Long-term evolution of the force-free twisted magnetosphere of a magnetar

We study the long-term quasi-steady evolution of the force-free magnetosphere of a magnetar coupled to its internal magnetic field. We find that magnetospheric currents can be maintained on long timescales of the order of thousands of years. Meanwhile, the energy, helicity and twist stored in the magnetosphere all gradually increase over the course of this evolution, until a critical point is reached, beyond which a force-free magnetosphere cannot be constructed. At this point, some large-scale magnetospheric rearrangement, possibly resulting in an outburst or a flare, must occur, releasing a large fraction of the stored energy, helicity and twist. After that, the quasi-steady evolution should continue in a similar manner from the new initial conditions. The timescale for reaching this critical point depends on the overall magnetic field strength and on the relative fraction of the toroidal field. The energy stored in the force-free magnetosphere is found to be up to $\sim 30\%$ larger than the corresponding vacuum energy. This implies that for a $10^{14}$ G field at the pole, the energy budget available for fast magnetospheric events is of the order of a few $10^{44}$ erg. The spindown rate is estimated to increase by up to $\sim 60\%$, since the dipole content in the magnetosphere is enhanced by the currents present there. A rough estimate of the braking index $n$ reveals that it is systematically $n < 3$ for the most part of the evolution, consistent with actual measurements for pulsars and early estimates for several magnetars.