= Homology relations for radiative stars
For chemically uniform, gas-pressure-supported stars with homologous dimensionless profiles, <mass conservation> and <hydrostatic pressure support equation> give $\rho_c\propto M/R^3$ and $T_c\propto M/R$. Suppose <opacity> scales as $\kappa\propto\rho^aT^b$ and specific nuclear energy generation as $\epsilon\propto\rho^cT^d$. Then
$$
L_{\rm nuc}\propto M^{1+c+d}R^{-3c-d},\qquad
L_{\rm rad}\propto M^{3-a-b}R^{3a+b}.
$$
The second scaling follows from <radiative diffusion in a star>, $L\propto RT_c^4/(\kappa_c\rho_c)$. Equating the two gives
$$
R\propto M^{(a+b+c+d-2)/(3a+b+3c+d)}
$$
when the denominator is nonzero. Fixed composition and self-similar profiles are essential; these are relations between idealized models rather than universal stellar laws.
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