Terrestrial atmospheric ion implantation occurred in the nearside lunar regolith during the history of Earth’s dynamo

Abstract

Light volatile elements in the lunar regolith are thought to be a mixture of the solar wind and Earth’s atmosphere, the latter sourced in the absence of geomagnetic field. However, the extent to which both the current and primitive geodynamo influence the transport of terrestrial ions still remains unclear, and this uncertainty is further complicated by the enigmatic composition and poorly constrained location of the Eoarchean exosphere. Here we use three-dimensional magnetohydrodynamic numerical simulations with contemporary magnetized and Archean unmagnetized atmospheres to investigate how Earth’s intrinsic magnetic field affects this transfer, aiming to constrain how and when the lunar isotopic signature provides a record of Earth’s paleoatmosphere. We find that atmospheric transfer is efficient only when the Moon is within Earth’s magnetotail. The non-solar contribution to the lunar soil is best explained by implantation during the long history of the geodynamo under present-day solar wind conditions, rather than by any brief, putatively unmagnetized epoch of the early Archean Earth. This further suggests the history of the terrestrial atmosphere, spanning billions of years, could be preserved in buried lunar soils. Our results indicate that the elemental abundances of Apollo samples are highly sensitive to Earth’s hydrodynamic escape boundary, which, at the time of ion implantation, was never smaller than 190 km.

Introduction

Elements such as H, C, N, and the light noble gases in lunar soils are essentially absent from lunar rocks but appear in the lunar regolith. This dearth of volatile elements in lunar rocks has been recognized since the Apollo missions and their exclusive presence in the lunar soil requires an extra-lunar source. One extra-lunar source is commonly recognized to be the solar wind (SW), which directly implants energetic ions on the Moon’s surface. The SW can account for some light volatile elements, but the overabundance of nitrogen compared to standard solar composition and the highly variable isotopic composition (¹⁵N/¹⁴N) has been a challenge to explain.

Lunar nitrogen conundrum

Possible explanations for the enrichment of N in the lunar regolith have included (i) elemental fractionation during lunar processes¹; (ii) N from a non-solar origin^2,3; and (iii) monotonic secular change in the SW due to the high-energy events near the solar surface over the last 2.5 billion years^4,5,6. Lunar soils seem to contain more N than what spallation processes could have generated from incoming cosmic rays^7,8, favoring that some lunar N is extra-solar. Proposed sources of N have included interplanetary dust⁷, outgassing from the lunar volcanoes, fractionated terrestrial atmosphere, and intensive meteorite, cometary or asteroid bombardment shortly after formation of the Moon^9,10,11.

Ozima et al.¹² attributed most of the extrasolar N, along with some other noble gases in lunar soil, to ion flows from Earth’s atmosphere during a putative period when the geomagnetic field was absent early in Earth’s history. Terada et al.¹³ also argued that the lunar soil oxygen isotope ratios are best explained by a contribution from Earth’s atmosphere. Poppe et al.^14,15 observed terrestrial ions in the Earth’s magnetotail, while Li et al.¹⁶ attributed the presence of lunar surface water primarily to the solar wind during the Moon’s passage through the magnetotail, also suggesting that the plasma sheet may play a significant role in its formation, whereas Wang et al.¹⁷ proposed the magnetospheric Earth wind as a potential exogenous source of lunar surface hydration. However, these studies failed to consider how the two end members—solar wind and escaping Earth atmospheric ions—interact and mix during their transport. Besides, these analyses did not include a dynamic model of how N or O ions escaped from the atmosphere, and thus, whether the absence of a geomagnetic field is essential for the transport of such ions from Earth’s atmosphere to the Moon was left unresolved.

Does the geomagnetic field affect the atmospheric mass loss?

The present intrinsic dipolar geomagnetic B-field interacts with the SW, forming a magnetospheric cavity that extends to the wind standoff location at ~8–11 Earth radii (R_⊕) on the dayside and to the distant neutral point at X ~100 R_⊕ on the nightside^18,19. There, the open field lines, which originated from dayside reconnection, again reconnect to form magnetospheric closed field lines (Fig. SD1). Lammer et al.²⁰ argued that such a configuration helps maintain habitability on Earth by protecting it against the deleterious effects of the SW, and reduces atmospheric escape by returning much of the ion outflow back to Earth via the Dungey cycle^21,22,23. This idea is further supported by Seki et al.²⁴, who concluded that the atmospheric loss of oxygen ions from Earth is substantially reduced by the return flux from the magnetosphere to the low-latitude ionosphere. At times, the Moon can be unprotected from the incoming wind near the midnight region due to a sharp deflection of the compressed magnetopause²⁵. But theoretical models have shown that the magnetic field need not always be protective, and Earth, Mars, and Venus have empirically similar mass loss rates despite the latter two lacking an intrinsic dipole magnetic field^26,27,28,29. Nevertheless, differences in atmospheric composition and their role in shaping the thermal structure of the upper atmosphere, which directly influences atmospheric escape, must be taken into account when comparing the loss rates among these planets³⁰. When the key question is habitability, planetary evolution must also be considered when assessing the influence of magnetic field³¹. The debate over the protective role of the magnetic field is likely to persist, given the uncertain composition of early planetary atmospheres, solar wind conditions, and the presently limited data from extrasolar examples that bear directly on early solar system conditions.

Here, we develop a dynamical model of the SW-Earth atmosphere interaction to determine whether Earth’s atmosphere is a viable extra-solar source of volatiles to the lunar regolith (Fig. 1). We first compare the net fraction of Earth’s atmospheric and SW flux carried to the Moon by the SW for both unmagnetized (no dynamo) and magnetized (with dynamo) Earth simulation cases. We then combine these results with photoionization models for the Eoarchean and present-day terrestrial atmospheres, respectively, to infer the relative ratios of these elements at Earth’s hydrodynamic escape boundary and predict their abundance ratios in lunar soil. We here address transport to the large regions on the lunar nearside surface without any magnetic anomalies; therefore, mini-magnetospheres of some small, discontinuous magnetized lunar crust are largely irrelevant. The implanted ions may vary spatially, exhibiting higher concentrations at the edges of the anomalies. However, these mini-magnetospheres do not necessarily fully shield the lunar surface from impinging ions; a higher ram pressure can allow more ions to penetrate, affecting any concentration gradient from the center to the edges³². We also assume that the Moon did not have an appreciable global or large-scale magnetosphere during the interval of study. This was assumed in previous modeling¹² and has been supported by more recent paleointensity data indicating the absence of any long-lived lunar core dynamo after ~4.36 Ga^33,34. As we will discuss below, our results can, in turn, serve as a check on this assumption.

Fig. 1: Schematic representation of the simulation framework for solar wind-Earth atmosphere interactions, incorporating terrestrial atmospheric profiles and the geodynamo model used in this study, along with the coordinate frames employed for analysis.

a Illustrated is the MHD computational domain used to model the interaction of the solar wind with the terrestrial atmosphere. Both the Earth and the Moon lie within the simulation box, while the solar wind impinges on the left boundary (y-z plane) with a velocity normal to the plane. The incoming wind is magnetized, with its field lines aligned along the y-axis. We used the Geocentric-Lunar Orbital Plane coordinate frame, which is tilted by the same angle as the Moon’s orbital plane, to denote the changing phase angle of the Moon in its orbit and to compute the solar and Earth wind number fluxes over a complete lunation. b Static isothermal atmosphere at the start of a run (Case–I) for a present-day magnetized Earth (refer to Equation 5) and an Archean unmagnetized Earth (see Equation 6). The interior of the planet is also assumed to be isothermal. The red and blue shaded regions represent the fixed total mass in the two subcases (~6.0 × 10¹⁸ kg), while the turquoise region illustrates the range of hydrodynamic escape boundary heights used in the binary mixing models. The kink in the red line denotes the ${\upbeta }_{{\rm{p}}}$ = 1 surface, outside of which the magnetic field governs the plasma outflow. c Left panel: Cross-section of the planetary magnetic field $({\overrightarrow{B}}_{\small{\!\!GF}})$ in the noon-midnight meridian plane used in the simulations at the beginning of the runs (t = 0). The color contour shows the log-normalized strength of the B-field (B_⊕ = 3.12 × 10⁴ nT), and the arrows denote the direction of field lines. We use a cutoff boundary of 20 R_⊕ beyond which the influence of the intrinsic field is essentially insignificant. The horizontal axis aligns with the ecliptic plane (marked by the solid black line), while the polar axis (indicated by the dashed red line) points from dusk to dawn. Right panel: Zoomed-in slice of the field structure in the interior of the planet. The solid thick circle represents the planet’s surface.

Results

Relative total terrestrial atmospheric flux implanted on the Moon for an Eoarchean unmagnetized and contemporary magnetized Earth

We analyze the planetary and SW components in the outflow using a single-fluid, multi-species MHD model dynamically, but with passive advective tracers that separately track Earth’s atmosphere material and SW material. The number flux of the terrestrial component is computed using the relation ${{\mathcal{F}}}_{\!\!\small{EW}}={n}_{\small{EW}}\left({\overrightarrow{v}}_{\!\!\!\small{EW}} \cdot {\widehat{n}}_{s}\right)$, while the number flux of the SW component is similarly expressed as ${{\mathcal{F}}}_{\!\!\small{SW}}={n}_{\small{SW}}\,\left({\overrightarrow{v}}_{\!\!\small{SW}} \cdot {\widehat{n}}_{s}\right)$. Here, ${\widehat{n}}_{s}$ is the inward lunar surface normal used to calculate the impinging fluxes, ${v}_{\!\small {SW}}$ and ${v}_{\!\small {EW}}$ stand for the SW flow speed upstream of the bow shock and the Earth wind (EW) speed, respectively. We use the term “Earth wind” to refer to the escaping atmospheric ions when the SW interacts with and flows past the terrestrial atmosphere. ${n}_{\small {SW}}$ denotes the number density of the SW, while ${n}_{\small {EW}}$ represents that of EW.

Here (and throughout the text), the term “magnetized case” is used for simulation runs in which the planet possesses an intrinsic dipolar field along with present-day atmosphere. In contrast, the “unmagnetized case” signifies the absence of such an intrinsic planetary magnetic field, as in the early Archean Earth. Case–I serves as the fiducial scenario, providing a reference for comparing the present-day Sun-Earth-Moon system with conditions during the Eoarchean era. In subcases where the geodynamo is active (denoted as the “Magnetized Earth”), the atmospheric surface density is obtained by integrating the current atmospheric profile over a single scale height (${H}_{\!\small {M}}$). All other cases are specifically utilized to estimate the analytic scaling relations that govern mass loss flux in the MHD regime. The label “Magnetized planet” applies to subcases where the atmospheric surface density, and therefore the total atmospheric mass, is lower than that in the fiducial case.

The mass flux received at the Moon over a full lunation (−π to π rad) exhibits a double-horned structure with global maxima at orbital phase angles ${\phi}_{\!\small {M}}$ ~−π/5 rad and ~π/5 rad in the magnetized subcase of Case–I (Table 1), and at phase angles ${\phi}_{\!\small {UM}}$ ~−π/9 rad and ~π/9 rad in the unmagnetized subcase (Case–I). These maxima coincide with lunar crossings of the respective bow shocks (Figs. 2 and SD2). We used the phase angle in the Geocentric-Lunar Orbital Plane (GLOP) coordinate frame to indicate the different locations of the tidally locked Moon around the Earth. The following sign convention is employed for the phase angle: counterclockwise angles are positive, while clockwise angles are negative. The phase angle starts at roughly −π rad near new Moon, when it is in the dayside upstream SW, and progresses to 0 rad at full Moon in the Earth’s magnetotail. The Moon then moves from 0 rad to near +π rad during the remainder of the cycle (Fig. 1).

Table 1 Stellar wind and planetary atmosphere parameters used in the simulations for the magnetized cases

Fig. 2: Orbital variations of species-total particle fluxes impinging on the lunar surface in the contemporary Earth’s magnetized case (Case−I).

Angular distributions of the (a) planetary atmosphere number flux, (b) solar wind number flux, (c) total particle flux (aggregate flux of the terrestrial atmosphere and solar wind), and (d) fractional contribution of the planetary wind to the total flux over a full lunation. At phase angles −π and π rad, the Moon is in the dayside upstream solar wind along the sub-solar line of Earth. The double-horned structures in the flux profiles result from shocked material located between the magnetosheath and the magnetotail lobe. Both solar wind and Earth wind densities increase sharply near the magnetopause boundary (the shocked region), resulting in increased solar wind and planetary atmosphere fluxes in that region compared to those near the X-line in the magnetotail lobe (at a phase angle of 0 rad). The solar wind number flux is further reduced inside the magnetosphere due to limited mixing with the terrestrial component, compared to the level of mixing at the boundary.

