On the force-free magnetosphere of an aligned rotator

We investigate in detail the properties of the stationary force-free magnetosphere of an aligned rotator assuming the last closed field line is lying in the equatorial plane at large distances from pulsar. The pulsar equation is solved numerically using a multigrid code with high numerical resolution, and physical properties of the magnetosphere are obtained with high accuracy. We found a set of solutions with different sizes of the closed magnetic field line zone and verify the applicability of the force-free approximation. We discuss the role of electron-positron cascades in supporting the force-free magnetosphere and argue that the closed field line zone should grow with time at a slower rate than the light cylinder. This yields a pulsar breaking index of less than 3. It is shown that models of an aligned rotator magnetosphere with a widely accepted configuration of the magnetic field, such as the one considered in this paper, have serious difficulties. We discuss the solutions of this problem and argue that in any case pulsar energy losses should evolve with time differently than is predicted by the magnetodipolar formula.