Wind-driven magnetic braking of a solar-type star ($\dot{J}\simeq -{\dot{M}}_w\Omega R_A^2$)

A magnetized stellar wind extracts angular momentum through matter transport and Maxwell stress tensor. The Alfvén radius supplies an effective lever arm larger than the stellar radius, with geometry factors absorbed into an averaged lever arm. Dynamo action couples the torque to rotation by supplying the magnetic field. For a simplified radial wind of fixed speed, $R_A^2$ scales as $B_{\ast }^2{R}_{\ast }^4/({\mu }_0{\dot{M}}_w{v}_w)$, so unsaturated $B_{\ast }\propto \Omega$ gives a torque proportional to $-{\Omega }^3$. Magnetic topology and wind thermodynamics can alter this scaling.

If the moment of inertia I changes slowly and wind-driven magnetic braking of a solar-type star has torque $-K{\Omega }^3$, then ${\Omega }^{-2}(t)={\Omega }^{-2}(t_0)+2K(t-t_0)/I$. At sufficiently late times within this unsaturated approximation, rotation scales as $t^{-1/2}$. This empirical main-sequence trend is not a universal law for saturated rapid rotators, changing stellar structure or every magnetic geometry.