Polytropic equation of state

A polytropic equation of state relates pressure and density by $p=K{\rho }^{\Gamma }$ with constant $K$ and exponent $\Gamma$. Writing $\Gamma =1+1/n$ defines the polytropic index $n$.

The coefficient $K$ in a polytropic equation of state $p=K{\rho }^{\Gamma }$. It is spatially constant for one simple polytrope, but need not stay constant between different equilibrium models. For a fixed-composition ideal gas with $\Gamma$ equal to its adiabatic index, $K$ parametrizes specific entropy: $s=c_v\ln K+\text{constant}$ with a suitable dimensional reference inside the logarithm.