Potential softening and eccentricity dynamics in razor-thin, nearly-Keplerian discs

Abstract

In many astrophysical problems involving discs (gaseous or particulate) orbiting a dominant central mass, gravitational potential of the disc plays an important dynamical role. Its impact on the motion of external objects, as well as on the dynamics of the disc itself, can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in calculation of the orbit-averaged potential --- disturbing function --- of a razor-thin disc using classical Laplace-Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the correct eccentricity dynamics in the disc potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behavior exactly, approximately or not converging at all. We also develop a general framework for computing secular disturbing function given an arbitrary softening prescription for a rather general form of the interaction potential. Our results demonstrate that numerical treatments of the secular disc dynamics, representing the disc as a collection of $N$ gravitationally interacting annuli, are rather demanding: for a given value of the (dimensionless) softening parameter, $\varsigma \ll 1$, accurate representation of eccentricity dynamics requires $N\sim C{\varsigma }^{-\chi }\gg 1$, with $C\sim O(10)$, $1.5\lesssim \chi \lesssim 2$. In discs with sharp edges a very small value of the softening parameter $\varsigma$ ($\lesssim 10^{-3}$) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.

arXiv commentFinal version accepted for publication in MNRAS: 23 Pages, 9 Figures, 2 Tables

References in corpus (21)

• Resonant relaxation and the warp of the stellar disc in the Galactic centre
• Gauss's method for secular dynamics, softened
• Growth of eccentric modes in disc-planet interactions
• Orbital evolution of a planet on an inclined orbit interacting with a disc
• Slow modes in Keplerian disks
• Planet formation in binaries: dynamics of planetesimals perturbed by the eccentric protoplanetary disk and the secondary
• The Secular Evolution of the Primordial Kuiper Belt
• Apse Alignment of Narrow Eccentric Planetary Rings
• On the mass and origin of Chariklo's rings
• A Simple Family of Models for Eccentric Keplerian Fluid Disks
• Resonant relaxation in protoplanetary disks
• Shepherding in a Self-Gravitating Disk of Trans-Neptunian Objects
• Planetary Rings
• Circumbinary Planet Formation in the Kepler-16 System. II. A Toy Model for In-situ Planet Formation within a Debris Belt
• Long-lived lop-sided modes of annular disks orbiting a central mass
• Secular transport during disk dispersal: the case of Kepler-419
• Eccentric Modes in Disks With Pressure and Self-Gravity
• Counter-rotating stellar discs around a massive black hole: self-consistent, time-dependent dynamics
• The Doubling of Stellar Black Hole Nuclei
• Secular Evolution Driven by Massive Eccentric Disks/Rings: An Apsidally Aligned Case
• Unstable $m=1$ modes of counter-rotating Keplerian discs

Cited by in corpus (5)

• Formation of Gaps in Self-Gravitating Debris Disks by Secular Resonance in a Single-planet System I: A Simplified Model
• A Simplified Model for the Secular Dynamics of Eccentric Discs and Applications to Planet-Disc Interactions
• The vertical structure of debris discs and the role of disc gravity: A primer using a simplified model
• Self-gravity of debris discs can strongly change the outcomes of interactions with inclined planets
• Resonant Dynamical Friction Around a Super-Massive Black Hole: Analytical Description