Viscous propagation of mass flow variability in accretion discs

TL;DR

This paper models how viscosity dampens mass flow variability in accretion discs, deriving analytical solutions for infinite and finite discs, and applies findings to different astrophysical systems.

Contribution

It provides a detailed analytical description of viscous damping of mass flow variability in accretion discs, including Green's functions for both infinite and finite disc models.

Findings

Infinite disc solution overestimates damping compared to finite disc.

Power-law decay for infinite disc; exponential decay for finite disc.

Damping is stronger when viscous time exceeds variability timescale.

Abstract

We study mass flow rate through a disc resulting from a varying mass supply rate. Variable mass supply rate occurs, e.g., during disc state transitions, and in interacting eccentric binaries. It is, however, damped by the viscosity of the disc. Here, we calculate this damping in detail. We derive an analytical description of the propagation of the flow rate using the solution of Lynden-Bell & Pringle, in which the disc is assumed to extend to infinity. In particular, we derive the accretion-rate Green's function, and its Fourier transform, which gives the fractional damping at a given variability frequency. We then compare this model to that of a finite disc with the mass supply at its outer edge. We find significant differences with respect to the infinite disc solution, which we find to overestimate the viscous damping. In particular, the asymptotic form of the Green's function is power-law for the infinite disc and exponential for the finite one. We then find a simple fitting form for the latter, and also calculate its Fourier transform. In general, the damping becomes very strong when the viscous time at the outer edge of the disc becomes longer than the modulation time scale. We apply our results to a number of astrophysical systems. We find the effect is much stronger in low-mass X-ray binaries, where the disc size is comparable to that of the Roche lobe, than in high-mass binaries, where the wind-fed disc can have a much smaller size.