Eccentric Behavior of Disk Galaxies

A theory is developed for the dynamics of eccentric perturbations $[\propto\exp(\pm i\phi)]$ of a disk galaxy residing in a spherical dark matter halo andincluding a spherical bulge component. The disk is represented as a largenumber $N$ of rings with shifted centers and with perturbed azimuthal matterdistributions. Account is taken of the dynamics of the shift of the matter atthe galaxy's center which may include a massive black hole. The gravitationalinteractions between the rings and between the rings and the center is fullyaccounted for, but the halo and bulge components are treated as passivegravitational field sources. Equations of motion and a Lagrangian are derivedfor the ring + center system, and these lead to total energy and total angularmomentum constants of the motion. We study the eccentric dynamics of a diskwith an exponential surface density distribution represented by a large numberof rings. The inner part of the disk is found to be strongly unstable. Angularmomentum of the rings is transferred outward and to the central mass ifpresent, and a trailing one-armed spiral wave is formed in the disk. We alsoanalyze a disk with a modified exponential density distribution where thedensity of the inner part of the disk is reduced. In this case we find muchslower, linear growth of the eccentric motion. A trailing one-armed spiral waveforms in the disk and becomes more tightly wrapped as time increases. Themotion of the central mass if present is small compared with that of the disk.Comment: 18 pages, 14 figures, ApJ in pres