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A number of early type galaxies show a polar ring of gas, dust and stars lying roughly perpendicular to the apparent major axis of the central galaxy. We have studied the dynamics of a self-gravitating ring which is inclined to the principal planes of a triaxial galactic potential tumbling about its short axis; in a steadily-precessing equilibrium state, the precession rate of the ring in the potential must be equal to the tumbling speed of the triaxial figure. As in an oblate galaxy, both stable and unstable equilibria exist: in the tumbling triaxial potential, there are stable equilibria bending towards the equator, if the ring is light, and towards the pole, at higher ring mass. The former are similar to the `anomalous retrograde orbits', while the latter resemble the stable equilibria for a self-gravitating ring in an oblate potential. Following the time evolution of unstable polar rings shows that in an oblate galaxy potential, even if the ring is not sufficiently massive to be stabilised, self-gravity can still cause the characteristic warp up towards the pole. In the triaxial potential, when the inclination of the polar ring is not such that its precession rate matches the galaxy tumbling speed, the ring can wobble gently in a quasiperiodic manner if it is massive enough, but is disrupted if its mass is too low.
Some polar rings are not mirror-symmetric about the center of the galaxy, but have a banana or `C' shaped bend: these distortions may represent axisymmetric bending modes. The sideways displacement of an isolated ring or disk represents a neutrally stable mode of zero frequency; in the central galaxy potential, this can become an m=0 bending mode, which remains discrete when the galaxy is not too massive in relation to the ring. The curvature of the mode is sensitive to the core radius of the galaxy halo. The mode is not spontaneously unstable, so the bending must be excited by an encounter, or during the accretion of ring material.
\noindent Send preprint requests to sparke@madraf.astro.wisc.edu
