Mercury's Circumsolar Dust Ring as an Imprint of a Recent Impact

Various observations and models show that 5%–10% of the inner ZC cross section is contained inside 1 au (Hahn et al. 2002; Nesvorny et al. 2010). Let us assume that the total cross section of the ZC is 1–2 × 10¹⁷ m² (Gaidos 1999; Nesvorný et al. 2011a), which provides approximately 0.5–2 × 10¹⁶ m² for the ZC cross section below a heliocentric distance of 1 au. The particle density of the inner ZC in the ecliptic n, scales with the heliocentric distance R as n ∝ R^−1.3 (Leinert et al. 1981) and we assume the same scaling for the particle cross section, Σ.

Based on Stenborg et al. (2018b), the observed dust ring is located at the solar elongation region between ε = 21°.3 ± 1°.3, i.e., between heliocentric distances of r₁ = 0.356271 au and r₂ = 0.400308 au. Using the total particle cross-sectional scaling, $\sigma \propto {r}_{\mathrm{hel}}^{-1.3}$, we can calculate the ratio between the ZC cross section inside the boundaries set by Mercury's dust ring and the ZC cross-sectional interior to 1 au as follows:

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{ring}}=\displaystyle \frac{{\int }_{{r}_{1}}^{{r}_{2}}{r}_{\mathrm{hel}}^{-1.3}{\rm{d}}{r}_{\mathrm{hel}}}{{\int }_{0.05\mathrm{au}}^{1\mathrm{au}}{r}_{\mathrm{hel}}^{-1.3}{\rm{d}}{r}_{\mathrm{hel}}}=0.03215,\end{eqnarray} \tag{ 1 }$$

where r₁ = 0.356271 au, r₂ = 0.400308 au, and we use r_hel = 0.05 au as the boundary of the dust-free zone (Stenborg et al. 2021b). Using the average relative brightness increase from Stenborg et al. (2018b) of ≈1.5% for Mercury's dust ring, we estimate the dust cross section of the entire ring to be Σ_ring = 2.4–9.6 × 10¹² m²; i.e., approximately 1/2000 of the total ZC cross section inside 1 au.

We can convert the approximate dust cross section of Mercury's dust ring into a ring mass assuming some size–frequency distribution (SFD). Following the equations from e.g., Pokorný et al. (2014) or Pokorný & Kuchner (2019), we convert between the ring dust cross section Σ_ring and ring dust mass M_ring:

$$\begin{eqnarray}&&{M}_{\mathrm{ring}}=\displaystyle \frac{6-2\alpha }{12-3\alpha }\rho {D}_{\max }{{\rm{\Sigma }}}_{\mathrm{ring}}\displaystyle \frac{1-{\left(\tfrac{{D}_{\min }}{{D}_{\max }}\right)}^{4-\alpha }}{1-{\left(\tfrac{{D}_{\min }}{{D}_{\max }}\right)}^{3-\alpha }},\end{eqnarray} \tag{ 2 }$$

where α is the differential SFD index, ρ is the bulk density of dust particles in the ring, D_min is the smallest particle diameter in the ring, and D_max is the largest particle diameter. D_min and D_max denote the range where the SFD follows a single power law with an index of α; i.e., the number of particles larger than D follows dN(>D) ∝ D^−α dD. Since we do not have any constraints for any of the parameters, we adopt α = 3.5 used for populations in collisional equilibrium (Dohnanyi 1969), ρ = 2000 kg m⁻³, ${D}_{\min }=10\,\mu {\rm{m}}$ = 10⁻⁵ m, and ${D}_{\max }=1\,\mathrm{cm}$ = 10⁻² m. This gives us a total mass for Mercury's dust ring of M_ring = 1.02–4.05 × 10¹² kg, equivalent to the mass of a single asteroid with a diameter of ${D}_{\mathrm{sphere}}=990\mbox{--}1570$ m. This is a mass comparable to that of Venus's circumsolar dust ring, as estimated in Pokorný & Kuchner (2019). Our estimated value of M_ring can easily vary by 1–2 orders of magnitude depending on the free parameter setup, such as changing α by unity, or by increasing D_min to 100 μm.

To reproduce the Stenborg et al. (2018b) observation of Mercury's dust ring, we created a simplified simulator of the H i-1 instrument on board STEREO A. The level of processing that Stenborg et al. (2018b) implemented is beyond the scope of this article and is not necessary because we can simulate each dust population separately. Moreover, any noise present in any F-corona/ZC model would further decrease the fidelity of our results. Therefore, we try to keep our synthetic telescope model as simple as possible.