The primordial Sun (prior to ~4.2 Ga) likely emitted a stronger SW³⁵. Obtaining data on Earth’s earliest magnetic field is arguably the most challenging of paleomagnetic measurements, and uncertainties will likely always remain for the Eoarchean to Hadean eras (see Supplementary Materials)³¹. Nevertheless, consistent records of zircon magnetism from two cratonic nuclei suggest the existence of a geodynamo operating as early as 4.2 Ga^36,37. It is possible that Earth was unmagnetized prior to this time. We therefore vary the SW parameters between the magnetized and unmagnetized cases, with the latter representing ancient SW conditions (Tables 1 and SD1). However, in Case–III (Table 2), representing a period when the existence of a dynamo is uncertain, we use identical SW pressure for both the magnetized and unmagnetized subcases to isolate the influence of the field on atmospheric escape. The surface density of the terrestrial atmosphere is not fixed between the magnetized and unmagnetized subcases, but instead emerges from assuming the same total atmospheric mass in both cases. For the subcases in Case–I (“Contemporary Earth” and “Early Archean Earth”), we used the present-day terrestrial atmospheric mass of ~6.0 × 10¹⁸ kg.

Table 2 Orbit-averaged values of solar wind and Earth wind number fluxes reaching the Moon

The fluxes are computed at the nearside sub-Earth point on the Moon. The Apollo soil data to which we compare our results are also from that vicinity^38,39. For over half of its orbit, the Moon is in the SW plasma on the sunlit side of its orbit, and outside of the bow shock. For both the magnetized and unmagnetized cases, we therefore observe that the SW flux gradually increases when the Moon is at first quarter and falls to zero after the third quarter when the sub-Earth point is on the night side. As the Moon ingresses or egresses the magnetosphere, the SW number density abruptly increases at the bow shock, producing a double-peaked structure. Within the magnetotail, the SW flux drops by an order of magnitude. On the other hand, the EW, originating from the terrestrial atmosphere, only adds to the flux within the magnetotail region.

The orbit-averaged species-total SW and terrestrial atmospheric fluxes impinging upon the lunar surface are tabulated in Table 2. In each case, the total atmospheric mass is kept the same to isolate the comparison of SW interaction with a magnetic versus non-magnetic Earth. The Moon is believed to have become tidally locked to Earth shortly after its formation and subsequently drifted away slowly from Earth, from ~15 R_⊕ to the current distance of ~60 R_⊕⁴⁰. We used the current Earth-Moon separation of 60 R_⊕ to calculate SW and EW fluxes for the magnetized cases, denoting the present time, and a lunar distance of 40 R_⊕ for the unmagnetized cases. This lunar distance was attained around 4.0 Ga, which falls within the constraints on the earliest presence of the geodynamo: ~3.5 Ga, based on paleomagnetic studies of extant rocks^41,42,43, and ~4.2 Ga, based on paleointensity data from detrital zircons of Australia and South Africa^36,44,45,46.

For the fiducial case (Case–I), we utilized the present SW velocity and density values for the magnetized sub-case, along with the current terrestrial atmospheric profile of the contemporary Earth. For the unmagnetized sub-case, we used the ancient SW parameters⁴⁷, representing the ancient Sun-Earth-Moon system, and determined the surface density of the atmosphere by conserving the total atmospheric mass from the magnetized case.

The Earth’s atmospheric contribution averaged over one lunar orbit (referred to as “planetary flux”) for the current Earth, with an intrinsic dipole field, is comparable to that observed when there is no intrinsic magnetic field. This results because the magnetic field distends the same amount of atmosphere mass onto a power-law tail rather than an exponential one, allowing more of Earth’s atmosphere to survive at larger radii from stripping, which compensates for any protection the field might provide. This also adds to the reasons why a magnetic field need not always be protective^26,28. In Case–III, however, the ram pressure of the SW, albeit stronger than in Case–I, is kept constant by fixing the values of SW number density and velocity. In Case–III, we isolated the effect of the magnetic field. The EW flux is an order of magnitude smaller in the magnetized subcase, suggesting an overall modest net protection effect. This further suggests that the higher SW flux in the Eoarchean era was comparatively more influential than the presence or absence of an intrinsic magnetic field during that period (Table 2).

In Case–IV, the atmosphere surface density for a magnetized Earth is reduced by five orders of magnitude while maintaining the same SW values as in Case–I. Since this reduction decreases the atmospheric mass available for pickup by the SW, we observe a corresponding decrease in the escaping EW flux. However, the decrease in flux is not proportional to the reduction in base density. This is because the atmosphere’s surface density does not scale linearly with the outflowing planetary flux, as discussed further in the next section.

By further reducing the surface densities in both magnetized and unmagnetized scenarios in Case–II, and increasing the ram pressure of the incoming wind, we observe that the planetary flux is 2.41 × 10⁸ m⁻² s⁻¹ for a magnetic Earth and 3.10 × 10⁹ m⁻² s⁻¹ for a non-magnetic Earth, which are higher than those in any of the previously discussed cases, suggesting that the SW ram pressure plays a more significant role in enhancing the value of the EW flux than the terrestrial mass density in the interaction region close to the respective plasma or magnetopauses. This is more clearly elucidated in Case–V, where the Earth’s atmosphere surface density is increased back to its fiducial value, along with an increase in both SW density and velocity, resulting in a higher average EW flux.

Overall, based on our simulation data in Table 2, we observe that the relative fraction of EW flux compared to the SW flux is not strongly dependent on the magnetic field in most cases, except for Case–III. In that case, where we have isolated the effect of the magnetic field change only, we see an order-of-magnitude reduction in planet flux (column 6).

Specific ion species implanted from Earth’s atmosphere

We estimate the contribution of N and light noble gas fluxes to the lunar soil from both SW and EW using the photoionization model (Tables SD2–SD4). The model calculates ion escape rates by integrating the dayside production above the hydrodynamic escape boundary, assuming that ions produced above this boundary are advected by the SW and escape the planet. The “hydrodynamic escape boundary” defines the altitude limit that the magnetized solar wind must penetrate to extract ions exhibiting a specific ion-composition ratio, based on a pair of isotope ratios from the end-members in each mixing plot. By comparing curves for various ions in the magnetized and unmagnetized cases (Figs. 3 and 4), we infer that the unmagnetized phases do not make a substantial additional contribution to EW deposition compared to the magnetized ones. The various cases for magnetic and non-magnetic Earths are summarized in Table 2.

Fig. 3: Altitude profiles of production rates of volatile ions by photoionization for present-day Earth (with a geodynamo) and Archean Earth (lacking a geodynamo).

Top panel (a): Ionization rates of ${{\rm{H}}}^{+}$ (sky blue), ${{\rm{He}}}^{+}$ (forest green), ${{\rm{N}}}^{+}$ (crimson), ${{{{\rm{N}}}}}_{2}^{+}$ (cerulean blue), ${{\rm{O}}}^{+}$ (turquoise), ${{{{\rm{O}}}}}_{2}^{+}$ (sky magenta), ${{\rm{Ne}}}^{+}$ (dandelion yellow), and ${{\rm{Ar}}}^{+}$ (orange) ions, estimated from a semi-theoretical photoionization model (EUVAC) and an empirical model (NRLMSISE-00) of the neutral densities of the constituents in the terrestrial atmosphere under normal solar EUV flux conditions. Bottom panel (b): Similar to panel (a), but also including ${{\rm{CO}}}^{+}$ (lavender) and ${{{{\rm{CO}}}}}_{2}^{+}$ (coral) for the unmagnetized Earth with an Archean terrestrial upper atmosphere, representing the primordial Earth case. We adopted the homopause composition of volatile species and the solar XUV flux for the Eoarchean era from Johnstone et al.⁷⁷, and incorporated these parameters into our photoionization model of Earth’s early atmosphere.

Fig. 4: Hydrodynamic escape boundary profiles of volatile species escape rates due to the collisionless nature of plasma for present-day Earth (with a geodynamo) and Archean Earth (lacking a geodynamo).

Top panel (a): Ion escape rates (secondary horizontal axis, top x-axis) of different constituent species for magnetized Earth, calculated using the NRLMSISE00 model of the present terrestrial neutral upper atmosphere and ion production rates under normal solar activity condition (as described in Fig. 3) assuming that ions produced above the hydrodynamic boundary¹² escape from the planet. Escape fluxes (primary horizontal axis, bottom x-axis) of ions in the nightside tail region are computed by normalizing the escape rates by the magnetotail cross-sectional area at the apogee point of the Moon’s orbit. Our calculated O⁺ escape rates for present-day Earth show excellent agreement with the estimates provided by Seki et al.²⁴. Bottom panel (b): Same as panel (a), but for unmagnetized Earth with the Archean terrestrial upper atmosphere, representing the primordial Earth case. Both plots directly present the escape flux and escape rate at each escape boundary location, calculated by integrating the ion production rate from that altitude up to the lunar orbit.

The SW effectively captures and mixes ions in the EW-SW interaction region without fractionation by mass, because the ion gyro-radii are small compared to global scales. The pick-up EW ions thereby undergo rapid acceleration, all reaching the same average velocity as the SW. Some of these accelerated ions implant in the lunar soil. Since all ions enter at the same velocity as the SW and the plasma is collisionless before impact, heavier ions from both EW and SW origins may penetrate more deeply than lighter ions, but EW and SW components would not be separately segregated by depth alone. The depth profile may, however, reflect the evolution of Earth’s atmosphere on geological timescales^7,48.

Implications of the variation in isotopic composition of lunar soil due to implanted solar and Earth winds

In order to determine whether the non-solar components of the lunar soil have a terrestrial origin attributable to the EW, we compare the theoretically calculated implanted EW with the observed non-solar fluxes for both the present-day magnetized Earth and the non-magnetized early Earth. Our MHD simulations for these two phases of Earth suggest that mixing between two primary end members, solar and planetary, controls the lunar geochemical variation (Figs. 5 and SD3). The non-solar contribution to the lunar soil is then estimated and compared with the EW flux computed using our theoretical model, as tabulated in Table 3. Theoretical mixing diagrams (Figs. 6 and SD4) characterize the relative proportions of various isotopes sourced from the SW and a non-solar component (NSC), as well as the elemental abundances of different species. The construction and interpretation of these diagrams are explained in detail in the Methods section. For each mixing diagram, exemplified in Fig. 6, we specifically constrain the pairwise hydrodynamic escape boundary (PHE) for that particular combination of isotope ratios, based on the mean molecular mass of each of the two species and the temperature, by identifying the best fit to the data. The overall hydrodynamic escape boundary (OHE) for the atmosphere, and thus the employed lower height bound for our flux integration, is the minimum value that emerges from the set of PHEs across the different mixing diagrams.

Fig. 5: Results from our 3D-MHD simulation model for the case in which the planet possesses an intrinsic dynamo (Case–I).

a Total number density plot depicting the large-scale structure of the wind-atmosphere interaction up to the lunar orbit in the quasi-steady state. b Polar-cut view of the topology of the planet’s magnetosphere when it is impinged by magnetized solar wind. Quivers represent the magnetic field. c 2D slice of the normalized solar wind number flux in the lunar orbital plane. The flux drops by ~4 orders of magnitude inside the magnetotail compared to the shocked solar wind region between the bow shock and the magnetopause boundary, where it exhibits a sharp gradient. d Terrestrial atmospheric number flux in the magnetotail region. The quiver arrows denote the plane-projected bulk velocity component, indicating the escaping pathways. The filled white circle representing the planet’s core is not included in the passive advective tracers.

Table 3 Solar, terrestrial atmosphere, and non-solar fluxes in the lunar nearside regolith for both present-day and primordial conditions

Fig. 6: Mixing diagrams for contemporary Earth with a core dynamo.

