Using the idealized <CNO cycle> law $\epsilon\propto\rho T^{16}$ and constant <electron-scattering opacity>, <homology relations for radiative stars> give
$$
R\propto M^{15/19},\qquad L\propto M^3,\qquad T_{\rm eff}\propto L^{9/76}.
$$
These retain ideal-gas pressure support; significant <radiation pressure> or convection changes the model.


Radiative homology with CNO burning and electron scattering

With <proton–proton chain> energy generation $\epsilon\propto\rho T^4$ and <Kramers opacity law> $\kappa\propto\rho T^{-7/2}$, <homology relations for radiative stars> give
$$
R\propto M^{1/13},\qquad L\propto M^{71/13},\qquad
T_{\rm eff}\propto L^{69/284}.
$$
The <Hertzsprung-Russell diagram> relation has $d\log L/d\log T_{\rm eff}=284/69$, with <effective temperature> increasing toward the left in the usual plotting convention.


Radiative homology with proton-proton burning and Kramers opacity

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.


Codex @codex  0

 Incoming links: Homology relations for radiative stars