# Magma Ocean, Water, and the Early Atmosphere of Venus

Arnaud Salvador, Guillaume Avice, Doris Breuer, Cédric Gillmann, Helmut Lammer, Emmanuel Marcq, Sean N Raymond, Haruka Sakuraba, Manuel Scherf, M J Way

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## To cite this version:

Arnaud Salvador, Guillaume Avice, Doris Breuer, Cédric Gillmann, Helmut Lammer, et al., Magma Ocean, Water, and the Early Atmosphere of Venus. Space Science Reviews, 2023, 219, pp.51-10.1007/s11214-023-00995-7. insu-04213184

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Submitted on 21 Sep 2023

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# Magma Ocean, Water, and the Early Atmosphere of Venus

Arnaud Salvador¹,²,³ ID · Guillaume Avice⁴ ID · Doris Breuer⁵ ID · Cédric Gillmann⁶ ID ·

Helmut Lammer⁷ ID · Emmanuel Marcq⁸ ID · Sean N. Raymond⁹ ID ·

Haruka Sakuraba¹⁰ ID · Manuel Scherf⁷,¹¹,¹² · M.J. Way¹³,¹⁴ ID

Received: 22 February 2023 / Accepted: 10 August 2023

© The Author(s) 2023

## Abstract

The current state and surface conditions of the Earth and its twin planet Venus are drastically different. Whether these differences are directly inherited from the earliest stages of planetary evolution, when the interior was molten, or arose later during the long-term evolution is still unclear. Yet, it is clear that water, its abundance, state, and distribution between the different planetary reservoirs, which are intimately related to the solidification and outgassing of the early magma ocean, are key components regarding past and present-day habitability, planetary evolution, and the different pathways leading to various surface conditions.

In this chapter we start by reviewing the outcomes of the accretion sequence, with particular emphasis on the sources and timing of water delivery in light of available constraints, and the initial thermal state of Venus at the end of the main accretion. Then, we detail the processes at play during the early thermo-chemical evolution of molten terrestrial planets, and how they can affect the abundance and distribution of water within the different planetary reservoirs. Namely, we focus on the magma ocean cooling, solidification, and concurrent formation of the outgassed atmosphere. Accounting for the possible range of parameters for early Venus and based on the mechanisms and feedbacks described, we provide an overview of the likely evolutionary pathways leading to diverse surface conditions, from a temperate to a hellish early Venus. The implications of the resulting surface conditions and habitability are discussed in the context of the subsequent long-term interior and atmospheric evolution. Future research directions and observations are proposed to constrain the different scenarios in order to reconcile Venus’ early evolution with its current state, while deciphering which path it followed.

**Keywords** Venus · Interior evolution · Atmosphere · Degassing

## 1 Introduction

Water is not only an essential element for the chemical and biological processes responsible for the emergence and development of life and living organisms (e.g., Westall and Brack 2018; Hoehler et al. 2018). Its abundance and distribution within and between the different

Venus: Evolution Through Time

Edited by Colin F. Wilson, Doris Breuer, Cédric Gillmann, Suzanne E. Smrekar, Tilman Spohn and

Thomas Widemann

Extended author information available on the last page of the article
reservoirs of a rocky planet, from the deep interior to the exosphere, is also of fundamental importance in controlling their properties and thus the physical and chemical processes at play within and between these reservoirs, making it a crucial factor in setting the conditions suitable for the development of life and for the overall planetary evolution (e.g., Kasting and Catling 2003; Hirschmann 2006; Karato 2015; Foley and Driscoll 2016; Korenaga et al. 2017; Ohtani 2020). Understanding the mechanisms shaping the water budget within the planets and how they evolve through time is thus required to build a consistent picture of planetary evolution and characterize the potential habitability of terrestrial planets (classically defined as the ability of a planet to sustain liquid water at its surface; e.g., Kasting et al. 1993b).

From that perspective, the early stages of planetary evolution appear to be of first importance. Indeed, during this relatively short but intense period of time, the combination of different heat sources, in particular the energy delivered by successive collisions of planetary building-blocks during planetary accretion and growth, is believed to substantially melt the planetary body (e.g., Safronov 1978; Kaula 1979; Tonks and Melosh 1993; Nakajima et al. 2021), resulting in the creation of a so-called “magma ocean” (“MO”): a possibly global, from mantle-thick to near-surface shell of magma of low, water-like viscosity (hence “ocean”; e.g., Warren 1985). During this magma ocean stage, because of the molten surface, the absence of a thick, long-lasting boundary layer between the planetary interior and the atmosphere allows for free and extremely efficient thermal and chemical exchanges between these reservoirs.

The modern concept of magma ocean originally took root in the study of the mineralogy of the Moon’s surface from the lunar samples brought back to Earth in the seventies. The formation of the bright anorthositic crust, making up the ancient lunar highlands covering most of the surface (e.g., Ji et al. 2022), was proposed to result from the flotation of light Ca- and Al-rich anorthite minerals (plagioclase feldspar) over a crystallizing, denser molten silicate layer, hence requiring the upper part of the lunar mantle to be initially molten (e.g., Smith et al. 1970; Wood et al. 1970; Warren 1985; Binder 1986). Early and extensive melting of the surface and subsurface was confirmed by trace element analysis and experimental petrology (Schnetzler and Philpotts 1971; Taylor 1986). The origin of the lunar magma ocean was then tied to the formation process of the Moon itself and in particular to the energetics of collisional accretion (e.g., Wetherill 1976; Taylor 1986; Matsui and Abe 1986c; Hartmann 1986; Cameron 1986), where conversion of kinetic energy into heat from impacts could increase the temperature up to the melting point of silicates and potentially melt the surface and the interior of the impacted body to an extent depending primarily on the impactor size and resulting heat burial.

Reciprocally, the Earth was thought to be significantly heated and molten as a result of the Moon-forming event(s) and more generally out of the accretion sequence (Safronov 1978; Kaula 1979; Wetherill 1985), thought to be of increasing violence with time, with the increasing size of impactors. By considering each plausible combination of heating mechanisms involved in melting the Moon, Hostetler and Drake (1980) extended the analysis on the early thermal state of the Earth and other terrestrial planets and suggested that early global melting (and differentiation) was also likely for all terrestrial planets (e.g., Drake 2000). The hypothesis of a single moon-forming giant impact gained popularity and became the overarching paradigm for lunar origin (e.g., Hartmann and Davis 1975; Cameron and Ward 1976; Benz et al. 1986; Boss 1986; Taylor 1986; Newsom and Ross Taylor 1989; Drake 2000; Canup and Asphaug 2001). The reciprocal idea that the Earth experienced a planetary-scale global magma ocean, possibly melting the entire mantle molten (Wetherill 1985), as a result of such collision (or from violent accretion) gained interest (Melosh 1990;
Tonks and Melosh 1993). The existence of a molten stage on rocky planets was also indepen-
dently confirmed by early melting in planetesimals triggered by radiogenic heating from the
decay of radioactive elements (e.g., Urey 1955; Fish et al. 1960; Lee et al. 1976). It is now
generally accepted that all terrestrial planets, including Venus, experience at least a transient
magma ocean stage early in history but the number of occurrences, timing, timescale and
depth of the molten layer may significantly vary from one planet to another (see Elkins-
Tanton 2012; Schaefer and Elkins-Tanton 2018, for reviews).

In this phase, some of the major processes shaping the conditions at the surface and
in the interior of the planet take place, and set the initial conditions for the later long-
term state and evolution of the planet (e.g., Schaefer and Elkins-Tanton 2018). The volatile
and “atmophile” (i.e., air-loving) species dissolved within the molten mantle are outgassed
and build-up the atmosphere, the rapid segregation of heavy metallic iron (and associated
“siderophile”, i.e., metal-loving, elements) and lighter silicate (and “lithophile”, i.e., stone-
loving, elements) favored by the low viscosity of the melt is responsible for the chemical
differentiation of the planet. The distribution of water between the different reservoirs, the
amount reaching the surface as well as the early climate and associated liquid water sustain-
ability at the surface are direct outcomes of the magma ocean solidification. The resulting
internal structure and elemental distribution thus set the starting point for the onset of man-
tle convection and for the long-term evolution of the planet. These early stages of planetary
evolution are thus of fundamental importance in controlling the potential past and present-
day habitability of a planet and in understanding the pathways to the present-day surface
conditions.

For a planet to be habitable, two sine qua non conditions must be achieved. First, water
needs to reach and be abundant enough at the surface of the planet, and second, the surface
conditions (pressure and temperature) must allow for the water to condense. These two
conditions are often addressed separately, while implicitly assuming that the other condition
is satisfied. One positive condition does not imply the other, yet they are intricately linked.
As mentioned above, these two requirements strongly rely on the outcomes of the magma
ocean phase. On the other hand, maintaining a temperate and stable climate with sustainable
water oceans long enough for life to emerge then relies on additional processes such as
atmospheric escape and geochemical cycles, which affect the planetary environment over
longer timescales, throughout the entire evolution of the planet, including when the internal
heat is no longer climatologically significant.

Given the fact that planetary surface conditions are essentially controlled by atmospheric
and climate dynamics, habitable zone and habitability-related studies have been historically
conducted from an atmospheric point-of-view, using atmospheric models as main tools to
assess the surface temperature and pressure and infer the presence of liquid water at the plan-
etary surface (e.g., Kasting et al. 1993b; Selsis et al. 2007; Kopparapu et al. 2013; Ramirez
2018). Despite being highly influential, the early evolution stages have been neglected in
habitable zone studies, thus possibly missing important pieces of the puzzle. Only recently
has the importance of interior-atmosphere feedback on habitability been recognized (e.g.,
Zahnle et al. 2007; Noack et al. 2014; Foley and Driscoll 2016; Ramirez et al. 2018; Dehant
et al. 2019).

The case of Venus is still puzzling and these neglected components might hold the key.
Given their vicinity in the inner Solar System, the Earth and Venus likely experienced many
similarities during their formation, including analogous accretion sequences, chemical el-
ements endowments and volatile delivery. Their nearly identical sizes and bulk densities
suggesting similar bulk compositions (e.g., Smrekar et al. 2018) are also indicative of a
common past. Yet, Earth's present-day liquid water- and life-sustaining temperate climate
differs strikingly from the mostly dry and hellish Venusian surface conditions. Whether these
differences are directly inherited from the tumultuous magma ocean stage or appeared later
in history remains unclear. Studying the planetary evolution and in particular this common
molten past, at the earliest ages of the Solar System, may be one of the keys to resolving
this paradox and unveil the evolutionary pathway(s) leading to a habitable and inhabited
planet. This is the object of the present chapter, with particular emphasis on the role and
feedback of and between water and interior-atmosphere interaction when the planet was
molten. Throughout the chapter, we will then address the following key questions:

* What are the initial conditions and water content of the molten interior of Venus? (Sect. 2)

* What processes affect the thermal evolution of the magma ocean and the concurrent atmospheric formation? (Sect. 3)

* How is water distributed between the solidifying mantle, the molten interior, and the atmosphere during magma ocean evolution? (Sect. 3)

* What are the outcomes of the magma ocean phase and the corresponding implications for the habitability and long-term evolution of the planet? (Sect. 4)

* When and how did the Earth and Venus surface conditions diverge? (Sect. 4)

* What are the next steps to achieve towards a better understanding of early Venusian evolution? (Sect. 5)

To do so, the chapter is organized as follows: Sect. 2 describes the initial conditions of the molten stage evolution, by focusing on its initial thermo-chemical state resulting from the accretion sequence, and on the initial amount of water accreted as a function of the different sources and processes of volatiles delivery. Section 3 focuses on the processes at play during magma ocean thermal evolution and concurrent formation of the atmosphere, including the feedback between the interior and the atmospheric reservoirs. Section 4 examines the possible outcomes of the magma ocean cooling sequence and how the different evolutionary paths could influence the subsequent surface conditions and long-term evolution of the planet. Finally, Sect. 5 discusses how further research directions combined with future observations and measurements could provide clues to decipher the early evolution of Venus and choose between the different scenarios to eventually reconstruct a consistent picture of the history of the planet.

## 2 Water Delivery and Initial Conditions for the Molten Mantle Evolution: What do We Start with?

Near the end of planetary accretion, the planet's water inventory and thermal state play the two key roles in its subsequent evolution. Yet, even for Earth, these parameters are highly unconstrained. In this section, we provide an overview of the processes that could be responsible for water delivery on early Venus. We estimate the amount of water delivered, and describe the early energy sources that control the initial heat budget of the planet. In Sect. 2.1 we discuss the possibility for Venus to produce water from the nebular gas gravitationally captured within the protoplanetary disk. We then describe the outcomes of the accretion sequence as a function of the different planetary formation scenarios in Sect. 2.2. While the sources, timing, and amount of water delivered to Earth are still debated, in Sect. 2.2.1 we discuss how the different planetary formation scenarios and their implications for water delivery apply to Venus. We also describe the different heat sources at play and the corresponding thermal state of the mantle resulting from the accretion sequence (Sect. 2.2.2). We then examine scenarios of water delivery, budget, and fate in light of available constraints from isotopic ratios in Sect. 2.3.
## 2.1 Proto-Venus and H₂/He Envelope: Did Early Venus Produce Water from the Protoplanetary Nebula?

If proto-Venus accreted a sufficient mass within the protoplanetary disk lifetime, the growing planet could have captured an H₂-dominated primordial atmosphere by accreting nebular gas from the circumstellar disk, which could have then been quickly lost by Extreme UltraViolet-driven (EUV) hydrodynamic escape after the disk dissipated (e.g., Hayashi et al. 1979; Stökl et al. 2016; Sharp 2017). If this was the case, the efficient hydrodynamic hydrogen flow dragged heavier elements such as noble gas isotopes with it, leading to modifications of their initial isotope ratios (Pepin 1991).

Recently, Lammer et al. (2020b) applied a hydrodynamic upper atmosphere and a Smooth Particle Hydrodynamics (SPH) impact model for the loss calculations of captured H₂-dominated envelopes. This was done for various protoplanetary masses and a wide range of possible EUV-activity evolution tracks of the young Sun and initial atmospheric compositions based on mixtures of captured nebula gas, outgassed and delivered materials. As shown in Fig. 1, these authors could reproduce the present Venus atmospheric ³⁶Ar/³⁸Ar, ²⁰Ne/²²Ne isotope ratios if proto-Venus accreted a mass of ~0.84–1.0 M<sub>Venus</sub> (~0.68–0.81 M<sub>⊕</sub>) during the disk lifetime of ~3–4.5 Myr (Bollard et al. 2017; Wang et al. 2017) after the origin of the Sun.

Lammer et al. (2020b) also showed that proto-Venus could have captured primordial H₂-dominated envelopes with hydrogen surface partial pressures between ~40–1000 bar (Fig. 1). The blanketing effect of such dense and opaque atmospheres would be responsible

<figure><img src="image_3.png" /><figcaption>**Fig. 1** a) Successful reproduction attempt of the Pioneer Venus, Venera 11 and 12 ³⁶Ar/³⁸Ar, and Venera 13 and 14 ²⁰Ne/²²Ne noble gas isotope ratios within the measured error bars if Venus accreted its final mass during the disk lifetime. In such a case the planet would have lost, a hydrogen-dominated primordial atmosphere with ~1000 bar partial pressure after ≥ 200 Myr if the young Sun was a weakly to moderately active young G-star. b) Successful reproduction attempt of the Venera 11 and 12 ³⁶Ar/³⁸Ar, and Venera 13 and 14 ²⁰Ne/²²Ne noble gas ratios within the measured error bars for a proto-Venus with ~0.84 M<sub>Venus</sub> (~0.68 M<sub>⊕</sub>). We use a 100% EC-like building block composition that accreted about 17% of the planet mass after a primordial atmosphere with a hydrogen partial pressure of ~40 bar was lost until ~7 Myr after the origin of the Sun and ~3 Myr after the disk evaporated (after Lammer et al. 2020b)</figcaption></figure>
for extensive mantle melting (e.g., Hayashi et al. 1979; Olson and Sharp 2019). Then, if proto-Venus grew partly within the gas disk (Ikoma and Genda 2006; Ikoma et al. 2018), the primordial atmospheric H₂ can be oxidized by interactions between the gas and the underlying molten surface. This requires surface temperatures above the melting point of silicate (~1400 K), thereby producing water that originates from the protoplanetary nebula. Hydrogen and water would thus be ingassed into the molten interior while some other fraction of hydrogen would possibly be sequestered into the core (e.g., Sharp 2017; Wu et al. 2018; Olson and Sharp 2018, 2019; Young et al. 2023). How much water can be produced from the nebula depends both on the mass of the surrounding H₂-envelope and on the oxidation state of the magma ocean, characterized by the available iron oxides (e.g., wüstite and magnetite) and fayalite (Fe₂SiO₄) in the magma ocean (Ikoma and Genda 2006). Logically, the amount of water produced from primordial H₂ increases with the increasing chemical potential of oxygen (i.e., oxygen fugacity; see Sect. 3.2.5), as more oxygen atoms available combine with hydrogen atoms to produce more water. Indeed, according to Ikoma and Genda (2006), at 1500 K the mass ratio of produced H₂O to the primordial atmospheric H₂, M<sub>H₂O</sub>/M<sub>H₂</sub>, increases from approximately 0.49 to 0.88, and to 24.02 for increasingly oxidizing environments, corresponding to the buffer assemblages of quartz-iron-fayalite (QIF):

$$2 \underset{\text{iron}}{\mathrm{Fe}} + \underset{\text{quartz}}{\mathrm{SiO}_2} + \mathrm{O}_2 \rightleftharpoons \underset{\text{fayalite}}{\mathrm{Fe}_2\mathrm{SiO}_4}, \qquad (\mathrm{R1})$$

iron-wüstite (IW):

$$2 \underset{\text{iron}}{x \mathrm{Fe}} + \underset{\text{wüstite}}{\mathrm{O}_2} \rightleftharpoons 2 \underset{\text{magnetite}}{\mathrm{Fe}_x \mathrm{O}}, \qquad (\mathrm{R2})$$

and wüstite-magnetite (WM):

$$\underset{\text{wüstite}}{6/(4x-3)} \mathrm{Fe}_x \mathrm{O} + \underset{\text{magnetite}}{\mathrm{O}_2} \rightleftharpoons \underset{\text{magnetite}}{2x/(4x-3)} \mathrm{Fe}_3 \mathrm{O}_4 \qquad (\mathrm{R3})$$

respectively (e.g., Frost 1991). Note that the side of the reaction that has greater entropy,<fnref n="1" /> that is the side with oxygen, is stable at higher temperature than the other side (e.g., Fig. 1 in Frost 1991). Ikoma and Genda (2006) used values of x = 0.947 for IW and x = 0.974 for WM taken from Robie et al. (1978).

Note that the oxidation state of the magma ocean, and therefore water production efficiency, relates to the timing and location of metal-silicate separation. If metallic iron has not yet segregated from the magma ocean and coexists with the molten silicate, as believed for the earliest evolution steps, the magma ocean is thought to be reducing. It would be at the QIF or IW redox buffer rather than at the WM buffer, thus producing less water than if the metallic iron has already segregated into the core (see Sect. 3.2.5 for definitions of mineral redox buffers and discussion regarding the magma ocean oxidation state). However, the picture might be more complicated, making the influence of the magma ocean oxidation state on the process of hydrogen ingassing and water production less straightforward. For instance, the ingassing hydrogen may cause Fe reduction, followed by iron removal to the core. In this case, Sharp (2017) suggested that far larger amounts of H₂ could be incorporated into the melt, which could in turn enhance water production even with abundant metallic iron in a reducing environment.

<footnote n="1">¹By convention reactions are written with the high entropy side on the right. Here we kept it on the left, so that the oxidation reads from left to right.</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 2** The amount of Earth oceans (green line) that may be produced by a captured H₂-dominated primordial atmosphere (blue line) due to gas-melt interactions between the atmosphere and a magma ocean according to Ikoma and Genda (2006). The vertical and horizontal dashed-lines correspond to the possible proto-Venus mass and corresponding primordial atmospheric mass ranges where Venus' present-day Ar and Ne isotope ratios can be reproduced within the error bars</figcaption></figure>

If one applies the same assumption as Ikoma and Genda (2006) for the hydrogen envelopes as simulated by Lammer et al. (2020b), shown in Fig. 1, and assumes that such oxygen buffers (i.e., QIF, IW, and WM) are available, a proto-Venus with a mass of ~0.84 M<sub>Venus</sub> could produce a water reservoir on early Venus from the captured H₂-envelope of at least ~2 Earth oceans, as shown in Fig. 2. In case proto-Venus accreted its final mass during the disk lifetime, then a nebular-based H₂O amount of at least ~8 Earth oceans could have been produced (see Fig. 2). The probability of terrestrial planet formation within the nebular gas is still debated. This is because the dissipation mechanisms remain uncertain. Yet, the capture of other nebular gases such as Ne followed by their incorporation into molten planetary interiors has further been inferred from mantle isotopic ratios (Williams and Mukhopadhyay 2019). It supports the idea that terrestrial planets may partly form before the nebular gas dissipated. One should note that molten silicate-metallic core separation may not be complete by the time the nebular gas fully vanishes, suggesting a rather reducing magma ocean and associated relatively low water production (e.g., at the water production rates imposed by the QIF or IW buffers).

In comparison, Lammer et al. (2020a) found that Earth's present-day atmospheric ³⁶Ar/³⁸Ar, ²⁰Ne/²²Ne, and ³⁶Ar/²²Ne isotope ratios could only be reproduced if the protoplanet grew a mass of ≤ 0.6 M⊕ within the solar nebula. It implies that, contrary to Venus, only ~2% of the current value of Earth's seawater could have been produced from the protoplanetary nebula. This low value for the Earth lies within the error bars of the D/H seawater-carbonaceous chondrite "match" of 150±10 × 10⁻⁶ (Robert et al. 2000; Pahlevan et al. 2019), and is in agreement with measurements of the D/H ratios in glassy melt inclusions in two deep mantle basaltic rock samples (Hallis et al. 2015). This suggests that only a small fraction of Earth's water was derived from nebular accretion (e.g., Wu et al. 2018). Nevertheless, Young et al. (2023) recently emphasized that primordial H₂ atmospheres re-
main an influential and promising mechanism to explain fundamental chemical features of terrestrial planets, such as Earth's water content, core density, and oxidation state.

