Secular Dynamics around a Supermassive Black Hole via Multipole Expansion

TL;DR

This paper introduces improved computational methods for studying long-term stellar orbital dynamics around supermassive black holes, ensuring conservation laws, reducing costs, and providing new integrators, validated by analytical results.

Contribution

The study presents three novel enhancements to the secular approximation method, including reformulation for conservation, multipole expansion for efficiency, and new integrators for accurate long-term simulations.

Findings

Conservation laws are exactly satisfied in the new formulation.

Computational costs are significantly reduced via multipole expansion.

Simulation results match analytical diffusion coefficients of stellar eccentricities.

Abstract

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 $K$ nodes along each orbit. In this study we propose 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 up to order $\ell_{\mathrm{max}}$, reducing the overall computational costs from $O(N^2K^2)$ to $O(NK\ell_{\mathrm{max}}^2)$. 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.