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: use the