Forming Mercury by Giant Impacts

To model the two-body collisions we use the smoothed particle hydrodynamics (SPH) code Gasoline (Wadsley et al. 2004), a modern SPH implementation that was adapted for planetary collisions (Reinhardt & Stadel 2017). The colliding bodies are assumed to be composed of condensed materials that are modeled with the equation of state (EOS) of Tillotson (1962). While the Tillotson EOS lacks a thermodynamically consistent treatment of mixed phases and phase transitions,¹ it has a simple analytical form that can easily be implemented in SPH simulations (e.g., Benz et al. 1987; Canup et al. 2001; Marinova et al. 2011; Genda et al. 2012; Burger et al. 2018). In addition, the Tillotson EOS shows good agreement with measured data (Brundage 2013) as well as with more thermodynamically consistent EOSs such as ANEOS (Thomson 1972). The good agreement, however, is limited to relatively low-velocity collisions where only a small fraction of the material is close to the vaporization heat of the rocks (Benz et al. 1989; Canup 2004; Emsenhuber et al. 2018), where the proper treatment of phase transitions can affect the thermal pressure. While these differences in EOS are important for the Moon-forming collisions, where one is interested in the detailed physical states of the orbiting material, in the case of a Mercury-stripping impact, where we concentrate on the total mass, these differences are expected to have a small influence on the results. The bodies are assumed to be fully differentiated with a chondritic abundance, i.e., an iron core (30%) and a basalt mantle (70%). The particle representations of the planets are generated as described in Reinhardt & Stadel (2017). In order to allow for multi-component bodies the procedure was slightly modified as described below. First a 1D equilibrium model is obtained by solving the structure equations with boundary conditions $M(r={R}_{\mathrm{CMB}})={M}_{\mathrm{core}}$, M(r = R_p) = M_p, and ρ(r = R_p) = ρ₀ where R_CMB and R_p are the radius of the core–mantle boundary (CMB) and that of the planet, respectively. At the CMB we assume temperature and pressure to be continuous. The thermal profile is adiabatic. For an initial guess of the density and internal energy in the core, ρ_c and u_c are varied until the above boundary conditions are satisfied. Then the SPH particles are arranged on concentric shells, where each shell is divided using an equal-area tessellation of the sphere in order to obtain a very uniform distribution. The particle distribution is generated separately for each material in order to properly capture the transition between them. The resulting initial conditions closely follow the model and show very low noise, thus the particles are very close to the equilibrium configuration. Because standard SPH cannot properly capture the density discontinuity at the CMB (e.g., Canup et al. 2001) we still evolve the planets in isolation for a few hours in simulation time until the rms velocity is below 50 m s⁻¹. We show an example of proto-Mercury in Figure 9. We use an intermediate resolution (80k to 270k particles in total) in order to explore a large parameter space. The resolution of proto-Mercury is kept constant (55k particles) while the impactor's resolution is adapted such that all SPH particles have the same mass. All simulations lasted for 2.2 days in simulation time until the fragment mass converged. SPH results depend on the numerical resolution (e.g., Hosono et al. 2017; Reinhardt & Stadel 2017). In our simulations, the masses of the post-impact bodies converge with high resolution N = 10⁶ to within ∼5%–10%.

A giant impact is characterized by the following parameters: the impact parameter b, the relative velocity between the two bodies v_imp, and the masses of the target and the impactor. We focus on the regime in the parameter space that leads to mantle stripping. Our baseline models begin with a proto-Mercury with a mass of ≥2.25 M${\mbox{\mercury}}\equiv {M}_{{mpM}}$,² which is the minimal mass assuming chondritic abundance needed to obtain Mercury's current core mass. Since we find that some of the iron from the core can be lost we also consider cases with higher masses, of 1.1 and 1.2 M_mpM. We consider various masses for the colliding body: 0.1, 0.3, 0.4, 0.5, 2, 3, 4.4 M_mpM. The relative velocities are in the range v = [10, 60] km s⁻¹, typically a few times the escape velocity of the system. For a pairwise collision, the escape velocity is given by

$$\begin{eqnarray}&&{v}_{\mathrm{esc}}=\sqrt{\displaystyle \frac{2G(M+m)}{R+r}},\end{eqnarray} \tag{ 1 }$$

where M, R, and m, r are the mass and radius of the first and second bodies, respectively. For reference, Mercury's escape velocity is ∼4 km s⁻¹. The impact parameter b is taken to be between 0.1 and 0.7. A very small b is not considered since in that case either the energy is too low to eject material from the target and the impactor is eventually accreted, or the energy is too high and both bodies are destroyed. A large b > 0.7 is also irrelevant since such grazing impacts do not efficiently strip the mantle.

We denote as "Case-1" the collisions where proto-Mercury is the target and is hit by a smaller body (such as in Benz et al. 2007). "Case-2" refers to the hit-and-run case, where proto-Mercury is actually the impactor and collides with a larger body (such as in Asphaug & Reufer 2014), which no longer resides in the solar system. To determine Mercury's mass after the impact, we use the clump finder SKID (Stadel 2001), which uses a friends-of-friends algorithm to determine whether a given SPH particle is part of a clump bound by gravity. For an individual clump, this is equivalent to testing whether the surrounding SPH particles' velocities are smaller than the escape velocity. Depending on the simulation, Mercury is either the largest remnant (Case-1) or the second largest (Case-2).

