A Comprehensive Model of the Meteoroid Environment around Mercury

Mutual collisions between dust grains effectively shape the characteristics of the entire meteoroid cloud. Collisional fading is more significant for meteoroids with longer dynamical lifetimes, thus affecting mostly larger meteoroids. This is because their dynamical evolution is much slower under the influence of PR drag, which is inversely proportional to the particle's size. In this manuscript, we use a method developed by Steel & Elford (1986, henceforth SE86) that uses several analytic expressions to estimate the collisional lifetime between a meteoroid and the meteoroid cloud in the solar system. To calibrate the density of the meteoroid cloud in the solar system used to estimate collisions, we use Long Duration Exposure Facility (LDEF) measurements that were recently reinvestigated by Cremonese et al. (2012). SE86 also discusses the mass ratio between two colliding meteoroids at characteristic impact velocities of 10 km s⁻¹ that can cause catastrophic disruptions. The authors concluded that a 5 × 10⁴ less massive body (30–40 times smaller in radius) caused catastrophic fragmentation of both projectile and target. In this manuscript we assume that meteoroids in our simulations are destroyed by projectiles 30 times smaller in radius or larger. Using results from Cremonese et al. (2012) and the previous assumption of catastrophic collisions, we parameterize the decadic logarithm of the spatial density ψ of projectiles at 1 au for a given diameter D in micrometers by the seventh-degree polynomial:

$$\begin{eqnarray}{\mathrm{log}}_{10}(\psi (D)) & = & {a}_{0}+{a}_{1}D+{a}_{2}{D}^{2}+{a}_{3}{D}^{3}\\ & & +{a}_{4}{D}^{4}+{a}_{5}{D}^{5}+{a}_{6}{D}^{6}+{a}_{7}{D}^{7},\end{eqnarray} \tag{ 1 }$$

with coefficients a₀ = 24.5740501811233, a₁ = −0.00273729871839826, a₂ = 2.42366145666713 × 10⁻⁰⁶, a₃ = −1.55353572997867 × 10⁻⁰⁹, a₄ = 6.15333399708131 × 10⁻¹³, a₅ = −1.42903699974705 × 10⁻¹⁶, a₆ = 1.76861418259828 × 10⁻²⁰, and a₇ = −8.9675339812721 × 10⁻²⁵, where ψ is in units number of particles per m² s, i.e., the particle flux in SI units.

As in SE86, the spatial density of meteoroids is assumed to decrease with heliocentric distance r as ψ ∝ r^−1.3 and varies with the ecliptic latitude β as $(1-\sin \beta )$. Recalling the conclusions of Nesvorný et al. (2011a) and Pokorný et al. (2014), who reported that the nominal collisional lifetime is too short for meteoroids from short-period comets to dynamically evolve enough to obtain the observed distribution of orbital elements, we adopt a fudge factor F_coll that extends the nominal collisional lifetime. In the following calculations we used six different values of ${F}_{\mathrm{coll}}=1,5,10,20,50,\,\mathrm{and}\,\infty$, where F_coll = 20 was previously used in Pokorný et al. (2014) and Pokorný et al. (2017), and values in the range F_coll = 20–50 are comparable to those in Nesvorný et al. (2011a), who used a different collisional approach.

Since meteoroids in our model have a wide variety of orbits, we must determine their collisional lifetimes at different positions of their orbits in order to take into account the effects of the heliocentric distance r and inclination I. Such an approach prevents us from under/overestimating the collisional lifetime for eccentric and/or high-inclination orbits where the instantaneous collisional lifetime at a meteoroid's pericenter might be an order of magnitude different from the value at aphelion. To estimate the collisional probability for each meteoroid, we use 50 points uniformly distributed in true anomaly f along the meteoroid's orbit. Then, we calculate a weighted average of the collisional lifetime considering nonuniform variation of the true anomaly f with time t, i.e., the average is weighted by df/dt. We select 50 points along the orbit for two reasons: the precision does not significantly increase for more points along the orbit, and the calculation would be more computationally demanding.

Collisions in our model are applied in a rather simple manner. At the beginning of the simulation each meteoroid is assigned a collisional weight W_c = 1. As time elapses, the weight of each meteoroid is reduced to ${W}_{c}^{\mathrm{new}}\,={W}_{c}\ast \exp (-{T}_{\mathrm{step}}/{T}_{\mathrm{coll}})$, i.e., a simple exponential decay, sometimes called collisional fading. We compared this method to that of Nesvorný et al. (2011a) and Pokorný et al. (2014) and found that the collisional fading used in this manuscript works better for smaller samples of simulated particles.

Despite the significant number of simulated meteoroids of various sizes, we record only a limited number of direct impacts on Earth and other planets in our simulations. This is not surprising, because the actual number of meteoroids currently orbiting in the solar system is many orders of magnitude higher than the number of particles we can simulate. In order to overcome this limitation, we implemented a method developed by Kessler (1981) that allowed us to calculate the intrinsic collisional probability of every meteoroid in our model at each time step with the target of interest, in this case Mercury. Kessler's method was selected because it grants us the freedom to choose the target's position and velocity vector, contrary to other methods (Öpik 1951; Wetherill 1967; Greenberg 1982; Vokrouhlický et al. 2012; Pokorný & Vokrouhlický 2013). As pointed out by Rickman et al. (2014), Kessler's method contains singularities for certain orbital configurations, leading to overestimation of the collisional probability between the target and projectile. However, the high computational speed of Kessler's purely analytic method, combined with the high statistics in our model, outweighs these limitations.

An additional improvement from the version of the model presented in Pokorný et al. (2017) and Janches et al. (2018) is that we include the effect of gravitational focusing on the collisional cross sectional area σ between the target and the projectile (Equation (3) in Kessler 1981):

$$\begin{eqnarray}&&\sigma =\pi {({r}_{1}+{r}_{2})}^{2}\left[1+\displaystyle \frac{{V}_{\mathrm{esc}}^{2}}{{V}_{\mathrm{rel}}^{2}+{\epsilon }^{2}}\right],\end{eqnarray} \tag{ 2 }$$

where r₁ and r₂ are the average radii of the target and the projectile, respectively, V_esc is the escape velocity of the target, V_rel is the relative velocity of the target and the projectile, and is the softening parameter. We chose  = 0.1 km s⁻¹ to avoid extreme values of σ for individual meteoroids with V_rel close to 0, e.g., MBA meteoroids crossing Earth's orbit at very low eccentricities. Gravitational focusing is very important for the correct determination of the meteoroid mass flux on Earth, since meteoroids originating from MBAs and JFCs have in general V_rel smaller than V_esc of Earth. Due to Mercury's lower escape velocity V_esc = 4.25 km s⁻¹ and larger orbital velocities of all meteoroid populations in Mercury's vicinity, this effect has a smaller impact for the Hermean meteoroid environment. Therefore, this effect was found to significantly affect (by a factor of two) the relative numbers of MBA and JFC meteoroids intercepted by Earth and Mercury and influenced the calculation of total influx and vaporization rates for Mercury.

In this manuscript we investigated meteoroid impacts on Mercury for one Hermean year (∼88 days) from 2013 February 17 at 00:00 UTC (TAA = 3594) until 2013 May 16 at 00:00 UTC (TAA = 3595). Impacts of meteoroids were calculated every 12 hr along Mercury's orbit, thus sampling the environment at 177 uniformly separated positions of Mercury in time. Mercury's orbital elements were taken from JPL HORIZONS Web-Interface for Target body: Mercury (199), Center body: Sun (10), DE431mx. We evaluated the collisional probability of each meteoroid in our model using Kessler's method for each of the 177 positions of Mercury separately to precisely capture changes of the Hermean meteoroid environment.

The SFD or mass–frequency distribution (MFD) of meteoroids is an essential characteristic of the meteoroid environment, yet for many source populations only a handful of independent measurements are available (as explained in this section). In this work, we treat SFD/MFD as a free parameter in order to investigate how sensitive or robust our resulting model is to variations of the assumed SFD/MFD. One of the simplest descriptions of SFD/MFD is a single power-law distribution, where the number of meteoroids, dN, with diameter between D and D + dD is expressed as

$$\begin{eqnarray}&&{dN}\propto {D}^{-\alpha }{dD},\end{eqnarray} \tag{ 3 }$$

where the exponent α is the differential size index. We can rewrite Equation (3) for the mass instead of size as

$$\begin{eqnarray}&&{dN}\propto {M}^{-\beta }{dM}\propto {D}^{-3\beta }{D}^{2}{dD},\end{eqnarray} \tag{ 4 }$$

where the exponent β is the differential mass index with the conversion formula α = 3β − 2. Integrating Equations (3) and (4) leads to the total number of meteoroids with diameters/masses in selected diameter/mass ranges (D₁, D₂) or (M₁, M₂)

$$\begin{eqnarray}&&N(D,\alpha )\propto {\int }_{{D}_{1}}^{{D}_{2}}{D}^{-\alpha }{dD}\propto \displaystyle \frac{1}{\alpha +1}[{D}_{2}^{-(\alpha -1)}-{D}_{1}^{-(\alpha -1)}],\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&N(M,\beta )\propto {\int }_{{M}_{1}}^{{M}_{2}}{M}^{-\beta }{dM}\propto \displaystyle \frac{1}{\beta +1}[{M}_{2}^{-(\beta -1)}-{M}_{1}^{-(\beta -1)}].\end{eqnarray} \tag{ 6 }$$

In order to simplify our calculation and to decrease the size of the large free parameter space, we treated all SFD/MFDs in this manuscript as single power-law distributions. Although more complex distributions, such as broken power-law distributions, have been used to describe dynamical models in the past (e.g., Nesvorný et al. 2011a; Pokorný et al. 2014; Poppe 2016), recent observations of the Earth meteoroid complex by radar and optical systems (Blaauw et al. 2011; Pokorný & Brown 2016) suggest that a single power law may provide a good approximation of the meteoroid complex SFD/MFD for the diameter range considered in this manuscript (D = 10–2000 μm).

