Tidal pull of the Earth strips the proto-Moon of its volatiles

While it is widely accepted that the Moon formed after a giant impact, the mechanisms by which it acquired its current physical state and composition remain debated (Canup, 2004; Ćuk et al., 2016; Nakajima and Stevenson, 2014; Lock et al., 2018; Cameron and Benz, 1991). Lunar mare basalts have refractory lithophile element ratios akin to terrestrial basalts but are invariably depleted in volatile and moderately volatile elements by a factor of $\sim 5$ (for alkali metals) to $\sim 50$ (highly volatile elements like Zn, Ag and Cd) relative to their terrestrial counterparts (Wolf and Anders, 1980; Ringwood et al., 1987; O’Neill, 1991; Lodders and Fegley, 2011; Day and Moynier, 2014). This depletion occurs in concert with enrichment in the heavy isotopes of moderately volatile elements including Zn (Paniello et al., 2012; Kato et al., 2015), K (Wang and Jacobsen, 2016; Tian et al., 2020), Cl (Sharp et al., 2007; Boyce et al., 2015; Gargano et al., 2020), Ga (Kato and Moynier, 2017), and Rb (Pringle and Moynier, 2017), indicative of an evaporative origin for volatile element loss from the Moon. Conversely, the enrichment of the lighter isotopes of Cr in lunar magmatic rocks compared to Earth’s mantle suggests devolatilisation occurred near equilibrium and under low temperatures ($\leq$ 1800 K) (Sossi et al., 2018), far below those expected during a giant impact event.

Even under the extreme temperatures reached as a result of a giant impact (from $\sim 2000$ to $\sim 10000$ K Canup (2004); Ćuk et al. (2016); Nakajima and Stevenson (2014), the removal of volatile elements from the Earth-Moon system by thermal escape is inhibited because of (i) the Earth’s substantial gravitational field and (ii) the high molecular masses of the moderately volatile elements ($\geq 20$ g/mol). Both cases are aggravated by the low abundance of hydrogen in the proto-Earth, preventing loss through aerodynamic drag (Nakajima and Stevenson, 2018). As such, recent models attempt to devolatilize Moon material prior to, or during, its assembly close to the Roche Limit within the post-impact debris disk (the ”protolunar disk”, a partially- to fully- vaporized disk). In the protolunar disk, volatile depletion by incomplete condensation and liquid-vapour partitioning may result from viscous separation of vapour from liquid (Charnoz and Michaut, 2015), or gravitational torque from the proto-Moon exerted on the vapour (Canup et al., 2015) or separation of growing melt droplets from the vapour due to gas-drag (Lock et al., 2018). However, these scenarios are critically dependent on the structure of the protolunar disk, which remains poorly understood, mostly due to its multi-phase nature and the unknown mixing states between gas and condensed phases (Thompson and Stevenson, 1988; Machida and Abe, 2004; Ward, 2012; Gammie et al., 2016; Charnoz and Michaut, 2015; Salmon and Canup, 2012; Lock et al., 2018). In some of these models, the protolunar disk is contained within the Earth’s Roche Limit, whereas numerical simulations of giant impacts show that most material ($>70\%$ by mass) is ejected beyond the Roche Limit (Nakajima and Stevenson, 2014) permitting volatile depletion beyond this point. The Roche Limit is the distance within which orbiting material cannot accrete into a single body due to disruption by the planets’ tides and is located at 3 Earth radii for silicate material orbiting the Earth.

