Dynamical Stability ofN‐Body Models for M32 with a Central Black Hole

We study the stability of stellar dynamical equilibrium models for M32. Kinematic observations show that M32 has a central dark mass of ~3 × 10 6 M ☉ , most likely a black hole, and a phase-space distribution function that is close to the "two-integral" form f = f ( E , L z ). M32 is also rapidly rotating; 85%-90% of the stars have the same sense of rotation around the symmetry axis. Previous work has shown that flattened, rapidly rotating two-integral models can be bar-unstable. We have performed N -body simulations to test whether this is the case for M32. This is the first stability analysis of two-integral models that have both a central density cusp and a nuclear black hole. Particle realizations with N = 512,000 were generated from distribution functions that fit the photometric and kinematic data of M32. We constructed equal-mass particle realizations and also realizations with a mass spectrum to improve the central resolution. Models were studied for two representative inclinations, i = 90° (edge-on) and i = 55°, corresponding to intrinsic axial ratios of q = 0.73 and q = 0.55, respectively. The time evolution of the models was calculated with a "self-consistent field" code on a Cray T3D parallel supercomputer. We find both models to be dynamically stable. This implies that they provide a physically meaningful description of M32 and that the inclination of M32 (and hence its intrinsic flattening) cannot be strongly constrained through stability arguments. Previous work on the stability of f ( E , L z ) models has shown that the bar mode is the most common unstable mode for systems rounder than q ≈ 0.3 (i.e., E7) and that the likelihood for this mode to be unstable increases with flattening and rotation rate. The f ( E , L z ) models studied for M32 are not bar-unstable, and M32 has a higher rotation rate than nearly all other elliptical galaxies. This suggests that f ( E , L z ) models constructed to fit data for real elliptical galaxies will generally be stable, at least for systems rounder than q ≳ 0.55, and possibly for flatter systems as well.