It is important to distinguish between SFD/MFD at the source (i.e., where meteoroids are released) and SFD/MFD measured at different locations in the solar system. Due to complex dynamical evolution, scaling of the effects of radiation pressure, and PR drag with meteoroid diameter, the SFD/MFD of different populations might vary throughout the solar system. Discussions about the values of α and β have appeared before. Dohnanyi (1969) reported that systems in collisional equilibrium have α = 3.5, β = 11/6, while a population with evenly distributed mass per logarithmically spaced mass bins has α = 4.0, β = 2.0. Grun et al. (1985) compiled all measurements available at the time to produce a comprehensive empirical model for the SFD/MFD observed at 1 au, and they reported β = 2.35 for the size range considered in this manuscript. Cremonese et al. (2012) revisited the original LDEF results from Love & Brownlee (1993) and inferred a varying SFD/MFD that can be approximated by a broken power law with β = 1.6 for D < 200 μm and β = 3.6 for D > 200 μm. The sporadic meteoroid background, as observed at Earth from the latest meteor radar and optical measurements reported by Blaauw et al. (2011), Pokorný & Brown (2016), suggests β = 2.17 ± 0.07 and β = 2.10 ± 0.08, respectively, which translates to α = 4.51 ± 0.21 and α = 4.30 ± 0.28. Nevertheless, all these values are referring to the SFD/MFD of a dynamically and collisionally processed meteoroid population that is a mixture of meteoroids from different sources.

The SFD/MFD at the source of each meteoroid population, i.e., the required model input, is more difficult to assess. The in situ observations made by the Rosetta mission in the coma of 67P/Churyumov–Gerasimenko reported the SFD/MFD to vary with time and to be rather complex but approximated by a double broken power-law function (Rotundi et al. 2015; Fulle et al. 2016). For meteoroids with D > 1 mm the differential SFD/MFD index was α = 4, β = 2, while for D < 1 mm the differential index varied from α = 2, β = 1.33 at comet's aphelion to α = 3.7, β = 1.9 at the comet's perihelion. Measurements from Giotto (1P/Halley—HTC; Fulle et al. 1995) and Stardust (81P/Wild—JFC; Green et al. 2007) provided similar values for the differential SFD/MFD indices.

In our models we perform all meteoroid integrations for each meteoroid diameter separately and apply the SFD/MFD at the source regions afterward. This gives us the flexibility to evaluate the effect of a wide range of differential SFD/MFD indices where we selected the ranges α = (1, 7), β = (1, 3), covering all currently available measurements.

MBA meteoroids impacting Mercury were the subject of several studies in the past decade. Marchi et al. (2005) investigated the flux of MBA meteoroids on Mercury's surface for meteoroids with D > 1 cm, for which we can assume that PR drag is negligible. Due to the absence of PR drag and its induced dynamical evolution, Marchi et al. (2005) placed MBA meteoroids into the ν₆ secular resonance and the 3:1 mean motion resonance (MMR) with Jupiter. They found that the impact velocity of MBA meteoroids at Mercury ranges widely between 15 and 80 km s⁻¹, the impacts are much faster at Mercury's perihelion, while the dawn–dusk asymmetry is more pronounced at perihelion and almost disappears at aphelion, and that both the ν₆ secular resonance and the 3:1 MMR with Jupiter had comparable efficiency of perturbing MBA meteoroids into crossing orbits with terrestrial planets. In contrast, Borin et al. (2009, 2016b) focused on smaller meteoroids with diameters in the range D ∈ (20, 200) μm with semimajor axes, a, randomly selected between 2.1 and 3.3 au, eccentricities e < 0.4, and inclinations I < 20°. Their findings differ from those of Marchi et al. (2005) since their velocity distribution is much narrower with a peak at 17 km s⁻¹, which is similar to the results of Cintala (1992).

MBA meteoroids in our model (D < 2 mm) originate from the largest MBAs, and thus their initial orbits are predominantly distributed close to the ecliptic and have low eccentricity. Hence, there are no initial intersections between the young MBA meteoroid population and Mercury. MBA meteoroids are shepherded to the inner solar system by PR drag, where the dynamical timescale of the meteoroid transport from the original location toward Mercury crossing orbits is a linear function of meteoroid diameter D (Equation (47) in Burns et al. 1979). For meteoroids on initially circular orbits, the time before meteoroids reach the Sun is τ_PR = 400 R²/β_PR, where R is the heliocentric distance in au and β_PR = 5.7 × 10⁻⁵/(ρs), where ρ and s are the meteoroids' density and radius, respectively both in cgs units (Equation (50) in Burns et al. 1979). Using these expressions, we recognize that MBA meteoroids in our model have a wide range of dynamical timescales, ranging from τ_PR ≈ 60 ky for meteoroids with D = 10 μm to τ_PR ≈ 12 My for D = 2 mm meteoroids (here we assumed that meteoroids start at heliocentric distances R = 3 au). Hence, not only must MBA meteoroids be dynamically evolved in order to decrease their perihelion distances to be able to reach Mercury, but also the planet generally encounters more evolved MBA meteoroids at its perihelion, while the less evolved meteoroids are able to reach the planet even at its aphelion.

First, we explore the influence of the collisional lifetime multiplier F_coll on MBA meteoroid population with access to Mercury. Figure 1 shows the semimajor axis, eccentricity, inclination, and impact velocity of meteoroids with D = 10, 400, and 2000 μm impacting Mercury's surface at perihelion.

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We readily recognize from this figure that the collisional lifetime is not a significant agent in the dynamical evolution of the smallest MBA meteoroids, D = 10 μm (Figure 1, left column). This is a consequence of their shorter PR evolution time, τ_PR, which is approximately 40 times shorter than for D = 400 μm meteoroids, leaving a much shorter time for mutual collisions to influence the meteoroid population. The smallest MBA meteoroids have a very narrow distribution of eccentricities, with a peak at e = 0.02 at Mercury's perihelion, which has a direct influence on the distribution of semimajor axes of impactors of this size, making them concentrated around Mercury's heliocentric distance. The almost circularized nature of orbits of impacting meteoroids with D = 10 μm is also a direct consequence of their short dynamical lifetime since these particles effectively pass through MMRs, secular resonances, and areas of close encounters with terrestrial planets. The inclination distribution of MBA meteoroids is fairly similar along Mercury's whole orbit, where the only significant difference is the minimum inclination cutoff caused by the nonzero inclination of Mercury's orbit (I = 7005 with respect to the ecliptic) for different TAAs. The impact velocities are fairy low, in the range of V_imp = 5–30 km s⁻¹, where V_imp is higher when Mercury is closer to the pericenter.

MBA meteoroids with D = 400 μm (Figure 1, middle column) arrive with low eccentricities, with a median value close to e = 0.1, regardless of Mercury's true anomaly and the collisional lifetime multiplier F_coll. The eccentricity distribution is wider than for D = 10 μm meteoroids, mostly due to eccentricity pumping when passing through interior MMRs with Jupiter and exterior MMRs with Earth. The inclination distribution is similar to that of the smallest MBA meteoroids. The impact velocity is correlated with the meteoroid eccentricity and is broader than for smaller MBA meteoroids. When no collisions are assumed (${F}_{\mathrm{coll}}=\infty$), the relative flux of asteroidal meteoroids of this size is 3–4 orders of magnitude larger than for the nominal collisional lifetime F_coll = 1. Despite such a rapid decay of the meteoroid population with shorter characteristic collisional lifetimes, it is still possible to transport material from the main belt to Mercury. However, shorter collisional lifetimes might require unrealistically high production rates of the source population in order to achieve the same volume of accreted mass or flux in the inner solar system.

The orbital characteristics of the largest MBA particles in our sample, D = 2000 μm, are significantly different from their smaller counterparts. Such large meteoroids evolve, on average, over intervals 200 times longer than the smallest grains considered here, D = 10 μm, before they arrive at Mercury. Because of that, collisions and mean motion and secular resonances shape the orbital distribution of larger MBA meteoroids impacting Mercury's surface (Figure 1, right column). Even for the longest collisional lifetime considered here, F_coll = 50, the comparison with the collisionless population shows a severe decay where more than 95% of meteoroids were destroyed in collisions. MMRs, secular resonances, and close encounters with terrestrial planets (mainly with Earth) effectively excite the eccentricity of impactors that is now exceeding the original population, allowing meteoroids to access very eccentric orbits resembling trends reported by Marchi et al. (2005). The tail in their eccentricity distribution allows them to reach Mercury from larger semimajor axes; however, the majority of impacts have a close to the heliocentric distance of Mercury. Inclinations and impact velocities of impactors with D = 2000 μm are fairly immune to the dynamical influence from other planets, skewing the distributions of inclinations and impact velocities toward I = 40° and V_imp = 40 km s⁻¹.

