Abstract
Context. The modelling of stationary galactic stellar populations can be performed using distribution functions. Aims. This paper aims to write explicit integrals of motion and distribution functions. Methods. We propose an analytic formulation of the integrals of motion with an explicit dependence on potential. This formulation applies to potentials with rotational symmetry or triaxial symmetry. It is exact for Stäckel potentials and approximate for other potentials. Results. Modelling a stationary stellar population using these integrals of motion allows the force field to be found with satisfactory accuracy. On the other hand, the mass density distribution that generates the force field and the gravitational potential is recovered with less accuracy due to lower precision in modelling box-type orbits.
Highlights
It is well established that orbits in galactic potentials generally admit three integrals of motion (Contopoulos 1960; Ollongren 1962) with the possible presence of ergodic orbits (Hénon & Heiles 1964)
This is done using an approximation of orbits in the frame of Stäckel potentials
Kent & de Zeeuw (1991) already noted that an accurate approximation of a quasiinvariant integral can be achieved by adjusting a Stäckel potential for each of the orbits of a galactic potential
Summary
It is well established that orbits in galactic potentials generally admit three integrals of motion (Contopoulos 1960; Ollongren 1962) with the possible presence of ergodic orbits (Hénon & Heiles 1964). Our results are based on the same principles as those of Sanders (2012) and Sanders & Binney (2014), save that the search for integrals is analytic and not numerical These integrals of motion are exact for Stäckel triaxial potentials and their expressions depend explicitly on the potential. With this coordinate system, the equations of motion separate and it can be shown that the Stäckel potentials admit three independent integrals. O. Bienaymé: Integrals of motion for non-axisymmetric potentials three positions on two planes and one axis of symmetry (for all the expressions below, it is no longer necessary that the potential be zero at the origin):. We note that it is possible to write expressions depending on more than three positions
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