We use the spacecraft orbital elements a = 0.9618 au, e = 0.00583, I = 0126, Ω = 2138, and ω = 938 to generate 360 heliocentric position vectors along the orbit uniformly distributed with respect to the spacecraft's ecliptic longitude λ. From each of these positions, we calculate the relative brightness of each dust particle as:

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal B } & = & \displaystyle \frac{\pi {D}^{2}}{4}\displaystyle \frac{1}{\parallel {{\boldsymbol{r}}}_{\mathrm{par}}-{{\boldsymbol{r}}}_{\mathrm{SC}}{\parallel }^{2}}\displaystyle \frac{1}{\parallel {{\boldsymbol{r}}}_{\mathrm{par}}{\parallel }^{2}}{ \mathcal P }(\cos \gamma )\\ & = & \displaystyle \frac{\pi {D}^{2}}{4}\displaystyle \frac{1}{\parallel \vec{{\rm{\Delta }}}{\parallel }^{2}}\displaystyle \frac{1}{\parallel {{\boldsymbol{r}}}_{\mathrm{par}}{\parallel }^{2}}{ \mathcal P }(\cos \gamma ),\end{array}\end{eqnarray} \tag{ 3 }$$

where D is the particle diameter, r _par is the heliocentric position vector of the particle, r _SC is the heliocentric position vector of the spacecraft, ${ \mathcal P }$ is the scattering phase function, γ is the Sun–particle–observer angle, $\vec{{\rm{\Delta }}}={{\boldsymbol{r}}}_{\mathrm{par}}-{{\boldsymbol{r}}}_{\mathrm{SC}}$ is the vector pointing from the spacecraft to the particle, and ∥ denotes the length of the vector. Note that all particles in our model are much larger than the wavelength range observed by the H i-1 instrument; 630–730 nm.

The scattering phase function ${ \mathcal P }$ is calculated following Table 1 from Hong (1985), who used a linear combination of three Henyey–Greenstein functions as:

$$\begin{eqnarray}&&{ \mathcal P }(\cos \gamma )=\displaystyle \sum _{i=1}^{3}\displaystyle \frac{{w}_{i}}{4\pi }\displaystyle \frac{1-{g}_{i}^{2}}{{\left(1+{g}_{i}^{2}-2{g}_{i}\cos \gamma \right)}^{3/2}},\end{eqnarray} \tag{ 4 }$$

where w₁ = 0.665, w₂ = 0.330, w₃ = 0.005, g₁ = 0.70, g₂ = −0.20, and g₃ = −0.81. The scattering angle γ is calculated as:

$$\begin{eqnarray}&&\gamma =\arccos \left(\displaystyle \frac{\vec{{\rm{\Delta }}}\cdot {{\boldsymbol{-r}}}_{\mathrm{SC}}}{\parallel \vec{{\rm{\Delta }}}\parallel \parallel {{\boldsymbol{r}}}_{\mathrm{SC}}\parallel }\right).\end{eqnarray} \tag{ 5 }$$

Finally, we calculate the particle solar elongation in the ecliptic for each particle as the angle between the vernal equinox and the particle's position in the ecliptic:

$$\begin{eqnarray}&&\epsilon =\arccos \left(\displaystyle \frac{\vec{{{\rm{\Delta }}}_{\mathrm{ecl}}}\cdot {{\boldsymbol{-r}}}_{{\mathrm{SC}}_{\mathrm{ecl}}}}{\parallel \vec{{{\rm{\Delta }}}_{\mathrm{ecl}}}\parallel \parallel {{\boldsymbol{r}}}_{{\mathrm{SC}}_{\mathrm{ecl}}}\parallel }\right),\end{eqnarray} \tag{ 6 }$$

where the vector projection to the ecliptic is calculated as $\vec{{{\rm{\Delta }}}_{\mathrm{ecl}}}=\vec{{\rm{\Delta }}}-(\vec{{\rm{\Delta }}}\cdot {{\boldsymbol{n}}}_{z}){{\boldsymbol{n}}}_{z}$, where n _z = (0, 0, 1) is the normalized vector normal to the ecliptic plane. This is γ projected to the ecliptic. We also need to calculate the direction/sign of the elongation, ${ \mathcal D }$, in order to distinguish westward/eastward pointing. The direction, ${ \mathcal D }$, is calculated as:

$$\begin{eqnarray}&&{ \mathcal D }=\mathrm{sign}({{\boldsymbol{r}}}_{{\mathrm{SC}}_{\mathrm{ecl}}\perp }\cdot \vec{{{\rm{\Delta }}}_{\mathrm{ecl}}})=\mathrm{sign}(-{r}_{{\mathrm{SC}}_{y}}{{\rm{\Delta }}}_{x}+{r}_{{\mathrm{SC}}_{x}}{{\rm{\Delta }}}_{y}).\end{eqnarray} \tag{ 7 }$$

The sporadic meteoroid and dust background is the main component of the ZC (Koschny et al. 2019). Numerous modeling (Wiegert et al. 2009; Nesvorny et al. 2010) and observational studies (Campbell-Brown 2008; Janches et al. 2015; Carrillo-Sánchez et al. 2020; Rojas et al. 2021) showed that three main components dominate the inner solar system dust and meteoroid budget with diameters between several micrometers and several millimeters. The main belt asteroids supply the ZC component closest to the ecliptic (Nesvorny et al. 2010), the short-period comets/JFCs make up the majority of the observable ZC with 80%–90% of the total mass and particle cross section (Nesvorný et al. 2011a), while the long-period comets (LPCs) supply the broad envelope and particles on retrograde orbits (Nesvorný et al. 2011b; Pokorný et al. 2014). The outer Solar sources of dust with perihelion distances beyond Jupiter do not significantly contribute to the inner solar system budget due to Jupiter's gravitational barrier (Poppe 2016, 2019).

