The Solar Wind Prevents Reaccretion of Debris after Mercury's Giant Impact

The picture of the solar wind we employ considers the wind plasma to expand radially outwards along the stellar equator. Its outward motion is governed by both pressure and magnetic stresses, but its azimuthal acceleration is dominated by magnetic torques. A pathway toward a solution of this "Weber–Davis" model has been presented in numerous works (Weber & Davis 1967; Hartmann & MacGregor 1982; Lamers et al. 1999) and so we only provide a brief outline of the solution here. Closely following Hartmann & MacGregor (1982), we assume that both the magnetic field B and wind velocity are axisymmetric, and possess only radial (v_r, B_r) and azimuthal (v_ϕ, B_ϕ) components. The condition ${\rm{\nabla }}\cdot {\boldsymbol{B}}=0$ in spherical coordinates demands that

$$\begin{eqnarray}&&{B}_{r}={B}_{r,0}{\left(\displaystyle \frac{{R}_{\star }}{r}\right)}^{2},\end{eqnarray} \tag{ 6 }$$

where B_r,0 is the magnetic field strength at the stellar photosphere, taken to be when r = R_⋆, the stellar radius. Note that all calculations are carried out in the equatorial plane.

If Faraday's law is combined with the assumptions of ideal MHD (i.e., infinite conductivity), we obtain a steady-state relationship of

$$\begin{eqnarray}&&{B}_{\phi }={B}_{r}\left[\displaystyle \frac{{v}_{\phi }-{{\rm{\Omega }}}_{\star }r}{{v}_{r}}\right],\end{eqnarray} \tag{ 7 }$$

where the stellar angular velocity ${{\rm{\Omega }}}_{\star }$ has been introduced. For simplicity, we assume that the wind is isothermal, with isothermal sound speed ${c}_{s}={({kT}/{m}_{H}\mu )}^{1/2}$ given in terms of the photosphere temperature T = 10⁶ K, the mean molecular number μ = 0.6, the mass of a proton m_H, and the Boltzmann constant k (Lamers et al. 1999).

Each parcel of plasma carries with it a constant specific angular momentum L and energy E, which are derived from the steady-state, azimuthal and radial components of the momentum equation respectively:

$$\begin{eqnarray}\begin{array}{rcl}r\,:0 & = & \displaystyle \frac{1}{2}\displaystyle \frac{d}{{dr}}\left[{v}_{r}^{2}+{v}_{\phi }^{2}\right]+{c}_{s}^{2}\displaystyle \frac{d}{{dr}}\left[\mathrm{ln}({\rho }_{\mathrm{sw}})\right]+\displaystyle \frac{{{GM}}_{\star }}{{r}^{2}}\\ & & -\displaystyle \frac{{v}_{\phi }}{r}\displaystyle \frac{d}{{dr}}\left[{{rv}}_{\phi }\right]+\displaystyle \frac{{B}_{\phi }}{{\mu }_{0}\rho r}\displaystyle \frac{d}{{dr}}\left[{{rB}}_{r}\right]\end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\phi \,:0=\displaystyle \frac{{B}_{r}}{{\mu }_{0}}\displaystyle \frac{d}{{dr}}\left[{{rB}}_{\phi }\right]-\rho {v}_{r}\displaystyle \frac{d}{{dr}}\left[{{rv}}_{\phi }\right],\end{eqnarray} \tag{ 9 }$$

where μ₀ = 4π × 10⁻⁷ Hm⁻¹ is the vacuum permeability.

