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

Abstract

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.