The spin of accreting stars: dependence on magnetic coupling to the disc

We formulate a general, steady-state model for the torque on a magnetized star from a surrounding accretion disc. For the first time, we include the opening of dipolar magnetic-field lines due to the differential rotation between the star and disc, so the magnetic topology then depends on the strength of the magnetic coupling to the disc. This coupling is determined by the effective slip rate of magnetic-field lines that penetrate the diffusive disc. Stronger coupling (i.e. lower slip rate) leads to a more open topology and thus to a weaker magnetic torque on the star from the disc. In the expected strong coupling regime, we find that the spin-down torque on the star is more than an order of magnitude smaller than calculated by previous models. We also use our general approach to examine the equilibrium (‘disc-locked’) state, in which the net torque on the star is zero. In this state, we show that the stellar spin rate is roughly an order of magnitude faster than predicted by previous models. This challenges the idea that slowly-rotating, accreting protostars are disc locked. Furthermore, when the field is sufficiently open (e.g. for mass accretion rates ≳5 × 10−9 M⊚ yr−1, for typical accreting protostars), the star will receive no magnetic spin-down torque from the disc at all. We therefore conclude that protostars must experience a spin-down torque from a source that has not yet been considered in the star-disc torque models - possibly from a stellar wind along the open field lines.