Numerical Simulations of Collisional Cascades at the Roche Limits of White Dwarf Stars

We consider the long-term collisional and dynamical evolution of solid material orbiting in a narrow annulus near the Roche limit of a white dwarf. With orbital velocities of 300 {km} {{{s}}}^-1, systems of solids with initial eccentricity e≳ {10}^-3 generate a collisional cascade where objects with radii r ≲ 100{--}300 {km} are ground to dust. This process converts 1-100 km asteroids into 1 μm particles in 10²-10⁶ yr. Throughout this evolution, the swarm maintains an initially large vertical scale height H. Adding solids at a rate \dot{M} enables the system to find an equilibrium where the mass in solids is roughly constant. This equilibrium depends on \dot{M} and {r}₀, the radius of the largest solid added to the swarm. When {r}₀ ≲ 10 km, this equilibrium is stable. For larger {r}₀, the mass oscillates between high and low states; the fraction of time spent in high states ranges from 100% for large \dot{M} to much less than 1% for small \dot{M}. During high states, the stellar luminosity reprocessed by the solids is comparable to the excess infrared emission observed in many metallic line white dwarfs.