= Solution
The <transonic accretion in a dipolar flux tube> values are
$$
\boxed{r_s=\frac{GM_\star(7-5\gamma)}{6c_0^2},\qquad
c_s=c_0\sqrt{\frac2{7-5\gamma}},\qquad
\rho_s=\rho_0\left(\frac2{7-5\gamma}\right)^{1/(\gamma-1)}.}
$$
The <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> gives
$$
\boxed{\dot M=A_\star\left(\frac{r_s}{R_\star}\right)^3\rho_s c_s.}
$$
For two equal polar caps with surface angular radius $\theta_\star$, $A_\star\simeq2\pi R_\star^2\theta_\star^2$, so the total <mass accretion rate> is
$$
\boxed{\dot M=\frac{2\pi\theta_\star^2\rho_0(GM_\star)^3}{27R_\star c_0^5}
\left(\frac2{7-5\gamma}\right)^{1/(\gamma-1)-5/2}.}
$$
For 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 $\gamma\to1$ has $r_s=GM_\star/(3c_0^2)$, $c_s=c_0$, and $\rho_s=e^{5/2}\rho_0$.
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