The diffusion of stars through phase space

Deviations of the potentials of stellar systems from integrability cause stars to diffuse through three-dimensional orbit space. The Fokker–Planck equation that describes this diffusion takes a particularly simple form when actions are used as orbit-space coordinates. The rate of diffusion is governed by a vector $$\overline {\Delta}$$ and a tensor $$\overline {\Delta^2}$$, which according to the circumstances of a particular problem should be calculated either from kinetic theory or from Hamiltonian perturbation theory. In many astrophysically interesting circumstances $$\overline {\Delta}$$ is related to the divergence of the more readily calculated tensor $$\overline {\Delta^2}$$. In addition to being computationally handy, this relationship ensures that the orbital diffusion described by the Fokker–Planck equation causes the system's entropy, derived from any H-function, to increase whenever the system is interacting with a hotter system of scatterers. An investigation of the heating of stellar discs in the light of these general results yields the following conclusions: (i) a population of stars with an initially Maxwellian peculiar-velocity distribution will remain Maxwellian as it diffuses through orbit space only if $$\overline {\Delta^2}$$ is proportional to epicycle energy and the population's velocity dispersion grows as √t; (ii) the self-similar distribution functions that are the end-points in two and three dimensions of the star-cloud scattering process proposed by Spitzer and Schwarzschild, predict neither Max-wellian velocity distributions nor σ ∝ √t; (iii) scattering by ephemeral spiral waves can account for the observed kinematics of the solar neighbourhood only if the waves have wavelengths in excess of 9 kpc and constantly drifting pattern speeds. However, even such frequency-modulated, global spirals cannot maintain σ ∝ √t above σ_R 40 km s⁻¹, and they can account for the increase in the vertical dispersion with time only with the assistance of clouds. Only scatterers that are not confined to the Galactic disc are capable of simultaneously increasing the vertical and radial dispersions as √t to dispersions in excess of 40 km s⁻¹.