The Axisymmetric Pulsar Magnetosphere

TLDR

We present the first axisymmetric force‑free magnetosphere model of an aligned rotating dipole under ideal MHD conditions, and propose it as a basis for future studies of pulsar magnetospheres when those conditions break down. We numerically determine the unique current distribution along open field lines that ensures a continuous, smooth solution, and then compute the resulting MHD wind dynamics from the derived field geometry. The magnetosphere’s relativistic outflow fails to reach gamma‑ray–producing energies, and the most probable locations for a breakdown of the ideal MHD condition are the polar cap, zero‑charge surface crossings, and return‑current boundary, rather than the light cylinder.

Abstract

We present, for the first time, the structure of the axisymmetric force-free magnetosphere of an aligned rotating magnetic dipole, in the case in which there exists a sufficiently large charge density (whose origin we do not question) to satisfy the ideal MHD condition, E · B=0, everywhere. The unique distribution of electric current along the open magnetic field lines that is required for the solution to be continuous and smooth is obtained numerically. With the geometry of the field lines thus determined, we compute the dynamics of the associated MHD wind. The main result is that the relativistic outflow contained in the magnetosphere is not accelerated to the extremely relativistic energies required for the flow to generate gamma rays. We expect that our solution will be useful as the starting point for detailed studies of pulsar magnetospheres under more general conditions, namely, when the force-free and/or the ideal MHD condition, E · B=0, are not valid in the entire magnetosphere. Based on our solution, we consider that the most likely positions of such an occurrence are the polar cap, the crossings of the zero space charge surface by open field lines, and the return current boundary, but not the light cylinder.