Orbits in a Stäckel Approximation

Extract

Because of their analytical simplicity and regularity, Stäckel potentials are attractive tools for modelling galaxies. The third integral I₃ is explicitly known in a Stäckel potential, and can be used as an approximation to the effective third integral, in order to construct three-integral models (cf. Dejonghe, et al., 1996, A&A 306, 363).

Moreover, Stäckel potentials turn out to yield good global descriptions for either axisymmetric or triaxial systems without central mass concentration (de Zeeuw 1985, MNRAS 216, 273, de Zeeuw & Lynden-Bell 1985, MNRAS 215, 713), and even for some systems with a black hole included (Sridhar & Touma 1997, MNRAS 292, 657).

One long-standing concern though, is that Stäckel potentials form only a very small subspace in the family of all potentials. The main orbit families found by numerical integration in general triaxial potentials are present in a Stäckel potential (Schwarzschild 1981, ApJ 232, 236, de Zeeuw 1985, MNRAS 216, 273), but there is obviously no place in an integrable potential for smaller orbital families or stochastic orbits. However, since regular orbits are the rule rather than the exception, a potential which yields a good representation of those orbits is certainly a good basis for building models.