Eccentric Disks with Self-gravity

Abstract Can a disk orbiting a central body be eccentric, when the disk feels its own self-gravity and is pressureless? Contradictory answers appear in the literature. We show that, to linear order in eccentricity, such a disk can be eccentric, but only if it has a sharply truncated edge: the surface density Σ must vanish at the edge, and the Σ profile must be sufficiently steep at the point where it vanishes. If either requirement is violated, an eccentric disturbance leaks out of the bulk of the disk, into the low-density edge region, and cannot return. An edge where Σ asymptotes to zero but never vanishes, as is often assumed for astrophysical disks, is insufficiently sharp. Similar results were shown by Hunter & Toomre for galactic warps. We demonstrate these results in three ways: by solving the eigenvalue equation for the eccentricity profile; by solving the initial value problem; and by analyzing a new and simple dispersion relation that is valid for any wavenumber, unlike WKB. As a byproduct, we show that softening the self-gravitational potential is not needed to model a flat disk, and we develop a softening-free algorithm to model the disk’s Laplace–Lagrange-like equations. The algorithm is easy to implement and is more accurate than softening-based methods at a given resolution by many orders of magnitude.