Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 4 / d

The transonic accretion in a dipolar flux tube values areThe mass density follows from $c^2/c_0^2=(\rho /{\rho }_0{)}^{\gamma -1}$ for the same polytropic equation of state. If $A_{\star }$ is the total loaded area at the surface, mass conservation at the sonic point givesFor two equal polar caps with surface angular radius ${\theta }_{\star }$, $A_{\star }\simeq 2\pi R_{\star }^2{\theta }_{\star }^2$, so the total mass accretion rate isFor only one loaded polar cap, divide this expression by two. Expressing the result first in $A_{\star }$ also covers a bundle whose loading angle is defined at another radius. The smooth isothermal limit $\gamma \to 1$ has $r_s=G{M}_{\star }/(3c_0^2)$, $c_s=c_0$, and ${\rho }_s=e^{5/2}{\rho }_0$.