Local and global dynamics of eccentric astrophysical discs

Abstract

We formulate a local dynamical model of an eccentric disc in which the dominant motion consists of elliptical Keplerian orbits. The model is a generalization of the well known shearing sheet, and is suitable for both analytical and computational studies of the local dynamics of eccentric discs. It is spatially homogeneous in the horizontal dimensions but has a time-dependent geometry that oscillates at the orbital frequency. We show how certain averages of the stress tensor in the local model determine the large-scale evolution of the shape and mass distribution of the disc. The simplest solutions of the local model are laminar flows consisting of a (generally nonlinear) vertical oscillation of the disc. Eccentric discs lack vertical hydrostatic equilibrium because of the variation of the vertical gravitational acceleration around the eccentric orbit, and in some cases because of the divergence of the orbital velocity field associated with an eccentricity gradient. We discuss the properties of the laminar solutions, showing that they can exhibit extreme compressional behaviour for eccentricities greater than about 0.5, especially in discs that behave isothermally. We also derive the linear evolutionary equations for an eccentric disc that follow from the laminar flows in the absence of a shear viscosity. In a companion paper we show that these solutions are linearly unstable and we determine the associated growth rates and unstable modes.

An affine model of the dynamics of astrophysical discs

“…This Schrödinger-like dispersive wave equation agrees exactly with the linear equation for the secular evolution of eccentricity in a 3D adiabatic disc, as found in equation (2) of Teyssandier & Ogilvie (2016) or equation (176) of Ogilvie & Barker (2014).…”

“…which gives a precise meaning to the scaleheight as the standard deviation of the density distribution. Simple examples of solutions of these equations (Ogilvie & Barker 2014) are the isothermal structure,…”

Magnetorotational Instability in Eccentric Disks

“…The second difficulty is with the simulation setup. One may think that the setting of localized perturbations in Chan et al (2018) lends itself naturally to shearing-box simulations (e.g., Ogilvie & Barker 2014;Wienkers & Ogilvie 2018). Drawing inspiration from circular disks, one may imagine shearing boxes in eccentric disks to have edges running along curves of constant semilatus rectum and constant azimuth.…”

Focusing of non-linear eccentric waves in astrophysical discs

“…where F (lin) is for a 3D adiabatic disc in linear theory, F (2D) and F (3D) are for the 2D and 3D adiabatic discs and F (iso) is for an isothermal disc. Here h(q, cos E) is the dimensionless scale height (equal to the scale height divided by the orbit averaged scale height); this varies around the orbit owing to the presence of the disc breathing mode which is forced by the variation of the vertical gravity around the orbit (Ogilvie (2001), Ogilvie (2008), Ogilvie & Barker (2014)). At lowest order there is no difference between the 3D and 2D isothermal discs, as the breathing mode is independent of q owing to the vertical specific enthalpy gradient being independent of horizontal compression.…”