Nonaxisymmetric Instabilities in Self-Gravitating Disks. II Linear and   Quasi-Linear Analyses
  We studied global nonaxisymmetric hydrodynamic instabilities in an extensive collection of hot, self-gravitating polytropic disk systems, systems that covered a wide expanse of the parameter space relevant to protostellar and protoplanetary systems. We examined equilibrium disk models varying three parameters: the ratio of the inner to outer equatorial radii, the ratio of star mass to disk mass, and the rotation law exponent $q$. We took the polytropic index $n$ = 1.5 and examined the exponents $q =$ 1.5 and 2, and the transitional one $q$ = 1.75. For each of these sets of parameters, we examined models with inner to outer radius ratios from 0.1 to 0.75, and star mass to disk mass ratios from 0 to 10$^3$. We numerically calculated the growth rates and oscillation frequencies of low-order nonaxisymmetric disk modes, modes with azimuthal dependence $\propto$ e$^{im\phi}$. Low-$m$ modes are found to dominate with the character and strength of instability strongly dependent on disk self-gravity. Representatives of each mode type are examined in detail, and torques and mass transport rates are calculated. 
Kathryn Z. Hadley, Paul Fernandez, James N. Imamura, Erik Keever,   Rebecka Tumblin, and William Dumas
astro-ph.SR