Isothermal Bondi equation

For inward speed $u>0$, steady spherical mass conservation and radial momentum balance give

With $x=r/r_s$ and $\mathcal{M}=|u_r|/c_s$, the radial isothermal equation integrates to ${\mathcal{M}}^2/2-\ln \mathcal{M}=2/x+2\ln x+C$. A regular transonic branch has $C=-3/2$, and the reservoir-fed accretion branch crosses $(x,\mathcal{M})=(1,1)$ with slope $-1$.

The isothermal Bondi sonic point is $r_s=GM/(2c_s^2)$, where the regular transonic solution has $u=c_s$.

At the critical point, both the coefficient of $du/dr$ and the remaining right-hand side of the Isothermal Bondi equation vanish, allowing a finite derivative through the sonic transition.