 Solution

In the stated gravity-dominated approximation, neglect the pressure force in the radial momentum equation. A steady outward stellar wind then obeysAt $R_e=R_{\mathrm{esc}}$, the escape velocity is $v_e=\sqrt{2G{M}_{\ast }/R_e}$, so $C=0$. The ballistic marginal-escape stellar wind hasThe outward flow decelerates and approaches zero speed at infinity; it is exactly at the escape threshold in this pressure-negligible model. An initial speed above that threshold gives a nonzero asymptotic speed.

Adiabaticity alone would instead conserve the full Bernoulli equation $v^2/2+h-G{M}_{\ast }/r=B$. If finite pressure is retained and $v_e$ equals the mechanical escape speed, then $B=h_e$, not zero. With $p=K{\rho }^{\gamma }$ and conservation of mass, the more general implicit solution isIt needs $h_e$ and $\gamma$ in addition to the escape-speed datum. The boxed explicit power law is the leading gravity-dominated result, not an exact solution for an arbitrary warm adiabatic wind.