# Sodium Enrichment of Mercury's Subsurface Through Diffusion

Sébastien Verkercke, François Leblanc, Jean-Yves Chaufray, Liam S Morrissey, Menelaos Sarantos, Parvathy Prem

► To cite this version:

Sébastien Verkercke, François Leblanc, Jean-Yves Chaufray, Liam S Morrissey, Menelaos Sarantos, et al. Sodium Enrichment of Mercury's Subsurface Through Diffusion. Geophysical Research Letters, 2024, 51 (21), pp.e2024GL109393. 10.1029/2024GL109393. insu-04773844

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Distributed under a Creative Commons Attribution 4.0 International License
# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL109393

**Key Points:**

* Subsurface diffusion of sodium forms shallow reservoir in the first meter of regolith, which is denser at the cold longitudes

* Greater surface binding energies increase the amount of sodium retained in the regolith per synodic cycle

* Convergence of the sodium accumulation should happen on timescales of billions of years, which could lead to the saturation of the regolith

**Correspondence to:**

S. Verkercke,
sebastien.verkercke@latmos.ipsl.fr

**Citation:**

Verkercke, S., Leblanc, F., Chaufray, J.-Y., Morrissey, L., Sarantos, M., & Prem, P. (2024). Sodium enrichment of Mercury's subsurface through diffusion. *Geophysical Research Letters*, *51*, e2024GL109393. https://doi.org/10.1029/2024GL109393

Received 28 MAR 2024

Accepted 1 AUG 2024

**Author Contributions:**

**Conceptualization:** S. Verkercke, F. Leblanc, J.-Y. Chaufray, L. Morrissey

**Methodology:** S. Verkercke, F. Leblanc, J.-Y. Chaufray, L. Morrissey

**Software:** S. Verkercke

**Supervision:** F. Leblanc, J.-Y. Chaufray, L. Morrissey, M. Sarantos, P. Prem

**Validation:** F. Leblanc, J.-Y. Chaufray, L. Morrissey, M. Sarantos, P. Prem

<figure><img src="image_2.png" /></figure>

## Sodium Enrichment of Mercury's Subsurface Through
Diffusion

S. Verkercke¹ Ⓓ, F. Leblanc² Ⓓ, J.-Y. Chaufray¹ Ⓓ, L. Morrissey³ Ⓓ, M. Sarantos⁴ Ⓓ, and P. Prem⁵ Ⓓ

¹LATMOS/CNRS, Université Versailles Saint Quentin, Guyancourt, France, ²LATMOS/CNRS, Sorbonne Université, Paris, France, ³Memorial University, St John's, NL, Canada, ⁴Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA, ⁵Johns Hopkins Applied Physics Laboratory, Laurel, MD, USA

**Abstract** Mercury's surface undergoes large temperature gradients between day and night, which repeats periodically over the same longitudes due to its 3:2 spin-orbit resonance. This effect combined with the orbit's eccentricity, creates hot and cold geographic longitudes. The planet is covered with a highly porous regolith, allowing exospheric atoms to diffuse in depth. By using a 1-D diffusion model, we studied the subsurface precipitation of gas over the cold and hot longitudes to understand gas retention. This work identifies the cold longitudes as favorable regions to form subsurface reservoirs closer to the surface. Moreover, subsurface reservoirs of adsorbates increase two to three times faster over cold longitudes than over hot longitudes, depending on the surface binding energy distribution of the atoms. We suggest that this result may be related to the observation that Mercury's sodium exosphere persists at later local times over the cold pole.

**Plain Language Summary** The surface of Mercury experience large temperature differences between day and night, and this repeatedly occurs over the same geographic positions as an orbit of the planet around the Sun lasts exactly one and half Mercury day. Additionally, Mercury's orbit is highly elliptic, causing the planet-Sun distance to vary. This creates cold and hot geographic longitudes on the planet surface. Mercury's surface is porous and covered with very fine grains which allows gas to diffuse through it. By using a one dimensional model, we study how gases can penetrate Mercury's surface depending on the geographic position. We identify that colder regions are favorable to the formation of shallow gas reservoirs, which grow two to three times faster than over the hottest regions, depending on the gas-surface interactions. This result may be related to the higher measurements of sodium over the cold geographic longitudes of Mercury, which last through each local afternoon.