If proto-Venus grew partly within the gas disk and produced nebula-based water, its initial D/H ratio of $21\pm5 \times 10^{-6}$ could then resemble that of nebular gas which is ~7 times lower than Earth's seawater (Geiss and Gloeckler 1998; Robert et al. 2000) and CCs (e.g., Marty 2012) measured D/H ratios. If this was the case, then Venus' present-day D/H ratio of $160\pm20 \times 10^{-4}$ (Donahue et al. 1982) would be ~762 times higher than initially instead of ~106 times, if the planets initial H$_2$O inventory were CC-based as on Earth. In such a case atmospheric escape must have fractionated D/H much more significantly than expected in previous studies (Donahue 1999).

Note that many stars are surrounded by gaseous and dusty debris disks. Compared to the protoplanetary disks surrounding young newly formed stars, those long-lasting (possibly for hundreds of millions of years; Wyatt 2008; Matrà et al. 2017) debris disks can be found and remain after planets have already formed. They result from the collisions and destruction of asteroids, comets, planetesimals, and dwarf planets (rather than being of primordial origin as the younger protoplanetary disks; e.g., Dent et al. 2014; Hughes et al. 2018; Matrà et al. 2019). Rather than being dominated by hydrogen and helium, the gas of those disks is mainly composed of CO, carbon, and oxygen (e.g., Cataldi et al. 2014; Kral et al. 2017). Terrestrial planets could form massive atmospheres from the capture of the gas contained in those debris disks (Kral et al. 2020). Yet, the amount of water produced and incorporated into the planet from such accreted atmospheres has not been assessed. It may be limited compared to protoplanetary disks due to the absence of hydrogen in their carbon and oxygen rich compositions (e.g., Cataldi et al. 2014; Kral et al. 2017).

## 2.2 Outcomes of Planetary Formation Scenarios

### 2.2.1 Water Delivery

Understanding the growth of the rocky planets is essential because it sets their volatile inventories. Yet our picture of the growth of the rocky planets remains incomplete. This comes in large part from a lack of hard constraints during the era of accretion. The initial conditions for Solar System formation are likely represented in the circumstellar disks that are directly observed around nearby young stars, for instance with the ALMA telescope (e.g., ALMAPartnership et al. 2015; Andrews et al. 2018). The population of known exoplanets represent thousands of outcomes of planet formation (e.g., Winn and Fabrycky 2015). At intermediate size- and time scales between young circumstellar disks and mature planetary systems, empirical constraints are hard to find.

The current paradigm of Solar System formation involves a series of different stages of growth (for a review, see Raymond and Morbidelli 2022). First, micron-sized dust grains collide and coagulate until they reach one of several growth barriers at roughly mm-sizes (Blum and Wurm 2008; Zsom et al. 2010). These large dust grains are commonly called “pebbles”, and have sufficient inertia that their orbits are mainly Keplerian (see Johansen and Lambrechts 2017). However, the gas orbits slightly slower than the Keplerian speed because of partial pressure support. Pebbles therefore feel a headwind from the gas, which causes them to lose orbital energy and spiral inward – or rather, along the local pressure gradient – on short timescales (Weidenschilling 1977; Haghighipour and Boss 2003). Pebbles generally drift inward, but can become concentrated in regions in pressure bumps within the gas disk (Birnstiel et al. 2012). When pebbles are sufficiently concentrated in any given region, they can be further concentrated by the streaming instability and directly clump into
10–100 km-scale planetesimals (Youdin and Goodman 2005; Johansen et al. 2009, 2014). These are generally considered the “building blocks” of the planets. While the classical model assumes that planetesimals formed uniformly across the disk (e.g., Raymond et al. 2014), new models propose that planetesimals form in different regions at different times, often at (moving) condensation fronts (Armitage et al. 2016; Drążkowska and Alibert 2017; Drążkowska and Dullemond 2018; Morbidelli et al. 2022; Izidoro et al. 2022). Planetesimals continue to grow by both collisions between planetesimals and by accreting pebbles that continue to drift within the disk (Johansen and Lambrechts 2017; Ormel 2017). The giant planet cores are thought to have rapidly accreted by accreting pebbles (Lambrechts and Johansen 2014; Levison et al. 2015). In contrast, the terrestrial planets are generally thought to have mainly grown by planetesimal accretion (Raymond et al. 2009; Chambers 2016; Izidoro et al. 2021b), although a new class of models proposes that pebble accretion may have played a role (Johansen et al. 2021). Next, planetesimals grow into terrestrial planet embryos and giant planet cores. In the terrestrial planet-forming region, planetary embryos were perhaps a Mars-mass (~10% of an Earth-mass; Morbidelli et al. 2012), whereas the giant planets’ cores are thought to have been ~10–20 $M_⊕$. The final assembly of the terrestrial planets is thought to have involved giant impacts between planetary embryos as well as a sweep-up of leftover planetesimals (for reviews, see Morbidelli et al. 2012; Raymond et al. 2014; Jacobson and Walsh 2015; Raymond and Morbidelli 2022).

Because it is smaller and faster orbiting, Venus is significantly more likely to have experienced more “hit-and-run” (bouncing; e.g., Asphaug et al. 2006; Reufer et al. 2012) collisions than Earth (Emsenhuber et al. 2021). In addition, Emsenhuber et al. (2021) showed that Venus is expected to eventually accrete most of its runners while Earth loses about half, with many of those ending up at Venus. Generally, Venus serves as a sink of runners emerging from farther out collisions and is more likely to have accreted a massive outer Solar System body. This means that despite their proximity, the collisional histories of the two planets were not identical.

The first quantitative model for terrestrial planet formation, developed from the 1970s to the 2000s, is commonly referred to as the “classical model”, and was built upon the work of several pioneers in the field, notably that of George Wetherill (key papers include Safronov 1972; Greenberg et al. 1978; Wetherill 1978, 1985, 1990). Yet, the classical model – which assumes that a smooth disk of planetesimals existed between the present-day orbits of Venus and Jupiter upon dissipation of the Sun’s gaseous disk – systematically fails to reproduce the planets’ orbital architecture, forming Mars analogs that are as massive as Earth (Wetherill 1991; Raymond et al. 2009; Lammer et al. 2020a). In recent years, three planetesimal accretion-based solutions have been proposed to this “small Mars” problem and are illustrated in Fig. 3 (for details, see Raymond et al. 2020). The Low-mass asteroid belt model proposes that Venus and Earth grew within a narrow ring/annulus of planetesimals, but few planetesimals ever formed in the primordial asteroid belt. Mars was stranded outside this ring and its growth was stunted (Hansen 2009; Drążkowska et al. 2016; Raymond and Izidoro 2017b; Izidoro et al. 2022). The Grand Tack model starts from the same extended planetesimal disk assumed in the classical model, but Mars’ feeding zone was strongly depleted by Jupiter’s gas-driven orbital migration, which first drove the gas giant inward to 1.5–2 AU, then back outward past 5 AU (Walsh et al. 2011; Jacobson and Morbidelli 2014; Raymond and Morbidelli 2014; Brasser et al. 2016). The third model is based on the giant planets’ instability, which has been invoked by many studies to explain the giant planets’ orbits and populations of small Solar System bodies (for a review, see Nesvorný 2018). While it was originally invoked as a late instability (Gomes et al. 2005), re-analysis of the dynamics and constraints suggest that it instead took place within 100 million years
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Cartoon depiction of three global models of terrestrial planet formation that can match the inner Solar System (adapted from Raymond et al. 2020)</figcaption></figure>

of the start of planet formation (Zellner 2017; Nesvorný et al. 2018; Morbidelli et al. 2018; Mojzsis et al. 2019), and perhaps during the dispersal of the gas disk itself (Liu et al. 2022b). An early instability naturally depletes the planetesimal disk exterior to Earth's present-day orbit and can match the terrestrial planets and asteroid belt (Clement et al. 2018, 2019,b, 2021). Finally, two recent models have proposed that pebble accretion and gas-driven migration may have been the dominant processes in terrestrial planet formation (Johansen et al. 2021; Brož et al. 2021), with just a single late giant impact to explain the formation of the Moon (Ćuk and Stewart 2012; Canup 2012).

Recent measurements have uncovered a dichotomy in the isotopic signatures of meteorites (Warren 2011; Kleine and Walker 2017; Kruijer et al. 2020). These signatures are apparent in a number of different elements with different chemical characteristics. This has led to a change in paradigm regarding the source material that built the planets: rather than a smooth gradient of chemical characteristics across a broad planetesimal disk, it is thought that two distinct reservoirs of planetesimals formed concurrently during the gas disk lifetime in different regions within the Solar System (e.g., Budde et al. 2016). In fact, recent analysis of samples of the asteroid Ryugu, returned from JAXA's Hayabusa2 mission, indicates that the meteorite isotopes actually represent a “trichotomy,” with three distinct clusters such that there may have been three distinct planetesimal reservoirs in the Sun's planet-forming disk (Hopp et al. 2022). Exactly how the different reservoirs were kept apart is debated: the flux of inward-drifting pebbles may have been cut off by Jupiter's growing core (Kruijer et al. 2017) or by a pressure bump in the disk (Brasser and Mojzsis 2020; Izidoro et al. 2021a) or by something else (Lichtenberg et al. 2021b; Liu et al. 2022). The two main types of meteorites are called non-carbonaceous and carbonaceous. The parent bodies of non-carbonaceous meteorites are thought to have originated in the inner Solar System, whereas carbonaceous meteorites likely originated beyond Jupiter's orbit. The present-day orbital distributions of the different types of meteorites overlap in the asteroid belt (DeMeo and Carry 2014), but they almost certainly formed far apart (Raymond and Izidoro 2017b; Raymond and Nesvorný 2022).

Meteorite isotopes constrain the origin of Earth's water, and although we do not have comparison samples from Venus, we can use dynamical arguments to extrapolate to Venus. It has long been known that bulk Earth's D/H and ¹⁵/¹⁴N ratios are well-matched by carbonaceous chondrite meteorites (Marty and Yokochi 2006; Marty 2012). For the past 20
years it has been generally accepted that most of Earth's water was delivered from a carbonaceous source (Morbidelli et al. 2000; Meech and Raymond 2020). To match the isotopic signatures of a range of meteorites, Earth must have accreted almost entirely from non-carbonaceous (Enstatite chondrite-like) bodies indigenous to the inner Solar System, but with a ~5% contribution from carbonaceous material, assumed to have originated beyond the orbit of Jupiter (Burkhardt et al. 2021). This small "pollution" of carbonaceous chondrite-like planetesimals represents the source of Earth's water in this context. However, recent measurements of Enstatite chondrites revealed that these objects contain far more water than previously thought, and with D/H and ¹⁵/¹⁴N ratios consistent with inner Earth (Piani et al. 2020). Measurements and analysis of zinc isotopes in different types of meteorites can concurrently match the isotope ratios of hydrogen, nitrogen and zinc if roughly two-thirds of Earth's volatiles were sourced locally and one-third was delivered from a carbonaceous source (Steller et al. 2022; Savage et al. 2022). Future isotopic measurements of other volatiles will certainly refine this analysis and give stronger constraints on the relative fraction of Earth's water that was sourced locally vs. delivered.

The question remains: what was the source of the carbonaceous component of Earth's water? In the classical model, Earth's feeding zone was broad enough that it extended past 2.5 AU, where it is assumed that water-rich planetesimals and planetary embryos originated (Morbidelli et al. 2000; Raymond et al. 2006). This mechanism delivered a modestly-smaller amount of water to Venus than Earth, simply because the tails of feeding zones of closer-in planets are less extended (Raymond et al. 2004). However, the initial conditions of the classical model are suspect, as it assumes that carbonaceous and non-carbonaceous planetesimals formed as close neighbors, which would predict a gradient in isotopic compositions rather than the observed dichotomy. Indeed, the present-day orbits of water-rich asteroids cannot represent the correct initial conditions for Solar System formation. Rather, carbonaceous asteroids were likely implanted from beyond Jupiter's orbit during the gas giants' growth (Raymond and Izidoro 2017) or migration (Walsh et al. 2011; Raymond and Izidoro 2017). During this implantation, a large population of scattered carbonaceous planetesimals would also have crossed the orbits of the terrestrial planets (Walsh et al. 2011; O'Brien et al. 2014; Raymond and Izidoro 2017; O'Brien et al. 2018). Given their source beyond Jupiter, scattered planetesimals would more efficiently deliver water to rocky planets at larger orbital radii. Earth (or its building blocks) would have accreted modestly more scattered planetesimals than Venus.

In the pebble accretion framework, water can be delivered by water-rich pebbles but only to planets orbiting beyond the snow line (Sato et al. 2016; Ida et al. 2019). As the disk cools in time, the snow line moves inward, and most models find a snow line interior to 1 AU during the late phases of the disk (Lecar et al. 2006; Kennedy and Kenyon 2008; Martin and Livio 2012; Bitsch et al. 2015). If the flux of carbonaceous pebbles was not blocked at this time, then Earth could have accreted water from drifting pebbles (as in the models of Johansen et al. 2021; Brož et al. 2021). Earth and Venus (or their building blocks) would have accreted similar amounts of water unless the snow line happened to lie in between the planets' orbits, in which case Earth could have accreted much wetter than Venus. While this is a narrow range of orbital radii, that range falls within the potential values for the snow line's position for reasonable disk parameters (Martin and Livio 2012; Bitsch et al. 2015). However, the relative contribution of carbonaceous material could not have been more than ~5% according to meteorite isotopic studies that include a number of different elements (Burkhardt et al. 2021). This makes it difficult to reconcile the pebble accretion scenario with the terrestrial planets' growth and water delivery.

We can make an educated guess as to Venus' initial water content. Let us first assume that two-thirds of Earth's water was indeed sourced locally, as suggested by recent zinc isotopic
analyses (Savage et al. 2022; Steller et al. 2022), and that the rest was delivered by scattered carbonaceous planetesimals (Walsh et al. 2011; Raymond and Izidoro 2017b). Given their near-identical feeding zones (Izidoro et al. 2022), Venus would presumably have the same concentration of locally-sourced water as Earth. However, water delivery from scattered planetesimals has a radial gradient in efficiency such that Venus was likely delivered somewhat less water from carbonaceous sources than Earth (O'Brien et al. 2014; Raymond and Izidoro 2017b), implying an overall amount of water delivery within a factor of a few between the two planets, with Earth being modestly more water-rich. While there is still some variation in water contents in different Enstatite chondrites (Piani et al. 2020), it is hard to imagine that Venus could have formed completely dry.

The relative amount of incorporated water within Earth and Venus sensitively depends on the delivery mechanism, as well as their relative growth rates and the behavior of the underlying gaseous disk. Models for the source of Earth and Venus' water are clearly heavily influenced by isotopic measurements of different types of meteorites. We expect that new measurements that include a wider range of elements – in particular, more and more volatile elements and elements with different chemical properties – will guide our future thinking.

### 2.2.2 Energy Delivery, Heat Sources, and Mantle Initial Thermal State

In addition to water and other volatiles, the accretion process delivers a substantial amount of energy to the growing planets through impacts. Whether due to the accumulation of accretional energy after successive impacts (e.g., Safronov 1978; Tonks and Melosh 1993) or resulting from the thermal blanketing effect of a proto-atmosphere (e.g., Hayashi et al. 1979; Abe and Matsui 1985), the early stages of planetary evolution have long been thought to be punctuated by one or several events of surface and mantle melting (see Elkins-Tanton 2012, for a review). Yet, the extent and timing of this molten phase remains unclear. We review below the early heating mechanisms at play during planetary formation and discuss how they may affect the initial thermal state of the Venusian mantle.

**Accretional Heating** Several processes likely contribute significantly to increasing the protoplanet temperature and thus favor melting. During the accretion phase, planetary growth by impacts is thought to provide a large amount of kinetic energy. This is dissipated in the form of heat upon impact and responsible for local to global melting of the surface, extending throughout the mantle (Safronov 1978). The size/mass of colliding bodies, the impact velocity, angle, and frequency are key parameters controlling the time- and spatial-extent of melting (e.g., Nakajima et al. 2021).

During the so-called runaway accretion phase (e.g., Kokubo and Ida 1996; Kortenkamp et al. 2001), frequent successive/continuous collisions and associated heat accumulation are thought to provide an important heat source leading possibly to global-scale melting events. At the end of the accretion phase, planetary system formation models (e.g., N-body simulations) predict that the size of impactors increases so that collisions, although less frequent and well separated in time, become more and more energetic (e.g., Kokubo and Ida 1998, 2000; Chambers 2010; Morbidelli et al. 2012). Several tens of Mars-sized protoplanets are thought to form from planetesimal accretion, thus leading to several giant impacts between these protoplanets in the late accretion stages (e.g., Quintana et al. 2016). This ultimately implies melting episodes of greater extent (Kaula 1979; Tonks and Melosh 1993; Nakajima and Stevenson 2015), but likely more isolated in time. Giant impacts, followed by isostatic readjustment are even expected to entirely melt the mantle (Canup 2008; Nakajima and Stevenson 2015). However, in these late accretion stages, it has been suggested that the
energy of the giant impacts could be quickly (~1 Myr) radiated to space because of their low frequency (Zahnle et al. 2007). Other studies suggest that the accretional heat generation rate does not significantly change with time, i.e., planetary heating is almost constant with/insensitive to planetary growth time, as longer time implies higher relative velocities, larger planetesimals and thus deeper heat burial (Kaula 1979). Furthermore, a remnant or outgassing steam atmosphere could sustain a runaway greenhouse state (e.g., Hamano et al. 2013) and significantly slow down the cooling of the protoplanet, and thus possibly maintaining a molten surface and mantle between successive giant collisions.

On early Earth, the hypothesis of a moon-forming single giant impact is often referred to as being responsible for the last global-scale mantle melting event (Canup 2004; Nakajima and Stevenson 2015). On Venus, the apparent absence of a moon and associated late giant impact may challenge the occurrence of such a deep and global magma ocean. Yet, it should be noted that the standard model of a single giant moon-forming impact is still debated for the Earth (e.g., Asphaug 2014). Alternative scenarios, involving smaller multiple impacts have been proposed and may better explain the compositional similarity of the Earth and Moon (Rufu et al. 2017). Regardless, the absence of a giant moon-forming impact cannot completely discard the likelihood of a fully or large-scale molten mantle stage, neither on Earth nor on Venus. The frequency of impacts and thus the accretion time scale must be considered. The collisional histories of the two planets may not have been identical, with Venus experiencing more hit-and-run collisions (that could explain its lack of satellite; Emsenhuber et al. 2021). Yet, giant impacts and associated tremendous energy delivery remain ubiquitous in inner Solar System planetary accretion (Quintana et al. 2016), thus suggesting that magma oceans were common events in the early evolution of terrestrial planets (e.g., Elkins-Tanton 2012; Schaefer and Elkins-Tanton 2018).

However, the absence of a Venusian moon certainly rules out the possibility of substantial tidal heating (solid-body tidal dissipation from the Sun alone could still slow Venus' rotation rate and induce tidal heating; Way and Del Genio 2020). This additional heat source affects solid materials more than the melt and thus is likely concentrated at the bottom of the crystallizing mantle (Zahnle et al. 2007). As mentioned previously, bulk similarities between the Earth and Venus, in terms of sizes and densities, suggest that the two planets have similar bulk compositions (e.g., Smrekar et al. 2018). Combined with their vicinity in the Solar System, and although no moon orbits Venus, it seems reasonable to think that they are composed of similar materials. As discussed above they likely accreted in a relatively similar way with associated accretional heating and therefore analogous thermal histories including substantial interior melting.

**Radiogenic Heating** The presence of radioactive elements, generating heat with decay, is also thought to be a significant *internal* heat source of rocky planets, thus contributing to melting of the planetary interior (e.g., Elkins-Tanton 2012; Solomatov 2015). Active on a timescale of about one million years and thus affecting the earliest evolutionary stages (within the first few million years after CAI formation<fnref n="2" />), radioactive decay of short-lived radionuclides, such as now extinct ²⁶Al and ⁶⁰Fe isotopes (half-lives of 0.717 Myr and 2.61 Myr, respectively), contributed to the heating and differentiation of early forming and growing bodies: planetary embryos, planetesimals, and proto-planets, starting before the end of the accretion phase (Urey 1955; Merk et al. 2002; Yoshino et al. 2003; Elkins-Tanton et al. 2011; Šrámek et al. 2012; Fu et al. 2017; Bhatia 2021). ²⁶Al decay dominates the heat budget while the decay of other radionuclides such as ¹⁰Be, ⁵³Mn, ¹⁴⁶Sm, ¹⁸²Hf, or ²⁴⁴Pu have

<footnote n="2">²Calcium-Aluminum-rich Inclusions are believed to be the earliest formed materials in the Solar System and thus define its formation age, ~4.568 billion years ago (Bouvier et al. 2007; Burkhardt et al. 2008).</footnote>
a minor contribution (Lee et al. 1976; Chaussidon and Gounelle 2007; Elkins-Tanton 2012; Fu et al. 2017; Lugaro et al. 2018). To a lesser extent but on longer timescales and still active today (e.g., Schubert et al. 2001; Jaupart et al. 2015), long-lived radioactive elements $^{238}$U, $^{235}$U, $^{40}$K, and $^{232}$Th (half-lives ranging between 0.7 and 15 Gyr) were more abundant early in the Solar System and also took part in the early heat budget (Urey 1955; Lebrun et al. 2013; Nikolaou et al. 2019).

The amount of radiogenic heat produced relates directly to the abundance of the radioac-
tive elements. As mentioned above the proximity of Earth and Venus combined with their
similar bulk compositions (e.g., Smrekar et al. 2018), also suggest that they have accreted
roughly similar materials with similar endowments of radioactive elements. Both planets
have thus likely experienced similar amounts of radiogenic heating and resulting extent of
interior melting.