Top panel (a) [δ¹⁵N – δD mixing diagram]: A mixing curve between the solar wind component (orange square) and the Earth wind component (navy square) in the magnetized case (Case–I) is constructed using estimated parameters that depend on the end-member isotopic compositions and the curvature parameter (K_c = [N/H]_SW${\displaystyle{/}}$[N/H]_EW). The elemental ratios of the end members are inferred from fluxes given in Tables SD2 and SD3. The mixing hyperbolae are for different probable hydrodynamic escape boundary heights for a magnetized Earth. The solid line indicates the profile for the current PHE at ~ 401 km. Observational data for lunar ilmenite samples from Apollo-17 breccias are plotted as squares and circles, respectively, along with their corresponding error bars. Solid symbols represent mean values of surface grains taken from the depth profile at < 100 nm depth, whereas open symbols represent mean values of whole grains for the entire depth profile (~ 500 nm). For solar wind isotope ratios, we used ¹⁵N/¹⁴N = 2.32 × 10⁻³ and D/H = 1.01 × 10⁻⁵ following Hashizume et al.⁷. Following Ozima et al.¹², we used present-day terrestrial atmosphere isotope ratios, ¹⁵N/¹⁴N = 3.68 × 10⁻³ and D/H = 1.56 × 10⁻⁴, for the Earth wind composition. Middle panel (b) [δ ³He − δ ²⁰Ne mixing diagram]: Mixing curves were generated for different hydrodynamic escape boundary altitudes with respective solar and planetary abundances estimated from Tables SD2 and SD3. Data for ilmenite grains in lunar soils (solid markers) and regolith breccias (open markers) were obtained from Apollo samples. For the solar wind composition (orange square), we used ³He/⁴He = 4.26 × 10⁻⁴ and ²⁰Ne/²²Ne = 13.78¹⁰⁷. For the Earth wind isotopic ratios (navy square), we used the commonly accepted present atmospheric values, ³He/⁴He = 1.38 × 10⁻⁶ and ²⁰Ne/²²Ne = 9.8¹⁰⁸. Bottom panel (c) [δ ³He − δ ⁴⁰Ar mixing diagram]: Six mixing curves are plotted with different elemental abundance ratios but fixed isotopic ratios assumed for the end members. The only variable to construct a unique mixing curve connecting the two end members with fixed isotopic ratios is the elemental abundance ratio, K_c = [He/Ar]_SW${\displaystyle{/}}$[He/Ar]_EW. We chose the generally accepted values of ³He/⁴He = 4.26 × 10⁻⁴ and ⁴⁰Ar/³⁶Ar = 3 × 10⁻⁴ for the solar wind ratio (orange square)¹⁰⁸. For the Earth wind components (navy square), we used ³He/⁴He = 1.38  × 10⁻⁶ and ⁴⁰Ar/³⁶Ar = 296.16 to construct the mixing curves¹⁰⁸. Colored dots in each subplot denote the corresponding increments in ${{\mathcal{f}}}_{\!\!\small{SW}}$.

The respective color bars in the mixing plots (Figs. 6 and SD4) show increments of the SW mass fraction in the mixture. Stated differently, they represent the weight fraction due to the contributions of the two elements from the two end members. Comparing the corresponding subplots in Figs. 6 and SD4 suggests that the SW mass fraction decreases from unity more rapidly for an unmagnetized Earth with an early Archean atmosphere (and less steeply for the contemporary terrestrial atmosphere) as one traverses away from the SW component point, indicated by the orange square. This, in turn, implies that in the present-day atmosphere case, there is a significant amount of mixing between the SW and the non-solar component. Conversely, the mixture in the early Archean Earth case is predominantly solar. Using the computed non-solar fraction for each species, we estimated the non-solar flux by multiplying it by the SW flux values. This estimated non-solar flux—which is a measure of the observed species abundances from the Apollo sample data—was then compared with the EW flux computed using the combined MHD-ionization model for both the magnetized and unmagnetized Earth phases. The combined model explains the data only for the contemporary magnetized Earth case when the EW flux value surpasses the recorded non-solar flux (Table 3).

We utilized isotope ratio data of N and H in lunar regolith grains from Hashizume et al.⁷ to compare with our theoretical mixing curves between the two fixed end-members, SW and EW, for different hydrodynamic escape boundaries in both magnetized and unmagnetized cases. Hashizume et al.⁷ attributed the isotopic variation in lunar nitrogen, and the depletion of ¹⁵N relative to terrestrial nitrogen, to the mixing of a solar and a non-solar component. They assigned the latter to presolar interstellar solids that underwent ¹⁵N enrichment through isotopic fractionation. However, our combined 3D-MHD and ionization model predicts that this implanted non-solar N could have originated from the terrestrial atmosphere if the OHE altitude was below 300 km. This is evident from the mixing plot, where the majority of lunar N and H isotope sample data lie close to the curves for PHE heights less than 300 km (Figs. 6a and SD4a).

The data for other light volatile elements, such as He, Ne, and Ar in lunar ilmenites and regolith breccias, were obtained from Heber et al.⁴⁹, who focused on the temporal variation of these isotopic compositions and revealed a strong correlation among these ratios. Given the absence of known processes capable of temporally altering the Ne isotopic composition in the SW, they argued that the variation in the He isotopic ratio cannot be explained by incomplete H burning and mixing in the Sun’s outer convective zone, which closely aligns with the protosolar composition. Heber et al.⁴⁹ thus attributed these variations to secondary processes, such as grain surface erosion and diffusion. Once again, our theoretical calculations show that these implanted light-element gases can also be well explained by mixing between the SW and EW. Our model best matches the experimental data from lunar soil if the PHE height is ~250 km in Case–I for a magnetic Earth (refer to Fig. 6a, b) and ~275 km in Case–I for a non-magnetic Earth (see Figs. SD4a, b).

Similarly, the observed correlation between He and Ar isotopic ratios⁴⁹ can also be accounted for by both our present-day magnetized (Fig. 6c) and Eoarchean unmagnetized (Fig. SD4c) atmosphere models, which include partly SW- and partly EW-transported constituents, provided the OHE height is close to 190 km. The lunar data also match our binary mixing curves for N-H isotope mixing, exhibiting greater curvature compared to that for He-Ne and He-Ar isotope mixing, as indicated by the mixing diagrams.

From columns 5 and 6 of Table 3, we infer that all species, except H, can be accounted for by terrestrial components transported in the form of EW in the magnetized present-day case but not in the Eoarchean case (Case–I). The Eoarchean model predicts too dominant a SW contribution to species-specific abundance compared to the data due to the intense SW that increases the ratio of SW to EW components around 3.8 Ga ago (refer to columns 10 and 11 of Table 3). That neither model matches the observed lunar H implies that the observed lunar H is non-terrestrial and could very well have a solar or extrasolar origin. Our model’s mixing ratio curves (Figs. 6 and SD4) are strongly sensitive to the PHE height. Although this height is difficult to measure empirically, the sensitivity of the mixing ratio observables to it serves as a theoretical constraint on its evolution.

Discussion

Our findings reveal that Earth’s atmosphere contributes significantly to light volatile elements in the lunar regolith when the Moon is situated in the magnetotail. The calculations, using a combination of 3-D MHD numerical simulations of magnetized SW-Earth atmosphere interactions and a semi-analytic ionization model, demonstrate that material from the present-day Earth’s atmosphere can indeed account for the non-solar contributions to N and noble gas constituents in lunar soil, offering a plausible explanation for the observed isotope ratio differences between solar and non-solar sources.

Only a primitive Earth could have been unmagnetized, a period when the solar mass-loss rate was likely higher and SW ram pressure much stronger. But we find that the relative SW and EW abundances fractions are underpredicted by such an early non-magnetic Earth, primarily due to increased SW power at early times. Our model predictions for the lunar soil, therefore, suggest that its composition reflects a dominant acquisition during the active dynamo phase of Earth. This, in turn, is consistent with the flatter temporal evolution profiles of the SW mass-loss rate⁵⁰, where the time-integrated mass deposition is governed by present-day conditions.

Ozima et al.¹² assigned an age of 3.8–3.9 Ga to the ilmenites used for volatile measurements. This is a common age for some high-Ti lunar basalts that might have been the original hosts of the ilmenites and overlaps with some estimates of the Late Heavy Bombardment (LHB)⁵¹, which could have disrupted the hosts and placed the ilmenites on the surface for ion implantation. However, the occurrence and even the existence of the LHB remain highly uncertain, with ongoing debates regarding both the nature of the event and when it took place^52,53,54,55. Our results indicate that if the implantation ages are this old, the Moon did not have a core dynamo or magnetosphere, as this would have blocked the terrestrial atmospheric transfer needed to exemplify the ilmenite volatile data (Figs. 5, 6, SD3, and SD4). Nevertheless, our model is also consistent with much later implantation ages.

We also isolated the effect of a magnetic field to assess its influence when the presence or absence of a dynamo is unknown. Keeping a fixed total atmospheric mass and fixed SW properties, we found that the magnetic field reduces the escaping EW fraction by about an order of magnitude for a very high SW flux. However, the magnetic field also distends the atmosphere, increasing its surface area exposed to the influence of the SW. This underscores the subtle interplay between the SW, Earth’s dynamo, and the Moon’s orbital characteristics when assessing the net influence of the magnetic field. The competing effects of SW conditions and dynamo strengths warrant further investigation. In the case involving a high-density SW at 1 AU and a distended atmosphere (Table 2, Case–III), the magnetic field provides only modest protection compared to the unmagnetized subcase, indicating that the SW flux played a significant role in driving atmospheric loss during early Earth.

In the standard reconnection model, plasma motion pushes oppositely directed magnetic fields together, leading to reconnection at the distant-tail X-line. During this process, magnetic energy is converted into plasma kinetic energy, thereby accelerating the plasma and producing both Earthward and tailward ion flows in the magnetotail. The Earthward flow travels toward the lunar farside^56,57,58. At these large distances, the SW would dominate the Earthward-directed plasma. Over time, we may expect some mixing between the tailward flow that overshoots the Moon outward and the Earthward flow that overshoots the Moon inward, but still likely leaving the farside deposition more solar-like than the nearside. The orbit-averaged total SW flux to the farside would be similar to that of the nearside, but more dominant than the EW flux.

During any unmagnetized phase, an induced magnetosphere (or, more precisely, a magnetosheath) would form, draping around the Earth, and the field lines might sink into the tail to create an induced magnetotail. However, due to the absence of any terrestrial field during the hypothetical unmagnetized Archean epoch, no reconnection with the draped ambient SW field occurs on the dayside. On the nightside, the plasma pressure is weak enough that the X-line would lie well within the Moon’s orbit. We therefore infer that, during the hypothetical unmagnetized early Earth, there was no appreciable Earthward ion flow capable of causing substantial implantation on the farside. Volatile implantation on the farside would thus have occurred primarily during Earth’s contemporary magnetized phase.

We conclude that the delivery of EW ions, sourced from the terrestrial atmosphere, to the Moon occurred during Earth’s extended magnetized phase with present-day solar wind parameters, as opposed to any brief, putative non-magnetic phase during Archean eon. Our results show that the observed soil isotope mixing ratio data lie on hyperbolae of fixed pairwise hydrodynamic escape boundaries, with slightly different values for different element ratios, but with widely varying total EW-to-SW fractions. This suggests that the latter varied over Earth’s history more than the former. Comparing the model to the data suggests that the lunar soil never acquired substantial material when the OHE boundary was less than 190 km.

Methods

Wind-atmosphere MHD interaction model and computational methodology

We model the intrinsic magnetic field of Earth by the far-field solution of an Earth-centered point-like dipole with dipole moment 8.07 × 10¹⁵ T m³. In order to preclude the magnetic field from diffusing into the highly conductive plasma (in accordance with ideal MHD), the dipole field is confined in the initial frame (near t = 0). The confined magnetic vector potential, ${\overrightarrow{A}}_{\!\!\small {GF}}$ (where ${\overrightarrow{B}}_{\!\! \small {GF}}={\overrightarrow{\nabla }} \times {\overrightarrow{A}}_{\!\!\small {GF}}$), in the initial (t = 0) frame is defined as

$${\overrightarrow{A}}_{\!\!\small {GF}}=\left\{\begin{array}{ll}8\,{B}_{\oplus }\,r\,\sin \theta \,{\left[1-\dfrac{r}{\widetilde{R}}\right]}^{2}\,\widehat{\phi },\hfill&{{{\rm{if}}}}\,r\,\le \,\dfrac{{R}_{\oplus }}{2}\\ {B}_{\oplus }\,r\,\sin \theta \,{\left(\dfrac{{R}_{\oplus }}{r}\right)}^{3}\,{\left[1-\dfrac{r}{\widetilde{R}}\right]}^{2}\,\widehat{\phi },&{{{\rm{if}}}}\,r\, > \,\dfrac{{R}_{\oplus }}{2}\end{array}\right.$$

(1)

where $\widetilde{R}=20\,{R}_{\oplus }$ is the initial B-field cutoff boundary, R_⊕= 6.371 Mm is Earth’s radius, and B_⊕ is the mean value of the magnetic field at the magnetic equator on Earth’s surface. Beyond this boundary, the solar wind ram pressure, together with the thermal and magnetic pressures of the ambient (interplanetary) plasma, exceeds the energy density of the dipole magnetic field. Because the SW is magnetized, the outside ambient field completely drapes around the magnetopause, and the magnetosphere becomes surrounded by the ambient wind magnetic field. As the simulation progresses, the field lines are compressed on the dayside by both the SW dynamic and thermal pressures, while on the nightside they extend into an open, magnetotail-like configuration (Fig. SD1).