The conserved angular momentum is extracted via integration as

$$\begin{eqnarray}&&L={{rv}}_{\phi }-\left[\displaystyle \frac{{{rB}}_{r}{B}_{\phi }}{{\mu }_{0}\rho {v}_{r}}\right]={r}_{{\rm{A}}}^{2}{{\rm{\Omega }}}_{\star },\end{eqnarray} \tag{ 10 }$$

where the Alfvén radius r_A is defined as the point where the radial wind velocity is equal to the Alfvén velocity, defined as ${v}_{{\rm{A}},r}\equiv {B}_{r}/\sqrt{{\mu }_{0}{\rho }_{\mathrm{sw}}}$ (Lamers et al. 1999; Bellan 2008). Likewise, integration yields the conserved energy

$$\begin{eqnarray}&&E\equiv \displaystyle \frac{1}{2}\left[{v}_{r}^{2}+{v}_{\phi }^{2}\right]+{c}_{s}^{2}\mathrm{ln}({\rho }_{\mathrm{sw}})-\displaystyle \frac{{{GM}}_{\star }}{r}-\displaystyle \frac{r{{\rm{\Omega }}}_{\star }{B}_{r}{B}_{\phi }}{{\mu }_{0}{\rho }_{\mathrm{sw}}{v}_{r}}.\end{eqnarray} \tag{ 11 }$$

Finally, rearranging the momentum equation yields to the relationship

$$\begin{eqnarray}&&\displaystyle \frac{r}{{v}_{r}}\displaystyle \frac{{{dv}}_{r}}{{dr}}=\displaystyle \frac{\left({v}_{r}^{2}-{v}_{{\rm{A}},r}^{2}\right)\left(2{c}_{s}^{2}+{v}_{\phi }^{2}-{{GM}}_{\star }/r\right)+2{v}_{r}{v}_{\phi }{v}_{{\rm{A}},r}{v}_{{\rm{A}},\phi }}{\left({v}_{r}^{2}-{v}_{{\rm{A}},r}^{2}\right)\left({v}_{r}^{2}-{c}_{s}^{2}\right)-{v}_{r}^{2}{v}_{{\rm{A}},\phi }^{2}},\end{eqnarray} \tag{ 12 }$$

where we define ${v}_{{\rm{A}},\phi }\equiv {B}_{\phi }/\sqrt{{\mu }_{0}{\rho }_{\mathrm{sw}}}$.

By inspection of Equation (12), any solution passing through a point where the denominator vanishes must also cause the numerator to vanish. This situation occurs at 2 radii, which are r_f (the fast point) and r_s (the slow point), and physically correspond to the transition of the radial wind velocity through the fast and slow magnetosonic points (Weber & Davis 1967; Hartmann & MacGregor 1982; Lamers et al. 1999; Bellan 2008).

The requirement of vanishing numerator and denominator at both points constitutes a system of four simultaneous equations. Two further equations arise from the constancy of the energy E at the stellar surface r = R_⋆ and at the Alfvén radius r = r_A. Altogether, these six equations must be solved for unknowns, which include the three radii $\{{r}_{s},{r}_{f},{r}_{{\rm{A}}}\}$, the values of radial wind velocity at r_f and r_s, and finally the velocity of the wind at the stellar photosphere. We obtain these six parameters by way of a Newton–Rhapson solver, from which the solar wind properties, and thus the ejecta drift timescales, may be obtained as a function of radius.

Illustrative radial profiles of v_r and v_ϕ are presented in Figure 3 at three epochs—3, 10, and 30 Myr. The radial velocity increases monotonically with distance, reaching a constant as $r\to \infty$. In contrast, the azimuthal velocity displays an extremum in velocity slightly interior to the Alfvén radius (vertical dotted lines). Physically, this peak arises because close to the star the magnetic field is strong enough to accelerate an outwardly expanding ring of plasma over a timescale that is short compared to the radial expansion timescale of the plasma. Approximately, this strong coupling locks plasma to solid-body rotation close to the star's surface (denoted by the slanting, dashed red line). At larger distances (r > r_A), magnetic coupling weakens, such that the expanding ring of plasma no longer receives significant acceleration from the weakening magnetic stresses. In order to approximately conserve its angular momentum v_ϕ approximately falls as 1/r (Carolan et al. 2019).

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At times earlier than about 10 Myr, a substantial range of radii exist where v_ϕ exceeds the Keplerian velocity (the black line in the bottom panel). We discuss the significance of this in greater detail below, but qualitatively it suggests that debris occupying Mercury's orbit at earlier times are more likely to experience orbital expansion than at later epochs, when orbital contraction is favored.