The radiant distributions of impacting MBA meteoroids are reflecting the distribution of orbital elements for different sizes and different collisional lifetimes. For the smallest meteoroids, D = 10–50 μm, the collisional lifetime plays a negligible role and the radiant distribution is fundamentally driven by the orbital evolution of MBA meteoroids. Figure 2 shows radiant distributions for six different phases of Mercury's orbit for D = 10 μm meteoroids. At perihelion (TAA = 0°) MBA meteoroids impact predominantly from the apex direction (local time, or LT, of 6 hr), with a significant concentration at high northern latitudes (panel (a) in Figure 2). This north/south asymmetry in the radiant distribution at perihelion is caused by Mercury's nonzero velocity with respect to the ecliptic, due to Mercury's argument of pericenter value ω = 2917 and inclination I = 7005. This fact causes the planet to move through the meteoroid cloud upward and thus enhances the north hemisphere contribution. The radiant map at perihelion would be symmetric around the apex if the planet's inclination were zero or if its argument of pericenter were ω = 90° or 270°. Ten days later, Mercury approaches TAA = 60° and the radiant distributions of MBA meteoroids change significantly, with the majority of impacts now coming from the nightside (at 1.5 hr LT). This effect is caused by the change of direction of Mercury's velocity vector with respect to the radial vector. At perihelion the angle between these two vectors is always 90° simply because the change of the heliocentric distance with respect to time dr/dt is zero; however, for eccentric bodies, once the body leaves perihelion (or aphelion), the 90° symmetry is disturbed. At TAA = 60° Mercury's velocity vector is pointing 100° from the Sun; thus, the radiant flux is skewed toward the nightside. Interestingly, at TAA = 60° Mercury reaches the highest point of its orbit above the ecliptic, and its vertical velocity is close to 0, resulting in a symmetric radiant distribution with respect to the ecliptic/Mercury's equator (panel (b) in Figure 2). Twenty-four days after perihelion passage (TAA = 120°), the smallest MBA meteoroids impact mainly around midnight (i.e., from the antihelion direction), and the overall flux decreases compared to the flux during the perihelion passage (panel (c) in Figure 2). The flux of MBA meteoroids decreases to the minimum at Mercury's aphelion and is focused in the southern part of the anti-apex region (18 hr LT, −75° latitude). This is caused by the lower orbital velocity of Mercury in its aphelion compared to orbital velocities of surrounding MBA meteoroids on nearly circular orbits. At aphelion Mercury is moving slower than the circularized MBA meteoroid cloud; thus, the planet is being impacted from behind with respect to its velocity vector. Due to low relative impact velocities, the overall flux of impacting meteoroids is also minimized. Once Mercury leaves its aphelion, the MBA meteoroids impact mostly from the dayside (12 hr LT, panel (e) in Figure 2), where the situation for TAA = 240° reflects the meteoroid environment for TAA = 60° but now mirrored to the dayside. Closer to perihelion the overall flux increases, and once again it is concentrated in northern latitudes (panel (f) in Figure 2).

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The radiant distribution of the largest MBA meteoroids, D = 2000 μm, is more complex than for their smaller cousins owing to their more complex dynamical evolution. The higher eccentricity achieved by eccentricity pumping in MMRs during their path from the main belt allows larger MBA meteoroids to impact Mercury from a variety of directions (Figure 3). At perihelion (panel (a) in Figure 3) the north/south asymmetry is still present; however, the pattern is similar to that of the toroidal structure observed at Earth (e.g., Pokorný et al. 2014). The closer the meteoroid radiants are to the ecliptic, the higher is their eccentricity, which introduces an additional effect when the collisions are taken into account. Since meteoroids with higher eccentricities are able to impact Mercury sooner after their release from the source population, collisions are more effective at removing low-eccentricity meteoroids, and thus for short collisional lifetimes the MBA meteoroid environment for populations affected by collisions will be more concentrated around the ecliptic. The MBA meteoroid environment for the largest grains follows the same pattern as the smallest meteoroids along Mercury's orbit, where the smallest particles represent the low-eccentricity core and the broader structures of larger meteoroids reflect meteoroids with more eccentric orbits.

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To combine all MBA meteoroids and quantify their flux at Mercury, we must know (1) the production rate and SFD/MFD of MBA meteoroids and (2) the relative efficiency of transport of MBA meteoroids from Earth to Mercury. Because no in situ observations of the MBA meteoroid production rates exist currently, for condition 1 we use constraints on the MBA particle flux at Earth's proximity (this is addressed in the next paragraph of this section). For condition 2, hereby we compare the modeled ratio between the MBA meteoroid flux averaged over Mercury's orbit and the MBA meteoroid flux at Earth, which we will call the transport rate (Figure 4). First, we see that without collisions (${F}_{\mathrm{coll}}=\infty$) different sizes have different transport rates, where at Mercury the meteoroid flux is lower than at Earth with the exception of D = 2000 μm meteoroids. This is due to the effect of gravitational focusing, which is stronger at Earth than at Mercury for two reasons: (a) Earth has a deeper gravitation well and a higher escape velocity V_esc = 11.2 km s⁻¹ as compared to Mercury with V_esc = 4.2 km s⁻¹ (see Equation (2)), and (b) the relative velocity of MBA meteoroids is smaller at Earth than at Mercury. The increasing transport rate for larger meteoroids reflects a higher fraction of high-eccentricity meteoroids at Earth and then subsequently higher relative velocities and smaller gravitational focusing effects. We note that the total accreted mass will only be <1/7 compared to that at Earth, because Mercury's surface area is smaller by 1/7. Once collisions are taken into account, the transfer rate of meteoroids with larger diameters decreases rapidly, emphasizing the significant role that collisions have on shaping the MBA meteoroid environment around Mercury. Without addressing collisions accurately, one can easily overestimate the contribution of larger meteoroids by a factor of 2–3. Figure 4 allows us to calculate the total mass influx on Mercury if we have an estimated value from Earth and we know the SFD/MFD of the meteoroid population.

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Quantifying the mass influx of MBA meteoroids at Earth has proven to be rather challenging owing to their small relative velocity with respect to Earth (Carrillo-Sánchez et al. 2016). Despite the advancements in detection techniques and our current understanding of the chemical processes that meteoroids experience when entering Earth's atmosphere, the mass influx of MBA meteoroids is uncertain, where the most recent estimate is 3.7 ± 2.1 t day⁻¹. In this section, we assume that regardless of the collisional lifetime or the SFD/MFD of the MBA meteoroid population, the mass influx in the modeled size range is 1000 kg day⁻¹ over the whole surface of Earth. We opt to use this number for scalability across different populations. Finally, by combining the transport rates from Figure 4, considering all sizes together, and taking into account the surface area of Earth and Mercury, we obtain the total mass of MBA impactors on Mercury's surface (Figure 5). As expected, applying no collisions results in the largest amount of material accreted at Mercury, which decreases when the smaller meteoroids' contribution in SFD/MFD is increased (i.e., larger values of α and β). The maximum mass influx obtained from MBA meteoroids at Mercury is approximately 17% of that at Earth, but the median from our model is around 7%. Interestingly, in Figure 5 we can observe that even for the differential size–frequency index α ∼ 4, β ∼ 2 the accreted mass is fairly constant for different collisional lifetimes, where we can see the gradual decrease in accreted mass with decreasing differential size–frequency/mass index. This is a consequence of collisional breakup of larger MBA meteoroids during their dynamical evolution from Earth-crossing orbits toward Mercury-crossing orbits, since the dynamical evolution of MBA meteoroids before they can impact Earth is not reflected in Figure 5.

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In summary, our MBA meteoroid model provides similar orbital element distributions to those reported by Borin et al. (2009, 2016b) and velocities like those of Cintala (1992). Hence, the vaporization rates from this population will be similar to their estimates per unit mass. However, we additionally report a tail for V_imp > 25 km s⁻¹ that becomes more prominent for larger MBA meteoroids, D > 200 μm. Owing to their low V_imp and the nonzero eccentricity of Mercury, MBA meteoroids experience significant motion for their radiant locations along Mercury's orbit. Mutual collisions with the zodiacal cloud are important for shaping the meteoroid environment around Mercury for D > 200 μm. The mass accretion of MBA meteoroids at Mercury is at most 17% compared to values at Earth, but it could decrease down to 5% for shallow SFDs/MFDs (α ≤ 2.8, β ≤ 1.6) and shorter collisional lifetimes F_coll ≤ 20.

Unlike MBA meteoroids, JFC meteoroids can cross Mercury's orbit almost immediately after their ejection from their parent bodies, resulting in particles with a mixture of ages impacting Mercury's surface. Consequently, in this population, dynamically young meteoroids with large eccentricities are unaffected by collisions, while dynamically older meteoroids undergo collisional decay. Figure 6 shows the distributions of orbital elements of D = 10, 500, and 2000 μm JFC meteoroids at Mercury's perihelion.

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The smallest JFC meteoroids considered in our model (D = 10 μm) are unaffected by collisions and impact Mercury with preferentially lower eccentricities with a distribution that peaks around e = 0.08 (Figure 6, left column). However, due to the wide range of eccentricities of JFC meteoroids, the semimajor axis distribution is wider than for MBA meteoroids. JFC meteoroid inclinations are slightly higher compared to those of MBA meteoroids. The velocity distribution reflects higher intrinsic eccentricities of JFC meteoroids, peaking at 15–20 km s⁻¹.

For larger JFC meteoroids, D = 500 μm (Figure 6, middle column), the meteoroid eccentricity is significantly influenced by the mutual collisions with the zodiacal cloud. If no collisions are assumed, JFC meteoroids impact Mercury preferentially with lower eccentricities, resulting in a distribution that peaks around e = 0.08, similar to the smallest JFC meteoroids. By increasing the rate of collisions (decreasing F_coll), the low-eccentricity portion of this population wanes while the high-eccentricity portion becomes more prominent. Due to a wide range of eccentricities, the semimajor axis distribution is wider than for MBA meteoroids. The meteoroid population still peaks at the heliocentric distance of Mercury, similarly to MBA meteoroids; however, for shorter collisional lifetimes a significant fraction of impacting meteoroids arrives with a > 0.5 au. The inclination distribution is unaffected by the collisional lifetime in terms of the overall shape but demonstrates very clearly the decay of the relative flux with shorter collisional lifetimes. A negligible fraction of JFC meteoroids was perturbed into retrograde orbits by close encounters with Jupiter (a weak signal around I ∼ 160°). Finally, the velocity distribution reflects higher intrinsic eccentricities, and while peaking around 15–20 km s⁻¹, a significant portion of JFC meteoroids arrive with velocities over 50 km s⁻¹ for TAA = 0° and over 35 km s⁻¹ for TAA = 180°.