Due to the sporadic nature of ZC particles in the inner solar system, a detection of any significant enhancement would require a strong concentration in semimajor axis a, as shown for the circumsolar dust ring co-orbiting with Venus (Pokorný & Kuchner 2019). As shown in Sommer et al. (2020), meteoroids migrating via PR drag are likely to get captured in external MMRs with Venus and Earth and are unaffected by or quickly migrate to their internal MMRs. This effect also diminishes Venus' external MMR capture rates due to the interference of internal MMRs with Earth. Nevertheless, Sommer et al. (2020) showed no or negligible efficiency of capture of any MMRs near the orbit of Mercury.

To test the hypothesis that the Mercury dust ring is a natural consequence of ZC particles trapped in Mercury's MMRs, we expanded the numerical models used in Pokorný & Kuchner (2019) and Pokorný et al. (2018, 2021). We analysed dust and meteoroid semimajor axis distributions near Mercury's orbit for 26 different dust/meteoroid sizes with diameters of D = 0.6813 μm to D = 6813 μm originating from MBAs, JFCs, and LPCs, thus expanding the models used by Pokorný & Kuchner (2019) and Pokorný et al. (2018, 2021). For D > 1000 μm we integrated 5000 individual particles and for D ≤ 1000 μm we integrated 5 × 10⁶/D (μm) individual particles to reflect the faster particle migration via PR drag. A summary of this analysis is shown in Figure 2. For all analysed sizes and populations, there is a maximum ∼200% increase in the number of particles for main belt and JFC meteoroids with D ≥ 1000 μm related to their temporary capture in several exterior MMRs with Mercury (e.g., the 3:4 MMR for main belt asteroids and the 15:17 MMR for JFCs) and around interior 2:1 MMR with Venus for JFCs. The number of particles temporarily caught in MMRs is negligible in comparison to the total number of particles in Mercury's neighborhood and is only emphasized by the bin size of da = 0.0003 au. Moreover, all these temporary captures occur beyond the maximum heliocentric distance that STEREO A H i-1 can observe R_hel < 0.407 au. For these reasons, we conclude that the sporadic meteoroid background is not a viable source of Mercury's dust ring.

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In the previous section, we showed that the sporadic meteoroid background does not show any signature of clumping close to Mercury's orbit. The sporadic background represents asteroidal and cometary particles that dynamically evolved for thousands to millions of years and lost their connection to their parent bodies. In this section, we focus on freshly ejected dust and meteoroids from active asteroids and comets that can form concentrated streams of matter, called meteoroid streams (Jenniskens 1998).

To analyse the potential of solar system small bodies to reproduce Mercury's dust ring, we downloaded the latest version of the MPCORB database, ⁵ which contains more than 1.2 million entries for asteroids and 942 entries for comets. For each of these bodies, we ran a minimum orbit intersection distance (MOID) analysis with respect to 360 points along Mercury's orbit using J2000 coordinates. These points were spaced uniformly in time, sampling one complete orbit of Mercury. In our analysis, we calculated the number of points on Mercury's orbit that have MOID <0.05 au with each object in our database. We assumed this threshold from the estimated width of Mercury's circumsolar dust ring based on the Stenborg et al. (2018b) observation, where the ring was observed between solar elongations of 20° < < 23°; i.e., 0.356 < R_hel < 0.407 au. We did an additional analysis with a threshold of MOID <0.10 au to explore a potentially more diffuse dust ring that may be beyond the field of view of the STEREO A H i-1 instrument. The results of our analysis showing the five closest asteroids, five closest asteroids with an absolute magnitude of H < 16 (approximately 2.2 km in diameter), and five closest comets are found in Table 1.

The results in Table 1 show that the smallest asteroids (H > 23 or D < 100 m) have the greatest similarity with Mercury's orbit. Their MOID (<0.05 au) is ∼27%, so each asteroid's orbit overlaps with approximately one quarter of Mercury's orbit. Therefore, even if we assume that the orbits of these asteroids do not overlap close to Mercury's orbit, the current longitudinal extent of Mercury's dust ring requires at least 3–4 of these asteroids. However, their sizes are orders of magnitude smaller than the amount of dust in the currently observed dust ring (Section 2.1). When we consider larger asteroids with absolute magnitudes of H < 16 or D ≳ 2 km (second section in Table 1), the orbital coverage of Mercury's ring by individual asteroids drops significantly, requiring >10 objects to cover the longitudinal profile of the ring fully. This is assuming no overlap between the dust streams generated by these objects. Figure 3(A), which presents a face-on view of the inner solar system and the orbits of the three best candidates from all asteroids (thin solid lines) and asteroids with H > 16 (thick solid lines), shows that this is not the case.

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Similar to recent asteroid break-ups, recent comet activity or disruptions are unlikely to be the source of Mercury's dust ring. Our analysis shows that to cover the longitudinal structure of Mercury's dust ring fully, >10 comets would have needed to have fortuitously aligned orbits and have significant activity/established meteoroid streams (third section in Table 1). As shown in Figure 3(B), this is not the case. Moreover, the cometary orbits shown in Figure 3(B) extend closer to the Sun with the exception of 2P/Encke, which is not observed in the STEREO observations. We also emphasize that in Figure 3 we show orbits projected into the ecliptic plane disregarding the inclination of each comet. Since most comets have inclinations significantly different from Mercury's inclination, including any latitudinal alignment condition would make the cometary origin even less probable.

