Isothermal Bondi accretion

Isothermal Bondi accretion has $p=c_s^2\rho$ with constant sound speed. Its regular transonic solution crosses the Bondi sonic point and approaches free fall near the accretor.

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

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.