13. Orbits in triaxial mass distributions and surfaces of section¶
Orbits in axisymmetric potentials are relatively simple: because of the conservation of the \(z\) component of the angular momentum, \(L_z\), all orbits with non-zero \(L_z\) loop around the \(z\) axis. Orbits with \(L_z = 0\) are polar orbits, which reach their maximum height \(z_\mathrm{max}\) at \(R=0\). In Chapter 9.2, we argued that orbits in galactic axisymmetric potentials typically satisfy a third integral \(I_3\) in addition to \(E\) and \(L_z\); this implies that the orbits are all highly regular.
Orbits in triaxial mass distributions are more complicated than those in axisymmetric distributions. For a general triaxial mass distribution, no component of the angular momentum is conserved and orbits with non-zero initial angular momentum therefore do not have to maintain a well-defined sense of rotation. For a static triaxial potential, only the energy is a classical integral of the motion that restricts an orbit’s trajectory to a five-dimensional volume of phase-space. In general, we expect orbits to fill this entire five-dimensional subspace, unless there are additional, non-classical integrals that restrict the motion further—like the third integral for axisymmetric potentials.
In this chapter, we discuss the types of orbits that exist in static triaxial gravitational potentials. Once we lift the restriction of axisymmetry, the types of orbits that can exist and their structure rapidly becomes more complex. To better study orbits, we therefore first introduce a more sophisticated tool than the simple orbit projections that we have relied on so far: surfaces of section. Along the way, we will also return to a question that we posed in Chapter 9.2 : Do all orbits in an axisymmetric potential have a third integral of motion?
.
.
.