Our analysis of the orbital similarities between the orbits of various small bodies and Mercury's dust ring demonstrates that recent break-ups or cometary activity of known small bodies in the solar system are unlikely to be the source of Mercury's circumsolar dust ring. Even if we found several small bodies that would have a significant orbital similarity with Mercury's orbit (such as asteroids locked in an 8:7 MMR with Mercury), the dust would either need to be dynamically fresh or it would require some mechanism to keep it close to its source region. This is due to the strong effect of PR drag, which is able to change the semimajor axis of D = 1 mm particles from 0.45 to 0.3 au within 80,000 yr. The only mechanisms that can delay the orbital decay via PR drag are MMRs, which we investigate in the next section.

In this section, we analyse the ring-generating potential of dust particles released from the external MMRs of Mercury. We simulated a sample of 26 different meteoroid/dust grain diameters ranging from D = 0.6813 to D = 6813 μm with an equal logarithmic spacing. We released the particles from the vicinity of the following external MMRs of Mercury: 1:2, 2:3, 3:4, 4:5, 5:6, 6:7, and 7:8; i.e., from the semimajor axis range 0.423 < a < 0.614 au, where for each MMR we used its corresponding value of a. All particles were released with initial eccentricities of 0 < e < 0.2, inclinations 0 < i < 10°, and randomly generated longitudes of the ascending node Ω, argument of pericenter ω, and mean anomaly M. These initial orbital elements represent values before the effect of radiation pressure is accounted for, since we want to simulate the ejection of dust grains from larger parent bodies. For each particle diameter–MMR combination, we simulated the dynamical evolution of 1000 particles using the methods described in Section 4. In total, we simulated the evolution of 182,000 unique particles.

A summary of the contributions of the source populations near these external MMRs with Mercury is presented in Figure 4. Particles smaller than D < 100 μm do not interact with any of the MMRs of Mercury. This is expected due to the strong effect of PR drag (Figure 4(A)) and the large value of the semimajor axis drift da/dt that allows such particles to skip easily any MMR with Mercury. As the particle size increases, the semimajor axis drift da/dt due to PR drag decreases and the simulated particles spend more time (∼1–10 kyr) in various resonances, as shown in Figure 4(B) for D = 681.3 μm. However, this resonant capture is not efficient enough and we see no strong clumping in any of the MMRs. Moreover, particles originating in all analysed MMRs effectively skip the 1:1 MMR with Mercury (dips at a = 0.3871 au). For even larger particles where D = 2154 μm, the particles show temporary residence in their initial MMRs (Figure 4(C)). For the largest particle diameters analysed here (D = 6813 μm), this residence time can reach ∼1 million years.

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However, from the STEREO data and the stability of the longitudinal structure of the ring (Section 2), we can assume that the particle clumping necessary to reproduce the ring shape requires particles being efficiently trapped in some MMR and secular apsidal resonance with Mercury (Murray & Dermott 1999). Such a capture does not efficiently occur for any combination of the MMR order and particle size that we modeled once they leave their source regions. Additionally, with increasing particle size, the particles are more likely to skip over the 1:1 MMR with Mercury, thereby creating a gap or deficiency, in contrast to the significant clustering required to explain the dust ring.

Regardless of particle size, dust and meteoroids migrating from source populations located near the external MMRs with Mercury do not get captured at or close to the 1:1 MMR with Mercury, and their capture in the exterior MMRs of Mercury is inefficient. Smaller particles (D ≤ 100 μm) smoothly drift past Mercury whereas larger particles skip the co-orbital resonance. For these reasons, we conclude that any particle population migrating from a region exterior to the 1:1 MMR with Mercury is unlikely to reproduce a circumsolar dust ring close to Mercury's orbit.

In the previous section, we showed that dust and meteoroids released from source bodies locked in the 1:1 MMR with Mercury produce an eccentric circumsolar ring of dust that is aligned with Mercury's orbit, and its shape is stable over millions of years. To compare our results to available data, we use the simple STEREO A simulator described in Section 4.1. As shown in Figure 4 and Section 3.1.2 in Stenborg et al. (2018b), STEREO A observes light scattered by the dust ring particles at longitudes shifted by ≈70° because the line of sight tangentially traverses the ring at elongations of ≈ 20°. Therefore, STEREO A observing at an ecliptic longitude of λ = 340° detects the strongest brightness increase from the ring sections with an ecliptic longitude of λ = 270°. The H i-1 instrument is pointing in the eastward direction (behind the spacecraft with respect to its direction of motion). The westward pointing detector would experience a similar shift but in the positive direction. To accommodate this 70° shift in λ and to make it easier to compare to our dynamical model from Section 6, we use λ = (λ_S/C − 70°) in all subsequent figures in this article.

