text:
\section{Introduction}

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  \citep{Canup_2004, Cuk_Stewart_2012, Nakajima_Stevenson_2014, Lock_2018, Cameron_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 \citep{Wolf_Anders_1980, Ringwood_1987, ONeil_1991, Lodders_Fegley_2011, Day_Moynier_2014}. This depletion occurs in concert with enrichment in the heavy isotopes of moderately volatile elements including Zn \citep{Paniello_2012, Kato_et_al_2015}, K \citep{Wang_Jacobsen_2016, Tian_et_al_2020}, Cl \citep{Sharp_2007, Boyce_2015, Gargano_et_al_2020}, Ga \citep{Kato_Moynier_2017},  and Rb \citep{Pringle_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 ($\le$ 1800 K) \citep{Sossi_Moynier_VanZuilen_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 \cite{Canup_2004, Cuk_Stewart_2012, Nakajima_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 ($\ge 20$ g/mol).  Both cases are aggravated by the low abundance of hydrogen in the proto-Earth, preventing loss through aerodynamic drag \citep{Nakajima_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 \citep{Charnoz_Michaut_2015}, or gravitational torque from the proto-Moon exerted on the vapour \citep{Canup_2015} or separation of growing melt droplets from the vapour due to gas-drag \citep{Lock_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 \citep{Thompson_Stevenson_1988, Machida_Abe_2004, Ward_2012, Gammie_2016, Charnoz_Michaut_2015, Salmon_Canup_2012, Lock_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 \citep{Nakajima_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 \ref{section_considerations} we examine, using simple arguments, under what conditions material may escape from the proto-Moon surface under the effect of tides. In section \ref{section_physics_hydro_escape} we investigate two models of hydrodynamic escape assisted by tides and show that under all reasonable assumptions, escape is inevitable. In section \ref{section_orbital_expansion} we discuss the orbital evolution of the proto-Moon, which sets an upper time limit for hydrodynamic escape to occur. In section \ref{section_core_text_evaporation_section} 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 \ref{section_discussion_conclusion}.

\section{A conceptual framework for atmospheric loss from the proto-Moon at the Roche Limit}
\label{section_considerations}

 Models of lunar assembly suggest that the Moon's building blocks coalesce just beyond the Roche Limit \citep{Canup_2015,Takeda_2001}. Traditionally the Roche Limit for a fluid body $R_{\rm R}$, is  \citep{Roche_1873, Canup_Esposito_1995} :

\begin{equation}
    R_{\rm R}=2.456 \times R_{\rm p} \left( \frac{\rho_p}{\rho_{\rm m}} \right)^{1/3}
\end{equation}

   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 $R_{\rm R} \simeq 3 R_\oplus$. Aggregates can begin to form at the edge of the proto-lunar disk around $3 R_\oplus$ but the fully-grown Moon might appear further away, around $5R_{\oplus}$ \citep{Canup_2015,Salmon_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 $3 R_{\oplus}$), 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 \ref{SOM_potential_energy}. As the L1 and L2 Lagrange points are the points on the Hill's sphere closest to the Moon's surface (Appendix \ref{SOM_potential_energy}), escape of the gas is most readily achieved via passage through L1 and L2, because it requires the least energy \ref{Figure_energy}). The kinetic energy required can be converted into a gas temperature  \ref{Figure_T_escape}. For this fiducial case, we assume a molar mass equal to $20$ g/mol, as a proxy for an atmosphere consisting of sodium \cite{Visscher_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 \ref{Figure_T_escape}). 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 $\lesssim 5 R_\oplus$. 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 (Figure\ref{Figure_T_escape}. 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 \citep{Murray_Dermott_SS_Dynamics} 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 \ref{fig_orbits} (left) and Figure \ref{figure_potential_gas_drag} 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) \citep{Canup_2004, Cuk_Stewart_2012, Nakajima_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 \citep{Hyodo_2018}  (Figure \ref{fig_orbits}, right and Figure \ref{figure_potential_rad_press}). 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 \ref{SOM_RAD_PRESSURE}).

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.

\section{Tidally-assisted hydrodynamic escape}
\label{section_physics_hydro_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 \ref{XUVearth}), 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 \ref{SOM_hydro_escape_adiabatic}). 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 \ref{hydro_escape_wet_model}). 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.

\subsection{Dry expanding atmosphere}
\label{SOM_hydro_escape_adiabatic}
Due to the intrinsic complexity of the physical process we aim to investigate, it is not feasible to treat the full problem self-consistently. We are mindful that, due to the temperature diminution with altitude, some fraction of the gas may recondense, and thus may not behave adiabatically. This case is treated in Section \ref{hydro_escape_wet_model}. However, it is of primary importance to first understand the physics of hydrodynamic escape above the lunar magma ocean by solving the fully adiabatic approximation, as it is the original (and natural) framework of the theory of hydrodynamic escape \citep{Parker_1963, Parker_1965}, and hence the crux of the present paper. Since the atmosphere is thin (at most 100 Pa) and composed largely of monatomic gases, it is reasonable to start with the adiabatic approximation, since acceleration is strong at the base of the atmosphere. However, for the sake of completeness, isothermal solutions are also presented in Appendix \ref{isothermal_appendix}.

\subsubsection{Mathematical treatment}
The derivation of the mass flux in the dry model is solved within the adiabatic approximation, thus we ignore here any condensation process during the escape of the gas from the proto-Moon's surface, as well as heat transfer with the surroundings. We solve the Euler equations (conservation of mass and momentum + adiabatic equation of state) and calculate the 1D flow along the line joining the lunar surface to the Lagrange L1 or L2 points (located at the Hill radius). The gas follows the adiabatic equation of state, so that
\begin{equation}
P=\alpha \rho^{\gamma}
\label{eq_adiab_P}
\end{equation}
where P is the total gas pressure, $\alpha$ is a positive constant, $\gamma$ is the adiabatic index ( $\simeq 1.5$, corresponding to a gas with a mixture of monatomic -$\gamma$ = 5/3- and diatomic species $\gamma$ = 7/5), where P is given by the sum of partial pressures of a gas produced in equilibrium with a magma of Bulk Silicate Earth (BSE) - like composition \citep{Palme_Oneil_2003}.  These properties are related to the temperature via the adiabatic relation:
\begin{equation}
    T=\frac{\mu \alpha \rho^{\gamma-1}}{k}
    \label{Eq_temp_adiabatic}
\end{equation}
where $\mu$ and k are the mean molecular weight and Boltzmann's constant, respectively. The molar mass of the gas is assumed to be $\simeq 35g/mol$ (so $\mu \sim 5.81\times 10^{-26} kg)$. This corresponds roughly to a mixture of Na, O$_2$ and K that are the dominant species for $T<2000K$. At steady state, the conservation of mass reads :

\begin{equation}
\frac{\partial \rho v r^2 }{\partial r}=0
\label{eq_adiab_ass}
\end{equation}

Conservation of momentum for a steady-state flow reads:
\begin{equation}
\rho v \frac{\partial v }{\partial r}=-\frac{\partial P}{\partial r}+\rho A(r)
\label{eq_adiab_momentum}
\end{equation}

where A(r) is the local acceleration in the non-inertial frame rotating with the Moon (Equation \ref{equation_accel} in Appendix \ref{SOM_potential_energy}).
Equations \ref{eq_adiab_P} to \ref{eq_adiab_momentum} form a closed set of ODEs. The famous Bernoulli equation for an adiabatic gas is recovered by integrating equation \ref{eq_adiab_momentum} from the surface (subscript S) to the Hill sphere (subscript H) to obtain:

\begin{equation}
    \frac{V_S^2}{2}+\frac{\alpha\gamma}{\gamma-1}\rho_S^{\gamma-1}+Ep_S=\frac{V_H^2}{2}+\frac{\alpha\gamma}{\gamma-1}\rho_H^{\gamma-1}+Ep_H
\end{equation}

where $Ep$ is the potential energy in the frame rotating with the Moon. In the above Bernoulli equation, V and $\rho$ are linked through the mass conservation equation so that $V_S r_S^2\rho_S=V_H r_H^2\rho_H$. Replacing $\rho_H$ in the Bernoulli equation one finally obtains:

\begin{equation}
    \frac{V_S^2}{2}+K\rho_S^{\gamma-1}+Ep_S=\frac{V_H^2}{2}+V_H^{1-\gamma}K(V_S\rho_Sr_S^2/r_H^2)^{\gamma-1}+Ep_H
    \label{eq_adiab_vs_vh}
\end{equation}

where $K=\alpha\gamma/(\gamma-1)$. Note that Equation \ref{eq_adiab_vs_vh} can also be written as follows :
\begin{equation}
    \frac{V_S^2}{2}+\frac{\gamma}{\gamma-1} \frac{kT_S}{\mu}+Ep_S=\frac{V_H^2}{2}+\frac{\gamma}{\gamma-1} \frac{kT_H}{\mu}+Ep_H
    \label{eq_adiab_vs_vh_T}
\end{equation}
with $T_S$ and $T_H$ standing for the gas temperature at the surface and at the Hill sphere, by inserting Equation \ref{Eq_temp_adiabatic} into Equation \ref{eq_adiab_vs_vh}. It shows that the velocity field is \textit{independent} of the surface pressure and is controlled only by the surface temperature and the gravity field. So for a given surface velocity $V_S$ and temperature $T_S$, the velocity at the crossing of the Hill sphere $V_H$ can be determined by numerically solving equation \ref{eq_adiab_vs_vh_T}. This is a well known problem for the case of a point-source gravity field and has been extensively treated by E.N. Parker in a series of papers on solar wind outflows (see e.g. Parker 1963).

The Bernoulli equation gives rise to several families of solutions extensively discussed in \cite{Parker_1963} and \cite{Parker_1965}. The only physical solutions are those which start from the surface with flow velocities smaller than the sound speed. Then the velocity increases with height, passing through the \textit{transonic point} beyond which the flow velocity becomes supersonic (but without a shock). All other solutions correspond either to inflow solutions (subsonic at large distance, supersonic on the ground) or do not connect the surface with infinity, or have finite pressure at infinity (see Parker 1965 for a detailed discussion).

The location of the transonic point is found as follows: inserting equation \ref{eq_adiab_ass} into equation \ref{eq_adiab_momentum} and replacing $\partial P/\partial r$ by $\partial P/\partial\rho \times \partial\rho/\partial r$, and using Equation \ref{Eq_temp_adiabatic} one obtains:

\begin{equation}
\frac{\partial v }{\partial r}=\frac{2 \gamma T k/(\mu r) +A(r)}{v(1- \gamma T k/(\mu v^2))}
\label{eq_adiab_dvdr_sonic}
\end{equation}

For v(r) to be physical, the denominator and numerator must cancel simultaneously (this is the transonic point). Finding the zero of the denominator gives v at the sonic point ($v_{transonic}$), and finding the zero of the numerator, and noting that $v^2/2+\gamma/(\gamma-1) kT/\mu+Ep(r)=cst$ along a current line, gives the location ($r_{transonic}$) and temperature of the sonic point. Once $r_{transonic}$ and $v_{transonic}$ are determined, the velocity at the Moon's surface is determined by solving equation \ref{eq_adiab_vs_vh} between the sonic point and the surface, and the velocity at the Hill sphere is obtained by solving the Bernoulli equation between the sonic point and the Hill sphere. One unknown still needs to be determined: the gas density at the sonic point. In order to do so, we prescribe the surface temperature. Since the velocity field is independent of density (Equation \ref{eq_adiab_vs_vh_T}) V can be computed everywhere starting from the sonic point. Once the velocity field is found, the density and pressure at the surface of the magma ocean must be determined (see next section), in order to obtain the net escaping flux.
\\
\subsubsection{Determining the gas density at the evaporating surface}
\label{vs_vt}

We now focus on the determination of the surface flux, which depends on the evaporative flux at the ocean magma surface (+) and on the return flux back to the magma ocean (-), with the difference being the net flux away from the surface. The evaporative flux can be determined by the Hertz-Knudsen-Langmuir (HKL) equation (e.g. Knudsen, 1909) integrated over the surface area of the Moon, and is proportional to the equilibrium partial pressure of the gas ($P_{sat}$) that can be calculated thermodynamically. The subsequent transport of this gas away from the surface occurs by molecular motion through two distinct mechanisms; advection and diffusion, or a combination thereof. If this transport rate is slow relative to the evaporation rate, there is a 'return flux' to the surface, which, in turn, results in a build up of the partial pressure (or density) at this surface ($P_{s}$) owing to mass conservation (\cite{Richter_et_al_2002, Young_2019, Tang_Young_2020}). This is a classical feature of the HKL equation (the $P_{sat} - P_{s}$ term), though it does not explicitly prescribe how $P_{s}$ is calculated.
An expression for calculating the return flux (and thus the $P_{s}$ term) is proposed in \cite{Young_2019, Tang_Young_2020}, following the \cite{Richter_et_al_2002} approach. Because \cite{Young_2019, Tang_Young_2020} assume there is a surface boundary layer directly above the magma ocean that is well-mixed and in hydrostatic equilibrium, diffusion is the only possible mode of mass transport within this layer.  In this approach, diffusive transport is derived from Fick's Law, in which the diffusion coefficient for a binary gas is given by the Chapman-Enskog equation. This model always results in $P_{s}/P_{sat}$ values that are very close to unity (i.e., equilibrium between the magma and the gas) because diffusion is exceedingly slow when integrated over the Moon's surface area.  While self-consistent, such an approach is not applicable for cases in which advection occurs at the surface, as described here. A suitable expression for an advected gas is derived in Appendix \ref{Appendix_Pressure_surface}, where we compute the return flux, as well as pressure and density of a gas at the surface of an evaporating surface, taking into account advection. We use a kinetic approach very similar to the original Hertz-Knudsen theory. We show that surface pressure and density depend only on the ratio of the gas velocity at the surface $V_S$ to the local thermal velocity $V_t$. It is found that the gas density above the fluid surface ($\rho_s$) is (Equation \ref{equation_Ps_Psat} in appendix \ref{Appendix_Pressure_surface}) :

\begin{equation}
\rho_s/\rho_{sat} = \left(  \frac{V_S}{V_t} \sqrt \pi(2- erfc(\frac{V_S}{V_t})) + e^{\frac{-V_S^2}{V_t^2}}
\right)^{-1}
\label{equation_rhos_rhosat}
\end{equation}

where $\rho_{sat}$ is the gas density at saturation and at temperature $P_{sat}$ is the saturating vapour pressure of a magma of BSE composition at temperature $T_S$ given in \cite{Visscher_Fegley_2013}. We start by computing the velocity profile using the Bernoulli equation. This permits the surface velocity to be calculated, and the density and pressure at the surface are then computed using Equation \ref{equation_rhos_rhosat}. The escaping flux is given as $F=\rho_sV_sR_{\rm m}^2$. Here we assume for simplicity that the solid angle of gas emission is 1 steradian (compared to $4\pi$ for a full sphere). Finally, one obtains a velocity profile with a mass flux that is perfectly conserved, allowing a precise estimate for the return flux to the surface to be made.

\subsubsection{Results}
Figure \ref{Figure_Vsurface_adiab_1Mm} shows the ratio of surface velocity to thermal velocity, for different surface temperatures, as a function of the Earth-Moon distance. We see that $V_s/V_t<0.4$ for all cases considered and thus the surface pressure is always $> 0.6  P_{sat}$ (see appendix \ref{Appendix_Pressure_surface}), but for $T<1800K$ (our preferred case), $V_s/V_t$ never exceeds $0.2$, meaning the surface pressure is always $> 0.8 P_{sat}$. Therefore, the actual surface pressure is found to lie close to gas-liquid equilibrium.  In our case, and contrary to comets, the atmosphere expands at velocities much lower than the thermal velocity. Values reported in Figure \ref{Figure_Vsurface_adiab_1Mm} also imply that the net escaping gas flux from the proto-Moon is small compared to the free evaporative flux from the magma ocean (the ratio of the two being displayed in Figure \ref{figure_flux_ratio}), apart from cases in which $T>2000K$ and $a<3.5 R_{\oplus}$ where the escaping flux becomes about half of the free evaporative flux.