**Thermal Blanketing Effect of a Primordial Atmosphere** Whether the energy sources primar-
ily heat up the surface and extend downward through the mantle (accretional heating) or
primarily affect the interior (radiogenic heating), as soon as the planet is heated, the ex-
cess energy must escape to reach thermal equilibrium with the relatively cold surroundings.
Then, whole planet cooling involves heat being radiated to space. This is particularly impor-
tant when the surface is molten.

Without an atmosphere, the planet radiates directly into interplanetary space as a black body with the temperature of the planet's surface, making radiation of energy very efficient (dashed purple line, Fig. 4). In such a case, the cooling stops when the planet reaches thermal equilibrium, i.e., when the surface temperature reaches the equilibrium temperature of the planet (that is the temperature at which the power supplied by the star is equal to the power emitted by the planet).

In the presence of an atmosphere, radiative dissipation of energy from the surface is buffered by the thermal blanketing effect of the surrounding atmosphere and heat is retained. This can maintain the surface temperature above the rock melting point. Several works have reviewed the different stages of the evolution, corresponding to different types of early/pri-mordial atmosphere (i.e., not outgassed from the interior during the solidification process) and related forming-mechanisms.

In the earliest disk-embedded stages of planetary evolution (if the proto-planet accreted enough mass within the disk lifetime), the gravitational capture of gas from the surrounding solar nebula (see Sect. 2.1) and the blanketing effect of the accreted opaque atmosphere is itself thought to keep the surface temperature of the proto-planet above the silicate melting point (e.g., Hayashi et al. 1979; Ikoma and Genda 2006; Olson and Sharp 2019). Ikoma and Genda (2006) have shown that starting from 0.5 Earth masses, proto-planets surrounded by such nebula-captured atmospheres are expected to have a molten surface. If the surface is already molten because of the previously discussed energy sources, such an atmosphere would undoubtedly favor a long-lasting molten layer. Depending on the thermal opacity of the primordial atmosphere and on the prior thermal state of the interior, the molten shell is thought to extend more or less deeply into the mantle. The interior could thus be main-tained in a partially or fully molten state throughout the protoplanetary disk lifetime, until dissipation of the solar nebula and the primordial atmosphere.

In the eighties, Abe & Matsui pioneering magma ocean-related studies addressed the evo-
lution of an impact-generated steam atmosphere and its implications for the early thermal
history of the Earth (Matsui and Abe 1984; Abe and Matsui 1985; Matsui and Abe 1986a,b;
Abe and Matsui 1988). Such atmospheres result from the devolatilization of incoming plan-
etesimals upon impact, thus releasing volatile species, such as H₂O (due to shock dehydra-
tion; e.g., Benlow and Meadows 1977; Boslough et al. 1980; Lange and Ahrens 1982) and
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Outgoing thermal flux emitted at the top of the atmosphere ($F_{IR}$, black line) as a function of surface temperature for a steam (H₂O-dominated) atmosphere (see Fig. 13 for CO₂-dominated atmospheres). Global radiative equilibrium and associated climate stability are reached on either sides of the radiation limit of steam atmospheres: the asymptotic Simpson–Nakajima limit ($F_{SNL} \approx 280 \text{ W.m}^{-2}$, dashed red line), when the thermal emissions balance the absorbed insolation ($F_{IR} \approx F_{abs}$). This radiation limit is responsible for a runaway greenhouse. At the ERCS (i.e., when the heat flux from the interior of the planet becomes negligible compared to the net absorbed stellar flux; see Sect. 3.3), climate stability can be reached either with temperate surface conditions and a water ocean (Type I planets, e.g., blue crosses), for planets whose net absorbed flux is below the radiation limit ($F_{abs} < F_{SNL}$), or with a molten surface (Type II planets, e.g., red crosses), for planets whose absorbed flux exceeds the radiation limit ($F_{abs} > F_{SNL}$). Crosses indicate the surface conditions reached at the ERCS for early Venus (i.e., at Venus' orbital distance around the young Sun) and considering different Bond albedo values ($\alpha = [0.2, 0.3, 0.5, 0.7, 0.77]$ from right to left). These crosses correspond to the ones shown in the lower part of Fig. 12. Depending on its Bond albedo at the end of the rapid cooling stage, early Venus can either be habitable or not. Reciprocally, these surface conditions would be reached on a planet absorbing (and emitting) the corresponding amount of solar radiation, i.e., located at various orbital distances, while considering a constant Bond albedo. The orange dot denotes present-day Venus</figcaption></figure>

CO₂, directly into the atmosphere. Alongside other early studies (Zahnle et al. 1988; Kasting 1988), they showed that once such an impact-generated atmosphere is formed, its thermal blanketing effect is responsible for heating the surface of the accreting Earth up to its melting point (~1400 K), thus forming a magma ocean. This may also apply to Venus. Matsui and Abe (1986b) emphasized that the distance from the star and the heat flux equilibria at the end of the magma ocean stage might explain the current differences between the Earth and Venus, thus demonstrating the importance of the early evolution of terrestrial planets in setting their potential habitability. These results were revisited almost 30 years later in the light of a more sophisticated molten interior–atmosphere coupled model (Hamano et al. 2013, see Sects. 3 and 4). Note that a hybrid-type proto-atmosphere, made of a mixture of both solar nebula and impact degassing products is also thought to be thermally opaque enough to produce a deep magma ocean, and hence sustain an already existing one (Abe et al. 2000; Saito and Kuramoto 2017).

The increase of atmospheric water vapor concentration, or the increase of heat to be radiated through a steam atmosphere after an energy delivery (such as after an impact), results in high surface temperatures triggered by the runaway greenhouse effect of steam dominated atmospheres (e.g., Simpson 1927; Kasting 1988; Nakajima et al. 1992). This runaway be-
**Fig. 5** Schematic cross section of a planet in the magma ocean stage surrounded by an atmosphere and associated heat fluxes. The vigorous convection of the molten mantle provides a substantial heat flux at the surface of the planet, $F_{\text{conv}}$ (dotted red arrows). A fraction $\alpha$, the Bond albedo, of the total solar radiation received by the planet $F_{\odot}$ (plain lines at the top of the atmosphere) is reflected back into space. The remaining flux is absorbed by the atmosphere ($F_{\text{abs}}$, dashed black arrows entering the atmosphere). These two incoming heat fluxes balance the outgoing infrared/thermal flux radiated at the top of the atmosphere, $F_{\text{IR}}$ (blue arrows), that ultimately control the cooling of the whole planet

<figure><img src="image_7.png" /></figure>

behavior is due to the existence of a nearly-asymptotic radiation limit in the thermal (infrared) radiation a planet can emit as a function of its temperature (Fig. 4). This asymptotic radiation limit is often referred to as the Simpson–Nakajima limit (see Goldblatt and Watson 2012, for a review), whose corresponding flux value is $F_{\text{SNL}} \approx 280 \text{ W.m}^{-2}$. In planetary atmospheres, steady-state global radiative balance, and thus climate stability, is achieved by the equilibrium between the net, absorbed incoming stellar radiation, $F_{\text{abs}}$, and the outgoing thermal radiation ($F_{\text{IR}}$) emitted by the atmosphere (also referred to as the outgoing longwave radiation, OLR) (Fig. 5), as a response to the heating of the surface and atmosphere resulting from their absorption of the stellar flux (sunlight). The net, absorbed incoming stellar radiation is:

$$F_{\text{abs}} = (1 - \alpha) \times F_{\odot}, \qquad (1)$$

where $\alpha$ is the fraction of the total incident solar radiation scattered back out into space in all directions and integrated spatially over the whole globe (spherically) and spectrally over all wavelengths, i.e., the global bolometric or Bond albedo (e.g., Hanel et al. 2003). $F_{\odot}$ is the total incident stellar flux received at the top of the atmosphere (TOA) of a planet orbiting the host star at a distance $D$ (in AU) and averaged over the planetary surface area:

$$F_{\odot} = \frac{1}{4} \frac{S_{\odot}^{\text{N}}}{D^2}. \qquad (2)$$

$S_{\odot}^{\text{N}} = 1361 \text{ W.m}^{-2}$ is the total solar irradiance, i.e., the spectrally integrated, mean total electromagnetic energy received from the Sun at a distance of 1 AU per unit area (Kopp and Lean 2011; Prša et al. 2016). The factor 1/4 comes from the ratio of the planet's cross-sectional area to its surface area: the planet intercepts the incident solar beam as a disk but radiates as a sphere over its entire surface (e.g., Pierrehumbert 2010). Thus, if there is a heat
supply, through an increase of the absorbed stellar flux or after an impact, the planet warms and consequently emits more thermal radiation to space to maintain energy balance. In the steam atmosphere case, most of this emission occurs through the water vapor infrared atmospheric “window,” between ~8–13 µm. It thus matches the Wien peak in surface thermal emission at 10 µm; the spectral region essentially transparent (with relatively little absorption) to infrared radiation. However, when the heat flux reaches the critical radiation limit, or when atmospheric water vapor becomes too abundant as a result of oceans evaporation or following impact-degassing, the moist atmosphere becomes optically too thick for heat to be radiated. The outgoing thermal flux can no longer increase and lose heat despite an increasing temperature, i.e., surface warming no longer leads to more thermal emission. This inevitably leads to an increase in the surface temperature $T_{surf}$, hence a runaway greenhouse. This remains until the planet is warm enough to radiate heat in the 4 µm water vapor window, when $T_{surf}$ reaches about 1600 K (which is above the surface melting point), thus restoring radiative balance and climate stability (e.g., Goldblatt and Watson 2012; Goldblatt et al. 2013, see Fig. 4, Sects. 3 and 4 for more details and implications for habitability). This explains the abrupt transition between these extreme stable climatic states on either side of the radiation limit (Fig. 4): temperate or relatively cold surface conditions on one side ($F_{IR} < F_{SNL}$), and a molten surface on the other side ($F_{IR} > F_{SNL}$). Conversely, a molten surface state can only transition to a temperate climate by two means. Through removing water vapor from the atmosphere by photo-dissociation and hydrogen escape, thus decreasing atmospheric opacity. Alternatively it is possible via a decrease of the thermal radiation: if the additional heat supply stops and the absorbed stellar flux remains below the radiation limit for instance.

However, it is important to note that the runaway greenhouse effect and associated abrupt climate transition occurs only for H₂O-dominated atmospheres, as their main component is both a condensable and thermal absorber species. Its triggering radiation threshold value may be sensitive to the atmospheric relative humidity and planetary mass (e.g., Ishiwatari et al. 2002; Goldblatt 2015). For other atmospheric compositions, the radiation limit, if relevant, is not that steep and radiative balance and thus stable climates can be reached for intermediate conditions where the surface is not molten (e.g., Type III planets in Salvador et al. 2017, see Fig. 13 and Sect. 4 for a detailed discussion). In the presence of such “moderately-blanketing” primordial atmospheres, the atmosphere may not melt the surface by itself, but would still buffer the dissipation of heat resulting from other processes, thus contributing to maintaining a molten surface.

Overall, the blanketing effect of a proto-atmosphere, regardless if it is nebula-captured, impact-degassed, or a mixture of both, can melt the planetary surface. Consequently, if the surface is already molten, it will buffer radiative dissipation and maintain the surface temperature above the melting point. If not, it will enhance the thermal effect of any heating mechanism, significantly increasing the likelihood and subsistence of a magma ocean. Yet, it is important to note that any pre-existing primordial atmosphere is subject to depletion, either through the disk dissipation for nebula-captured atmospheres, or through impact erosion. In the case of a nebula-captured atmosphere, the timescale of disk dissipation and following atmospheric escape must be compared to the energy delivery timescale (accretion + radioactive decay timescales) to evaluate the thermal contribution of the atmosphere in maintaining the molten surface.

For an impact-generated atmosphere, if the atmospheric mass added by impact degassing overcomes or balances atmospheric losses due to impact erosion, the atmosphere keeps being replenished after successive collisions or keeps a roughly constant mass and its thermal
blanketing effect is maintained over time. In such a scenario, the atmosphere helps to main-
tain the molten surface between successive impacts and the heat generated keeps accumu-
lating. This likely keeps the planet molten during the entire accretion sequence. Conversely,
if the atmosphere loses mass after each collision, its blanketing power would decrease ac-
cording to the accretion time scale and resulting erosion rate until it becomes too thin to
retain heat. Then, heat would more efficiently dissipate and the atmosphere would not be re-
sponsible for melting the surface or keeping it molten during the entire accretion sequence.
If the heating rate due to the other mechanisms is lower than the dissipation rate, the planet
would only experience transient and local magma oceans/ponds. Depending on the dissipa-
tion timescale of the proto-solar nebula relative to Venus' accretion timescale, and in case
a protoatmosphere survives atmospheric erosion by late impacts, the blanketing effect pro-
vides an additional heat source to extend and enhance the existing ones.

Based on element partitioning analysis, Sakuraba et al. (2019) suggested that the sur-
vival of the primordial atmosphere through the late accretion may partially account for the
present-day atmosphere of Venus. This implies that such an early and long-lasting atmo-
sphere maintained a strong greenhouse effect during the accretion of Venus. Thus it would
significantly retain any heat produced – strongly supporting the idea that Venus’ evolution
started with a molten surface and an extensively molten mantle.

**Metal–Silicate Differentiation and Core Formation** The early thermal state and melting ex-
tent of the mantle is tightly linked to metal–silicate segregation and metal downwards mi-
gration that relate to core formation. Due to the strong fractionation of lithophile (silicate-
loving) Hf from siderophile (metal-loving) W during core formation, the ¹⁸²Hf–¹⁸²W system
(¹⁸²Hf decays to ¹⁸²W with a half life of 8.9 Ma, which can be used to date processes in the
first ~60 Myr) is particularly well suited to trace and date metal–silicate differentiation (Lee
and Halliday 1995; Harper and Jacobsen 1996; Lee and Halliday 1997; Horan et al. 1998;
Quitté et al. 2000; Jacobsen 2005; Rubie et al. 2015b).

Early work on ¹⁸²Hf–¹⁸²W chronometry of meteorites indicate that an idealized, instant core formation in the terrestrial planets must have been completed during the first ~30 Myr of the Solar System history (Kleine et al. 2002; Yin et al. 2002). Yet, this simplistic instantaneous core formation model only provides a lower time constraint and does not apply for larger, Earth-sized bodies, where core formation did not occur as a single event but rather as a combination of discrete delivery of metallic components to the deep interior (e.g., Kleine et al. 2009; Deguen et al. 2014; Rubie et al. 2015b; Nimmo and Kleine 2015).

Accounting for the different mechanisms of core formation in more realistic scenarios
leads to estimates ranging between 30 to 200 Myr (e.g., Chambers 2001) at the very most
(see Rubie et al. 2015b; Nimmo and Kleine 2015, for extensive reviews). Similarly, the
more radiogenic W isotopic composition of the silicate Earth compared to that of chondritic
meteorites demonstrates that Earth has differentiated into a mantle and core within the life-
time of ¹⁸²Hf (~30–50 Myr; e.g., Jacobsen 2005). Rapid terrestrial planet formation and
early metal–silicate separation are also consistent with rapid planetesimals formation and
differentiation informed from ⁵³Mn–⁵³Cr (Lugmair and Shukolyukov 1998) and ²⁶Al–²⁶Mg
(Srinivasan et al. 2000) systematics (Nyquist et al. 2001; Wadhwa et al. 2006; Nyquist et al.
2009), and observations of other planetary systems (Briceño et al. 2001; Bodenheimer and
Lin 2002).

Core formation is a complex, multi-stage phenomenon most likely resulting from the
combination of different processes operating over different timescales such as percolation
of liquid metal through a compacting solid silicate matrix, iron droplets sinking through
a magma ocean, negative diapirism generated by Rayleigh-Taylor instability, or hydraulic
fracturing (e.g., Ricard et al. 2009; Deguen et al. 2014; Rubie et al. 2015b; Landeau et al. 2016; Wacheul and Le Bars 2018; Clesi et al. 2020; Landeau et al. 2021). Because the relative contribution and timing of each process is still unclear, large uncertainties remain. In any case, mechanisms of core formation depend on the thermal state of the planetary body and require some extent of partial melting (e.g., Rubie et al. 2015b). Given its suggested relatively short timescale, core formation is also generally thought to require a mechanism for efficiently segregating metal (Stevenson 1990; Yoshino et al. 2003). The low viscosity of the melt and high temperatures facilitate material deformation needed for mass redistribution (e.g., Šrámek et al. 2010, 2012; Nimmo and Kleine 2015). Therefore, large extent of mantle melting is thought to facilitate metal-silicate segregation and metal migration. Thus, a great extent of mantle melting is believed to favor rapid core formation suggested by geochemical evidence. This view is also supported by the mantellic abundance of moderately siderophile elements (MSEs; e.g., Fe, Ni, Co, W, and Mo) that may be explained by a final metal-silicate equilibration at very high *P* and *T*, i.e., at the base of a deep magma ocean (e.g., Li and Agee 1996; Righter and Drake 1997, 1999; Rubie et al. 2003, 2015b). Note however that alternative mechanisms, such as percolation of iron-rich melt through solid silicate, have been proven to result in rapid core formation without necessarily involving a substantial amount of mantle melting (Ghanbarzadeh et al. 2017; Berg et al. 2018). The respective contribution and timing of the different mechanisms involved in core formation need to be further considered to draw a realistic picture of this complex, multi-stage phenomenon, accounting for both geochemical and geophysical constraints (e.g., Badro et al. 2015).

Other terrestrial isotopic data analysis suggest that the mantle experienced early and global/major chemical differentiation ($^{146}$Sm-$^{142}$Nd system; Caro et al. 2003, Boyet et al. 2003, Boyet and Carlson 2005), and that the growth of continental crust began early ($^{176}$Lu-$^{176}$Hf and $^{147}$Sm-$^{143}$Nd; e.g., Bennett et al. 1993, Amelin et al. 1999, Bizzarro et al. 2003, Harrison et al. 2005), within 30 Myr and in the first 200 Myr of evolution, respectively, which both might require a global, deep magma ocean (Harrison 2009; Caro 2011; Solomatov 2015). Yet, one should note that these results could reflect multiple possible scenarios given the uncertainties affecting the isotopic measurements themselves, their representativeness and the various fractionation processes. One promising way to overcome individual limitations of each isotopic system and to provide tight constraints on the timescale of accretion and early differentiation is to build scenarios that satisfy several different isotopic systems simultaneously and that are consistent with the results of dynamical models (e.g., Halliday 2004; Wood and Halliday 2005; Nimmo and Kleine 2015).

Overall, geochemical constraints may indicate that core formation and large-scale mantle differentiation occurred at similar timescales. These are likewise comparable to the accretion timescales constrained by planetary formation models. If planetary accretion, core formation, and mantle differentiation occurred concurrently and on short timescales, the rapidity of the processes themselves would be indicative of a significant degree of mantle melting. The amount of energy involved would induce large-scale melting, thus likely suggesting the existence of a global and deep magma ocean (Solomatov 2015).

Core formation is not only linked to the degree of mantle melting through the ease of metal-silicate segregation and iron migration but has also important energetic implications for the mantle thermal state (e.g., Jaupart et al. 2015; Solomatov 2015). Indeed, the release of gravitational energy from the gravitational differentiation between metallic and silicate materials during core formation, is dissipated by viscous heating in both the iron and silicate phases. This supplies an additional and substantial source of heat supporting a whole molten silicate layer scenario (e.g., Solomon 1979; Sasaki and Nakazawa 1986; Ricard et al. 2009; Monteux et al. 2009; Šrámek et al. 2010; Samuel et al. 2010; Rubie et al. 2015b; Landeau et al. 2016). The energy released by rapid core formation, itself favored by extensive
mantle melting, could be responsible for heating the entire Earth to ~1700 K (Tozer 1965; Flasar and Birch 1973). Additional release of gravitational energy due to the redistribution of chemical and thermal heterogeneities in the mantle and segregation of leftover iron (density heterogeneities) following core formation is also thought to increase the temperature by several hundred degrees (Tonks and Melosh 1992; Solomatov 2015).

Numerical simulations of giant impacts emphasize that during collisions of large differentiated bodies the iron core of the impactor is the most heated material (Canup 2004). After its presumably rapid segregation into the impacted protoplanetary core, a substantial fraction of the associated gravitational potential energy might be retained within the segregated iron, resulting in an early superheated core (Stevenson 2001; Solomatov 2015). The heat transferred to the mantle may induce additional mantle melting or sustain its molten state and extend the magma ocean lifetime (Ke and Solomatov 2006, 2009).

Similarly and over longer timescales, once the core is formed, the release of heat from the core at the core-mantle boundary (CMB) constitutes an additional heat source. While little is known about early magma ocean-core interactions, they certainly affect each other's evolution. Indeed, the core constitutes an evolving boundary condition of the mantle and contributes to the heat budget of the interior. It thus certainly affects the cooling timescale and crystallization pattern of the magma ocean. In turn, the lower mantle conditions resulting from the magma ocean evolution may buffer the heat loss from the core and provide additional heat. This could thus affect the thermal state and dynamics of the core and influence the resulting dynamo and potential magnetic field strength with time (e.g., Stevenson 2001; Ke and Solomatov 2009). However, in the absence of evidence of a present or past magnetic field on Venus, that could inform the thermal and dynamical state of its core, the core-mantle interactions and their effects on the thermal history of the planet remain highly unconstrained. For the early Earth, estimates of the initial CMB temperature are above the mantle melting temperature and thus indicate extensive lower mantle melting at that time, therefore implying a deep magma ocean (e.g., Andrault et al. 2016, see Fiquet 2018 for a review). Relics of this early magma ocean retained at the CMB may exist today as a hypothetical deep molten zone on top of the Earth's core that might explain the current zones/patches of extremely low seismic velocities (ultra-low velocity zones, ULVZs) shown by seismology. On Venus, the supposedly less efficient cooling of the interior relative to Earth due to the absence of plate tectonics might favor the existence of a thick basal magma ocean today (O'Rourke 2020), inherited from the primordial heat accumulated. Testing this hypothesis might provide constraints on Venus early state and accretion vigor.