## 1. Introduction

Being the closest planet to the Sun, having a low mass and a low atmospheric source rate, Mercury can not sustain a collisional atmosphere. This allows the solar wind and micro-meteoroids to directly interact with Mercury's surface, ejecting atoms into its exosphere. This was discovered by Mariner 10 spacecraft, which observed some traces of gaseous H, He and potentially O (Broadfoot et al., 1976). More recently, BepiColombo, during its first Mercury flyby, also observed H and He (Quémerais et al., 2023), which were believed to have a solar origin. Mercury experiences a large temperature gradient with a surface temperature ranging from approximately 90 K on the nightside to 700 K on the dayside (Chase et al., 1976; Morrison, 1970) over a 176 Earth day diurnal cycle.

The evidence of exospheric atoms released from Mercury's surface was first detected by ground based observations, years after the Mariner 10 fly-bys, when Potter and Morgan (1985) observed strong sodium emission features. Later on, other species such as potassium, calcium and iron were identified in Mercury's exosphere by ground based observations (Bida et al., 2000; Bida & Killen, 2017; Doressoundiram et al., 2009; Potter & Morgan, 1986). Between 2007 and 2011, the MErcury Surface, Space ENVironment, GEochemistry, and Ranging (MESSENGER) spacecraft performed three fly-bys of Mercury followed by 4 years in orbit around it. The Mercury Atmospheric and Surface Spectrometer (MASCS) consisted of an UltraViolet and Visible Spectrometer (UVVS) and a Visible and InfraRed Spectrograph (VIRS) fed by the same Cassegrain telescope (McClintock & Lankton, 2007). UVVS performed exospheric "limb scans" from South to North, as well as other observational modes (e.g., tail). From these, the radiance value along its line of sight was computed (Cassidy et al., 2015; McClintock et al., 2018). Over the mission, MASCS detected magnesium, aluminum and manganese

<footnote>© 2024. The Author(s).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</footnote>
(McClintock et al., 2018) alongside some of the previously mentioned species. Calcium and magnesium have also been observed by BepiColombo during its recent Mercury flybys (Robidel et al., 2023).

Mercury's spin-orbit resonance causes the same longitudes to reach the subsolar point when reaching aphelion or perihelion forming respectively cold and hot longitudes (Leblanc et al., 2023; Soter & Ulrichs, 1967). Such high temperatures on the dayside favor the ejection of different chemical elements from Mercury's surface through thermal stimulated desorption (TSD). These atoms populate the planet's surface bounded exosphere (Leblanc & Johnson, 2003; McClintock et al., 2018). This resonance is supposed to have occurred early in the Solar System formation so that this state influenced Mercury's exosphere and its subsurface over geological times (Noyelles et al., 2014). Other ejection processes responsible for the presence of atoms in the Hermean exosphere are the photon stimulated desorption (PSD), the ions sputtering (IS) and the micro-meteoroid impact vapourization (MMIV) (Chaufray et al., 2022; Gamborino et al., 2019; Leblanc & Johnson, 2003; Mura et al., 2023; Shemansky & Morgan, 1991; Wurz et al., 2010; B. Yakshinskiy & Madey, 1999). The ejection rates of these processes depend on the True Anomaly Angle (TAA), the position on the planet surface and are species dependent (Cassidy et al., 2016; Killen et al., 2007, 2022; Leblanc & Johnson, 2003).

Since its discovery, sodium remains the most studied species in Mercury's exosphere. Many observations have demonstrated that Mercury's sodium exosphere is evolving dynamically according to the orbital position of the planet while exhibiting an inter-annual variability (Mura et al., 2023). Some features of the Na-exosphere, such as the dawn-dusk asymmetry, have been partially explained through the introduction of the concept of a surface reservoir. The surface is replenished as atoms are adsorbed on the nightside cold surface (Leblanc and Johnson, 2003, 2010). Only considering such reservoir seems, however, inefficient in explaining the sodium enhancement measured over the cold longitudes by MESSENGER and reported by Cassidy et al. (2016). With actual treatments of the ejections processes in numerical models, a surface reservoir ends up rapidly depleted after dawn (Leblanc & Johnson, 2010). This is inconsistent with MESSENGER observations exhibiting an increase of sodium column density over the cold longitudes from dawn to noon. At noon, it reaches a maximum, and then decreases until TAA 300° where the sodium ejection stops (Cassidy et al., 2016). Such disagreement between the models and observations highlights the need to include a more realistic description of the surface-exosphere interactions.