The resulting escape fluxes are displayed in \ref{Figure_escape_adiab_1Mm}. The flux of escaping gas drops with increasing Earth-Moon distance up to the point where escape becomes negligible because of the diminishing gravitational influence of the Earth that otherwise promotes escape. For a lunar surface temperature of 2200 K, the atmosphere of the proto-Moon is efficiently lost provided the proto-Moon is located  inward of 5$R_{\oplus}$, with a flux varying between $8\times10^{-5}$ Moon masses/year at to the Roche limit, down to $10^{-8}$ Moon mass/year at 5.5$R_{\oplus}$. For $T=1600$ K the flux is much lower, between $3 \times 10^{-7}$ and $6 \times 10^{-9}$ Moon mass/year at 3.7 $R_{\oplus}$ only, due to the lower internal energy of the escaping gas. Note that the fluxes reported above may be underestimated, as we assume (for simplicity) the gas emitting surface is only 1 steradian, one order of magnitude smaller than a fully degassing spherical surface ($4\pi$ steradians). This conservative lower bound is justified due to the uncertainty of the areal extent of a lunar magma ocean, together with the fact that the mathematical treatment outlined above strictly applies only along the Earth-Moon line (L1 and L2 Lagrange points).

If we consider potential lunar buildings blocks with half a lunar mass, the flux is more intense by about a factor 2 to 5 (Figure \ref{Figure_escape_adiab_05Mm}), and, interestingly escape is arrested, on average, at a distance about twice as far from the Earth compared to a fiducial lunar mass. Thus, hydrodynamic escape from a 0.5 lunar mass proto-Moon at 1600 K occurs up to about 6.4 $R_{\oplus}$ (compared to 3.7 $R_{\oplus}$ for a 1 lunar mass object), and beyond 10  $R_{\oplus}$ for a surface temperature of 2000K.

\subsection{Wet model with condensation}
\label{hydro_escape_wet_model}

The dry model presented above neglects any condensation or drag of silicate and oxide droplets. Condensation may affect the escape flux due to internal energy transfer within the atmosphere associated with the latent heat of condensation. To account for this, we have developed a \textit{wet} model that includes gas condensation.  We calculate the composition of the vapour phase presuming it is in equilibrium with a liquid of Bulk Silicate Earth (BSE) composition whose renewal timescale at the surface of the magma ocean is very short ( $< 50$ days, see Appendix \ref{magma_renewing}) relative to the cooling timescale.

\subsubsection{Mathematical treatment}
We introduce a method to compute the velocity field and composition of the escaping atmosphere. The treatment is not fully self-consistent as we treat each species as pure (i.e. they have the same speciation in both the condensed and the gas phase), however this approach still represents a significant advance compared to the state-of-the-art, as we consider a multicomponent atmosphere in dynamical flow following a moist-adiabat. Whereas the computation of a steady-state atmospheric structure taking into account a moist adiabat is common, the present work is, to our knowledge the first work that explicitly computes the composition of the atmosphere during its escaping trajectory and for a mixture of species.

\subsubsection{Calculation of the (P,T) structure}
During gas expansion leading to escape, we assume that the mixture (comprised of gas and condensates) outgassed from the Moon's surface follows a \textit{moist adiabat}, that is, the mixture cools adiabatically with condensates remaining bound to the gas (as justified in Appendix \ref{SOM_drag_droplets}), leading to a specific (P,T) or ($\rho$,T) relation that depends only on the initial entropy and composition of the mixture at the Moon's surface. Here we calculate the (P,T) and ($\rho$,T) relations for a given mixture composition and temperature at the Moon's surface. We consider a mixture including N species that can be either in gaseous or condensed form. We consider two stages during hydrodynamic escape of the gas. First, the temperature drops isentropically from that of the magma ocean, $T_S$, to some temperature, $T_0$ , with no heat exchange with the surroundings. When the gas has cooled sufficiently to reach a lower threshold temperature $T_0$, we assume that the mixture of gas and dust is heated by the Sun in order to maintain a constant temperature. $T_0$ is therefore simply assumed to be the equilibrium temperature at the location of the Earth-Moon system in the absence of greenhouse heating: 250 K. This is an approximation; in order to compute the true temperature self-consistently, we would need to couple our hydrodynamic/chemical system to a radiative transfer model which is beyond the scope of this work.
This 250 K can be considered as a lower bound for the wet model, because it neglects heating from the Moon's hot surface and from radiative heating by the Earth. The "upper bound" is given by the purely isothermal case in which $T_0$ = $T_S$. This case is described in Appendix \ref{isothermal_appendix}. During the isentropic stage (from $T_S$ down to $T_0$), the decrease of temperature leads to a variation in the partial pressures of every species, as well as phase-changes. For each species, $i$, the number of moles ($n_i$) present in one volume unit of the system at any location is $n_i=(\rho_{i,g}+\rho_{i,c})/\mu_i$ where  $\rho_{i,g}$ and $\rho_{i,c}$ are the densities of species $i$ in gaseous and condensed form. $\mu_i$ is the molar mass.

Let consider a transformation of this mixture. The total entropy variation of one volume unit is  $dS=\Sigma(dS_i)$ where $dS_i$ is the entropy variation of species $i$. Each $dS_i$ is given as \citep{Ingersoll_1969, Atmos_Thermo_1998, Bechtold_2009}:

\begin{equation}
    \text{if species $i$ is not saturated\\}
dS_i=Cp_{i,g}\rho_{i,g}\frac{dT}{T}-\rho_{i,g}\frac{R}{P_i}dP_i
    \label{Eq_ds_not-saturared}
\end{equation}

\begin{equation}
\text{if species $i$ is  saturated\\}
dS_i=Cp_{i,g}\rho_{i,g}\frac{dT}{T}-\frac{ \rho_{i,g} L_i}{T^2} dT-\frac{L_i}{T}d\rho_{i,c}+\rho_{i,c}Cp_{i,c}\frac{dT}{T}
\label{Eq_ds_saturared}
\end{equation}

where $Cp_{i,g}$  and $Cp_{i,c}$ are the heat capacity of species $i$ at constant pressure for its gaseous and condensate form respectively, and $L_i$ is the latent heat. Note that these three quantities depend on the local temperature. A species is considered as saturated if $P_i > P_{sat,i}$ where $P_{sat,i}$ is the vapour pressure of species $i$ at the local temperature T. When this condition is met, we set $P_i= P_{sat,i}$ and the density of formed condensates $d\rho_{i,c}$, is found by solving $dS=\Sigma(dS_i)=0$. Since we assume (for simplicity) that each species has the same speciation in both the condensate and the gas (neglecting solid solutions or complex solid phases) $P_{sat,i}$ only depends on the local temperature. This is a necessary simplification for the moment to make the calculation tractable when coupled to the computation of the gas-velocity field (next section).

For a given temperature change, $dT$, the new partial pressure $P_i$ and condensate density $\rho_{i,c}$ of each species $i$ is found by solving $dS=\Sigma(dS_i)=0$. To close the system of equations we assume that the mole fraction of every species $i$ is constant with T and height above the surface. We include O$_2$, Zn, Fe, Cu, SiO, Mg and Na. All thermodynamic values are taken from the python library \textit{Thermo}  (Chemical properties component of Chemical Engineering Design Library (ChEDL)
https://github.com/CalebBell/thermo). The calculation is performed as follows. A surface temperature $T_S$ and a gas surface composition are calculated in section \ref{section_core_text_evaporation_section}, in which all species are present in gaseous form (by definition) with partial pressures $P_i$ at the surface. Then temperature is incremented by a small quantity dT ($dT << 1K$ in general) and the new partial pressures and densities are found by solving Equations \ref{Eq_ds_not-saturared}, \ref{Eq_ds_saturared} simultaneously (using a Newton-Raphson method) and constraining the molar fraction of every species to be constant. With this procedure isentropic (P,T) and ($\rho$, T) curves are tabulated. A final check of entropy variations shows that the total entropy of the system changes by less than $10^{-3}$ relatively (at worst) over the course of the calculation.

