Numerical models: The Monte Carlo method

We have recently developed a new Monte Carlo (MC) code to follow the long term evolution of galactic nuclei (Freitag & Benz, 2002; Freitag & Benz, 2001; Freitag, 2001). This tool is based on the scheme first proposed by Hénon (Hénon, 1973) to simulate globular clusters but, in addition to relaxation, it also includes collisions, tidal disruptions, stellar evolution and captures. The MC technique assumes that the cluster is spherically symmetric and represents it as a set of particles, each of which may be considered as a homogeneous spherical shell of stars sharing the same orbital and stellar properties. The number of simulated stars may be larger than the number of particles, provided the number of stars per particle is constant. Another important assumption is that the system is always in dynamical equilibrium so that orbital time scales need not be resolved and the natural time-step is a fraction of the relaxation (or collision) time. The relaxation is treated as a diffusive process (Binney & Tremaine, 1987). The MC code offers a unique combination of physical realism (collisions, mass segregation, velocity anisotropy...are easily taken into account) and computational efficiency ( $T_\mathrm{CPU}\propto N_\mathrm{part} \ln(N_\mathrm{part}$), compared to $T_\mathrm{CPU}\propto N_\mathrm{part}^{2-3}$ for direct $N$-body codes) which allows Hubble-time simulations to be carried on standard PCs with multi-million particle resolution.

We assume that a star is irremediably captured by the MBH if it gets on an orbit with a time scale for shrinkage by emission of gravitational radiation (Peters, 1964) shorter than the time over which 2-body relaxation could significantly modify the pericenter distance (Ivanov, 2002; Sigurdsson & Rees, 1997), $T_{\mathrm{mod}}\simeq \theta^2 T_\mathrm{rel}$. $T_\mathrm{rel}$ is the usual relaxation time (Binney & Tremaine, 1987) and $\theta$ is the angle between the trajectory and the direction to the centre.

We concentrate here on a model set to represent the nucleus of the Milky Way (Genzel et al., 2000), with a central BH of mass $2.6\times 10^6 M_\odot$ and a total mass in stars of $8.67\times 10^7 M_\odot$. A stellar population with a broad mass spectrum (Kroupa, 2001) is assumed. There is no initial mass segregation. The number of particles is $6\times 10^6$.