Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2015 / Paper 1 / 2 / c

Hydrostatic equilibrium and dimensional scaling give characteristic stellar pressure $p\sim G{M}^2/R^4$ 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 relationThe 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 $R\propto M^{-1/3}$ 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 $k_{B}T\sim G\bar{m}M/R$. 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 $K$.