C. Hunter

Triaxial galaxy models with thin tube orbits

It is shown how to construct self-consistent distribution functions for triaxial galaxy models with Stackel potentials. The Stackel potentials cause the dynamics to be simple, with all the stellar orbits regular and belonging to one of four families. By also restricting each of the three families of tube orbits to be infinitesimally thin and with no radial epicyclic motion, we are able to obtain explicit expressions for their phase-space distribution functions, and then their densities and velocity moments throughout space

General solution of the Jeans equations for triaxial galaxies with separable potentials

The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system. For general three-dimensional stellar systems, there are three equations and six independent moments. By assuming that the potential is triaxial and of separable Stackel form, the mixed moments vanish in confocal ellipsoidal coordinates. Consequently, the three Jeans equations and three remaining non-vanishing moments form a closed system of three highly symmetric coupled first-order partial differential equations in three variables. These equations were first derived by Lynden-Bell, over 40 years ago, but have resisted solution by standard methods. We present the general solution here. We consider the two-dimensional limiting cases first. We solve their Jeans equations by a new method which superposes singular solutions. The singular solutions, which are new, are standard Riemann–Green functions. The resulting solutions of the Jeans equations give the second moments throughout the system in terms of prescribed boundary values of certain second moments. The two-dimensional solutions are applied to non-axisymmetric discs, oblate and prolate spheroids, and also to the scale-free triaxial limit. There are restrictions on the boundary conditions that we discuss in detail. We then extend the method of singular solutions to the triaxial case, and obtain a full solution, again in terms of prescribed boundary values of second moments. There are restrictions on these boundary values as well, but the boundary conditions can all be specified in a single plane. The general solution can be expressed in terms of complete (hyper)elliptic integrals, which can be evaluated in a straightforward way, and provides the full set of second moments that can support a triaxial density distribution in a separable triaxial potential.

Chaos in orbits due to disk crossings.

Abstract: We study orbits of halo stars in simple models of galaxies with disks and halos to see if the cumulative effects of the sudden changes in acceleration that occur at disk crossings can induce chaos. We find that they can, although not in all orbits and not in all potentials. Most of the orbits that become chaotic stay relatively close to the disk and range widely in the radial direction. Heavier disks and increased halo flattening both enhance the extent of the chaos. A limited range of experiments with a three‐component model of the Milky Way with an added central bulge finds that many chaotic disk‐crossing orbits can be expected in the central regions, and that prolateness of the halo is much more effective than oblateness in generating chaos.

Bifurcations of Periodic Orbits in Axisymmetric Scalefree Potentialsa

ABSTRACT: We study orbits in potentials with central cusps, emphasizing the spheriodal equidensity (SED) potentials generated by mass distributions with spheroidal equidensity surfaces. The most prominent bifurcations are those related to 1:1 and 4:3 resonances between radial motions and motions perpendicular to the central plane. We find that 1:1 resonances can cause the thin tube orbit, as well as the equatorial plane orbit, to become unstable. We concentrate on period‐tripling bifurcations because they appear to be the least understood. We study them via a class of analytic maps. This study suggests that stable period‐three orbits generally arise de novo in stable and unstable pairs via a turning‐point bifurcation, and not through a bifurcation from the thin tube at a 120° rotation angle. The stable period‐three orbits typically have only a short span of existence before becoming unstable to a period‐doubling instability through a supercritical pitchfork bifurcation.