For the largest JFC meteoroids in our sample, D = 2000 μm, collisions heavily shape the orbital element distributions (Figure 6, right column). It is apparent that even for larger values of the collisional lifetime (F_coll = 20), the collisional rates are too high for meteoroids to reach lower eccentricities before they are destroyed in our model. Overall, the largest JFC meteoroids, D = 2000 μm, resemble D = 500 μm, but they require F_coll four times higher in order to experience similar dynamical evolution since their PR drag timescale is four times longer.

The radiant distributions for JFC meteoroids with D = 10 μm (see Figure 7) are similar to those of larger MBA meteoroids (see Figure 3), a finding that can be attributed to having similar distributions of orbital elements. However, some differences remain that are caused mainly by the resulting wider range of eccentricities and inclinations of JFC meteoroids. At perihelion the meteoroid flux is concentrated around (6 hr, 60°) as a result of the nonzero inclination of Mercury's orbit and the orientation of the Hermean velocity vector (Figure 7). Similarly to MBA meteoroids, there is a shift in the radiant distribution of JFC meteoroids as Mercury moves toward or away from the Sun, caused by the nonzero eccentricity and inclination of Mercury's orbit and a consequent drift of the planet's velocity vector from the ecliptic plane and its perpendicular orientation with regard to the radial vector.

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When no collisions are applied, the changes in the radiant distribution of JFC particles with Mercury's TAA are similar for all particle sizes. The radiant distribution becomes broader and less concentrated compared to smaller meteoroids. However, when we apply collisions, the directionality of JFC impacts changes. Figure 8 shows an extreme example of a collisionally evolved (F_coll = 1) JFC meteoroid population at Mercury with D = 2000 μm. The most striking feature in the collisionally groomed JFC populations is the lack of high-latitude particle impacts that was characteristic of the collisionless case. The low-eccentricity JFC meteoroids were collisionally destroyed in this case, and only the high-eccentricity/high-velocity (the original) portion of the JFC population remains. Another consequence of collisions is a much smaller helion/antihelion asymmetry with very subtle variations along Mercury's orbit.

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Due to higher initial eccentricities of JFC meteoroids, the transport process is generally more efficient compared to MBA meteoroids (Figure 9). Consequently, even for the shortest collisional lifetimes (F_coll = 1) and larger sizes considered in the model, the flux of JFC meteoroids at Mercury is larger than at Earth. This result is a consequence of the combination of two phenomena: (a) the high-eccentricity JFC meteoroids are able to impact both Mercury and Earth even though their low-eccentricity counterparts are destroyed, and (b) Mercury and Earth are swiping through a different volume with a different velocity, which results in an increase of the collisional probability of the meteoroid cloud with Mercury. For MBA meteoroids the transport rate rapidly decreases by a factor of 10 with increasing meteoroid diameter D for shorter collisional lifetimes (F_coll = 1, 5, 10, 20). Such an effect is not as pronounced for JFC meteoroids, yielding at most a factor of 3 decrease.

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Figure 10 shows that for each 1000 kg of JFC mass accreted at Earth approximately 200–300 kg of JFC meteoroids are accreted at Mercury for a wide variety of collisional lifetimes and SFDs. Similarly to MBA meteoroids, the JFC meteoroids provide more mass accreted on Mercury for shallower SFDs/MFDs. This is because of the gravitational focusing effect at Earth that is more pronounced for smaller meteoroids in our model, which are not that susceptible to eccentricity pumping from MMRs. Even for the most extreme values considered in Figure 10, the range of accreted mass is quite narrow where the difference between the maximum and minimum is only a factor of 2. This suggests an overall insensitivity of JFC meteoroids to SFD/MFD and collisional lifetimes considered in our model.

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In summary, the dynamics of JFC meteoroids are more complex compared to their MBA counterparts owing to the influence of close encounters with Jupiter, MMRs, and secular resonances. Collisions with the zodiacal cloud become significant for larger particles, D ≥ 200 μm, but even for the shortest collisional lifetimes considered here, the mass influx at Mercury does not become negligible owing to the low perihelion initial distances of a portion of JFC meteoroids. When comparing the mass influx at Earth and Mercury, JFC meteoroids are less affected than MBA meteoroids by uncertainties in the SFD/MFD and collisional lifetime.

To simulate HTC meteoroids, we adopt the Pokorný et al. (2014) model, even though the most recent observations of HTCs suggest that they are more uniformly distributed in inclination space (Wang & Brasser 2014; Nesvorný et al. 2017), while in the Pokorný et al. (2014) model HTCs had inclinations with a median value of I = 55°. We tested weighting meteoroids in our model based on their initial inclination to reflect the more recent HTC observations, and the results did not significantly change.

HTC meteoroids are dynamically quite different from MBA and JFC meteoroid populations because they originate beyond Jupiter/Saturn. All meteoroids in our model start with very high eccentricities (e > 0.9) and must undergo significant dynamical evolution before their orbits are circularized and thus accreted more effectively by Mercury. The orbits of the smallest HTC meteoroids considered in our model, D = 10 μm, are circularized regardless of the collisional lifetime (Figure 11, left column). As explained before, this occurs because PR drag is highly efficient for these sizes. For larger HTC meteoroids, D = 100–400 μm, only the shortest collisional lifetime F_coll = 1 is effectively preventing the particles from circularization. With further increases in meteoroid size, longer collisional lifetimes are required for particles to undergo circularization under the influence of PR drag before being collisionally disrupted. For the largest meteoroids considered in our model, D = 2000 μm, even the longest collisional lifetime assumed here, F_coll = 50, is not sufficiently long enough to ensure full circularization for most of the modeled meteoroids. The inclination distribution of HTC meteoroids (Figure 11, third row) results in two different regimes: a more dominant prograde population (I < 90°) and a smaller retrograde population (I > 90°). For D = 400 μm meteoroids the prograde portion of the HTC population always dominates over the retrograde one regardless of the collisional lifetime factor used, whereas for D = 10 μm the dominance of the prograde trajectories is less pronounced. This dichotomy translates into a bimodal distribution of impact velocities of HTC meteoroids, where the prograde/retrograde transition happens at V_imp = 80 km s⁻¹ at perihelion (Figure 11, bottom row). This transition shifts to lower V_imp as Mercury is moving toward its aphelion, where the transition occurs around V_imp = 60 km s⁻¹ owing to Mercury's smaller orbital velocity.

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The largest HTC meteoroids in our model, D = 2000 μm, are significantly influenced by mutual collisions with the zodiacal cloud. The dynamical evolution via PR drag is 200 times slower than for the smallest meteoroids in our model. This implies that larger HTC meteoroids have effectively more time to be collisionally destroyed while evolving from high-eccentricity orbits. Hence, even for the collisionless case, ${F}_{\mathrm{coll}}=\infty$, the largest HTC meteoroids always have a higher excess of high-eccentricity orbits (e > 0.95) compared to their smaller counterparts. This results from an effective barrier posed by giant planets (mostly by Jupiter), scattering particles into hyperbolic orbits or accreting them during frequent close encounters, thus preventing them from entering the inner solar system. For smaller meteoroids D ∼ 20 μm the PR drag is able to decrease the meteoroid semimajor axis quickly and thus avoid frequent close encounters with Jupiter and Saturn. Larger meteoroids spend more time in Jupiter- and Saturn-crossing orbits and thus are more susceptible to planetary scattering. This effect influences not only the eccentricity of HTC meteoroids but also their inclination, and subsequently the impact velocity distribution. The dichotomy is still apparent; however, the inclination distribution of prograde particles is more dominant, suggesting that the giant planet barrier is far more effective in shielding the inner solar system from retrograde HTC particles. Once meteoroids are decoupled from Jupiter (i.e., their aphelion distance is well below the perihelion distance of Jupiter, ∼4.5 au), the effect of F_coll becomes more important and is similar to what we report for JFC meteoroids, where larger meteoroids are more affected by collisions than smaller meteoroids.

The inclination dichotomy, the presence of the retrograde particles, and the higher median inclination value of the prograde portion of HTC meteoroids result in unique patterns of radiant distributions unlike the radiant distributions of MBA and JFC meteoroids (Figure 12 for D = 10 μm). The first unique feature is the strong concentration around Mercury's apex direction/dawn terminator (6 hr, 0°). This concentration is well known from the Earth meteor radar observation as the apex source (e.g., Campbell-Brown 2008; Janches et al. 2015) and is solely populated by retrograde meteoroids with high impact velocities (Pokorný et al. 2017). The apex source is persistent during the entire orbital cycle, but the flux decreases with increasing TAA, reaching the minimum at aphelion (TAA = 180°). The second feature is a north/south concentration centered at the dawn terminator when Mercury is at perihelion. This population of radiants is fed by prograde low-eccentricity meteoroids and follows a similar motion to that of MBA and JFC meteoroids, moving toward the nightside (antihelion source) as Mercury moves from perihelion toward aphelion, becoming centered at the dawn terminator when Mercury arrives at aphelion, and moving toward the dayside when Mercury is moving toward perihelion.

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For the largest HTC meteoroids D = 2000 μm, the prograde population dominates the mass flux and the giant planet barrier prevents meteoroids from efficient circularization, resulting in generally more eccentric orbits (Figure 11, right column). This translates into a wider spread of radiants and lower flux in the apex sources (Figure 13). The absence of low-inclination HTC meteoroids effectively translates to an absence of particles with equatorial radiants up to latitudes around ±30°. The variations of the radiant distributions with TAA show a similar behavior to that previously reported for MBA and JFC meteoroids.

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The trend of the transport rate is different from MBA and JFC meteoroids (Figure 14). Higher relative velocities of HTC meteoroids at Earth and Mercury lead to smaller gravitational focusing effects for both planets. For the smaller HTC meteoroids (D < 100 μm) the collisional lifetime uncertainty yields no changes in transport rates. The effect of the giant planet barrier is noticeable as a decrease in the transport rate for larger meteoroids D > 200 μm and is accentuated by shorter collisional lifetimes. Though not completely intuitive, a fraction of HTC meteoroids can access Earth while their perihelion distance is beyond Mercury's aphelion. Overall, the transport rates for smaller HTC particles (D = 10–200 μm) are around 500%, they significantly decrease for larger HTC meteoroids and shorter collisional lifetimes, and they are mostly larger than those of JFC meteoroids.

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Transport rates significantly larger than those of MBA and JFC meteoroids result in close-to-equal Earth versus Mercury mass accretion rates (700–800 kg at Mercury per 1000 kg at Earth) for larger mass indices α > 4, β > 2 (Figure 15). With the decreasing differential mass index of the HTC meteoroid population, the accreted mass ratio decreases, reflecting the lower transport rates of larger HTC meteoroids. The overall decrease is significant even for longer collisional lifetimes (factor of 2–3 for F_coll = 20, 50) and increases for shorter collisional lifetimes. The area centered around F_coll = 20, α = 4, β = 2 is more sensitive than for the case of JFC meteoroids, but it varies by just 20%.

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In conclusion, HTC meteoroids follow a more complex dynamical evolution compared to MBA and JFC meteoroids. Due to the presence of the giant planet barrier, the larger HTC meteoroids are easily scattered, which prevents them from circularization due to PR drag. HTC meteoroids impact Mercury from both the apex and toroidal directions. The apex direction is populated solely by retrograde HTC meteoroids whose presence, in addition to prograde meteoroids, creates bimodal inclination and impact velocity distributions. HTC meteoroids thus yield much more energy per mass unit than both JFC and MBA meteoroids. Moreover, for each kilogram of HTC meteoroids accreted on Earth a similar amount of meteoroids is accreted on Mercury for a wide spectrum of possible configurations of collisional lifetimes and SFDs.

The dynamical evolution of HTC meteoroids shows that the giant planet barrier is difficult to overcome and its effectiveness is closely related to the meteoroid diameter D. In our model OCC meteoroids start with an initial semimajor axis a_init > 300 au before they embark on their long journey to the inner solar system (see Section 2). Their characteristic dynamical timescales, i.e., the time period during which the number of meteoroids in the model decreases to 1 – 1/e = 36.8%, are approximately 0.58 Myr for D = 10 μm OCC meteoroids and approximately 1.7 Myr for those with D = 100 μm. The dynamical timescale of OCC meteoroids is on average 30% shorter for larger initial semimajor axes (in our model a_init = 1000, 3000 au) owing to higher initial eccentricities and more efficient scattering by giant planets. OCC meteoroids with initial a_init = 300 au are scattered the least, and the dynamical timescale is more strongly driven by PR drag. In our model the same weight is given to all OCC meteoroids regardless of their initial semimajor axis. Due to the easier access of a_init = 300 OCC meteoroids, the resulting flux at Mercury is mostly composed of these meteoroids, whereas meteoroids with a_init = 1000 au contribute on average a factor of 5 less in terms of flux than a_init = 300 au particles, and a_init = 3000 au meteoroids have a factor of 10 lower flux compared to a_init = 300 au.

Figure 16 shows the semimajor axis, eccentricity, inclination, and impact velocity distributions for three different sizes of OCC meteoroids; D = 10, 400, and 1200 μm. The tracing of OCC meteoroids results in noisier distributions than other simulated populations because only a very small fraction of these meteoroids can penetrate the inner solar system and evolve into orbits with accountable collisional probability with Mercury. This effect is more evident for larger diameters, while for D = 10 μm the statistics are better. Similar results were reported previously by Nesvorný et al. (2011b). However, those authors simulated an even smaller statistical sample (1000 meteoroids per size bin) and thus motivated us to expand our models to 20 times more meteoroids. The low number of meteoroids able to decouple from Jupiter results in a limited reservoir of possible orbital configurations (records in our model) that undergo further orbital evolution with higher probabilities. In reality, the amount of OCC meteoroids is many orders of magnitude higher than our current modeling capabilities; thus, such spikiness is just a model artifact due to low statistics.

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Many features of orbital element distributions of OCC meteoroids are similar to HTC meteoroids. While OCC meteoroids start with much higher eccentricities, their impact probabilities with Mercury are extremely low as a result of their large values of semimajor axis. Once OCC meteoroids assume orbits with lower semimajor axes, their eccentricity is actually less extreme than for HTC meteoroids, resulting in fewer meteoroids with e > 0.9. Results for the smallest OCC meteoroids in our model, D = 10 μm, are very similar to those for HTC meteoroids of the same size, with the only difference being that the ratio of prograde to retrograde orbits is in favor of the latter, and consequently the distribution of impact velocities for OCCs is dominated by V_imp > 80 km s⁻¹. The difference stems from HTC meteoroids having initially preferentially prograde orbits compared to an isotropic initial distribution of OCC meteoroids. With increasing size, the overall trends seem to follow a similar distribution to the smallest grains. For D = 400 μm meteoroids the effect of collisions starts becoming important, where for the shortest collisional lifetimes F_coll = 1, 5 OCC meteoroids do not survive long enough to be driven to low eccentricities by PR drag. The ratio between prograde and retrograde meteoroids does not change regardless of the collisional lifetime even for the largest grains, D = 1200 μm, which is different from HTC meteoroid behavior, where the retrograde population was decreasing in importance with increasing particle diameter. This might be an important factor in distinguishing HTC and OCC meteoroid populations since their orbital element distributions are very similar.

The orbital element distribution of OCC meteoroids impacting Mercury suggests that the majority of meteoroids will impact from the apex region (6 hr, 0°). This is true for all OCC meteoroid sizes and collisional lifetimes considered in this manuscript. Figure 17 shows the normalized distribution of impacting meteoroid radiants for D = 10 μm. The apex source dominates the total flux regardless of Mercury's TAA. Furthermore, the north and south toroidal sources (panel (a) in Figure 17) experience the same motion along Mercury's orbit as previously shown for HTC meteoroids. The apex source, while seemingly always centered at LT = 6 hr, experiences a subtle ±30-minute motion in local time during Mercury's orbital cycle following the direction of the more pronounced motion of lower-velocity meteoroids.

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Due to very similar orbital distributions of OCC meteoroids impacting Mercury, the radiant distributions are very similar for all sizes and collisional lifetimes. The only exception is the most extreme case for the shortest collisional lifetime and the largest meteoroids, D = 1200 μm and F_coll = 1, shown in Figure 18. Collisions of OCC meteoroids with the zodiacal cloud effectively prevent all meteoroids from being circularized below e = 0.9. This results in a very broad distribution of OCC meteoroid radiants with a still prominent apex source but also with a significant contribution from other sources except for the anti-apex source (18 hr, 0°). Note that the collisions in this case decrease the flux of OCC meteoroids at Mercury by more than 95%.

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Due to the very high eccentricities and low perihelion distances of OCC meteoroids in the initial distributions, both Earth and Mercury are being impacted by OCC meteoroids since the beginning of their dynamical evolution. The fact that Mercury is orbiting faster and through a more dense medium than Earth means that more OCC meteoroids impact on Mercury (Figure 19), similar to other meteoroid populations. For grains smaller than D = 100 μm, the transport rate between Earth and Mercury is fairly constant, resulting in seven times higher flux of OCC meteoroids impacting Mercury as compared to Earth, regardless of the collisional lifetime multiplier F_coll. With increasing size the collisions decrease the transport rates by a factor of 2–3. The low number statistics cause noticeable fluctuations of the transport rate values for larger sizes, D = 800, 1200 μm, particularly compared to the smooth progression seen for HTC meteoroids (Figure 14).

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By comparing the accreted fluxes at Earth and Mercury, it becomes evident that OCC meteoroids are very effective in transporting meteoroid mass to the inner solar system. For a large spectrum of size–frequency and mass indices, α ≥ 4, β ≥ 2, the amount of mass accreted at Mercury is actually comparable to that at Earth despite the planet's seven times smaller surface area (Figure 20). The mass accretion rate is also insensitive to changes of the collisional lifetime, where the mass accretion values are fairly constant for F_coll ≥ 20. Only for shorter collisional lifetimes F_coll = 1, 5 and large-particle-dominated SFDs/MFDs (α < 2.2, β < 1.4) is the mass accreted at Mercury significantly smaller when compared to Earth.

In summary, our results show that OCC meteoroids are very similar to HTC meteoroids where the only significant difference is their preferably retrograde orbits in comparison to preferentially prograde HTC meteoroids. Regardless of their size or the collisional lifetime, OCC meteoroids preferentially impact Mercury from the apex region, showing only a subtle motion during Mercury's orbital path. Consequently, OCC meteoroids have the highest average impact velocities for all meteoroid sources considered here. A very complex dynamical evolution, mainly overcoming the Jupiter barrier, prevents a large fraction of OCC meteoroids from decoupling from Jupiter and obtaining higher impact probability with inner solar system objects (see, e.g., Nesvorný et al. 2011b). Nevertheless, when OCC meteoroids effectively penetrate the inner solar system, the amount of material accreted onto Earth and Mercury is comparable for a large variety of possible modeled parameters based on variations of SFD/MFD and the collisional lifetime.