Similar to the Moon, the top surface Hermean regolith presents a fairy-castle structure with a high porosity. Its porosity is thought to be similar or smaller than the lunar regolith (Veverka et al., 1988), which is believed to range between 0.7 and 0.99 (Szabo et al., 2022; Verkercke et al., 2023), and to decrease with depth under the compression due to its own weight. As porous structures are favorable to subsurface diffusion of exospheric atoms (R. M. Killen & Morgan, 1993; Sprague, 1990; Teolis et al., 2023), these could be adsorbed and desorbed successively, leading to a net diffusion in depth, similarly to what was suggested by Reiss et al. (2021) for water on the Moon. However, such a scenario would be highly sensitive to the local temperature in the near surface region and thus would only occur on the dayside as sufficiently high temperatures are required to enable the desorption of these atoms (Leblanc & Johnson, 2010; Sarantos & Tsavachidis, 2020, 2021; Sprague et al., 1995). While diffusing, the regolith's temperature is decreasing with depth, thus reducing the diffusion coefficient and forming a subsurface reservoir of adsorbed atoms (Sarantos & Tsavachidis, 2021; Yan, 2006). The reservoir's depth depends on the regolith's temperature profile and on the surface binding energy (SBE) distribution of the species. Low temperatures would thus potentially favor the formation of a reservoir close to Mercury's surface. Such consideration is a novelty with respect to (Reiss et al., 2021) regarding subsurface diffusion. New models would be needed to explore SBE influence on the subsurface diffusion.

In this study, a simple 1-D diffusion model is considered relying on a 1-D thermal model for Mercury, which are both described in Section 2. The diffusion at both the cold and hot longitudes is simulated subject to a constant Na-influx and the model's outputs are discussed in Section 3. The conclusions are presented in Section 4.

## 2. Modeling

This Section describes both the temperature and the subsurface diffusion 1-D models used in this study. In both models, the subsurface consists of 1 m of depth described as 50 layers, with the first one having a thickness of 50 µm, close to the mean size of a regolith grain (McKay et al., 1991; Wurz et al., 2010). The layers' sizes increase downwards with the last layer having a thickness of 5 cm.
## 2.1. Thermal 1-D Model

The thermal profile of the regolith is constrained by the solar radiation received at the surface, the geothermal heat flux, the radiative loss and the thermal conduction (Rognini et al., 2022; Yan et al., 2006). Supposing that the heat flux is dominated by its vertical component, we used a 1-D heat diffusion model. The equation describing the temperature temporal evolution is:

$$ \rho C_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) \quad (1) $$

where $\rho$ is the density, $k$ the thermal conductivity and $C_p$ the specific heat, which all vary with $z$, the depth, and $T$, the temperature. We suppose that the regolith is in thermodynamic equilibrium. The contact thermal conductivity is considered as the sum of a contact ($k_c$) and a radiative thermal ($k_r$) conductivity expressed as:

$$ k_c = 1 \times 10^{-3} Wm^{-1}K^{-1}/(2P) \quad (2) $$

with $P$ the porosity, coherent with Apollo samples thermal conductances (Cremers, 1972). The radiative conductivity ($Wm^{-1}K^{-1}$) is expressed as (Hale & Hapke, 2002; Leblanc et al., 2023; Rognini et al., 2022; Yan et al., 2006):

$$ k_r = k_c X(T/350)^3 \quad (3) $$

With $X$ being the ratio between the radiative and contact conduction at 350 K, taken equal to 2P. The specific heat expression is taken from Rognini et al. (2022).

According to Mariner 10 observations, the Hermean regolith can be considered similar to the lunar regolith (Chase et al., 1976). The regolith is approximated as entirely composed of SiO$_2$, as this is the most abundant component of the lunar regolith (McKay et al., 1991; Wurz et al., 2010). Its surface porosity is considered to be large, with a fairy-castle structure composed of very fine grains, below which it decreases with depth (Szabo et al., 2022; Verkercke et al., 2023). The minimum density of the fine regolith at the surface is ~1,200 kg/m$^3$ and the maximum density of the regolith at 1 m deep is ~1,950 kg/m$^3$ (Mitchell & de Pater, 1994; Rognini et al., 2022). Reporting these values to the SiO$_2$ bulk density, this leads to a porosity $P$ = ~0.5 close to the surface and decreasing to ~0.25 in depth. The porosity of the uppermost fairy-castle structure is likely to be higher than the near-surface average value of ~0.5 (Szabo et al., 2022). This study thus considers the porosity of the uppermost layer as 0.75 and decreasing to 0.5 at 600 µm deep. The porosity values then decreases to 0.3 at 1 m deep.