To obtain a relative brightness increase of our model ring with respect to the ZC and data in Figure 1, we perform two additional operations with our brightness model. First, we multiply the brightness of each particle in our model ${ \mathcal B }$ by a factor of ^2.35 to capture the brightness increase of the smooth ZC background with as discussed in Stenborg et al. (2018a). We also tested different scaling factors from ^2.30 up to ^2.35 and our analysis yielded similar results. Second, we subtract the median brightness at an elongation of = 18° from the entire model to remove the contribution of the dust that blends in with the smooth ZC background to replicate the approach of Stenborg et al. (2018b). We set all negative values of the relative brightness increase after subtracting the median value to zero.

Figure 7(A) shows the best fit of our dust ring model to the data and 7(B) shows the model residuals. The quality of the fit was determined by minimizing the rms deviation, where the free parameters were the SFD index α_ring and the maximum relative brightness increase of the ring F_M . The best fit was achieved for α_ring = 4.03 and F_M = 1.93. Considering the fact that the observational data set from Stenborg et al. (2018b) is longitudinally averaged with an angle of 40° and consists of more than six years of data, we find that the model fits the ring well. There are two features that our current model cannot reproduce: (1) an enhancement in brightness located at = 23° and λ = 360° that could be attributed to the longitudinal averaging, and (2) a broad enhancement at = 225 and λ = 120° that is attributed to dust from comet 2P/Encke. We explore the contribution of 2P/Encke in the following section.

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Figure 7 showed that our model is unable to reproduce the significant relative brightness increase located at = 225 and λ = 120°. Stenborg et al. (2018b) suggested that 2P/Encke's orbit is in a favorable configuration that could explain this enhancement. Moreover, models of the impact-driven calcium exosphere at Mercury show that 2P/Encke plays a major role in the calcium production rate when Mercury is close to its pericenter (Christou et al. 2015; Killen & Hahn 2015; Pokorný et al. 2018). To explore the potential 2P/Encke contribution, we created a simple dynamical model. First, we backward integrated 1000 clones of 2P/Encke for 5000 yr, where the orbital elements and the covariance matrix were obtained from the JPL Small-Body Database Lookup API for 2P/Encke [2015] (K235/2) - default 2. ⁶ The 5000 year time span is based on the conclusions of Egal et al. (2021) that the 2P/Encke meteoroid stream (Taurid complex) might be the result of the fragmentation of a larger body 5000–6000 yr ago using orbit convergence methods. The clones were generated using a standard multivariate normal distribution available in NumPy (Harris et al. 2020). Then we simulated the dynamical evolution of particles with diameters from D = 10.0 μm to D = 4642 μm (in nine logarithmically spaced bins), assuming a bulk density of ρ = 2000 kg m^–3, released in 100 yr intervals and recorded their positions in the year 2020. We used the same integration methods as in the previous sections described in Section 4.

To obtain data comparable to the observations of Stenborg et al. (2018b), we adopted the same methods described in Section 6.1; i.e., we normalized the brightness of each particle by its elongation and then subtracted the median value at = 18°. Furthermore, we included an additional filter for the latitudinal extent of the STEREO field of view. Stenborg et al. (2018b) does not provide any information about the latitudinal extent of the ring due to the fact that the nature of their measurement "precludes an analysis of its latitudinal extent". The STEREO H i-1 instrument has an angular field of view of 20° (Figure 4 in Eyles et al. 2009) and thus we apply an 8° cutoff for the maximum absolute value of the ecliptic latitude of particles that contribute to our model. The 8° cutoff corresponds to the height of a triangle with a 10° hypotenuse and a 6° base; the STEREO field of view is centered at = 14° and the ring structure is ≈6° away. The contributions of particles of all sizes are combined using a single power-law SFD (Equation (8)) with the differential size index α_comet. To obtain the best model fit for just the cometary contribution, we fit only the region between 110° < λ < 135° and 214 < < 230.

Figure 8(A) shows our best fit of the 2P/Encke model to the selected data region using a differential size index of α_comet = 3.02. The cometary contribution to the relative brightness increase is asymmetric in λ with respect to the longitude of periapsis of 2P/Encke ϖ = 161°.1 due to the latitudinal filter that we applied to simulate the STEREO H i-1 field of view and the fact that a portion of particles seen at λ > 150° is outside the field of view. When neglecting the latitudinal extent filter, the asymmetry of the brightness increase is negligible. This asymmetric shape of the brightness increase is a potential solution for filling the gap that our model of Mercury's dust ring cannot reproduce (Section 6.1). However, as shown in Figure 8(A), the contribution to λ > 150° is not negligible and is ∼50% of the maximum relative brightness increase in the model. Therefore, filling the gap by combining our dust ring model from the previous section and our 2P/Encke model does not result in a significantly better fit. Either the comet's contribution is too strong and distorts the rest of the ring, or the comet provides only a fraction of the missing signal at 110° < λ < 135° and 214 < < 230. The best model fit combining both the ring and comet models has the following parameters: α_ring = 4.28, α_comet = 2.71, F_M(ring) = 1.74, and F_{M(comet)=0.49}, where the goodness of fit increases only by 7%. We show this best-fit model in Figure 9(B) and provide a direct comparison with the dust ring only model in Figure 9(A).

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Figure 8(B) shows the 2P/Encke relative brightness increase for a wider range of elongations ε and the entire range of longitudes λ. The 2P/Encke stream extends beyond the dust sourced by bodies in the 1:1 MMR with Mercury and should be detectable by remote sensing similar to the discovery of Mercury's dust ring (Stenborg et al. 2018b). Such an observation would provide an independent constraint for determining the contributions of the dust co-orbiting with Mercury and the 2P/Encke meteoroid stream.