In the isothermal region of the atmosphere ( when T crosses $T_0$) the gas is assumed to depressurise isothermally (due to geometrical expansion of the gas), and the new composition is simply computed by assuming that the total mole fraction of each species (including gas and condensate phases) is constant.

An example of gas transformation following the above thermodynamic path is provided in Figure \ref{Figure_PT_adiababatic}, for a mixture with the elemental composition of a vapour derived from the BSE at 1600 K (see section \ref{section_core_text_evaporation_section}). One observes that, during the isentropic cooling process from 1600 K down to 250 K, Cu and Fe are the first to condense. At $\sim 575K$ Zn condenses, then Na at about $\sim 490K$. K condenses in the isothermal regime for T = 250 K and $P<10^{-3}$ Pa.

\subsubsection{Calculation of the velocity field}

The velocity field of the escaping atmosphere is computed as follows: the atmospheric gas (composed of different species, either in gaseous or in solid form) has a total local density $\rho(r)$, a total pressure P(r) and a local temperature T(r). Starting from an initial composition at the lunar surface at temperature $T_S$ ( Appendix \ref{section_core_text_evaporation_section}) we assume that the gas follows a moist adiabat along its escaping trajectory such that the mixture follows the P(T) relation computed in the previous section. In spherical geometry the steady-state conservation of mass reads :
\begin{equation}
    \frac{1}{r^2}\frac{\partial}{\partial r}\left( \rho v r^2\right)=0.
    \label{Eq_flux}
\end{equation}

The conservation of linear momentum gives:
\begin{equation}
    \frac{1}{r^2}\frac{\partial \rho v vr^2}{\partial r} = \rho v \frac{\partial v}{\partial r}=-\frac{\partial P}{\partial r}+\rho A(r)
    \label{Eq_momentum}
\end{equation}
The gas velocity field depends on its surface velocity $V_S$. The two above equations are solved simultaneously along the line joining the Earth-Moon's centers, while considering the $P(\rho)$ function to be isentropic. It is solved using a simple finite difference method. To find the transonic solution (the one that has zero pressure at infinity) we adopt a shooting method: we perform an iterative search on $V_S$. Starting from very low $V_S$ we progressively increase its value and determine when the the solution to Equations \ref{eq_flux} and \ref{Eq_momentum} switches abruptly from the "breeze" solution (low $v$ and finite pressure at infinity) to the transonic solution (supersonic $v$ and zero pressure at infinity). This procedures allow to find $V_S$ within $10\%$ error quite rapidly ($<$ 100 iterations). However the computation of the full velocity, pressure and temperature profiles is much more demanding. Indeed the transonic solution is the threshold solution between the four different families of solutions of the Parker Wind. To obtain a reasonable velocity profile after integration of the partial differential equations, $V_S$ must be determined to very high precision, which requires more than $10^3$ and sometime more than $10^4$ iteration steps on $V_S$.

Note that for the wet model the decrease in surface pressure relative to that at equilibrium caused by the hydrodynamic wind (see Appendix \ref{Appendix_Pressure_surface}) is not taken into account. To consider this effect properly, the iterative search on the surface velocity (to find for the transonic solution) must be also coupled to a search on the surface gas density (to take into account the density correction due to flux of material returning to the surface). This would necessitate a 2D iterative search rather than a 1D search. As the determination of surface velocity is computationally intensive (see above), this 2D search could not be conducted. This results in an overestimation of the net escaping flux of the order of the surface velocity divided by the thermal velocity (Appendix \ref{Appendix_Pressure_surface}) for the wet model. Some posterior checking reveals that for $T_s> 2000K$ and for A=Earth-Moon distance $< 3.5 R_{\oplus}$ our net flux is overestimated by less that $30\%$, and falls to less than $10\%$ for  $A > 5 R_{\oplus}$. For $T_s$ in the range of 1600-1800K (our preferred range) the flux is overestimated by less than $15\%$ and falls to less than $2\%$ for  $A > 5 R_{\oplus}$. This error is small, and compensated for, by other approximations in the development of our model. In particular, we consider that the evaporating surface of the Moon is only 1 steradian in solid angle (rather than $4\pi$ at maximum), because of lack of knowledge on the actual state of the Moon's surface.

\subsection{Results for the wet model}

The resulting atmospheric structure for a 1 lunar mass proto-Moon located at $4 R_{\oplus}$ is displayed in Figure \ref{Figure_atmo_struc} for $T_S=1600K$ and  $T_S=2000K$. While the $T_S=2000K$ velocity profile is typical of a transonic adiabatic flow with a linear increase of velocity with altitude, the $T_S=1600K$ shows more peculiar features. A strong velocity bump is visible at 1.8 $R_m$, corresponding to $T \sim 575K$. Figure \ref{Figure_PT_adiababatic} reveals that this is the effect of Zn condensation, that leads to a step dP/dT, and thus strong dP/dr (because dP/dr=(dP/dT) $\times$ (dT/dr) ), leading to strong gas acceleration. A similar, but more pronounced effect is also visible at R=2.1 $R_m$ and $T\sim 490 K$ corresponding to Na condensation, due to its being the most abundant gas species. Condensation causes a steep pressure drop that, in turn, induces an acceleration of the gas and leads to a higher flux. This process was already identified in atmospheric dynamics, where \cite{Makarieva_et_al_2013} suggest that gas condensation acts to release of some internal potential energy that then becomes available to accelerate the gas.

The gas composition as a function of altitude is displayed in Figure \ref{gas_compo_1600K_wet_model_escape}, for $T_S=1600K$ and for a proto-Moon located at 4 Earth Radii. In our model atmosphere, most of the gas species present at the surface re-condense at higher levels in the atmosphere prior to reaching the Hill Sphere (2.45 Moon radii). Only potassium and oxygen remain in gaseous form when crossing the Hill Sphere. Although condensation does occur along the \textit{moist adiabat}, they remain in the atmosphere and are nevertheless dragged outward with the gas-flow provided the grains or droplets into which they condense remain small. This is justified because, when the condensates form by momentum conservation, they do so together with the velocity vector of the surrounding gas. By inertia they continue their travel upward and do not fall immediately to the ground. Later, as they are less subject to buoyancy forces than the gas itself, they decelerate more rapidly than the surrounding gas and gas-drag may occur. Our analysis in Appendix \ref{SOM_drag_droplets} shows that the droplets stay well coupled to the gas and are efficiently dragged outward during hydrodynamic escape, provided their size is $< 1$ mm  and that the condensate mass fraction relative  $<0.5$ at the surface of the magma ocean. This is because the Moon's gravitational field drops with altitude, enabling more efficient coupling of particles with the gas at higher altitudes, despite the gas' decreasing density.

The escaping fluxes for different surface temperatures and various Earth-Moon distances are displayed in Figure \ref{fig_flux_vs_Moon_distance_wet_model}, a key result of our study. It illustrates that escaping fluxes drop with increasing Earth-Moon distance, and increase with surface temperature. Comparing this behaviour with the dry adiabatic case (Figure \ref{Figure_escape_adiab_1Mm}) shows that the fluxes for the wet model are, on average, 10- to 50 times higher. This difference is due to the release of latent heat during condensation, which provides additional energy to drive the escape. Neglecting the return flux to the magma-ocean may amount to only about a $20\%$ error at most in the calculated fluxes for a Moon < $4R_{\oplus}$. For a proto-Moon at 3.5 $R_{\oplus}$ and $T_S=1600K$ the flux is about $ 8 \simeq 10^{-5}$ Moon mass/year but drops almost to 0 beyond $6R_{\oplus}$, while for  $T_S=2000K$ it is still about
from $ 10^{-4}$ Moon mass/year at $6R_{\oplus}$.
These fluxes may be under under-estimated considering (1) that the solid angle of the escaping flux might increase up to $4\pi$ and (2) that a partially-formed Moon, or moons building blocks with individual masses smaller than the fully grown Moon would experience more efficient atmospheric loss because of lower mass.

