Characterizing the gravitational instability in cooling accretion discs

In this paper we perform a systematic analysis of the structure induced by the onset of gravitational instabilities in cooling gaseous accretion discs. It is well known that for low enough cooling rates the disc reaches a quasi-steady configuration, with the instability saturating at a finite amplitude such that the disc is kept close to marginal stability. We analyse the dependence of the saturation amplitude on the imposed cooling rate, and we find that it scales with the inverse square root of the cooling parameter beta = t(cool)/t(dyn). This indicates that the heating rate induced by the instability is proportional to the energy density of the density waves excited by the disc self-gravity. In particular, we find that at saturation the energy dissipated per dynamical time by weak shocks due to the gravitational perturbations is of the order of 20 per cent of the wave energy. We further perform a Fourier analysis of the disc structure, and subsequently determine the dominant radial and azimuthal wavenumbers of the density waves. While the number of spiral arms (corresponding to the azimuthal wavenumber) is fairly constant with radius, we find that the disc displays a range of radial wavenumbers whose mean increases with increasing radius. The dominant modes closely match the locally most unstable wavelength as predicted by linear perturbation analysis. As a consequence, we numerically demonstrate that the density waves excited in relatively low-mass discs M(disc)/M(star) similar to 0.1 are always close to corotation, deviating from it by approximately 10 per cent. This result can be understood in terms of the constancy of the Doppler-shifted phase Mach number of the flow-the pattern speed self-adjusts so that the flow into spiral arms is always sonic. This has profound effects on the degree to which the extraction of energy and angular momentum from the mean flow through density waves can be modelled as a viscous process. Our results thus provide (i) a detailed description of how the self-regulation mechanism is established for low cooling rates, (ii) a clarification of the conditions required for describing the transport induced by self-gravity through an effective viscosity, (iii) an estimate of the maximum amplitude of the density perturbation before fragmentation takes place and finally (iv) a simple recipe to estimate the density perturbation in different thermal regimes.