Barotropic spherical accretion equation ($(u-c_s^2/u)u^{\prime }=2c_s^2/r-{\Phi }^{\prime }$)

Steady spherical continuity equation gives $r^2\rho u=\text{constant}$. Combining this with Euler equations for an inviscid fluid and $c_s^2=dp/d\rho$ gives the displayed equation. Smooth transonic passage requires $u^2=c_s^2$ and ${\Phi }^{\prime }=2c_s^2/r$ simultaneously. The barotropic enthalpy function gives the integral $u^2/2+h+\Phi =\text{constant}$.

For $p=K{\rho }^{1+1/m}$, the barotropic enthalpy function is $m{c}_s^2$. In $\Phi =-\lambda /R^2-\mu /R$, a regular sonic point obeys $c_{s,s}^2=\lambda /R_s^2+\mu /(2R_s)$, and its Bernoulli integral yields the displayed quadratic. For $K,\lambda ,\mu >0$ and the usual positive polytropic index, its unique positive root exists precisely when $m>1/2$. The mass accretion rate is $4\pi R_s^2{\rho }_{\infty }{c}_{s,s}(c_{s,s}^2/c_{\infty }^2{)}^m$. If indices with $0<1+1/m<1$ are admitted, $m<-1$ also permits a positive sonic root; negative compressibility indices do not.

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