Integrated isothermal Bondi flow relation
= Integrated isothermal Bondi flow relation
{title2=$\mathcal M^2/2-\ln\mathcal M=2/x+2\ln x-3/2$}
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 $-1$.