Figure 4 shows the final clump mass versus the impact parameter and the escape velocity in Case-1. We notice a correlation between b, M_clump, and Z_Fe: the more head-on the collision is, the more mass is stripped, leading to a higher final iron mass fraction. The chances of collision are limited because of the small size of the impactor: since the collisional cross section is as large as the impactor, this limits the maximum energy that is transferred to the target. For a given velocity, the outcome of the collision can be reduced to a geometry problem: more head-on collisions lead to deeper penetration of the impactor toward the target iron core, and more mantle stripping. Some of the iron in the core can be lost in such cases, requiring us to increase the initial target mass by 10% to 20% in order to achieve the desired final Z_Fe.

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To reproduce Mercury's properties, collisions must be quite head-on (0.3 < b < 0.55) with velocities at least five times the typical escape velocity of the system, or v_imp ∼ 30 km s⁻¹, which is notably high. However, for very head-on collisions, lower velocities are required (3–4 v_esc, i.e., 20 km s⁻¹) to prevent destruction of proto-Mercury and lead to the right Z_Fe. For lower velocities (∼2 v_esc), the mantle can be stripped but the impactor is reaccreted by the target (not shown in Figure 4).

3.1.1. Sensitivity to the Impactor's Composition

In the previous simulations, both the target and impactor were massive enough to be differentiated. We next consider a baseline case where proto-Mercury is hit by a small impactor (0.1 M_mpM) that could be undifferentiated. We consider impactors with various initial Z_Fe as well as pure iron and pure rock impactors. We then investigate the sensitivity of M_clump and Z_Fe to the impactor's composition. The impactor's mass is kept constant but the size changes in accordance with the assumed composition. The collisions occur with velocity ${v}_{\mathrm{imp}}=7.5\,{v}_{\mathrm{esc}}$ (i.e., 30 km s⁻¹) and b = 0.3. Figure 5 shows M_clump and the core-to-clump ratio ${M}_{\mathrm{core},\mathrm{Fe}}/{M}_{\mathrm{clump}}$ versus the initial Z_Fe, where M_core,Fe is the iron mass in the core. Z_Fe is defined as the total iron mass fraction, and therefore does not discriminate between the condensed iron in the core and the liquid/gaseous iron in the atmosphere/mantle.

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Interestingly, all the final clumps have the same mean Z_Fe of 0.6 but the distribution of iron within the planet is different. The inferred Z_Fe value is the same because of two competing effects. (1) The denser the object is, the more it can penetrate through the planetary body and eject more material, both in the core and in the mantle. In our case the iron sphere induces a final clump that is 30% lighter than its differentiated counterpart. On the other hand, rocky impactors lead to final objects that are slightly heavier. Note that this effect is strictly different from that of geometry, where a smaller object at a smaller angle has a smaller impact surface and cannot strip as much material as a bigger object. Here we actually see the opposite effect. (2) Since the iron impactor can only contaminate the target with iron, it increases the final Z_Fe, while rocky impactors disrupt the iron core less, leaving proto-Mercury with a larger core, and they can only contribute rock to the mantle.

We find that the final planetary mass and the core's Z_Fe (and the core mass fraction) depend on the impactor's composition. Denser impactors lead to a smaller mass of the clump and a smaller Z_Fe in the core, i.e., more iron is present in the mantle and/or is vaporized. M_clump and Z_Fe in the core do not follow a well-defined function of the initial Z_Fe. Our inferred iron distributions correspond to times shortly after the collision and they are likely to change with time. When the mantle is still in a magma state, the iron droplets in the outer parts are expected to settle to the core without leaving a signature on Mercury's surface (e.g., Elkins-Tanton 2012). Once the mantle solidifies, the iron droplets remain in the mantle (and even on the surface). Therefore, current-state observations of Mercury's iron distribution cannot be used to discriminate among the different cases.

Figure 6 shows the final clump mass versus the impact parameter and the escape velocity in Case-2. In the hit-and-run scenario there are more possible combinations: a large Z_Fe can be achieved with various impact parameters. However, we find that in these collisions more mass is removed from proto-Mercury than in Case-1. This is because in this configuration the interacting area and mass are larger. For a given b the impactor transfers more energy than in Case-1, which affects the results. For example, collisions with b = 0.3 and a velocity of  20 km s⁻¹ (3 v_esc) can lead to Mercury's Z_Fe, unlike in Case-1. This also holds for impacts with b = 0.6–0.65 that are energetic enough (with v_imp = 30 km s⁻¹, ∼4–5 v_esc). These energetic collisions can also entirely destroy proto-Mercury: in several cases, the collisions were found to be disruptive. Therefore we find that in Case-2 we must start with a larger initial mass of 3 to 5 M${\mbox{\mercury}}$, while we performed our simulations with initial masses of 2.475 to 3.375 M${\mbox{\mercury}}$. This is in agreement with the most promising result of Asphaug & Reufer (2014) with a proto-Mercury of 4.5 M${\mbox{\mercury}}$. We conclude that more configurations in Case-2 can lead to a Mercury-like planet. This case is also characterized by a lower ratio of v_imp/v_esc than for Case-1. This is the consequence of using the same chosen velocity range as in the standard case with more massive systems, which leads to escape velocities higher than in Case-1.

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First, we study how many impacts are required to strip proto-Mercury's mantle. We consider two scenarios: (i) an angle of 45°, i.e., b = 0.7, the most probable angle, and an impact velocity of 3–4 v_esc, which is required to prevent mergers. The impactor's mass is 0.2, 0.3, or 0.5 M_mpM; (ii) an impact angle of b = 0.5, 0.7, or 0.8, with an impact velocity of 3–4 v_esc. The impactor's mass is 0.5 M_mpM. For the subsequent collisions, the target is modeled with the updated Z_Fe with the impactor always taken to be a new object.

