Critical-point slope of barotropic spherical accretion
= Critical-point slope of barotropic spherical accretion
{title2=$C=\rho\,d(c_s^2)/d\rho$}
At a regular <sonic point>, differentiating the vanishing coefficient and numerator yields $(2+C/c_s^2)(u^{\prime})^2+4Cu^{\prime}/(ru)+(4C+2c_s^2)/r^2+\Phi^{\prime\prime}=0$. A real slope compatible with the desired branch is needed in addition to the sonic equalities. For constant <sound speed> this reduces to $(u^{\prime})^2=-c_s^2/r^2-\Phi^{\prime\prime}/2$.