The issue of fragmentation in self‐gravitating gaseous accretion discs has implications both for the formation of stars in discs in the nuclei of active galaxies, and for the formation of gaseous planets or brown dwarfs in circumstellar discs. It is now well established that fragmentation occurs if the disc is cooled on a time‐scale smaller than the local dynamical time‐scale, while for longer cooling times the disc reaches a quasi‐steady state in thermal equilibrium, with the cooling rate balanced by the heating due to gravitational stresses. We investigate here how the fragmentation boundary depends on the assumed equation of state. We find that the cooling time required for fragmentation increases as the specific heat ratio γ decreases, exceeding the local dynamical time‐scale for γ= 7/5. This result can be easily interpreted as a consequence of there being a maximum stress (in units of the local disc pressure) that can be sustained by a self‐gravitating disc in quasi‐equilibrium. Fragmentation occurs if the cooling time is such that the stress required to reach thermal equilibrium exceeds this value, independent of γ. This result suggests that a quasi‐steady, self‐gravitating disc can never produce a stress that results in the viscous α parameter exceeding ∼0.06.

We study the efficiency and dynamics of supermassive black hole binary mergers driven by angular momentum loss to small‐scale gas discs. Such binaries form after major galaxy mergers, but their fate is unclear since hardening through stellar scattering becomes very inefficient at subparsec distances. Gas discs may dominate binary dynamics on these scales, and promote mergers. Using numerical simulations, we investigate the evolution of the semimajor axis and eccentricity of binaries embedded within geometrically thin gas discs. Our simulations directly resolve angular momentum transport within the disc, which at the radii of interest is likely dominated by disc self‐gravity. We show that the binary decays at a rate which is in good agreement with analytical estimates, while the eccentricity grows. Saturation of eccentricity growth is not observed up to values e≳ 0.35. Accretion on to the black holes is variable, and is roughly modulated by the binary orbital frequency. Scaling our results, we analytically estimate the maximum rate of binary decay that is possible without fragmentation occurring within the surrounding gas disc, and compare that rate to an estimate of the stellar dynamical hardening rate. For binary masses in the range 105≲M≲ 108 M⊙ we find that decay due to gas discs may dominate for separations below a∼ 0.01–0.1 pc, in the regime where the disc is optically thick. The minimum merger time‐scale is shorter than the Hubble time for M≲ 107 M⊙. This implies that gas discs could commonly attend relatively low‐mass black hole mergers, and that a significant population of binaries might exist at separations of a few hundredths of a parsec, where the combined decay rate is slowest. For more massive binaries, where this mechanism fails to act quickly enough, we suggest that scattering of stars formed within a fragmenting gas disc could act as a significant additional sink of binary angular momentum.

We study the evolution of disk accretion during the merger of supermassive black hole binaries in galactic nuclei. In hierarchical galaxy formation models, the most common binaries are likely to arise from minor galactic mergers, and have unequal mass black holes. Once such a binary becomes embedded in an accretion disk at a separation a ∼ 0.1 pc, the merger proceeds in two distinct phases. During the first phase, the loss of orbital angular momentum to the gaseous disk shrinks the binary on a timescale of ∼ 10 7 yr. The accretion rate onto the primary black hole is not increased, and can be substantially reduced, during this disk-driven migration. At smaller separations, gravitational radiation becomes the dominant angular momentum loss process, and any gas trapped inside the orbit of the secondary is driven inwards by the inspiralling black hole. The implied accretion rate just prior to coalescence exceeds the Eddington limit, so the final merger is likely to occur within a common envelope formed from the disrupted inner disk, and be accompanied by high velocity (∼ 10 4 kms −1 ) outflows.

We study protoplanetary disc evolution assuming that angular momentum transport is driven by gravitational instability at large radii, and magnetohydrodynamic (MHD) turbulence in the hot inner regions. At radii of the order of 1 AU such discs develop a magnetically layered structure, with accretion occurring in an ionized surface layer overlying quiescent gas that is too cool to sustain MHD turbulence. We show that layered discs are subject to a limit cycle instability, in which accretion onto the protostar occurs in bursts with an accretion rate of 10^{-5} solar masses / yr, separated by quiescent intervals where the accretion rate is 10^{-8} solar masses / yr. Such bursts could lead to repeated episodes of strong mass outflow in Young Stellar Objects. The transition to this episodic mode of accretion occurs at an early epoch (t < 1 Myr), and the model therefore predicts that many young pre-main-sequence stars should have low rates of accretion through the inner disc. At ages of a few Myr, the discs are up to an order of magnitude more massive than the minimum mass solar nebula, with most of the mass locked up in the quiescent layer of the disc at around 1 AU. The predicted rate of low mass planetary migration is reduced at the outer edge of the layered disc, which could lead to an enhanced probability of giant planet formation at radii of 1-3 AU.Comment: MNRAS, in pres