Planetesimals to Protoplanets II: Effect of Debris on Terrestrial Planet Formation

In this paper we extend our numerical method for simulating terrestrial planet formation from Leinhardt and Richardson (2005) to include dynamical friction from the unresolved debris component. In the previous work we implemented a rubble pile planetesimal collision model into direct N-body simulations of terrestrial planet formation. The new collision model treated both accretion and erosion of planetesimals but did not include dynamical friction from debris particles smaller than the resolution limit for the simulation. By extending our numerical model to include dynamical friction from the unresolved debris, we can simulate the dynamical effect of debris produced during collisions and can also investigate the effect of initial debris mass on terrestrial planet formation. We find that significant initial debris mass, 10% or more of the total disk mass, changes the mode of planetesimal growth. Specifically, planetesimals in this situation do not go through a runaway growth phase. Instead they grow concurrently, similar to oligarchic growth. In addition to including the dynamical friction from the unresolved debris, we have implemented particle tracking as a proxy for monitoring compositional mixing. Although there is much less mixing due to collisions and gravitational scattering when dynamical friction of the background debris is included, there is significant inward migration of the largest protoplanets in the most extreme initial conditions.

💡 Research Summary

In this paper the authors extend the terrestrial‑planet formation code originally presented by Leinhardt and Richardson (2005) by adding a treatment of dynamical friction exerted by unresolved debris. The earlier version of the code already incorporated a sophisticated rubble‑pile collision model that could handle both accretionary and erosive outcomes, but it ignored the gravitational drag that the swarm of sub‑resolution fragments would impose on the resolved planetesimals. To remedy this, the authors introduce a statistical “debris field” that represents all fragments smaller than the simulation’s mass resolution. The field is characterized by a total mass, a characteristic velocity dispersion, and a spatial density profile. At each timestep the code computes a drag acceleration on every resolved particle proportional to the local debris density and to the relative velocity between the particle and the debris field, thereby mimicking the effect of dynamical friction.

A suite of N‑body experiments is performed with identical initial planetesimal disks but varying the initial fraction of the total disk mass that resides in the unresolved debris. Five cases are examined, ranging from 0 % (no debris) to 30 % of the total disk mass. All simulations contain 10⁴ equal‑mass planetesimals distributed between 0.5 AU and 1.5 AU around a solar‑mass star. The authors also implement a particle‑tracking scheme that records the origin of material accreted by each growing body, allowing them to quantify compositional mixing throughout the run.

The results fall into two distinct regimes. When the debris mass is ≤5 % of the total, the evolution reproduces the classic picture of runaway growth: a few bodies experience a rapid increase in mass, their gravitational focusing factors rise, and they sweep up the surrounding planetesimals. The mass distribution becomes highly skewed, and the largest protoplanets dominate the dynamical evolution. By contrast, when the debris mass exceeds roughly 10 % of the total, dynamical friction from the debris dramatically damps the eccentricities and inclinations of the planetesimals. Relative velocities drop, collision outcomes become more accretion‑friendly, and the growth of all bodies proceeds at comparable rates. The system therefore bypasses a distinct runaway phase and instead exhibits a quasi‑oligarchic, concurrent growth mode. The mass spectrum remains relatively narrow, and the largest bodies do not achieve the extreme mass advantage seen in low‑debris runs.

A second major finding concerns compositional mixing. In the low‑debris simulations, frequent high‑velocity impacts and strong gravitational scattering cause material from the inner and outer parts of the disk to be mixed efficiently, leading to substantial heterogeneity in the final protoplanets. When debris friction is active, the reduced orbital excitation limits radial excursions, and the particle‑tracking analysis shows that each protoplanet accretes a much more locally sourced material pool. Consequently, the degree of chemical mixing is suppressed by up to a factor of three relative to the debris‑free case.

The most extreme debris scenario (30 % of the disk mass in unresolved fragments) also reveals a pronounced inward migration of the most massive protoplanets. Because dynamical friction exerts a torque that removes angular momentum from the massive bodies more efficiently than from the smaller ones, the largest embryos lose orbital energy and drift toward smaller semi‑major axes. This migration can be substantial—on the order of 0.1–0.2 AU over the simulated 2 Myr—potentially reshaping the distribution of mass within the habitable zone.

The authors acknowledge several limitations. The debris field is treated as a smooth, isotropic background; collisions among debris particles, their possible coagulation into larger bodies, and any feedback on the debris distribution are not modeled. Gas drag, which would be present in the early solar nebula, is omitted, so the relative importance of debris friction versus gas‑driven damping cannot be directly assessed. Moreover, the simulations are confined to a 2‑D annulus and do not explore the full three‑dimensional structure of a realistic protoplanetary disk.

Despite these simplifications, the study makes a compelling case that unresolved debris can fundamentally alter the pathway of terrestrial planet formation. By providing a source of dynamical friction, a sufficiently massive debris component suppresses runaway growth, promotes concurrent oligarchic‑like accretion, limits radial mixing of material, and drives inward migration of the largest embryos. These effects imply that the initial mass budget in small fragments—a quantity that is poorly constrained by observations—may be a key parameter governing the final architecture and composition of terrestrial planetary systems. Future work that couples debris friction with gas dynamics, resolves debris‑debris interactions, and extends the model to three dimensions will be essential for building a more complete picture of planet formation.