Figure 7 shows the mass loss from proto-Mercury after repeated collisions for the two scenarios. As expected, a more massive and faster impactor strips away more material. In scenario (i), Mercury can reach its current mass after six impacts with v_imp ∼ 4 v_esc or alternatively with ten impacts with v_imp ∼ 3 v_esc, with the impactor's mass being 0.5 M_mpM. On the other hand, if the impactor has a mass of 0.2 M_mpM, more than a dozen impacts are needed, up to 20 in the least optimistic case. Even with an impactor with half the target's mass, the number of impacts required is high. We therefore conclude that this formation scenario for Mercury is rather unlikely.

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In the second scenario, Mercury can reach its current Z_Fe after two impacts with b = 0.5 and v_imp ∼ 4 v_esc, or alternatively after four impacts with v_imp ∼ 3 v_esc. We find that colliding an impactor of 0.5 M_mpM at impact parameter b = 0.8 is equivalent to an impactor of 0.2 M_mpM at an impact parameter of b = 0.7.

We conclude that Mercury could form as a result of several collisions with less extreme conditions than the giant impact of Case-1 or Case-2 presented in Sections 3.1 and 3.2. The exact number of collisions depends on the impactor's mass, the collision angle, and the impact velocity. It is more likely that Mercury formed via two collisions with b ∼ 0.5 at moderate velocities of v ∼ 20 km s⁻¹ than via 10 collisions at the most probable angle.

In the case of multiple collisions the time between collisions and the state of the target must be considered. A subsequent collision can occur at any time after the first one. Collisions that occur shortly after each other (i.e., close together in time) are characterized by a non-spherical and hot target, while in the case of collisions that are well separated in time the body is more likely to be relaxed and differentiated. We therefore consider different thermal profiles, corresponding to different cooling times, to model the post-impact target. In some cases, the planet is still surrounded by a hot cloud of low-density material. We begin with one collision and then explore how the assumed thermal state affects the subsequent collision outcomes. The first collision is between a proto-Mercury of 2.25 M${\mbox{\mercury}}$ and a body of 1.125 M${\mbox{\mercury}}$, at b = 0.7 and v = 4 v_esc, and the subsequent proto-Mercury has a mass of 1.97 M${\mbox{\mercury}}$. The second collision is with another body of 1.125 M${\mbox{\mercury}}$ at the same angle and velocity. Figure 8 shows the different thermal states we consider.

The first case we consider (simulation setup A) is when the second collision occurs right after the first collision. Then, the object is still hot, mostly evaporated, and is non-spherical. This state corresponds to a time of ∼4–5 hr between the two collisions, i.e., 4–5 hr after the initial impact. While this scenario is somewhat unlikely because it requires two nearly simultaneous collisions, it is an interesting case to consider since it provides an upper bound to the envelope stripping in the multiple-collision scenario. We take the original simulation output to represent the target and use it directly in the subsequent collision. After the first collision, we include the particles that compose the gravitationally bound clump, with all stored quantities (e.g., density, internal energy, physical state of the material) and use them as an input for the second collision.

In simulation setup B, we assume that the time between collisions is intermediate: the target is spherical because it has a moderate rotation period and its hot cloud has recondensed. The atmosphere is expected to cool down quickly by radiation. However, if the surface of the body is still hot, the particles in the atmosphere could be heated and evaporate again. As a result, we estimate the cooling time as the time it takes the planet to radiate the energy associated with the impact and reach Mercury's equilibrium temperature of ∼600 K. With this simple consideration, which neglects the existence of an atmosphere, the cooling time is estimated to be ∼10⁴ yr. This time estimate is appropriate until the rheological transition from liquid to solid phase is achieved, which is at ∼1300 K for silicates. A typical timescale for the surface temperature to drop below the melting temperature for half the Earth's mass is 10³–10⁴ yr (Bower et al. 2017; Bonati et al. 2018). To account for this effect, we use a condensed-state structure but we increase the internal energy such that the surface temperature is ∼1200 K. Although the Tillotson EOS is not explicitly temperature-dependent, we can estimate T_s using the internal energy u. As there is no pressure, the contribution from the molecular interactions in the material also drops, which leaves u = u_thermal = c_vT. Assuming that the heat capacity c_v does not change between 500 K and 1200 K, we can estimate T_s and derive the new structure of the target.

The last case we consider, simulation setup C, is when the target has cooled down completely between two collisions. In this case the target has a surface temperature of a few hundred kelvin, near the equilibrium temperature assuming there is no significant internal heat source or mechanism to retain the heat. Models of thermal evolution suggest that a terrestrial planet could reach thermal equilibrium after billions of years, or on geological timescales (e.g., de Pater & Lissauer 2010; Stamenkovic & Breuer 2014). For this case, we use the standard way to describe the target. However, since the target has acquired some angular momentum in the collision we must consider its spin. To obtain the target's post-impact rotation period we bin the angular momentum of the SPH particles as a function of radius and determine the angular velocity by fitting a solid-body rotation to the data. The outermost layers, represented by only a few particles, are noise-dominated and are therefore excluded. Under these assumptions, we get a period of P ∼ 30 hr. This is a lower bound since the outermost layers can contribute their (large) angular momentum as the target cools down. Possible deviation from solid-body rotation introduces an additional challenge in determining the rotation period accurately. For comparison, we also consider intermediate and fast spinning bodies with P ∼ 10 and 5 hr (the break-up speed for a planet of Mercury's density equates to a period of P ∼ 1.5 hr). We then consider, for the same absolute value of angular momentum, both prograde spins (C1, C3, C5) and retrograde spins (C2, C4, C6). We assume that the subsequent collision occurs in the same plane as the first one.