Up until recently, in situ measurements of the solar wind's azimuthal velocity were unavailable at distances comparable to Mercury's orbit, and so the theoretically expected peak in v_ϕ could not tested. However, the Parker Solar Probe recently detected azimuthal winds, increasing toward the Sun at distances inward of 0.2 au (Kasper et al. 2019). Surprisingly, these winds possessed azimuthal components of up to 40 km s⁻¹, at least an order of magnitude above those predicted for the Sun at its current age. Indeed, these speeds are more consistent with the winds we compute at an age of 30 Myr (bottom panel of Figure 3). It is impossible to tell as of yet how these theoretical underestimations propagate to younger, more active stars, and thus whether or not our calculations serve as lower limits of the early wind strength. Undoubtedly, the upcoming closer passes of the Solar Probe to the Sun will help shed light on this unfolding mystery.

The epoch of giant impacts persisted throughout approximately the first 100 million years of solar system history (Raymond et al. 2018), though the exact time of Mercury's hypothesized impact is uncertain (see Section 6.1). As mentioned above, young stars typically possess spin periods between 1 and 10 days and magnetic fields between 0.1 and 1 kGauss (Bouvier et al. 2014; Vidotto et al. 2014). A less well-constrained aspect lies in choosing the mass-loss rate $\dot{M}$, alongside its time-dependence. Given the lack of direct measurements, previous work has typically resorted to 3D magnetohydrodynamic models from which the mass loss emerges as an output of the model (Cohen & Drake 2014; Ó Fionnagáin & Vidotto 2018).

Unfortunately, these computational approaches do not approach a consensus, with estimates varying widely, though generally the periods P_⋆, $\dot{M}$, and B_⋆ all decrease with time. We adopt the time-scalings presented in Ó Fionnagáin & Vidotto (2018),

$$\begin{eqnarray}\begin{array}{rcl}\dot{M}(t) & = & {\dot{M}}_{0}{\left(\displaystyle \frac{t}{\mathrm{Myr}}\right)}^{-3/4}\\ {B}_{\star }(t) & = & {B}_{0}{\left(\displaystyle \frac{t}{\mathrm{Myr}}\right)}^{-1/2}\\ {P}_{\star }(t) & = & {P}_{0}{\left(\displaystyle \frac{t}{\mathrm{Myr}}\right)}^{1/2},\end{array}\end{eqnarray} \tag{ 13 }$$

with magnitudes that are chosen to match observed T Tauri magnetic fields and spin periods, as well as allowing the wind to pass through $100{\dot{M}}_{\odot }$ during the period of terrestrial planet formation:

$$\begin{eqnarray}&&{\dot{M}}_{0}={10}^{3}{\dot{M}}_{\odot }\,\,{B}_{0}=0.1\,\mathrm{kG}\,\,\,{P}_{0}=1\,\mathrm{day}.\end{eqnarray} \tag{ 14 }$$

Though the uncertainty in the above parameters is large, it is important to point out that the time at which Mercury's purported giant impact occurred is currently unconstrained. Thus, even if an exact knowledge existed of the young Sun's stellar parameters, the conditions persisting during Mercury's giant impact would remain with large uncertainties. Accordingly, our goal is to explore whether a broad range of stellar parameters and impact times are consistent with removal of ejected material.

In addition to the above parameters, the stellar radius itself would be changing throughout the terrestrial planet formation period. Specifically, the Sun contracts along the fully convective Hayashi track, entering the Henyey track after several tens of Myr (Shu et al. 1987; Maeder 2008). Using stellar models from the online database of Siess et al. (2000), we computed the time evolution of a Sun-like star's radius. We find that this contraction is well-approximated by the equation

$$\begin{eqnarray}&&{R}_{\star }(t)={R}_{\odot }\left[0.9+\exp \left(-\displaystyle \frac{t}{10\,\mathrm{Myr}}\right)\right],\end{eqnarray} \tag{ 15 }$$

where R_⊙ is the present-day Sun's radius. The decay of the stellar radius provides a significant reduction in the effect of the wind, because of the ${R}_{\star }^{2}/{r}^{2}$ dependence of B_r.

