Static stars : Some mathematical curiosities

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Abstract
The equations of structure of static Newtonian and general relativistic stars are investigated. By using Lie group theory, it is shown that, in each case, the condition that there should exist a simple ’’homologous’’ family of similar solutions necessitates precisely those equations of state for the stellar matter that are usually invoked by means of extraneous physical arguments. In the relativistic case, a diagram which depicts these families is drawn, using the qualitative theory of differential equations. This vividly exhibits the nature of the general soutions, and the exceptional character of the Misner–Zapolsky solution. This diagram is contrasted with similar ones obtained by Chandrasekhar in the Newtonian case.

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