Accretion-powered Stellar Winds. III. Spin-Equilibrium Solutions

We compare the stellar wind torque calculated in a previous work (Paper II) to the spin-up and spin-down torques expected to arise from the magnetic interaction between a slowly rotating (~10% of breakup) pre-main-sequence star and its accretion disk. This analysis demonstrates that stellar winds can carry off orders of magnitude more angular momentum than can be transferred to the disk, provided that the mass outflow rates are substantially greater than the solar wind. Thus, the equilibrium spin state is simply characterized by a balance between the angular momentum deposited by accretion and that extracted by a stellar wind. We derive a semianalytic formula for predicting the equilibrium spin rate as a function only of the ratio of Ṁw/Ṁa and a dimensionless magnetization parameter, Ψ ≡ B2*R2*(Ṁavesc)−1, where Ṁw is the stellar wind mass outflow rate, Ṁa is the accretion rate, B* is the stellar surface magnetic field strength, R* is the stellar radius, and vesc is the surface escape speed. For parameters typical of accreting pre-main-sequence stars, this explains spin rates of ~10% of breakup speed for Ṁw/Ṁa ∼ 0.1. Finally, the assumption that the stellar wind is driven by a fraction of the accretion power leads to an upper limit to the mass flow ratio of Ṁw/Ṁa ≲ 0.6.