The statistical mechanics of self-gravitating Keplerian discs

Abstract

We describe the dynamics and thermodynamics of collisionless particle discs orbiting a massive central body, in the case where the disc mass is small compared to the central mass, the self-gravity of the disc dominates the non-Keplerian force, and the spread in semi-major axes is small. We show that with plausible approximations such discs have logarithmic two-body interactions and a compact phase space, and therefore exhibit thermodynamics that are simpler than most other gravitating systems, which require a confining box and artificial softening of the potential at small scales to be thermodynamically well-behaved. We solve for the microcanonical axisymmetric thermal equilibria and demonstrate the existence of a symmetry-breaking bifurcation into lopsided equilibria. We discuss the relation between thermal and dynamical instability in these systems and draw connections to astrophysical settings, as well as to the wider subject of the statistical mechanics of particles with logarithmic long-range interactions, such as point vortices in two-dimensional fluids. © 2014 IOP Publishing Ltd.