**Young Sun** Stars are expected to be significantly more active during their early evolution, and as a result, stellar evolution models predict that both the X-ray-EUV emissions and the magnetic field strength of the Sun were larger in the past (e.g., Bahcall et al. 2001; Ribas et al. 2005; Tu et al. 2015; Johnstone et al. 2021). On top of affecting planets' atmospheric properties and habitability (e.g., Gallet et al. 2017), the energetic environment in which forming planets and planetesimals are embedded can be responsible for significantly heating and possibly melting their interiors. In particular, the strong magnetic field and enhanced solar wind of young stars generates electromagnetic induction heating that could substantially melt planetary mantles and result in the formation of magma oceans (Sonett et al. 1968; Herbert et al. 1977; Kislyakova et al. 2017; Kislyakova and Noack 2020; Noack et al. 2021). While this energy source was likely effective during the early stages of planetary formation and certainly affected proto-Venus, its contribution remains unconstrained. Finally, despite a relatively weaker contribution compared to the other aforementioned mechanisms, the enhanced high energy X-ray and EUV emissions of the young Sun (e.g., Ribas et al.
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Sketches of the hypothetical initial and final states usually considered in magma ocean-atmosphere thermal evolution numerical studies. Initially, as the aftermath of a hypothetical last giant impact, the silicate mantle is assumed to be fully molten, meaning that the magma ocean extends throughout the entire mantle. Any primary, captured atmosphere (if one existed at all) has been lost. All volatile species are then assumed to be initially dissolved within the molten mantle. Heavier, core-forming elements, such as metallic iron, are thought to have already fully segregated to the core, meaning that the metallic core is already formed. During the cooling and crystallization of the magma ocean, volatile species initially dissolved within the melt are exsolved and outgassed, thus progressively building-up the atmosphere. One should distinguish the end of the molten stage, that is when the mantle has fully solidified, from the end of the rapid cooling stage (ERCS; see Sect. 3.3). The latter is reached when the heat flux from the interior of the planet becomes negligible compared to the net absorbed stellar flux ($F_{abs}$) that thereafter balances the thermal emissions at the top of the atmosphere, $F_{IR}$ (i.e., the planet has reached global radiative equilibrium), and controls the surface conditions. Depending on the net absorbed flux, the planet can achieve global radiative balance with a fully or partially solidified mantle at the ERCS. Once the ERCS is reached, the evolution is no longer controlled by the molten mantle (if still molten) but rather by the balance between the absorbed stellar flux and the thermal emissions</figcaption></figure>

2005; Johnstone et al. 2021) may also induce additional heating of the planetary surface and atmosphere, and should then be accounted for in the thermal budget.

**Initial Thermal State** To conclude, the respective contribution and timing of each individual heating source to the warming and melting of the terrestrial planets' mantle remain unclear. Their combination and temporal overlap imply that generated temperatures were high enough to melt a large part – if not all – of the silicate mantle. Furthermore, the different heat sources involved are likely common and inherent to the early history of rocky planets. Thus, it seems reasonable to assume that, like Earth, Venus experienced at least one global magma ocean phase extending throughout the entire mantle during its early evolution.

For the sake of simplicity, thermal evolution studies of the early magma ocean phase generally model the cooling of a hypothetical last episode of mantle global-scale melting. By cooling down and crystallizing, the molten, liquid-state mantle evolves toward a solid-state mantle, which represents the starting point for the long-term evolution of the planet (Fig. 6; see Sects. 3 and 4). Note that depending on the absorbed stellar flux, global radiative equilibrium can be reached with a molten or solidified surface (see Sect. 3.3). For the Earth, such a global melting event often refers to the result of the late, giant moon-forming impact scenario. Yet, as discussed above, it seems reasonable to follow the same approach to address the early evolution of Venus (and other Earth-sized rocky planets) even in the absence of a Venusian moon. It should be noted that subsequent events such as late giant impacts may substantially affect, at least locally, the heat budget, interior stratification, and volatile distribution of the solidifying planet (e.g., Gabriel and Cambioni 2023). The classically described early thermal evolution sequence remains a simple and theoretical view.

As mentioned above, isotopic signatures can provide more specific constraints on the timescales and energetic of the accretion phase and early evolution and thus on the resulting
early thermal state of rocky planets. Such future measurements on Venus would provide
helpful constraints to decipher its evolutionary pathway and understand why it ended up
being so different from Earth (see Sect. 5 and Widemann et al. 2023, in this collection).

**2.3 Constraints from Isotopes**

The very high D/H ratio of hydrogen in the Venus atmosphere (157 times higher than ter-
restrial water; Von Zahn et al. 1983) has often been interpreted as evidence that Venus un-
derwent intense episodes of hydrogen escape (Donahue et al. 1982). These escape episodes
would be responsible for the low abundance of water on Venus and would have influenced
the entire geological evolution of the planet (Baines et al. 2013). However, fractionation
of D from H isotopes during atmospheric escape of hydrogen depends on the regime and
parameters of atmospheric escape (Hunten et al. 1987). Some escape regimes of hydrogen
create only a limited isotopic fraction of the remaining hydrogen (Zahnle et al. 1990). Thus
the D/H ratio gives only partial information and it remains unclear how much water really
escaped from Venus. One could imagine a scenario in which considerable amounts of water
escaped but that the isotopic fractionation was relatively inefficient. Conversely, the planet
could have accreted almost dry and lost only a small amount of water but with a regime of
escape leading to intense isotopic fractionation.

The initial D/H ratio of water is an important unknown in models of hydrogen escape from Venus. This ratio was probably lower than it is today but by how much remains unclear. It cannot be excluded that Venus' water was initially sourced by comets which can show elevated D/H ratios (e.g., 3 times the terrestrial value for water emitted from comet 67P/Churyumov-Gerasimenko; Altwegg et al. 2015). Interestingly, there is a high abundance of neon in the atmosphere of Venus and its isotopic composition seems intermediate between solar and meteoritic components (see Avice et al. 2022, this collection). Comets are likely to be devoid or highly depleted in neon (whose T<sub>condensation</sub> ≈ 20 K). The presence of neon in its atmosphere likely indicates that comets cannot be the sole contributors to volatile elements on Venus. Cometary water could have been brought to Venus and added to an original solar- or chondritic-derived atmosphere, or primordial neon could have been degassed from the mantle through time. The value of the Ar/Ne elemental ratio of Venus' atmosphere seems high compared to what would be expected from a pure mixture between solar and chondritic neon. While the large error bars on the Ar/Ne ratio does not allow one to draw any firm conclusions, a high Ar/Ne would be compatible with a cometary contribution to the Venus atmosphere. Such cometary contribution has been identified for Earth, based on the isotopic composition of primordial Earth's atmospheric xenon (Pepin 1991; Marty et al. 2017; Avice et al. 2017). However, the D/H ratio of Earth's water is close to a chondritic value, meaning that comets probably contributed less than 1 percent of Earth's total water budget (Fig. 7; Marty et al. 2016, Bekaert et al. 2020). Yet, this estimate is only valid if data collected for 67P/C-G are representative of cometary diversity. The possibility of an asteroid–comet continuum makes definitive statements difficult given the few available measurements and their uncertainties (Gounelle 2011). Furthermore, D/H ratio interpretations remain problematic given the number, uncertainties, and complexity of the processes affecting them (Stephant et al. 2016, 2018; Piani et al. 2020; Stephant et al. 2020).

It must be noted that elements such as noble gases can also escape from planetary atmo-
spheres during episodes of hydrogen escape and that such escape would lead to elemental
and isotopic fractionation of these elements (Hunten et al. 1987; Zahnle et al. 2019). Future
data on the elemental and isotopic composition of noble gases collected in the atmosphere
of Venus, for example during the recently selected DAVINCI (NASA) mission (Garvin et al.
<figure><img src="image_9.png" /><figcaption>**Fig. 7** $^{36}$Ar/H$_2$O versus D/H mixing diagram between cometary (represented by 67P/C-G) and chondritic end-members. The mixing curves depend on the elemental and isotopic composition of the end-members. Comets have high Ar/H$_2$O ratios implying that more than 90% of Ar could of cometary origin while only about 1% of Earth's water would have been brought by comets (without altering the initial chondritic-like D/H ratio). HA, DM and MA are the bulk Earth estimates from Halliday (2013), Dauphas and Morbidelli (2014) and Marty (2012), respectively. Figure modified after Marty et al. (2016, see also refs. therein)</figcaption></figure>

2020, 2022), will certainly help to put constraints on models attempting to track the joint history of water and noble gases on Venus and to compare the case of Venus with other terrestrial planets (Avice and Marty 2020).

# 3 Magma Ocean Evolution and the Formation of the Secondary Atmosphere

## 3.1 Magma Ocean Cooling and Solidification

There is no strict definition of what is referred to as “magma ocean”. The term “magma” refers to a multiphase mixture of liquid (melts), solid (crystals), and gas (bubbles) composing a molten rock (e.g., Lesher and Spera 2015) and “ocean” was used to reflect the low, water-like viscosity of the mixture and the fact that the system is virtually 100% liquid (e.g., Warren 1985; Solomatov 2015). The magma ocean must then behave rheologically as a liquid, with a crystal fraction small enough that crystals are suspended within the melt without being interconnected. In addition, previous definitions include a criterion regarding the extent of melt, such that the magma must encompass a substantial fraction (more than 10%) of the body (Taylor and Norman 1992).

This general definition can be confusing as it may encompass a large variety of cases. These include drastically different scenarios such as a periodically molten planet as a result of tidal heating in an eccentric orbit, and a planet *ad-aeternum* molten owing to its proximity to the star. A fundamental distinction can be made compared to the early transient
molten stage. The latter is likely inherent to the early evolution of rocky planets. This is in contrast to the long-lasting, steady-state molten stage experienced by planets in close-in orbits around their star; sometimes referred to as “lava worlds”. Here we focus on the former, which is likely experienced by most terrestrial planets due to the combination of the heat sources detailed in Sect. 2.2.2. Compared to the steady-state, sustained molten stage of lava worlds, early transient magma oceans are not in thermal equilibrium with their colder stellar environment. They are cooling down, losing excess heat to reach equilibrium. As a fluid-like material, the magma ocean can efficiently transport and lose heat by advecting matter. Its dynamics and cooling are thus controlled by thermal convection. More precisely, as the molten mantle loses its internal heat while being cooled from above, it corresponds to the classical problem of Rayleigh-Bénard convection.

In such a case, the Navier-Stokes equations are used to describe the motion and evolution of the fluid. Using appropriate characteristic scales, two key dimensionless parameters regarding the fluid convective dynamics arise from the non-dimensional form of the momentum conservation equation (itself originating from Newton’s second law): the Rayleigh number, Ra, and the Prandtl number, Pr. The thermal Rayleigh number,

$$Ra = \frac{\theta g \Delta T L^3}{\kappa \nu}, \qquad (3)$$

compares the thermal buoyancy force that drives convection to the advection-resisting effects of thermal and momentum diffusion (viscous dissipation). The motion-driving parameters are the thermal expansion coefficient, $\theta$, gravity, $g$, the superadiabatic temperature difference $\Delta T$ across the convective layer of thickness $L$ (here being the magma ocean thickness). The parameters buffering advection are the thermal diffusivity, $\kappa$, and the kinematic viscosity, $\nu = \eta/\rho$, where $\eta$ is the dynamic viscosity, and $\rho$ is the fluid density. The Rayleigh number is thus a measure of the convective vigor. It is estimated to be up to $Ra = 10^{31}$ for a typical magma ocean encompassing an Earth-sized mantle (e.g., Solomatov 2000). It decreases with magma ocean crystallization (the present-day Earth’s mantle is estimated to be between $10^6$ and $10^8$; e.g., Ricard 2015). The Prandtl number is

$$Pr = \frac{\nu}{\kappa}, \qquad (4)$$

and expresses the ratio of momentum to heat diffusion. For the present-day solid-state mantle of Solar System terrestrial planets $Pr \gg 1$ ($Pr > 10^{23}$ on Earth where mantle convection is essentially a buoyancy-driven Stokes flow; e.g., Schubert et al. 2001, Ricard 2015), which implies that fluid motions stop as soon as the heat source disappears because of negligible inertial effects compared to viscous effects. Conversely, inertia cannot be neglected in the frame of vigorously convective, low viscosity magma oceans. In that case $Pr \approx 1$ for a global Earth-sized mantle magma ocean and progressively increases with solidification. These two parameters characterize the convective dynamics.

The heat loss is a function of the convective vigor so that the dimensionless heat flux, i.e., the Nusselt number, $Nu$, which is the ratio of convective to conductive heat transfer ($Nu = 1$ when heat is dissipated by conduction only, and exceeds unity as soon as convection starts), is representative of the heat flux ($F_{conv}$) leaving the magma ocean through the top boundary of the convective layer (i.e., the surface). $F_{conv}$ scales as a power-law of the Rayleigh number (e.g., Jaupart et al. 2015):

$$Nu = \frac{F_{conv} L}{k \Delta T} = a_{Nu} Ra^{\gamma Nu}, \qquad (5)$$
where the scaling law pre-factor $a_{Nu}$ and exponent $\gamma_{Nu}$ values depend on the convective regime and patterns (e.g., Malkus 1954; Grossmann and Lohse 2000; Chillà and Schumacher 2012; Stevens et al. 2013, 2018). For extremely vigorous and turbulent convection relevant to magma ocean parameters, the convective heat flux is usually parameterized using $a_{Nu} = 0.089$ and $\gamma_{Nu} = 1/3$ (e.g., Siggia 1994, Solomatov 2015; regime $IV_u$ in Grossmann and Lohse 2000). This implies that the heat loss is governed solely by local instabilities of the upper thermal boundary layer and is independent of the magma ocean thickness (e.g., Lebrun et al. 2013; Salvador et al. 2017; Nikolaou et al. 2019).

Given its low viscosity and the large temperature difference between the base of the MO and the surface, the convection is highly vigorous. The temperature across the molten layer is nearly adiabatic and isentropic (e.g., Abe 1997; Solomatov 2015). For a one-phase system, such as for the completely molten or completely solid and homogeneous layer, it can be expressed as:

$$ \frac{dT}{dP} = \frac{\alpha T}{\rho c_p}, \qquad (6) $$

where $P$ is the pressure and $c_p$ is the isobaric specific heat. In two-phase systems, i.e., where the melt and crystals co-exist, the equation for the adiabat is modified to account for the effects of phase changes (e.g., Abe 1997; Solomatov and Stevenson 1993; Solomatov 2015). The secular cooling of the magma ocean is then often computed by integrating the equation of conservation of energy over the evolving magma ocean thickness (e.g., Abe 1997; Lebrun et al. 2013; Jaupart et al. 2015).

Starting from a magma ocean extending throughout the entire mantle, the depth at which crystallization starts is the depth at which the adiabat first intersects the liquidus (above which temperature the material is completely liquid). Where it starts and how it proceeds with cooling thus firstly depends on the respective slopes of the melting curves (in particular that of the liquidus) and the adiabat (Fig. 8e). While the liquidus is well constrained over the first 1000 km of Earth's mantle, it remains uncertain at lower depth (e.g., Andrault et al. 2017; Fiquet 2018; Andrault 2019, for reviews). If the $P-T$ slope of the adiabatic temperature of the magma ocean is flatter than the liquidus (such as for a chondritic composition; Andrault et al. 2011), crystallization would start at the base of the mantle and proceeds from the bottom upward, as generally assumed in magma ocean studies (Fig. 8a-c, e and f). But this may vary if the liquidus profile presents a strong curvature at mid-lower mantle depths (such as for a fertile peridotite; Fiquet et al. 2010; Thomas et al. 2012), resulting in a crystallization starting at mid-mantle depths, where a growing shell of solid mantle material would form. Such a scenario would induce the formation of a persistent molten layer at the base of the mantle, a so-called “basal magma ocean” (Labrosse et al. 2007), and an early thermochemical separation between the upper and lower parts of the molten mantle. Whether the differences between these two crystallization scenarios arise from the different compositions considered or from experimental uncertainties remains unclear.

The depth at which crystallization starts is insufficient information to describe the subsequent solidification pattern of the magma ocean. Of equal importance is the relative buoyancy and thus the density difference between crystals and melts, which controls the way solid-liquid segregation proceeds (e.g., Boukaré and Ricard 2017; Caracas et al. 2019; see Andrault et al. 2017; Asimow 2018, for detailed reviews). At low pressures, forming crystals are typically assumed to be denser than the surrounding melt and thus sink to the bottom. Yet, the larger compressibility of liquids compared to that of crystals may induce the latter to be buoyant with respect to silicate melts in the lower mantle (Nomura et al. 2011;
Andrault et al. 2012; Thomas et al. 2012). Changes of Fe melt-crystal partition coefficient may also result in denser (and sinking) Fe-rich melts than co-existing crystals (e.g., Tateno et al. 2014; Andrault et al. 2017). This scenario contrasts with the classical view of crystals settling, compaction, and melt ejection in a crystallizing magma ocean (Solomatov 2007; Boukaré et al. 2015), and could also produce an early, isolated basal magma ocean above the CMB. Accounting for the effect of convective dynamics on the settling behavior of crystals (resulting from the competition between crystals settling and entrainment within the convective flow; e.g., Maas and Hansen 2015, 2019; Patočka et al. 2020, 2022) as a function of their evolving properties such as their shape, size, density, and distribution may also significantly affect the solidification and differentiation pattern (e.g., Solomatov 2015). The location of initial crystallization as well as the subsequent solidification pattern have critical implications regarding both the chemical and thermal stratification of the mantle, and may significantly affect the distribution of volatile species within the planetary interior and their outgassing. Despite being an open question whose influence is still poorly constrained, it might profoundly affect the early and long-term evolution of terrestrial planets.

As a result of cooling and crystallization, the melt fraction of the magma, $\phi_{melt}$, decreases. Conversely its crystal fraction, $\phi_{crystal} = 1 - \phi_{melt}$, increases. It depends on the temperature at which crystals start to form, i.e., the liquidus $T_{liq}$, and on the temperature below which the material is completely solid, i.e., the solidus $T_{sol}$. In a linear and simplified form it can be written (Abe 1997):

$$ \phi_{melt} = \frac{T - T_{sol}}{T_{liq} - T_{sol}}. \qquad (7) $$

Because the temperature, $T$, is a function of depth and time, $\phi_{melt}$ varies accordingly and globally decreases with magma ocean cooling from $\phi_{melt} = 1$ ($T \ge T_{liq}$), when no crystals are present ($\phi_{crystal} = 0$), to $\phi_{melt} = 0$ ($T \le T_{sol}$), when the magma is fully crystallized ($\phi_{crystal} = 1$).

The viscosity of the magma and associated rheological behavior greatly varies as a function of the temperature/crystal content and controls virtually all dynamic processes in the magma ocean. For instance, it strongly affects the convective vigor, the associated heat redistribution, and thus, the amount of heat the mantle can release. Throughout the magma ocean lifetime, viscosity changes over several orders of magnitude (Fig. 8d). Where and when the magma temperature is above liquidus, the dynamic viscosity of the silicate melt can be as low as the viscosity of water at ambient conditions ($\eta \approx 10^{-3}$ Pa.s; e.g., Urbain et al. 1982, Solomatov 2007, Kono 2018). A rheological transition associated with an abrupt viscosity increase occurs when the crystal fraction becomes large enough such that the connectivity of the crystals hampers efficient, liquid-like motion (e.g., Lejeune and Richet 1995; Abe 1993b, 1997). The mantle depth at which the rheological transition occurs is called the rheological front and separates the upper, low viscosity, vigorously convective, melt-dominated-motion mantle from the lower, rheologically solid-like mantle. The critical melt fraction at which this transition occurs is classically taken to be $\phi_{melt,c} \approx 0.4$ (Marsh 1981; Abe 1993b), but other mechanisms, such as diffusion creep and melt contiguity (Faul and Jackson 2007; Takei and Holtzman 2009a), may keep the viscosity low and fluid-like even at significantly lower melt fractions (from $\phi_{melt,c} \approx 0.019$ for 15 µm grain size and down to $\phi_{melt,c} \approx 10^{-4}$ for typical mantle grain size of 3 mm; Takei and Holtzman 2009b). Assuming that the solidification of the magma ocean proceeds from the bottom upward, with crystals efficiently segregating from the melt and accumulating downward, two layers of very distinct rheological behaviors exist from either side of the rheological front (Fig. 8b).
<figure><img src="image_10.png" /><figcaption>**Fig. 8** (a–c) Magma ocean schematic evolution sequence from (a) a fully ($\phi = 1$)/essentially ($\phi_c < \phi < 1$) molten stage of global extent ($L = R_{\text{planet}} - R_{\text{core}}$), to (b) an intermediate stage where an essentially molten upper unit coexists with a lower mushy unit, to (c) a globally mushy stage where the liquid fraction is below the critical value of the rheological transition everywhere. (d) Viscosity and (e) temperature profile of the silicate mantle as a function of its evolving temperature/melt fraction. (f) Typical thermal evolution of an Earth-like magma ocean overlaid with an outgassing H$_2$O–CO$_2$–N$_2$ atmosphere (modified from Salvador et al. 2017)</figcaption></figure>

Here we consider that the magma ocean is rheologically liquid-like, either fully ($\phi_{\text{melt}} = 1$) or essentially molten, with a melt fraction above $\phi_{\text{melt,c}}$ ($\phi_{\text{melt,c}} < \phi_{\text{melt}} \le 1$), while the “mushy” layer below ($0 \le \phi_{\text{melt}} \le \phi_{\text{melt,c}}$) convects less vigorously and behaves rheologically more like the present-day solidified terrestrial mantle, with a melt fraction-dependent viscosity. Starting from a fully molten mantle, the magma ocean thus extends from the sur-
face to the rheological front (Figs. 5 and 8a and b) and with time, it shrinks until the rheo-logical front reaches the surface (Fig. 8c). From then on, the mantle dynamics and mixing is much less vigorous and the associated convective flux is much lower: the end of the transient magma ocean stage is reached.

Note that global thermal equilibrium can be reached independently of the magma ocean end: the former does not necessarily imply that the magma ocean phase is over. Indeed, global thermal equilibrium and climate stability depend on the radiative balance between the net absorbed insolation ($F_{abs}$) and the outgoing radiation ($F_{IR}$), and can thus be achieved at surface temperatures higher than the surface melting point. This can be the case for close-in planets that can keep their planetary surface molten indefinitely (e.g., lava worlds). For early Venus, depending on the atmospheric composition and on the absorbed solar flux resulting in part from the cloud–albedo feedback, climate stability may have been achieved with a molten or solidified surface (see Sects. 3.3 and 4, and Fig. 12).

## 3.2 Water/Volatiles Partitioning and Outgassing

At a given magma ocean crystallization stage, water and other volatile species are distributed between the different planetary reservoirs, from the deep interior to the atmosphere. These volatile species are chemical compounds with a low boiling point that thus vaporize readily and partition preferentially into a fluid or gas phase (e.g., Chap. 7 in White 2013). We first describe how this distribution proceeds at a given crystallization stage and then how it evolves with magma ocean cooling and solidification.

### 3.2.1 Melt-Crystal Volatile Partitioning

Within the molten mantle, the partitioning of a species between the co-existing melt (silicate liquid) and solid phase (crystals) is determined through partition coefficients that dictate how much of a given volatile species dissolved in the melt can be incorporated into a given mineral phase at given thermodynamic conditions. This is sometimes referred to as the storage capacity (e.g., Hirschmann et al. 2005). “Water” is not incorporated as an integral part of the crystal structure. It occurs mostly in the form of hydroxyl groups (–OH) in hydrous minerals, or as hydrogen atoms decorating point defects of nominally anhydrous minerals, the so-called NAMs, and bonded to structural oxygen, but rarely as molecular water. For extensive reviews regarding hydrogen/“water” in the solid mantle, refer to Hirschmann et al. (2005), Hirschmann (2006), Keppler and Bolfan-Casanova (2006), Demouchy and Bolfan-Casanova (2016), Peslier et al. (2017). Most coupled magma ocean-atmosphere thermal evolution studies consider a single solid-melt partition coefficient ($\psi_{vol}$) throughout the entire mantle (Hamano et al. 2013, 2015; Schaefer et al. 2016; Bower et al. 2019; Barth et al. 2021), usually taken as the partition coefficient of perovskite ($\psi_{H_2O} = 10^{-4}$ for water; except in Schaefer et al. (2016) and Barth et al. (2021), where $\psi_{H_2O} = 10^{-2}$ is close from the partition coefficient of lherzolite). Or they consider the partition coefficient of lherzolite for the upper mantle ($\psi_{H_2O} = 1.1 \times 10^{-2}$), and of perovskite for the lower mantle ($\psi_{H_2O} = 10^{-4}$; after Bolfan-Casanova et al. 2003), thus accounting for differences due to pressure and associated stable mineral phases (Elkins-Tanton 2008; Lebrun et al. 2013; Salvador et al. 2017; Nikolaou et al. 2019).

In addition to water partitioning between silicate melts and cumulates (i.e., crystals accumulated by gravitational settling in a magma), another process responsible for the incorporation of water into solidifying silicates is the interstitial trapping of water-enriched melt (Elkins-Tanton 2008; Hamano et al. 2013, 2015; Ikoma et al. 2018). Some models assume that 1% of the melt mass is incorporated into cumulates through interstitial trapping
throughout solidification (Elkins-Tanton 2008; Hamano et al. 2013, 2015). Yet, as discussed in Hamano et al. (2013), estimates for magma ocean solidification remain highly unconstrained. They are based on the degree of melting at which melt separation starts for abyssal peridotites (Johnson and Dick 1992) and likely overestimates the mass fraction of interstitially trapped melt. Furthermore, interstitial trapping would only store significant amounts of water into cumulates in the presence of water-rich melts. The latter correspond to evolved magmas found at late solidification stages where water exsolution and outgassing are highly efficient (e.g., Elkins-Tanton and Grove 2011; Lebrun et al. 2013; Salvador et al. 2017; Nikolaou et al. 2019; Bower et al. 2019) and thought to be the water-draining dominant processes. Yet, it has been recently suggested that this interstitial trapping and melt retention process might be highly effective at the magma ocean freezing front (i.e., at the rheological transition) and may retain a substantial fraction of volatile-rich melt at the end of magma ocean evolution (Hier-Majumder and Hirschmann 2017; Miyazaki and Korenaga 2021, 2022). Volatile species that are highly soluble in the melt, such as water, would be very sensitive to this process and largely incorporated into the primordial solidified mantle. As this relates to volatile partitioning during the entire magma ocean evolution and crystallization sequence rather than being specific to a given crystallization stage, we discuss it in more detail in Sect. 3.2.4.

If one does not take into account volatile trapping in the porous melt phase, the incorporation of volatile species into the solidifying mantle is negligible compared to its partitioning into the melt during the magma ocean crystallization. Volatile trapping in the solidified mantle might thus be ignored to first order when addressing magma ocean and atmospheric thermal evolution (Lichtenberg et al. 2021). Yet this very process is of major importance when the magma ocean is eventually fully solidified, and when addressing the long term evolution of the planet and the dynamics of the solidified mantle.

Water has a strong effect on solid-state mantle creep strength (Hirth and Kohlstedt 1995; Mei and Kohlstedt 2000; Hirth and Kohlstedt 2004; Demouchy and Bolfan-Casanova 2016), melting relations (Hirth and Kohlstedt 1996; Asimow and Langmuir 2003), shear zones development (e.g., Skemer et al. 2013), viscosity and density (e.g., Lange 1994), and phase transitions (Wood 1995). Its partition between the melt and the solidifying mantle during the crystallization of the magma ocean will be of fundamental importance for the style and strength of convective flow during the following stages of planetary evolution (e.g., Schaefer and Elkins-Tanton 2018). Namely, the distribution of water within the planet has been proposed to resolve the distinct tectonic behaviors between the Earth's wet mantle and that supposedly dry of Venus (Mackwell et al. 1998; Richards et al. 2001). This water could be directly inherited from the magma ocean stage in the case of a gravitationally stable stratified crystallization sequence, or acquired later after mantle global overturn (e.g., Tikoo and Elkins-Tanton 2017). Furthermore, by decreasing the solidus temperature and thus facilitating melting at greater depth and greater extent (because water lowers the Gibbs free energy of the melt), water abundance in the solid mantle is a crucial parameter for Earth's mantle melting and volcanism. Particularly at subduction zones, mid-ocean ridges and for intraplate hotspot volcanism (e.g., Hirschmann 2006; Litasov and Ohtani 2007; Ni et al. 2016). The resulting interior melting and volcanism can in turn affect the atmospheric composition, climate, volatile cycling, and consequently, the surface conditions and habitability on rocky planets (e.g., Foley and Driscoll 2016; Dehant et al. 2019).

### 3.2.2 Melt-Gas Volatile Partitioning

Compared to crystals (i.e., magma's solid phase), the melt (i.e., magma's liquid phase) can contain a large amount of volatile species, including water, that are dissolved within the
latter. Water is sometimes said to be more soluble in the melt than in the crystals/solid phase. This even though, technically, water is rather incorporated in the solid phase than dissolved within it. Thus, compared to the crystallizing solid mantle, the molten mantle has a larger volatile storage capacity. Unsurprisingly, the thicker the magma ocean, the more water can be dissolved into it.

Yet, while water is highly soluble in the silicate melt, the molten mantle still has a limited maximum storage capacity for dissolved water. This is also referred to as the so-called water solubility (see Moore 2008; Baker and Alletti 2012, for reviews). The solubility of volatile species in melts has been constrained both experimentally (e.g., Holloway and Blank 1994) and theoretically (e.g., Ottonello et al. 2018). It depends primarily on the volatile species considered, on the melt composition, and pressure, but also changes according to temperature, and volatile species mixture (H₂O only or H₂O + CO₂ for instance; see Papale 1997, 1999, Papale et al. 2006).

It is important to note that rather than giving the actual water content, solubility studies constrain the storage capacity of the mantle. This is the maximum amount of water that may be stored, which is the mantle saturation limit (e.g., Keppler and Bolfan-Casanova 2006). It does not give any constraint on its saturation state, i.e., if it is actually volatile-saturated or not. It means that the solubility of a given volatile species into the melt does not provide any information regarding the amount of this very species delivered to the planet. It is rather a bulk property of the molten mantle, so that one should always distinguish the actual water content of the magma ocean, inherited from the planetary formation, from its intrinsic storage capacity.

For a given melt composition and mass fraction of water, the higher the pressure, the larger amount of a volatile species can be maintained dissolved within the melt (Fig. 9). In the deep molten mantle, i.e., at high pressure, the silicate melt can retain more than 10 wt% of water and carbon dioxide in solution (e.g., Papale 1997). Thus, the decrease of ambient overburden pressure experienced by a fluid parcel rising up through convective motions will be responsible for a decrease of the melt storage capacity (Fig. 9), similarly to magma ascent in volcanic systems (e.g., Chap. 5 in Parfitt and Wilson 2008; Sparks et al. 1994; Gonnermann 2015; Wallace et al. 2015). This overburden pressure is the lithostatic load, that is the pressure at a given depth below the surface due to the weight of the overlying layers of silicate material. When the water-dissolved maximum storage capacity is reached, the molten mantle is water-saturated, meaning that all supplementary water exceeding the saturation limit cannot remain dissolved within the liquid phase and is thus exsolved out of the melt to form gas bubbles (i.e., decompression-induced degassing; Figs. 9 and 10; e.g., Edmonds and Woods 2018). Due to their low density compared to the ambient liquid, gas bubbles naturally tend to rise up through the fluid towards the surface, unless they are entrained downwards by the convective currents. Mantle degassing and atmosphere formation and replenishment then occur when gas bubbles reach and burst at the surface of the magma ocean, thus transferring volatile species from the gas phase of the magma to the atmosphere.

In volcanic systems, ascent of gas bubbles is a complex process. Whether they are able to reach and burst at the surface involves the interplay between diffusion, bubbles coalescence, growth, and re-entrainment, which are influenced by several parameters such as magma viscosity and melt-bubble relative density, themselves controlled by the amount of dissolved/exsolved volatiles (e.g., Berlo et al. 2011). Thus far this has been poorly investigated in the context of magma ocean outgassing. For volcanic eruptions, it is believed that rapid growth and ascent of gas bubbles is favored by the high temperatures and low viscosity of the magma, which allows gas to easily segregate from the melt in the form of small bubbles that can themselves easily merge leading to larger bubbles with a faster ascending
speed (Sparks 1978; Sparks et al. 1994; Lesher and Spera 2015). The dynamics of a magma ocean may significantly differs from that of a volcanic erupting system. Yet, it is generally assumed that as soon as bubbles are formed, they also rapidly reach the surface without being re-entrained downwards by convective current, albeit vigorous, thus assuming optimistic efficient outgassing.

Instead of outgassing after bubbles formation and ascent to the surface, water can also be released to the atmosphere by molecular diffusion through the upper thermal boundary layer (e.g., Hamano et al. 2013, 2015; Massol et al. 2016; Ikoma et al. 2018). This degassing process dominates if bubbles are too small to rise up and reach the surface before being transported back downwards by vigorous convective flow. Such a diffusion-limited volatile flux provides a minimum estimate for the degassing rate. Using an experimentally-derived molecular diffusion coefficient for basaltic melts (Zhang et al. 2007), Hamano et al. (2013) found that even with a modest degree of supersaturation, this mechanism can still be responsible for the formation of a massive steam atmosphere, by keeping its water content in solution equilibrium with that of the silicate melt.

### 3.2.3 The Core as an Additional Reservoir of Hydrogen/Water

The metallic core may also be an additional reservoir of water (in the form of hydrogen) ($M_{\text{H}_2\text{O}}^{\text{core}}$ in Eq. (8)), potentially storing several oceans' mass equivalents (e.g., Genda 2016; Peslier et al. 2017). The core density deficit and sound velocity excess (compared to those of pure iron) inferred from seismology (e.g., Birch 1952; Poirier 1994) suggest that the inner core contains a few percents of light elements (see Hirose et al. 2021, for a recent review). Similarly, the density of Earth's outer core is estimated to be 5–10 wt% lower than that of pure iron. Approximately 10 wt% light alloys in the core have been suggested (e.g., Birch 1952; Hirose et al. 2013). Furthermore, some outer core temperature estimates, about 400 K lower than previously thought, would also support a similar fraction of elements lighter than iron to maintain its liquid state (Nomura et al. 2014).

During the differentiation, core-forming metal transports hydrogen and other volatile species from the magma ocean to the core (e.g., Rubie et al. 2015b; Dasgupta and Grewal 2019). Iron-loving elements are prone to be incorporated with metal droplets suspended in the magma ocean, and then partitioned into the core. A large amount of H may be partitioned into the metallic iron at high pressure and be incorporated to the core this way (e.g., Kuramoto and Matsui 1996; Okuchi 1997; Abe et al. 2000; Shibazaki et al. 2009; Li et al. 2020). Umemoto and Hirose (2015) found that ~1 wt% of H in the outer core (equivalent to the hydrogen content of ~130 oceans of water) can reproduce its lower density and faster bulk compressional sound velocity. More conservative estimates suggest an equivalent of 0.2 to up to 90 Earth oceans mass if H is the only volatile in the core (see Peslier et al. 2017, and references therein). The partitioning process is controlled by the chemical equilibration between metal liquids and silicate melt. The partitioning coefficients depend on the P–T conditions at which silicate melt–metal equilibration and core sequestration occur, and on the redox state of the magma ocean (e.g., Li and Agee 1996; Okuchi 1997; Badro et al. 2015).

H has been proposed as a strong candidate because of its abundance in the Solar System and in planetary interiors and its high incompatibility in Fe liquid metal (Okuchi 1997; Abe et al. 2000; Iizuka-Oku et al. 2017). Recent ab initio molecular dynamics calculations have suggested that at the P–T conditions of both magma ocean and the current Earth's CMB, H₂ and H₂O are both more compatible in the core than in silicate melts (Yuan and Steinle-Neumann 2020; Li et al. 2020). This implies that the Earth's core can potentially
act as a large reservoir of water. This is also supported by recent experimentally determined partitioning of hydrogen between liquid Fe and molten silicate under high pressures and temperatures relevant to the conditions of core-forming metal segregation from silicate (Tagawa et al. 2021).

However, other recent experimental results contradict H strong siderophile behavior (Clesi et al. 2018; Malavergne et al. 2019), favoring alternative light elements such as O, S, C, or Si (e.g., Helffrich and Kaneshima 2004; Badro et al. 2015; Shahar et al. 2016). The long-standing question of hydrogen in the Earth's core still remains an unresolved topic. Determining if this discussion applies to Venus would first require estimates of its core structure and density, that may be assessed through future ionospheric and/or ground-based seismology and would certainly improve our understanding of current and early interior processes. As a first approximation it seems reasonable to think that such process may also apply to Venus. This relies on Earth and Venus having similar bulk density and composition, and if one considers that they experienced roughly similar accretion sequences due to their vicinity (which might be challenged; see Emsenhuber et al. 2021), Although little is known about Venus' interior today, future exploration missions, such as NASA's VERITAS (Smrekar et al. 2022) and ESA's EnVision are promising to provide some constraints on its physical properties (see Widemann et al. 2023, this collection).

### 3.2.4 Evolution During Magma Ocean Solidification

**Distribution Between the Different Reservoirs** As the magma ocean crystallizes, each volatile species is distributed between the different and evolving reservoirs of the magma ocean-atmosphere system: the solid mantle, the remaining melt (molten mantle), and the atmosphere. As discussed above, some amount of volatile species may also be incorporated into the core during the segregation of metallic iron to the core if metal-silicate separation is not completed. This distribution can be described by the mass conservation equation. For water, it is (following Lebrun et al. 2013; Schaefer et al. 2016; Salvador et al. 2017):

$$M_{H_2O}^{\text{MO},t=0} = M_{H_2O}^{\text{melt}} + M_{H_2O}^{\text{solid}} + M_{H_2O}^{\text{atm}} + M_{H_2O}^{\text{core}}, \quad (8)$$

where the initial and total mass of water of the system, dissolved within the molten mantle at $t=0$, $M_{H_2O}^{\text{MO},t=0}$, is distributed between the molten part of the mantle, the solidifying mantle, and the atmosphere, whose water masses are $M_{H_2O}^{\text{melt}}$, $M_{H_2O}^{\text{solid}}$, and $M_{H_2O}^{\text{atm}}$, respectively. The mass of water transferred to the core, $M_{H_2O}^{\text{core}}$, is often neglected as it is generally assumed that metallic iron has fully segregated into the core by the time of the last hypothetical global magma ocean, and that chemical exchanges between the mantle and the core are then limited. Before metallic iron has fully segregated to the core, a substantial amount of hydrogen may be incorporated into the core that could thus possibly contain a significant amount of water initially accreted by the planet (Sect. 3.2.3). Here, $t=0$ represents the starting point of the cooling of the last and global magma ocean stage following a giant impact, whose timing is uncertain (e.g., Elkins-Tanton 2012), and implicitly relies on the accretion timeline.

The magma ocean initial water content is usually expressed as a mass fraction of water, $X_{H_2O}^{\text{MO},t=0}$, relative to the initial magma ocean mass, $M_{\text{MO},t=0}$:

$$M_{H_2O}^{\text{MO},t=0} = X_{H_2O}^{\text{MO},t=0} \times M_{\text{MO},t=0}. \quad (9)$$

This initial water content delivered to the planet is highly unconstrained. For Earth, it is estimated to be $18_{-15}^{+81}$ times the equivalent mass of the oceans (Peslier et al. 2017), and implicitly
relies on assumptions regarding the accretion sequence (see Sect. 2). This includes, but is not limited to the timing and dynamics of planetary formation, both affecting the sources and the processes of water delivery (e.g., Morbidelli et al. 2012; Marty 2012; O’Brien et al. 2018). It is generally assumed that any pre-existing atmosphere has been blown away due to the energetics of the giant impact such that water and other volatile species are initially dissolved within the molten interior. Yet, atmospheric erosion may not be as efficient (e.g., Genda and Abe 2003; Pham et al. 2011; Schlichting et al. 2015) and a fraction of accreted volatiles may be released directly into the atmosphere upon impact (e.g., Lange and Ahrens 1982; Matsui and Abe 1986b). $M_{H_2O}^{\text{melt}}$ is the mass of water dissolved within the molten part of the mantle and staying in the residual liquid as solidification proceeds. It is a mass fraction of the rheologically liquid-like magma ocean, $X_{H_2O}^{\text{melt}}$, of mass $M_{\text{melt}}$, and is given by:

$$M_{H_2O}^{\text{melt}} = X_{H_2O}^{\text{melt}} \times M_{\text{melt}} \quad (10)$$

The mass of water stored within the solidifying mantle depends both on the melt-crystals partition coefficient, $\psi_{H_2O}$, and on the mass fraction of water incorporated through interstitial melt trapping, $X_{H_2O}^{\text{inter}}$, following:

$$M_{H_2O}^{\text{solid}} = \psi_{H_2O} X_{H_2O}^{\text{melt}} M_{\text{solid}} + X_{H_2O}^{\text{inter}} M_{\text{melt}} \iota, \quad (11)$$

where $\iota$ is the mass fraction of melt incorporated into the solid mantle through interstitial trapping. It is relative to the melt mass, $M_{\text{melt}}$, taken as $\iota = 1\%$ of $M_{\text{melt}}$ in Elkins-Tanton 2008, Hamano et al. 2013, 2015. Note that most coupled magma ocean-atmosphere studies do not consider any incorporation of water through interstitial trapping of water-enriched melt (i.e., $\iota = 0$), so that the corresponding last term in the right-hand side of Eq. (11) is neglected.

The partition coefficient is either computed as a mantle-averaged value and taken as such (e.g., Hamano et al. 2013, 2015; Schaefer et al. 2016; Bower et al. 2019; Barth et al. 2021), or written as (Lebrun et al. 2013; Salvador et al. 2017):

$$\psi_{H_2O} = \frac{M_{\text{perov}}^{\text{solid}} \psi_{H_2O}^{\text{melt-perov}} + M_{\text{lherz}}^{\text{solid}} \psi_{H_2O}^{\text{melt-lherz}}}{M_{\text{perov}}^{\text{solid}} + M_{\text{lherz}}^{\text{solid}}}, \quad (12)$$

when considering distinct and specific melt-solid partition coefficients for the lower and upper mantle mineral phases, such as perovskite and lherzolite (e.g., Elkins-Tanton 2008; Lebrun et al. 2013; Salvador et al. 2017; Nikolaou et al. 2019). In such a case, $M_{\text{perov}}^{\text{solid}}$ and $M_{\text{lherz}}^{\text{solid}}$ stand for the evolving masses of the perovskite and lherzolite mineral phases, and $\psi_{H_2O}^{\text{melt-perov}}$ and $\psi_{H_2O}^{\text{melt-lherz}}$ their melt-solid partition coefficients. Because of its dependence on the mass of crystallized phases, $\psi_{H_2O}$ changes with solidification.

