Stochastic tidal heating by random interactions with extended
  substructures
  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 4-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 & 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.
