Electric Field Screening by Pairs in the Presence of Returning Positrons

We solve the one-dimensional Poisson equation along a magnetic field line, both analytically and numerically, for a given current density incorporating effects of returning positrons. We find that the number of returning positrons per one primary electrons should be smaller than unity, and the returning of positrons occurs only in a very short braking distance scale. As a result, for realistic polar cap parameters, the accelerating electric field will not be screened out; thus, the model fails to be self-consistent. A previous belief that pair creation with a pair density higher than the Goldreich-Julian density immediately screens out the electric field is unjustified. We suggest some possibilities to resolve this difficulty.