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isothermalwind

-              v2
+              v3
 $\left.\frac{du}{dt}\right|_{r=r_c}=\frac{\pm 2a^2}{GM}$
+Remark: use the
+Remark: We can use mass conservation law to get $\rho(r)$ profile.
+$\dot{M}=4\pi \rho u r^2$
+In subsonic regime, this profile is very much alike the \rho(r) profile in hydrostatic limit. This is because $u\cdot \nabla u is small compared to pressure gradient and the momentum equation is simply reduced to the hydro-equilibrium case (which is also the solution of isothermal polytrope but there should be another long story in connecting the surface density and the wind density). The match of two density profile in subsonic regime deems appropriate.