Our model for 2P/Encke was simplified for computational reasons as we originally thought that it is a peculiar effect intended for future work. Recently, comprehensive models for the 2P/Encke meteoroid stream and its effect on Earth, observed as the Taurid complex, were published by Egal et al. (2021) and Egal et al. (2022). Egal et al. (2022) shortly discuss impacts of 2P/Encke meteoroids on Mercury, but do not discuss their potential signature in the STEREO data. We will seek the implementation of these new comprehensive models in future work.

The previous sections showed that the only potential source of the currently observed brightness increase is a population of dust particles locked in the 1:1 MMR with Mercury. Pokorný & Kuchner (2019) showed that Venus co-orbital asteroids could be primordial and survive 4.5 Gyr in the 1:1 MMR with Venus. However, the search for such asteroids has been unsuccessful so far (Pokorný et al. 2020; Sheppard et al. 2022) and Pokorný & Kuchner (2021) showed that the Yarkovsky effect force can hinder the primordial origin hypothesis. Mercury, being closer to the Sun and less massive than Venus, is not the most viable candidate for keeping its primordial co-orbital small body population. Nevertheless, we decided to test the primordial origin hypothesis as well as the dynamic stability of particles of various sizes by analysing the co-orbital residence time of test particles of different sizes with diameters from D = 0.1 mm to D = 100 mm using the same simulation scheme described in Section 4; i.e., PR drag and the solar wind are added as additional drag forces. Additionally, we performed a simulation with gravity only to obtain the maximum residence time for any small body in the 1:1 MMR with Mercury. The co-orbital residence time T_res is calculated as the time when the number of particles in the 1:1 MMR decreased below 2% of the initial number of simulated particles. To determine if the particles are in the 1:1 MMR, we evaluate whether the particle semimajor axis a is within 1% of Mercury's instantaneous a_Mer in the simulation (e.g., 0.3832 < a < 0.3910 au for a_Mer = 0.3871 au). Note that the half width of the 1:1 MMR with Mercury is 0.38% (Murray & Dermott 1999) and using this cutoff provides almost identical results to our simulation.

Figure 10 shows that the co-orbital residence time T_res increases with particle diameter D and ranges from thousands of years to tens of millions of years. We identified three different regimes of T_res behavior depending on D: (a) an unstable regime for D < 1 mm, where the drag forces are too strong for particles to get efficiently captured and particles migrate quickly from the 1:1 MMR; (b) a linear log-log increase for 1 ≤ D < 10 mm, showing ${T}_{\mathrm{res}}\propto {D}^{1.2}$ behavior that is expected from the decreasing efficacy of PR drag with increasing D; and (c) a residence time plateau for D ≥ 10 mm, where T_res stops following an exponential growth with increasing D and plateaus at around 15 Myr. Our test without additional drag forces showed ${T}_{\mathrm{res}\,({\rm{D}}=\infty )}=16.3\,\mathrm{Myr}$, which is comparable to our D = 100 mm particle simulation (${T}_{{\rm{r}}{\rm{e}}{\rm{s}}({\rm{D}}=100\,{\rm{m}}{\rm{m}})}=15.9$ Myr).

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From our analysis we conclude that no primordial small body is able to survive more than 20 Myr in the 1:1 MMR with Mercury and thus the source of the current ring must be provided by a rather recent event due to the maximum co-orbital residence time ${T}_{\mathrm{res}}\lt 20\,\mathrm{Myr}$. Tabachnik & Evans (2000) analysed the stability of Mercury's co-orbitals and for some orbital configurations the co-orbital stability in horseshoe orbits extended to 100 Myr. This would open a longer window for a recent impact but still prevents a primordial origin of the dust ring. We calculated that the currently observed dust would be equivalent to a several-kilometer-diameter sphere of material, which leads to two potential sources: (1) a captured asteroid, and (2) a recent impact on Mercury. Greenstreet et al. (2020) showed that the capture of Centaur asteroids into the 1:1 MMR with Jupiter is possible but inefficient, where approximately 1 in 100,000 asteroids gets temporarily captured. They also find that the maximum residence time is <100,000 yr. Since Mercury is ∼5700 times less massive than Jupiter, we assume that the resonant capture efficiency is negligible and not viable as a potential progenitor of Mercury's dust ring. Therefore, to support the recent impact hypothesis, we analyse whether a recent impact on Mercury is able to transport the material into a 1:1 MMR with Mercury and maintain the ring structure until today.

Mercury's surface shows definite markings of small body impacts in the geologically recent past. Kinczyk et al. (2020) identified 22 "Class 5" craters with diameters larger than D > 40 km, with two of these craters being larger than 100 km in diameter. Class 5 craters are the freshest craters classified on Mercury's surface, and are interpreted to have formed during the Kuiperian cratering era, which spans from <300 Ma through today (Banks et al. 2017). This could potentially mean that some of the freshest Class 5 craters have ages <20 Ma, i.e., our estimated lifespan of Mercury's circumsolar ring (Section 6.3). To evaluate whether the formation of any of these young craters could explain the observed dust ring, we first estimate the mass of material that can be transferred from the surface into the co-orbital resonance with Mercury during an impact event.

