Eccentric behavior of disk galaxies

A theory is developed for the dynamics of eccentric perturbations (proporti onal to exp +/- i phi) 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 azimutha l matter distributions, Account is taken of the dynamics of the shift of th e 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 treat ed as passive gravitational field sources. Equations of motion and a Lagran gian are derived for the ring + center system, and these lead to total ener gy and total angular momentum constants of the motion. We first study the eccentric motion of a disk consisting of two rings of di fferent radii but equal mass, M-d/2. For small M-d, the two rings are stabl e, 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 centr al mass. In this case instability sets in above a threshold value of the ce ntral mass (for a fixed ring mass), and it acts to increase the angular mom entum 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 surfac e density distribution represented by a large number of rings. The inner pa rt of the disk is found to be strongly unstable. Angular momentum of the ri ngs is transferred outward and to the central mass if present, and a traili ng one-armed spiral wave is formed in the disk. Fourth, we analyze a disk w ith a modified exponential density distribution where the density of the in ner part of the disk is reduced. In this case we iind much slower, linear g rowth of the eccentric motion. A trailing one-armed spiral wave forms in th e disk and becomes more tightly wrapped as time increases. The motion of th e central mass if present is small compared with that of the disk.