The top and bottom boundary conditions are respectively ruled by the solar and infrared radiations and the geothermal heat flux (Yan et al., 2006). At the surface, this gives respectively on the day and night sides:

$$ \rho C_p \frac{\partial T_s}{\partial t} = \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right)_s + \frac{\sigma \epsilon}{d_s} (T_{sky}^4 - T_s^4) + D_s \frac{1 - A}{d_s} \frac{F_S}{R^2} \cos(SZA), \text{Dayside} \quad (4) $$

and

$$ \rho C_p \frac{\partial T_s}{\partial t} = \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right)_s + \frac{\sigma \epsilon}{d_s} (T_{sky}^4 - T_s^4), \text{Nightside} \quad (5) $$

The right-hand side is composed of three terms which are, from left to right, the heat conduction flux, the radiated energy flux and, for the dayside, the solar flux absorbed by the surface. In this equation, $d_s$ is the thickness of the surface layer of 50 µm, $\sigma$, the Stefan-Boltzman constant, $\epsilon = 0.9$, the emissivity (Chase et al., 1976), $T_s$ and $T_{sky}$ are respectively the surface and sky temperatures, with $T_{sky} = 3$ K (Yan et al., 2006), A is the Bond albedo, with $A = 0.12$ (Veverka et al., 1988), $F_S$ is the solar constant equal to 1367 W m$^{-2}$, R is the heliocentric distance in AU and SZA is the solar zenith angle. The solar disk appears larger in Mercury's sky, with an apparent size of ~1.5° (against only 0.5° on Earth). This affects the terminator and poles temperatures. This is considered in this model
with the factor $D_s$, the apparent fraction of the solar disk (Salvail & Fanale, 1994). The last term of (4) is null on the nightside leading to Equation 5. At the bottom boundary, the heat equation can be expressed as:

$$ \rho C_p \frac{\partial T_b}{\partial t} = \left( \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) \right)_b + \frac{J_b}{d_b} \quad (6) $$

with the geothermal heat flux $J_b = 2 \cdot 10^{-2} \text{ W m}^{-2}$ (Yan et al., 2006) and $d_b$ is the deepest layer's thickness.

## 2.2. Subsurface Diffusion 1-D Model

For the subsurface atomic diffusion in the regolith, each particle makes numerous collisions with the regolith grains (Sarantos & Tsavachidis, 2021). Since Mercury's atmosphere is tenuous, the collisions between particles can be neglected, allowing diffusion to be described via Knudsen Theory (Sarantos & Tsavachidis, 2021).

The diffusion coefficient $D$ depends on the regolith structure and can be expressed as (Reiss, 2018; Yan, 2006):

$$ D = \frac{2 Pr_0}{3 \tau} \sqrt{\frac{8k_b T}{\pi m}} \quad (7) $$

with $\tau$ the regolith conduit's tortuosity taken as 5 (Reiss, 2018), $r_0$ the mean radius of a conduit, $m$ the mass of a sodium atom, $k_b$ the Boltzmann constant and $T$ the temperature of the regolith layer (Yan, 2006). $r_0$ is taken of the same order of size than a grain ($10^{-5} \text{ m}$).

During a time interval $\Delta t$, an atom collides with many grains, which can be considered as a 1-D random walk. The 1-D mean displacement $l$ during $\Delta t$ can thus be expressed as (Berg, 1993; Einstein, 1906):

$$ l = \sqrt{2D\Delta t\sin(\theta)} \quad (8) $$

with $\theta$, the angle between the displacement and the horizontal of the 1-D regolith.