Finally, the mass of outgassed atmospheric water ($M_{H_2O}^{\text{atm}}$) when considering a pure steam atmosphere is given as (e.g., Elkins-Tanton 2008; Schaefer et al. 2016):

$$M_{H_2O}^{\text{atm}} = 4\pi R_p^2 \frac{P_{H_2O}^{\text{atm}}}{g}, \quad (13)$$

where $R_p$ is the planetary radius, $P_{H_2O}^{\text{atm}}$ is the atmospheric partial pressure of water (at the planetary surface), and $g$ is the gravity. Importantly, one should note that when considering multi-species atmospheres, the molar mass needs to be considered to relate the partial pressure of each volatile species to its atmospheric mass (e.g., Pierrehumbert 2010; Bower et al.
2019):

$$M_{\text{vol}}^{\text{atm}} = 4\pi R_p^2 \left(\frac{\mu_{\text{vol}}}{\bar{\mu}}\right) \frac{P_{\text{vol}}}{g}, \quad (14)$$

where $M_{\text{vol}}^{\text{atm}}$ is the mass of the volatile species vol in the atmosphere, $\mu_{\text{vol}}$ is its molar mass, $P_{\text{vol}}$ its atmospheric partial pressure at the planetary surface, and $\bar{\mu}$ is the mean molar mass of the atmosphere. As discussed above, the atmospheric partial pressure of a volatile species at the surface comes from its outgassing from the interior. Its efficiency depends mainly on its abundance and solubility within the melt, as well as on the convective style and crystallization sequence of the molten mantle.

Because water is an incompatible element, it partitions preferentially into the remaining silicate melt relative to the silicate solid when the magma crystallizes. Therefore, as magma ocean crystallization proceeds, the residual and thinning melt layer becomes more and more enriched in water. Its mass fraction of dissolved water (relative to the magma ocean mass), $X_{\text{H}_2\text{O}}^{\text{melt}} = M_{\text{H}_2\text{O}}/M_{\text{MO}}$, increases with time (e.g., Elkins-Tanton 2008; Massol et al. 2016; Ikoma et al. 2018). The magma ocean mass, $M_{\text{MO}}$, decreases with mantle solidification while the mass of dissolved water $M_{\text{H}_2\text{O}}$ remains similar due its preferential partitioning into the melt (unless a significant amount of water goes into the solid phase). This facilitates the water saturation of the melt with magma ocean solidification (Fig. 9). Yet, a parcel of magma still needs to reach shallow depths and pressures low enough for saturation and exsolution to occur (Figs. 9 and 10).

Given its low, liquid-like viscosity (~0.1 Pa.s) and associated vigorous convection, magma ocean large convective velocities ($v_{\text{conv}}$, on the order of 0.5 m.s⁻¹; see Solomatov 2000, 2015; Elkins-Tanton 2008) are expected to rapidly carry volatile-bearing melt close to the surface (black arrows in Fig. 10). There exsolution can occur at the exsolution depth, $d_{\text{exsol}}$; blue dashed line in Fig. 10. The complete circulation of a deep magma ocean would be achieved in about 1 to 3 weeks (Elkins-Tanton 2008). Compared to the solidification timescale, this is fast enough for the entire magma ocean volume to reach the exsolution depth and outgas its volatiles before the solidification front substantially rises-up. Its velocity is $v_{\text{sol}}$; see yellow arrow in Fig. 10. Furthermore, this 1–3 weeks timescale being smaller than computational time-steps, magma ocean-atmosphere models generally assume that at each time-step, all the magma ocean volume has “seen” the surface, reached volatile saturation, and has thus virtually instantaneously outgassed all its volatiles in excess of saturation at each modeling time-step (Zahnle et al. 1988; Elkins-Tanton 2008; Hamano et al. 2013; Lebrun et al. 2013; Hamano et al. 2015; Schaefer et al. 2016; Salvador et al. 2017; Nikolaou et al. 2019; Bower et al. 2019; Lichtenberg et al. 2021; Barth et al. 2021; Krissansen-Totton et al. 2021). Based on this optimistic and efficient outgassing assumption, these models simulate volatile degassing by considering the pressure equilibrium of the gas phase between the magma and the atmosphere (i.e., on either side of the planetary surface). All volatiles in excess of saturation are thus directly transferred to the atmosphere. Solubility laws are then used to relate the atmospheric partial pressure of each volatile, $P_{\text{vol}}^{\text{atm}}$, to its abundance in the melt, $X_{\text{vol}}^{\text{melt}}$, according to (e.g., Elkins-Tanton 2008; Lebrun et al. 2013; Massol et al. 2016):

$$P_{\text{vol}}^{\text{atm}} = \left( \frac{X_{\text{vol}}^{\text{melt}}}{\zeta_{\text{vol}}} \right)^{\beta_{\text{vol}}}, \quad (15)$$

where $\zeta_{\text{vol}}$ and $\beta_{\text{vol}}$ are volatile-specific solubility constants taken from experimental data (such as Pan et al. 1991; Papale 1997). In magma ocean-atmosphere models, the atmospheric pressures are generally computed out of the volatile species abundances in the melt based on their solubility.
**Typical Outgassing Sequence** Under this assumption and outgassing treatment, typical coupled magma ocean–atmosphere models describe the formation of an H₂O–CO₂ atmosphere through magma ocean degassing as follows<fnref n="3" /> (e.g., Elkins-Tanton 2008; Lebrun et al. 2013; Salvador et al. 2017; Nikolaou et al. 2019; Bower et al. 2019). Given the low solubility of CO₂ in silicate melts, it exsolves from magma at much greater depths beneath the surface than H₂O and outgasses early. Most of the carbon dioxide is outgassed to the atmosphere in the very first time-steps, resulting in high CO₂ atmospheric partial pressures, PₐCO₂, at the earliest evolutionary stages.

Water being much more soluble in the melt, its outgassing occurs later on (starting after ~10⁴ years of cooling), as it requires a more advanced crystallization stage for its concentration in the melt to be high enough to reach saturation. Once it starts, H₂O outgassing and resulting atmosphere build-up reflects the progressive replacement of melt mantle volume with solid mantle volume that has a smaller volatile storage capacity (e.g., Lebrun et al. 2013; Massol et al. 2016; Nikolaou et al. 2019). As an example, Fig. 11 shows the progressive partitioning of water between the different reservoirs during a typical magma ocean outgassing sequence (derived from Salvador et al. 2017 simulations). Interestingly, because the exsolution of a given volatile species depends on its concentration within the melt, lower initial abundances of water within the magma ocean may result in larger relative amounts of water stored in the planet's interior after the magma ocean ends (Nikolaou et al. 2019).

Compared to the H₂O–CO₂ atmospheres typically considered, Lichtenberg et al. (2021) has comprehensively studied how alternative atmospheric components may individually affect the radiative transfer of the atmosphere and the cooling of the magma ocean. How the combination of these individual components in more complex and realistic multi-species atmospheres would affect the heat transfer and magma ocean cooling rate has recently been investigated in a model framework considering variable oxidation states of the magma ocean (e.g., Katyal et al. 2020; Bower et al. 2022; Gaillard et al. 2022a; Maurice et al. 2023, see Sect. 3.2.5 for a detailed discussion).

**Solubility Laws** The implicit and model-dependent choice of solubility law has strong implications regarding magma ocean outgassing efficiency. That is because it ultimately dictates the outgassing rate, timescale, and the volatile amounts that can be dissolved in/exsolved out of the melt. Yet, this is poorly emphasized in the literature. Solubility laws obtained for basaltic systems are generally used (e.g., Elkins-Tanton 2008; Lebrun et al. 2013; Massol et al. 2016) as they are more appropriate for magma oceans since basalts, i.e., relatively low silica content melts, are high temperature and low viscosity melts (e.g., Sparks et al. 1994; Lesher and Spera 2015). Yet, they also have high gas diffusivity and tend to have more modest volatile contents than more viscous, silicic melts such as rhyolitic (as considered in Lichtenberg et al. 2021) or dacitic melts. In the case of water, at a given depth, a silicate liquid of basaltic composition can contain less dissolved water than one of rhyolitic composition (Fig. 9). Reciprocally, for a given water content, basaltic melts become saturated and exsolve H₂O at greater depths, i.e., at larger confining pressures, compared to rhyolitic melts. This facilitates exsolution and outgassing in basalt compared to rhyolite as the magma reaches saturation deeper and does not have to travel to shallower depths, that are closer to the surface (Fig. 9).

<footnote n="3">³Recall that coupled magma ocean–atmosphere models that consider the outgassing from the molten interior (in contrast to impact-generated atmosphere) generally assume that any prior atmosphere has dissipated after a giant impact responsible for melting the entire mantle, so that no pre-existing atmosphere is considered at the beginning of the simulations.</footnote>
<figure><img src="image_11.png" /></figure>

**Fig. 9** Water solubility in basalt (red) and rhyolite (blue) as a function of the pressure acting on the magma (i.e., confining pressure) and corresponding depth beneath the surface. The top axis indicates the amount of water dissolved within a fully molten Earth-sized mantle corresponding to the mass fractions of dissolved water of the bottom axis. For a given water content, the saturation curve gives the location of the exsolution depth, *d*<sub>exsol</sub>, at which gas bubbles start to form (Fig. 10, blue dashed line). At a given cooling stage, a parcel of magma ascending from a depth corresponding to point A undergoes decompression and would become saturated in H<sub>2</sub>O at point B if it were a basalt and would start exsolving the excess of water, if any. If it were a rhyolite, it would not become saturated until it had ascended to the lower pressure marked by point C. Reciprocally, the saturation curve indicates the maximum amount of water that can be dissolved within the melt at a given pressure. As solidification proceeds, water partitions readily into the silicate melt, whose volume decreases owing to crystallization, such that evolving magma ocean liquids become more and more enriched in water. With magma ocean crystallization, a melt starting at a mass fraction of dissolved water corresponding to point A would see its water concentration increase and would become saturated in H<sub>2</sub>O at point D. Once saturation is reached, the melt's mass fraction of dissolved water would evolve along the saturation curve. Solubility laws taken from Parfitt and Wilson (2008) and Carroll and Holloway (1994)

However, it should be noted that additional complexities may arise. For instance, the viscosity of mafic and ultramafic melts (such as basalt and komatiite, respectively) is comparatively much less affected by the dissolved water content than that of polymerized, low temperatures rhyolitic or dacitic melts (e.g., Fig. 5.5 in Lesher and Spera 2015, and references therein). In turn, viscosity influences magma motion and transport and may affect the amount of melt reaching the exsolution depth.

Furthermore, even for a given silicate melt composition, several solubility laws are available and their differences will affect outgassing efficiency as described before. These differences become more and more important when decreasing the mass fraction of dissolved water and are thus larger for low water contents and increase with planetary size for a given absolute water content (see Appendix A in Salvador and Samuel 2023, for a detailed discussion). Predictions regarding magma ocean outgassing, both in terms of degassing timescale and amount of outgassed volatiles, are thus ultimately influenced by the generally implicit and model-dependent choice of solubility law. In turn, the solubility law choice will alter the cooling rate of the magma ocean through the amount of radiative energy that can be dissipated to space, which is dictated by the outgassed atmospheric composition.

In addition, while the solubility of each volatile species is often treated separately when several volatile species are present in the magma ocean (for instance, H<sub>2</sub>O and CO<sub>2</sub>), solubility functions are actually more complex in such a case. This is in fact virtually always
<figure><img src="image_12.png" /><figcaption>**Fig. 10** Schematic of the magma ocean degassing process and associated velocities. $v_{sol}$ is the upward solidification velocity associated with the uprising rheological front where $\phi = \phi_c$ (assuming that solidification proceeds from bottom-up). $v_{conv}$ is the liquid-like convection velocity and $v_{perc}$ is the percolation velocity of the gas bubbles rising up to the surface. $d_{exsol}$ is the exsolution depth above which oversaturated volatiles are exsolved out of the melt and start forming gas bubbles. In a rapidly convecting magma ocean, volatile-bearing melt is assumed to reach the exsolution depth much faster than solidification proceeds and loses its volatiles in excess of saturation on that occasion. The outgassing is thus often assumed to be efficient. Adapted from Salvador and Samuel (2023)</figcaption></figure>

the case: the gas phase is never just pure H₂O or CO₂. Then, chemical interactions between the volatiles themselves and the magma affect the solubility (e.g., Papale 1997; Papale et al. 2006; Massol et al. 2016; Edmonds and Woods 2018). For instance, as soon as the least soluble volatile starts to exsolve and form gas bubbles, some small amounts of most of the other species present will also diffuse into those bubbles (e.g., Parfitt and Wilson 2008). Such migration of the molecules of volatiles into the gas bubbles will increase the concentration gradient in the surrounding liquid, which will enhance the diffusion of more molecules toward the growing bubble. The non-ideal mixing behavior of the volatile species in the vapor phase is also responsible for the curvature of the saturation curves (see Papale et al. 2006; Moore 2008; Berlo et al. 2011, for detailed discussions).

**Limits of the Efficient Degassing Assumption** Stemming from the vigorous convection of the magma ocean, the efficient degassing scenario, relying on one a priori robust – yet debatable – assumption is mainly considered. Yet, despite high convective velocities, several mechanisms may buffer the outgassing efficiency of the magma ocean (see Table 1 for a summary). Alternatively, minimal to no degassing scenarios may be considered (e.g., Ikoma et al. 2018), where a substantial fraction of volatiles is retained in the melt on longer timescales and possibly ends up being trapped in the solidified mantle.

Although vigorous convection is prone to efficient and rapid magma ocean mixing, the fluid motions at relevant magma ocean dynamics can be organized according to large-scale circulations (e.g., Krishnamurti and Howard 1981; Castaing et al. 1989; Siggia 1994). This
<figure><img src="image_13.png" /><figcaption>**Fig. 11** Evolution of the distribution of water between the molten (red) and solidified (orange) mantle, the atmosphere (plain blue line for the total amount of water outgassed, including both vapor and liquid phase) and the ocean (dashed blue) during magma ocean solidification of an Earth-sized planet, while assuming efficient outgassing. The vertical dashed line indicates the time at which the End of the Rapid Cooling Stage is reached, i.e., when the planet has lost its excess of internal heat and reached global radiative balance with the net absorbed solar flux. Derived from Salvador et al. (2017) simulations</figcaption></figure>

may isolate a significant part of the magma ocean from the surface in the center of large-scale convection rolls (e.g., Fig. 1 in Banzhaf 2003), thereby limiting the amount of silicate melt reaching low pressures at which volatiles are exsolved, and thus, tamping down magma outgassing. Using numerical fluid dynamics experiments, a recent study investigated how the detailed convective patterns may influence the time-dependent ability of the convecting fluid to reach shallow depths at which volatile exsolution and degassing proceeds (Salvador and Samuel 2023). They showed that vigorous convection and associated mixing does not ensure magma ocean chemical homogeneity. Rather, vortex-like small-scale features entrained in large-scale convective currents might generate local chemical heterogeneities. Then, despite high convective velocities and effective mixing, additional convective cycles and longer timescales would be required to erase those small-scale heterogeneities and to bring the volatile-rich melt blobs close to the surface where exsolution and degassing can occur. In some cases, these timescales may be comparable to or even larger than the magma ocean time spent in a given dynamic regime. Accumulated over the whole magma ocean lifetime, the amount of outgassed water may substantially decrease. There may be cases resulting in no outgassing of water initially dissolved deeper than the exsolution depth.

The crystallization sequence of the magma ocean may also significantly affect volatile distribution and outgassing efficiency. If the crystallization of the magma ocean starts from mid-mantle depths (Fiquet et al. 2010), rather than proceeding from bottom-up as it is commonly assumed in coupled magma ocean–atmosphere models, two distinct and separate molten layers may exist on either side of the crystallization front (e.g., Labrosse et al. 2007). While the upper molten layer would remain in contact with the surface where exsolution and outgassing may proceed as described above, the lower, deep basal magma ocean may keep its volatile species dissolved thanks to the large pressures at depth. The two molten layers would have diverging compositions, with the upper one being progressively depleted in
volatiles and dehydrated through outgassing, while the primordial lower one would become more and more volatile-rich with crystallization due to volatiles' melt affinity. Depending on the volume of this lower molten layer, dictated by the depth at which crystallization would start, a substantial amount of volatile species might remain isolated from the surface and thus unable to be outgassed.

In a similar fashion, melt-crystals relative density and resulting segregation (e.g., Boukaré and Ricard 2017; Caracas et al. 2019) may prevent possibly volatile-rich melt to reach the surface. If the melt becomes more dense than coexisting minerals at pressures near the base of the mantle (Mosenfelder et al. 2007; Asimow 2018), denser melts and their dissolved volatiles would be entrained downwards. Solidification would then proceed downward from a septum, and the dissolved volatiles would be trapped in the deep interior rather than rising-up throughout the melt and being released in the atmosphere (Elkins-Tanton 2011). Finally, even within a given reservoir of magma, the mode of crystallization, and more specifically melt-crystals segregation is believed to affect the amount of exsolved volatiles. In particular, the fraction of exsolved volatiles from a given magma is enhanced if the evolved liquids and exsolved volatiles are efficiently segregated from their crystalline products (e.g., Edmonds and Woods 2018).

If the magma ocean crystallization results in a gravitationally stable mantle stratification (e.g., Solomatov 2015), volatile species that have not been outgassed may remain in the solidified mantle and mantle chemical heterogeneities may persist until the present-day. On the other hand, if the crystallization leads to an unstable cumulate density stratification, the solidified mantle may overturn to a stable configuration (e.g., Elkins-Tanton et al. 2003). Mantle overturn may induce late episodes of outgassing through the melting of remnant volatile-rich mantle reservoirs, thus possibly releasing to the atmosphere volatile species that have not been outgassed during magma ocean solidification.

The planetary rotation rate may also affect the chemical structure of the crystallizing magma ocean and favor the development of chemical heterogeneities (Maas and Hansen 2015, 2019). In particular, for moderate and high rotation rates, the resulting crystallization pattern may induce an inhomogeneous magma ocean solidification with changes in depth and latitude. Major chemical segregation in distinct reservoirs might isolate a substantial magma ocean fraction from efficiently and rapidly reaching shallow depths where volatile exsolution occurs. Similarly to a basal magma ocean, these molten layers would likely retain their dissolved volatiles and outgassing may not be as efficient as commonly modeled. Yet, if the rotation rate of Venus during magma ocean crystallization was similar to its present-day value, this process would most likely not induce significant latitudinal crystal dynamics variations and would thus be negligible.

Another mechanism, that may significantly alter magma ocean outgassing efficiency and contradicts the classical efficient degassing assumption, is the efficient retention of interstitial, volatile-rich melt at the freezing front of a rapidly crystallizing magma ocean (Hier-Majumder and Hirschmann 2017; Miyazaki and Korenaga 2021, 2022). It stems from the fact that, if crystal-melt separation, matrix compaction, and melt upwards percolation are not efficient, a substantial fraction of melt and dissolved volatiles could be trapped within the crystallizing magma ocean.

The crystallizing melt is a solid-dominated, yet porous matrix with interstitial melt. McKenzie (2011) emphasized that depending on the competition between compaction within the freezing front and the rate of freezing, a substantial volume of the melt could be trapped during melt crystallization in a magma chamber, given no horizontal variations. Compaction within the freezing front expels out trapped melt, while rapid crystallization of the front freezes the melt in place and thus inhibits expulsion. Using a two-phase flow (of
melt and matrix) model, Hier-Majumder and Hirschmann (2017) showed that for magma ocean crystallization characteristic timescales, compaction within the moving freezing front is inefficient. Hence melt keeps being trapped at the rheological front while solidification proceeds. Then, if melt percolates upward through the solidified mantle insufficiently fast, Rayleigh-Taylor instabilities would continuously transport the newly formed solid matrix downwards (Miyazaki and Korenaga 2021, 2022), thus entraining the trapped melt and its dissolved volatiles at depth. Volatiles would thus also be trapped within the solidifying mantle and then sequestered in the deep mantle, rather than enriching the remaining melt. Volatile trapping would be enhanced by the fact that their solubility is greater at depth, where magma ocean crystallization is often thought to start, thus trapping melts with high volatile storage capacity and potential high water content. This way, up to 77% H₂O and 12% CO₂ of the total accretion-delivered volatile content could have been trapped in the mantle during magma ocean crystallization (Hier-Majumder and Hirschmann 2017). Yet, the low volatile storage capacity of solid mantle mineral phases (e.g., Bolfan-Casanova et al. 2003; Shcheka et al. 2006) do not support such large amounts of remnants water and carbon. To reconcile their results with such a low volatile storage capacity, Hier-Majumder and Hirschmann (2017) suggest different storage sites for the magma ocean crystallization leftover volatiles. Mantle carbon would be stored in solid carbonates and carbonate-rich melts in the upper mantle, and in diamond and metal carbides in the mid and lower mantle. Volatile-rich accessory minerals and water-enriched melt remaining along grain boundaries or as hydrous fluids or melts would act as storage sites of mantle water. They argue that such a buffered magma ocean outgassing reconciles the fact that the Earth's deep interior contains significant reservoirs of volatiles despite their incompatible nature. This process would be more effective in the case of equilibrium crystallization, where crystals and liquid remain in equilibrium (i.e., in contact) throughout crystallization. As opposed to fractional crystallization where melt-crystal separation is efficient. Assuming that 1% of the melt mass is interstitially trapped, lower estimates of the initial amount of water retained in the solid mantle range from 3% (Hamano et al. 2013) to 13% (Elkins-Tanton 2008; Hier-Majumder and Hirschmann 2017).

Note that in several cases, the amount of remnant water in the solidified mantle would mainly depend on the crystal-melt partition coefficients and on its solubility in the melt. This would include cases of high compaction and resulting efficient expulsion of melt from the porous matrix, or cases of slow magma ocean solidification relative to the compaction rate. It would also apply to cases of efficient crystal-melt separation at the rheological front. The latter owing to effective fractional crystallization or even tearing off and re-entrainment of the newly formed solid matrix within the molten mantle by vigorous convection currents and subsequent settle down of individual crystals.

