Magnetic Braking Formulation for Sun-like Stars: Dependence on Dipole Field Strength and Rotation Rate

We use two-dimensional axisymmetric magnetohydrodynamic simulations to compute steady-state solutions for solar-like stellar winds from rotating stars with dipolar magnetic fields. Our parameter study includes 50 simulations covering a wide range of relative magnetic field strengths and rotation rates, extending from the slow- and approaching the fast-magnetic-rotator regimes. Using the simulations to compute the angular momentum loss, we derive a semi-analytic formulation for the external torque on the star that fits all of the simulations to a precision of a few percent. This formula provides a simple method for computing the magnetic braking of Sun-like stars due to magnetized stellar winds, which properly includes the dependence on the strength of the magnetic field, mass loss rate, stellar radius, surface gravity, and spin rate, and which is valid for both slow and fast rotators.