Homologous Stars

Question:
Many stellar properties can be understood by assuming that stars are homologous. Homologous transformations are essentially self-similar and require that the mass profile scale with respect to radius. For example, if 1/2 of the mass of a star is located within the first quarter of its radius, then this ratio will be true for all other homologous stars. Let the homology variable ξ be defined as:

where primes denote properties of the second star and M and M′ represent the masses of the two stars, respectively. Also, 0 ≤ ξ ≤ 1. Therefore

In terms of the homologous variables, show that

[Note that you should start with m being the independent variable in the differential equations of stellar structure.]

You will first have to show that the density scales as the mass divided by the volume [i.e., (ρ′/ρ) = (M′/M )(R′3/R3)−1]. Further show that the pressure scales as mass squared over radius to the fourth power. Assuming an ideal gas equation of state, determine how the temperature scales. By scaling the density ρ and radius r appropriately, is it possible to make a polytrope homologous?

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