Miyazaki and Korenaga (2021, 2022) have studied how such hampered magma ocean degassing and subsequent solidified mantle evolution and degassing patterns might affect the formation of early water oceans on rocky planets. Hamano et al. (2013) suggested that the rapidly-outgassed, steam atmosphere of Venus had prevented its climate from being temperate owing to the strong thermal blanketing effect. Miyazaki and Korenaga (2021) have suggested that, even for inefficient water degassing during the magma ocean phase, Venus may have continued to lack oceans throughout its history whereas Earth was able to form water oceans since the solidification of the mantle surface (Miyazaki and Korenaga 2022). In particular, if no ocean were formed upon solidification of the surface on early Venus, comparable rates of subsequent solid-state mantle degassing and water loss (through hydrogen escape) would not supply enough water to allow for later ocean formation.

Overall, any process that significantly reduces the amount of melt reaching the low pressures at which volatile exsolution (i.e., melt-gas partitioning) occurs during magma ocean
<table><caption>**Table 1** Mechanisms possibly stifling magma ocean outgassing by isolating volatile-rich melt from the surface and contrasting with the classical efficient outgassing scenario</caption><thead><tr><td>Mechanism</td><td>Reference(s)</td></tr></thead><tbody><tr><td>Magma ocean crystallization starting at mid-mantle depth</td><td>Labrosse et al. (2007), Fiquet et al. (2010)</td></tr><tr><td>Planetary rotation and asymmetric crystal settling</td><td>Maas and Hansen (2015, 2019)</td></tr><tr><td>Melt-crystal density-driven segregation</td><td>Boukaré and Ricard (2017), Caracas et al. (2019)</td></tr><tr><td>Interstitial melt retention at magma ocean freezing front</td><td>Hier-Majumder and Hirschmann (2017), Miyazaki and Korenaga (2021)</td></tr><tr><td>Magma ocean convective dynamics</td><td>Salvador and Samuel (2023)</td></tr></tbody></table>

crystallization would alter outgassing efficiency and contrast with the classical view (Table 1). Then, if the melt exsolves volatiles, gas bubbles further need to reach the surface. If not, this would also stifle magma ocean outgassing. The global contribution of these combined mechanisms is still poorly constrained but the amount of volatile species remaining in the solidified mantle during the magma ocean solidification may have strong implications regarding the early and long-term evolution of the planet and should be accounted for in the future generation of models.

### 3.2.5 Influence of the Mantle Redox State

**Generalities** Molten mantle chemistry plays an important role in magma ocean outgassing. This is less by affecting its efficiency and the *amount* of outgassed species, such as the physical mechanisms mentioned above, than by controlling the *type* of outgassed species (see Table 2 for a summary of the different early outgassed atmospheres and underlying assumptions/constraints). In particular, the oxidation (redox) state of the mantle is critical. It has been interpreted in different ways but can be fundamentally defined by its availability of electrons and the resulting oxidation/valence state of its constituent elements (e.g., McCammon 2005a,b).

Oxidation-reduction reactions are chemical reactions generally<fnref n="4" /> involving a transfer of electrons between two species. One is an electron donor (a chemical reductant or reducing agent, which readily gives up electrons) that becomes oxidized and the other an electron acceptor (a chemical oxidant or oxidizing agent, which possesses a strong affinity for electrons) that becomes reduced (e.g., Chap. 3 in White 2013; Kasting 2014). Oxidation thus refers to the loss of one or more electrons and the increase in the oxidation state/number. Reduction<fnref n="5" /> is the gain of one or more electrons.

To maintain conservation of electrons (i.e., electrical neutrality), the oxidation of one compound necessarily implies the reduction of another, as electrons are transferred from one species to another. While originally used to refer to the reaction of a compound with oxygen (oxygen being the most common electron acceptor), hence oxidation, the term oxidation has

<footnote n="4">⁴While the transfer of electrons will always cause a change in oxidation state, there are many reactions that are classified as redox reactions even though no electron transfer occurs, such as those involving covalent bonds.</footnote>

<footnote n="5">⁵The term reduction originates from the loss in weight upon heating a metallic ore, such as a metal oxide, to extract the metal: the ore was then “reduced” to the metal. Its meaning has then been generalized to any processes involving a gain of electrons.</footnote>
been generalized to any electron-transfer reactions, without necessarily involving oxygen, after realizing that electron loss was the core of a substance being oxidized.

Yet, as it is the most common electron acceptor, the abundance of oxygen usually controls the oxidation state of a system, which is the case for the mantle, but this need not be the case. The oxidation state of the mantle is controlled by its chemical potential of oxygen, commonly referred to as the “oxygen fugacity”, $f_{O_2}$. It is the equivalent of the partial pressure of oxygen in a particular environment corrected for the non-ideal character of the gas. It describes the formal abundance of $O_2$ gas in units of atmospheres (although often not representing an actual gas phase). Given its strong oxidizing power, a high oxygen fugacity/abundance corresponds to an oxidized environment while a low oxygen fugacity is associated with abundant reduced species in a reduced environment.

While most major elements constituting the mantle have only one stable oxidation state (e.g., Si, Al, Mg, O, Ca, Na, Ti), others, like Fe and several minor elements (e.g., C, S, Ni, Co, V) exhibit several stable oxidation states. Their geochemical behavior is thus affected by oxidation-reduction processes and reciprocally, their distribution, behavior, and abundance affect the oxidation state of their mantellic environment (e.g., Wood et al. 1990; Mallmann and O'Neill 2009). Because of their abundance, iron- and carbon-bearing solid species play an important role in controlling the oxygen fugacity.

**Oxidation States of Iron and Redox Buffers** From its most reduced to most oxidized state, iron most commonly occurs as metallic iron (oxidation state 0, noted $Fe^0$, $Fe(O)$, or simply Fe), ferrous (divalent) iron (oxidation state of +2, meaning that it lost 2 electrons in the oxidation process; noted $Fe^{2+}$ or $Fe(II)$), and ferric (trivalent) iron (oxidation state of +3 after the loss of 3 electrons; noted $Fe^{3+}$ or $Fe(III)$). Iron is, by mass, the most abundant element on Earth (followed by oxygen). It is the most abundant element with a variable oxidation state. Iron's relative abundance in different oxidation states is also indicative of that of the mantle.

Furthermore, some specific iron-bearing (but not only) mineral assemblages, are in chemical equilibrium at a certain oxygen fugacity. For these two reasons, such mineral assemblages, called redox buffers (e.g., Reactions (R1), (R2), and (R3)), are used as references to constrain the oxygen fugacity of the mantle, hence its oxidation state (e.g., Frost 1991; Hirschmann 2021). For instance, in a given set of physicochemical conditions, at a given temperature, the oxidation of fayalite (F, which contains two ferrous ions $Fe^{2+}$) to magnetite (M, which contains one $Fe^{2+}$ and two $Fe^{3+}$) and quartz (Q) imposes or “buffers” a specific value of $f_{O_2}$ at equilibrium when all three mineral species are present. The reaction is (e.g., Frost 1991):

$$3 \underset{\text{fayalite}}{\text{Fe}_2\text{SiO}_4} + \underset{\text{magnetite}}{\text{O}_2} \rightleftharpoons 2 \underset{\text{quartz}}{\text{Fe}_3\text{O}_4} + 3 \underset{\text{quartz}}{\text{SiO}_2} \quad (\text{R4})$$

This particular quartz-fayalite-magnetite (QFM) redox buffer is representative of relatively oxidized conditions that are typical of modern Earth's volcanic degassing and presumably of Earth's upper mantle. It is important to understand that the oxygen fugacity is fixed at a given value in a given oxygen fugacity-pressure-temperature space (e.g., Wood et al. 1990). This is simply by the equilibrium coexistence of fayalite, magnetite, and quartz. The oxygen fugacity does not depend on the proportion of these minerals. This is why it is appropriately called a buffer. This can be illustrated by considering a magma containing some amount of fayalite, magnetite, quartz, and oxygen. If the oxygen fugacity is increased by adding oxygen to the system, equilibrium in Reaction (R4) is driven to the right until the log of the oxygen fugacity returns to the given value (which is temperature and pressure dependent).
Only when all fayalite is converted to magnetite (and quartz) can the oxygen fugacity rise (e.g., Chap. 3 in White 2013). A decrease of oxygen fugacity would be buffered in exactly the opposite way until all magnetite and quartz were gone.

Another common buffer of oxygen fugacity is the iron-wüstite (IW) buffer (Reaction (R2)), which describes the coexistence of metallic (Fe) and ferrous ($Fe^{2+}$, which combines with $O_2^-$ to form FeO) iron and indicates a highly reducing environment. Here, an increase in $f_{O_2}$ favors the reaction from left to right, which stabilises iron in its higher oxidation state (wüstite), whereas low $f_{O_2}$ stabilises iron in its metallic, lower oxidation state.

**Geochemical Implications, Speciation, and Outgassed Composition** Like minerals, gas species can exist in various oxidation/valence states depending on the oxygen fugacity. The redox state of gases in equilibrium with rocks is often described by the mineral buffers that govern the capacity of the rock to consume or release oxygen (e.g., Zahnle et al. 2020). As mentioned above, oxidation-reduction processes affect the geochemical behavior of elements of several stable oxidation states, and consequently, that of the other elements interacting and having some chemical affinities with them (e.g., siderophile elements for iron). Overall, the redox state of the mantle (and its oxygen fugacity) controls the molecular speciation of C-H-O-N-S-bearing fluids and melts (French 1966; Frost 1979; Holloway 1981; Wood et al. 1990; Holloway and Blank 1994; Hirschmann et al. 2012; Hirschmann 2012; Grewal et al. 2020). It influences the solid-melt-gas partitioning of volatile species (Holloway and Blank 1994; Libourel et al. 2003; Miyazaki et al. 2004; Hirschmann 2012; Armstrong et al. 2015; Boulliung et al. 2020; Gaillard et al. 2021) and trace elements in the silicate magma – and thus, the composition of the atmosphere –, their partitioning among silicate melt and molten or solid metallic iron (Drake et al. 1989; Hillgren 1991; Holzheid et al. 1994; Righter et al. 1997; Gaillard et al. 2022b), as well as the mineral saturation and bulk mineralogy (e.g., Feig et al. 2010, and references therein). Indeed, given an elemental inventory of C-H-O-N-S, the abundance of oxygen, $f_{O_2}$, critically affects the identity of molecular species outgassed and controls the relative abundances of the oxidized and reduced form of major gases (e.g., H$_2$/H$_2$O, CO/CO$_2$).

The question of atmospheric oxidation state (hence, of its composition) is thus inex-
tricably interlinked to the mantle redox evolution (e.g., Kasting et al. 1993a; Lécuyer and
Ricard 1999). Essentially, an oxidized magma, i.e., of high oxygen fugacity (for instance at
the QFM buffer, consistently with present-day upper-mantle conditions), induces outgassing
of oxidized volatile species, such as H$_2$O, CO$_2$, SO$_2$, and N$_2$. Conversely, a magma of re-
duced composition (of low oxygen fugacity) releases reduced volatile species, such as CO,
H$_2$, CH$_4$, NH$_3$, and H$_2$S (e.g., Holland 1962; Frost 1979; Wood et al. 1990; Kasting 1993;
Hirschmann 2012). This speciation process can be explained by the fact that an environment
of high oxygen fugacity/abundance makes it readily available to bond with C-H-O-N-S
fluids and produce oxygen-bearing compounds. Note however that redox changes during
magma ascent (Sato 1978; Burgisser and Scaillet 2007), mixing (Ueki et al. 2020), and de-
gassing (Mathez 1984; Humphreys et al. 2015) may occur, making the link between the
redox state of magmas and their outgassed products less straightforward than commonly
assumed.

**Present-Day Redox State** The Earth's present-day atmosphere is highly oxidized as its second most abundant constituent is O₂, which is a powerful electron acceptor. However, early magma ocean and subsequent long-term volcanic degassing only cannot explain this current characteristic feature, which certainly results from the complex interplay between geochemical and biotic processes (e.g., Holland 2006; Kasting 2013; Catling 2014; Catling and Zahnle
2020). Yet, at present-day, H₂O and CO₂ are the two most abundant gases released by volcanic degassing, which indicates that the Earth's present-day upper mantle is in an oxidized state (e.g., Frost and McCammon 2008). With CO₂ its most abundant constituent, Venus' atmosphere is currently relatively oxidized, despite being considered possibly weakly reducing in its lower part (e.g., Catling 2015). Surface-atmosphere interactions likely result in an oxidizing surface with a high oxygen fugacity at about the Magnetite-Hematite (MH) redox buffer (e.g., Fegley et al. 1997; Zolotov 2018). The mean FeO/MnO ratio measured at Venus' surface by Venera 13, 14, and Vega 2 is slightly less than that of the bulk silicate Earth (~52 for Venus' surface, Surkov et al. 1984, 1986; Volkov et al. 1986; and ~60 for the BSE). Used as a proxy of the planet's oxidation state (Wänke et al. 1973), it also suggests a rather oxidized surface and interior. If the present-day Venusian SO₂ atmospheric inventory originates from volcanic outgassing, it could indicate that at least some parts of the Venusian mantle are highly oxidized, at about the QFM buffer.

**Early Oxidation State** While present-day volcanic outgassing can be related to the current magma sources redox state, the earliest oxidation state of planetary mantles is much more speculative and little is known about the magma ocean initial and evolving redox conditions. The oldest clues available on Earth come from measurements on Hadean zircons, recording oxygen fugacity of Hadean magmatic melts identical to those in the present-day mantle, thus indicative of an oxidized upper mantle (at about QFM buffer) as early as 4.35 Gyr ago (Trail et al. 2011). Earth's oldest igneous rocks also suggest that the mantle redox state was only weakly reduced and has not varied significantly over at least the past 3.9 Gyr of Earth's history (Canil 1997; Delano 2001; Canil 2002; Lee et al. 2003; Li and Lee 2004; Rollinson et al. 2017; Nicklas et al. 2018). As today, the redox state was then likely set by the QFM buffer, so that the resulting volcanically-outgassed atmosphere would have been mostly made of H₂O, CO₂, and N₂, with small amounts of CO and H₂ (Holland 1962; Abelson 1966). There is no such records available for Venus and whether these conditions are systematic outcomes of planetary early evolution and can be generalized to early Venus or are Earth-specific is unclear.

**Constraints from Prebiotic Chemistry** Prebiotic chemistry can provide independent but indirect constraints on the environment at the surface of early Earth. Yet, the prevalence of an early oxidized environment suggested by geochemical evidence is hard to reconcile with the reduced conditions requirements inferred from chemical studies on the origin of life. Indeed, prebiotic chemistry experiments, starting from the famous Miller-Urey experiments in the early 1950s (Miller 1953, 1955; Miller and Urey 1959), have shown that amino acids and a wide number of other building blocks of living organisms can be synthesized from electrical discharges in highly reduced conditions with the presence of methane and ammonia and where free oxygen is absent (see Oro et al. 1990; McCollom 2013; Benner et al. 2020, for reviews). The fact that such conditions result in efficient abiotic synthesis of biologically important organic compounds suggested that the origin of life required a very reducing environment. Hence hinting that this was the case for the earliest terrestrial atmosphere overlying a solidified surface.

It should be noted that bridging the gap between the non-agnostic conditions under which prebiotic chemistry experiments are successful in producing life-essential compounds and actual early Earth's conditions where life emerged remains challenging and is still an extensive area of study (e.g., McCollom 2013). For instance, Cleaves et al. (2008) have suggested that yields of amino acids observed in more neutral (N₂/CO₂-dominated) or oxidized environment experiments may have been biased low/underestimated owing to oxidation of the
acids during sample processing. Then, using oxidation inhibitors during sample processing or adding CaCO$_3$ to buffer the pH of the aqueous phase would significantly increase the yields.

Energy sources other than electrical discharges/lightnings have also been invoked to syn-
thesized complex biological molecules without necessarily requiring highly reducing envi-
ronments (e.g., Chyba and Sagan 1992; Kurosawa et al. 2013). In particular, coronal mass
ejection events from the active young Sun could trigger nitrogen fixation and ignite reac-
tive chemistry yielding complex molecules in less reducing environments (Airapetian et al.
2016, 2020).

Based on the abundance of organic molecules in the Solar System, alternative scenarios have invoked exogenic sources of organics for the origin of life rather than an in-situ production (e.g., Anders 1989; Chyba et al. 1990; Chyba and Sagan 1992; Whittet 1997; Matthews and Minard 2006). This abundance of organic molecules may imply that they are not that hard to make (see Zahnle et al. 2010 and references therein for a thorough discussion). Yet, the extraterrestrial delivery of complex and organic materials by comets, asteroids, and interplanetary dust particles and their implications for the emergence of life is challenged by their survival upon impact (e.g., Whittet 1997; Pierazzo and Chyba 1999; Pasek and Lauretta 2008).

The apparent contradiction between the reducing conditions required by prebiotic chemistry and the likelihood of an oxidized early atmosphere (and mantle) might also be resolved by alternative scenarios involving an oxidizing surface environment. Such an environment may result from magma ocean degassing. But the oxidizing environment might experience a perturbation making it temporarily and/or locally reducing (Airapetian et al. 2016; Itcovitz et al. 2022). In fact, such transient conditions may actually be the most favorable for prebiotic chemistry (Benner et al. 2020).

**Impact-Induced Degassing and Transient Reducing Environments** To produce such a perturbation and reconcile geological evidence of an oxidizing past with the requirements for life's emergence, impact-induced degassing has been proposed. Gases are then directly released into the atmosphere on impact (Benlow and Meadows 1977; Jakosky and Ahrens 1979; Lange and Ahrens 1982; Matsui and Abe 1984, 1986b). It is an efficient mechanism to release strongly reduced species into the early atmosphere, and thus to provide favorable conditions for the origin of life despite a rather oxidized, volcanically-produced, environment (e.g., Zahnle et al. 2010, 2020). It is thought to proceed as follows: due to the energetics of impact, all of the volatiles carried by the impactors are expected to be vaporized, to equilibrate chemically with the other materials of the impactor, and to directly enter the atmosphere. Atmospheres resulting from impact degassing would then tend to have a composition reflective of the impacting bodies (rather than the proto-planet mantle). These are mostly volatile-rich and much more reduced than the mantle (e.g., Zahnle et al. 2010). Note that additional and more complex reactions between the impactor components and the impacted body's atmosphere or crust can increase the production rates of reduced species (e.g., Kasting 1990).

Models of impact-generated atmospheres generally suggest that the outgassing of reduc-
ing species dominates for a variety of meteoritic materials (e.g., carbonaceous, ordinary, and
enstatite chondrites; Kasting 1990; Schaefer and Fegley 2007; Hashimoto et al. 2007; Sugita
and Schultz 2009; Schaefer and Fegley 2010). If the gases of Earth's earliest atmosphere had
equilibrated with material like that of primitive meteorites, the resulting atmosphere would
be much more reduced than present-day volcanic gases (e.g., Zahnle et al. 2010). This is due
to the reducing power of the substantial amounts of metallic iron and iron sulfides contained
in many meteorites, including ordinary and enstatite chondrites. Even for CI carbonaceous chondrites the resulting gases are rather strongly reducing owing to the reducing power of the abundant carbon contained in CI chondrites and present in reduced form (Hashimoto et al. 2007; Zahnle et al. 2010). These are the most highly oxidized meteorites (e.g., Urey and Craig 1953, Rubin et al. 1988, Krot et al. 2014; and Fig. 5.2 in Trieloff et al. 2006) and thus expected to generate the most highly oxidized gases among the different meteoritic sources. Note however that Schaefer and Fegley (2010) found that carbonaceous chondritic material such as CI, CM, and CV chondritic materials would rather produce oxidized atmospheres.

Yet, mixtures of solid meteoritic material (from highly oxidized carbonaceous chondritic material to highly reduced enstatite chondritic or eucritic materials) may also produce more oxidizing compositions (Schaefer and Fegley 2017). They are relevant to the mixing of materials from different locations in the solar nebula and thus representative of the primitive solar nebular materials. Indeed, because of variations in the abundance of buffering phases (such as metallic iron) and volatiles within the solid materials, gases produced out of their mixtures are not a straightforward linear mixture of the gases produced by individual materials. As a result, adding only a few percent of carbonaceous material to a differentiated body may be sufficient to raise the $f_{O_2}$ of the mixture by several log units. Given the stochastic nature of accretion (Sossi et al. 2022) and since the oxygen fugacities differ between individual materials and their mixtures, the behavior in mixtures remains uncertain (Schaefer and Fegley 2017). Schaefer and Fegley (2017) also emphasized that other processes affecting the redox state of the atmosphere, such as hydrogen escape, photochemistry, or surface-atmosphere interactions, would certainly induce additional complexities and should be further investigated. For instance, they have found that oxidized materials become more oxidized by hydrogen loss whereas reduced materials do not. While $NH_3$- and methane-rich atmospheres are prone to efficient photochemical dissociation by solar ultraviolet radiation (e.g., Leighton and Steiner 1936; Lasaga et al. 1971; Kuhn and Atreya 1979; Kasting et al. 1983) and thus unlikely stable as steady states, impact-induced outgassing of infalling meteoritic material could repeatedly create transients and life emergence-favorable conditions many times.

**Initial Redox State of the Magma Ocean and Its Evolution** Based on the aforementioned direct and indirect constraints and processes, a consensual view may emerge. It suggests that the mantle oxidation state was relatively oxidized and close to its current state early in history, once the surface was solidified (i.e., post-magma ocean). The overlying atmosphere was also likely oxidized and composed dominantly of $H_2O$ and $CO_2$. In such an oxidizing environment, impact-induced degassing transiently produced reduced conditions favorable for the development of life on Earth. Before the surface was solid, prior to the available geological records and to the setting of life-sustainable conditions on a solidified planet, the oxidation state of the earlier evolution stages, such as the magma ocean phase itself and its evolution upon cooling, is highly unconstrained. In particular, whether oxidized conditions are a direct/intrinsic consequence of the magma ocean evolution or inherited from other mechanisms related to planetary formation remains unclear. To address this issue, we discuss below the processes thought to affect the redox state of the earliest stages of planetary evolution, with particular emphasis on the magma ocean stage, and how they may be reconciled with the available constraints of the subsequent stages.

