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

Authors
  • Sefilian, Antranik A.1
  • Rafikov, Roman R.1, 2
  • 1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
  • 2 Institute of Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
Type
Published Article
Journal
Monthly Notices of the Royal Astronomical Society
Publisher
Oxford University Press
Publication Date
Sep 02, 2019
Volume
489
Issue
3
Pages
4176–4195
Identifiers
DOI: 10.1093/mnras/stz2412
PMID: 33384461
PMCID: PMC7734392
Source
PubMed Central
Keywords
Disciplines
  • Article
License
Green

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, ς ≪ 1, accurate representation of eccentricity dynamics requires N ∼ C ς−χ ≫ 1, with C ∼ O (10), 1.5 ≲ χ ≳. In discs with sharp edges a very small value of the softening parameter ς (≲ 10−3) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.

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