Polytropic Parker wind

A polytropic Parker wind is isentropic with $p=K{\rho }^{\gamma }$. Its sound speed changes with density, and a wind reaching infinity with positive terminal kinetic energy requires $1<\gamma <5/3$.

For radial speed $u$, sound speed $c_s$, and central mass $M$, steady spherical mass and momentum conservation giveAt a regular sonic point, $u=c_s$ and $r=GM/(2c_s^2)$ simultaneously.