Each time an atom collides with a grain, it has a probability to be adsorbed on the grain surface which can be expressed as (Leblanc & Johnson, 2003):

$$ \text{Stick} = A \exp(B/T) \quad (9) $$

and

$$ p_{ad} = \frac{\text{Stick}}{1 - (1 - \text{Stick}) P} \quad (10) $$

with Stick, the probability of an Na to adsorb on a surface at a temperature $T$, and $p_{ad}$, the effective probability of an atom to be adsorbed by the regolith with a porosity $P$. A and B are fit parameters to experimental measurements (Leblanc & Johnson, 2003). Once adsorbed, an atom has a probability to desorb, which can be expressed as (Grava et al., 2021; Hodges, 1980; Leblanc & Johnson, 2003):

$$ p_{des} = t_{res} C^{-1} T^2 \exp(-U/(Tk_b)) \quad (11) $$

where $t_{res}$ is the time since the atom was adsorbed, $C$ is a constant including the sodium vibration frequency on the grain ($10^{-11} \text{ sK}^2$), and $U$ is the SBE to the grain, corresponding to the strength of the bond with the grain. Yakshinskiy et al. (2000) reported values of SBE between 1.4 and 2.7 eV for sodium atoms deposited on a $\text{SiO}_2$ film. However, the $\text{SiO}_2$ surface is only an approximation for Mercury (Nittler et al., 2017), a fact which could influence the SBE distribution toward higher SBE values (Morrissey et al., 2022). According to MESSENGER observation, forsterite ($\text{Mg}_2\text{SiO}_4$), entstatite ($\text{Mg}_2\text{Si}_2\text{O}_6$ + metals) or albite (NaAlSi$_3$O$_8$) are the most likely minerals on Mercury (Vander Kaaden et al., 2017), which could have more open O bonds and lead to higher SBE.
Note that all these minerals are SiₓOᵧ-based, thus our approximation to SiO₂. Even if other ejections processes might exist at the top surface, this work only considers the thermal contribution for the sake of simplicity.

This model describes the subsurface diffusion through a Monte-Carlo test-particles approach. To simplify the boundary condition, the particles are injected at the top surface with a constant rate of 0.2/s (corresponding to a 2·10⁶ atoms/cm²/s flux over a 10 µm² surface (Yan, 2006)), independently from the position or local time. Injected particles start as adsorbed with a random position in the regolith's top layer. The process of desorption, diffusion and adsorption are treated successively at each time step, as in Sarantos and Tsavachidis (2021). For each adsorbed atom, a random SBE is picked out from a Gaussian distribution of mean 1.85 eV and σ = 0.4 eV. Once a particle desorbs, it moves according to (8) with a random displacement angle θ. After a time step, the adsorption probability of the moving atoms is computed accordingly to their positions following (9). Particles crossing the regolith upper limit are deleted from the simulation and considered as ejected. Similarly, the desorption probability is computed for all adsorbed atoms. The thermal profile evolves according to the thermal model presented above.

### 3. Results

The above 1-D subsurface model was run for the hot and cold longitudes, at the equator (respectively longitude 0° and 270°). The simulation was run over 4 Hermean synodic rotations (8 Mercury's orbits) in order to observe the temporal evolution of the subsurface density. Figures 1a and 1b show respectively the temperature profiles for the hot and cold longitudes at midnight, dawn, noon and dusk, in such chronological order.

For the hot longitudes in Figure 1a, noon always happens at perihelion. At this TAA (TAA = 0°), Mercury is at its closest point to the Sun, and the rotation of the planet is counterbalanced by its orbital rotation, briefly changing the apparent movement of the Sun in Mercury's sky (Leblanc et al., 2023). This extends the time during which the hot longitude is at the sub-solar point, explaining the high surface temperature (green line). At aphelion, the hot longitude is at dusk, where the surface temperature drops rapidly while the heat wave propagates downward (red line). From dusk to dawn, the surface temperature continues to drop as it only radiates energy (blue line). At dawn, the surface temperature rises quickly at the surface while the heat wave propagating in the subsurface attenuates (orange line). For cold longitudes, perihelion corresponds to either dawn or dusk. The apparent change in Mercury's rotation thus makes the Sun rise and set twice at these longitudes, which maintains the surface temperature between 100 and 200 K, as shown in Figure 1b. At aphelion, the cold longitudes are at noon or midnight. At noon, the surface temperature of the cold longitude (~580 K) is lower than at the hot longitude at noon (~700 K) because of the high eccentricity of Mercury's orbit. At 1 m depth, the temperature at hot longitudes is constant ~420 K and is 100 K larger than the temperature at 1m of the cold longitudes. These simulated profiles are in agreement with the simulated profiles by Hale and Hapke (2002) and Yan et al. (2006).

