Spin equilibrium in strongly-magnetized accreting stars

The spin rate of a strongly-magnetized accreting star is regulated by the interaction between the star's magnetic field and the accreting gas. These systems are often hypothesized to be in `spin equilibrium' with their surrounding accretion flows such that the net spin change of the star as a result of accretion is very small. This condition requires that the accretion rate changes more slowly than it takes the star to reach spin equilibrium. However, this is not true for most magnetically accreting stars, which have strongly variable accretion outbursts (by one to many orders of magnitude) on timescales much shorter than the time it would take to reach spin equilibrium. This paper examines how accretion outbursts affect the time a star takes to reach spin equilibrium and its final equilibrium spin period. I consider several different models for angular momentum loss -- where angular momentum is carried away in an outflow (the standard `propeller', centrifugally-launched outflow), where most angular momentum is lost via a stellar wind, and where it is mainly transferred back to the accretion disc (the `trapped disc'). For transient sources, the `propeller' scenario leads to significantly longer times to reach spin equilibrium (often 10x), and shorter equilibrium spin periods (typically a factor of a few) than would be expected from spin equilibrium arguments, while the `trapped disc' does not. Accretion outbursts also show a smaller effect on spin down due to a stellar wind, mainly because in this case spin-down occurs for all accretion rates. The difference between a trapped disc and `propeller' scenario arises mainly from the ability of a trapped disc to efficiently spin the star down during quiescence. The results suggest that disc trapping plays a significant role in the spin evolution of strongly magnetic stars.