When we consider the rotation acquired, the mass difference from the standard (non-rotating) case is about 1%. We also cannot distinguish the prograde and retrograde cases. Overall this effect is small because the spinning velocities, respectively 1.2, 0.6, and 0.2 km s⁻¹ for cases C5 (C6), C3 (C4), and C1 (C2) are much smaller than the impact velocity of ∼17.4 km s⁻¹, and do not contribute much angular momentum to the system. We therefore conclude that mantle stripping is relatively insensitive to the target's rotation period in the regime of high-velocity collisions, because the impact velocity plays a more critical role. We expect this effect to be profound in the low-velocity regime, where the impact velocity is of the order of the spinning velocity. To summarize, we find that the multiple-collision scenario has different outcomes depending on the time interval between successive collisions. If the target has enough time to cool and recondense, less material can be stripped away. For a given collision, up to 23% of the planet's mass can be lost when the planet is still hot compared to 13% for a compact and cold target. This indicates that the required number of impacts can be reduced if they are close together in time. Table 1 summarizes the different cases for the multiple-collision scenario.

4.2.1. Impact Timing

An important aspect in giant impacts is the evolution of the ejected material after the collision. The projectile deposits up to 70% of its kinetic energy on the target (e.g., de Pater & Lissauer 2010). This is large enough to melt large parts of the targeted body: typically the kinetic energy is of the order a few times the energy required to melt and evaporate a basalt sphere of the target's mass. In such conditions, the mantle is expected to be in a magma phase while the ejected material is mostly heated and vaporized (e.g., Tonks & Melosh 1993 for quantitative modeling). After the collision the material can cool down and recondense directly or undergo the droplet regime before becoming solid again. Based on this simple thermodynamic argument, Benz et al. (2007) showed that the droplets or solid fragments are of the millimeter–centimeter size and are therefore subject to Poynting–Robertson drag. This effect, however, is expected to remove the particles only after ∼2.5 Myr, with 35%–40% of particles being reaccreted by Mercury. Gladman & Coffey (2009) suggested that the ejecta material would spread around the Sun in a thin ring that is optically thick, which would reduce the drag. They estimated that the opacity can be larger than one in some cases. Both the debris size distribution and dynamical behavior post-impact are critical to estimating the final fraction of the ejecta that will reaccrete—aspects that are not considered in this work. The clump's mass is estimated by gravitational binding without explicit size considerations, and could be considered as an upper limit because some of the fragments can be dragged toward the Sun. However, if a consequent fraction of the material reaccretes, higher post-impact values of Z_Fe than those presented in this work are needed to correctly predict Mercury's current Z_Fe.

Another important aspect is the fate of the impactor after the collision. When the collision energy is high, part of the colliding material can be unbound and distributed in the neighboring region. This debris is expected to cool and condense, and can cross Mercury's (or another planet's) orbit and be reaccreted later on. If the impactor is larger than Mercury, it likely survives the collision with little mantle stripping. Unless its orbit is extremely eccentric, which could lead to inward scattering and possibly falling into the Sun's gravitational well, it is expected to survive. Therefore it seems that an additional mechanism is required to remove the impactor from Mercury's orbit, and we plan to investigate the dynamical evolution of the debris in future work.

We performed simulations with initial conditions from Benz et al. (2007) and Asphaug & Reufer (2014) to compare with their results. For run-6 of Benz et al. (2007) with M_tar = 2.25 M${\mbox{\mercury}}$, M_imp = 0.225 M${\mbox{\mercury}}$, v_imp = 20 km s⁻¹, θ = 0, a Mercury analog of 0.92 M${\mbox{\mercury}}$ and Z_Fe = 0.61 was inferred, while we obtain 0.74 M${\mbox{\mercury}}$ and Z_Fe = 0.67. The most successful collision in Asphaug & Reufer (2014) (M₁ = 0.85 R_⊕, M₂ = 4.52 M_☿, v_imp = 3.25 v_esc, θ = 34°) results in 1.0 M_☿ and Z_Fe = 0.70, while our simulations imply a Mercury analog of 1.07 M_☿ and Z_Fe = 0.76. For Case-2 we find agreement at the 5% level while Case-1 agrees within 20%.

Overall, good agreement is found between our simulations and those of Asphaug & Reufer (2014). There are several possible reasons for the differences between our simulations and those of Benz et al. (2007). First, Benz et al. (2007) used ANEOS to model the mantle, which leads to a higher reference density than that in the Tillotson EOS used in this work (3.32 g cm⁻³ for dunite, but 2.7 g cm⁻³ for basalt). This makes the initial bodies ∼10% smaller and more condensed; as a result they have a higher gravitational binding energy and are harder to disrupt. In order to investigate this effect, we simulate run-6 of Benz et al. (2007) with a modified Tillotson EOS where we substitute the reference density of basalt by that of dunite. We find a body of 0.81 ${M}_{\mbox{\mercury}}$ with Z_Fe of ∼0.76. Thus the choice of mantle material can explain up to 5%–6% of the mass difference. Second, as discussed above, disruptive collisions are highly energetic and a significant part of the material can undergo phase transitions. Accounting for the latent heat thus can change the thermal pressure of the material. For the Tillotson EOS (which neglects phase transitions), we obtain a hotter and more pressure-supported material, allowing for more disruption. Finally, differences in the numerical treatments and analysis methods can also affect the results by a few per cent (e.g., Genda et al. 2015; Suetsugu et al. 2018).

Figure 9 shows the density profile of proto-Mercury of 2.25 M${\mbox{\mercury}}$.