Figures 1c and 1d show the vertical profiles of the sodium content in the subsurface normalized by the total number of injected atoms over a synodic rotation, respectively for the hot and the cold longitudes, and at times corresponding to Figures 1a and 1b. One can notice that the subsurface density profiles exhibit a maximum in depth, but that the peak forms globally deeper at the hot longitudes than at the cold ones. Over the course of a day the reservoir peaks oscillate from 10 cm at noon to 15 cm at dusk at hot longitudes, and from 6 cm at dawn to 7 cm at dusk at the cold longitudes. This effect is due to the temperature of the regolith increasing as the heat wave propagates and dissipates. These values stabilize after a few cycles as the probability to further diffuse is too small, even on timescales of multiple synodic cycles. These depths represent the position where the temperature allows an absorption probability significantly higher than the desorption probability ($p_{ad} > 2 p_{des}$). For the mean of the SBE distribution presented above, this temperature is ~430 K, according to Equations 10 and 11. The reservoirs' positions depend on the SBE distribution but the cold longitudes always exhibit reservoir building up closer to the surface than the hot longitudes because of the colder temperatures at the cold longitudes. This shows that the reservoir might be more accessible than under the hot longitudes. Micro-meteoroids should at least impact the first mm of regolith (Moroni et al., 2023), ejecting the trapped volatiles and exposing deeper layers to the solar wind and radiation. Solar protons can also penetrate the regolith from a depth of a few grains' size to a millimeter, depending on the porosity (Leblanc et al., 2023; Verkercke et al., 2023).

Over the course of a synodic rotation, one can also observe that the variation of the Na-content in the regolith is greater at the cold longitudes than at the hot ones. This phenomenon combined with the shallower reservoir shows that the cold longitudes could, over geological timescales, have accumulated sodium in its subsurface. This could
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Regolith temperature profiles for the hot (a) and cold (b) longitudes. Number of Na atoms normalized by the number of injected atoms over a synodic rotation as a function of depth for the hot (c) and cold (d) longitudes, over a synodic rotation, with SBE following a Gaussian distribution of mean $\mu$ = 1.85 eV and $\sigma$ = 0.4 eV. (e, f) are respectively similar to (c, d) but with SBE following a Gaussian distribution of mean $\mu$ = 2.5 eV and $\sigma$ = 0.3 eV. In the latters, “Midnight” line is over-plotted by the “Dawn” line. The gray dashed vertical lines represent the layer's delimitation. In panel (c), the green line exhibit a squared bump at 1 mm depth which origin is purely statistical and is not relevant.</figcaption></figure>

be relevant for the contribution of MMIV to the exosphere. The impactors' sizes influence the craters' depths. In order to reach several centimeters of depth, the impactor itself must be centimeteric. This type of impactor is less likely than a millimeteric impactor which would partially reach the reservoir. Micrometeric impactors are even more likely (Grotheer & Livi, 2014), and can remove layers of surface grains, exposing the reservoir. Larger impact would increase the amount of energetic Na in the exosphere while smaller impacts could expose Na to the other means of ejections. Over long periods, a larger part of the injected atoms are trapped in depth at the cold longitudes than at the hot longitudes. The amounts of retained atoms are dependent on the SBE. Additionally, as most atoms accumulate at a specific depth, a steady increase of the Na-content would lead to a saturation (i.e., full
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Total number of Na atoms in the regolith, normalized by the total amount of injected atoms during one synodic cycle, as a function of the TAA for the two studied SBE distributions. The red and green dashed lines show the quasi-steady increase of the minimum amount of atoms in the regolith during each synodic cycle for the cold and hot longitudes respectively. The abscissa of the top panel is dashed (solid) during daytime (nighttime) of hot longitudes, and similarly in the bottom panel for the cold longitudes.</figcaption></figure>

coverage of grains) of the layer which could lead to a decrease of the SBE (Yakshinskiy et al., 2000) as the sites enabling strong bonds would be filled. However, the influence of coverage on SBE is not yet well understood. Lower SBEs at depth could potentially favor the formation of deeper enriched layers, as well as more diffusion toward the regolith surface as illustrated in Figure 1.