Here, we present an alternative scenario for atmospheric loss from the proto-Moon assembling at Earth’s Roche limit. It is underpinned by the result that the influence of Earth’s tides lower the kinetic energy required for a gas to thermally escape from the proto-Moon (see next section), relative to that for the Moon considered in isolation in space. As a conceptual investigation of this process, model approximations were made in order to illustrate the first-order influence of tidally-assisted atmospheric escape on the chemical composition of the Moon. For all realistic model parameters, intense hydrodynamic escape is found to be inevitable from the proto-Moon, or its building blocks, provided it accreted at a distance of a few Earth’s radii such that Earth’s tides were sufficiently strong to lower the Moon’s escape velocity. We describe tidally-assisted thermal escape and show that, if the gas at the surface of the proto-Moon is hot enough to accelerate upwards to its Lagrange points, then this gas would be lost from the Moon, either by ejection into space, or by injection into orbit around the Earth. We explore two end-member scenarios, a ’wet’ and a ’dry’ model of the proto-lunar atmosphere, and show that, in both cases, the gas accelerates sufficiently through tidally-assisted thermal expansion to be able to escape. The gas velocity slows as the Moon’s orbital expansion causes the escaping flux to diminish. Over time, atmospheric loss via hydrodynamic escape produces a net depletion in the volatile element abundances in the Moon, consistent with measurements of lunar mare basalts.

The paper is organized as follows. In section 2 we examine, using simple arguments, under what conditions material may escape from the proto-Moon surface under the effect of tides. In section 3 we investigate two models of hydrodynamic escape assisted by tides and show that under all reasonable assumptions, escape is inevitable. In section 4 we discuss the orbital evolution of the proto-Moon, which sets an upper time limit for hydrodynamic escape to occur. In section 5 we compute the composition of the exsolved gas and the resulting melt composition. We show that it is possible to explain Na and K abundances in lunar records. We discuss our results and conclude in section 6.

Models of lunar assembly suggest that the Moon’s building blocks coalesce just beyond the Roche Limit (Canup et al., 2015; Takeda and Ida, 2001). Traditionally the Roche Limit for a fluid body $R_{\rm R}$, is (Roche, 1873; Canup and Esposito, 1995) :

where $R_{\rm p}$ is the planet radius, $\rho_{p}$ is the planet density, and $\rho_{\rm m}$ is the orbiting material density. For the case of the Earth, $\rho_{p}=5500~{\rm kg/m}^{3}$, and assuming the orbiting material has the same average density as the present-day Moon, we choose $\rho_{\rm m}=3300~{\rm kg/m}^{3}$. These numbers give $R_{\rm R}\simeq 3R_{\oplus}$. Aggregates can begin to form at the edge of the proto-lunar disk around $3R_{\oplus}$ but the fully-grown Moon might appear further away, around $5R_{\oplus}$ (Canup et al., 2015; Salmon and Canup, 2012). The Moon’s gravitational attraction sphere is called the Hill Sphere and its radius is $R_{\rm h}=a(M_{\rm m}/3M_{\oplus})^{1/3}$ ($a$ is the Earth-Moon distance, $M_{\rm m}$ and $M_{\oplus}$ are the masses of the Moon and Earth, respectively). We assume that the proto-Moon is hot and is surrounded by gas evaporated from its surface. If the proto-Moon’s atmosphere is contained within its Hill sphere, then it may be hydrostatically stable, like it is for the present-day atmospheres of the terrestrial planets. However, when the proto-Moon (or its building blocks) is located close to the Roche Limit, its Hill sphere has a size comparable to its physical radius ($R_{\rm h}\simeq 2500$ km for a Moon located at $3R_{\oplus}$), meaning the atmosphere is hydrostatically unstable, and thereby liable to escape. Specifically, the Earth’s tidal pull lowers the minimum energy required for a particle to escape the proto-Moon’s surface (relative to the case for the Moon considered in isolation). The computation of the potential energy at the Moon’s surface, under Earth’s tidal field is detailed in Appendix A. As the L1 and L2 Lagrange points are the points on the Hill’s sphere closest to the Moon’s surface (Appendix A), escape of the gas is most readily achieved via passage through L1 and L2, because it requires the least energy 1). The kinetic energy required can be converted into a gas temperature 2. For this fiducial case, we assume a molar mass equal to $20$ g/mol, as a proxy for an atmosphere consisting of sodium Visscher and Fegley (2013)) . A well-mixed, static and transparent atmosphere in thermal equilibrium with the Moon’s hot surface at temperature $T_{s}$ (which is treated as a free parameter) is assumed, for simplicity, to produce a temperature profile that scales with $T_{s}(r/R_{m})^{-1/2}$, where $r$ is the distance to the Moon’s center and $R_{\rm m}$ is the radius of the Moon (red lines in Figure 2). This scaling is simply obtained by equalizing the cooling power of a gas parcel, $P_{cool}=\sigma T^{4}$ to the radiative power of a hot sphere at at distance $r$ : $P_{rad}=L/(4\pi r^{2})$ where L is the luminosity of the hot sphere (this is, for example, a common scaling in passive protoplanetary disk studies).