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In this section we compare our results to scaling laws found by previous studies (e.g., Benz & Asphaug 1999; Genda et al. 2012; Leinhardt & Stewart 2012). A universal scaling law to determine the largest remnant mass after a collision was proposed by Leinhardt & Stewart (2012):

$$\begin{eqnarray}&&{M}_{\mathrm{lr}}/{M}_{\mathrm{tot}}=-0.5{Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * }+1,\end{eqnarray} \tag{ 1 }$$

where Q_R is the specific total kinetic energy in the center-of-mass frame and ${Q}_{\mathrm{RD}}^{{\prime} * }$ is the specific energy where half of the target's mass is disrupted (corrected for the interacting mass).

For super-catastrophic collisions, i.e., M_lr/M_tot < 0.1, laboratory experiments and simulations indicate that the remnant's mass deviates from the universal scaling law (Leinhardt & Stewart 2012). This regime is better described by the following power law:

$$\begin{eqnarray}&&{M}_{\mathrm{lr}}/{M}_{\mathrm{tot}}=\displaystyle \frac{0.1}{{1.8}^{\eta }}{({Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * })}^{\eta },\end{eqnarray} \tag{ 2 }$$

where η is $\sim -1.5$. Usually one first derives the characteristic energy of the system ${Q}_{\mathrm{RD}}^{{\prime} * }$ (Marcus et al. 2009; Leinhardt & Stewart 2012). The calculated ${Q}_{\mathrm{RD}}^{{\prime} * }$ values following both prescriptions differ from our results by a factor of ∼2. The fit to ${Q}_{\mathrm{RD}}^{{\prime} * }$ cannot be used because of differences in the numerical treatment, confirming that the critical disruption of a body depends on the details of the numerical method and numerical parameters (Genda et al. 2015; Suetsugu et al. 2018). We can estimate ${Q}_{\mathrm{RD}}^{{\prime} * }$ from our data for each available subsets of collisions with the same target mass, mass ratio, and impact angle. We linearly fit the ratio M_lr/M_tot as a function of the specific energy Q_R, and infer the corresponding specific energy Q_R such that M_lr/M_tot = 0.5. Subsets with ${M}_{\mathrm{lr}}/{M}_{\mathrm{tot}}\lt 0.3$, and where the available maximum and minimum M_lr/M_tot differ by less than 10%, are excluded in order to avoid spurious extrapolation. For the hit-and-run collisions we replace the mass ratio M_lr/M_tot by M_2lr/M_pM, which is the mass ratio between the second largest remnant and the protoplanet's initial mass (Leinhardt & Stewart 2012).

For the inferred Z_Fe, Marcus et al. (2009) proposed the following power law:

$$\begin{eqnarray}&&{M}_{\mathrm{Fe}}/{M}_{\mathrm{lr}}=0.3+0.25{({Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * })}^{1.65},\end{eqnarray} \tag{ 3 }$$

where the numerical factor can be scaled down to fit our initial core fraction of 0.3 instead of 0.33 (Carter et al. 2018). The upper panels of Figure 10 show M_clump/M_tot and M_2lr/M_pM as a function of the specific energy normalized by the characteristic specific energy ${Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * }$ for Case-1 and Case-2, respectively. The blue line represents the universal scaling law of Leinhardt & Stewart (2012). The lower panels of Figure 10 show Z_Fe as a function of ${Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * }$ for Case-1 (circles) and Case-2 (squares).

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For low energies, we indeed find that the mass ratio is linearly dependent on the energy ratio. For ${Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * }\gt 1.7$, we find a significant scatter, especially for Case-1. This might be due to the transition to the super-catastrophic regime. We also note that our results for Case-1 have a more moderate slope because we do not probe the low range of energies, which would correct the extrapolated ${Q}_{\mathrm{RD}}^{{\prime} * }$. We also note some deviations from the universal slope between subsets depending on the impact angle, especially for more oblique collisions in agreement with Leinhardt & Stewart (2012). We also find good agreement with the power law of Marcus et al. (2009) where the mass ratio is described by a linear relation with ${Q}_{R}/{Q}_{\mathrm{RD}}^{{\prime} * }$. This is especially noticeable for Case-2, which follow both scaling laws except for a few outliers. For higher energies, we find a similar trend in the scatter in Z_Fe to the scatter in the mass ratio. While the precise value of ${Q}_{\mathrm{RD}}^{{\prime} * }$ depends on several numerical parameters, we find general agreement with the proposed scaling laws once ${Q}_{\mathrm{RD}}^{{\prime} * }$ is obtained from simulation data.

A.3.1. Case-1

Table 2 shows the initial conditions and outcomes of the simulation for all the Case-1 collisions.

Table 2.  Initial Conditions (Initial Proto-Mercury's and Impactor's Masses, ${M}_{p{\mbox{\mercury}}}$, M_imp and Resolutions ${N}_{p{\mbox{\mercury}}}$, N_imp, Impact Parameter b, Impact Velocity v_imp) and Outcomes (Mass of the Largest Fragment M_lr and its Iron Mass Fraction Z_Fe) for Case-1 Collisions

