An oblique pulsar magnetosphere with a plasma conductivity

An oblique pulsar magnetosphere with a plasma conductivity is studied by using a pseudo-spectral method. In the pseudo-spectral method, the time-dependent Maxwell equations are solved, both electric and magnetic fields are expanded in terms of the vector spherical harmonic (VSH) functions in spherical geometry and the divergencelessness of magnetic field is analytically enforced by a projection method. The pulsar magnetospheres in infinite (i. e., force-free approximation) and finite conductivities are simulated and a family of solutions that smoothly transition from the Deutsch vacuum solution to the force-free solution are obtained. The $\sin^2\alpha$ dependence of the spin-down luminosity on the magnetic inclination angle $\alpha$ in which the full electric current density are taken into account is retrieved in the force-free regime.