The magnetosphere of an oscillating neutron star. Non-vacuum treatment

We generalize a formula for the Goldreich-Julian charge density (rho(GJ)), originally derived for a rotating neutron star, for arbitrary oscillations of a neutron star with an arbitrary magnetic field configuration under the assumption of low current density in the inner parts of the magnetosphere. As an application, we consider the toroidal oscillation of a neutron star w ith a dipole magnetic field and calculate the energy losses. For some oscil lation modes, the longitudinal electric field cannot be cancelled by puttin g charged particles in the magnetosphere without the presence of a strong e lectric current ( j similar or equal to rho(GJ)cxc/omega r). It is shown th at the energy losses are strongly affected by plasma in the magnetosphere, and cannot be described by vacuum formulae.