${M}_{p{\mbox{\mercury}}}$ (M${}_{{mercury}}$)	${N}_{p{\mbox{\mercury}}}$	Mimp (M${\mbox{\mercury}}$)	Nimp	b	vimp (km s−1)	Mlr (M${}_{{mercury}}$)	ZFe
2.25	57877	0.675	17473	0.4	15	1.72	0.39
2.25	57877	0.675	17473	0.4	20	1.35	0.49
2.25	57877	0.675	17473	0.48	30	1.08	0.62
2.25	57877	0.675	17473	0.5	20	1.65	0.41
2.25	57877	0.675	17473	0.5	30	1.16	0.58
2.25	57877	0.675	17473	0.6	20	1.90	0.36
2.25	57877	0.675	17473	0.6	25	1.71	0.39
2.25	57877	0.675	17473	0.6	30	1.50	0.45
2.25	57877	0.9	20953	0.48	30	0.97	0.66
2.25	57877	0.9	20953	0.49	30	0.99	0.65
2.25	57877	0.9	20953	0.5	30	1.05	0.63
2.25	57877	0.9	20953	0.6	20	1.83	0.37
2.25	57877	0.9	20953	0.6	30	1.40	0.48
2.25	57877	1.125	21205	0.1	15	0.86	0.50
2.25	57877	1.125	21205	0.2	10	2.32	0.42
2.25	57877	1.125	21205	0.2	15	1.11	0.51
2.25	57877	1.125	21205	0.2	17	0.73	0.58
2.25	57877	1.125	21205	0.25	20	0.60	0.68
2.25	57877	1.125	21205	0.3	15	1.07	0.55
2.25	57877	1.125	21205	0.3	20	0.81	0.64
2.25	57877	1.125	21205	0.4	20	1.15	0.55
2.25	57877	1.125	21205	0.5	16.5	1.70	0.40
2.25	57877	1.125	21205	0.5	20	1.43	0.47
2.25	57877	1.125	21205	0.5	27	1.08	0.61
2.25	57877	1.125	21205	0.5	30	0.95	0.67
2.25	57877	1.125	21205	0.51	30	0.96	0.66
2.25	57877	1.125	21205	0.6	15	1.32	0.51
2.25	57877	1.125	21205	0.7	30	1.74	0.39
2.25	57877	1.125	21205	0.7	45	1.26	0.51
2.25	57877	1.125	21205	0.7	50	1.08	0.56
2.25	57877	1.125	21205	0.7	60	0.64	0.62
2.475	58645	0.675	17473	0.45	30	1.18	0.62
2.475	58645	0.675	17473	0.6	20	2.15	0.35
2.475	58645	0.675	17473	0.6	25	1.95	0.38
2.475	58645	0.9	20953	0.4	25	1.13	0.63
2.475	58645	0.9	20953	0.4	30	0.87	0.76
2.475	58645	0.9	20953	0.42	28	1.04	0.68
2.475	58645	0.9	20953	0.42	29	0.98	0.71
2.475	58645	0.9	20953	0.42	30	0.93	0.73
2.475	58645	0.9	20953	0.43	30	0.95	0.72
2.475	58645	0.9	20953	0.435	30	0.97	0.71
2.475	58645	0.9	20953	0.437	30	0.97	0.71
2.475	58645	1.125	21205	0.2	17	1.00	0.56
2.475	58645	1.125	21205	0.43	27	1.02	0.68
2.475	58645	1.125	21205	0.45	27	1.10	0.65
2.475	58645	1.125	21205	0.45	29	0.96	0.71
2.475	58645	1.125	21205	0.45	30	0.92	0.73
2.475	58645	1.125	21205	0.47	30	0.98	0.69
2.475	58645	1.125	21205	0.5	30	1.13	0.64
2.7	68905	1.125	21205	0.1	15	1.70	0.46
2.7	68905	1.125	21205	0.1	17	0.96	0.54
2.7	68905	1.125	21205	0.1	20	0	0
2.7	68905	1.125	21205	0.15	17	1.04	0.55
2.7	68905	1.125	21205	0.15	18	0.70	0.59
2.7	68905	1.125	21205	0.2	15	1.56	0.48
2.7	68905	1.125	21205	0.2	17	1.21	0.54
2.7	68905	1.125	21205	0.2	18	0.99	0.59
2.7	68905	1.125	21205	0.2	20	0.68	0.65
2.7	68905	1.125	21205	0.3	20	1.18	0.58
2.7	68905	1.125	21205	0.3	23	0.90	0.67
2.7	68905	1.125	21205	0.3	25	0.68	0.74
2.7	68905	1.125	21205	0.43	30	1.01	0.73
2.7	68905	1.125	21205	0.45	30	1.09	0.69
2.7	68905	1.125	21205	0.5	30	1.32	0.61
2.7	68905	1.125	21205	0.7	30	1.41	0.53

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A.3.2. Case-2

Table 3 shows the initial conditions and outcomes of the simulation for all the Case-2 collisions.

Table 3.  Initial Conditions (Initial Proto-Mercury's and Impactor's Masses ${M}_{p{\mbox{\mercury}}}$, M_imp and Resolutions ${N}_{p{\mbox{\mercury}}}$, N_imp, Impact Parameter b, Impact Velocity v_imp) and Outcomes (Mass of the Second Largest Fragment M_2lr and its Iron Mass Fraction Z_Fe) for Case-2 Collisions

