Elliptical galaxies with separable potentials

The perfect ellipsoid is presented: an inhomogeneous triaxial mass model with a gravitational potential that is of Stäckel form. The equations of motion are separable in ellipsoidal coordinates and all stellar orbits have three isolating integrals that are known explicitly. The orbital structure in the perfect ellipsoid is generic for all centrally concentrated triaxial mass models with a finite central density and a Stäckel potential, and it is tractable by analytic means. The orbital shapes are classified in terms of the values of the integrals of motion. This classification becomes particularly transparent in terms of the action integrals. Individual orbit densities can be computed without integration of the equations of motion, and some examples are shown. There are four families of general orbits: boxes, inner and outer long axis tubes and short axis tubes. These are identical to the major orbit families that occur in Schwarzschild’s ellipsoid and are thought to be of prime importance for the structure of elliptical galaxies. Special values of the axis ratios of the perfect ellipsoid lead to a simpler orbital structure. The prolate, oblate and spherical limits are treated, as well as the elliptic and circular discs and the needle. The existence of the perfect ellipsoid shows that ellipsoidal coordinates are the natural coordinates for the description of triaxial systems. The relevance of mass models with Stäckel potentials for the construction of self-consistent models of elliptical galaxies is discussed.