With the above prescriptions, the stellar wind properties as a function of r and t are fully specified. We now turn to the influence this wind has upon material ejected from Mercury's surface.

All simulations were carried out using the REBOUND symplectic integration package, where the hybrid integration scheme "Mercurius" was employed (Rein & Liu 2012; Rein & Tamayo 2015). This integration scheme conducts a WHFast algorithm (Wisdom & Holman 1991), where our chosen timestep was 0.2 days, transitioning to an IAS15 algorithm to deal with close encounters (Rein & Spiegel 2014). Multiple timesteps were checked, ranging from 2 to 0.2 days, but there was no systematic dependence observed between collision rate and step size. In order to model solar wind-induced drag, we prescribed exponential evolution of both the eccentricities and semimajor axes of ejecta particles by way of the additional forces options within the REBOUNDx package (Tamayo et al. 2020).

We began with a control simulation that did not include the solar wind, similarly to Gladman & Coffey (2009). These integrations were performed for 10 million years each, and 2 particle ejection velocities (the velocity of the ejecta relative to Mercury) were chosen, v_ej = {4, 10} km s⁻¹. In parallel, we simulated these two launch velocities under the action of the solar wind. In these simulations, eccentricities and semimajor axes ejecta orbits were forced to exponentially decay over respective timescales of τ_e and τ_a, computed according to Equation (19).

Given that eccentricity-damping timescales computed above are relatively fast, and for the sake of computational speed, we employed Equation (19) to lowest order in eccentricity, such that

$$\begin{eqnarray}&&2{\tau }_{e}={\tau }_{a}=\displaystyle \frac{8\pi {\rho }_{s}\,{{sa}}^{2}}{3{C}_{D}\dot{M}},\end{eqnarray} \tag{ 21 }$$

and assumed that s = 1 cm, the typical size of debris computed by Benz et al. (2007). Accordingly, the drift times are

$$\begin{eqnarray}&&2{\tau }_{e}={\tau }_{a}\approx 0.23{\left(\displaystyle \frac{a}{0.4\mathrm{au}}\right)}^{2}\left(\displaystyle \frac{100{\dot{M}}_{\odot }}{\dot{M}}\right)\mathrm{Myr},\end{eqnarray} \tag{ 22 }$$

and were updated with the particles' new semimajor axes after each 1000 yr interval throughout the simulation. We simulated two cases for each of v_ej = 4 km s⁻¹ and v_ej = 10 km s⁻¹, with $\dot{M}=\{10,100\}{M}_{\odot }$. Cumulatively, we ran three cases for each of the two launch velocities, with these three cases denoted as "weak wind," "strong wind," and "wind-free" in Figure 6.

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Each simulation began with 110 particles, ejected isotropically from Mercury. During the 10 Myr simulation these particles were removed if either (1) they collided with one of the six massive bodies (including the Sun), or (2) their apocenter fell below 0.2 au (which we considered as safely removed from Mercury's vicinity). Our earlier, analytic treatment assumed coplanarity between the ejecta orbits and Mercury's orbit. This assumption is easily lifted within N-body simulations. If the planets and ejecta were assumed coplanar, reaccretion timescales would be deceptively short, owing to the enhanced probability of collision between coplanar orbits as compared to mutually inclined orbits (Farinella & Donald 1992).

We initialize the planetary orbits with similar eccentricities, inclinations, and semimajor axes to their modern values, but with uniformly randomized mean anomalies, arguments of pericenter, and longitudes of ascending node. The exception is Mercury itself, which we began with mean anomaly, argument of pericenter, and longitude of ascending node all set to zero, i.e., the planet began at y = 0 on our coordinate grid (Figure 1). This set-up was simply to aid in prescribing the initial particle orbits relative to Mercury, which we now describe.

