Main-sequence mass-radius relation ($R_{\ast }\propto M^q$)

An approximate relation between the mass and radius of main-sequence stars in a specified composition and mass range. It is not determined by hydrostatic equilibrium alone. For gas-pressure stellar homology with Kramers' opacity law, radiative diffusion gives $L\propto {\mu }^{15/2}{M}^{11/2}{R}_{\ast }^{-1/2}$. Substitution of $R_{\ast }\propto M^q$ gives $L\propto M^{(11-q)/2}$ at fixed composition, explaining precisely which extra radius assumption produces any claimed exponent.