= Isothermal dipolar accretion equation
{title2=$(v-c_s^2/v)v\prime=3c_s^2/z-GM/z^2$}
A narrow <dipolar flux-tube area> $A(z)\propto z^3$ gives $A\rho v=\dot M$ for inward speed $v>0$. The <isothermal magnetic Bernoulli integral> in $\Phi=-GM/z$ then gives $(v-c_s^2/v)v\prime=3c_s^2/z-GM/z^2$. A smooth <transonic branch> crosses at $z_s=GM/(3c_s^2)$ with $v_s=c_s$, $\rho_s=e^{5/2}\rho_\infty$ and $\dot M=\rho_sc_sA(z_s)$. The crossing must lie in the domain; subsonic branches need not attain it.
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