Isothermal atmosphere model

In a frame rotating with a constant angular velocity, $\overrightarrow{\Omega}$, the fully compressible, inviscid momentum equation (Euler form) can be written as

$$\dfrac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v} \cdot \overrightarrow{\nabla })\,\overrightarrow{v} =\dfrac{\partial \overrightarrow{v}}{\partial t}+\overrightarrow{\nabla } \left(\dfrac{{v}^{2}}{2}\right)\,-\,(\overrightarrow{v} \times \overrightarrow{\nabla } \times \overrightarrow{v})\\ =-2\,\overrightarrow{\Omega} \times \overrightarrow{v}-\overrightarrow{\Omega } \times \overrightarrow{\Omega } \times \overrightarrow{r}-\dfrac{\overrightarrow{\nabla }p}{\rho }+\overrightarrow{g}$$

(2)

Here $\overrightarrow{v}$ is the velocity in the rotating frame, $\overrightarrow{r}$ is the radius vector, p is the pressure, ρ is the mass density of the atmosphere, and $\overrightarrow{g}$ is the surface gravity. The quantities appearing after the second equality are source and sink terms that provide the acceleration. A useful solution of Equation 2 can be obtained by approximating the atmosphere as being in hydrostatic equilibrium, which is valid for highly subsonic atmospheric flow. Then Earth’s gravity and atmospheric pressure gradient are the dominant balancing forces. For an isothermal atmosphere obeying the ideal gas law, with gravity acting radially inward $\left(\overrightarrow{g}\!=-g\hat{r}\right)$ and no angular dependence of density or pressure, we have

$$\overrightarrow{\nabla }p=-\rho g\hat{r}\Rightarrow \dfrac{p}{{p}_{\circ }}=\dfrac{\rho }{{\rho }_{\circ }}={e}^{-\frac{r-{R}_{\small{\oplus }}}{H}}$$

(3)

where ${p}_{\circ }$ and ${\rho}_{\circ }$ are the surface pressure and density, respectively, and H is the scale height.

The Earth’s magnetic field of order 50 μT near the surface produces a magnetic pressure p^(Mag) of ~10⁻³ Pa (≃10⁻⁸ atm). At the outer reaches of the dayside magnetosphere, where the magnetic pressure begins to dominate the Earth’s atmospheric pressure, the nearly vacuum dipole magnetic field determines the global plasma structure, and the characteristic exponential falloff of the isothermal density profile is no longer valid. In this low plasma-beta (${\upbeta }_{{\rm{p}}}$) regime, where ${\upbeta }_{{{{\rm{p}}}}}$ is the ratio of thermal to magnetic pressure, we model the atmospheric density as a power law ρ ∝ r^−ξ with index ξ > 0. Specifically, by equating the two pressures we obtain the radius $({r}_{{\small \beta }_{ p}})$ where ${\upbeta }_{{{{\rm{p}}}}}$ = 1,

$${p}^{(Atm)}={p}^{(Mag)}\Rightarrow {p}_{\circ }\,{e}^{-\frac{{r}_{{\small{\beta }}_{\small {p}}}-{R}_{\small {\oplus }}}{H}}=\dfrac{{p}_{\small {1}}}{{\left[\dfrac{r}{{r}_{{\small {\beta }}_{ p}}}\right]}^{\xi }} \approx \dfrac{{B}_{\oplus }^{2}}{2\,{\mu }_{\circ }}{\left(\dfrac{{R}_{\oplus }}{r}\right)}^{\!\!6}$$

(4)

${p}_{\small {1}}$ represents the atmospheric thermal pressure at that transition radius. We use the following pressure profile for the atmosphere in the magnetized case:

$${p}_{\!\small {M}}^{(Atm)}=\left\{\begin{array}{ll}{p}_{\circ , {\small {M}}}\,{e}^{-\frac{r-{R}_{\small {\oplus }}}{{H}_{\small {M}}}},&{{{\rm{if}}}}\,{R}_{\oplus } < \,r\le \,{r}_{{\small {\beta }}_{p}}\\ \dfrac{{p}_{\small {1,M}}}{{\left[\dfrac{r}{{r}_{{\small {\beta }}_{p}}}\right]}^{\xi }},\hfill&{{{\rm{if}}}}\,{r}_{{\small \beta }_{p}} < \,r\le {r}_{\small {MP}}\,\end{array}\right.$$

(5)

Here ${H }_{\!\small {M}}$ is the atmospheric scale height during the current magnetized Earth phase. In the case without a planetary dipole field, the atmospheric pressure is assumed to vary as

$${p}_{\!\small {UM}}^{(Atm)}={p}_{\circ ,{\small{UM}}}\,{e}^{-\,\frac{r-{R}_{\small {\oplus }}}{{H}_{UM}}},\,{{{\rm{for}}}}\;{R}_{\oplus } < \,r\le \,{r}_{\!\small {PP}}$$

(6)

${H }_{\!\small {UM}}$ is the scale height of the Archean atmosphere during the unmagnetized era. The plasmapause boundary, which represents the interface between Earth’s plasma and the magnetized SW, is denoted by ${r }_{\!\small {PP}}$. This boundary, along with the pressure at the base of the planet’s atmosphere ($\,{p}_{\circ , {\small{UM}}}$) in the unmagnetized case, is determined by equating the total atmospheric mass to that in the magnetized case.

$${{{{\mathcal{M}}}}}_{\!\small {M}} = \, {\rho }_{\circ ,{\small {M}}} \int\limits_{{R}_{\oplus }}^{{r}_{{\small {\beta }}_{\small {p}}}}{r}^{2}\,{e}^{-\frac{r-{R}_{\small {\oplus }}}{{H}_{\small {M}}}}\,dr\,+\,{\rho }_{\small {1,M}}\int\limits_{{r}_{{\!\small{\beta }}_{\small {p}}}}^{{r}_{\!\small {MP}}}{r}^{2-\xi }\,{r}_{{\!\small{\beta }}_{p}}^{\xi }\,dr\, \\ = \,{\rho }_{\!\circ , {\small {UM}}} \int\limits_{{R}_{\oplus }}^{{r}_{\small {PP}}}{r}^{2}\,{e}^{-\frac{r-{R}_{\small {\oplus }}}{{H}_{\small {UM}}}}\,dr={{{{\mathcal{M}}}}}_{\small {UM}} $$

(7)

The surface density of the atmosphere for the contemporary magnetized Earth case, ${\rho }_{\circ ,{\small {M}}}$, is obtained by computing the total atmospheric mass from the COESA data^59,60 using the following relation

$${\rho }_{\circ ,{\small {M}}} \int\limits_{{R}_{\oplus }}^{{R}_{\oplus }+{H}_{\circ }}\!\!{r}^{2}\,{e}^{-\frac{r-{R}_{\small{\oplus }}}{{H}_{\!\small {M}}}}\,dr\,=\,{H}_{\circ }\,\left\langle {\rho }_{\!\small {COESA}}\right\rangle$$

(8)

Here $\left\langle {\rho }_{\small {COESA}}\right\rangle$ is the altitude-averaged mass density over ${H}_{\circ}$ = 1000 km, assuming that there is no appreciable mass above this height. Balancing the atmospheric thermal pressure with the SW ram and thermal pressures yields

$${r}_{\small {PP}}={R}_{\oplus }+{H}_{\!\small {UM}}\,{\log }_{e}\,\left(\dfrac{{p}_{\circ ,{\small {UM}}}}{{\rho }_{\!\small {SW}}\,{v}_{\!\small {SW}}^{2}/2\,+\,{n}_{\small {SW}}\,{k}_{\small {B}}\,{T}_{\small {SW}}}\right)$$

(9)

We integrate the last term in Equation 7 analytically. Simplifying the expression results in

$$\Xi \,-\,{\varsigma }_{\small {PP}}\,=\,{\log }_{e}\varpi$$

(10)

where

$$\Xi = \,\,{\log}_{e}\!\left(\dfrac{{{{{\mathcal{M}}}}}_{\small {M}}\,{k}_{\small {B}}\,{T}_{\circ }}{4\pi \,{\mu }_{\small {A}}\,{m}_{\small {H}}\,{R}_{\oplus }^{2}\,{H}_{\!\small {UM}}} \times \dfrac{1}{{\rho }_{\small {SW}}\,{v}_{\small {SW}}^{2}/2\,+\,{n}_{\small {SW}}\,{k}_{\small {B}}\,{T}_{\small {SW}}}\right)\\ {\varsigma }_{\small {PP}} = \,\,\dfrac{{r}_{\!\small {PP}}}{{H}_{\small {UM}}}\\ \varpi = \,{\left(\dfrac{{H}_{\small {UM}}}{{R}_{\oplus }}\right)}^{2}\left[2-{e}^{-{\varsigma }_{\small {PP}}}\left({\varsigma }_{\small {PP}}^{2}+2\,{\varsigma }_{\small {PP}}+2\right)\right]\,\\ +\;\dfrac{2\,{H}_{\small {UM}}}{{R}_{\oplus }}\left[1-{e}^{-{\varsigma }_{\small {PP}}}\left(1+{\varsigma }_{\small {PP}}\right)\right]\,+\,\sinh {\varsigma }_{\small {PP}}\,-\,\cosh {\varsigma }_{\small {PP}}+1\\ $$

(11)

${\mu}_{A}$ denotes the mean molecular mass, and ${m}_{\small {H}}$ is the hydrogen mass. The transcendental equation above is solved numerically to obtain ${r }_{\!\small {PP}}$, ${p }_{\circ ,{\small {UM}}}$, and ${\rho }_{\circ ,{\small {UM}}}$.

Parametric analysis of the magnetosphere standoff distance

The magnetosphere standoff distance can be computed from the pressure balance equation by equating the ram pressure of the SW, plus its magnetic and thermal pressures, to the thermal pressure of the Earth’s atmosphere and the magnetic pressure of the dipolar geomagnetic field. Explicitly, this gives

$$\dfrac{{\rho }_{\small {SW}}\,{v}_{\small {SW}}^{2}}{2}\,+\,\dfrac{{B}_{\small {SW}}^{2}}{2\,{\mu }_{\circ }}\,+\,\dfrac{{\rho }_{\small {SW}}\,{v}_{\small {Th,SW}}^{2}}{2}\,=\dfrac{{p}_{\small {1}}}{{\left[\dfrac{r}{{r}_{{\small {\beta }}_{p}}}\right]}^{\xi }}\,+\,\kappa \dfrac{{B}_{\small {GF}}^{2}}{2\,{\mu }_{\circ }}$$

(12)

where ${\rho }_{\small {SW}} = {n }_{\small {SW}}\, {m}_{\small {P}}$ is the mass density of the SW, ${v}_{\small {Th,SW}}$ is the SW thermal velocity, ${B}_{\small {SW}}$ is the magnetic field strength of the incoming wind, and ${B}_{\small {GF}}$ (Equation 1) denotes the geomagnetic field strength. Here, ${r}_{\!\small {MP}}$ stands for the location of the terrestrial magnetopause, and κ ≈ 2 is the dipolar magnetic field compression factor⁶¹.

Solar and planetary parameters

The parameters characterizing the stellar and planetary outflows are the escape velocity parameter and the plasma beta in the wind-atmosphere interaction region. The atmospheric escape speed is given by ${\left[2G{M}_{\oplus }/{R}_{\oplus }\right]}^{\frac{1}{2}}$, while the sound speed at the base of the planet’s atmosphere is ${c}_{s}^{(Atm)}={\left[\gamma\, {p}^{(Atm)}(r){\displaystyle{/}}{\rho }^{(Atm)}(r)\right]}^{\frac{1}{2}}={\left[\gamma\, {k}_{B}{T}_{\circ }{\displaystyle{/}}{m}_{mol}^{(Atm)}(r)\right]}^{\frac{1}{2}}$. The escape velocity parameter can then be defined as

$${{\mbox{Ж}}}^{(Atm)} = {\left[{v}_{Esc}^{(Atm)}\big/{c}_{s}^{(Atm)}\right]}^{2} = {\rm Ma_{p}^2}$$

(13)

where Ma_p is the plasma Mach number. The plasma beta characterizes the dynamical significance of the magnetic field and is given by

$${\upbeta }_{{{{\rm{p}}}}}^{(Atm)}=2\,{\mu }_{\circ }\,{p}^{(Atm)}(r)/{B}_{\!\small {GF}}^{2}(r)$$

(14)

Similarly, the corresponding expressions for ${v}_{Esc}^{\small {(SW)}},\,{c}_{s}^{\small {(SW)}}$, ${{\mbox{Ж}}}^{(SW)}$, and ${\upbeta }_{{{{\rm{p}}}}}^{\small {(SW)}}$ are used for the SW. Table 1 lists the basic model parameters used in the simulations for the contemporary Sun and Earth.

