= Solution
<Hydrostatic equilibrium> and dimensional scaling give characteristic stellar pressure $p\sim GM^2/R^4$ and <mass density> $\rho\sim M/R^3$. For a fixed-index <stellar polytrope> with $p=K\rho^{1+1/n}$, eliminating density gives the <polytropic mass-radius relation>
$$
\boxed{R\propto\left(\frac KG\right)^{n/(3-n)}M^{(1-n)/(3-n)},\qquad n\ne3.}
$$
The dimensionless coefficient is determined by the corresponding <Lane-Emden equation> and boundary conditions. Both the index and $K$ must be held fixed to infer this mass dependence.
Nonrelativistic electron degeneracy has $n=3/2$, hence \b[$R\propto M^{-1/3}$] for cold fixed-composition <white dwarfs>: adding mass makes the object smaller. Relativistic electron degeneracy approaches $n=3$; the mass scale becomes proportional to $(K/G)^{3/2}$ independently of the radius, giving the ideal <Chandrasekhar limit>. The limiting model signals loss of the ordinary stable cold sequence; it does not determine a unique radius at that mass.
For a gas-pressure-supported star, the <virial theorem> instead gives characteristic temperature $k_BT\sim G\bar mM/R$. Specifying a common characteristic temperature gives $R\propto M$, but stellar temperatures are not generally identical. A <main-sequence star> also requires nuclear energy generation and radiative or convective transport. Consequently there is no universal mass-radius exponent applicable to every main-sequence mass and composition. An isentropic convective model and a cold degeneracy-supported model can share index $3/2$ while having quite different physical meanings and values of $K$.
Back to article page
