Equation of State

Consider a composite polytrope composed of \(\nreg\) regions extending from the origin out to the stellar surface. In the \(i\)’th region (\(1 \leq i \leq \nreg\)), the pressure \(P\) and density \(\rho\) are related by the polytropic equation-of-state

(22)\[\frac{P_{i}}{P_{i,0}} = \left( \frac{\rho_{i}}{\rho_{i,0}} \right)^{(n_{i} + 1)/n_{i}}\]

where the normalizing pressure \(P_{i,0}\) and density \(\rho_{i,0}\), together with the polytropic index \(n_{i}\), are constant across the region but may change from one region to the next. At the \(\nreg-1\) boundaries between adjacent regions, the pressure and interior mass \(M_{r}\) are required to be continuous, but the density may jump.