et al., 2013; Salvador et al., 2017).

Hamano et al. (2013) also included loss of water associated with hydrodynamic escape of hydrogen driven by stellar EUV radiation (see Kasting & Pollack, 1983; Watson et al., 1981) in their model. At orbits inside the innermost edge of the HZ, the net stellar radiation that the planet receives exceeds the runaway threshold so that the steam atmosphere can stay in a runaway greenhouse state by stellar radiation alone. In this case, the planetary surface would never solidify until the steam atmosphere is thin enough (Figure 4). Hamano et al. (2013) proposed that the atmospheric escape would desiccate the planetary surface and interior during solidification. The actual degree of desiccation in the interior would depend on the transport efficiency of water to the surface. They also showed that the crystallization time of the magma ocean can be approximated by the time required for total loss of water, with a larger initial amount of water leading to a longer crystallization time (Figure 5).

These theoretical studies indicated that, as a result of a small difference in the orbital distance, terrestrial planets can have totally different crystallization ages and water budgets during their solidification stage. In the previous studies, there seemed to be little objection to the idea that Earth would correspond to the former case and would have cooled rapidly, whereas the early evolution of Venus is still controversial because of its location in the vicinity of the innermost edge of the HZ. In terms of atmospheric science, the main cause of the uncertain prediction is the effects of clouds on greenhouse effects and planetary albedo. If Venus followed an early evolution similar to Earth’s, it should have cooled on a timescale of less than several million years, irrespective of the water inventory, leading to the rapid formation of early oceans. Hashimoto and Sugita (2003) and Mueller et al. (2008) reported that there is a general trend of low $1μm$ emissivity in highlands on Venus and the observed emissivity values are consistent with felsic rocks. If primordial felsic crusts formed early in Venus’ history, part of them might have been preserved as the low emissivity regions. On the other hand, the latter desiccation scenario provides a good explanation for the fate of leftover oxygen, which is produced by dissociation of $H2O$ molecules into $H$ and $O$ atoms in the upper atmosphere, followed by preferential escape of $H$ (Hamano et al., 2013). $O$ atoms can be dragged into space along with escaping $H$ (Schaefer et al., 2016) or they can be consumed by reaction with reduced gaseous species, such as $H2S$ or $CO$. Even if these $O$ loss processes are inefficient, the remaining oxygen would not accumulate in an atmosphere overlying a magma ocean, as long as $Fe2+$ in the surface magma efficiently reacts with the atmosphere (Gillmann, Chassefière, & Lognonné, 2009). To explain the absence of an Earth-like geodynamo on Venus, Jacobson, Rubie, Hernlund, Morbidelli, and Nakajima (2017) recently proposed that Venus might have formed by late accretion of planetesimals, with few giant impacts at an early stage. A different formation history could have resulted in a different mode of metal–silicate separation, affecting the abundance of siderophile elements in the mantle. These distinct evolution scenarios predict different characteristics for Venus’s mantle, such as the crystallization time, the degree of differentiation, the oxidation state, and the abundance of siderophile elements. Sampling of Venus’s rocks should be key to understanding Venus’s early history, although that will be a challenging task.

### In the Context of Exoplanet Science

Although the theories of planet formation and evolution predict the formation of a magma ocean and steam atmosphere at some stages of accretion, essentially we have no direct record of the earliest history of the terrestrial planets in the solar system. On the other hand, beyond our solar system, 106 exoplanets with a mass below 10 Earth masses and radius below 2 Earth radii are already known to exist (The Extrasolar Planets Encyclopaedia). Some of these small, low-mass exoplanets will be rocky planets and have different ages, and their numbers will increase as detection technology and data analysis techniques advance. Observing terrestrial exoplanets orbiting a young host star may provide us some insight into what young planets and their atmospheres are like. Thermal emission spectra from hot terrestrial planets have already been explored for a wide range of atmospheric compositions (Lupu et al., 2014; Miller-Ricci, Meyer, Seager, & Elkins-Tanton, 2009). The studies showed that thermal emissions from a hot surface would be obscured by a thick atmosphere, but part of it can leak through near-IR atmospheric windows. In addition to the thermal spectra, their variations along the atmospheric evolution during crystallization of a magma ocean have also been predicted by using an atmosphere-magma-ocean coupled model (Bower et al., 2019; Hamano et al., 2015; Katyal et al., 2019). The formation of an outgassed atmosphere can prolong the duration of a magma ocean phase and can increase the occurrence rate of molten terrestrial planets, especially around the innermost edge of the HZ (Hamano et al., 2015). The thermal emissions at the near-IR windows may be faint, but they could still be detectable by future large telescopes. It is also worth considering that multiple giant-impact events would further increase the occurrence of hot molten planets (Bonati, Lichtenberg, Bower, Timpe, & Quanz, 2019). Observing hot molten planets will provide us conclusive evidence on the hot origins of terrestrial planets, as well as some constraints on their solidification timescales and early atmospheres. Turbet, Ehrenreich, Lovis, Bolmont, and Fauchez (2019) have indicated that a transition from a moderate climate to a runaway greenhouse state may cause a quick inflation of the planetary radius. Measuring the radius inflation, as well as the atmospheric structure in a runaway greenhouse state, may further our understanding of the runaway greenhouse concept and its onset conditions.

Recently, a new mechanism for accelerating planetary accretion has been proposed, namely pebble accretion. Compared with planetesimals, millimeter-to-centimeter-size pebbles can accrete onto a planetary body more efficiently, due to the presence of effective gas drag. In addition, pebbles can drift from the outer orbit, which continuously replenishes solid materials around the growing body. As a result, the accretion rate of protoplanets by pebble accretion can be one order of magnitude higher than that by the planetesimal accretion assumed in the previous studies on the thermal evolution and water budget of protoplanets (see Lambrechts & Johansen, 2014). The evolution and thermal state of protoplanets would depend on the water and volatile contents and the redox state of pebbles drifting from the outer orbits, as well as the accretion history of the pebbles. Because these characteristics of pebbles probably depend on the evolution of a protoplanetary disk and giant planet formation, fully formed terrestrial planets and their atmospheres may also become diverse in different planetary systems. Examining the interior structure and atmosphere of protoplanets growing by pebble accretion has just started (see Brouwers & Ormel, 2019) and there is much that remains to be explored within the new framework of planet formation.

### Acknowledgments

The author thanks two anonymous reviewers for their careful reading and constructive comments. This work was funded by KAKENHI grant numbers JP18K13603 from JSPS and JP17H06457 from MEXT.