General solution of the Jeans equations for triaxial galaxies with separable potentials
Glenn van de Ven, Chris Hunter, Ellen Verolme, Tim de Zeeuw
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 Staeckel 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 which we discuss in detail. We then extend the method of singular solutions to the triaxial case, and obtain a full solution, also in terms of prescribed boundary values of second moments. There again 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 which can support a triaxial density distribution in a triaxial separable potential.