We now combine all modeled meteoroid populations to describe the characteristics of the Hermean meteoroid environment as a whole. In the previous sections we demonstrated that choices regarding the collisional lifetime and SFD/MFD can somewhat influence the access of different populations to Mercury while maintaining the same amount of meteoroid mass impacting Earth. The total mass flux at Earth and its partitioning to different populations are themselves uncertain. In this section we use the latest meteoroid influx calibration at Earth reported by Carrillo-Sánchez et al. (2016), who presented estimates for each meteoroid population assumed here: MBA = 3.7 ± 2.1 t day⁻¹, JFC = 34.6 ± 13.8 t day⁻¹, LPC = 5.0 ± 2.7 t day⁻¹, where LPC stands for long-period comets. Carrillo-Sánchez et al. (2016), who focused on the atmospheric effects produced by the meteoroid flux, assumed that LPC = HTC because differences between considering HTC or OCC meteoroids are negligible since both meteoroid populations at Earth have atmospheric entry velocities large enough (V_imp > 25 km s⁻¹) to efficiently ablate all metals in the atmosphere (Vondrak et al. 2008; Janches et al. 2009). Here we will assume a 1:1 ratio between HTC and OCC meteoroids since there are undeniably sources of OCC meteoroids observed in the solar system (Marsden 2005), and recently this ratio was used to explain the dust ejecta cloud observed around the Moon (Janches et al. 2018). Since this choice is rather arbitrary, we also tested the sensitivity of our solution for different mixing ratios of HTC and OCC meteoroids.

Besides the influx at Earth, two additional adjustable parameters of our model are the SFD/MFD of meteoroids at their sources characterized by the indices α, β and the collisional lifetime characterized by F_coll. Recent dynamical modeling investigations suggest that the F_coll must be much higher than unity in order to plausibly explain many features observed at Earth (Nesvorný et al. 2011a; Pokorný et al. 2014). In this section we will use F_coll = 20 as our preferred solution based on Pokorný et al. (2014) and similar to Nesvorný et al. (2011a) and let F_coll vary between 10 and 50 for all meteoroid populations. Selecting the correct SFD of each meteoroid population is a challenging task since in situ measurements of SFDs are scarce and limited. In recent years, progress has been made in estimating the SFD/MFD of impact cratering observations on the Moon (Suggs et al. 2014), sporadic meteor background measurements at Earth (Blaauw et al. 2011; Pokorný & Brown 2016), and the infrared observations of the zodiacal cloud (Planck Collaboration et al. 2014), suggesting various values for the sporadic meteoroid environment. In this work we choose α = 4, β = 2 as our preferred solution for all meteoroid populations, and we will investigate the sensitivity of our solution for the following ranges of SFD/MFD indices: $\alpha =[3.4,4.6],\beta =[1.8,2.2]$, which cover a wide suite of possible solutions and bracket the observed meteoroid environment at Earth (Pokorný & Brown 2016). Our model has 4 × 3 = 12 free parameters (the mass influx at Earth, SFD/MFD, F_coll for each of four meteoroid populations), which even for three different values for each parameter leaves us with a rather large number of model outcomes (531,441). Thus, we will show the preferred solution and then the most extreme deviations from the preferred solution in order to keep the manuscript at a publishable length.

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With the mixing ratio proposed by Carrillo-Sánchez et al. (2016), as well as the estimates of mass transfer rates between Earth and Mercury reported in previous sections, JFC meteoroids represent the majority of the meteoroid mass accreted on Earth; however, as we showed in previous sections, the accretion efficiency of HTC and OCC meteoroids is much higher than those of MBA and JFC meteoroids. The actual mass accreted on Mercury averaged over its whole orbit in our preferred solution is MBA_M = 0.26 ± 0.15 t day⁻¹, JFC_M = 7.84 ± 3.13 t day⁻¹, HTC_M = 1.69 ± 0.91 t day⁻¹, and OCC_M = 2.37 ± 1.38 t day⁻¹. This results in a mass influx ratio of JFC/(OCC+HTC) ∼ 2, which is much lower than that at Earth (∼7; Carrillo-Sánchez et al. 2016). Figure 21 shows the orbital elements and velocities of impactors accumulated over the entire Mercury year and provides an averaged mass flux in kg day⁻¹. For this last calculation we divided the total accumulated mass by a Mercury year, i.e., 88 days. The daily mass influx varies with TAA as will become evident in later figures. The preferred solution is shown by the thick solid black line, and the minimum and maximum values recorded for all possible permutations of our free parameters are shown as thin solid black lines. The gray solid area thus represents our confidence interval. We see several trends in the orbital element distributions. The a distribution shows that the mass flux is dominated by perihelion particles and decreases toward aphelion, with the wider distribution of semimajor axes reflecting Mercury's range of heliocentric distances. The eccentricity distribution is dominated by low-eccentricity meteoroids, with a plateau from e = 0.3 to 0.9. Interestingly, the confidence interval allows for an almost flat distribution in e, however none of the individual solution curves are in fact this flat. Cases with lower size/mass index, α = 3.4, β = 1.8, have an increased contribution to the high-eccentricity part of the meteoroid populations, while cases with higher size/mass indices emphasize low-eccentricity orbits. The inclination distribution is dominated by the JFC meteoroid portion, resulting in a predominantly prograde distribution (10.3 t day⁻¹ of prograde meteoroids vs. 1.8 t day⁻¹ of retrograde meteoroids). Only the SFD/MFD has a measurable influence on this ratio: for α = 3.4 the mass of prograde meteoroids increases to 10.6 t day⁻¹, whereas for α = 4.6 the prograde mass decreases to 10.1 t day⁻¹. A similar dichotomy that is directly tied to the meteoroid inclination is evident in the impact velocity distribution. Notice that the impact velocity distribution is broader than the meteoroid inclination. This is caused by the changes in Mercury's orbital velocity during its orbit, which has no effect on meteoroid inclinations.

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The orbital elements of the preferred solution are within the confidence region in Figure 21 for all possible HTC-to-OCC ratios within LPC = 5 t day⁻¹ range, with the inclination and impact velocity distributions being the two exceptions. For the OCC-dominating solution, LPC = OCC = 5 t day⁻¹ at Earth, there is a 60% increase in impacting mass from meteoroids with I > 90° and 90 km s⁻¹ < V_imp < 120 km s⁻¹, whereas for the HTC-dominating solution, LPC = HTC = 5 t day⁻¹, the model results in a 40% decrease for the same inclination and impact velocity range.

Variations of the impactor velocity with Mercury's TAA, which are introduced by the planet's eccentricity, are one of the most interesting outputs of the Hermean meteoroid environment model. Figure 22 shows the distribution of meteoroid impact velocities for our preferred solution for all TAAs color-coded in terms of their impacting mass. That is, each impact velocity bin (2 km s⁻¹) represents the mass flux from a particular bin in kg day⁻¹. Thus, the integral over the whole velocity spectrum gives the total mass influx rate in kg day⁻¹. The bimodal distribution of impact velocities is present for all TAAs and follows the expected trend where the separation impact velocity (prograde vs. retrograde orbits) decreases with TAA from V_imp = 80 km s⁻¹ at perihelion (TAA = 0°) to V_imp = 65 km s⁻¹ at aphelion (TAA = 180°). The mass flux provided by both fast and slow populations decreases when Mercury is moving toward its aphelion, with the local maximum around TAA = 150° for the slower impact velocities. Such a local maximum is not present for faster meteoroids. The global minimum for the faster meteoroids occurs at aphelion (TAA = 180°), while for slower meteoroids it occurs after the aphelion passage at TAA = 210°. The global maximum for slower meteoroids is at TAA = 326°, while faster meteoroids have the maximum mass influx rate at perihelion.

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One of the goals of this work is to provide the absolute impact vaporization flux. Figure 22 provides the highest-resolution data set for estimating the absolute impact vaporization flux reported to date. By simply applying a function that transforms the impacting meteoroid mass/volume and the impact velocity V_imp into the impact vaporization mass/volume, we can obtain the absolutely calibrated impact vaporization flux for any segment of Mercury's orbit. Moreover, our fine binning in the impact velocity allows us to estimate the effect that velocity cutoffs/thresholds have on vaporization efficiency rates to simulate the behavior of different chemical species. Here, we consider a rather simple but still applicable relation between the impacting volume, impact velocity, and amount of vaporized volume, given by Cintala (1992):

$$\begin{eqnarray}&&{{ \mathcal V }}_{\mathrm{vapor}}={{ \mathcal V }}_{\mathrm{Met}}(c+{{dV}}_{\mathrm{imp}}+{{eV}}_{\mathrm{imp}}^{2}),\end{eqnarray} \tag{ 7 }$$

where ${{ \mathcal V }}_{\mathrm{vapor}}$ is the volume vaporized by a meteoroid with volume ${{ \mathcal V }}_{\mathrm{Met}}$, and c = −3.33, d = −0.0102 km⁻¹ s, and e = 0.0604 km⁻² s² are empirically determined constants from Cintala (1992). These constants are valid for an iron projectile on a target substrate with a temperature of 400 K. The variation with different target temperatures is insignificant (a few percent) according to Cintala (1992). Changing the projectile material to diabase yields on average 10% differences depending on the meteoroid impact velocity, where diabase provides more impact vaporization for V_imp < 50 km s⁻¹. By assuming a bulk density of the vaporized surface ρ_vapor = 2000 kg m⁻³, we can determine the impact vaporization rate as ${{ \mathcal R }}_{\mathrm{vapor}}=\sum {{ \mathcal V }}_{\mathrm{Met}}{\rho }_{\mathrm{vapor}}(c+{{dV}}_{\mathrm{imp}}+{{eV}}_{\mathrm{imp}}^{2})$, where the sum is over all recorded meteoroids for a particular time period. Consequently, the impact vaporization flux is the impact vaporization rate divided by a surface area A, ${{ \mathcal F }}_{\mathrm{vapor}}={{ \mathcal R }}_{\mathrm{vapor}}/A$. If not specified differently, we consider A = 7.477 × 10¹⁷ cm², the surface area of Mercury. To simulate a threshold for vaporization, which is comparable to the sound speed in the medium, we consider that impactors with V_imp < 7.5 km s⁻¹ have zero contribution to the total vapor (Cintala 1992).

