Seminars ASD

Secular evolution of resonant dynamical systems is one of the most
difficult problems in celestial mechanics. However, it can be efficiently
cured via the introduction of the so-called pendulum model. In this talk
I will review the background of this theory and introduce an alternative
solution that may suit better to some numerical applications.

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.