Several simplifications were made in reaching this result (Appendix \ref{SOM_approximations}). Most importantly, neither radiative heating from the lunar surface nor radiative cooling were considered, as this would require the development of a full radiative transfer model with a chemical kinetics code. Note that calculation of the atmospheric optical depth (Appendix\ref{SOM_optical_depth}) shows that the atmosphere may become transparent for  $T_{\rm s} < 2000$ K. This would facilitate heating (thereby increasing the escaping flux) of the Moon's surface on its near-side by thermal emission from the Earth as well as heating and cooling of the expanding atmosphere. Figure \ref{Figure_T_escape} shows that, even for exact radiative balance of droplets in a transparent atmosphere, the minimum escaping temperature is always reached. So either the escape is fully dry (our dry model) or implies recondensation (our wet model), we find in these two scenarii that intense hydrodynamic escape of the atmosphere above a lunar magma ocean occurs. Future studies should incorporate a self-consistent Gibbs Free Energy minimisation code with a more realistic condensate mineralogy. This would notably include feldspars and olivine that are shown to be stable phases for a cooling nebular gas. That these are not considered in the present study means that the condensation behaviour of the elements in the wet model should only be taken as instructive rather than definitive.
Because of Earth-Moon tides, the proto Moon will inevitably migrate outwards, lowering the escaping flux over time, a phenomenon that is discussed in the next section.

\section{Effect of the Moon's orbital expansion}
\label{section_orbital_expansion}
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. \cite{Goldreich_Soter_1966}):

\begin{equation}
  \frac{da}{dt}=\frac{3 k_{2e} M_{\rm m} R_e^5 \sqrt{G}}{Q_e \sqrt{M_\oplus a^{11}}}
   \label{Equ_tidal_evol}
\end{equation}

The resulting semi-major axis evolution of the Moon is reported in figure Fig \ref{Figure_tidal_evol}. 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 \cite{Cuk_Stewart_2012, Chen_Nimmo_2016}, noting that the present-day value is $Q_e \simeq 10 $ (\citep{Goldreich_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 \ref{Figure_escape_adiab_1Mm} and Figure \ref{Figure_escape_adiab_05Mm}). 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 $7.5 R_\oplus$ in $10^{4-5}$ years. Moon-disk interaction shortens this timescale by only 1000 years \citep{Salmon_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 \ref{Figure_atmo_struc}, 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 \ref{isothermal_appendix}. 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.

\section{Behaviour of moderately volatile elements during evaporation of the Moon }
\label{section_core_text_evaporation_section}

\subsection{Vapour pressures of metal-bearing gases above the silicate Moon}

The range of fluxes and tidal expansion timescales computed above permits evaluation of how the residual composition of the Moon evolves as a result of atmospheric escape. Given that the initial bulk composition of the Moon is likely similar to that of the bulk silicate Earth (BSE) \citep{Ringwood_1987}, and presuming that the composition of the escaping vapour is that of the vapour at the liquid-gas interface (neglecting molecular diffusion in the outgoing flux, that is negligible in a hydrodynamic wind, see \cite{Tang_Young_2020}), the effect of atmospheric loss on the composition of the Moon may be estimated.

The composition of the gas at the surface of the magma is computed using a thermodynamic approach (Appendix \ref{SOM_THERMO}) taking advantage of recent laboratory measurements of chemical activities of melt oxide species (\cite{Sossi_Fegley_2018, sossi2019}). The equilibrium partial pressure may be calculated for any gas species containing a metal, \(M\), according to the generalised congruent vaporisation reaction:

\begin{equation}
\left( M^{x+n}  O_ \frac{x+n}{2} \right) ({l})=\left( M^{x} O_\frac{x}{2} \right) ({g})+\frac{n}{4}O_2 ({g})
\label{Eq_reaction_MO_vap}
\end{equation}

where $l$ denotes the liquid phase, $g$ the gas phase, $x$ the oxidation state of the metal in the gas phase and $n$ the number of electrons exchanged (an integer value). At equilibrium, the partial vapour pressure of species $M^x O_{x/2}$ (where the melt oxide species is designated as species $i$ for short) is obtained:

\begin{equation}
p \left( M^x O_{\frac{x}{2}} \right)=\frac{K X(i) \gamma(i)}{f(O_2)^{n/4}}
\label{Eq_partial_pressure_equil}
\end{equation}
where $K$ is the equilibrium constant of reaction, $X(i)$ is the mole fraction of species $i$ (see Table \ref{table_fO2}), $\gamma(i)$ its activity coefficient and $f(O_2)$ the oxygen fugacity. The $f(O_2)$ is calculated according to the model of \cite{Visscher_Fegley_2013} for their BSE composition and lies close to the fayalite-magnetite-quartz (FMQ) buffer, as indicated by thermodynamic and experimental studies for the evaporation of silicates \citep{Costa_et_al_2017, Sossi_Fegley_2018}. Activity coefficients are calculated according to enthalpies of solution determined in experimental studies (\cite{sossi2019, charles1967}). It is emphasised that the activity coefficients of trace elements in silicate melts are accurate only to within a factor of ~2 to 3 for basaltic melts (\cite{sossi2019}). Moreover, because these compositions are similar, but not identical to those expected for a lunar magma ocean, this introduces an additional uncertainty in their application to the evaporation of lunar compositions. Therefore the associated partial pressures of these elements may be conservatively estimated to vary by up to an order of magnitude. This translates into an equivalent temperature uncertainty of about $\pm 100$ K. However, the relative partial pressures of these elements are more robust as systematic errors in the measurement of activity coefficients cancel. Therefore, the total mass loss from the Moon required to account for elemental depletion is less well constrained than are the conditions leading to the \textit{relative} chemical fractionation between them. Thus, element ratios of Na, K and Zn are more exacting than are their abundances for constraining the conditions of volatile loss. Keeping these caveats in mind, the partial pressures of the gas species are computed for all elements for which laboratory measurements in lunar rocks are available. The resulting partial pressures of the major gas species are plotted in Figure \ref{Figure_partial_pressure}, and show that Na dominates the composition of the vapour phase with Zn, Cu and K as minor metal-bearing gas species. The major components, Mg, Si and Fe, despite their high mole fractions in the BSE, constitute a relatively smaller fraction of the gas phase owing to their low volatilities over the temperature range 1500 to 2000 K  (\cite{Sossi_Fegley_2018}). The total pressure obtained (solid black line) is very close to the one provided by \cite{Visscher_Fegley_2013} (dashed line) using their MAGMA code.

 \subsection{Influence of evaporation on the Na, K and Zn abundances in the Moon}
 \label{SOM_Evaporation_K_Zn}
Highly- and moderately volatile elements are particularly exacting records of lunar volatile loss, because they still remain in abundances that are measurable in lunar rocks \citep{Wolf_Anders_1980}. Such volatile metals in the Moon define two distinct plateaus relative to the BSE (Figure \ref{Fig_Earth_Moon_abundances}). The moderately volatile elements (including the alkalis and Ga) show only modest depletion (a factor of 5) whereas highly volatile elements in the Moon (e.g. Zn, Cd, In, Bi) are depleted by a factor of ~50, and, importantly, define constant chondrite-normalised abundances, leading some authors \citep{ONeil_1991,Taylor_et_al_2006,Wolf_Anders_1980} to suggest that they were accreted later than the alkali metals, by addition of chondritic material. However, the heavy stable isotopic composition of Zn and K attest instead to their evaporative loss at an early stage during lunar formation \citep{Wang_Jacobsen_2016, Paniello_2012}. The alkali metals and Zn, in particular, are unequivocal probes of vapour loss because they do not appreciably partition into the metallic phase during lunar core formation. Indeed, the chemical similarities among alkali metals have been used to constrain likely conditions of volatile loss from the Moon \citep{Kreutzberger_et_al_1986}. \\

Although the temperature of evaporation is often treated as a free parameter in lunar formation models \citep{Canup_2015, Lock_2018}, the similarity of the abundance and isotopic composition of lithium in lunar mare basalts relative to terrestrial basalts, in contrast to the near-quantitative loss of Na in lunar rocks, sets limits on the likely  temperatures under which volatile depletion occurred \citep{ONeil_1991, Taylor_et_al_2006, Magna_et_al_2006}.
 In order to vaporise 80 percent of the Na whilst preserving 90 percent of the Moon's Li budget requires temperatures below 1800 K, considering estimates for the activity coefficients of these two elements in silicate melts \citep{Sossi_Fegley_2018, ONeil_1991}. These temperatures are in good agreement with those deduced from Cr isotope systematics between lunar and terrestrial basalts. The light composition of the former require temperatures $>1600$ K (to vaporise sufficient Cr) but $<1800$ K to produce sufficient isotope fractionation \citep{Sossi_Moynier_VanZuilen_2018_} where near-equilibrium conditions had to have prevailed during vaporisation. In addition, the Li/Na and Cr constraints yield similar temperatures to those obtained by thermal models of the near-surface melt  (Tang and Young 2020).

