Tilting of a Disk of 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. A new constant of the motion of $N$ tilted rings is identified (in addition to the energy). 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 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 ($j=1..N$). We numerically solve the equations for $\Theta_j$ 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 toa tilted halo potential, and (4) warp evolution resulting from an initially tilted disk plane.