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Origins, Structure, and Inflows of m=1 Modes in Quasi-Keplerian Disks

Simulations show eccentric disks (m=1 modes) forming around quasi-Keplerian potentials, a topic of interest for fueling quasars, forming super-massive BHs, planet formation and migration, explaining the origin and properties of nuclear eccentric stellar disks like that in M31, and driving the formation of the obscuring AGN torus. We consider the global, linear normal m=1 modes in collisionless disks, without the restriction that the disk mass be negligible relative to the central (Keplerian) mass. We derive their structure and key resonance features, and show how they arise, propagate inwards, and drive both inflow/outflow and eccentricities in the disk. We compare with hydrodynamic simulations of such disks around a super-massive BH, with star formation, gas cooling, and feedback. We derive the dependence of the normal mode structure on disk structure, mass profiles, and thickness, and mode pattern speeds and growth rates. We show that, if the disk at some radii has mass of >~10% the central point mass, the modes are linearly unstable and are self-generating. They arise as 'fast modes' with pattern speed of order the local angular velocity at these radii. The characteristic global normal modes have pattern speeds comparable to the linear growth rate, of order (G*M_0*R_0^{-3})^{1/2}, where M_0 is the central mass and R_{0} is the radius where the enclosed disk mass ~M_{0}. They propagate inwards by exciting eccentricities towards smaller and smaller radii, until at small radii these are 'slow modes.' With moderate amplitude, the global normal modes can lead to shocks and significant gas inflows at near-Eddington rates at all radii inside several ~R_0.