Here we exploit calculations of equilibrium partial pressures to determine whether the observed relative depletion factors between Na, K and Zn are consistent with a hydrodynamic escape scenario over a range of temperatures between 1200 and 2200 K. Considering a conservative possible range of lunar abundances for Na, K, Zn relative to the BSE from Figure \ref{fig_visscher}, reported in Table \ref{Table_minmax_K_NA_ZN}, we investigate the temperature at which the modelled abundances are consistent with observations.

\begin{table}[h]
\begin{tabular} {|l|l|l|l|}
  \hline
  \textbf{Moon/BSE abundance}  & \textbf{Na} & \textbf{K} & \textbf{Zn}  \\
  \hline
  \textbf{Maximum} & 0.63   &  0.8 & 0.093  \\
    \hline
  \textbf{Minimum} & 0.11   & 0.11  &  0.02 \\
  \hline

\end{tabular} \\
\caption{Minimum and maximum values for the mass fraction of each element measured in the silicated Moon divided by that in the silicate Earth ($f$ in the main text). Values extracted from Figure \ref{fig_visscher} \citep{Visscher_Fegley_2013}. In Figures \ref{Figure_vaporize_to_Na}, \ref{Figure_vaporize_to_K} and \ref{Figure_vaporize_to_Zn} we show that we can match simultaneously the range of measurements for Na and K, but never for Zn}
\label{Table_minmax_K_NA_ZN}
\end{table}

Given the mixing ratios of these three elements in the atmosphere, it is possible to uniquely determine the mass of vapour that must have escaped from the Moon, $M_v^L$, at a temperature T, in order to match the  abundance of a given element in lunar mare basalts (and by extension the bulk Moon).  Due to the simplicity of the atmospheric structure assumed here, these calculations are designed to provide first-order estimates. Consider a element, whose Lunar abundance / BSE abundance is denoted, $f$, then, the mass that must be evaporated and subsequently lost from the Moon is computed as follows (see derivation in Appendix \ref{SOM_escaped_mass}):

  \begin{equation}
      M_v^L=\frac{(f-1)M_M}{1-\mu_i^v/\mu_i^E}
     \label{eq_mass_removed}
  \end{equation}
 where $M_M$ is the mass of the Moon, $\mu_i^v$ and $\mu_i^E$ are the mass fractions of element $i$ in the vapour phase and in the BSE respectively.
 We assume that the escaping gas has the composition of the vapour in equilibrium with the magma at the lunar surface (Figure \ref{Figure_partial_pressure}) at temperature T. Assuming that the proto-Moon starts with the composition of the BSE we consider two modes of evaporation:

 \begin{itemize}
     \item \textit{Equilibrium evaporation}, the magma vapour composition does not change as material is removed, so $\mu_i^v$ is constant with time in Equation. \ref{eq_mass_removed}.
     \item \textit{Fractional evaporation}: the vapour composition changes progressively as vapour is removed from the system at steps of $10^{-6}$ lunar masses per increment. After each step of vapour removal, the compositions of the magma and the vapour are re-calculated.
 \end{itemize}

Applying the two procedures described above, the Na (Figure \ref{Figure_vaporize_to_Na}), K (Figure \ref{Figure_vaporize_to_K}), and Zn abundances (Figure \ref{Figure_vaporize_to_Zn}) are treated successively as target quantities, with the other two elements calculated at the temperature listed in the figure caption.

The total mass of material that must escape from the Moon in order to match the mean abundance of K measured in lunar mare basalts is in listed Table \ref{table_evap_K} for different surface temperatures. The same calculation is presented but for Na in table \ref{table_evap_Na}. For a temperature around 2000 K, both abundances could be matched almost simultaneously. However, there is a range of measured values for K and Na that allows a larger range of possible temperatures matching both abundances, reported as blue boxes in  \ref{Figure_vaporize_to_Na} and \ref{Figure_vaporize_to_K}. In these figures we see that it is possible to match both Na and K composition (within measured ranges) via either equilibrium or fractional evaporation between 1600 K and 2000 K. Inspection of Tables \ref{table_evap_K} and \ref{table_evap_Na} shows that it requires from $0.1\%$ to $0.6\%$ of a lunar mass to evaporate to reach the observed depletion. For a small escaping surface flux $\sim 10^{-6}$ lunar mass/year (Figure \ref{fig_flux_vs_Moon_distance_wet_model}), this requires between $\sim 1000$ and $\sim 6000 $ years. For an  escaping surface flux $\sim 10^{-4}$ lunar mass/year, these timescales are reduced by a factor of 100.

However all three of Na, K and Zn are never adequately fit simultaneously because Zn is quantitatively vaporised for even minor evaporation of Na and K, in disagreement with their abundances in lunar basalts. Possible solutions to this conundrum may be deviations in activity coefficients of these elements relative to those measured in experimental studies. Alternatively, evaporation occurred at higher oxygen fugacities than predicted according to \cite{Visscher_Fegley_2013}.  Higher \(fO_2\) promotes alkali vaporisation relative to Zn because Zn vaporises according to an $n$ = 2 stoichiometry compared to $n$ = 1 for the alkali metals (Equation \ref{Eq_reaction_MO_vap}). A similar exercise performed with a greater suite of elements, combined with a more detailed atmospheric model would further aid in constraining the conditions of vapour loss from the Moon, but is beyond the scope of this work.

\begin{table*}
\begin{tabular} {|l|l|l|l|l|}
  \hline
  \textbf{Temperature (K)}  &  \textbf{1600} & \textbf{1800} & \textbf{2000}  & \textbf{2200} \\
  \hline
  \textbf{Lunar mass fraction evaporated}  &  0.088 \% & 0.197 \% & 0.33\% & 0.55 \%    \\
  \textbf{for equilibrium evaporation}      & & &  &  \\
    \hline
  \textbf{Lunar mass fraction evaporated} &  0.083 \% & 0.115 \% & 0.154\% & 0.223 \%    \\
  \textbf{for fractionated evaporation/}      & & &  &  \\
  \hline

\end{tabular} \\
\caption{Mass that must be evaporated to reproduce the mean lunar K abundance, for different temperatures of evaporation.}
\label{table_evap_K}
\end{table*}

\begin{table*}
\begin{tabular} {|l|l|l|l|l|}
  \hline
  \textbf{Temperature (K)}  &  \textbf{1600} & \textbf{1800} & \textbf{2000}  & \textbf{2200} \\
  \hline
  \textbf{Lunar mass fraction evaporated}  &  0.33 \% & 0.48 \% & 0.58\% & 0.73 \%    \\
  \textbf{for equilibrium evaporation}      & & &  &  \\
    \hline
  \textbf{Lunar mass fraction evaporated} &  0.24 \% & 0.23 \% & 0.24\% & 0.27 \%    \\
  \textbf{for fractionated evaporation/}      & & &  &  \\
  \hline

\end{tabular} \\
\caption{Mass that must be evaporated to reproduce the mean lunar Na abundance, for different temperatures of evaporation.}
\label{table_evap_Na}
\end{table*}

 \subsection{Isotopic fractionation during hydrodynamic escape}

 The observation that the stable isotope compositions of moderately volatile elements, such as K, Zn and Cr, are distinct in lunar mare basalts compared to their terrestrial equivalents \cite{Wang_Jacobsen_2016,Paniello_2012,Sossi_Moynier_VanZuilen_2018_} may be used to place constraints on the potential conditions under which volatile loss occurred from the Moon. While K and Zn isotopes are heavier in the Moon than in the Earth, Cr isotopes are lighter, implying that, should this variation be due to volatile loss, the vapour reached equilibrium with the lunar surface. Moreover, there could not have been any process in the atmosphere capable of separating isotopes from one another prior to escape. For atmospheric loss driven by a hydrodynamic wind, as detailed here, there is a bulk outflow of mass in which no mass separation can occur in the atmosphere. Therefore, no isotopic fractionation results during this process.\\

