60 - Orbits and third integrals

Summary

If you ask the special function

Of our never-ceasing motion,

We reply, without compunction,

That we haven't any notion!

Gilbert and Sullivan

Few orbits in smooth axisymmetric galaxies are exactly circular, and no orbits in unsymmetric force fields are regular. Some orbits, like the denizens of Arcadia, move ergodically hither and thither, described by no special function. Others, more constrained, are partly predictable.

In this section we explore some general properties of orbits in the smooth force field of a flattened galaxy. After a galaxy has formed, the two-body relaxation timescale τ_R usually becomes very long compared to the dynamical crossing time τ_c or even to the Hubble time. Therefore, the collisionless Boltzmann equation provides a good description of the internal dynamics. We saw from (7.15) that this description is equivalent to knowing the orbits of stars in the smooth mean field. Thus it is possible to invert the approach of Section 7 by starting from the orbits and building up a self-consistent distribution function (see Section 63). Orbits in unsymmetric systems are naturally very complicated. Finding them usually requires extensive numerical integrations which are highly specific to individual models. Some types of orbits, however, are common to a reasonable range of idealized models, so we may consider them here.

After circular orbits, the next simplest are motions across the symmetry plane z = 0. If these orbits do not stray far above or below the plane, they respond primarily to the local density ρ(r,z) rather than to the overall galactic field.