Alfv\'en-wave driven magnetic rotator winds from low-mass stars I: rotation dependences of magnetic braking and mass-loss rate

Observations of stellar rotation show that low-mass stars lose angular momentum during the main sequence. We simulate the winds of Sun-like stars with a range of rotation rates, covering the fast and slow magneto-rotator regimes, including the transition between the two. We generalize an Alfv\'en-wave driven solar wind model that builds on previous works by including the magneto-centrifugal force explicitly. In this model, the surface-averaged open magnetic flux is assumed to scale as $B_\ast f^{\rm open}_\ast \propto {\rm Ro}^{-1.2}$, where $f^{\rm open}_\ast$ and ${\rm Ro}$ are the surface open-flux filling factor and Rossby number, respectively. We find that, 1. the angular momentum loss rate (torque) of the wind is described as $\tau_w \approx 2.59 \times 10^{30} {\rm \ erg} \ \left( \Omega_\ast / \Omega_\odot \right)^{2.82}$, yielding a spin-down law $\Omega_\ast \propto t^{-0.55}$. 2. the mass-loss rate saturates at $\dot{M}_w \sim 3.4 \times 10^{-14} M_\odot {\rm \ yr^{-1}}$, due to the strong reflection and dissipation of Alfv\'en waves in the chromosphere. This indicates that the chromosphere has a strong impact in connecting the stellar surface and stellar wind. Meanwhile, the wind ram pressure scales as $P_w \propto \Omega_\ast^{0.57}$, which is able to explain the lower-envelope of the observed stellar winds by Wood et al. 3. the location of the Alfv\'en radius is shown to scale in a way that is consistent with 1D analytic theory. Additionally, the precise scaling of the Alfv\'en radius matches previous works which used thermally-driven winds. Our results suggest that the Alfv\'en-wave driven magnetic rotator wind plays a dominant role in the stellar spin-down during the main-sequence.