The rate of secular evolution in elliptical galaxies with central masses

We study a series of $N-$body simulations representing elliptical galaxies with central masses. Starting from two different systems with smooth centres, which have initially a triaxial configuration and are in equilibrium, we insert to them central masses of various values. Immediately after such an insertion a system presents a high fraction of particles moving in chaotic orbits, a fact causing a secular evolution towards a new equilibrium state. The chaotic orbits responsible for the secular evolution are identified. Their typical Lypaunov exponents are found to scale with the central mass as a power law $L\propto m^s$ with $s$ close to 1/2. The requirements for an effective secular evolution within a Hubble time are examined. These requirements are quantified by introducing a quantity called \emph{effective chaotic momentum} $\mathscr{L}$. This quantity is found to correlate well with the rate of the systems' secular evolution. In particular, we find that when $\mathscr{L}$ falls below a threshold value (0.004 in our $N-$body units) a system does no longer exhibit significant secular evolution.