= Solution
The <dipole magnetic-field line> geometry gives the angular radius at the <sonic point> as $\theta_s\simeq\theta_\star\sqrt{r_s/R_\star}$. The sonic region must remain a narrow <flux tube> for the one-dimensional <transonic branch> equations to apply. Thus a necessary small-angle constraint is
$$
\boxed{\theta_\star\ll\sqrt{\frac{R_\star}{r_s}}
=\sqrt{\frac{6c_0^2R_\star}{GM_\star(7-5\gamma)}}.}
$$
This also makes the transverse sound-crossing time $r_s\theta_s/c_s$ small compared with the radial flow time $r_s/c_s$. The restriction must hold at any larger matching radius as well. To match to a reservoir with negligible <gravitational potential> within the same narrow dipolar approximation, one needs an overlap region
$$
\frac{GM_\star}{c_0^2}\ll r_L\ll\frac{R_\star}{\theta_\star^2}.
$$
Hence the stronger useful sufficient scaling is $\theta_\star^2\ll R_\star c_0^2/(GM_\star)$. The small-angle dipolar model should not be extrapolated to literal infinity, where its boundary <magnetic field line> turns toward the equatorial plane.
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