Three-Dimensional Stationary Spherically Symmetric Stellar Dynamic Models Depending on the Local Energy

Abstract

Three-Dimensional Stationary Spherically Symmetric Stellar Dynamic Models Depending on the Local Energy. Juergen Batt, Enno Joern, Alexander L. Skubachevskii The stellar dynamic models considered here are triples (f,rho,U) of three functions: the distribution function f=f(r,u), the local density rho=rho(r) and the Newtonian potential U=U(r), where r:=|x|, u:=|v| ((x,v) in R^3xR^3 are the space-velocity coordinates), and f is a function q of the local energy E=U(r)+u^2/2. Our first result is an answer to the following question: Given a (positive) function p=p(r) on a bounded interval [0,R], how can one recognize p as the local density of a stellar dynamic model of the given type ("inverse problem")? If this is the case, we say that p is "extendable" (to a complete stellar dynamic model). Assuming that p is strictly decreasing, we reveal the connection between p and F, which appears in the nonlinear integral equation p=FU[p] and the solvability of Eddington's equation between F and q. Second, we investigate the following question ("direct problem"): Which q induce distribution functions f of the form f=q(-E(r,u)-E0) of a stellar dynamic model? This leads to the investigation of the nonlinear equation p=FU[p] in an approximate and constructive way by mainly numerical methods. The paper extends preceding work on flat galaxies to the three-dimensional case. In particular, the present answer to the extendability problem is completely different as in [1]. The present paper also opens the way to further explicit solutions of the Vlasov-Poisson system beyond the classical known examples which are for instant given in [4]. Keywords: Vlasov-Poisson system, stationary solutions, numerical approximation, mathematical physic, galaxy astrophysics.