With respect to the ejected particles, we adopted a uniform distribution of ejecta angles over 4π steradians. Specifically, we chose N = 11 angles, uniformly spaced in ranges 0 < ϕ_i ≤ 2π and N − 1 angles from $-1\lt \cos ({\theta }_{j})\lt 1$, where θ_i and ϕ_j take their usual definitions within a spherical coordinate system centered upon Mercury⁴ and are defined as

$$\begin{eqnarray}\begin{array}{cc}{\phi }_{i}=\displaystyle \frac{i}{N}2\pi & i\in \{1...N\}\\ {\theta }_{j}=\arccos \left[-1+2\displaystyle \frac{j}{N}\right] & j\in \{1...N-1\},\end{array}\end{eqnarray} \tag{ 23 }$$

yielding N(N–1) = 110 ejected particles.

Given the above choices of angles and ejecta velocities, we set up the particles on heliocentric orbits. Having chosen the angles θ_i and ϕ_j, we initialize each particle with a position

$$\begin{eqnarray}&&{{\boldsymbol{x}}}_{{ij}}={x}_{0}\{\sin ({\theta }_{i})\cos ({\phi }_{j}),\sin ({\theta }_{i})\sin ({\phi }_{j}),\cos ({\theta }_{i})\}\end{eqnarray} \tag{ 24 }$$

relative to Mercury, such that the initial heliocentric position is ${{\boldsymbol{x}}}_{{ij}}+{{\boldsymbol{x}}}_{M}$ where x_M is Mercury's initial position. Likewise, the initial velocity of the particle relative to Mercury is given by

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{{ij}}={v}_{0}\{\sin ({\theta }_{i})\cos ({\phi }_{j}),\sin ({\theta }_{i})\sin ({\phi }_{j}),\cos ({\theta }_{i})\}.\end{eqnarray} \tag{ 25 }$$

To ensure that the particle is well clear of Mercury at the beginning of the integration, we set x₀ = 0.006 au, which is approximately 4 Hill radii from Mercury. The value of v₀ took either of $\{4,10\}\,$ km s⁻¹.

The cumulative number of collisions onto Mercury within our numerical simulations is presented in Figure 6. The wind-free examples agree reasonably well with Gladman & Coffey (2009), in that typical reaccretion timescales are 10 million years. Note that we underestimate the number of collisions arising from 4 km s⁻¹ ejections (∼20 % versus ∼40% after 10 Myr). The reason for this discrepancy likely arises owing to the stochasticity inherent to the problem, together with the smaller number of ejected particles (110) used in the present study, as compared to the 7000 used in Gladman & Coffey (2009). For the purposes of the present study, a comparison between the cases with and without wind-induced drag is the primary focus.

The reaccretion rates occurring under the influence of wind-induced drag were also presented in Figure 6. Both $\dot{M}=100{\dot{M}}_{\odot }$ (strong wind) and even $\dot{M}=10{\dot{M}}_{\odot }$ (weak wind) were sufficient to prevent most reaccretion, with the former allowing only one or two collisions upon Mercury, as opposed to ∼20−30 with no drag. These numerical results confirm the expectation from our analytics that drag timescales corresponding to the early solar wind prevent most of the reaccretion of material ejected from Mercury's surface.

In conclusion, within the scope of our analysis, the giant impact origin for Mercury's large core is consistent with the dynamical trajectory taken by material ejected from the primordial Mercury within the early solar system, permeated by the young Sun's enhanced wind.

Age constraints upon the final formation times of the inner terrestrial planets are largely based upon isotopic data of both Earth rocks and Martian meteorites (Raymond et al. 2018). In particular, the radioactive decay system of W-Hf suggests that Earth's formation time spanned 10 s of millions of years, with its final giant impact perhaps occurring upwards of 100 Myr after the beginning of planet formation (Kleine et al. 2009; Fischer & Nimmo 2018). Mars, in contrast, likely formed substantially earlier, within a few million years and before the dissipation of the protoplanetary nebula gas (Dauphas & Pourmand 2011).