**Hydrogen-Rich and Reducing Early Atmospheres** A primordial $H_2$-He atmosphere captured from the solar nebula (see Sect. 2.1) would induce a reduced start. Before being lost after the
dissipation of the solar nebula (e.g., Zahnle et al. 2007), its reducing power could have affected both the surface and the mantle of the proto-planet. Similarly the H₂-He solar nebula could have reduced the planetesimals that subsequently encountered the planet, thus being a source of reduced materials (Zahnle et al. 2010). Tian et al. (2005) have suggested that in a hydrogen-rich early atmosphere, the escape rate of hydrogen would be energy-limited due to lower exospheric temperatures rather than diffusion-limited and thus operating much slower. In such a case, a primordial hydrogen-rich atmosphere and its reducing power would have been sustained longer and maintained after dissipation of the solar nebula. Without necessarily involving the solar nebula, on longer timescales, the balance between slow hydrogen escape and hydrogen outgassing (requiring a reducing upper mantle) could have preserved/induced an atmospheric hydrogen mixing ratio of up to 30% (Tian et al. 2005; Liggins et al. 2020) and sustained a reducing environment. Note however that Tian et al. (2005) hydrogen escape rate estimates may be biased low and are controversial (Catling 2006; Tian et al. 2006; Kuramoto et al. 2013; Tian 2015).

**Link with Core Formation and Evidence for a Reduced Start** The depth, timing, and timescale of metal-silicate equilibration, separation, and resulting core formation, play a critical role in determining the oxidation state of the magma ocean and crystallizing mantle. These processes are likely occurring throughout the accretion sequence and possibly extend through the different magma ocean stages. Because of the reducing power of metallic iron, metal-silicate equilibration necessarily implies a reducing environment (of low oxygen fugacity, near the IW buffer) imposed by the coexistence of metallic iron in equilibrium with silicate (Reaction (R2)), thought to significantly reduce the silicate melt (O'Neill 1991; Wade and Wood 2005; Frost and McCammon 2008; Rubie et al. 2011). One may then expect that the magma ocean was initially in such a reduced state, prior to the complete metal segregation to the core. Importantly, the redox state of the upper part of the mantle, where the gases originate, would mainly matter for the oxidation state of contemporary outgassed species.

The segregation of metallic iron through the mantle, as well as the pressure-temperature dependence of metal-silicate partition coefficients (e.g., Wade and Wood 2005) could induce a vertical zonation/gradient of the redox state. However, vigorous convection and mixing of the molten mantle could also buffer metallic iron sinking through Rayleigh-Taylor instabilities and could repeatedly bring reducing iron to shallow depths. In addition, effective convective re-entrainment could slow down iron accumulation at the base of the molten layer (alike crystals re-entrainment; e.g., Solomatov 2015). It could also buffer the formation of metal diapirs large enough to sink to the core, therefore delaying metal-silicate separation. Thus, the upper mantle may have remained in such a reduced state prior to at least advanced planetary differentiation and core formation (e.g., Catling and Claire 2005). Here the metallic iron would have sunk deep enough to no longer affect the upper mantle redox state, and every time the metallic iron of accreting planetesimals would merge into the mantle of the growing proto-planet. Yet, the latter effect strongly relies on the details of metallic iron distribution upon impact, itself influenced by impact parameters and mantle mixing (Deguen et al. 2014; Landeau et al. 2016, 2021; Maas et al. 2021; Itcovitz et al. 2022).

Keppler and Golabek (2019) suggested that, in the case of a highly reduced mantle resulting from metal-silicate equilibration, most of the carbon could be reduced to graphite, which is less dense than a peridotite melt. In such a case, regardless of metal-silicate equilibration depth, as long as such highly reduced conditions prevail somewhere within the magma ocean, graphite flotation and accumulation on top of the magma ocean may have imposed low $f_{O_2}$ conditions in its shallow parts given the CCO buffer ($C + O_2 = CO_2$).
These mechanisms favor a reduced start, but predicting the early oxidation state of the shallow part of the magma ocean and the resulting atmospheric composition remains challenging. In part because linking core formation to the mantle oxidation state is not straightforward. Yet also because core formation itself is complex, and likely results from the interplay of different processes with lots of remaining uncertainties (e.g., Rubie et al. 2015b).

Changes of partition coefficients, phase separation and phase changes (and resulting density contrasts; e.g., Boukaré and Ricard 2017) under extreme and varying $P/T/f_{O_2}$ conditions relevant to magma ocean cooling could induce unknown yet possibly significant feedback that need more investigations. Furthermore, additional complexities regarding the fate of iron in the post-impact mantle and the details of metal-silicate equilibration in a turbulent molten mantle, as well as their influence on the mantle redox state have been poorly studied. Deguen et al. (2014) have shown that the turbulent mixing of a vigorously convecting magma ocean enhances equilibration. By considering the effect of rotation in the convective magma ocean in spherical geometry, Maas et al. (2021) emphasized that metal dispersion and settling also depend on the impactor's target latitude. Citron and Stewart (2022) and Itcovitz et al. (2022) recently showed that the impact parameters strongly influence the distribution, mixing, and availability of the reduced phases of the impactor (e.g., metallic iron) to potentially reduce the atmosphere. They also investigated the interactions between the post-impact atmosphere and the impact-generated melt phase. Importantly, the manner in which iron is accreted by and mixed with the target, and the timing and depth at which metal-silicate equilibration occurs requires further investigation to constrain the oxidation state of the shallow magma ocean layers upon impacts. It would also determine the oxidation state of the outgassed species, and hence the composition of the resulting atmosphere. Because most of the projectile iron (more than 70%) is deposited in the crust and upper mantle, where it is not available to reduce surface water, reducing environments could be less likely to arise after large impacts than previously suggested.

In any case, the fact that the molten mantle must have been in equilibrium with the metallic iron now forming the core, and hence, characterized by low oxygen fugacity, is hard to reconcile with the prevailing early oxidized conditions of the solidified surface, by the time minerals and rocks were formed as described before. From a reduced start, some mantle oxidation mechanisms must then have been in place to resolve this paradox.

**Oxidation Mechanisms from a Reduced Start** To explain the formation of the core in agreement with the abundance of siderophile elements of the mantle and their experimentally-determined partition coefficients, several models have been suggested. These have implications for the oxidation state of the mantle (e.g., Li and Agee 1996; Wood et al. 2006; Fischer et al. 2015). From a reduced start, several oxidation mechanisms have been proposed to increase the oxygen fugacity of the mantle to the QFM buffer (e.g., McCammon 2005b; Wood et al. 2006; Frost et al. 2008; Frost and McCammon 2008; Wood et al. 2009; Schaefer and Elkins-Tanton 2018; Armstrong et al. 2019).

Photo-dissociation of $H_2O$ followed by the escape of $H_2$ from the atmosphere (Hunten 1993) may have caused oxidation of both the atmosphere and mantle by the leftover oxygen (Kasting et al. 1993a; Sharp et al. 2013). This mechanism likely occurred to some extent. The composition of martian meteorites (Dreibus and Wänke 1987) and the elevated abundance of light elements in its core (Khan et al. 2022) suggest that Mars was possibly more volatile-rich than Earth. If hydrogen escape was the main oxidizing mechanism, it is unclear why Mars produced basalts more reduced than MORB and indicative of a relatively more reduced (Herd et al. 2002; Wadhwa 2008) and FeO-rich (Wänke et al. 1988) mantle (at about the IW buffer), despite efficient atmospheric water loss through hydrodynamic escape
(Pepin 1991; Jakosky and Phillips 2001). Furthermore, while Venus likely experienced massive loss of hydrogen, its silicate portion is not significantly more oxidized than the Earth (Schaefer and Fegley 2017).

An alternative solution involves the increase of oxidation state of accreting bodies during late accretion (Wänke et al. 1988; O'Neill 1991; Wood et al. 2009; Rubie et al. 2011, 2015; Fischer et al. 2015; Shi et al. 2022), also referred to as heterogeneous accretion. This allows trace element partitioning models to match the present-day mantle FeO abundance, but fails to predict the current redox state-governing $Fe^{3+}/\Sigma Fe$ (which is the proportion of the Fe that is ferric) and the final oxygen fugacities of these models is generally still too low (typically at IW-2, which is about $5 \log f_{O_2}$ units below the Earth's present-day value; e.g., Schaefer and Fegley 2017; Schaefer and Elkins-Tanton 2018).

Another possibility is that the mantle self-oxidized as a result of perovskite crystallization (Mao and Bell 1977; Frost et al. 2004; Wade and Wood 2005; Galimov 2005; Frost et al. 2008; Frost and McCammon 2008; Hirschmann 2012; Armstrong et al. 2019; Hirschmann 2022). At depth, the growth of silicate perovskite (bridgmanite), the dominant lower-mantle mineral, forced the disproportionation<fnref n="6" /> of ferrous iron ($Fe^{2+}$) into ferric iron ($Fe^{3+}$) (dissolved in perovskite; McCammon 1997) plus iron metal (segregated to the core; Frost et al. 2004): $3FeO \rightarrow Fe^{0} + Fe_{2}O_{3}$ (or $4FeO \rightarrow Fe^{0} + Fe_{3}O_{4}$). Repeated crystallisation and dissolution of $Fe^{3+}$-containing perovskite at the base of the magma ocean acted as an “oxygen pump” (Wade and Wood 2005). This raised the oxygen fugacity of the growing mantle by releasing ferric ($Fe^{3+}$) iron to the magma ocean, while the produced reduced metallic iron segregated to the core (e.g., Armstrong et al. 2019). Infalling metallic iron from subsequent impactors would have been oxidized by the ferric iron, therefore raising the FeO content of the silicate magma and the oxygen fugacity of core separation. Mantle convection and homogenization would further raise the oxygen fugacity of the upper mantle to its present-day value toward the end of core formation and by the time of Hadean zircon formation (e.g., McCammon 2005a; Frost and McCammon 2008). This self-oxidation process relies on the ability of Earth's mantle to stabilize substantial amounts of perovskite at depth. It is essentially planetary size-dependent and could have thus operated similarly on Venus (assuming it experienced a deep enough magma ocean) but not on Mars (e.g., Wood et al. 2006). Note that Schaefer and Elkins-Tanton (2018) proposed that the disproportionation of $Fe^{2+}$ would occur directly in the silicate melt phase in the magma ocean during equilibration with the sinking core-forming metal delivered by impactors, rather than during crystallization (see also Hirschmann 2022). This alternative scenario allows immediate separation of the metallic liquid, leaving the oxidized mantle material behind without requiring additional mixing nor complex scenarios of repeated crystallisation and melting of bridgmanite.

Thus, despite a favored early reduced start for the magma ocean, these oxidation mechanisms may have brought the molten mantle to an oxidized state before its complete solidification. If the planet is still molten once core formation is complete and the planet has a differentiated Fe metal and FeS core, Lupu et al. (2014) showed that the equilibrium of a molten surface of either bulk silicate Earth or continental crust composition results in the formation of $H_{2}O-CO_{2}$-dominated (i.e., oxidized) atmospheres. This corresponds to a case where metal-silicate segregation occurs before the last magma ocean episode or rapidly compared to its solidification.

Finally, an increasing amount of evidence has suggested that magma ocean evolution itself might be the determining factor in the evolution of the oxidation state of the magma

<footnote n="6">⁶A disproportionation reaction is a redox reaction in which the initial material of intermediate oxidation state undergoes oxidation as well as reduction and converts simultaneously to two compounds, one more oxidized and one more reduced.</footnote>
ocean–atmosphere system (Armstrong et al. 2019; Solomatova and Caracas 2021). It may also reconcile the two favored end-members: a reduced start, owing to metal–silicate equilibration, and an oxidized end, as demonstrated by geological records. For instance, Maurice et al. (2023) suggested that the crystallization sequence of the magma ocean itself could be an oxidizing process, because of the incompatibility of ferric iron. This would imply that the shift towards an oxidized atmosphere would be a systematic outcome of the magma ocean evolution.

**Indirect Evidence of Magma Ocean Redox State** Different attempts recently aimed at assessing the oxidation state of the molten mantle prior to the aforementioned available direct or indirect constraints (Pahlevan et al. 2019; Deng et al. 2020; Thompson et al. 2021; Solomatova and Caracas 2021). Thompson et al. (2021) recently conducted laboratory investigations of meteorite outgassing on CM-type carbonaceous chondrites (i.e., undifferentiated), as representative samples of the bulk composition of material in the solar nebula. This was in order to assess the composition of resulting atmospheres. However, one should note that their experimental setup and procedure is more representative of outgassing from the molten interior of a planet (of bulk chondritic composition) rather than impact-induced degassing as modeled in the aforementioned studies. Indeed, by simply heating chondrites and measuring the abundances of released volatiles, they reproduce the conditions expected for the outgassing from a molten planet and resulting secondary atmosphere formation. Here shock-induced devolatilization does not occur. Yet, devolatilization along with atmospheric entry and structural changes upon impact most likely altered the chondrite that was sampled and its overall representativeness. They found that water would makeup most of such an outgassed atmosphere (66%), followed by CO (18%) and CO₂ (16%), thus producing a rather oxidizing atmosphere (Thompson et al. 2021).

Pahlevan et al. (2019) used hydrogen isotopic records (the deuterium content) of Earth's hydrosphere to independently assess the oxygen fugacity of terrestrial magma ocean outgassing. They emphasized that the preservation of a carbonaceous chondritic (i.e., undifferentiated) D/H signature in the terrestrial oceans requires a hydrogen-poor (relative to water, i.e., H₂/H₂O < 0.3) Earth's outgassed envelope. This is indicative of oxidizing conditions (log fO₂ > IW + 1) for last atmospheric equilibration with the magma ocean. In agreement with the earliest geological record (Trail et al. 2011), they concluded that oxidation of the silicate Earth occurred during the crystallization of the (last) magma ocean, and may not require later geological processes such as subduction (and associated oxidation through slab dehydration for instance; e.g., Wood et al. 1990; Kasting et al. 1993a) to reach the oxidized present-day state.

Earth and chondrites similar volatile isotopic compositions suggest that the source of terrestrial volatiles is chiefly chondritic (Marty 2012). Yet, the elemental composition of the bulk silicate Earth is depleted in C, N, and H relative to chondritic material. Accretion of some non-chondritic materials has thus been proposed to account for this depletion (Hirschmann 2016). Sakuraba et al. (2021) modeled the evolution of volatile abundances during the accretion by considering elemental partitioning, including the effect of the magma ocean redox state on volatile solubility and metal–silicate partition coefficient, and impact erosion. They found that the BSE depletion pattern can be reproduced from continuous accretion of chondritic bodies only, without the need of non-chondritic materials, but requires a relatively oxidized magma ocean (log fO₂ ≈ IW + 1). In good agreement with Pahlevan et al. (2019), their results indicate a relatively oxidizing magma ocean at the end of the accretion stage.
Based on first-principles molecular dynamics calculations, Solomatova and Caracas (2021) showed that during the first stages of magma ocean evolution, C is rapidly de-volatilized (in the form of CO₂). Yet H remains dissolved in the magma for longer and is volatilized (as H₂O) during later magma ocean stages. This would have resulted in the out-gassing of a CO₂-rich and H₂O-depleted atmosphere during the early stages of the magma ocean. H₂O would have been outgassed later on with solidification, thus supporting the clas-sically modeled outgassing sequence of coupled magma ocean–multi-species atmosphere models (e.g., Elkins-Tanton 2008; Lebrun et al. 2013; Salvador et al. 2017; Nikolaou et al. 2019; Bower et al. 2019). They also suggested that, despite the uncertain oxidation state after a giant impact, the amount of oxygen available in the system influences the speciation of the vaporized carbon. Hence more oxidized systems favor the release of more oxidized vapor species, i.e., CO₂ outgassing rather than CO. The composition of the melt itself also affects the composition of the vapor phase. In addition to increasing with the oxidation state, they demonstrated that the relative proportion of CO₂ to CO increases with decreasing den-sity and temperature, thus suggesting that the abundance of atmospheric CO₂ increased with time and magma ocean cooling.

Deng et al. (2020) combined first-principle molecular dynamics simulations with thermo-dynamic modeling at relevant P/T conditions. They studied the redox controlling reactions in magma oceans. In agreement with Armstrong et al. (2019) high pressure experiments, they found that compared to ferrous iron (Fe²⁺), ferric iron (Fe³⁺) becomes increasingly energetically favorable with pressure because of its small partial molar volume in silicate melts under large compression (Fe³⁺-bearing melts are more compressible). In a convec-tively homogenized magma ocean, this would produce a vertical gradient in oxygen fugac-ity (as proposed by Hirschmann 2012). Here the upper part being relatively more oxidized than the deeper part where redox values were taken to be representative of local equilibrium with reduced molten iron ponds, i.e., assuming that core–mantle equilibration occurred at the bottom of a relatively deep magma ocean (consistent with metal–silicate partitioning experiments; e.g., Li and Agee 1996; Fischer et al. 2015). Yet, before gradually decreasing downwards throughout the rest of the mantle, the relative oxygen fugacity first increases slightly with pressure in the uppermost mantle. The increasing trend was also reported pre-viously in experiments establishing the effect of pressure on the Fe³⁺/ΣFe ratio of silicate melts (Zhang et al. 2017). Yet, as Zhang et al. (2017) experiments were restricted to rela-tively shallow depths (up to 7 GPa), they concluded that oxygen fugacities at the surface of shallow magma oceans are more reduced than at depth. They also found that when ex-trapolated to higher pressures relevant for magma oceans on Mars- or Earth-/Venus-sized planets, atmospheres overlying magma oceans should be highly reduced and rich in H₂ and CO. Importantly, Deng et al. (2020) showed that this trend actually reverses at larger depth and emphasized that an oxidized upper mantle is a natural consequence of a magma ocean due to the increasing stability of ferric (oxidized) iron with depth, even at relatively reduced conditions, which raises the Fe³⁺/ΣFe ratio of silicate melts in equilibrium with metal al-loy. By calculating the speciation of volatiles to determine the composition of an outgassed atmosphere at chemical equilibrium with the underlying magma ocean, they found that the resulting early atmosphere of Earth-/Venus-sized planets (i.e., with deep magma oceans) should be enriched in H₂O (~70 mol%) and CO₂ (~15 mol%), and therefore, be highly oxidized.

However, Righter and Ghiorso (2012) used a different approach to estimate $f_{O_2}$ and de-spite agreeing with a similar vertical zonation in magma oceans (from shallow oxidized portions to deep reduced portions), they argued that metal–silicate experiments would bias $f_{O_2}$ estimations such that none of the magma ocean part would be more oxidized than the
IW buffer. Conversely to what is assumed in accretion models described previously, increasing the FeO content of planetary mantles during accretion may not lead to oxidation. It would rather lead to significant reduction relative to the IW buffer, therefore decreasing $f_{O_2}$ from high to low during accretion. This view indicates that further investigations are needed but remains challenged by the apparent prevalence of oxidized conditions indicated in rock records. It also contrasts with the studies mentioned above (Pahlevan et al. 2019; Deng et al. 2020; Thompson et al. 2021; Solomatova and Caracas 2021), all suggesting that the degassing of the upper magma ocean produced oxidized species and was thus already in a relatively oxidized state early on.

**Influence of the Magma Ocean Redox State on the Thermo-Chemical Evolution** Because of the uncertainties regarding the magma ocean redox state, it is important to assess its influence on the thermo-chemical evolution of the magma ocean-atmosphere system. This has been the focus of recent studies that simulated the solidification and/or outgassing of the magma ocean while considering $f_{O_2}$ as a free parameter, thus considering both reduced and oxidized conditions (Katyal et al. 2020; Bower et al. 2022; Gaillard et al. 2022a; Maurice et al. 2023).

Katyal et al. (2020) investigated the effect of mantle oxidation state on volatile outgassing and chemical speciation at the surface at specific conditions relevant to different steps of magma ocean evolution for various redox states. They showed that the oxidation state of the magma ocean, through the composition of the outgassed atmosphere, affects the atmospheric thermal opacity, surface pressure, and scale height, and thus, ultimately, the cooling rate of the magma ocean. Because of its relatively low mean molecular weight, a H₂/CO-dominated reduced atmosphere outgassed from a reduced magma ocean has a larger scale height than a H₂O/CO₂-dominated oxidized atmosphere (outgassed from a oxidized mantle, which is consistent with experimental determinations; e.g., Grewal et al. 2020) containing heavier species. The higher thermal blanketing effect of the latter dense and small scale height atmosphere made of efficient greenhouse gases is expected to slow down the cooling of the magma ocean. Conversely, a reduced mantle surrounded by reduced outgassed species is expected to cool down faster. Yet, the greenhouse effect of reduced species is far from being negligible. In particular for dense atmospheres, the collision-induced absorption (CIA) of reduced species such as H₂ and CH₄ can extend the outer edge of the habitable zone (Pierrehumbert and Gaidos 2011; Ramirez and Kaltenegger 2017, 2018) and maintain ancient Mars temperate (Ramirez et al. 2014; Wordsworth et al. 2017; Turbet et al. 2019, 2020; Godin et al. 2020). Note that both the atmospheric thermal opacity and scale height affect the thermal emission and transmission spectra that could be used to infer the oxidation state of an extrasolar planetary interior and constrain its evolution with time.

Bower et al. (2022) reached similar conclusions while considering additional C/H ratios and hydrogen budgets. As shown in earlier coupled magma ocean-atmosphere studies for oxidized species (Salvador et al. 2017; Nikolaou et al. 2019; Bower et al. 2019), because of their larger solubility in silicate melts the H-bearing species are outgassed later than C-bearing ones. This also applies to reduced conditions. Because of the combined influence of volatile solubility and redox reactions, Bower et al. (2022) emphasized that an atmosphere may evolve from dominantly CO-rich and reducing to H₂O-rich and oxidizing with cooling. They further suggested that the mode of crystallization (i.e., equilibrium versus fractional) would induce additional complexity. For example, the formation of an early surface lid (at around 30% of remaining melt), as a result of faster magma ocean freezing under equilibrium crystallization, could delay or even prevent subsequent outgassing of highly soluble volatile species (such as water).