Planetary chaotic zone clearing: destinations and timescales

Abstract

We investigate the orbital evolution of particles in a planet's chaotic zone to determine their final destinations and their timescales of clearing. There are four possible final states of chaotic particles: collision with the planet, collision with the star, escape, or bounded but non-collision orbits. In our investigations, within the framework of the planar circular restricted three body problem for planet-star mass ratio $\mu$ in the range $10^{-9}$ to $10^{-1.5}$, we find no particles hitting the star. The relative frequencies of escape and collision with the planet are not scale-free, as they depend upon the size of the planet. For planet radius $R_p\ge0.001R_H$ where $R_H$ is the planet's Hill radius, we find that most chaotic zone particles collide with the planet for $\mu\lesssim10^{-5}$; particle scattering to large distances is significant only for higher mass planets. For fixed ratio $R_p/R_H$, the particle clearing timescale, $T_{cl}$, has a broken power-law dependence on $\mu$. A shallower power-law, $T_{cl}\sim \mu^{-{1/3}}$, prevails at small $\mu$ where particles are cleared primarily by collisions with the planet; a steeper power law, $T_{cl}\sim\mu^{-{3/2}}$, prevails at larger $\mu$ where scattering dominates the particle loss. In the limit of vanishing planet radius, we find $T_{cl}\approx0.024\mu^{-{3\over2}}$. The interior and exterior boundaries of the annular zone in which chaotic particles are cleared are increasingly asymmetric about the planet's orbit for larger planet masses; the inner boundary coincides well with the classical first order resonance overlap zone, $\Delta a_{cl,int}\simeq1.2\mu^{0.28}a_p$; the outer boundary is better described by $\Delta a_{cl,ext}\simeq1.7\mu^{0.31}a_p$, where $a_p$ is the planet-star separation.