Unfortunately, similar direct geochemical constraints are absent for Mercury (and Venus) owing to a lack of Mercurian meteoritic samples. The likely time of Mercury's mantle-stripping impact is therefore poorly constrained. Indirect inferences may be made using the Earth–Moon system, which likely formed from a similar giant impact (Canup & Asphaug 2001), hinting that Mercury's final giant impact may have occurred within a similar timeframe; within ∼100 Myr. In principle, a lower limit on the age of Mercury's surface might be deduced from crater size-frequency analysis (Strom et al. 2011). However, an absolute chronology for Mercury's cratering record is largely absent, such that it is difficult to rule out impacts that occurred within the first few million years after disk dissipation.

Uncertainty in the timing of the impact is not critical to our hypothesis in a qualitative sense; ejecta launched within the first ∼100 Myr is uniformly removed from Mercury's orbit. However, the decaying strength of the wind suggests that if the impact occurred earlier than ∼10 Myr, debris would be pushed outward to wider orbits than Mercury, potentially becoming part of Venus or Earth (see Figure 5). This contrasts with debris launched later, which is more likely to simply decay onto the Sun's surface. If the final impact occurred earlier than 10 Myr, it would suggest that Mercury's formation time is more in-line with Mars (Dauphas & Pourmand 2011). Mars is consistent with a planetary embryo, which had its growth halted, possibly owing to the inward migration of Jupiter (Walsh et al. 2011; Batygin & Laughlin 2015). Interestingly, Mercury's suspected initial mass, at 2.25 times its current value (Benz et al. 2007) is only about 10% larger than Mars' current mass. This raises the possibility that Mercury, too, might have been a stranded embryo, later altered during the phase of giant impacts (Raymond et al. 2006, 2018).

Ultimately, there is no specific reason to favor late or early impact scenarios for Mercury, owing to incomplete age-constraints upon its surface. However, large uncertainties likewise exist in the early wind's properties, and so even with accurate ages for Mercury's impact it may still be difficult to say with great confidence whether the debris ended up in the Sun or in the other terrestrial planets. Nevertheless, our modeling here indicates that at all times earlier than 100 Myr, and subsequent to disk dissipation, the Sun's wind was likely strong enough to remove ejected material from Mercury's orbital vicinity and prevent substantial reaccretion.

Throughout this work, we implicitly assumed that the majority of the mass of ejected particles existed as centimeter-sized spheres. This assumption was chosen in light of thermodynamic arguments presented in Benz et al. (2007). All timescales computed in our work depend linearly upon particle sizes, and thus it is important to briefly return to the robustness of our assumption of 1 cm particles. In particular, Benz et al. (2007) followed the thermodynamic evolution of material that is initially shocked to a supercritical fluid state, before subsequently cooling along an isentrope. As it cools, the ejecta eventually becomes subcritical, with the most energetic material initially condensing to a vapor, while less energetic material first reaches a liquid state (Johnson & Melosh 2012). A full treatment of the cooling process must apply different approaches for each of these thermodynamic outcomes. The work of Benz et al. (2007) used classical nucleation theory (Raizer 1960; Zel'Dovich & Raizer 2002) to model the highest-energy material, but the lower energy material required a different approach (Grady 1982; Melosh & Vickery 1991).

A more recent treatment of the condensation of ejected material was presented in Johnson & Melosh (2012), but within a different impact regime—that of the 10 km impactor that marked the end of the Cretaceous Period on Earth (Alvarez et al. 1980). Their model consisted of a hemispherical, expanding plume of vapor, out of which solid spherules condensed. The size distribution of condensed material was strongly peaked, with a mean value that scales approximately linearly with the impactor radius. Additionally, the average size of debris exhibited strong variation with respect to the impactor's velocity. Specifically, for an impactor of 1000 km in diameter, the size of debris ranges from roughly 0.01 cm at impact velocities of 15 km s⁻¹, rising to ∼10 cm for 25 km s⁻¹, but falling gradually again to 0.01 cm at speeds of 50 km s⁻¹.