Planet interior model

The planet’s surface is treated as an internal boundary, and the planet’s interior is not dynamically evolved during the simulation runs. The interior parameter values are kept fixed by overwriting them at each time step. Since the interior does not interact with the SW, its physical conditions are computationally unimportant, and we model it as an isothermal sphere. The gravitational acceleration due to the planetary core is then given by

$$\overrightarrow{g}(x,y,z)=-\dfrac{4}{3}\pi \,G\,{\rho }_{\oplus }\,r\,\hat{r}=-\dfrac{4}{3}\pi \,G\,{\rho }_{\oplus }\,\left[x\hat{x}+y\hat{y}+z\hat{z}\right]$$

(15)

Assuming hydrostatic equilibrium, the pressure p_⊕ within the spherical volume due to gravitational compression can be expressed as a function of the radial distance r:

$$\dfrac{1}{{p}_{\oplus }(r)}=\dfrac{1}{{p}_{\circ }}+\dfrac{2}{3}\pi \,G\,{\left(\dfrac{{\mu }_{A}\,{m}_{H}}{{k}_{B}\,{T}_{\circ }}\right)}^{\!\!2} \left[{r}^{2}-{R}_{\oplus }^{2}\right]$$

(16)

where ${p}_{\circ }$ is the atmospheric surface pressure. Finally, for our magnetized planet runs, we approximate the Earth as a roughly uniformly magnetized sphere in the interior $\left(r\le {R}_{\oplus }/2\right)$ with a smooth magnetic field near the center, where singularities would otherwise occur, and retain the dipolar structure outside $r > {R}_{\oplus }/2$ (see Equation 1). At large distances where the planet’s intrinsic dipole field would be very weak, we impose a cutoff boundary ($\widetilde{R}=20\,{R}_{\oplus }$), beyond which the field vanishes.

On the MHD approximation and requisite resolution

Even though the Earth and SW plasmas are collisionless, the gyro-radii of both, using SW field strengths, are smaller than the macroscopic scales of interest, such as the distance from Earth to the bow shock. This crudely justifies simulating the interaction between the SW and atmosphere using the MHD approximation to gain some basic insights.

The numerical simulations must resolve a pressure scale height of the planetary atmosphere that is small compared to the dynamical scales of interest, but not so small as to present practical computational limitations. The pressure scale heights equal the density scale heights for the isothermal equations of state that we employ here. Since 3-D numerical MHD simulations are computationally expensive, we model the star-planet interaction using a grid with multiple levels of refinement, finely resolving only the regions of interest. We set up a static, locally refined grid in a planetocentric frame of reference so that the location of the planet remains fixed throughout the run. Working in the planet frame also simplifies the treatment of the planet’s core as a fixed, bounded region that does not evolve during the simulations.

We carry out our global MHD simulations with AstroBEAR^62,63, a massively parallelized adaptive mesh refinement (AMR) code that includes a variety of multiphysics solvers, such as self-gravity, magnetic resistivity, heat conduction, radiative transport, and ionization dynamics. AstroBEAR has been previously utilized, benchmarked, and tested for modeling interactions between planetary atmospheres and stellar winds^64,65,66. We validated our results by comparing them with those from other widely used codes for modeling star-planet interactions. For the magnetized cases, our simulation results align well with those from the BATS-R-US MHD code and the PLUTO code^67,68. Besides, the magnetic field strengths obtained in our models are also consistent with values reported in similar global MHD simulations of the solar wind-terrestrial magnetosphere interaction^69,70,71. We solve the MHD equations in the planetocentric reference frame on a Cartesian grid in the following conservative form:

$$\dfrac{\partial \rho }{\partial t}+\overrightarrow{\nabla } \cdot \left[\rho \overrightarrow{v}\right]=0$$

(17)

$$\dfrac{\partial }{\partial t}\left( \rho \overrightarrow{v} \right)+\overrightarrow{\nabla } \cdot \left[\rho\, ( \overrightarrow{v} \otimes \overrightarrow{v})-\overrightarrow{B} \otimes \overrightarrow{B}+\left( p+\dfrac{{B}^{2}}{2} \right) \overleftrightarrow{I} \right]=\rho \overrightarrow{g}$$

(18)

$$\dfrac{\partial E}{\partial t}+\overrightarrow{\nabla }\cdot \left[\left( E+p+\dfrac{{B}^{2}}{2}\right)\overrightarrow{v} - \overrightarrow{B}\left(\overrightarrow{B}\cdot \overrightarrow{v}\right)\right]=\rho\, \overrightarrow{v}\cdot \overrightarrow{g}$$

(19)

and

(20)

where ρ is the mass density, $\overrightarrow{v}\!$ is the fluid velocity, and the magnetic field is given by $\overrightarrow{B}\!$. Equations 17 to 20 are written in rationalized electromagnetic units (in which the magnetic permeability μ = 1), as defined by Cunningham et al.⁶². Equation 17 is the continuity equation, describing the conservation of plasma mass. The conservation of momentum density $(\rho \overrightarrow{v})$ in the fluid is given by Equation 18. The quantity ${P}_{\small {T}}=p+\frac{{B}^{2}}{2}$ denotes the total pressure, which consists of the thermal pressure of the gas (p) and the magnetic pressure. $\overrightarrow{g}\!$ is the gravitational acceleration experienced by the fluid due to the field of the planet. Equation 19 represents the evolution of the total energy density $E\,\left(E=\frac{p}{\gamma -1}+\frac{1}{2}\rho {v}^{2}+\frac{{B}^{2}}{2}\right)$, which is the sum of the internal, kinetic, and magnetic energy densities, respectively. Finally, the magnetic induction associated with fluid motion is described by Equation 20.

We solve these “ideal” MHD equations, i.e., without explicit viscous or resistive terms, to maximize the dynamic range of the simulations, because the phenomenon we are interested in, namely, the atmospheric escape due to electromagnetic star-planet interaction, is at least in principle not much affected by magnetic field diffusion into the upper atmosphere⁷². Throughout most of the magnetosphere, collision frequencies are sufficiently low to be negligible compared to the ion gyro-frequency, and an estimate for transport is given by the Bohm diffusion coefficient (D_B), which has the form

$${D}_{B} \sim n\,{T}_{i}^{-\frac{3}{2}}\,{r}_{g,i}^{\,\,2}\,\gg {\nu }_{m}=\frac{\eta }{{\mu }_{\circ }}$$

(21)

where r_g,i is the ion gyroradius, T_i denotes the ion temperature of the plasma, and ν_m is the classical magnetic diffusivity. This (Bohm) is the fastest possible diffusion, applicable only when the mean free path approaches the ion gyroradius. However, the magnetospheric plasma is essentially collisionless, with the ion mean free path being much larger than the typical size of the magnetosphere. The magnetic Reynolds number (the ratio between the advective and diffusive terms in the induction equation) provides an estimate of the comparative effects between the induction of the magnetic field by the motion of the conducting plasma and magnetic diffusion:

$$R{e}_{m} =\dfrac{\left| \overrightarrow{\nabla }\times (\overrightarrow{v} \times \overrightarrow{B})\right| }{\left| \dfrac{\eta \,{\nabla }^{2}\overrightarrow{B}}{{\mu }_{\circ }}\right| } \sim \dfrac{{\mu }_{\circ }\,{v}_{\!\small {f}}\,{L}_{\small {S}}}{\eta } \sim \dfrac{B\,{L}_{\small {B}}}{\eta }\sqrt{\dfrac{{\mu }_{\circ }}{\rho }}\\ \sim \dfrac{6.71 \times 1{0}^{12}}{\ln \Lambda }\,{\left(\dfrac{n}{1\,{{{{\rm{m}}}}}^{-3}}\right)}^{\!-\frac{1}{2}}\left(\dfrac{B}{1\,{{{\rm{nT}}}}}\right)\,\left(\dfrac{{L}_{\small S}}{1\,{{{\rm{Mm}}}}}\right)\,{\left(\dfrac{{T}_{i}}{1{0}^{5}\,{{{\rm{K}}}}}\right)}^{\!\frac{3}{2}}$$

(22)

Here, $\ln \Lambda$ is the Coulomb logarithm, which characterizes the effectiveness of Coulomb collisions between charged particles in a weakly coupled plasma; η is the Spitzer resistivity, and L_S denotes the typical length scale of the magnetospheric flow. In the second term on the right, we assume the outflow velocity (${v}_{\!\small {f}}$) to be the characteristic velocity scale for fully developed magnetohydrodynamic motion in the system. Typically, Re_m ~10¹⁵ (corresponding to L_S ~10 Mm and ${v}_{\!\small {f}}$ ~280 km/s) in the real magnetotail plasma, and ~ 10¹¹ (corresponding to L_S ~1 Mm and ${v}_{\!\small {f}}$ ~200 km/s) at the magnetopause. At such large Re_m (Re_m≫ 1), advection and induction dominate over diffusion, the magnetic field is effectively frozen into the plasma, and the diffusion term in Equation 20 can, in principle, be ignored.

However, the simulations are not actually devoid of diffusive effects and we are not truly solving ideal MHD because of numerical diffusion. The effective numerical value of Re_m is probably closer to ~10³, which is much less than the ~10¹¹–10¹⁵ range for the actual magnetospheric plasma estimated above. Numerical diffusion mediates reconnection in the thin current sheets at the center of the magnetotail and also in determining the thickness of the bow shock. Figure 1 illustrates the simulation domain and the Cartesian coordinates used. (i) Axes: The x-axis is taken along the line connecting the center of the planet and star with the positive direction directed towards the antipode of the substellar point. The y-axis points toward the north ecliptic pole, which also coincides with the planet’s spin-aligned magnetic dipole axis along the S to N zero-tilt magnetic poles. The z-axis completes the orthogonal right-handed coordinate system. (ii) Origin: All models share the same geometrical configuration, with a non-rotating planet located at the center of the coordinate system and assumed to orbit in the x-z plane.

The simulations were performed in a Cartesian mesh having a base grid of 55 × 48 × 48 cells and with a base resolution of 10⁴ km (≈ 6.68 × 10⁻⁵ AU) for the magnetized case. We applied three additional levels of AMR around the magnetotail and four additional levels around the planet and its atmosphere, yielding a resolution of 10⁴${\displaystyle{/}}$2³= 1.25 × 10³ km in the tail region and a resolution of 10⁴${\displaystyle{/}}$2⁴ = 6.25  ×  10² km in the atmosphere. For the case with a non-magnetic planet, we used a 50 × 20 × 20 cells base grid with a 10⁴ km resolution along with two additional levels (five in total) of AMR, providing a resolution of 312.5 km in the downtail region of plasma outflow. The planet itself, along with its atmosphere, was resolved with Δx, Δy, Δz ≃ 156.25 km in this case.

We initialized the entire box outside the planet’s core and atmosphere with a magnetized ambient medium, moving at a velocity equal to that of the SW. The latter is consistent with an isothermal Parker model⁷³ at 1 AU, associated with a given coronal temperature and dipolar magnetic field for a representative stage in Earth’s evolution. Except for the left boundary (y-z plane), through which the SW enters, all other external boundaries of the simulation box have outflow boundary conditions. The simulations presented here were evolved until they reach a quasi-steady state. The numerical results were determined to be in a quasi-steady state when the time scale over which the planet loses its atmosphere (t_Loss) far exceeds the simulation time (t_Sim). That is,

$${t}_{Loss}=\dfrac{{M}_{\!Atm}}{{\dot{M}}_{\!Atm}}\ \gg \ {t}_{Sim}$$

(23)

where ${\dot{M}}_{\!Atm}$ represents the mass loss rate of the atmosphere, and ${M}_{\!Atm}$ refers to the total atmospheric mass. Each simulation is run for a duration equal to two wind-crossing times, with the total run time being the same across all simulations. The time step in our simulations is limited by the Courant-Friedrichs-Lewy criterion and is on the order of ~10⁻³ s⁷⁴.

Solar wind-atmosphere interaction for an unmagnetized Earth

We also carried out runs for an unmagnetized Earth. Throughout Earth’s history, the SW has been both strongly supersonic and super-Alfvénic. The height above Earth’s surface where the wind plasma shocks occur is just outside the standoff location determined from pressure balance (refer to Equation 12) and is approximately determined by the point where the wind ram pressure equals the atmospheric thermal pressure for an unmagnetized Earth.