The amount of ejecta released during cratering events and the ejecta velocity distribution are not well constrained (Holsapple 1993; Holsapple & Housen 2007). Due to the size of the craters (D > 1 km), the impact-related calculations are strictly in the gravity regime (Holsapple 1993). Let us assume that an impactor with a radius of r_imp, bulk density of δ, and a normal impact velocity component of U impacts Mercury's surface. Using the relations from Holsapple & Housen (2007) we can estimate the mass of ejecta M_e produced by a point source impactor made of sand or cohesive soil with a velocity of v where the ejected mass has an ejecta velocity of v_ej larger than v as:

$$\begin{eqnarray}&&{M}_{e}({v}_{\mathrm{ej}}\gt v)={m}_{\mathrm{imp}}0.018{\left(\displaystyle \frac{v}{U}\right)}^{-1.23}{\left(\displaystyle \frac{\delta }{\rho }\right)}^{0.2},\end{eqnarray} \tag{ 10 }$$

where m_imp is the impactor mass and ρ is the impacted surface density. To estimate the impactor mass, we can employ the estimated crater radius R_c caused by our model impactor from Holsapple & Housen (2007):

$$\begin{eqnarray}&&{R}_{c}={r}_{\mathrm{imp}}1.03{\left(\displaystyle \frac{{{gr}}_{\mathrm{imp}}}{{U}^{2}}\right)}^{-0.17}{\left(\displaystyle \frac{\delta }{\rho }\right)}^{0.332},\end{eqnarray} \tag{ 11 }$$

where g = 3.7 m s⁻² is the gravitational acceleration on the surface of Mercury. Let us assume that the impactor and Mercury's surface have similar densities and δ/ρ ≈ 1. Then for an impactor with U = 30,000 m s⁻¹ (e.g., Marchi et al. 2009), δ = 3000 kg m⁻³, and the resulting crater diameter is 40 km we obtain the impactor radius to be r_imp = 2811 m and m_imp = 2.79 × 10¹⁴ km. The amount of ejecta with ejection velocities larger than Mercury's escape velocity of v_ej > 4220 m s⁻¹ is M_e (v_ej > 4220 m s⁻¹) = 5.60 × 10¹³ kg, which translates into a sphere of radius of 1646 m. Equations (10) and (11) show that for a fixed crater radius of R_c , the mass ejected is not proportional to the tangential impact velocity U because m_imp ∝ U^1.23 and the impact velocity term cancels out in Equation (10).

Continuing the impact example in the previous paragraph, after a recent impact of a ∼3 km object, approximately 1.25 × 10¹⁴ kg of material reaches the boundary of Mercury's Hill sphere. Ejecta released with velocities of v_ej < 4220 km s⁻¹ do not reach the Hill sphere boundary (r = 0.0012 au or r = 0.1753 × 10⁶ km) and likely fall back to the planet. We look for ejecta that reach Mercury's Hill sphere with a heliocentric velocity that puts the ejected particle into 1:1 MMR with Mercury, therefore we require that a_ej = 0.3871 au. If we permit a 1% variation in a_ej (0.3867 < a_ej < 0.3875 au), we can calculate the ejecta velocity range satisfying such a condition. We generated 360 × 180 particle positions uniformly on the surface of Mercury's hill sphere with 360 longitudinal and 180 latitudinal positions. Approximately 50% of ejection directions were feasible for both the perihelion and aphelion ejection.

Assuming the particles are leaving Mercury radially, we can calculate the velocity range that satisfies the 1% range for a_ej. We calculated the median δ v_ej = 4.7 m s⁻¹ for an impact occurring at Mercury's perihelion, and δ v_ej = 5.9 m s⁻¹ at Mercury's aphelion. Using Equation (10), we get approximately 0.2% for ejected material in such a velocity range; i.e., a factor of 500 material loss.

With a population of ejecta with orbits in the 1:1 MMR, we simulate their orbital evolution for 1000 yr using 0.1 day time steps while keeping the same settings shown in Section 4. The number of particles still trapped in the 1:1 MMR after 100 yr was approximately 0.1%; i.e., 1 in 1000 was stable.

This brings us to an estimate that 1 in 1,000,000 particles ejected from Mercury's surface are transferred into quasi-stable 1:1 MMR with Mercury. In this exercise, we neglected the effects of radiation pressure, magnetic forces, and solar wind, which add more complexity to this largely unconstrained scenario for having assumed only the gravity of the Sun and 8 solar system planets. Moreover, due to lack of constraints we were forced to generalize many assumptions, such as the location and direction of the impact.

Using Equation (10) we estimated that an event resulting in a 40 km crater in diameter ejects M_e = 5.60 × 10¹³ kg of material from which M_MMR = 5.60 × 10⁷ kg remains captured in the 1:1 MMR with Mercury. This value is of order four orders of magnitude smaller than the one we estimate from STEREO observations in Section 2.1. Looking for larger craters, we find that there are some fresh 100 km craters on Mercury's surface, such as Bartok and Amaral. Doubling the size of the source crater increases the ejected mass by a factor of ∼10.