Figure 23 shows the impact vaporization flux ${{ \mathcal F }}_{\mathrm{vapor}}$ in g cm⁻² s⁻¹ per velocity bin (2 km s⁻¹) for the whole range of TAAs. It is evident that the dependence of vaporization efficiency of the soil on the projectile velocity enhances the relative importance of the high-velocity component of the Hermean meteoroid environment. Even though the mass influx ratio between the slower and faster component is approximately 6:1, the higher impact velocities allow the faster meteoroids to dominate the total impact vaporization flux. Note that our calibration allows for less than 5 t day⁻¹ of retrograde meteoroids at Earth and yet, compared to more than 35 t day⁻¹ of slower meteoroids at Earth, the long-period comet meteoroids dominate the impact vaporization fluxes. The importance of this fast (retrograde) meteoroid component on the impact-driven portion of the Hermean exosphere was first emphasized by Pokorný et al. (2017).

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In our preferred solution the arrival direction of meteoroids is mainly influenced by JFC meteoroids with a smaller contribution from the apex source (Figure 24). At perihelion (TAA = 0°) the meteoroids are mostly arriving from the north toroidal source (associated with JFC meteoroids) and from the ring structure, with the mass influx in the south being less than the mass flux arriving from northern latitudes. At TAA = 60° the radiant distribution is almost perfectly symmetric over the ecliptic, with the nightside/antihelion source dominating the mass flux. The maximum mass flux is at latitudes around ±60°, whereas the apex source is weaker compared to its activity during perihelion passage. As Mercury moves farther away from perihelion, TAA = 120°, the original ring structure is pushed more toward the nightside and becomes weaker. At this point, the apex source is centered at LT = 5:40 hr and most of the dayside impacts disappear. At aphelion, TAA = 180°, the overall flux reaches its minimum value and is enhanced at southern latitudes. The apex flux is symmetric around LT = 6 hr. At TAA = 240°, the overall flux increases and the ring structure is almost entirely on the dayside and near the subsolar terminator (12 hr). As Mercury moves closer to perihelion, TAA = 300°, the north toroidal concentration on the dayside (10 hr, 60°) dominates once again the mass flux of meteoroids.

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Now, we combine the distribution of radiants with the impact velocities and calculate the dependence of the impact vaporization flux ${ \mathcal F }$ from Mercury's surface as a function of local time, latitude, and the planet's TAA. These maps were constructed by first converting the meteoroid flux from steradians (Figure 24) to the mass flux from apparent zenith per unit area by applying a factor $1/\cos (\mathrm{latitude})$, which accounts for the smaller surface area of patches at higher latitudes. The resulting zenith flux allows us to calculate the flux to each surface patch. We assume that the flux decreases as $\cos ({ \mathcal A })$, where ${ \mathcal A }$ is the angular distance from the zenith for a particular surface patch, and we only consider positive values of the flux (i.e., only points within the hemisphere focused on the zenith receive any meteoroids from this direction). The zenith flux does not diverge at the poles in our model since we recorded no meteoroids impacting exactly at either of the poles. With the meteoroid flux per surface area we incorporate the impactor velocity, and using Equation (7), we calculate the absolute impact vaporization flux for each Mercury surface patch. This calculation assumes that the incidence angle of a meteoroid with respect to the local vertical does not influence the total amount of vapor produced (i.e., only the flux is reduced from the zenith, not the vapor yield). The resulting impact vaporization flux on Mercury's surface is shown in Figure 25, where the color bar shows the absolute impact vaporization flux in g cm⁻² s⁻¹ × 10⁻¹⁶. We added 10% contour levels of the maximum value to emphasize the movement and shape of underlying features. A comparison with the radiant distribution of the meteoroid flux (Figure 24) shows that the impact vaporization flux ${{ \mathcal F }}_{\mathrm{vapor}}$ is closer to the apex region than the radiant distribution would suggest. This is a consequence of the impact velocity dichotomy, where the apex source is populated by meteoroids several times faster on average compared to other directions. The difference in vaporization flux between the dawn (6 hr) and dusk (18 hr) terminators is significant for all TAAs of Mercury's orbit, consistent with the clear dawn/dusk asymmetry in Mercury's exosphere observed by MESSENGER (Burger et al. 2014; Killen & Hahn 2015; Merkel et al. 2017).

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By integrating the impact vaporization flux over the whole Hermean surface and dividing its own surface area, we obtain the average impact vaporization flux ${ \mathcal F }$ (Figure 26). Our preferred solution (thick solid line in Figure 26) predicts a global maximum value at TAA = 337° and a global minimum at TAA = 188°. The global maximum and minimum are assumed for the same values of TAA for all solutions in our confidence interval (gray area with thin solid black lines as borders). The ratio between the maximum and minimum is ${{ \mathcal F }}_{\mathrm{TAA}={337}^{^\circ }}/{{ \mathcal F }}_{\mathrm{TAA}={188}^{^\circ }}=436/82=5.29\pm 0.07$, where the impact vaporization flux averaged over one Mercury orbit is ${{ \mathcal F }}_{\mathrm{orbit}}=(200\pm 16)\times {10}^{-16}$ g cm⁻² s⁻¹. The predicted impact vaporization flux has a similar dependence with TAA to source rates for Mercury's exospheric calcium presented by Burger et al. (2014), with a few exceptions: enhancements at TAA = 20° and TAA = 170° that are thought to originate from 2P/Encke (Christou et al. 2015; Killen & Hahn 2015), and TAA from 315° to 350° where our model predicts 30%–40% higher impact vaporization fluxes than Burger et al. (2014).

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In this section we explore the effect that the most extreme variations from the preferred solution have on the overall results. The mixing ratios for individual meteoroid populations at Earth are still being refined and tested when new measurements become available. The calibration by Carrillo-Sánchez et al. (2016) shows that individual populations have uncertainty of >50%, which would significantly alter the absolute calibration of our model. However, using results we provided for individual meteoroid populations, it is possible to reconstruct a majority of our results with a new calibration without performing a full-fledged dynamical model. A major conclusion of this manuscript is the overall insensitivity of individual meteoroid populations to changes in both collisional lifetime setting F_coll and the SFD for various combinations of free parameters. The radiant maps, orbital element distributions, and mass transportation ratios hold their characteristics very well for any combination of F_coll ≥ 10 and α > 3.4, β > 1.8. Moreover, no matter how we set the free parameters in our model, there will always be a strong dawn/dusk asymmetry, since the impacts around the dusk terminator are restricted only to meteoroids on circular orbits close to Mercury's aphelion (TAA = 180°), because Mercury's orbital velocity compared to the meteoroids' orbital velocity is smaller, while their velocity vectors are aligned.

To shed light on the sensitivity of the model, we calculated several most extreme solutions and compared them with our preferred solution in terms of the impact vaporization flux with respect to TAA (Figure 27). In the previous sections we showed that the mass accreted at Mercury and at Earth is fairly constant for individual populations for the differential SFD/MFD indices α ≥ 4, β ≥ 2. Unsurprisingly, when the meteoroid complex is dominated by the smallest meteoroids α = 7, β = 3, the impact vaporization fluxes are similar regardless of the collisional lifetime (orange and cyan solid lines in Figure 27). When, on the other hand, the SFD/MDF is dominated by the largest grains α = 1, β = 1, the collisional lifetime multiplier F_coll significantly shapes the results. When no collisions are assumed, ${F}_{\mathrm{coll}}=\infty$, the average impact vaporization flux increases above the confidence interval significantly. Around Mercury's perihelion the impact vaporization flux increases to 25% higher values as compared to the preferred solution. This is a consequence of higher mass transfer rates of JFC and OCC meteoroids for these SFD/MFD (solid magenta line in Figure 27). At aphelion the impact vaporization flux is a few percent above the confidence interval. The ratio between the maximum and minimum value along the whole orbit is ${{ \mathcal R }}_{\max }/{{ \mathcal R }}_{\min }=5.45$ (we omitted one outlying point in this analysis at TAA = 63). If, on the other hand, the shortest collisional lifetimes in our model are assumed, F_coll = 1, we obtain a much flatter solution than our preferred model (solid green line), where the ratio between the maximum and minimum impact vaporization flux is ${{ \mathcal F }}_{\max }/{{ \mathcal R }}_{\min }=3.22$.

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Note that for all settings of F_coll and α, β we keep the amount of mass accreted on Earth constant. Increasing the collisional lifetime and changing the SFD/MFD does not thus change the mass accreted on Earth; however, the production rates required for the source populations to generate the amount of accreted meteoroids change dramatically. For instance, for very short collisional lifetimes and large-grain-dominated SFD/MFD, the asteroid belt might require constant production of an extreme amount of meteoroids, which might not be possible based on our current knowledge. Since this issue greatly exceeds the focus of this paper, we leave it for future work.

