TILTING OF A DISK OF SELF-GRAVITATING RINGS
The present work represents an attempt to understand the ‘rules of behavior’ of observed warps in the HI disks of spiral galaxies found by Briggs (1990). In contrast with most earlier theoretical work, the present study investigates different initial value problems of a warped disk in an oblate (or prolate) halo potential, and it represents the disk warp in terms of N independently tilted, self-gravitating, concentric rings. This representation gives new insight into the disk warping. The phenomenon of phase-locking of the lines-of-nodes of nearby rings due to self-gravity is demonstrated. We consider the influence of dynamical friction due to ring motion through the halo matter as well as friction between gaseous rings with different vertical motions due to turbulent viscosity. We first consider the dynamics of one, two, and three tilted rings of different radii in a halo potential. We go on to develop dynamical equations for N−rings which are most simply expressed in terms of the complex tilt anglesΘj = θjexp(−iφj), whereθj is the actual tilt angle andφj the line-of-nodes angle for the j ring (j = 1..N ). Relatively small values of N ( ∼102) are sufficient to describe the warp evolution during the age of a galaxy during which time discrete modes are not important. The equations for Θj are solved numerically for four different types of initial conditions: (1) warp excitation by a passing satellite, (2) excitation by a sinking compact minor satellite, (3) warp evolution due to a tilted halo potential, and (4) warp evolution resulting from an initially tilted disk plane. The nature of the warps is most clearly shown by the polar plots of θj versusφj .