Gravitational aggregation regimes: critical dissipation threshold, optimal rigidity and fractal transition

TL;DR

This work tackles gravity-driven aggregation of granular particles by integrating Newtonian self-gravity with a linear visco-elastic contact law, tracking orientation via quaternions and employing a nondimensionalization based on $k^*$ and $t_{ff}$. It performs systematic campaigns over the dissipation parameter $\gamma_n$ and the normalized stiffness $\widetilde{k}_n$, revealing three aggregation regimes, a stiffness optimum near $\widetilde{k}_n \approx 10^6$, and a transition from compact to fractal morphologies described by the fractal dimension $F$. The findings quantify how microscopic contact physics governs both aggregation kinetics and microstructure, offering a predictive framework for planetesimal formation and a calibration pathway for DEM models against microgravity experiments (e.g., MEDEA). The results bridge laboratory-scale contact laws and astrophysical aggregation, with implications for modeling rubble-pile bodies and validating DEM implementations across disciplines.

Abstract

I present a three-dimensional Discrete Element Method study of self-gravitation and contact mechanics in cold granular assemblies. The model couples direct Newtonian attraction between every particle pair with a linear visco-elastic normal contact law. Particles are treated as non-cohesive spheres; the normal force is parameterized to reproduce a prescribed restitution coefficient. Rotations are integrated using quaternions to avoid singularities. By normalizing the stiffness kn by kstar = G*m^2/R^3 and time by the free-fall time t_ff, I perform systematic parameter campaigns over dissipation (gamma) and normalized stiffness ktilde = kn/kstar. Results reveal three aggregation regimes. For low gamma the particles remain largely dispersive; above a critical gamma of about 5e2 aggregation accelerates until plateaus are reached in the aggregation time T_agg divided by t_ff. For stiffness ktilde on the order of 1e6 the aggregation time reaches a clear minimum. The cluster fraction C/Ntot shows a non-monotonic dependence on ktilde, with optimal cohesion at intermediate rigidity and peripheral isolation at extreme stiffness. Mapping the fractal dimension F across (gamma, ktilde) demonstrates transitions from compact structures (F about 3) to ramified structures (F below 2). These findings quantify how microscopic contact laws govern both the kinetics and microstructure of gravity-driven aggregation, providing a predictive framework for planetesimal formation and for calibrating DEM models against laboratory and micro-gravity experiments.

Figures (9)

• Figure 1: Initially loose sample of spherical particles designed to reproduce values produced by Weidling et al. weidling2012free
• Figure 2: An initially loose sample of 650 spherical particles distributed in a cubic domain of $\mathbb{R}^3$.
• Figure 3: Normalized aggregation time $\frac{T_{\rm aggr}}{t_{\rm ff}}$ as a function of damping coefficient $\gamma_n$ in simulations where aggregation is observed. Opaque circles: average achievements; transparent circles: individual raw samples.
• Figure 4: Index $C/N_{tot}$ as a function of the damping coefficient $\gamma_n$. On the left, the final state of a simulation of the case $\gamma_n = 10$ and on the right a simulation with $\gamma_n = 10^4$. Opaque circles: average achievements; transparent circles: individual raw samples.
• Figure 5: Median evolution of the $\widetilde{K}/\widetilde{P}$ energy ratio over time for several $\gamma_n$ values. The transparent points represent the raw data for all simulations.