Self-gravitating response of a spherical galaxy to sinking satellites

I present a method for computing the response of a spherical stellar system to a periodic perturbation. The perturbation can be a normal mode or an externally applied force. Any physically realistic model [general distribution function |f(E,J)| with |ρ>0|] can be investigated. The technique is an adaptation of a method used by Kalnajs for studying discs. As an application, I derive an analytic solution for the orbital decay of a satellite in a self-gravitating galaxy. This problem is central to our understanding of galactic cannibalism. Strictly, the scenario is applicable to the merger of a dwarf and, for example, a cD galaxy. However, the analysis presented here should help provide a physical framework for understanding more complicated realistic systems. The central result is that the orbital decay time in the self-gravitating case is increased by a factor of 2–3 over the non-self-gravitating case. The response of the galaxy to the satellite is dominated by the dipole contribution; the self-gravity causes a phase shift which may decrease the orbital torque by an order of magnitude depending on galactocentric position. I present and contrast both the self-gravitating and non-self-gravitating wakes and clarify the importance of local and global effects. The results suggest that Chandrasekhar's formula will not adequately represent the orbital decay of an extended satellite in general.