Figures 1e and 1f exhibit similar evolution of the regolith sodium content for both hot and cold longitudes, but considering a SBE Gaussian distribution of mean $\mu$ = 2.5 eV and $\sigma$ = 0.3 eV. This distribution was selected as an extreme case for SBE, as SBE values greater than 2.7 eV do not allow desorption at the cold longitudes. This value is also the upper limit of SBE proposed by Yakshinskiy et al. (2000) for sodium deposited on SiO$_2$. One can observe that, for greater SBE, the reservoir's peak moves toward the regolith surface. Indeed, at the hot longitudes, the reservoirs' peak now oscillates between depths of 7.3 cm at dawn and 9.5 cm at dusk. At the cold longitudes, it keeps oscillating between 3.1 cm at dawn and 3.7 cm at dusk. For this SBE distribution mean, the critical temperature at which desorption overcomes adsorption significantly is ~490 K. As on the dayside, the temperature decreases with depth, the position where adsorption becomes more efficient than desorption moves upward for larger SBE values.

Figure 2 exhibits the total number of sodium atoms contained in the regolith normalized by the total amount of sodium injected over a synodic cycle, as a function of TAA over 4 synodic rotations. The first cycle is not studied as it is influenced by the initial conditions. The regolith sodium content is presented for both cold and hot longitudes while considering SBE Gaussian distributions of mean $\mu$ = 1.85 eV and $\sigma$ = 0.4 eV (top) and $\mu$ = 2.5 eV and $\sigma$ = 0.3 eV (bottom). The red and green dashed lines represent the minimum Na content of each synodic rotation for the cold and hot longitudes respectively. This minimum value corresponds to the Na-content remaining in the regolith over long time scales. Through these line slopes, one can observe that the sodium content is increasing more rapidly with time at the cold longitudes for both SBE distributions. The top panel presents the quasi-steady sodium content's increase of 9.7% and 4.4% of the injected atoms over a synodic
rotation, for the hot and cold longitudes respectively. For the second SBE distribution, this grows respectively to 22.5% and 7.2% of the injected atoms over a synodic rotation. For any distribution, the sodium increase is exponentially converging to a steady state when the escape and injection fluxes are equal. This convergence happens faster when less sodium is retained in the regolith, hence at the hot longitudes for lower SBE values (e.g., Figure 2a, blue curve). The convergence is reached if the number of all atoms injected during a cycle matches the number of atoms escaping. However, for both SBE distributions, some parts of the distribution give atoms residence time larger than $5 \cdot 10^9$ years. This residence time is computed through Equation 11, taking the largest temperature reached at the reservoir depth over a synodic cycle and atoms with SBE larger than ~2.7 eV. During such accumulation periods, Figure 2 shows that the cold longitudes are more favorable for the development of a subsurface reservoir. The SBE distributions scale the amount of sodium retained by the regolith but the greater subsurface sodium enhancement at the cold longitudes is solely due to the temperature profile of these longitudes. In the two cases presented, the sodium retained at the cold longitudes is respectively more than 2 and 3 times larger than at the hot longitudes. Larger SBEs values enhance the difference in retention rate between the hot and cold longitudes. More generally, one can also observe that higher SBE favors the formation of a shallower reservoir as more energy is needed to desorb the atoms.

The thermal model used above considered Mercury's surface in thermodynamic equilibrium. However, this is an approximation and solving the full radiative transfer equations shows that the top 300–500 µm of regolith exhibit a steep thermal gradient because of the difference in transmission of visual versus infrared radiation (Hale & Hapke, 2002; Prem et al., 2022). This thermal gradient reduces the top surface temperature by ~70 and ~50 K at respective noons of the hot and cold longitudes (Hale & Hapke, 2002). To test the influence of such gradients on the diffusion, we forced the above gradients over the first 500 µm at hot and cold longitude's noons and made these decrease as $cos(SZA)$ on the dayside, to match results from Hale and Hapke (2002). At the hot longitude while considering the second SBE distribution presented above, the diffusion is slightly delayed by such gradient as surface's atoms reach sufficiently high temperatures to desorb later in the day. However, this has a limited effect on the number of atoms retained in the subsurface with an increase of ~1% of retention of the injected sodium atoms per cycle. At the cold longitude, the diffusion pattern is slightly delayed but the retention is increased by ~17% of the injected atoms. This effect will be further investigated in a following work coupling an Exospheric Global Model (Leblanc & Johnson, 2003, 2010) with Prem et al. (2022) thermal outputs.