 The other possible locus at which isotope fractionation could develop is at the magma ocean-atmosphere interface. In section \ref{vs_vt}, the effect of a non-stationary gas on the pressure at the surface of the magma ocean relative to the equilibrium partial pressure ($P_s/P_{sat}$) was quantified, and shown to be \textless 1. This quantity is important in determining the degree of isotope fractionation that could result during the evaporation process, through the relation (\cite{Richter_et_al_2002}):

\begin{equation}
^{i/j}M\alpha_{gas-liq, net} = ^{i/j}M\alpha_{gas-liq, eq}+ ((M_j/M_i)^{0.5}-1)(1-P_s/P_{sat}))
\label{Eq_isotopes}
\end{equation}

Where $i$ and $j$ are the two isotopes of element with molar masses $M$, and for the Moon, liq = lunar magma and gas = lunar atmosphere, the subscripts 'eq' and 'net' denote the equilibrium isotope fractionation and net isotopic fractionation, respectively. As the former is proportional to 1/T$^2$ and temperatures are relatively high ($ > 1000 K$), $\alpha_{gas-liq, eq}$ is expected to be of the order of ~ 0.9999 for vaporisation reactions between minerals and monatomic gases at 1500 K (see \citep{Sossi_et_al_2020} and references therein). However, precise values for the relevant melt-oxide species of K and Zn and gas remain uncalibrated, precluding an accurate estimate of $\alpha_{gas-liq, eq}$ at present. This expression for calculating the fractionation factor neglects the effect of diffusion, because it is not the predominant mode of mass transport during hydrodynamic escape of the protolunar atmosphere, rather, it is advection. That is, the Peclet number of the flow is \textgreater \textgreater 1 and diffusion is unimportant. As such, the lower the value of $P_s/P_{sat}$, the larger the fractionation factor between vapour and liquid. The apparent fractionation factors between vapour and liquid deduced for Zn and K in lunar basalts with respect to the Earth's mantle can be used to place limits on $P_s/P_{sat}$. These values are obtained by assuming a Rayleigh fractionation process during evaporation and using the measured \textit{f} of Zn and K in the bulk Moon together with their isotopic compositions (see \citep{Sossi_et_al_2020}, their Table 4), assuming initial values reflected those of the present-day BSE. Substituting in the values obtained by this calculation, $(^{66/64}Zn)\alpha_{gas-liq}$ = 0.9996 $\pm{0.0001}$ and $(^{41/39}K)\alpha_{gas-liq}$ = 0.9998 $\pm{0.0001}$ for evaporation from the Moon into \ref{Eq_isotopes} yields values of $P_s/P_{sat}$ of 0.98$\pm0.01$ for Zn and 0.99$\pm0.01$ for K, both illustrating the necessity of near-equilibrium conditions to have prevailed between the liquid and the atmosphere. Inspection of Fig.\ref{Figure_Vsurface_adiab_1Mm} shows that this precludes scenarios in which $V_s/V_t$ $>$ 0.03 (see also \ref{equation_rhos_rhosat}) that are favoured during high temperatures and/or for low Earth-Moon distances. Thus, isotopic constraints require relatively low temperatures at moderate Earth-Moon distances such that the advective velocity does not exceed 1 \% of the gas thermal velocity.  This configuration is indeed achieved for T in the range 1600-1700 K and for Earth-Moon distances beyond $\sim 3.7 R_{\oplus}$ (Figure \ref{Figure_Vsurface_adiab_1Mm}) for the wet model. This combination of temperatures, $V_s/V_t$ values and Earth-Moon distances constrain maximum loss rates to be $\leq 10^{-5}$ (wet) or $\leq 10^{-7}$ (dry) lunar masses/yr, generally exceeding those calculated by \citep{Tang_Young_2020}, $7\times 10^{-8}$ lunar masses/yr.

\section{Implications for the early evolution of the Moon}
\label{section_discussion_conclusion}
Combining the low temperature range implied by isotopic data with best fits to the observed depletion factors of Na, K and Zn at $T = 1600 - 1800 K$, 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 $\sim 1 Kyr$. 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 \citep{Meyer_et_al_2010, Elkins-Tanton_ET_AL_2011}, crystallisation of a plagioclase lid may occur over much shorter timescales, around 1000 years \citep{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 \ref{fig_stagnant_lid_timescale}). 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 \textgreater 500 years and \textless 3000 years \citep{Meyer_et_al_2010, Elkins-Tanton_ET_AL_2011, Tang_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 \citep{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 $1600-1700 K$ 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 \cite{Sossi_Moynier_VanZuilen_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.

\section{Differences with previous work and conclusions}

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 \cite{Tang_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 (\cite{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 \textit{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 \ref{Appendix_Pressure_surface} 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 \ref{Appendix_Pressure_surface} by using an approach very similar to the original Hertz-Knudsen calculation. The effective pressure at the surface is given by Equation \ref{equation_Ps_Psat}, 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 \ref{Figure_partial_pressure}) while \textit{allowing a substantial net escaping flux}. Thus, the surface pressure is coupled to the velocity at the surface through Equation \ref{equation_Ps_Psat}.
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 \textit{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 \textgreater 250 K and isothermal for T \textless 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 \citep{Thompson_Stevenson_1988, Canup_2015, Charnoz_Michaut_2015, Lock_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.

\section{Acknowledgements}
SC acknowledges the support from IPGP and by the UnivEarthS Labex program of Sorbonne Paris Cit\'e, 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.

redpajama_set_name: RedPajamaArXiv


text:
staugadmin
Athletics, Students
SAU Basketball Student-Athlete Already a Book Author at Age 18 (Bre'cha Byrd Has the Write Stuff)
RALEIGH, N.C. – According to a website article, 80 percent of Americans aspire to pen a book and get it published within their lifetime. Few actually start or finish, however. Bre'cha Byrd bucked the odds and she's only 18 years old. A basketball student-athlete at Saint Augustine's University, Byrd is talented way beyond the basketball court. She is a published author and a motivational speaker. "Some people think you have to be 40 [years old] or 50 or 60 to become an author, but why not now," said Byrd, who majors in Business Administration. The title is Self Love…LOVESELF, a 33-page inspirational book for females. Its purpose is to uplift young girls and women in the journey of self-love. The book's inspiration comes from Byrd's parents Brenton and Chanique, her brother Breyden and a higher power. "God gave me the vision and my family supported me," Byrd said. "We live in a world where there is so much negativity. I want girls to know they are beautiful inside-and-out and they have worth. Their worth isn't based on what others think." Miss Lola Lakes is the main fictional character and much of her life is based on Byrd's past experiences and what she has seen others experience. Byrd was picked on and lost confidence, only to regain her self-esteem with help from her parents. She now oozes with confidence and wants to empower other females. "Growing up, I've been through a lot as far as learning to love me," Byrd said. "My self-awareness and self-worth were evident at an early age, and some people didn't know how to deal with me. My mindset was completely different. I saw things for what it was and called them out on it. I wouldn't comply to what everybody else thought. They couldn't put me in a box." "My parents always say, 'tell my story'. I love writing, so why not," said Byrd, who wrote songs and poems at 7 years old. "I want to be that positive difference. That book will always last forever, even when I am gone." A prolific writer, Byrd started her book three years ago at age 15. The book was published in November 2019 and released in January of this year. Self Love…LOVESELF is in paperback form and can be purchased on Amazon and Barnes & Noble websites.
Her English teachers and other English professionals edited the book before it went to publication.
Finding a publisher is usually a tough part of the process, but it didn't take long to secure the right deal. Her reaction when the book was published? "I said 'Thank you Jesus!'," Byrd said. "My family and I hugged each other. It was a beautiful day. It is a blessing to be a blessing. It was a different type of joy." A native of Surry County, Va., Byrd has become somewhat of a celebrity. Recently, she spoke virtually to several organizations as a guest motivational speaker. She was already known for her basketball exploits, leading Surry County High School to three state championship finals in four years. A 5-foot-9 shooting guard, Byrd was named All-State and All-Region four times during her prep career. She was a two-time All-Region Player of the Year. During her senior season, Byrd became a state champion and was named Virginia High School League Female Athlete of the Year. SAU Head Women's Basketball Coach Montel Jones says Byrd definitely fits the type of student-athlete he wants to represent the program. He says his prized freshman recruit is mature beyond her years. "It was amazing to a see a kid not in college yet who had already authored a book," Jones said. "You know she is a high IQ student who knows about more than basketball. It is simply amazing. She already has goals that she has set forth." "She will be successful in athletics and academics," Jones added. "She embodies all the things we are fighting for as far as equality and women's rights. Bre'cha is a strong, powerful black woman who is going after her goal regardless of what's in front of her." Byrd made an impression on Steven Artis, her assistant basketball coach at Surry County High School. Artis says the Lady Falcons got an impactful student-athlete who is a natural leader. "She was far ahead of the other girls in the thought process," Artis said about his former pupil. "She speaks as if she was an old head. Bre'cha had her stuff all mapped out, and she believes God is going to direct her to do great things. She was speaking about this on a high school level." "I was blown away. She was speaking as if she has been some places and seen some things," he added. Naturally, her qualities made Byrd a natural choice as team captain at Surry County High.
"I put a lot of demands on her in making everybody accountable," Artis said. "She said, 'Coach I am already on it.' She embraces leadership. When stuff was going down, she would have team meetings. When you see things like that coming from a young lady, that tells me a lot about her on and off the court." "As a basketball player, she is determined, feisty and very smart," Artis added. "She picks up things very well and has self-motivation which carries over in her writing a book." When asked what gives her more joy, basketball or writing, Byrd says the end result is what's important to her. "I live to inspire in everything I'm involved in," Byrd said. "I want to give the world the best of me." And she's doing it as a teenager. "You do not have to wait until you are older," Byrd said. "You can go after everything right now. I am a symbol of that. There are no limits. Each of you can inspire anyone, no matter the age. You are never too young to make a change."
For more information about Self Love…LOVESELF, contact Bre'cha Byrd on Instagram at @selfloveselfbook and @byrd_girl and Twitter at @_Brecha.
ABC11 also wrote a story on Bre'cha. Read here →
More on Athletics here →

