Secular Dynamics around a Supermassive black hole via Multipole Expansion

In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauss, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order l(max), reducing the overall computational cost from O((NK2)-K-2) to O(NKl(max)(2)). Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics.