Barotropic hydrostatic first integral
= Barotropic hydrostatic first integral
{title2=$\int^p\frac{d\widetilde p}{\rho(\widetilde p)+\widetilde p}+\log N=C$}
For a <barotropic fluid> in <hydrostatic equilibrium along a Killing vector>, assume $\rho+p>0$. Integrating $dp/(\rho+p)=-d\log N$ gives the displayed <first integral>, constant on each connected fluid region. For the <equation of state> $p=w\rho$ with constant $w>0$, it gives $\rho N^{(1+w)/w}=C_1$.