TILTING OF A DISK OF SELF-GRAVITATING RINGS

The present work represents an attempt to understand the 'rules of beh avior' of observed warps in the HI disks of spiral galaxies found by B riggs (1990). In contrast with most earlier theoretical work, the pres ent study investigates different initial value problems of a warped di sk in an oblate (or prolate) halo potential, and it represents the dis k warp in terms of N independently tilted, self-gravitating, concentri c 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 dyna mical friction due to ring motion through the halo matter as well as f riction between gaseous rings with different vertical motions due to t urbulent viscosity. We first consider the dynamics of one, two, and th ree tilted rings of different radii in a halo potential. We go on to d evelop dynamical equations for N-rings which are most simply expressed in terms of the complex tilt angles Theta(j) = theta(j)exp(-i phi(j)) , where theta(j) is the actual tilt angle and phi(j) the line-of-nodes angle for the j(th) ring (j = 1..N). Relatively small values of N (le ss than or similar to 10(2)) are sufficient to describe the warp evolu tion during the age of a galaxy during which time discrete modes are n ot important. The equations for Theta(j) are solved numerically for fo ur different types of initial conditions: (1) warp excitation by a pas sing satellite, (2) excitation by a sinking compact minor satellite, ( 3) warp evolution due to a tilted halo potential, and (4) warp evoluti on resulting from an initially tilted disk plane. The nature of the wa rps is most clearly shown by the polar plots of theta(j) versus phi(j) .