On the axisymmetric pulsar atmosphere

It is commonly assumed that a magnetized neutron star, with aligned magnetic and rotational axes, necessarily has a poloidal current extending to infinity. An attempt is made to better understand the physical necessity of such a current, and to estimate the polar region over which it is likely to extend. To this end the relativistic hydrodynamic equations are examined under the assumption that the plasma drift velocity has only a toroidal component which, however, may represent differential rotation. It is shown that, subject to reasonable boundary conditions for the field on the stellar surface, the integrals of these equations must have a singularity within the light cylinder if the electron region extends from the stellar surface to the light cylinder - demonstrating that a purely toroidal drift velocity is not possible for a cold atmosphere. To estimate the extent of the stellar surface which might have a poloidal current, an approximate solution of the magnetic field as predicted by a corotational atmosphere near the stellar surface is obtained. It is shown that such an approximation predicts the existence of detached as well as closed magnetic surfaces.