Modeling of Collisional Outcomes Based on Impact Simulations of Mars-sized Bodies

We investigate the outcomes of collisions between Mars-sized bodies through smooth particle hydrodynamics (SPH) simulations, focusing on the transitions among merging'', hit-and-run’’, and catastrophic disruption. By systematically varying impact velocity, angle, and mass ratio, we characterize the dependence of collision outcomes on geometric and energetic parameters. A new analytic model is developed using characteristic energies – particularly the energy deposited in overlapping regions of the colliding bodies – to accurately describe the mass of the largest and second-largest remnants. The model successfully reproduces simulation results across a broad range of impact conditions and improves on previous models by better capturing the transitions between merging'', hit-and-run’’, and disruption. We also derive outcome formulas averaged over impact-parameter-weighted angular distributions, enabling more realistic applications to integrated modeling of planet formation. The model further shows consistency with outcomes from dust aggregate collision simulations, highlighting its utility for modeling collisional processes not only for large planetesimals but also for smaller bodies.

💡 Research Summary

This paper presents a comprehensive study of collisions between Mars‑sized planetary embryos using high‑resolution smooth particle hydrodynamics (SPH) simulations and introduces a new analytic framework that accurately predicts the outcomes across the full spectrum of “merging”, “hit‑and‑run” (also called “skipping”), and catastrophic disruption.
The authors performed 96 SPH simulations varying three key parameters: the mass ratio of target to projectile (1, 3, 10, 30), the impact angle (30°, 45°, 60°), and the impact velocity normalized by the mutual escape velocity (v_imp/v_esc = 1.1–16). Each body was modeled with a 30 wt % iron core and a 70 wt % granite mantle using the Tillotson equation of state; 36 000 particles were employed (18 000 per body). The simulations were run on parallel supercomputers with the FDPS framework, and the post‑collision remnants were identified using a Friend‑of‑Friend algorithm at t = 2 × 10⁵ s.

Key diagnostics are the masses of the largest (M_lar) and second‑largest (M_s) remnants, expressed as (M_lar − M_tar)/M_pro and M_s/(M_tar + M_pro) respectively, plotted against the dimensionless impact energy E_imp/E_2B, where E_2B is the two‑body gravitational binding energy. The authors show that for a given mass ratio, low‑angle impacts (θ ≈ 30°) have larger geometric overlap, leading to stronger shock heating and earlier transition to erosion or vaporization, whereas high‑angle impacts retain more kinetic energy in the tangential direction and favor hit‑and‑run outcomes.

The novel analytic model is built on the concept of overlapping mass. The overlapping portions of the target and projectile (M_t,o and M_p,o) are calculated analytically from geometry (Appendix B). The kinetic energy deposited in these overlapping masses is E_over = γ′ E_imp, where γ′ depends on the fraction of mass that actually overlaps. Only the component of this energy normal to the contact surface contributes to shock heating, so the authors define a perpendicular overlapping energy E_over,⊥ = E_over cos²θ. The residual kinetic energy after shock dissipation is E_res = E_imp − E_over,⊥. If E_res ≲ E_2B the two bodies merge; if E_res ≫ E_2B they separate as a hit‑and‑run pair.

Mass loss is treated through two dimensionless energy ratios: (i) a normalized gravitational energy ˜E_g = E_g/E_over,⊥, where E_g is the self‑gravity binding energy of the relevant body (target, projectile, or merged mass), and (ii) a normalized vaporization energy ˜E_v = (M u_cv)/E_over,⊥, where u_cv is the specific energy required for complete vaporization of the mantle material. The authors adopt a simple functional form F_g(x) = 1 − (1 + x/0.15)⁻¹ to map ˜E_g onto the retained mass fraction. Consequently, the mass of the largest remnant is expressed as

• M_lar = (M_tar + M_pro) F_g(˜E_g,tot) for merging,
• M_lar = M_tar F_g(˜E_g,tar) + M_s,gain for hit‑and‑run,

where M_s,gain accounts for material transferred from the projectile to the target. A similar expression is given for the second‑largest remnant. The model reproduces the SPH results with high fidelity, outperforming earlier prescriptions such as those of Leinhardt & Stewart (2012) and Genda et al. (2017), especially near the transition regimes where shock heating and vaporization become important.

To make the model directly usable in planet‑formation N‑body codes, the authors derive angle‑averaged formulas. Assuming an isotropic distribution of impact parameters (probability ∝ sin 2θ), they integrate the outcome functions over θ, yielding closed‑form expressions for the mean largest and second‑largest remnant masses as functions of impact velocity and mass ratio alone.

The paper also discusses the broader implications. The angle‑averaged results indicate that hit‑and‑run collisions may account for roughly 30 % of all embryo‑embryo encounters, a higher fraction than previously estimated. This has consequences for the delivery of volatiles, the mixing of core‑mantle material, and the timing of giant impacts in terrestrial planet formation. Moreover, the same energy‑based framework successfully reproduces outcomes from laboratory dust‑aggregate collision experiments, suggesting that the model scales from sub‑kilometer aggregates up to planetary embryos.

In summary, the study provides a physically motivated, analytically tractable, and numerically validated description of collisional outcomes for Mars‑sized bodies. By explicitly incorporating overlapping geometry, perpendicular shock energy, and vaporization, it captures the subtle transitions between merging, hit‑and‑run, and disruption. The angle‑averaged formulas enable immediate implementation in large‑scale planet formation simulations, and the demonstrated scalability opens the door for unified modeling of collisional evolution across many orders of magnitude in size. Future work will extend the model to include rotation, heterogeneous compositions, and post‑impact thermochemical evolution, further bridging the gap between impact physics and the observable architecture of planetary systems.