We do not include the Sun in our simulations; rather, we model the direct exposure of the planetary atmosphere to the stellar wind, which leads to its depletion, and focus on the evolution thereafter. The interactions are shown in Figs. 5 and SD3, which provide face-on views of the lunar orbital plane. Both the stellar and planetary winds are depicted with arrows and the magnetic fields with solid lines. We note that only the approaching SW is supersonic, whereas the EW escaping from the terrestrial atmosphere and reaching the Moon (an orbit of radius ~60.21 R_⊕, centered on Earth) is subsonic.

When evaluating the average duration of the Moon passing through Earth’s tail in one orbit, we take into account the changing Earth-Moon distance over time. The gap between Earth and the Moon has been gradually widening due to tidal dissipation, a process that has been ongoing since the formation of the Earth-Moon system.

Reduction of chamberlain profiles to the barometric law

In quasi-collisionless terrestrial exospheres, where the only controlling factors are gravitational attraction and thermal energy conducted from below, atmospheric particles follow one of three distinct types of trajectories: ballistic, satellite, or escaping⁷⁵. Particles in ballistic orbits, characterized by negative total energy, are gravitationally bound to the planet. They rise from the classical exobase in elliptical orbits and eventually fall back to it if they do not undergo collisions. Satellite-orbit particles, like those in ballistic orbits, also have negative total energy but never cross the exobase. These neutral particles follow bound elliptical orbits with perigees above the exobase due to a balance between rare collisions, particularly close to the classical exobase, and destructive processes such as photoionization. Escaping neutral particles, on the other hand, have positive total energy, originate from the exobase, and follow hyperbolic trajectories. These particles can escape a planet’s gravitational field if their thermal velocities exceed the escape velocity (Jeans escape)⁷⁶. Chamberlain⁷⁵ developed an analytical framework to estimate the density of each population based on the invariance of the distribution function along a given orbit. Assuming a Maxwellian distribution at the classical exobase, the distribution is subsequently propagated to higher altitudes using Liouville’s theorem. The resulting total particle number density, ${n}_{\small {C}}(r)$, is then given by the product of the barometric profile and the total partition function, ${\zeta }_{\!\small {T}}$, of the three population classes.

$${n}_{\small {C}}(r)=\widetilde{{n}_{\small {C}}}(r={r}_{\!\small {Ex}})\,{e}^{-({\lambda }_{Ex}-\lambda )}\,{\zeta }_{\small {T}}(\lambda )$$

(24)

where λ = λ_Ex${\displaystyle{/}}$r is the ratio of the gravitational potential to thermal energy of the neutral particles, and $\widetilde{{n}_{\small {C}}}(r={r}_{\!\small {Ex}})$ is the number density at the exobase level. For an altitude-varying gravitational field, the gravothermal parameters in the exponent can be written as

$${\lambda }_{Ex}-\lambda =\dfrac{r}{{r}_{\!Ex}}\left(\dfrac{r-{r}_{\!Ex}}{{H}_{\!\small {UM}}}\right)$$

(25)

The total partition function is expressed as

$${\zeta }_{\small {T}} = \, {\zeta }_{\small {Bal}}+{\zeta }_{\small {Sat}}+{\zeta }_{\small {Esc}}\\ = \dfrac{1}{\sqrt{\pi }}\left\{\Gamma \left(\dfrac{3}{2}\right)+{\gamma }_{\small {C}}\left(\dfrac{3}{2},\,\lambda \right)-{e}^{-{\psi }_{\!\small {C}}}\,\sqrt{1-{\left(\dfrac{\lambda }{{\lambda }_{Ex}}\right)}^{2}}\,\left[\Gamma \left(\dfrac{3}{2}\right) -{\gamma }_{\small {C}}\left(\dfrac{3}{2},\,\lambda -{\psi }_{\!\small {C}}\right)\right]\right\}\\ = \dfrac{1}{2}+\dfrac{{P}_{\!\small {C}}\left(\dfrac{3}{2},\,\lambda \right)}{2}-\dfrac{{e}^{-{\psi }_{\!\small {C}}}}{2}\,{Q}_{\small {C}}\left(\dfrac{3}{2},\,\lambda -{\psi }_{\!\small {C}}\right)\,\sqrt{1-{\left(\dfrac{\lambda }{{\lambda }_{Ex}}\right)}^{2}}$$

(26)

Here, ${\psi }_{\!\small {C}}={\lambda }^{2}\left/\right.\left(\lambda +{\lambda }_{Ex}\right)$. Γ and ${\gamma }_{\!\small {C}}$ denote the complete and incomplete gamma functions, respectively. ${P}_{\!\small {C}}$ is the regularized lower gamma function, while ${Q}_{\small {C}} = 1 - {P}_{\!\small {C}}$ is the regularized upper gamma function.

To determine the planetary outflow flux in the unmagnetized case using Chamberlain profiles above the classical exobase for the Archean Earth, as described by Johnstone et al.⁷⁷, we begin by estimating the surface atmospheric density ${\rho }_{\circ ,{\small {C}}}$, the exobase pressure $\widetilde{{p}_{\!\small {C}}}$, and the location of the plasmapause boundary ${r}_{\small {PP,C}}$. At the standoff boundary, the pressure balance equation gives

$$\dfrac{{\rho }_{\small {SW}}\,{v}_{\small {SW}}^{2}}{2}\,+\,{n}_{\small {SW}}\,{k}_{\small {B}}\,{T}_{\small {SW}}=\widetilde{{p}_{\small {C}}}(r={r}_{\small {Ex}})\,{e}^{-({\lambda }_{Ex}-{\lambda }_{PP})}\,{\zeta }_{\small {T,PP}}(\lambda ={\lambda }_{\small {PP}})$$

(27)

In terms of the surface density, this is

$$\dfrac{1}{\rho }_{\circ ,{\small {C}}}=\dfrac{{k}_{B}\,{T}_{\circ }}{{\mu }_{A}\,{m}_{H}} \times \dfrac{{\zeta }_{\!\small {T,PP}}\,{e}^{-({\lambda }_{Ex}-{\lambda }_{PP})}}{{\rho }_{\!\small {SW}}\,{v}_{\small {SW}}^{2}/2\,+\,{n}_{\small {SW}}\,{k}_{\small {B}}\,{T}_{\!\small {SW}}}\,{e}^{-\frac{{r}_{\!\small {Ex}}}{{R}_{\small {\oplus }}}\left(\frac{{r}_{\!\small {Ex}}-{R}_{\small {\oplus }}}{{H}_{\!\small {UM}}}\right)}$$

(28)

Following Equation 7, mass conservation in the Archean case with Chamberlain profiles yields

$${{{{\mathcal{M}}}}}_{\small {M}}=4\pi \,{\rho }_{\circ ,{\small {C}}}\,{R}_{\oplus }^{2}\,{H}_{\!\small {UM}}\,{\varpi }_{\!\small {C}}\,+4\pi \,{\rho }_{\circ ,{\scriptsize {C}}}\,{e}^{-{\varsigma }_{\small {C}}\frac{{r}_{\small {Ex}}}{{R}_{\small {\oplus }}}}\int\limits_{{r}_{\small {Ex}}}^{{r}_{\small {PP,C}}}{r}^{2}\,{\zeta }_{\!\small {T}}\,{e}^{-\frac{r}{{r}_{\small {Ex}}}\left(\frac{r-{r}_{\small {Ex}}}{{H}_{\!\small {UM}}}\right)}\,dr$$

(29)

where

$$ {\varsigma }_{\small C} = \dfrac{{r}_{\small {Ex}} - {R}_{\oplus}}{{H}_{\small {UM}}}\\ {\varpi}_{\small {C}} = {\left(\dfrac{{H}_{\small {UM}}}{{R}_{\oplus }}\right)}^{\!\! 2} \int\limits_{0}^{\varsigma_{\small {C}}} {{{\mathcal{r}}}^{2}} {e}^{-{{\mathcal{r}}} - \frac{H_{\small {UM}}}{R_{\small\oplus}} {{{{\mathcal{r}}}^{2}}}} d{{\mathcal{r}}}\, +\,\dfrac{2\,H_{\small {UM}}}{R_{\oplus}} \int\limits_{0}^{\varsigma_{\small {C}}} {{\mathcal{r}}} \,{e}^{{-{\mathcal{r}}} - \frac{H_{{UM}}}{R_{\small\oplus}} {{\mathcal{r}}}^2} \, d{{\mathcal{r}}} \,+\, \int\limits_{0}^{\varsigma_{\small {C}}} {e}^{{-{\mathcal{r}}} - \frac{H_{\small {UM}}}{R_{\small\oplus}} {{\mathcal{r}}}^2} \, d{{\mathcal{r}}} $$

(30)

${{\mathcal{r}}}$ is a dummy variable. Equation 29 can be rearranged as

$$\dfrac{1}{{\rho }_{\circ ,{\small {C}}}}=\dfrac{4\pi \,{R}_{\oplus }^{2}\,{H}_{\small {UM}}\,{\varpi }_{\small {C}}}{{{{{\mathcal{M}}}}}_{\small {M}}}\,+\dfrac{4\pi \,{e}^{-{\varsigma }_{\small {C}}\frac{{r}_{\small {Ex}}}{{R}_{\small {\oplus }}}}}{{{{{\mathcal{M}}}}}_{\!\small {M}}}\int\limits_{{r}_{\!\small {Ex}}}^{{r}_{\small {PP,C}}}{r}^{2}\,{\zeta }_{\!\small {T}}\,{e}^{-\frac{r}{{r}_{\!\small {Ex}}}\left(\frac{r-{r}_{\small {Ex}}}{{H}_{\small {UM}}}\right)}\,dr$$

(31)

Substituting Equation 31 in Equation 28, we obtain

$$\dfrac{4\pi \,{R}_{\oplus }^{2}\,{H}_{\!\small {UM}}\,{\varpi }_{\small {C}}}{{{{{\mathcal{M}}}}}_{\!\small {M}}}\,+\dfrac{4\pi \,{e}^{-{\varsigma }_{\small {C}}\frac{{r}_{\small {Ex}}}{{R}_{\small {\oplus }}}}}{{{{{\mathcal{M}}}}}_{\small {M}}}\int\limits_{{r}_{\!\small {Ex}}}^{{r}_{\small {PP,C}}}{r}^{2}\,{\zeta }_{\small {T}}\,{e}^{-\frac{r}{{r}_{\small {Ex}}}\left(\frac{r-{r}_{\!\small {Ex}}}{{H}_{\!\small {UM}}}\right)}\,dr \\ =\dfrac{{k}_{B}\,{T}_{\circ }}{{\mu }_{A}\,{m}_{H}} \times \dfrac{{\zeta }_{\small {T,PP}}\,{e}^{-({\lambda }_{Ex}-{\lambda }_{PP})}}{{\rho }_{\small {SW}}\,{v}_{\small {SW}}^{2}/2\,+\,{n}_{\small {SW}}\,{k}_{\small {B}}\,{T}_{\small {SW}}}\,{e}^{-\frac{{r}_{\small {Ex}}}{{R}_{\small {\oplus }}}\left(\frac{{r}_{\small {Ex}}-{R}_{\small {\oplus }}}{{H}_{\!\small {UM}}}\right)}$$

(32)

Solving the above transcendental equation numerically gives ${r}_{\!\small {PP,C}}$ and ${\rho }_{\circ ,{\small {C}}}$. If the gravitational field does not vary significantly with altitude, the barometric profile follows an exponential decay. The total partition function in that case is obtained by replacing

$${\lambda }_{Ex}\longrightarrow \dfrac{r}{{H}_{\!\small {UM}}},\lambda \longrightarrow \dfrac{{r}_{\small {Ex}}}{{H}_{\!\small {UM}}}$$

(33)

The rest of the derivation follows as described above. We observe that, in both the varying and constant gravity scenarios, the Chamberlain models closely resemble the respective isothermal profiles (Fig. 7).

Fig. 7: Normalized density of various population classes in the Archean exosphere according to the Chamberlain model.

Top panel (a): Comparison of Chamberlain profiles across different population partitions with the barometric law, accounting for altitude-dependent gravity. Here, $\widetilde{{n}_{\small {C}}}$ represents the number density of neutral atmospheric particles at the exobase, located at an altitude of 1000 km. The atmospheric surface density and plasmapause location are determined by applying the pressure-balance condition at the stagnation point, assuming that the total atmospheric mass is conserved and equivalent to that of contemporary Earth. Bottom panel (b): Same as panel (a), but with constant gravity, resulting in an exponential falloff of the number density. In both scenarios, the net populations obtained using the Chamberlain method closely follow the barometric law.

Determining the flux of specific ion species

The model discussed so far has focused on deriving the relative flux of bulk SW and atmospheric deposition on the Moon. The MHD model does not distinguish between specific ion species; to achieve that, we introduce an ionization model to be combined with the calculations from the bulk flows.

