Barotropic spherical accretion equation
= Barotropic spherical accretion equation
{title2=$(u-c_s^2/u)u^{\prime}=2c_s^2/r-\Phi^{\prime}$}
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 $u^2/2+h+\Phi=\text{constant}$.