= Spherical magnetospheric pressure balance
{title2=$R_M\sim[B_*^4R_*^{12}/(\mu_0^2GM\dot M^2)]^{1/7}$}
A dipole's <magnetic pressure> scales as $B_*^2R_*^6/(\mu_0r^6)$. Spherical <free fall> with $\rho=\dot M/(4\pi r^2v)$ 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.
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