Eccentric Disks With Self-Gravity

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 such a disk can be eccentric, but only if it has a sharply truncated edge: the surface density $\Sigma$ must vanish at the edge, and the $\Sigma$ 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 $\Sigma$ asymptotes to zero but never vanishes, as is often assumed for astrophysical disks, is insufficiently sharp. Similar results were shown by Hunter & Toomre (1969) 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.