The differentially rotating force-free magnetosphere of an aligned rotator: analytical solutions in the split-monopole approximation

In this paper we consider the stationary force-free magnetosphere of an aligned rotator when the plasma in the open field-line region rotates differentially as a result of the presence of a zone with an accelerating electric field in the polar cap of the pulsar. We study the impact of differential rotation on the current density distribution in the magnetosphere. Using the split-monopole approximation we obtain analytical expressions for the physical parameters of the differentially rotating magnetosphere. We find the range of admitted current density distributions under the requirement that the potential drop in the polar cap is less than the vacuum potential drop. We show that the current density distribution could deviate significantly from the 'classical' Michel distribution and could be made almost constant over the polar cap, even when the potential drop in the accelerating zone is of the order of 10 per cent of the vacuum potential drop. We argue that the differential rotation of the open magnetic field lines could play an important role in adjusting the current density between the magnetosphere and the polar-cap cascade zone and could affect the value of the pulsar braking index.