In the case of Mercury, the impactor's diameter (assuming the single impact case Benz et al. 2007) likely exceeded 1000 km, perhaps approaching ∼6000 km (Chau et al. 2018). Furthermore, the most likely velocity of the impact has been suggested to lie within the range of 20–30 km s⁻¹ (Benz et al. 2007; Chau et al. 2018). Given that a 10 km impactor produces debris with characteristic size s = 350 μm, the larger impactor for Mercury will create correspondingly larger debris. For the expected range of impactor sizes, 1000–6000 km, and for an assumed impact speed of 30 km s⁻¹, we can scale up the results from Johnson & Melosh (2012) to estimate that the mean radius of debris would lie in the range s ≈ 3−20 cm. These values are roughly consistent with our assumed size (1 cm), but larger debris sizes are possible, and will depend upon the impact velocity and impactor size.

Despite the uncertainty in particle sizes, we may reframe the problem in terms of the largest particles s_max that survive after a time τ subsequent to the giant impact. Using Equation (4), we find

$$\begin{eqnarray}&&{s}_{\max }=\left(\displaystyle \frac{\dot{M}}{25{\dot{M}}_{\odot }}\right)\left(\displaystyle \frac{\tau }{\mathrm{Myr}}\right)\mathrm{cm},\end{eqnarray} \tag{ 26 }$$

which suggests that even if the typical size of particles was closer to 10 cm, as our extrapolations from Johnson & Melosh (2012) suggest, wind strengths of $100{\dot{M}}_{\odot }$ remain sufficient for removal. It is unclear what fraction of the total mass of ejected material would reside in the largest particles, and thus it is worth noting that all ejecta smaller than s_max would also be removed.

Observational constraints upon the remnants of giant impacts may become available in the form of debris disks around other stars (Genda et al. 2015), some of which are already interpreted as indicative of past giant impacts, for example, BD +20 307 (Weinberger et al. 2010) and HD 172555 (Lisse et al. 2009; Johnson et al. 2012). This latter system possesses a population of fine dust (s ≲ 100 μm) in addition to a population of larger dust inferred to match Spitzer data. There are few direct probes available of centimeter-sized populations around analogs of the young solar system, and thus it remains difficult to validate the assumption of centimeter sizes using existing data.

In addition to particle size, it is essential to consider whether an appropriate mass of silicates is removed from the system to account for Mercury's composition. While a full treatment of this problem is beyond the scope of the present investigation, we note that at the high energies required to match present-day Mercury the majority of escaping material is likely to become vaporized (Benz et al. 2007; Chau et al. 2018). The debris considered in this study condense from this expanding vapor cloud (Johnson & Melosh 2012), but eventually the condensation sequence is quenched. Accordingly, an uncertain fraction of the vapor column is converted into solid debris—the rest remains in vapor form and is likely removed by photodissociation on a short timescale (potentially ≲1 yr Johnson et al. 2012). Accordingly, the influence of solar wind-induced loss of solids upon Mercury's composition depends critically upon the ratio of vapor relative to condensates.

At least within the regime of a 10 km impactor, collision velocities of 30 km s⁻¹ appear to lead to a roughly equal split in mass between condensates and vapor (Johnson & Melosh 2012), but it is unclear whether a 50% vapor fraction is appropriate for Mercury-like giant collisions. In order to determine this fraction with greater precision, it is necessary in the future to undertake simulations of the giant impact itself, while tracking the debris with an appropriate equation of state for shocked silica (Pierazzo et al. 1997; Kraus et al. 2012). In tandem, imposing thermodynamical evolution models such as those of Johnson & Melosh (2012) on top of SPH simulations (Chau et al. 2018), promises to improve estimates of the particle size distribution. If particle sizes derived from these more involved treatments exceed several meters in radius, then unrealistically large solar winds may be required to remove the debris, and alternative pathways toward removal of material would be required.

Mercury's high density is anomalous in the solar system, but a growing number of exoplanetary density measurements are facilitating a comparison to exoplanetary analogs. While Mercury-sized planets have been detected around other stars (Barclay et al. 2013), few density estimates exist of this size class (Jontof-Hutter et al. 2016). The only current example, Kepler-138b, is a Mars-sized body with both mass and radius measurements, but does not show enhanced density and furthermore exists around an M-dwarf star (Jontof-Hutter et al. 2015).

