Transient particle acceleration in strongly magnetized neutron stars. II. Effects due to a dipole field geometry

Sheared Alfvén waves generated by nonradial crustal disturbances above the polar cap of a stronngly magnetized neutron star (B ≲ 10¹² gauss) induce an electric field component parallel to B. Earlier calculations assuming a uniform magnetic field showed that this mechanism could successfully accelerate particles to Lorentz factors γ ∼ (50 cm) m, where m (∼2-3 cm^-1) characterizes the degree of shearing. However, magnetic field gradients in a realistic dipole field geometry cannot be ignored. Our goal here is to determine the manner in which the strong radial dependence of B affects the propagation of these sheared Alfvén waves, and whether this magnetohydrodynamic process is still an effective particle accelerator. By comparison to the field gradients, the field-line curvature is small within a stellar radius of the polar cap, and we therefore use an approximate form of the dipole field that ignores the component perpendicular to the z-axis. We find that although the general field equation is quite complicated, a simple wavelike solution can still be obtained under the conditions of interest (i.e., B ≲ 10¹² G, and a particle number density n_e ≲ 10²⁶ cm^-3) for which the Alfvén phase velocity → c and n_e decouples from the wave equation. Our results may be applicable to γ-ray burst sources.