Nonlinear Outcome of Gravitational Instability in Disks with Realistic Cooling

By B. M. Johnson, Charles F. Gammie. 2003

151 citations recorded by the source corpus.

We consider the nonlinear outcome of gravitational instability in optically thick disks with a realistic cooling function. We use a numerical model that is local, razor thin, and unmagnetized. External illumination is ignored. Cooling is calculated from a one-zone model using analytic fits to low-temperature Rosseland mean opacities. The model has two parameters: the initial surface density Σ0 and the rotation frequency Ω. We survey the parameter space and find the following. (1) The disk fragments when ⟨⟨τc⟩⟩Ω ~ 1, where ⟨⟨τc⟩⟩ is an effective cooling time defined as the average internal energy of the model divided by the average cooling rate. This is consistent with earlier results that used a simplified cooling function. (2) The initial cooling time τc0 for a uniform disk with Q = 1 can differ by orders of magnitude from ⟨⟨τc⟩⟩ in the nonlinear outcome. The difference is caused by sharp variations in the opacity with temperature. The condition τc0Ω ~ 1 therefore does not necessarily indicate where fragmentation will occur. (3) The largest difference between ⟨⟨τc⟩⟩ and τc0 is near the opacity gap, where dust is absent and hydrogen is largely molecular. (4) In the limit of strong illumination the disk is isothermal; we find that an isothermal version of our model fragments for Q ≲ 1.4. Finally, we discuss some physical processes not included in our model and find that most are likely to make disks more susceptible to fragmentation. We conclude that disks with ⟨⟨τc⟩⟩Ω ≲ 1 do not exist.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/f6bb29351c26ec2aa48f563aaaf07ff33e88ce58

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Citations in the corpus

Dynamics of Protoplanetary Disks (2010, methodology): “Using a more realistic model, in which the cooling was calculated using a one-zone vertical model that took account of the optical depth, Johnson & Gammie (2003) found that strong variations in the opacity with temperature could result in large changes to the critical cooling time scale for…”
X-Ray Properties of Black-Hole Binaries (2006, background): “They will soon include dissipative processes (i.e., radiation; see Johnson & Gammie 2003), and it is hoped that they will then connect more directly with observation.”
Massive black hole binary mergers within subparsec scale gas discs (2008, methodology): “Moreover, the critical value of α can be much larger in this regime (e.g., Johnson & Gammie, 2003), rendering this analysis invalid.”
Investigating fragmentation conditions in self-gravitating accretion discs (2005, background): “This may be possible in AGN discs (Johnson & Gammie 2003), but appears unlikely in protostellar discs (Rafikov 2005).”
VARIABLE PROTOSTELLAR ACCRETION WITH EPISODIC BURSTS (2015, methodology): “…equation (3) is determined using the diffusion approximation of the vertical radiation transport in a one-zone model of the vertical disk structure (Johnson & Gammie 2003) Λ = Fcσ T 4mp τ 1 + τ2 , (4) where τ is the optical depth to the disk midplane, σ is the Stefan-Boltzmann constant, Tmp =…”
Accelerated planetesimal growth in self-gravitating protoplanetary discs (2004, background): “This again requires that the gas be cold and that it remain ‘almost isothermal’ (Pickett et al. 1998, 2000) or, equivalently, that any cooling mechanism is extremely efficient (Gammie 2001; Rice et al. 2003a; Johnson & Gammie 2003).”
Theory of Star Formation (2007, background): “For disks that are not externally illuminated, Johnson & Gammie (2003) performed two-dimensional simulations with realistic opacities (and a “one-zone” vertical radiative transfer approximation for cooling), and found that the transition between fragmentation and non-fragmentation lies in the range…”
THE BURST MODE OF ACCRETION AND DISK FRAGMENTATION IN THE EARLY EMBEDDED STAGES OF STAR FORMATION (2010, methodology): “We then make use of the diffusion approximation and link the effective surface temperature Teff with the midplane temperature of gas Tmp via the following relation T 4eff = 8T 4 mp/(3τ), where τ is the optical depth (Hubeny 1990; Johnson & Gammie 2003).”
Testing the locality of transport in self-gravitating accretion discs — II. The massive disc case (2003, methodology): “…0.25M , and to explore the results obtained with different initial conditions and cooling time-scale, which in the present work has been taken to be t cool = 7.5 −1), and toward a more realistic description of the cooling function; see, for example, Nelson et al. (2000) and Johnson & Gammie (2003).”
Pseudo-viscous modelling of self-gravitating discs and the formation of low mass ratio binaries (2009, background): “More serious is the argument of Johnson and Gammie (2003) that a given α (or cooling timescale) threshold is unreliable in the case that the cooling timescale is a strong function of temperature: they showed in the regime where dust sublimes (and the opacity is a very steeply decreasing function of…”
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 .”
Gravitational Instabilities in Circumstellar Disks (2016, background): “Studies have 16 Kratter & Lodato examined the role of the temperature dependence of cooling (Johnson & Gammie 2003; Cossins et al. 2010a), the role of the evolution of β in time at fixed radius (Clarke et al. 2007), and the role of irradiation (Boley et al. 2007; Stamatellos & Whitworth 2008; Rice…”
Clumps in the outer disk by disk instability: Why they are initially gas giants and the legacy of disruption (2009, background): “4 is required for an isothermal disk to fragment (e.g., Tomley et al., 1994; Nelson et al., 1998; Johnson and Gammie, 2003; Mayer et al., 2004; Durisen et al., 2007).”
The Evolution of Gravitationally Unstable Protoplanetary Disks: Fragmentation and Possible Giant Planet Formation (2003, background): “Recent local grid-based calculations of razor-thin self-gravitating disks also find that the disk fragments into clumps forQ 1:4 (Johnson & Gammie 2003).”
RESOLVING THE LUMINOSITY PROBLEM IN LOW-MASS STAR FORMATION (2011, methodology): “The latter is calculated using the diffusion approximation of the vertical radiation transport in a one-zone model of the vertical disk structure (Johnson & Gammie 2003): Λ = Fcσ T 4 τ 1 + τ2 , (4) where σ is the Stefan-Boltzmann constant, T is the midplane temperature of gas, and Fc = 2+20…”
Characterizing the gravitational instability in cooling accretion discs (2008, background): “…the thermodynamics of accretiondiscs is a complex undertaking and as such has received much attention, from the opacity-based treatment employed by Johnson & Gammie (2003) through to the various convective and radiative transfer models of Boss (2004), Boley et al. (2007), Mayer et al. (2007),…”