Despite 3−4 orders of magnitude discrepancy between our estimated mass captured in the 1:1 MMR with Mercury from a 40 km crater impact event and the estimated mass of Mercury's dust ring, we showed that there is a theoretical possibility to transfer a substantial amount of mass from Mercury's surface into the co-orbital resonance. Further developments in estimating impact yields for large craters on Mercury's surface, more complex simulations of ejecta transfer into co-orbital resonance, and additional observations of Mercury's dust ring might possibly narrow the gap between the model predictions and the current observations. In the next section we provide the last missing puzzle piece and estimate the ages for the largest fresh craters currently known on Mercury's surface.

The ages derived from the Poisson statistics suggest that all of the analysed craters most likely formed between ∼170 Ma and ∼260 Ma, with the Amaral crater forming most recently ${{ \mathcal T }}_{\mathrm{Pois}}=171\pm 0.02$ Ma. This is consistent with an analysis of the optical maturity properties that also suggests Amaral is among the youngest large craters on Mercury (Neish et al. 2013). Recall that the crater ages derived here are considered to be the maximum age estimates because the majority of superposing craters identified in our analysis are smaller than 7 km in diameter and therefore may be secondary impacts.

When considering the CSFD ages, Amaral appears to be the second youngest crater in our sample (${{ \mathcal T }}_{\mathrm{CSFD}}=54\pm 15$ Ma) and is surpassed only by Bash with ${{ \mathcal T }}_{\mathrm{CSFD}}=34\pm 14$. While these younger ages of craters look promising for our recent impact scenario described in Section 6, which requires an impact within the last 15 Myr, the ages derived from the CSFDs and Poission statistics do not overlap, even considering the larger uncertainties associated with the CSFD fitting. Due to the apparent advantages of Poisson statistics over CSFD fitting described in the previous section, we feel that more accurate constraints on crater ages are required to reject confidently the recent impact scenario for the origin of Mercury's dust ring. With the BepiColombo spacecraft on its way to Mercury, we can expect new exciting data sets in several years that may aid in further constraining the crater ages (Rothery et al. 2020).

Additional observations of Mercury's dust ring would provide important new data required for more comprehensive modeling and model fitting. Currently, there are several space missions capable of detecting Mercury's dust ring: (a) the Parker Solar Probe has completed its first 10 close encounters with the Sun (Fox et al. 2016), where each encounter passes through Mercury's orbit potentially giving an opportunity to observe Mercury's dust ring both from the outside and inside similarly to the all-sky observation of Venus's dust ring (Stenborg et al. 2021a); (b) the Solar Orbiter mission (Müller et al. 2020)—thanks to its unique, inclined orbit—might be able to observe Mercury's dust ring from a location away from the ecliptic and produce crucial data for characterization of Mercury's dust ring; and finally (c) BepiColombo will have a unique opportunity to detect Mercury's dust ring via remote sensing as well as in situ with the onboard dust detector Mercury Dust Detector (MDM) (Kobayashi et al. 2020) when it approaches and orbits the planet.

As we showed in Section 6.3, particles trapped in the 1:1 MMR with Mercury have a maximum residence time of ${T}_{\mathrm{res}}\sim 15\,\mathrm{Myr}$. All these trapped particles are continuously sweeping through the inner sections of the ZC and thus have the potential to be collisionally fragmented or destroyed before they become dynamically unstable. Using traditional methods to estimate the collisional lifetimes of particles sharing Mercury's orbit (Grun et al. 1985; Steel & Elford 1986), we estimate the collisional lifetime of particles to be ∼100× shorter than the dynamical lifetimes shown in Figure 10. Should these collisional lifetimes be valid, the age of Mercury's dust ring would be <1 Myr and thus very improbable based on our impact origin hypothesis. Fortunately, many recent dynamical model works have shown that the collisional lifetimes in the inner solar system for asteroidal and cometary meteoroids should be 20–100× longer (e.g., Nesvorný et al. 2011a; Pokorný et al. 2014; Soja et al. 2019) and based on our estimates the collisional lifetimes of meteoroids in the co-orbital resonance with Mercury are comparable to their residence times (Section 6.3). For these reasons, we think that collisions of particles locked in 1:1 MMR with Mercury with the inner ZC do not prevent the existence of Mercury's dust ring based on our impact origin hypothesis. ZC particles will erode and disrupt a certain portion of Mercury's dust ring and diminish its magnitude in time over timescales similar to the dynamical transport of particles from the co-orbital resonance with Mercury.

Starting with the crater database from Kinczyk et al. (2020) we provided age estimates for six Class 5 (fresh-looking) craters with D > 40 km. Many other fresh-looking craters smaller than 40 km are also present on the surface of Mercury. As described by the crater production function (e.g., Le Feuvre & Wieczorek 2011), the number of small craters that formed in Mercury's recent past is larger than the number of large craters, and therefore some of these smaller craters would have formation ages that are younger than those derived for the larger craters summarized in Table 2. While a single one of these craters cannot provide enough ejecta to form the currently observed dust ring in one event, it is possible that multiple smaller impacts close enough in time (∼10 Myr) may have contributed material that collectively amounts to the currently observed dust ring.