Solar extreme ultraviolet (EUV) radiation and soft X-rays shortward of ≲900 ${{\text{\AA}}}$ are predominantly absorbed within the Earth’s thermosphere, leading to the ionization of various atmospheric species. This not only forms the ionosphere but also photodissociates molecular components via direct interaction with long-wavelength photons and through indirect processes driven by ionization. The most important of these indirect processes is the generation of energetic electrons, which carry the surplus energy imparted by photoionization. These photoelectrons can further interact with neutral species, causing ionization, dissociation, and excitation, as well as a cascade of elastic and inelastic processes that transfer the initial photon energy into the kinetic energy of various thermospheric or ionospheric constituents. We calculate the orbit-averaged flux for each species by taking temporal averages.

Ionization model

For our calculations, we assume that the upper atmosphere has its base at the ionopause and extends upward to the radius where it merges with the stellar wind, or equivalently, with the ambient medium.

On Earth, at altitudes above ~80 km, the atmosphere becomes ionized by solar radiation, forming a dense plasma layer of ionized atmospheric gases and free electrons. Below this altitude, the principal atmospheric constituents are present in essentially the same relative abundances as at the surface of the Earth, but above this level, the abundances differ. Above the present-day homopause, at a geodetic altitude of ${z}_{\small {H}}$ ≈ 120 km, the atmosphere is in approximate collisional diffusive equilibrium up to the conventionally defined exobase, and the number density of each constituent species decreases exponentially with altitude. For our assumed isothermal model, this relationship follows:

$${n}_{i}(z)={n}_{i}({z}_{\!\small {H}})\,{e}^{-\frac{z-{z}_{\small {H}}}{{H}_{i}(z)}}$$

(34)

so that the constituents gets distributed vertically according to their individual molecular weights. Here, ${n}_{i}({z}_{\small {H}})$ is the neutral density of the i^th species at the homopause and H_i(z) is the neutral gas scale height defined as

$${H}_{i}(z)=\dfrac{{k}_{\!\small {B}}\,{T}_{ts}}{{m}_{i}\,g(z)}$$

(35)

where T_ts is the thermosphere temperature and g(z) is the altitude-dependent gravity. The radiative transfer calculations of stellar EUV energy deposition into the upper atmosphere are simplified due to the dominance of absorption. We also make the following simplifying assumptions:

• i. short-wavelength X-rays dominate photoionization,

• ii. the atmosphere consists only of those neutral absorbing species that are dominant at the homopause, whose number densities decrease exponentially with altitude, each following its respective scale height, and

• iii. the atmosphere is considered to be plane-parallel and horizontally stratified.

As the photon flux penetrates the atmosphere, it is attenuated by absorption. The solar actinic flux for each wavelength at different atmospheric levels is calculated by applying the Beer-Lambert law layer-by-layer:

$$\Phi (z,\,\lambda ,\,{\vartheta }_{s})={\Phi }_{\small {\infty }}(\lambda )\,{e}^{-{\tau }_{\small {\lambda }}}$$

(36)

where the monochromatic optical depth ${\tau }_{\small {\lambda }}$ for a multispecies atmosphere as a function of altitude z, wavelength λ, and the solar zenith angle ϑ_s is given by

$${\tau }_{\small {\lambda }}=\tau (z,\,\lambda ,\,{\vartheta }_{s})=\sec {\vartheta }_{s}\int\limits_{z}^{\infty }{\sum_i}{n}_{i}(z)\,{\sigma }_{i}^{(abs)}(\lambda )\,dz$$

(37)

Here, ${\sigma }_{i}^{(abs)}$ is the absorption cross section. Using the estimated neutral density of the constituents and their corresponding absorption cross sections above the homopause, the total ion production rate of each ion species is

$${S}_{tot,i}(z,\,{\vartheta }_{s})={n}_{i}(z) \int\limits_{0}^{{\lambda }_{c}}{\Phi }_{\!\small {\infty }}(\lambda )\,{e}^{-{\tau }_{\small {\lambda }}}\,{\sigma }_{i}^{(ion)}(\lambda )\,d \lambda$$

(38)

Importantly, the “hydrodynamic escape boundary” altitude, constrained from our mixing diagrams, is not a constraint on where the mean free path of atmospheric particles becomes large compared to the particle-specific atmospheric scale height, as in the traditionally defined exobase. Rather, it represents a constraint on the altitude that the magnetized SW must access to extract ions with the associated ion-composition ratio, given a pair of isotope ratios of the end-members in each mixing plot. For us, this determines the lower bound of the altitude range over which we integrate to compute the total fluxes arriving at the Moon. We refer to this as the “hydrodynamic escape boundary” to distinguish it from the usual exobase defined by the collisionality condition.

To calculate the escape flux of different species resulting from ion pick-up above the hydrodynamic escape boundary, we first determine the ion production rates above the homopause. We employ the NRLMSISE-00 model⁷⁸ as input. This empirical model characterizes the neutral densities of constituents in the terrestrial atmosphere. The EUVAC flux model⁷⁹, driven by variations in the 10.7 cm solar radio flux, is used for the reference solar spectrum. This model is based on the measured F74113 solar EUV reference spectrum⁸⁰ and provides fluxes across 37 wavelength intervals, covering a range from 50 to 1050 ${\text{{\AA}}}$. In the EUVAC model, the solar flux is obtained from the reference spectra using the following scaling relation:

$${\Phi }_{\!\small {\infty }}(\lambda )=\dfrac{{\Phi }_{ref}(\lambda )}{{d}^{2}}\,\left[1+{A}_{\lambda }\left(P-80\right)\right]$$

(39)

where Φ_ref (λ) is the binned reference flux, d is the distance of the planet from its host star in AU, A_λ is the scaling factor for each wavelength interval, and $P=\left({F}_{10.7}+\left\langle {F}_{10.7}\right\rangle \right){\displaystyle{/}}2$ accounts for variations in solar activity. Here, F_10.7 is the solar radio flux index, and $\left\langle {F}_{10.7}\right\rangle$ is its 81-day centered mean. We take ${F}_{10.7}=\left\langle {F}_{10.7}\right\rangle =150\,{{{\rm{W}}}}\,{{{{\rm{m}}}}}^{-2}\,{{{{\rm{Hz}}}}}^{-1}$, representing a typical EUV flux and level of solar activity. The ion production rates are then computed from the photo- and electron-impact ionization of neutrals by the incident stellar radiation (Fig. 3).

Assuming that ions produced above the hydrodynamic escape boundary are picked up by the SW and escape from the planet, both in the magnetized and unmagnetized cases, we estimated the total ion escape rate by integrating the ion production rate over the dayside planetary surface area above the escape boundary.

The picked-up terrestrial atmospheric ions flow with the SW and are carried anti-sunward down the magnetotail within the cross section of the magnetosphere. The roughly paraboloid-shaped magnetopause has an area of about 1.06 × 10¹¹ km² for magnetized Earth and 6.99 × 10⁹ km² in the unmagnetized case at the lunar orbit apogee location. Relative escape fluxes of different ion species in the nightside tail region are calculated by normalizing the ion escape rates by the magnetotail cross-sectional area. The estimated escape flux of each constituent as a function of geodetic altitude is displayed in Fig. 4.

Unmagnetized early Archean atmosphere composition

Standard solar evolution models suggest that the Sun’s total luminosity has increased by about 30–35% over the past ~4.6 billion years of geologic time^81,82, meaning that the Earth’s surface, with a present-day atmosphere, similar orbit, and comparable level of greenhouse forcing, would have below-freezing temperatures^83,84,85. In contrast, geological evidence indicates that the early terrestrial climate was warm and that liquid water was ubiquitously present at the surface from as early as the Hadean epoch through the Archean eon^86,87. The most widely accepted hypothesis that reconciles this apparent faint young Sun paradox involves an enhanced abundance of greenhouse gas to compensate for the lower intensity of primordial solar radiation³¹. Such forcing could occur if the homosphere of Archean Earth had more CO₂ or produced such gases (e.g., N₂O) from increased thermospheric heating caused by the more intense XUV radiation from the young Sun^77,88,89.

We utilize the model of the early Archean homosphere proposed by Johnstone et al.⁷⁷, which is composed of equal parts CO₂ and N₂ (49.95% each). The average Archean solar XUV flux was an order of magnitude higher than the present-day value^77,90,91,92. We therefore correspondingly increase the solar activity proxy (P in Equation 39) to scale the measured F74113 solar EUV reference and obtain the EUVAC solar irradiance spectra for a young Sun. The hydrogen abundance of the Archaean lower atmosphere, including its influence on Earth’s early climate as an indirect greenhouse gas, remains uncertain^93,94. But the Archean atmosphere was unlikely to have been H₂-rich, considering its low outgassing from Earth’s crust and microbial consumption, resulting in a low partial pressure^86,95. Following Kasting et al.⁹⁶, we adopt low H₂ mixing ratios on the order of 0.1% by volume. We used the homopause composition of various volatile species and the solar XUV flux for the Eoarchean era from Johnstone et al.⁷⁷, assuming the atmosphere was in approximate diffusive equilibrium, and incorporated these values into our photoionization model of the early terrestrial atmosphere. Studies of the composition of nitrogen and argon isotopes in fluid inclusions encapsulated in Archean hydrothermal quartz indicate N₂/³⁶Ar ratios of air-saturated water (ASW) comparable to those of present-day⁹⁷. We therefore assumed a constant concentration of atmospheric ³⁶Ar at the Archean homopause that matches the current ASW value of 1.0–1.3 × 10⁴. Ne also exhibits a similar trend; the atmospheric Ne composition during the early Earth is attributed to a mixing of mantle gases and chondritic gases delivered in a late veneer phase, and did not undergo much variation thereafter⁹⁸. We choose the atmospheric Ne/Ar ratio to be 0.051, akin to the present-day atmospheric value^98,99. Assuming complete degassing from the continental crust, an analogously steady He concentration, originating from the decay of radioactive elements and seeping into the atmosphere^100,101, also applies. These values are also consistent with the estimates of noble gas isotopic abundances in the early Earth’s atmosphere, which entered the atmosphere directly through meteorite or asteroid impact, or degassed from volatiles delivered to Earth after the loss of a primary atmosphere¹⁰².

Combining ionization model with total flux to predict lunar isotope ratios

To determine the relative elemental abundance of different major species in the SW and Earth’s atmosphere that are transported to the Moon, we combine results from our 3D-MHD model with the ionization model. We compute the impinging relative fluxes of H, N, and light noble gases by normalizing the flux of each species by the total escape flux of all constituents obtained from photoionization. We then scale these values so that the total escape flux equals the orbit-averaged planetary flux computed from the 3D-MHD simulations.

Such a scaling of the escaping flux is possible even if the plasma is collisionless, which it actually is above the exobase; then, the viscosity is anisotropic, with a lower value across the field lines due to the mean free path being approximately the same as that of the gyroradius of ions. Thus, the plasma is trapped by field lines perpendicular to the flow but moves more freely parallel to the magnetic field. On the nightside, the terrestrial ions are trapped by the magnetic field of the SW into the wind flow and move downtail with the bulk velocity, since the gyro-radius is small compared to macroscopic system scales. In particular, the characteristic gyro-radius for a typical magnetotail ion is given by

$${r}_{g,i}=10.44\left(\dfrac{{\mu }_{i}}{{Z}_{i}}\right)\left(\dfrac{{v}_{Bulk}}{1\,{{{\rm{km}}}}/{{{{\rm{s}}}}}}\right){\left(\dfrac{{B}_{\small {SW}}}{1\,{{{\rm{nT}}}}}\right)}^{\!-1}\,{{{\rm{km}}}}$$

(40)

where Z_i is the ion charge state and μ_i is the normalization factor used to normalize the ion mass by the proton mass. For a proton advecting with the bulk flow velocity (~268  km/s in Case–I) in the magnetotail, the gyroradius is ~560 km, indicating that it is roughly of the same order as the scale height used in our MHD simulation (~10³  km), thereby justifying our approximation of using a combined MHD-ionization model for all larger scales.

