Assume the wind is spherically symmetric. u is the radial velocity.

\frac{\partial u}{\partial t}+u\cdot\nabla u=-\nabla p-\frac{GM}{r^2} (1)

p=\frac{\rho RT}{\mu} (2)

\frac{RT}{\mu}=a^2 (3) is the sound speed

Substitute (3) and (2) into (1) and assume the wind is in steady state, we can get:

\frac{1}{u}\frac{du}{dt}=(\frac{2a^2}{r}-\frac{GM}{r^2})/(u^2-a^2)

u=a is a singular point and \frac{2a^2}{r}-\frac{GM}{r^2}=0 is critical point. If the wind is to experience constant acceleration, u will exceed a at some radius such that the equation become singular. Therefore the reasonable picture for the constant accelerating wind is that numerator goes to 0 as the denominator goes to 0.

Let r=r_c when \frac{2a^2}{r}-\frac{GM}{r^2}=0 therefore r_c=\frac{GM}{2a^2}

\frac{2a^2}{r}-\frac{GM}{r^2}<0 while u<a

\frac{2a^2}{r}-\frac{GM}{r^2}>0 while u>a

\frac{2a^2}{r}-\frac{GM}{r^2}=0 while u=a

Any other combination will not give constant accelerating wind but wind that experience deceleration in some region. Like supersonic wind in r<r_c region will decelerate and subsonic wind in r>r_c region will also decelerate.

Using De l'Hopital's rule at r=r_c

\frac{1}{u}\left.\frac{du}{dt}\right|_{r=r_c}=(-\frac{2a^2}{r_c^2}+\frac{2GM}{r_c^3})/(\frac{2udu}{dr})=\frac{a^2}{r_c^2}\left.\frac{udu}{dr}\right|_{r=r_c}

This also shows that numerator and denominator converge to 0 at the same rate. The mathematical appropriation of the isothermal wind is justified.

\left.\frac{du}{dt}\right|_{r=r_c}=\frac{\pm 2a^2}{GM}

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.