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

The stellar dynamic models considered here deal with 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 $$r: = \left| x \right|$$ , $$u: = \left| {v} \right|$$ ( $$(x,{v}) \in {{\mathbb{R}}^{3}} \times {{\mathbb{R}}^{3}}$$ are the space-velocity coordinates), and $$f$$ is a function $$q$$ of the local energy $$E = U(r) + \frac{{{{u}^{2}}}}{2}$$ . 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 $$q$$ (Theorem 4.1). Second, we investigate the following question (“direct problem”): Which $$q$$ induce distribution functions $$f$$ of the form $$f = q( - E(r,u) - {{E}_{0}})$$ of a stellar dynamic model? This leads to the investigation of the nonlinear equation $$p = FU[p]$$ in an approximative and constructive way by mainly numerical methods.