Formation of Rubble-Pile Asteroids from the Collapse of a Rotating Granular Cloud

Introduction

Many asteroids, especially near-Earth asteroids, are believed to be gravitationally bound granular aggregates (“rubble piles”) rather than monolithic rocks. Radar observations and spacecraft missions reveal irregular shapes, low bulk densities and surface morphologies consistent with such aggregates. One possible formation pathway is the collapse and re-accumulation of debris following, perhaps, a disruptive event. This is analogous to gravitational collapse processes discussed in the context of planetesimal formation (Goldreich & Ward 1973). In this scenario, a rotating cloud of fragments interacts through self-gravity and dissipative inter-grain collisions, gradually forming a bound aggregate. The evolution depends upon the system’s angular momentum, initial energy, and the collisional properties of its constituent grains. We investigate the collapse of an initially spherical rotating granular cloud under the influence of self-gravity and collisional energy dissipation. Because the system is assumed to be isolated, its total angular momentum is conserved throughout the evolution.

The cloud is composed of spherical, inelastic grains and its constitutive response is modeled, on a continuum scale within the framework of kinetic theory, as a thermally conducting nonlinear complex fluid. The dynamical evolution is determined systematically using the method of moments. The system is parameterized by the initial angular momentum, rotational kinetic energy and density of the cloud, together with the diameter, material density and coefficient of restitution of the grains.

We first consider axisymmetric deformation of the cloud, charting the change in its shape, rotation state, rate of energy dissipation, and internal structure. We  then investigate the granular cloud's stability to non-axisymmetric perturbations at different stages of its evolution. In particular, we identify the parameter regimes that permit the emergence of truly triaxial ellipsoidal shapes.

Mathematical model

We consider an isolated agglomerate of identical spherical grains. The grains interact through binary inelastic collisions characterized by a normal coefficient of restitution. The system is assumed dilute, so the dynamics are governed by the inelastic Boltzmann equation for the one-particle distribution function.

Macroscopic fields are defined as velocity moments of this distribution function. These include the bulk density, the mean velocity field, and the granular temperature, which is the fluctuational kinetic energy of the grains relative to their mean motion.

Velocity moments of the Boltzmann equation yield the hydrodynamic equations for a self-gravitating granular gas. This  yields  balance laws for mass, linear momentum and fluctuational energy. Together they describe the evolution of the density, velocity and granular temperature fields under the influence of pressure gradients, gravitational forces, internal stresses, heat transport and collisional dissipation.

On scales large compared with the grain size, the aggregate is modeled as a continuum granular medium whose constitutive response follows from kinetic theory (Jenkins & Savage 1983). In this, the pressure is proportional to the product of density and granular temperature. The stress tensor depends on velocity gradients through a viscosity determined by the grain size and temperature, while the heat flux arises from gradients in both temperature and density. Energy is dissipated through inelastic collisions between grains. The gravitational potential of the cloud is determined self-consistently through Poisson’s equation which contains the bulk density of the granular medium.

Homogeneous axisymmetric collapse

We solve the above set of equations using the method of moments (Chandrasekhar 1969, Sharma 2017). We begin by restricting the cloud to deform homogeneously so that, at any given time, the cloud may take only an ellipsoidal shape. Under this assumption, the governing equations reduce to a set of ordinary differential equations describing the evolution of the strain-rate tensor, the spin tensor, and the inertia tensor of the cloud.

The strain-rate and spin tensors characterize the homogeneous deformation and rotation of the cloud, while the inertia tensor describes the evolving mass distribution associated with the ellipsoidal geometry. The equations also incorporate the average stress tensor obtained from the constitutive relations, the effects of self-gravity through the gravitational shape tensor determined by the ellipsoid’s semi-major axes, and the rotation of the principal axes of the body.

The resulting equations are readily solved once the initial conditions of a rotating spherical granular cloud are specified. Because the cloud is initially spherical, we expect the early stages of the evolution to remain axisymmetric. This symmetry allows the governing equations to be simplified and integrated efficiently.

Once the axisymmetric evolution is known, we examine the stability of the cloud to non-axisymmetric perturbations. In particular, we investigate whether small departures from axisymmetry can grow, leading to bifurcations toward prolate or fully triaxial configurations. The bifurcation points depend on the physical parameters of the system, including the angular momentum of the cloud, its density, and the collisional properties of its constituent grains. These results therefore connect the emergence of non-axisymmetric rubble asteroids with the initial conditions of the collapsing granular cloud.

The present framework provides a continuum description linking granular kinetic theory with the dynamical evolution of self-gravitating aggregates. It offers a new pathway for understanding the formation and internal structure of rubble-pile asteroids and motivates future work on rotational fission, binary formation, triaxial shape instabilities and anisotropic internal pre-stresses in rubble asteroids.

References

Goldreich, P., and Ward, W. R. (1973). The formation of planetesimals. Astrophysical Journal, 183, 1051–1061.

Jenkins, J. T., and Savage, S. B. (1983). A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. Journal of Fluid Mechanics, 130, 187–202.

Chandrasekhar, S. (1969). Ellipsoidal Figures of Equilibrium. Yale University Press.

Richardson, D. C., Walsh, K. J., Murdoch, N., and Michel, P. (2013). Numerical simulations of asteroids modelled as gravitational aggregates. Planetary and Space Science, 107, 3–15.

Sharma, I., Jenkins, J. T., and Burns, J. A. (2006). Tidal encounters of ellipsoidal granular asteroids with planets. Icarus, 183, 312–330.

Sharma, I., Jenkins, J. T., and Burns, J. A. (2009). Equilibrium configurations of rubble-pile asteroids. Icarus, 200, 304–322.

Sharma, I. (2017). Shapes and Dynamics of Granular Minor Planets. Springer.