Characterising the Gravitational Instability in Cooling Accretion Discs


First Author:
Peter Cossins
Email: peter.cossins AT astro.le.ac.uk
University of Leicester
University Road
Leicester. LE1 7RH United Kingdom
Coauthors:
Lodato, Giuseppe, University of Leicester
Clarke, Cathie, IoA, University of Cambridge

Abstract

We use 3D global numerical simulations to perform a systematic analysis of the structure induced by the onset of gravitational instabilities in cooling gaseous accretion discs. For low enough cooling rates, discs reach a quasi-steady configuration, with the instability saturating at a finite amplitude where the disc is close to marginal stability. We analyse the dependence of the saturation amplitude on the cooling rate, and we find that it scales with the inverse square root of the cooling parameter beta = tcool / tdyn. This indicates that the heating rate induced by the instability is a fixed fraction of the energy density of the waves excited by the disc self-gravity. In particular, we find that at saturation the energy dissipated per dynamical time by shocks due to the gravitational perturbations is approximately 20 per cent of the wave energy. We perform a Fourier analysis of the disc to determine the dominant radial and azimuthal wavenumbers. While the azimuthal wavenumber is almost constant with radius, we find that the disc displays a large number of radial modes with wavelengths increasing with increasing radius. The dominant modes closely match the locally most unstable wavelength predicted by linear perturbation analysis. As a consequence, we demonstrate that the density waves excited in relatively low mass discs Mdisc / M* ~ 0.1 are always close to corotation, deviating from it by no more than 10%. This has profound effects on how accurately the extraction of energy and angular momentum from the mean flow can be modelled viscously. Our results provide (a) a detailed description of how the self-regulation mechanism is established for low cooling rates, (b) clarification of the conditions required for describing self-gravity induced transport with an effective viscosity, (c) an estimate of the maximum amplitude of the density perturbation before fragmentation takes place, and (d) a simple means of estimating the density perturbation in different thermal regimes.
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