Grid-based Boltzmann solution

The evolution of a perturbed equilibrium can be explored with a time-dependent perturbation theory (e.g. Weinberg 1994b, Murali & Weinberg 1997a). The physics behind this approach is as follows. The period of the LMC orbit is longer than the periods of many of the stellar orbits within the cloud and such orbits are adiabatically invariant to the time-dependent tidal forcing. However, in cases where the frequencies of the stellar orbit are commensurate with the forcing frequencies, the resulting degeneracy breaks the adiabatic invariant. The change in the gravitational potential causes the resonance to sweep through phase space as described in §3.1 and the direction of the passage determines the effect of the resonance on the conserved quantities. The net effect on the gravitational potential depends on the scale of the inhomogeneity in the phase-space distribution. The evolution equations, therefore, take the form of a collisional Boltzmann equation with the right hand side depending on a gradient of phase space. Numerically, we approximate the solution of this equation by a two step process:

1.

Update the phase-space distribution function in a fixed potential using the finite-difference representation for the evolution term on the right hand side. This term can be written in flux form as described below and therefore conserves density over one step.

2.

Hold the distribution function fixed as a function of actions and solve for a new equilibrium. For ease of solution, the potential is assumed to have spherical symmetry while the distribution itself may be generally axisymmetric (e.g. f=f(E, L, L_z)).