Characterizing the gravitational instability in cooling accretion discs

By P. Cossins, G. Lodato, C. Clarke. 2008

158 citations recorded by the source corpus.

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 β = 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 * ∼ 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.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/6164423608a84f3d5cd4194ad3a6e7b32b4eaff7

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Citations in the corpus

Pseudo-viscous modelling of self-gravitating discs and the formation of low mass ratio binaries (2009, background): “Simulations corroborate this tendency, in that deviations from local behaviour are more pronounced for more massive discs (Lodato & Rice 2005, Cossins et al 2009).”
Gravitational Instabilities in Circumstellar Disks (2016, methodology): “30) requires that the instability saturates at an amplitude that scales with the square root of the cooling time: δΣ Σ ∝ 1√ β , (31) a behavior confirmed by the numerical simulations of Cossins et al. (2009).”
Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics (2017, background): “…simulations of star formation (e.g. Price & Bate, 2007, 2009; Price et al., 2012; Lewis et al., 2015) and accretion discs (e.g. Lodato & Rice, 2004; Cossins et al., 2009), contains a rich array of physics necessary for star and planet formation including all of the above algorithms, but the legacy…”
An analytic model of angular momentum transport by gravitational torques: from galaxies to massive black holes (2010, background): “It is well-known, for example, that mode development depends significantly on gas thermodynamics in pure gas disks (e.g. Gammie 2001; Nayakshin et al. 2007; Cossins et al. 2009).”
THE RUNTS OF THE LITTER: WHY PLANETS FORMED THROUGH GRAVITATIONAL INSTABILITY CAN ONLY BE FAILED BINARY STARS (2009, background): “Cossins et al. (2009) have shown that even when the GI is non-axisymmetric, the most unstable axisymmetric wavelength λQ is one of the dominant growing modes, suggesting that this is a reasonable estimate.”
Rapid inward migration of planets formed by gravitational instability (2011, background): “Providing angular momentum transport driven by gravitoturbulence occurs locally, which is not necessarily the case (Balbus & Papaloizou 1999; Cossins et al. 2009), the alpha parameter associated with the angular momentum flux density is given by (Gammie 2001) α = 4 9 1 γ(γ − 1)β , (2) which…”
Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits (2010, background): “Possible solutions to reduce the drift rate include bumps in the radial pressure profile (see Kretke & Lin 2007; Brauer et al. 2008b; Cossins et al. 2009) or zonal flows (see Johansen et al. 2009).”
CHALLENGES IN FORMING PLANETS BY GRAVITATIONAL INSTABILITY: DISK IRRADIATION AND CLUMP MIGRATION, ACCRETION, AND TIDAL DESTRUCTION (2011, background): “On the other hand, Cossins et al. (2009) and Forgan et al. (2009) pointed out that with higher and higher disk masses, the assumption of local energy deposition gradually breaks down since lower mode spiral arms become strong.”