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 the N stars over their Keplerian orbits, using radial nodes
along the orbits. We will present three novel improvements to this
method. First, we re-formulate the discretisation 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 to reduce the overall
computational complexity. 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.
Federico Mogavero
