Electrodynamics of Magnetars IV: Self-Consistent Model of the Inner   Accelerator, with Implications for Pulsed Radio Emission
  We consider the voltage structure in the open-field circuit and outer magnetosphere of a magnetar. The standard polar-cap model for radio pulsars is modified significantly when the polar magnetic field exceeds 1.8x10^{14} G. Pairs are created by accelerated particles via resonant scattering of thermal X-rays, followed by the nearly instantaneous conversion of the scattered photon to a pair. A surface gap is then efficiently screened by e+- creation, which regulates the voltage in the inner part of the circuit to ~10^9 V. We also examine the electrostatic gap structure that can form when the magnetic field is somewhat weaker, and deduce a voltage 10-30 times larger over a range of surface temperatures. We examine carefully how the flow of charge back to the star above the gap depends on the magnitude of the current that is extracted from the surface of the star, on the curvature of the magnetic field lines, and on resonant drag. The rates of different channels of pair creation are determined self-consistently, including the non-resonant scattering of X-rays, and collisions between gamma rays and X-rays. We find that the electrostatic gap solution has too small a voltage to sustain the observed pulsed radio output of magnetars unless i) the magnetic axis is nearly aligned with the rotation axis and the light of sight; or ii) the gap is present on the closed as well as the open magnetic field lines. Several properties of the radio magnetars -- their rapid variability, broad pulses, and unusually hard radio spectra -- are consistent with a third possibility, that the current in the outer magnetosphere is strongly variable, and a very high rate of pair creation is sustained by a turbulent cascade. 
Christopher Thompson