Spherical magnetospheric pressure balance ($R_M\sim [B_{\ast }^4{R}_{\ast }^{12}/({\mu }_0^{2}GM{\dot{M}}^2){]}^{1/7}$)

A dipole's magnetic pressure scales as $B_{\ast }^2{R}_{\ast }^6/({\mu }_0{r}^6)$. Spherical free fall with $\rho =\dot{M}/(4\pi r^{2}v)$ and $v\propto (GM/r{)}^{1/2}$ has ram pressure proportional to $\dot{M}(GM{)}^{1/2}{r}^{-5/2}$. Equating them gives the seventh-root scaling. Disc geometry changes the coefficient; the estimate ceases to describe an exterior magnetosphere if the resulting radius lies inside the star.