Curvature and Acoustic Instabilities in Rotating Fluid Disks
Abstract
The stability of a rotating fluid disk to the formation of spiral arms is studied in the tightwinding approximation in the linear regime. The dispersion relation for spirals that was derived by Bertin et al. is shown to contain a new, acoustic instability beyond the Lindblad resonances that depends only on pressure and rotation. In this regime, pressure and gravity exchange roles as drivers and inhibitors of spiral wave structures. Other instabilities that are enhanced by pressure are also found in the general dispersion relation by including higher order terms in the small parameter 1/kr for wavenumber k and radius r. We identify two important dimensionless physical parameters: ε=2πGσ_{0}/(rκ^{2}), which is essentially the ratio of disk mass to total mass (disk and halo), and a/(κr), which is the ratio of epicyclic radius to disk radius (σ_{0} is the mass column density, κ is the epicyclic frequency, and a is the sound speed). The small term ζ=(k^{2}r^{2}+m^{2})^{1/2} is an additional parameter that is purely geometrical for number of arms m. When these terms are included in the dispersion relation, the oscillation frequency becomes complex, leading to the growth of perturbations even for large values of Toomre's parameter Q. The growth rate is proportional to a linear combination of terms that depend on ∊ and a/(κr). Instabilities that arise from ∊ are termed gravitationalcurvature instabilities because ∊ depends on the disk mass and is largest when the radius is small, i.e., when the orbital curvature is large. Instabilities that arise from a/(κr) are termed acousticcurvature instabilities because they arise from only the pressure terms at small r. Unstable growth rates are determined for these instabilities in four cases: a selfgravitating disk with a flat rotation curve, a selfgravitating disk with solid body rotation, a nonselfgravitating disk with solid body rotation, and a nonselfgravitating disk with Keplerian rotation. The most important application appears to be as a source of spiral structure, possibly leading to accretion in nonselfgravitating disks, such as some galactic nuclear disks, disks around black holes, and protoplanetary disks. All of these examples have short orbital times so the unstable growth time can be small, even when only terms of order ∊ contribute.
 Publication:

The Astrophysical Journal
 Pub Date:
 August 1999
 DOI:
 10.1086/307465
 arXiv:
 arXiv:astroph/9903413
 Bibcode:
 1999ApJ...520..592M
 Keywords:

 GALAXIES: KINEMATICS AND DYNAMICS;
 GALAXIES: SPIRAL;
 GALAXIES: STRUCTURE;
 HYDRODYNAMICS;
 Galaxies: Kinematics and Dynamics;
 Galaxies: Spiral;
 Galaxies: Structure;
 Hydrodynamics;
 Astrophysics
 EPrint:
 30 pages, 5 figures, scheduled for ApJ 520, August 1, 1999