Under these simplistic assumptions, we find that, if $T_{s}>1600K$ the atmosphere is hot enough to escape beyond the Hill sphere when the proto-Moon is located at distance $\lesssim 5R_{\oplus}$. Moreover a smaller proto-Moon (0.5 lunar mass), or its constituents, are more liable to atmospheric loss under the same conditions at same surface temperature (Figure2. b).

Whether the escaping atmosphere is permanently lost hinges upon the dynamics of the material after leaving the Moon’s Hill sphere. Classical celestial mechanics (Murray and Dermott, 1999) constrain outgoing particles to pass through either the L1 or L2 Lagrange points before being ejected into orbits around the Earth. The fate of this escaping material depends on the nature of the physical processes acting upon it. If the escaping material is gaseous, it does not re-accrete onto the Moon. That is, any dissipative process like gas viscosity or radiative cooling prevents the gas from re-entering the Hill Sphere because of a loss of mechanical energy. This is illustrated in Figure 3 (left) and Figure 4 where trajectories of particles leaving the Moon in a fictive gaseous disk were tracked and computed. It is observed that they do not fall-back onto the Moon, but rather migrate inward or outward, leaving the Moon permanently.

Alternatively, if the Earth’s surface is sufficiently hot during assembly of the proto-Moon ($>3000$ K) (Canup, 2004; Ćuk et al., 2016; Nakajima and Stevenson, 2014) with a photosphere around 2000 K, then black-body emission may induce radiation pressure on micrometer-sized particles (in addition to heating the near-side of the proto-Moon). This leads to their ejection from the Earth-Moon system (Hyodo et al., 2018) (Figure 3, right and Figure 5). This scenario of ejection strongly depends on the physical size of dusty-grains. Here it is noted that escape is efficient for particles in the micrometer range from a Moon surface heated to $>$ 1500K following a giant impact (Appendix B).

We conclude from the first-order considerations detailed above that it is reasonable to expect that tidal effects would (1) facilitate the escape of material from the Moon’s surface and (2) prevent its return to the lunar surface due to 3 body-effects. Although the above-mentioned scenarios (dissipative gas disk, radiation pressure) may prevent the return of escaping material onto the Moon’s surface, the ”bottleneck” is to understand how material can be transported from the proto-Moon’s surface (i.e., the locus of its evaporation) up to the L1/L2 Lagrange points at which this material can escape.

Mechanisms capable of carrying gas from the molten proto-lunar surface to the escape points are a product of the thermochemical structure of its atmosphere. We investigate the mode of atmospheric escape occurring under the influence of the tidal pull of the Earth, and derive expressions that permit calculation of the escaping flux. To do so, we formulate two end-member scenarios using a 1D approach and take into account (1) evaporation from a magma ocean at surface temperature $T_{\rm s}$ , and (2) the mutual gravitational attraction between the Earth and the Moon. Contrary to the abundant literature on planetary atmospheric escape, the atmosphere we consider here is not heated from above by an incoming XUV flux (see SI L), but rather from its base by a hot magma ocean, considered to be at constant temperature $T_{\rm s}$. In the first ”dry” model we assume a purely adiabatic gas evaporated from the magma ocean and neglect condensation during its adiabatic expansion (Section 3.1). The second ”wet” model assumes that individual species comprising the atmosphere may condense as liquid droplets depending on its pressure-temperature structure, and can be subsequently dragged along with the prevailing outflow (Section 3.2). Each of these scenarios involves simplifying assumptions to render calculations tractable, and the reality of atmospheric escape probably lies in between these two extremes. However, we use these two end-member scenarios as a proof-of-concept that tidally-driven thermal degassing is unavoidable under all reasonable assumptions, provided the proto-Moon assembles close to the Roche Limit.

