Long-term evolution of discs around magnetic stars

We investigate the evolution of a thin viscous disc surrounding a magnetic star, including the spin-down of the star by the magnetic torques it exerts on the disc. The transition from an accreting to a non-accreting state, and the change of the magnetic torque across the corotation radius r(c) are included in a generic way, the widths of the transition taken in the range suggested by numerical simulations. In addition to the standard accreting state, in which the star gradually moves into spin equilibrium with the disc, two more states are found. An accreting state can develop into a 'dead' disc state, with inner edge r(in) well outside corotation. More often, a 'trapped' state develops, in which r(in) stays close to corotation even at very low accretion rates. The long-term evolution of these two states is different. In the dead state the star spins down incompletely, retaining much of its initial spin. In the trapped state the star can asymptotically spin-down to arbitrarily low rates, with its angular momentum transferred to the disc. We identify these outcomes with respectively the rapidly rotating and the very slowly rotating classes of Ap stars and magnetic white dwarfs.