# Stochastic tidal heating by random interactions with extended substructures

## TL;DR

This paper models how random gravitational interactions with substructures cause stochastic tidal heating, leading to orbital diffusion and potential ejection of objects, with analytical and Monte Carlo methods applied to planetary systems.

## Contribution

It introduces a stochastic calculus framework for tidal heating, deriving analytical solutions and a Monte Carlo method for modeling orbital scattering in complex potentials.

## Key findings

- Stochastic heating causes orbital diffusion and evaporation of planetary bodies.
- Analytical expressions for escape rates and orbital distributions are derived.
- Dark microhaloes may influence Solar system objects through repeated interactions.

## 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 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.