A Survey of the Principal Modes of Nonaxisymmetric Instability in Self‐Gravitating Accretion Disk Models

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Abstract
We present a survey of the three principal nonaxisymmetric modes of instability (the so-called P-, I-, and J-modes) that have been found in rotating, self-gravitating, accretion disk models. The relevant equations that describe the equilibrium structure and the linear stability properties are formulated quite simply for what has turned out to be the "standard" idealized model of accretion tori orbiting a central star, the so-called slender incompressible torus with constant specific angular momentum. In this sense, our survey is a natural extension of the work done by Goodman & Narayan. We map out the three unstable nonaxisymmetric modes in this model, and we illustrate the regions of instability in various parameter spaces, the most important of which is the (MD/MC, T/|W|)-plane, where MD/MC is the torus/central star mass ratio and T/|W| is the ratio of the rotational kinetic energy of the torus to the total gravitational potential energy of the system. We place particular emphasis on this parameter space because the physical significance of the two ratios is easily understood and because of its potential importance for observers, having in mind protostellar star/disk systems for which the mass ratio of the two components can currently be determined and for which the ratio T/|W| may eventually be estimated from observations. We compare our solutions of the linearized stability equations with the results previously obtained from linear analyses of (in)compressible two-dimensional annular models and from nonlinear hydrodynamical simulations of compressible three-dimensional toroidal models. The I- and J-modes are primarily driven by self-gravity, so they appear in all models with nonzero torus mass that are sufficiently slender; the J-modes dominate over the I-modes only in the most slender of the unstable models; and the corresponding regions of instability are not expected to be strongly dependent on the assumption of incompressibility or on the particular choice of the angular momentum profile. The P-modes appear only at extremely low mass ratios (MD/MC ≤ 1.15 × 10-3) and only in extremely slender tori (aspect ratios ≤ 0.02733). Thus, they probably play no role in the dynamical evolution of realistic disk systems such as those around protostars and compact objects in active galactic nuclei—and the I-modes emerge as the most important and dangerous modes of instability for all self-gravitating accretion disks with small or moderate aspect ratios.