In the following, it is assumed that the proto-Moon (or its building blocks) is surrounded by an atmosphere. The surface of the proto-Moon is assumed to be always liquid, and, in contact with the gas. We examine under which conditions the atmosphere can escape from its Hill sphere by solving the hydrodynamic equations, and considering the Earth’s tidal force.

The tidal evolution of the early Moon is a matter of debate. Here we consider a simplified approach, where the Moon’s semi-major axis, $a$, depends only on the Earth’s $k_{2e}/Q_{e}$, where $k_{2e}$ is the Earth’s Love Number, and $Q_{e}$ is the Earth tidal dissipation quality factor. Both numbers are unknown for a molten Earth after the lunar giant impact. For simplicity, we neglect the Moon’s eccentricity, so that its orbital evolution is driven solely by the Earth’s dissipation. We solve the time evolution of the early Moon following (see e.g. Goldreich and Soter (1966)):

The resulting semi-major axis evolution of the Moon is reported in figure Fig 13. We assume that the Moon starts at the Earth’s Roche limit. Several values of $Q_{e}$ are explored from 10 to 1000, as in Ćuk et al. (2016); Chen and Nimmo (2016), noting that the present-day value is $Q_{e}\simeq 10$ ((Goldreich and Soter, 1966)). The $Q$ of a molten Earth is not known, but is expected to have been larger than today’s $Q$, because of lower internal dissipation.

As the Moon’s orbit expands, so does its Hill sphere, making hydrodynamic escape increasingly difficult and thereby inducing a decline of the escaping mass flux. Atmospheric escape shuts down between 3.5 to 5 $R_{\oplus}$ for the dry adiabatic regime for surface temperatures between 1400 K and 2200 K (Figure 7 and Figure 8). For the wet model, the escaping flux is still about $10^{-7}$ Moon masses/year, for surface temperatures between $\sim 1600$ K and $\sim 1800$ K in the same distance range. For conservative estimates of an entirely liquid Earth, its tidal dissipation factor is large ($Q>100$) and the Moon reaches $7.5R_{\oplus}$ in $10^{4-5}$ years. Moon-disk interaction shortens this timescale by only 1000 years (Salmon and Canup, 2012). This timescale is sufficient to allow atmospheric mass loss on the order of a percent of the Moon’s mass. We note also that, as the Moon’s orbit expands (or, alternatively, as the lunar surface cools down), degassing could switch to a Jeans escape rather than a hydrodynamic escape regime. Indeed the Knudsen number, $Kn$ (the molecular mean-free path divided by the atmosphere scale height; Jeans escape occurs for $Kn>1$), is found to increase as the Moon’s orbit expands and temperature drops (Figure 10, left column, third line). Therefore, while hydrodynamic escape would cease at larger distances, some residual atmospheric loss via Jeans escape, particularly for lighter molecules, may still occur. The escape process is, however, more efficient for an isothermal atmosphere. In the end-member case in which the atmospheric temperature is kept constant at the Moon’s surface temperature is displayed in Appendix E. Here the net escape flux varies smoothly with distance and is comparable to fluxes in the adiabatic case for an Earth-Moon distance about 3$R_{\oplus}$, that is, a very high value. The absence of a radiative transfer code to treat opacity increases due to droplet formation, prevents an accurate treatment of the thermal profile at this stage. In reality, a scenario falling in between the two end-member models adiabatic (lowest flux) and isothermal (highest flux) models is likely to have occurred.

