The Vogt-Russell Theorem, and New Results on an Old Problem

Abstract

Half a centry ago Henry Norris Russell and Heinrich Vogt independently made a conjecture concerning the structure of spherical stars which are in hydrostatic and thermal equilibrium (Russell, 1927; Vogt, 1926). This conjecture has later come to be known as the Vogt-Russell theorem and is usually formulated as follows: The structure of a star is uniquely determined by the mass and the composition. In other words, the statement claims the existence and uniqueness of a stellar equilibrium configuration for given parameters mass and composition, and you may find what is called a mathematical proof in many textbooks on stellar structure.

The Vogt-Russell Theorem, and New Results on an Old Problem

“…Half a centry ago Henry Norris Russell and Heinrich Vogt independently made a conjecture concerning the structure of spherical stars which are in hydrostatic and thermal equilibrium (Russell, 1927;Vogt, 1926). This conjecture has later come to be known as the Vogt-Russell theorem and is usually formulated as follows: The structure of a star is uniquely determined by the mass and the composition.…”

“…Next let me report about recent work which has given an exact local formulation of the Vogt-Russell theorem (Kahler, 1972;Kahler and Weigert, 1973). The problem is to give precise conditions for the local existence and uniqueness of stellar models, in the vicinty of a given model.…”

“…Accordingly i t seemed to be hopeless to look for general statements on the number and the stability of the different models. Surprisingly, recent work has shown that such statements are nevertheless possible when appropriate algebraical and topological methods are applied (Kahler, 1975). For this purpose two new properties of equilibrium models have been defined, the multiplicity m and the charge c.…”