Stochastic tidal heating by random interactions with extended substructures

Abstract

Gravitating systems surrounded by a dynamic sea of substructures experience fluctuations of the local tidal field which inject kinetic energy into the internal motions. This paper uses stochastic calculus techniques to describe ‘tidal heating’ as a random walk of orbital velocities that leads to diffusion in a four-dimensional energy–angular momentum space. In spherical, static potentials, we derive analytical solutions for the Green’s propagators directly from the number density and velocity distribution of substructures with known mass and size functions without arbitrary cuts in forces or impact parameters. Furthermore, a Monte Carlo method is presented, which samples velocity ‘kicks’ from a probability function and can be used to model orbital scattering in fully generic potentials. For illustration, we follow the evolution of planetary orbits in a clumpy environment. We show that stochastic heating of (mass-less) discs in a Keplerian potential leads to the formation, and subsequent ‘evaporation’ of Oort-like clouds, and derive analytical expressions for the escape rate and the fraction of comets on retrograde orbits as a function of time. Extrapolation of the subhalo mass function of Milky Way-like haloes down to the WIMP free-streaming length suggests that objects in the outer Solar system experience repeated interactions with dark microhaloes on dynamical time-scales.