Combining the low temperature range implied by isotopic data with best fits to the observed depletion factors of Na, K and Zn at $T=1600-1800K$, we find that between 0.1% and 0.6% of a lunar mass must have been evaporated and lost to explain its present composition. Whereas higher surface temperatures promote higher escaping fluxes, Moon orbital expansion leads to a decrease of flux distance. The rate of lunar orbital expansion increases with decreasing tidal $Q$ parameter, which is unknown for a molten Moon. So to explain the observed K and Na abundances in the Moon, both surface temperatures and Q must be sufficiently high to allow escape for an extended period of time. Solidification of the magma’s ocean surface also acts to dampen escape, as its solidification may occur in about $\sim 1Kyr$. To consider each possibility simultaneously, we have performed coupled tidal-evolution (with different Q) and evaporation simulations (using the wet model with constant surface temperature). In a first set of runs we assume that the magma ocean never solidifies, and that only the effect of the increasing Earth-Moon distance with time arrests hydrodynamic escape. The simulation ran for $10^{5}$ years, so that the tidal evolution of the Moon would always result in a distance sufficiently far from the Earth to shut down hydrodynamic escape. The final abundance of Na and K resulting from equilibrium or fractional evaporation in the Moon at the end of the run are then compared to observations. We find that the only combinations of (T,Q) that can simultaneously match lunar K and Na range from (T=1580K, Q=215) to (T=1690K, Q=46) for fractional vaporisation, whereas equilibrium vaporisation never leads to a satisfactory match within the allowed ranges. Within these ranges of T and Q, the resulting lunar abundances range from $\sim$ 250 to $400\mu g/g$ for Na and from $\sim$ 30 to $50\mu g/g$ for K. All other runs results either in too much or too little mass loss of Na, or K or both.

Finally, we consider an alternative case where the formation of a stagnant lid at the surface of the magma ocean abruptly shuts down escape. Indeed, while it may take about 200 Myrs for the complete solidification of the lunar magma ocean (Meyer et al., 2010; Elkins-Tanton et al., 2011), crystallisation of a plagioclase lid may occur over much shorter timescales, around 1000 years (Elkins-Tanton et al., 2011). Using our simulation with constant surface temperature, we compute the time at which the stagnant lid must form for resulting in Na and K abundances compatible with lunar measurement (Figure 19). We find a wider range of solution in terms of Q, but still a narrow temperature range. Keeping only solutions for which the stagnant lid forms >500 years and <3000 years (Meyer et al., 2010; Elkins-Tanton et al., 2011; Tang and Young, 2020) we find $100<Q<1000$ and $1600<T_{surf}<1700$. The remainder of mantle cooling under a conductive anorthositic lid may take $>100$ Myrs because of Earth-Moon tidal heating (Meyer et al., 2010; Elkins-Tanton et al., 2011) by which time the Moon would have migrated beyond any proximal $R_{\oplus}$ values, rendering loss by hydrodynamic escape ineffective.

In summary, independent of whether the Moon’s expansion or the formation of stagnant lid induce a shuts down in hydrodynamic escape, we find that the surface temperature must be close to $1600-1700K$ to explain the Na and K abundances, with $40<Q<300$. Interestingly, this range of temperatures is also compatible with the chromium isotopic fractionation observed in lunar basalts, if such an isotopic signature were due to the loss of $CrO_{2}$ vapour in equilibrium with the melt Sossi et al. (2018). It is therefore remarkable that a variety of independent observations and constraints (isotopic fractionation, amount of mass evaporated to reproduce K and Na, formation timescales of the stagnant lid and the tidal Q factor) all converge upon the conclusion that the surface temperature of during evaporative loss was in the range 1600-1700K. The ostensible coherence of these results may indicate that tidally-driven hydrodynamic escape is a key process governing the loss of volatile elements from the Moon.