Mp☿ (M${}_{{mercury}}$)	Np☿	Mimp (M${\mbox{\mercury}}$)	Nimp	b	vimp (km s−1)	M2lr (M${}_{{mercury}}$)	ZFe
2.475	58645	4.53	108901	0.2	10	5.83	0.35
2.475	58645	4.53	108901	0.3	20	0.54	0.70
2.475	58645	4.53	108901	0.4	10	1.83	0.37
2.475	58645	4.53	108901	0.4	17	1.16	0.54
2.475	58645	4.53	108901	0.4	20	0.75	0.68
2.475	58645	4.53	108901	0.4	30	0	0
2.475	58645	4.53	108901	0.5	10	1.97	0.37
2.475	58645	4.53	108901	0.5	15	0.54	0.47
2.475	58645	4.53	108901	0.5	20	1.08	0.56
2.475	58645	4.53	108901	0.5	25	0.49	0.82
2.475	58645	4.53	108901	0.6	25	1.16	0.56
2.475	58645	4.53	108901	0.6	30	0.85	0.65
2.475	58645	4.53	108901	0.6	40	0	0
2.475	58645	4.53	108901	0.7	30	1.50	0.49
2.475	58645	4.53	108901	0.7	40	1.06	0.64
2.475	58645	6.75	167329	0.4	10	0	0
2.475	58645	6.75	167329	0.4	20	0	0
2.475	58645	6.75	167329	0.5	10	1.44	0.38
2.475	58645	6.75	167329	0.5	17	0.58	0.71
2.475	58645	6.75	167329	0.5	20	0.93	0.50
2.475	58645	6.75	167329	0.6	20	1.30	0.52
2.475	58645	6.75	167329	0.6	25	0.96	0.60
2.475	58645	6.75	167329	0.7	30	1.35	0.47
2.475	58645	10	238777	0.4	10	0	0
2.475	58645	10	238777	0.5	10	0	0
2.475	58645	10	238777	0.5	20	0	0
2.475	58645	10	238777	0.6	20	1.04	0.54
2.475	58645	10	238777	0.6	22	0.68	0.71
2.475	58645	10	238777	0.6	30	0	0
2.475	58645	10	238777	0.7	30	1.16	0.59
2.7	68905	4.53	108901	0.4	30	0	0
2.7	68905	4.53	108901	0.5	30	0	0
2.7	68905	4.53	108901	0.6	25	1.39	0.56
2.7	68905	4.53	108901	0.6	27	1.13	0.60
2.7	68905	4.53	108901	0.7	40	1.33	0.60
2.7	68905	6.75	167329	0.5	10	1.89	0.37
2.7	68905	6.75	167329	0.5	20	1.03	0.58
2.7	68905	6.75	167329	0.6	10	2.11	0.36
2.7	68905	6.75	167329	0.6	20	1.48	0.50
2.7	68905	6.75	167329	0.6	27	0.99	0.62
2.7	68905	6.75	167329	0.6	30	0.71	0.69
2.7	68905	6.75	167329	0.7	30	1.54	0.51
2.7	68905	10	238777	0.6	20	1.32	0.52
2.7	68905	10	238777	0.6	30	0	0
2.7	68905	10	238777	0.7	30	1.37	0.56
3.375	85573	6.75	167329	0.5	10	2.70	0.35
3.375	85573	6.75	167329	0.5	20	1.58	0.54
3.375	85573	6.75	167329	0.5	30	0	0
3.375	85573	6.75	167329	0.6	20	2.09	0.47
3.375	85573	6.75	167329	0.6	30	1.35	0.61
3.375	85573	6.75	167329	0.7	30	2.17	0.46
3.375	85573	10	238777	0.5	10	0	0
3.375	85573	10	238777	0.5	20	0	0
3.375	85573	10	238777	0.5	30	0	0
3.375	85573	10	238777	0.6	20	1.84	0.50
3.375	85573	10	238777	0.6	30	0.73	0.68
3.375	85573	10	238777	0.7	30	1.96	0.50

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A.3.3. Case-3

Table 4 shows the initial conditions and outcomes of the simulation for all the Case-3 collisions.

Table 4.  Initial Conditions (Initial Proto-Mercury's and Impactor's Masses ${M}_{p{\mbox{\mercury}}}$, M_imp and Resolutions ${N}_{p{\mbox{\mercury}}}$, N_imp, Impact Parameter b, Impact Velocity v_imp) and Outcomes (Mass of the Largest Fragment M_lr and its Iron Mass Fraction Z_Fe) for Case-3 Collisions

