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 $r:=|x|$, $u:=|v|((x,v)\in\mathbb{R}^3\times\mathbb{R}^3$ 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.