Eccentric Behavior of Disk Galaxies Authors: Lovelace R. V. E.; Zhang L.; Kornreich D. A.; Haynes M. P. A theory is developed for the dynamics of eccentric perturbations (~exp+/i&phis;) of a disk galaxy residing in a spherical dark matter halo and including a spherical bulge component. The disk is represented as a large number N of rings with shifted centers and with perturbed azimuthal matter distributions. Account is taken of the dynamics of the shift of the matter at the galaxy's center which may include a massive black hole. The gravitational interactions between the rings and between the rings and the center is fully accounted for, but the halo and bulge components are treated as passive gravitational field sources. Equations of motion and a Lagrangian are derived for the ring+center system, and these lead to total energy and total angular momentum constants of the motion.We first study the eccentric motion of a disk consisting of two rings of different radii but equal mass, M_d/2. For small M_d the two rings are stable, but for M_d larger than a threshold value the rings are unstable with a dynamical timescale growth. For M_d sufficiently above this threshold, the instability acts to decrease the angular momentum of the inner ring, while increasing that of the outer ring. The instability results from the merging positive and negative energy modes with increasing M_d. Second, we analyze the eccentric motion of one ring interacting with a radially shifted central mass. In this case instability sets in above a threshold value of the central mass (for a fixed ring mass), and it acts to increase the angular momentum of the central mass (which therefore rotates in the direction of the disk matter), while decreasing the angular momentum of the ring. Third, we study the eccentric dynamics of a disk with an exponential surface density distribution represented by a large number of rings. The inner part of the disk is found to be strongly unstable. Angular momentum of the rings is transferred outward and to the central mass if present, and a trailing onearmed spiral wave is formed in the disk. Fourth, we analyze a disk with a modified exponential density distribution where the density of the inner part of the disk is reduced. In this case we find much slower, linear growth of the eccentric motion. A trailing onearmed spiral wave forms in the disk and becomes more tightly wrapped as time increases. The motion of the central mass if present is small compared with that of the disk.

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