Our calculated lunar loss rates, between $10^{-5}$ to $10^{-7}$ lunar masses/yr, are on the upper end- or higher than those presented in Tang and Young (2020) (TY20 hereafter), $7\times 10^{-8}$ lunar masses/year. On this basis, TY20 claim that hydrodynamic escape from the early Moon is insufficient to account for neither the chemical depletion of moderately volatile elements, nor their isotopic fractionation in the Moon. The discrepancy in loss rates arises due to the very different physical descriptions of the atmospheric structure imposed. In the present work, the escaping atmosphere is described as a single layer, globally accelerated upward. In detail, the gas begins to accelerate away from just above the liquid surface (z = 0), with an upward-directed velocity at z = 0. The atmospheric structure is then calculated self-consistently using Euler’s hydrodynamic equation, but neglecting radiative transfer to make the computation tractable.
In contrast, in TY20, the atmospheric structure is assumed to be stratified just above the surface, inspired by results on hydrostatic atmospheres (Lebrun et al. (2013)). In their case the atmospheric structure is not resolved as a single entity, rather, escaping fluxes are computed for each layer individually and balanced to determine a steady-state. Just above the magma ocean there is a hydrostatic layer, separated from an isothermal, hydrodynamically-escaping (i.e., non-hydrostatic) layer above it by the homopause. In TY20 hydrodynamic escape takes places above the hydrostatic layer, which means that the advective flow is zero at the surface of the magma ocean. TY20 state (correctly) that the net escaping flux at the transonic point ($F_{E}$) must be equal to the free evaporative flux at the liquid’s surface ($F_{F}$) minus the return flux to the liquid ($F_{R}$) (see our Appendix D for more details). To compute the return flux down to magma ocean they use the formalism of R02 (their Equation 6 precisely). It describes the transport of the evaporating species by molecular diffusion within a globally static surrounding gas (the R02 equation is simply Fick’s law in spherical coordinates). In the R02 approach the net evaporative flux is inversely proportional to the gas pressure, meaning transport is more efficient when the pressure is low. When the diffusive transport flux from the hydrostatic layer is fast enough (proportional to 1/P) to balance the escape flux by hydrodynamic escape at the transonic point (proportional to P) the ’cross-over pressure’ at the homopause (which is at a height above the melt surface given by that at which the adiabatic temperature in the diffusive layer declines to the skin temperature calculated by $2^{-Cp/(4R)}$) is obtained, which is equal to $10^{-8}$ bar in TY20. This value is far lower than that calculated for a BSE-derived gas at 2000 K ($10^{-3}$ bar) and hence results in a significantly lower escaping flux.
Our hypothesis differs in that we assume that the atmosphere starts to accelerate at the magma ocean surface (there is no hydrostatic layer), so that, in our study, the vapour is advected rather than diffused in the thin layer just above the magma ocean. Advective transport allows the density of the evaporating species to remain high because there is no need for a concentration gradient to sustain the flux - the flux is simply sustained by inertia. Our return flux, which is computed while taking advection into account, is detailed in Appendix D by using an approach very similar to the original Hertz-Knudsen calculation. The effective pressure at the surface is given by Equation D14, and only depends on the ratio of the velocity at the surface to the thermal velocity ($V_{s}/V_{t}$). We show that when the surface velocity is small compared to the thermal velocity, the pressure above the surface remains very close to $P_{sat}$ (about $10^{-3}$ bar, depending on temperature, see 14) while allowing a substantial net escaping flux. Thus, the surface pressure is coupled to the velocity at the surface through Equation D14. The pressure at the transonic point is then obtained by solving Euler’s hydrodynamic equations (as we do in both the ’wet’ and ’dry’ models) that describe advective transport. In our dry adiabatic model, these two equations are fully coupled, and the flux is perfectly conserved (including the return flux to the magma ocean). In our wet model this coupling could not be done because of computer limitations, but we show that the error on the flux is small ($20\%$ in the worst case, and $<1\%$ in general) because of small surface velocities compared to thermal velocities. At the transonic point, we calculate pressures between $10^{-2}$ and $10^{2}$ Pa ($10^{-7}$ and $10^{-3}$ bar), that are much larger than $10^{-7}$ Pa ($10^{-12}$ bar) obtained in TY20, which results in a much higher escaping flux. Nevertheless, solutions that are consistent with the lack of isotopic fractionation of moderately volatile elements in the Moon require near-equilibrium values of $P_{s}$ at the surface, and relatively low temperatures, such that total loss rates were likely no higher than $10^{-7}$ lunar masses/yr for the ’dry’ model and $10^{-5}$ lunar masses/yr for the ’wet’ model. Such loss rates, combined with the total lunar mass loss required to explain the Na and K depletion (0.1 to 0.6 $\%$) suggest that hydrodynamic escape in the tidal field of the Earth is a plausible mechanism for atmospheric loss from the early Moon over $<1$ kyr timescales.