If the stellar wind is instrumental in removing debris from giant impacts, we would make the qualitative prediction that closer-in rocky planets are capable of reaching densities that exceed those of more distant planets. This prediction follows from the expectation that subsequent giant impacts selectively remove silicates (Marcus et al. 2009; Bonomo et al. 2019), but only if the planet is close enough to the host star for the material to be removed. Such a trend has recently been revealed (Swain et al. 2019), where hotter terrestrial planets reach higher densities than cooler examples. Nevertheless, such a general prediction is confounded by the unknown degree to which stellar winds vary from system to system, and the enhanced impact speeds and reduced reaccretion times expected at shorter orbital periods (Volk & Gladman 2015).

Extrasolar material in the more exotic form of polluted white dwarfs (Jura & Young 2014) provide additional constraints upon the occurrence of Mercury-like planetary compositions. Specifically, white dwarfs present in their spectra evidence of rocky material recently and continuously falling onto their surfaces. It recently became possible to analyze the composition of this material and determine the Fe:Si ratios and oxygen fugacities they possess (Doyle et al. 2019). Although the parent bodies and dynamical histories of the white dwarf pollution remain unidentified, even among this set of cosmochemical measurements Mercury's oxygen fugacity and Fe:Si ratio stand as anomalies.

The giant impact scenario for Mercury's formation has existed for decades and taken numerous forms (Chapman 1988; Benz et al. 2007; Asphaug & Reufer 2014; Chau et al. 2018). In this paper, we have focused upon one specific concern with this model, and that is the likelihood that any material blasted off Mercury is likely to reaccrete over a short timescale. We have shown that the solar wind constitutes a viable pathway to removing this ejecta, and thereby alleviating this problem with the giant impact hypothesis.

The MESSENGER mission has afforded a higher level view of the evidence than merely the Si:Fe mass ratio (Solomon et al. 2018). In addition to Mercury's large iron content, the planet's surface was revealed to possess an abnormally low oxygen fugacity with respect to the other terrestrial planets (Nittler et al. 2011; McCubbin et al. 2012). These chemical constraints are difficult to compare with the giant impact framework, in part owing to computational limitations related to following individual chemical species within the SPH simulations typically applied to giant impact investigations (Lock & Stewart 2017; Chau et al. 2018), and furthermore due to the difficulty in addressing the degree of equilibration between silicates and iron within each body (Dahl & Stevenson 2010).

An additional revelation, that the surface of Mercury is volatile-rich, was once considered as inconsistent with a giant impact model (Peplowski & Evans 2011). The idea that giant impacts preferentially remove volatiles has typically been motivated by low volatile contents of the Moon (Stewart et al. 2016), itself an outcome of a giant impact. However, the volatile depletion of the Moon remains difficult to replicate within giant impact simulations, requiring fractionation between a Moon-forming disk and the proto-Earth (Canup et al. 2015). In other words, the volatile depletion seen on the Moon occurred as a result of it being the remnant of the impact, forming out of circumplanetary material that experienced volatile fractionation (Lock et al. 2016, 2018; Lock & Stewart 2017; Nakajima & Stevenson 2018). With respect to Mercury's high volatile content, then, it is more appropriate to compare Mercury to the Earth and not to the Moon (Stewart et al. 2016).

A final point regarding volatiles is that giant impacts constitute a robust expectation during the latter stages of terrestrial planet formation generally, across a range of models (Stewart et al. 2016; Raymond et al. 2018). Therefore, it is likely that Venus, Earth, and even Mars experienced giant collisions and yet exhibit similar volatile contents to Mercury (Ebel & Stewart 2018). Accordingly, the high volatile content of Mercury no longer appears inconsistent with the planet having experiencing a giant impact at the end of its formation epoch.

The feasibility of the giant impact hypothesis for the origin of Mercury's anomalously large core remains contentious. Nevertheless, the problem of reaccretion of ejected material becomes significantly reduced when the influence of the solar wind is taken into account. Thus, in early planetary systems, the stellar wind may play an instrumental, yet underappreciated role in the final architectures of planetary systems. Within this context, Mercury's anomalous density stands as the nearest, and most familiar example of the role of the Sun's wind in the early solar system.