Steady State of a Planet-Scattering Debris Disk

The dynamical evolution of massless objects under the gravitational scattering of a Sun-orbiting planet is called the scattering problem. Yabushita (1980) modeled the planetary scattering of comets as a random walk in energy space, whose steady-state solution approaches a dN/da∝a^-1.5 power law distribution. In this work, I try to tackle this problem from the perspective of flyby dynamics. I show that in a patched-conic model, the a^-1.5 power law is a natural outcome after multiple flybys of the perturbing planet homogenize the directions of planetary relative-velocity vectors (v infinity). The scattering problem can thus be modeled as a random walk in θ, the angle between the planetary velocity vector and v infinity. As opposed to estimating the diffusion coefficient numerically as in Duncan et al. (1987), this model allows one to analytically compute the diffusion coefficient in θ in closed form. The scattering timescale (which is the relaxation timescale for the diffusion problem) is found to be dependent on the planetary mass and orbital period, as well as the Tisserand parameter of small bodies. By carrying out numerical simulations, I confirm the a^-1.5 power law is a universal steady state regardless of initial conditions, and validate the analytically-computed timescale. This patched-conic approach (which applies to closely-coupled cases) is complementary to the diffusion approach (which best works for non-crossing orbits). Together, these viewpoints can be applied to a wide variety of dynamical problems, including the formation of the Solar System’s scattering disk, the evolution of highly-inclined/retrograde asteroids, and the dynamical interaction between exoplanets and debris disks.