The main uncertainty in the present work is the temperature profile of the escaping gas. In order to answer this question, the next step would be to couple a model for radiative transfer and a physical model for droplet growth to the existing chemical and hydrodynamic model. In the current work, we have considered two end-member temperature profiles: adiabatic and isothermal, with or without condensation (wet vs. dry). We have shown that in all cases considered, even for the ”coolest” model (adiabatic expansion for T >250 K and isothermal for T <250 K) hydrodynamic escape always occurs (it is stronger in the isothermal case than in the adiabatic case).

Assuming that the acceleration of the atmosphere starts at the surface of the magma ocean, we conclude that tidally-assisted hydrodynamic escape from an evaporating proto-Moon is an unavoidable outcome of lunar assembly and can explain the measured lunar abundances of K and Na (at least). However, under the oxygen fugacities presumed here, for any reasonable losses of Na and K, Zn is entirely evaporated, indicating either different accretion histories for the moderately- and highly-volatile elements, or different thermodynamic conditions than those modelled here. Although several simplifications were made in order to understand the magnitude and mechanisms of hydrodynamic escape from the Moon, that tides lower the energy required for thermal escape according to the proximity of the Moon from the Earth’s Roche Limit is founded on robust physical principles. The present paper is a first attempt to quantify this process. It is an independent and complementary process to devolatilisation through incomplete condensation occurring within a protolunar disk (developed in recent lunar accretion models (Thompson and Stevenson, 1988; Canup et al., 2015; Charnoz and Michaut, 2015; Lock et al., 2018)). Our study should encourage the development of more refined models in the future, including radiative transfer with full coupling with chemistry and magma ocean model, to precisely quantify tidally-assisted Moon devolatilisation and to better asses the thermal structure of the escaping atmosphere.

SC acknowledges the support from IPGP and by the UnivEarthS Labex program of Sorbonne Paris Cité, project ExoAtmos (ANR-10-LABX-0023, ANR-11-IDEX-0005- 02) and by IdEx Université de Paris ANR-18-IDEX-0001. PAS was supported by the European Research Council under the H2020 framework program/ERC grant agreement 637503 (PRISTINE) at IPGP and by an SNF Ambizione Fellowship (180025) at ETH Zürich. YNL acknowledges funding from the Einstein young scholar fellowship program (MOST 108-2636-M-003-001) and the MOE Yushan young scholar program. R.H. acknowledges the financial support of JSPS Grants-in-Aid (JP17J01269, 18K13600). We also would like to thank E. Marcq for useful discussions about modelisation of multi-species adiabatic atmosphere, P. Tremblin for discussions on radiative transfer, as well as our reviewers (B. Fegley and E. Young) for their thoughtful comments that improved the quality of this paper.