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text:
Ikire este un oraș agricol din statul Osun, Nigeria. În 2006 avea 244.273 de locuitori.

Vezi și
 Lista orașelor din Nigeria

Orașe din Nigeria

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text:
Q: Load Failed error on a project (of multiple projects in a Solution), after Checking out a git branch from SourceTree I was trying to Checkout a remote branch (ModuleA) on Source-tree to my Local Repo. The checkout was successful, however one of the Projects of the Solution (the solution contains 17 projects in all) did not check out correctly. when I open the solution on my Visual Studio 2019, it shows the Project Load Failed. it is missing the .csproj file in the physical location. And when i try to Reload it says

"Value Does not fall within the expected range"

I tried to use the Pull and Fetch options again, but no luck. Is there any way I can pull that particular project from the remote? Is there a workaround?
I have tried various suggestions mentioned online, but it mostly asks me to Delete branch and try again. And I am new to SourceTree (and clearly do not like it, for all the unclear messages it shows, without hinting on how to fix it), So please ignore my lack of knowhow on this.
Note: If it helps, I was earlier having issues with this Branch (ModuleA), with Commit and Push of some code i had done, which was not important and had tried to discard / Reset also. But in vain, so I had deleted the whole Branch. And now when I'm trying to start afresh by checking out the branch from remote, I have this issue. Also Note I had two other branches, which were also deleted.

redpajama_set_name: RedPajamaStackExchange


text:
Santa Cruz's Coastal Watershed Council…
Santa Cruz's Coastal Watershed Council 'walks the walk' in relocating to San Lorenzo River
Coastal Watershed Council's Outreach and Development Manager Laurie Egan, River Scientist Alev Bilginsoy and Executive Director Greg Pepping in the organization's new office. (Shmuel Thaler -- Santa Cruz Sentinel)
Coastal Watershed Council Outreach and Development Manager Laurie Egan works at her desk in the environmental nonprofit's new office, just steps away from the San Lorenzo River. (Shmuel Thaler -- Santa Cruz Sentinel)
PUBLISHED: May 9, 2018 at 12:00 a.m. | UPDATED: September 11, 2018 at 12:00 a.m.
SANTA CRUZ >> The new Branciforte Creek Bridge lands one of its feet just outside the Coastal Watershed Council's new office windows.
The office, marked on the organization's Watershed Ranger kids' activity booklet, will be where area children are invited come to pledge to learn about, care for and share their knowledge about the San Lorenzo River — receiving a small wooden ranger badge in exchange.
On Wednesday, Coastal Watershed Council staff members were in the midst of unpacking and arranging their first-floor office space, located off Dakota Avenue in property owned by weekly publication Good Times. For the previous decade, the 23-year-old nonprofit had called the Santa Cruz Small Craft Harbor home.
The move, though unrelated, came shortly after the city of Santa Cruz finalized updates to its Downtown Plan zoning codes which require new riverfront development to connect to, rather than turn its back on, the San Lorenzo River. All new development is called on to dedicate and/or financially contribute to creation of public pedestrian throughways near the extensions of Cathcart, Maple and Elm streets. The plan also requires riverfront developers to fill space between their building wall and levee, plus negotiate easements for the filled area to create "a positive open space relationship with the Riverwalk."
"It's about all of us getting to the river. Our organization having an office here is part of that. It's an example of that. We're walking the walk," said Greg Pepping, the environmental nonprofit's executive director.
Santa Cruz Economic Development Director Bonnie Lipscomb said the Coastal Watershed Council's relocation is "absolutely" a plus for the city's overall economic development and riverfront use strategies.
"Our goal is to have an active riverfront that is a draw to the community and visitors alike for its natural beauty and as an amenity for recreation, birdwatching and other related activities," Lipscomb wrote in an email. "There is incredible potential for the riverfront to be a key component of overall downtown revitalization efforts and a link from the Tannery to the Boardwalk."
When the Coastal Watershed Council measures success in its mission to improve public perception and use of the river — whether the organization is directly responsible or not — it looks to such changes as the city zoning update, Pepping said. Other positive changes include events such as Senderos' relocation on May 20 of its annual Vive Oaxaca Guelaguetza cultural festival for the first time to San Lorenzo Park; the city's Food Trucks in San Lorenzo Park on May 25; and the Arts Council Santa Cruz County's upcoming annual Ebb and Flow River Arts Celebration on June 1 and 2.
Pepping said he first noticed the 107 Dakota Ave. property was available for lease and began exploring the move during a walking meeting along the Santa Cruz Riverwalk with a donor.
"There was an 'aha,' because it makes such perfect sense for our strategy," Pepping said of the move. "We're focused on the river, we're located on the river."
The nonprofit began taking a laser-focus on the portion of the river approximately running from the Tannery Arts Center to the Santa Cruz Beach Boardwalk about a year ago. Other groups, Pepping said, are working for the betterment of the river upstream, while the river watershed within the city of Santa Cruz has potential for improvement. The organization is in "research mode" on what programs, funding strategies and partners needed to improve river water quality, surrounding public space and overall vibe," Pepping said.


See other counties in North Dakota.
Primarily a resource player, its key assets consist of shale acreage in Woodruff County, Arkansas and McIntosh County, Oklahoma and it also owns a variety of non-producing assets in Alabama, Louisiana, Mississippi, and New Zealand.
Primarily a resource player, its key assets consist of shale acreage in Woodruff County, Arkansas and McIntosh County, Oklahoma, and it also owns a variety of producing and non-producing oil and gas assets in Texas, Alabama, Louisiana, Mississippi, and New Zealand.
McIntosh County, Oklahoma - Touchstone has spudded its initial test well on its Checotah Prospect in the Arkoma Basin.
The Jamie #1 is located in Section 14-10N-17E of McIntosh County, Oklahoma.
OTCBB:TSNU) ("Touchstone" or the "Company") announced today that it has closed a farmout agreement on sixteen thousand gross acres in McIntosh County, Oklahoma.