Eccentric Disks With Self-Gravity

TL;DR

This work addresses whether eccentric, self-gravitating, pressureless disks can sustain coherent eccentricity. It develops a master equation for the eccentricity profile in a continuum disk, then solves it numerically without smoothing to show that sustained eccentric modes require a sharply truncated edge, with $\Sigma(a)$ vanishing at the boundary and $\int^{a_{\rm edge}} \frac{da}{\Sigma(a)}$ converging. A simple dispersion relation $\omega = (\sigma/a) \mathcal{K}(k) + \nu(u)$ is derived, where $\mathcal{K}(k)$ is a universal function of wavenumber and $\nu(u)$ is a precession term, clarifying how edge sharpness controls trapping and precession; soft-edged disks fail to trap disturbances. The authors also present a softening-free, AMD-conserving discretization that achieves high accuracy ($\mathcal{O}(\Delta^2)$) and demonstrate, via eigenvalue analyses and time evolution, that sharp-edge edge-modes exist as robust, long-lived solutions. The results have broad implications for planetary rings, protoplanetary disks, and nuclear stellar disks in galactic nuclei, indicating that edge physics plays a decisive role in the persistence and structure of eccentric disks.

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 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 (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.

Figures (11)

• Figure 1: Kernels: The kernels are defined in Equation (\ref{['eq:km']}). They have a strong $1/v^2$ divergence at $|v|\ll 1$ (Equation (\ref{['eq:kdiv']})), and hence have been multiplied in the left panel by $\pi v^2$ to compensate. The blue dotted curves in the left panel are softened kernels 2003ApJ...595..531H with the same compensation by $\pi v^2$. The right panel shows the uncompensated kernels, with axes adjusted. The softened kernels have a rapid zero-crossing at $v\lesssim H$, and so numerical integrations that incorporate these softened kernels require extreme resolution.
• Figure 2: Comparison of Approximations to ${\cal I}$: The points connected by solid curves (orange and dark blue) show the error in the discretized approximation (Equation (\ref{['eq:idisc']})), versus grid spacing $\Delta$. The error is better than $\Delta^2$. The light blue circles (labelled "discrete, naive") repeat the dark blue circles, but with the factor of 1.5 in Equation (\ref{['eq:fjk']}) changed to unity. The light blue squares (labelled "soften") show the result with a softened kernel, integrated at infinite resolution ($\Delta\rightarrow 0$), and plotted versus $H$ rather than $\Delta$. This performs much worse than the dark blue circles. For a fairer comparison, the light blue squares should be discretized at $\Delta\ne 0$, and plotted versus $\Delta$; that would worsen the discrepancy with the dark blue circles.
• Figure 3: Disk Profiles and Eigenvectors
• Figure 4: Eigenvalues (mode frequencies) vs. grid spacing: For each profile ($\gamma=0.5$ and 1.5), the first 6 eigenvalues are shown. For the $\gamma=0.5$ case, the eigenvalues are well separated at high resolution. But for the $\gamma=1.5$ case, they become degenerate at high resolution.
• Figure 5: Time evolution: For the $\gamma=0.5$ case, the disturbance initially travels outward, and then bounces off of the outer edge. For $\gamma=1.5$, it never reaches the outer boundary, and instead becomes highly oscillatory.