To test the sensitivity of our preferred model for Mercury to uncertainties of the mass influx of different meteoroid populations at Earth given by Carrillo-Sánchez et al. (2016), we fix the free parameters F_coll = 20, α = 4, β = 2 and alternate the mixing ratios of different populations (Figure 28). The lowest mass influx on Earth within the Carrillo-Sánchez et al. (2016) range, MBA = 1.6 t day⁻¹, JFC = 20.8 t day⁻¹, HTC = 1.15 t day⁻¹, OCC = 1.15 t day⁻¹, is represented by the solid green line and gives on average 48% smaller impact vaporization flux compared to the preferred solution. The upper boundary, on the other hand, gives on average 33% higher impact vaporization flux than our preferred solution, MBA = 5.8 t day⁻¹, JFC = 48.4 t day⁻¹, HTC = 3.85 t day⁻¹, OCC = 3.85 t day⁻¹ (solid orange line). We also picked two intermediate states where the inner solar system sources, the MBA and JFC meteoroids, and the outer solar system sources, the HTC and OCC meteoroids, were given a combination of the lowest and highest permitted values MBA = 1.6 t day⁻¹, JFC = 20.8 t day⁻¹, HTC = 3.85 t day⁻¹, OCC = 3.85 t day⁻¹ (solid cyan line) and MBA = 5.8 t day⁻¹, JFC = 48.4 t day⁻¹, HTC = 1.15 t day⁻¹, OCC = 1.15 t day⁻¹ (solid orange line). We test only the sensitivity with respect to different abundances of individual meteoroid populations and their uncertainty intervals estimated by Carrillo-Sánchez et al. (2016), and we omit further sensitivity to other free parameters. We conclude that even for the extreme values permitted by Carrillo-Sánchez et al. (2016) our model does not vary by more than a factor of 2 with respect to our preferred solution.

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One of the outstanding questions about our choice of free parameters is whether they are degenerate or oversimplified. The value of the collisional lifetime multiplier F_coll is kept the same for all meteoroid populations in our model; however, this might not be entirely correct owing to the different nature of meteoroids from different sources, e.g., the MBA meteoroids might be stronger and harder to break compared to cometary meteoroids especially from the Oort Cloud. Introducing these and testing these factors is beyond the scope of this study mainly because F_coll is unconstrained for most meteoroid populations. Our selection of a single power-law SFD/MFD might seem to be an oversimplification given that recent dynamical models (Nesvorný et al. 2010, 2011a; Pokorný et al. 2014) were using three parametric broken power-law SFD/MFD. By investigating a wide range of the SFD/MFD single power-law indices, we show how the results change if the meteoroid population is dominated by smaller meteoroids (high values of α, β), and then gradually the importance lies in larger meteoroids (lower values of α, β). Due to mostly smooth transitions between larger/smaller modes, we assume that models with more complicated power laws would lie somewhere in the range of models presented here. Another simplification lies in keeping the SFD/MFD the same for all meteoroid populations. While the possibility of alternating SFD/MFD of individual populations in our model is viable, due to added complexity, we keep this element for future investigations.

Our calibrated model appears to grossly overestimate observed vaporization fluxes. To quantify by how much, we assumed an Mg fraction of f_Mg = 0.17 in the high-Mg regions of Mercury (Lawrence et al. 2017) and converted the peak vaporization flux, ${ \mathcal F }=436\times {10}^{-16}$ g cm⁻² s⁻¹, to the number of Mg atoms expected to be vaporized from the surface, ${{ \mathcal N }}_{\mathrm{Mg}}=1.8\times {10}^{8}$ Mg atoms cm⁻² s⁻¹, using ${{ \mathcal N }}_{\mathrm{Mg}}\,={N}_{{\rm{A}}}{ \mathcal F }{f}_{\mathrm{Mg}}/{m}_{\mathrm{Mg}}$, where N_A = 6.022 × 10²³ mol⁻¹ is the Avogadro constant and m_Mg = 24.305 is the atomic weight of magnesium. This estimate exceeds the inferred Mg source rate from MESSENGER measurements, 8 × 10⁵ Mg atoms cm⁻² s⁻¹ (Merkel et al. 2017), by more than a factor of 100, whereas the variation of Mg on the surface is only about a factor of two (Lawrence et al. 2017). Using results in Figure 23 and applying a simple impact velocity cutoff where the impact vaporization no longer contributes for V_imp > 17 km s⁻¹ provides ${{ \mathcal N }}_{\mathrm{Mg}}$ comparable to rates observed by MESSENGER for a wide range of TAAs. It seems clear that high-velocity impacts, especially due to impactors in retrograde orbits, lead to ionization, but the ionized fraction is uncertain. The impact charge generated by the incidence of a dust particle on a target increases as ${{mV}}_{\mathrm{imp}}^{3.5}$ (Auer 2001), which is faster than the increase with impactor velocity of the total impact vapor from Cintala (1992). Therefore, the ionized fraction will increase with impactor velocity V_imp. Choosing different parameters for impactor and target from Cintala (1992) does not yield significantly different results for the total vapor, on average less than 10%. Thus, since our transport rate calculations show that it is impossible to decrease the flux on Mercury by more than a factor of two for any reasonable combination of collisional lifetime and SFD/MFD and at the same time explain the arrival rates of meteoroids at Earth, our findings suggest that the vaporization function is questionable at v > 20 km s⁻¹, possibly due to losses to ionization and other phases like condensates (Berezhnoy & Klumov 2008).

In this manuscript, we presented modeled results of the orbital element distributions, impacting meteoroid radiant maps, transportation rates, and ratio between mass accreted on Mercury and Earth for meteoroids originating in the MBAs, JFCs, HTCs, and OCCs. Our model includes the effects of meteoroid collisions with the zodiacal cloud through the Steel & Elford (1986) collisional model refined for Cremonese et al. (2012) meteoroid flux estimates.

The MBA meteoroids and JFC meteoroids impact Mercury preferentially with lower eccentricities e < 0.2 and small inclinations I < 30°, which results in impact velocities V_imp < 70 km s⁻¹ at perihelion and V_imp < 50 km s⁻¹ at aphelion. Approximately 7% of mass accreted at Earth originating from MBAs is recorded at Mercury. This is due to low-eccentricity orbits of MBA meteoroids resulting in low relative impact velocities with both Earth and Mercury. Such low velocities are efficiently attracted by Earth's gravity, while this effect is much smaller for Mercury. JFC meteoroids have a broader distribution of eccentricities, which weakens the gravitational focusing and leads to higher mass accretion at Mercury as compared to that at Earth, ∼23%.

The meteoroids produced by HTCs and OCCs impact Mercury with a flat eccentricity distribution and a bimodal distribution of orbital inclinations of prograde and retrograde orbits. Retrograde meteoroids are predominantly impacting from the apex direction, have impact velocities 95 km s⁻¹ < V_imp < 120 km s⁻¹ at perihelion and 75 km s⁻¹ < V_imp < 90 km s⁻¹ at aphelion, and are less influenced by Mercury's orbital motion. Both long-period comet meteoroid populations have significantly higher Mercury-to-Earth mass accretion ratios as compared to JFC meteoroids, ∼70% for HTC meteoroids and ∼90% for OCC meteoroids,

Our preferred solution for a dynamical model of meteoroids at Mercury based on calibration from Carrillo-Sánchez et al. (2016) provides the following values of accreted mass averaged over the entire Hermean orbit: MBA meteoroids MBA_M = 0.26 ± 0.15 t day⁻¹, JFC meteoroids JFC_M = 7.84 ± 3.13 t day⁻¹, HTCs HTC_M = 1.69 ± 0.91 t day⁻¹, and OCCs OCC_M = 2.37 ± 1.38 t day⁻¹. This results in a mass influx ratio of short/long-period comet meteoroids of ∼2, which is much lower than that at Earth (∼7; Carrillo-Sánchez et al. 2016). The vaporization flux averaged over one Mercury orbit then yields ${{ \mathcal F }}_{\mathrm{orbit}}=(200\pm 16)\times {10}^{-16}$ g cm⁻² s⁻¹, with maximum at TAA = 337°, ${{ \mathcal F }}_{\max }=(436\pm 57)\times {10}^{-16}$ g cm⁻² s⁻¹ and minimum at TAA = 188°, ${{ \mathcal F }}_{\min }=(82\,\pm 12)\times {10}^{-16}$ g cm⁻² s⁻¹. Our model provides impact vaporization fluxes of the same order of magnitude as Borin et al. (2009, 2017). However, this similarity is misleading because we predict a lower mass flux of meteoroids at Mercury (Mercury/Earth ratio less than 1 vs. 35 in their model) accompanied by higher-speed distributions. The variations of the impact vaporization flux and the impact directionality also contradict the result of Borin et al. (2009, 2017) because our model predicts a larger perihelion-to-aphelion ratio in the impact vaporization rate. Furthermore, we predict that meteoroid impacts onto Mercury provide a significant source of surface ions. Only ∼1% of the estimated total vapor appears to contribute neutrals to Mercury's exosphere.

Our preferred solution shows a strong dawn/dusk asymmetry in both meteoroid impact direction distribution and the impact vaporization pattern on the surface. The impact vaporization pattern undergoes significant movements during Mercury's orbit, being centered at the dawn terminator (6 hr) at perihelion and aphelion and moving toward the nightside during the first leg (maximum displacement of the center is 3 hr) and being shifted toward the dayside in the second leg, when Mercury is moving back to its pericenter. The impact vaporization flux integrated over the whole surface has a similar pattern to source rates for calcium presented by Burger et al. (2014), with a few exceptions: enhancements at TAA = 20° and TAA = 170° that are thought to originate from 2P/Encke (Christou et al. 2015; Killen & Hahn 2015), and TAA from 315° to 350° where our model predicts 30%–40% higher relative impact vaporization flux than Burger et al. (2014).