${M}_{p{\mbox{\mercury}}}$ (M${}_{{mercury}}$)	${N}_{p{\mbox{\mercury}}}$	Mimp (M${\mbox{\mercury}})$	Nimp	b	vimp (km s−1)	Mlr (M${}_{{mercury}}$)	ZFe
2.25	57877	0.45	11653	0.7	13.6	2.22	0.31
2.22	58273	0.45	11653	0.7	13.5	2.14	0.32
2.14	77185	0.45	11653	0.7	13.4	2.08	0.32
2.08	55081	0.45	11653	0.7	13.4	2.03	0.33
2.03	52585	0.45	11653	0.7	13.2	1.97	0.34
1.97	49369	0.45	11653	0.7	13.1	1.92	0.35
1.92	49369	0.45	11653	0.7	13.1	1.86	0.36
1.86	45037	0.45	11653	0.7	12.9	1.81	0.37
1.81	45481	0.45	11653	0.7	12.8	1.75	0.39
1.75	40285	0.45	11653	0.7	12.7	1.69	0.40
1.69	40705	0.45	11653	0.7	12.6	1.64	0.41
1.64	41149	0.45	11653	0.7	13.4	1.58	0.43
1.58	39865	0.45	11653	0.7	13.4	1.53	0.44
1.53	40705	0.45	11653	0.7	13.4	1.48	0.46
1.48	35233	0.45	11653	0.7	13.4	1.43	0.47
1.43	36397	0.45	11653	0.7	13.4	1.38	0.49
1.38	32929	0.45	11653	0.7	11.9	1.33	0.51
2.25	57877	0.45	11653	0.7	18.1	2.15	0.31
2.15	50281	0.45	11653	0.7	17.9	2.05	0.33
2.05	51721	0.45	11653	0.7	17.7	1.94	0.35
1.94	49369	0.45	11653	0.7	17.4	1.84	0.37
1.84	45481	0.45	11653	0.7	17.1	1.74	0.39
1.74	40285	0.45	11653	0.7	16.9	1.63	0.41
1.63	42121	0.45	11653	0.7	16.6	1.53	0.44
1.53	40705	0.45	11653	0.7	16.3	1.44	0.47
1.44	35605	0.45	11653	0.7	16.1	1.33	0.52
1.33	32929	0.45	11653	0.7	15.7	1.24	0.54
1.24	32929	0.45	11653	0.7	15.5	1.16	0.58
1.16	28069	0.45	11653	0.7	15.2	1.08	0.62
1.08	29089	0.45	11653	0.7	15.1	1.01	0.67
2.25	57877	0.675	17473	0.7	13.8	2.19	0.31
2.19	54637	0.675	17473	0.7	13.7	2.11	0.32
2.11	55081	0.675	17473	0.7	13.6	2.03	0.34
2.03	52585	0.675	17473	0.7	13.5	1.86	0.37
1.86	45481	0.675	17473	0.7	13.2	1.78	0.38
1.78	40285	0.675	17473	0.7	13.0	1.70	0.40
1.70	40705	0.675	17473	0.7	12.9	1.62	0.42
1.62	39865	0.675	17473	0.7	12.7	1.54	0.44
1.54	40705	0.675	17473	0.7	12.6	1.46	0.47
1.46	35605	0.675	17473	0.7	12.4	1.39	0.49
1.39	32929	0.675	17473	0.7	12.3	1.31	0.52
1.31	32929	0.675	17473	0.7	12.2	1.24	0.55
1.24	32533	0.675	17473	0.7	12.0	1.16	0.59
1.16	28393	0.675	17473	0.7	11.8	1.09	0.62
1.09	29089	0.675	17473	0.7	11.8	1.02	0.66
2.25	57877	0.675	17473	0.7	18.4	2.11	0.57
2.11	55081	0.675	17473	0.7	18.2	1.96	0.34
1.96	49369	0.675	17473	0.7	17.9	1.81	0.37
1.81	45481	0.675	17473	0.7	17.5	1.67	0.40
1.67	42337	0.675	17473	0.7	17.1	1.53	0.44
1.53	40705	0.675	17473	0.7	16.8	1.40	0.48
1.40	36397	0.675	17473	0.7	16.5	1.28	0.53
1.28	31789	0.675	17473	0.7	16.2	1.16	0.58
1.16	28069	0.675	17473	0.7	15.9	1.06	0.64
2.25	57877	1.125	21205	0.8	19.0	2.17	0.31
2.17	54637	1.125	21205	0.8	18.9	2.07	0.33
2.07	55081	1.125	21205	0.8	18.8	1.97	0.34
1.97	49369	1.125	21205	0.8	18.5	1.87	0.36
1.87	45037	1.125	21205	0.8	18.3	1.77	0.38
1.77	40285	1.125	21205	0.8	18.1	1.68	0.40
1.68	44593	1.125	21205	0.8	18.0	1.59	0.43
1.59	39865	1.125	21205	0.8	17.8	1.50	0.45
1.50	40705	1.125	21205	0.8	17.6	1.41	0.48
1.41	35605	1.125	21205	0.8	17.4	1.33	0.52
1.33	32929	1.125	21205	0.8	17.2	1.24	0.55
1.24	32533	1.125	21205	0.8	16.9	1.16	0.58
1.16	27769	1.125	21205	0.8	16.7	1.10	0.62
2.25	57877	1.125	21205	0.8	14.2	2.19	0.31
2.19	58273	1.125	21205	0.8	14.2	2.12	0.32
2.12	50281	1.125	21205	0.8	14.1	2.06	0.33
2.06	55081	1.125	21205	0.8	14.0	2.00	0.34
2.00	52585	1.125	21205	0.8	14.0	1.96	0.34
1.96	49369	1.125	21205	0.8	13.9	1.90	0.35
1.90	49369	1.125	21205	0.8	13.8	1.83	0.37
1.83	42337	1.125	21205	0.8	13.7	1.77	0.38
1.77	40285	1.125	21205	0.8	13.5	1.71	0.40
1.71	40705	1.125	21205	0.8	13.5	1.64	0.41
1.64	44593	1.125	21205	0.8	13.4	1.58	0.43
1.58	39865	1.125	21205	0.8	13.3	1.53	0.44
2.25	57877	1.125	21205	0.7	19.0	2.04	0.33
2.04	51721	1.125	21205	0.7	18.7	1.82	0.37
1.82	45481	1.125	21205	0.7	18.1	1.58	0.43
1.58	39865	1.125	21205	0.7	17.8	1.38	0.49
1.38	35605	1.125	21205	0.7	17.3	1.20	0.56
1.20	32533	1.125	21205	0.7	17.0	1.05	0.64
2.25	57877	1.125	21205	0.7	14.2	2.13	0.32
2.13	50281	1.125	21205	0.7	14.1	2.00	0.33
2.00	52489	1.125	21205	0.7	14.0	1.86	0.36
1.86	45037	1.125	21205	0.7	13.8	1.73	0.39
1.73	42721	1.125	21205	0.7	13.6	1.63	0.41
1.63	39469	1.125	21205	0.7	13.4	1.51	0.45
1.51	40705	1.125	21205	0.7	13.2	1.38	0.49
1.38	32929	1.125	21205	0.7	13.0	1.27	0.53
1.27	32533	1.125	21205	0.7	12.8	1.16	0.58
1.16	28069	1.125	21205	0.7	12.7	1.06	0.64
2.25	57877	1.125	21205	0.5	14.2	1.85	0.37
1.85	42337	1.125	21205	0.5	13.7	1.47	0.46
1.47	37789	1.125	21205	0.5	13.1	1.17	0.57
1.17	33157	1.125	21205	0.5	12.5	0.97	0.67
2.25	57877	1.125	21205	0.5	19.0	1.51	0.45
1.51	40705	1.125	21205	0.5	17.6	1.04	0.63

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