Binary mixing model to infer the non-solar component flux

We apply a binary mixing model¹⁰³ to estimate contributions from the SW and NSC, using their distinct isotope ratios and absolute abundances of different elements. We generate the theoretical binary mixing models by varying the proportions of each end member in incremental steps. The mixing equation for the two end members, NSC and SW, is the average of the isotopic ratios for an element in the end members, weighted by their respective mass fractions in the mixture. If the NSC and SW components contain different concentrations of element ${{{\mathcal{X}}}}$, and if x/X represents the isotope ratio of ${{{\mathcal{X}}}}$, then the binary mixing equation¹⁰⁴ can be expressed as

$$\left( \dfrac{x}{X}\right)_{\small {M}} = \dfrac{\left( \dfrac{x}{X}\right)_{\small {SW}} \,{{\mathcal{X}}}_{\!\! \small{ SW}} \, \,{{\mathcal{f}}}_{\!\! \small{SW}} + \left( \dfrac{x}{X}\right)_{\!\!\small{ NSC}} \,{{\mathcal{X}}}_{\!\!\small{ NSC}} \, \,{{\mathcal{f}}}_{\!\! \small{ NSC}}}{{{\mathcal{X}}}_{\!\!\small{ SW}} \, \,{{\mathcal{f}}}_{\!\! \small {SW}} + {{\mathcal{X}}}_{\!\!\small{ NSC}} \, \,{{\mathcal{f}}}_{\!\!\small{ NSC}}}$$

(41)

where ${{\mathcal{f}}}_{\!\! \small{ SW}}$ and ${{\mathcal{f}}}_{\!\! \small{NSC}}$ stand for the mass fractions of SW and NSC, subject to the constraint ${{\mathcal{f}}}_{\!\! \small{SW}} + {{\mathcal{f}}}_{\!\! \small{NSC}} = 1$, and ${\left(\frac{x}{X}\right)}_{\!\! \small {M}}$ denotes the isotope ratio of X in a mixture of SW and NSC, while ${{{{\mathcal{X}}}}}_{\!\small {SW}}$ and ${{{{\mathcal{X}}}}}_{\!\small {NSC}}$ are the respective concentrations of ${{{\mathcal{X}}}}$ in SW and NSC. Thus, ${\left(\frac{x}{X}\right)}_{\!\small {SW}} {{{{\mathcal{X}}}}}_{\!\small {SW}}$ it is the species-specific isotope fraction of the total elemental abundance. Similarly, the isotope composition of the element ${{{\mathcal{Y}}}}$ in the suite of binary mixture of SW and NSC is given by:

$$\left( \dfrac{y}{Y}\right)_{\!\!\small{ M}} = \dfrac{\left( \dfrac{y}{Y}\right)_{\small SW} \,{{\mathcal{Y}}}_{\!\!\small{ SW}} \, \,{\Large {\mathcal{f}}}_{\!\!\small{ SW}} + \left( \dfrac{y}{Y}\right)_{\!\!\small{ NSC}} \,{{\mathcal{Y}}}_{\!\!\small{ NSC}} \, \,{{\mathcal{f}}}_{\!\!\small{ NSC}}}{{{\mathcal{Y}}}_{\!\!\small{ SW}} \, \,{{\mathcal{f}}}_{\!\!\small{ SW}} + {{\mathcal{Y}}}_{\!\!\small{ NSC}} \, \,{{\mathcal{f}}}_{\!\!\small{ NSC}}}$$

(42)

For example, in terms of the ordinate of the ¹⁵N/¹⁴N - D/H mixing diagram (Fig. 6), the equation above can be represented as:

$$\left( \dfrac{{\scriptstyle{{15}}\atop} \!{{{\rm{N}}}}}{{\scriptstyle{{14}}\atop} \!{{{\rm{N}}}}}\right)_{\small {M}} = \dfrac{\left( \dfrac{{\scriptstyle{{15}}\atop} \!{{{\rm{N}}}}}{{\scriptstyle{{14}}\atop} \!{{{\rm{N}}}}}\right)_{\small {SW}} \,{{\rm{N}}}_{\small {SW}} \, \,{{\mathcal{f}}}_{\!\!\small{ SW}} + \left( \dfrac{{\scriptstyle{{15}}\atop} \!{{{\rm{N}}}}}{{\scriptstyle{{14}}\atop} \!{{{\rm{N}}}}}\right)_{\!\!\small{ NSC}} \,{{\rm{N}}}_{\small {NSC}} \, \,{{\mathcal{f}}}_{\!\!\small{ NSC}}}{{{\rm{N}}}_{\!\!\small{ SW}} \, \,{{\mathcal{f}}}_{\!\!\small{ SW}} + {{\rm{N}}}_{\!\!\small{ NSC}} \, \,{\Large {\mathcal{f}}}_{\!\!\small{ NSC}}}$$

(43)

The mass fraction of the NSC component (${{\mathcal{f}}}_{\!\! \small{NSC}}$) in the binary mixture of solar and Earth winds can be calculated if the isotopic compositions and elemental abundances of both end-members, as well as the isotopic composition of the presumed mixture (represented by the above mixing equation), are known:

$${{\mathcal{f}}}_{\small \!\! NSC} = \dfrac{{{{\mathcal{X}}}_{\small \!\! SW}} \, \left[\left( \dfrac{x}{X}\right)_{\small \!\! SW} - \left( \dfrac{x}{X}\right)_{\small \!\! M} \right]}{\left[\left( \dfrac{x}{X}\right)_{\small \!\! M} \left({{{\mathcal{X}}}_{\small \!\! NSC}} - {{{{\mathcal{X}}}}_{\small \!\! SW}}\right)\right] \, - \, \left( \dfrac{x}{X}\right)_{\small {NSC}}\,{{{\mathcal{X}}}_{\small {NSC}}}\,+\, \left( \dfrac{x}{X}\right)_{\small {SW}}\,{{\mathcal{X}}}_{\small {SW}}}$$

(44)

The exact analytical solution for a binary mixture of two elements is obtained by combining Equations 41 and 42 and eliminating ${{\mathcal{f}}}_{\!\! \small{SW}}$. The non-solar mass fraction of an element (say ${{{\mathcal{X}}}}$) is the weight fraction of that element in the two end-member mixture. Mixing between the end members generally results in hyperbolic relationships^103,105 of the form

$$a\,{\left(\dfrac{x}{X}\right)}_{\!\!\small {M}} +\,b\,{\left(\dfrac{x}{X}\right)}_{\!\!\small {M}} {\left(\dfrac{y}{Y}\right)}_{\!\!\small {M}} +\,c\,{\left(\dfrac{y}{Y}\right)}_{\!\!\small {M}} +\,d=0$$

(45)

Thus, the mixing equation for the ¹⁵N/¹⁴N - D/H system (Fig. 6) can be expressed as

$$a\,{\left(\dfrac{{\small{{15}}\atop} \!{{{\rm{N}}}}}{{\small{{14}}\atop} \!{{{\rm{N}}}}}\right)}_{\!\!\small {M}} +\,b\,{\left(\dfrac{{\small{{15}}\atop} \!{{{\rm{N}}}}}{{\small{{14}}\atop} \!{{{\rm{N}}}}}\right)}_{\!\!\small {M}} {\left(\dfrac{{{{\rm{D}}}}}{{{{\rm{H}}}}}\right)}_{\!\!\small {M}} +\,c\,{\left(\dfrac{{{{\rm{D}}}}}{{{{\rm{H}}}}}\right)}_{\!\!\small {M}} +\,d=0$$

(46)

The coefficients of the binary mixing equation (Equation 45) are:

$$a = {\left(\dfrac{y}{Y}\right)}_{\!\!\small {NSC}}\,{{{{\mathcal{Y}}}}}_{\small {NSC}}\,{{{{\mathcal{X}}}}}_{\!\small {SW}}-{\left(\dfrac{y}{Y}\right)}_{\!\!\small {SW}}\,{{{{\mathcal{Y}}}}}_{\small {SW}}\,{{{{\mathcal{X}}}}}_{\small {NSC}}\\ b = {{{{\mathcal{Y}}}}}_{\small {SW}}\,{{{{\mathcal{X}}}}}_{\small {NSC}}-{{{{\mathcal{Y}}}}}_{\small {NSC}}\,{{{{\mathcal{X}}}}}_{\small {SW}}\\ c = {\left(\dfrac{x}{X}\right)}_{\!\!\small {SW}}\,{{{{\mathcal{Y}}}}}_{\small {NSC}}\,{{{{\mathcal{X}}}}}_{\small {SW}}-{\left(\dfrac{x}{X}\right)}_{\!\!\small {NSC}}\,{{{{\mathcal{Y}}}}}_{\small {SW}}\,{{{{\mathcal{X}}}}}_{\!\small {NSC}}\\ d = {\left(\dfrac{y}{Y}\right)}_{\!\!\small {SW}}\,{\left(\dfrac{x}{X}\right)}_{\!\!\small {NSC}}\,{{{{\mathcal{Y}}}}}_{\small {SW}}\,{{{{\mathcal{X}}}}}_{\!\small {NSC}}-{\left(\dfrac{y}{Y}\right)}_{\!\!\small {NSC}}\,{\left(\dfrac{x}{X}\right)}_{\!\!\small {SW}}\,{{{{\mathcal{Y}}}}}_{\!\small {NSC}}\,{{{{\mathcal{X}}}}}_{\small {SW}}$$

(47)

where, as briefly described earlier, ${{{{\mathcal{X}}}}}_{\small {SW}}$, ${{{{\mathcal{X}}}}}_{\small {NSC}}$, ${{{{\mathcal{Y}}}}}_{\small {SW}}$, and ${{{{\mathcal{Y}}}}}_{\small {NSC}}$ are elemental abundances of the end members, i.e., pure SW or pure NSC, and ${\left(\frac{x}{X}\right)}_{\!\small {SW}}$, ${\left(\frac{y}{Y}\right)}_{\!\small {SW}}$, ${\left(\frac{x}{X}\right)}_{\!\small {NSC}}$ and ${\left(\frac{y}{Y}\right)}_{\!\small {NSC}}$ are the end member isotope ratios.

A curvature function, K_c, characterizes the shapes of the theoretical mixing hyperbolas and is used to diagnose the degree of mixing between the end members and to estimate the fraction of NSC in the lunar samples. This K_c depends on the coefficient b and is defined as [N/H]_SW${\displaystyle{/}}$[N/H]_EW for the N-H isotope plot, [He/Ne]_SW${\displaystyle{/}}$[He/Ne]_EW for the He-Ne isotope plot, and [He/Ar]_SW${\displaystyle{/}}$[He/Ar]_EW for the He-Ar isotope plot, respectively.

Along each mixing curve, which corresponds to a particular PHE height, from one end-member (say NSC) to the other (SW), the mass fraction of the starting end-member decreases (here, ${{\mathcal{f}}}_{\!\! \small{NSC}}$), and the mass fraction of the terminating end-member increases (${{\mathcal{f}}}_{\!\! \small{SW}}$), maintaining the sum of fractions to unity along the path. Mixing between the NSC and SW with different isotopic compositions results in isotope fraction values, ${\left(\frac{y}{Y}\right)}_{\!\small {M}}\;{{{\rm{and}}}}\;{\left(\frac{x}{X}\right)}_{\!\small {M}}$, that are intermediate between the two end-members. For example, if our mixing diagram represents ¹⁵N/¹⁴N versus D/H (Fig. 6), then the mass fraction of NSC (${{\mathcal{f}}}_{\!\! \small{NSC}}$), which comprises isotopes of both the elements nitrogen and hydrogen, is unity at the terrestrial component point (denoted by a navy square) in the plot. The value of ${{\mathcal{f}}}_{\!\! \small{NSC}}$ decreases to zero along the mixing trajectory non-uniformly as one progresses towards the SW component point (denoted by an orange square). Conversely, ${{\mathcal{f}}}_{\!\! \small{SW}}$ increases along the path.

The non-uniform decrement of ${{\mathcal{f}}}_{\!\! \small{NSC}}$ or increment of ${{\mathcal{f}}}_{\!\! \small{SW}}$ per unit length along a curve is due to the curvature of the rectangular hyperbolas. The extent and direction of this curvature are functions of the concentration contrast between the quantities in the denominator of each isotope ratio under consideration (K_c), representing the relative isotopic abundances of the end-members in the mixture. K_c varies between zero and infinity. Only in the special case where the ratio of the two denominators in the two end-members is equal (K_c= 1) is the mixing curve a straight line. As K_c becomes progressively greater or less than unity, the hyperbolic concavity becomes more pronounced. In the ¹⁵N/¹⁴N - D/H mixing plot (Fig. 6), the hyperbolic curves for PHE heights of 221 km (solid line), 275 km (dash-dot line), 300 km (dashed line), and 325 km (densely dotted line) display a concave-down shape, signifying K_c ∈ (0, 1), whereas the profiles corresponding to PHE altitudes of 350 km (dotted line) and 400 km (dash-dot-dotted line) exhibit concave-up behavior, suggesting K_c> 1. When all data points lie along a single mixing curve, this indicates consistency with a specific PHE choice, with variation in ${{\mathcal{f}}}_{\small{\!\! NSC}}$ (or ${{\mathcal{f}}}_{\small{\!\! SW}}$).

Reporting summary

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