## 4. Conclusion and Perspectives

Mercury singular rotation and orbit induce a inter-annual variability of its exosphere (Killen et al., 2007; Mura et al., 2023). Using a 1-D thermal model coupled with a 1-D subsurface diffusion model, this study focused on the possible formation of a spatially resolved sodium reservoir on Mercury and its dependencies on the longitude and on the SBE distribution. This model only considers the thermal desorption for atoms ejection. This approximation is valid deep in the regolith, but is not close to the surface, where other ejection processes such as MMIV, PSD and SWS are also contributing to the desorption, sometimes more efficiently than TSD (Leblanc & Johnson, 2010). This implies that the release rate of atoms compute here could not explain MESSENGER observations as our model is based on too much assumptions to expect to find a fit to these observations. However, the subsurface is unaffected by those assumptions. For a similar SBE distribution and under uniform Na-influx, the regolith's subsurface is enriched at a higher rate at the cold longitudes than at hot longitudes by a factor 2 to 3, depending on the SBE distribution. Cold longitudes exhibit a sodium reservoir which is the closest to Mercury surface, making it available for any other processes of ejection that could reach the reservoir's depth. This work also highlights the importance of the considered SBE distribution, as larger SBE influence the sodium subsurface enrichment. Larger SBE values allow atoms to be more efficiently adsorbed, leading to a larger portion of the exospheric atoms to be retained in the regolith. Additionally, as these atoms are more tightly bound with the regolith, higher temperatures are needed for desorption. The peak of the subsurface reservoir is therefore shifted toward the surface. Atoms exhibiting high SBE values at the cold longitudes can form the shallowest reservoir available on Mercury, as these both inhibit diffusion in depth. This highlights the importance of SBE in subsurface diffusion and stresses the need for studies quantifying SBEs for Na adsorbed onto representative minerals.

In our model, this enriched subsurface is allowed to increase indefinitely. In reality, the subsurface should saturate, potentially leading to a decrease of the SBE values (Yakshinskiy et al., 2000). The saturation would occur faster at the cold longitudes than at the hot longitudes, creating Na-enriched regions with large SBE distribution. The porosity affects both the diffusion coefficient and the adsorption probability. In this study, the
porosity profile was assumed to be the same at every longitudes and constant in time. However, micro-meteoroids impacts and gardening can lead to a change of the surface structure, and even to a turn-over of the topmost regolith layer, resetting its volatile content. This study does not account for the role of surface diffusion on a grain. If an atom finds a position where the Knudsen diffusion becomes inefficient, other means of diffusion could take over, like diffusion on the grain surface or diffusion inside a grain, which are out of the scope of this study.

In this study, the injection of particles was considered uniform. In reality, atoms would preferentially move toward colder regions, such as the poles, the terminators and the nightside. Indeed, atoms ejected on the dayside would get pushed toward the nightside by the solar radiation pressure. The atom influx should thus be higher at the cold longitudes than at the hot ones (Cassidy et al., 2016), increasing their subsurface sodium contents. Finally, our model highlights that the top-surface temperature gradients can further enhance this differentiation by increasing the retention rate only at the cold longitudes. The arrival of BepiColombo in Mercury's vicinity in late 2025 offers a great opportunity to search for such surface composition differentiation as this process should affect most species found in Mercury's exosphere that can thermally desorb.

## Conflict of Interest

The authors declare no conflicts of interest relevant to this study.

## Data Availability Statement

Data are openly available at Verkercke, S. 462 (2024). Sodium enrichment of Mercury's subsurface through diffusion [Dataset]. In Geophysical Research Letter via Zenodo Verkercke (2024).

## Acknowledgments

S.V., F.L., and J.-Y.C. acknowledge the support of ANR, France of the TEMPETE project (Grant ANR-17-CE31-0016) and the support of CNES, France for the BepiColombo mission. This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #559 "Improving the Description of Exosphere Surface Interface." S.V. gratefully acknowledge the support of Catherine Dukes and of Adam Woodson (University of Virginia) in the writing of this letter.

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