Klassische und moderne Formulierungen des Vogt-Russell-Theorems.

The Vogt-Russell theorem on the uniqueness of local solutions to the stellar structure equations is reviewed in historical perspectives. The original results of Vogt and Russell were formulated precisely by Chandrasekhar (1938). The construction of multiple solutions, however, became possible with high-speed computers (Gabriel and Ledoux, 1967, Lauterborn et al., 1971, Kozlowaski, 1971 and Kozlowski and Paczynski, 1973). A formulation of local uniqueness in terms of abstract topological spaces might state that a solution (R, L), where R is radius and L is luminosity, is locally unique, if there exists an open neighborhood U(R, L) such that (R, L) is the only solution in U. As a matter of fact, stellar models constructed with realistic material functions have never been constructed that violate local uniqueness. This result is analyzed by considering the elimination process by which pressure is determined as a function of a trial pressure function and studying the solution possibilities. These are three, namely: (1) the solutions are discretely distributed, (2) there is an accumulation point of solutions, and (3) there is a continuum of solutions. Arguments are presented showing why the latter two cases, corresponding to multiple solutions, do not occur, although the case of an accumulation point of solutions is not strictly ruled out.