A dynamical systems approach to a thin accretion disc and its time-dependent behaviour on large length scales

TL;DR

This paper models thin accretion discs as dynamical systems, analyzing critical points and flow stability, revealing mechanisms for transonic flow selection and drawing parallels to acoustic black holes.

Contribution

It introduces a dynamical systems framework for analyzing accretion disc flow solutions, including the role of critical points and flow stability, with novel insights into transonic solution selection.

Findings

Inviscid disc has two critical points: saddle and centre.

Weakly viscous disc has four critical points, including saddle and spiral.

Temporal evolution can select transonic solutions in inviscid discs.

Abstract

Modelling the flow in a thin accretion disc like a dynamical system, we analyse the nature of the critical points of the steady solutions of the flow. For the simple inviscid disc there are two critical points, with the outer one being a saddle point and the inner one a centre type point. For the weakly viscous disc, there are four possible critical points, of which the outermost is a saddle point, while the next inner one is a spiral. Coupling the nature of the critical points with the outer boundary condition of the flow, gives us a complete understanding of all the important physical features of the flow solutions in the subsonic regions of the disc. In the inviscid disc, the physical realisability of the transonic solution passing through the saddle point is addressed by considering a temporal evolution of the flow, which is a very likely non-perturbative mechanism for selecting the transonic inflow solution from among a host of other possible stable solutions. For the weakly viscous disc, while a linearised time-dependent perturbation imposed on the steady mass inflow rate causes instability, the same perturbative analysis reveals that for the inviscid disc, there is a very close correspondence between the equation for the propagation of the perturbation and the metric of an acoustic black hole. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length scale is defined for the spatial extent of the inward rotational drift of matter.