Investigating fragmentation conditions in self-gravitating accretion discs

By W. Rice, G. Lodato, Philip Armitage. 2005

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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.

Subfields: Astronomy and Astrophysics

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CHALLENGES IN FORMING PLANETS BY GRAVITATIONAL INSTABILITY: DISK IRRADIATION AND CLUMP MIGRATION, ACCRETION, AND TIDAL DESTRUCTION (2011, background): “This is consistent with Rice et al. (2005).”
Characterizing the gravitational instability in cooling accretion discs (2008, background): “Fragmentation into bound objects occurs when the cooling dominates, whereas the disc can persist in a quasi-equilibrium marginally sta- ble condition if the cooling rate is sufficiently low (Gammie 2001; Rice et al. 2005; Lodato & Rice 2004).”
THE RUNTS OF THE LITTER: WHY PLANETS FORMED THROUGH GRAVITATIONAL INSTABILITY CAN ONLY BE FAILED BINARY STARS (2009, background): “While Rice et al. (2005) find that an α-threshold is more robust than one for tcool when the adiabatic index varies, they did not include irradiation, which we contend would reveal that cooling is ultimately the more physical criterion, but the issue is best settled by simulation.”
Pseudo-viscous modelling of self-gravitating discs and the formation of low mass ratio binaries (2009, background): “Similar delineation of the fragmentation boundary in terms of the ratio of cooling to dynamical timescale has been found in a range of other (global) simulations (e.g. Rice et al 2005, Clarke et al 2007), though the exact location of this boundary depends on the dimensionality of the simulations.”
THE TWO MODES OF GAS GIANT PLANET FORMATION (2009, methodology): “…(GIs) to form gas giants directly, the Toomre (1964) Q = csκ/πGΣ, where cs is the sound speed, κ is the epicyclic frequency, and Σ is the surface density, must approach unity and the local cooling time must be . the local orbit period (Gammie 2001; Rice et al. 2005; Rafikov 2005, 2007).”
ON THE ROLE OF DISKS IN THE FORMATION OF STELLAR SYSTEMS: A NUMERICAL PARAMETER STUDY OF RAPID ACCRETION (2009, background): “…at various stages of evolution, however most involve isolated disks, without infall at large radii (Laughlin & Bodenheimer 1994; Laughlin & Rozyczka 1996; Rice et al. 2005; Lodato & Rice 2005; Fromang & Nelson 2006; Shetty & Ostriker 2006; Boley et al. 2006; Lodato et al. 2007; Cai et al. 2008).”
Gravitational Instabilities in Circumstellar Disks (2016, background): “For low mass disks, www.annualreviews.org • GI DISKS 17 where GI-induced transport is local, Rice et al. (2005) have shown that the existence of a critical cooling rate for fragmentation translates, through Equation (30), into a more general upper limit of α ≈ 0.1.”
Rapid inward migration of planets formed by gravitational instability (2011, background): “1, depends on the disc’s adiabatic index (Rice et al. 2005), and on the local disc to primary mass ratio (Meru & Bate 2011b).”
How do massive black holes get their gas (2009, background): “Note that there are conditions under which even an extremely cold gaseous disc could avoid catastrophic fragmentation (see e.g. Lodato & Rice 2004; Rice et al. 2005; Boley Figure B2.”
FORMATION OF THE FIRST NUCLEAR CLUSTERS AND MASSIVE BLACK HOLES AT HIGH REDSHIFT (2008, background): “Rice et al. (2005) proposed that fragmentation in a thin disk sets in when the gravitationally induced stress exceeds a critical value.”
The properties of brown dwarfs and low-mass hydrogen-burning stars formed by disc fragmentation (2008, background): “Both theory and simulations (Gammie 2001; Johnson & Gammie 2003; Rice et al. 2003, 2005; Mayer et al. 2004; Mejia et al. 2005; Stamatellos et al. 2007b) indicate that for disc fragmentation to occur, the cooling time must satisfy t COOL < C(γ) t ORB , 0.5 .”
The impact of magnetic fields on single and binary star formation (2007, background): “First, if the gravitational instability is strong enough, the disc may fragment to form a companion (e.g. Bonnell 1994; Bonnell & Bate 1994a,b; Whitworth et al. 1995; Rice et al. 2005).”
Evolution of binary black holes in self gravitating discs. Dissecting the torques (2012, background): “To prevent it from fragmenting, we force the gas to cool slowly, setting β = tcool/tdyn = 10 (Gammie 2001; Rice et al. 2005).”
A dual-frequency sub-arcsecond study of proto-planetary disks at mm wavelengths: first evidence for radial variations of the dust properties (2011, background): “Competing models are the coreaccretion mechanism (e.g. Hubickyj et al. 2005), which faces apparent timescale difficulties, and the gravitational instability (e.g. Boss 1997; Rice et al. 2005), which requires massive disks.”