2003ApJ...591..334H

We investigate a stationary pair production cascade in the outer magnetosphere of a spinning neutron star. The charge depletion due to global flows of charged particles causes a large electric field along the magnetic field lines. Migratory electrons and/or positrons are accelerated by this field to radiate γ-rays via curvature and inverse Compton processes. Some of these γ-rays collide with the X-rays to materialize as pairs in the gap. The replenished charges partially screen the electric field, which is self-consistently solved together with the energy distribution of particles and γ-rays at each point along the field lines. By solving the set of Maxwell and Boltzmann equations, we demonstrate that an external injection of charged particles at nearly Goldreich-Julian rate does not quench the gap but shifts its position and that the particle energy distribution cannot be described by a power law. The injected particles are accelerated in the gap and escape from it with large Lorentz factors. We show that such escaping particles migrating outside of the gap contribute significantly to the γ-ray luminosity for young pulsars and that the soft γ-ray spectrum between 100 MeV and 3 GeV observed for the Vela pulsar can be explained by this component. We also discuss that the luminosity of the γ-rays emitted by the escaping particles is naturally proportional to the square root of the spin-down luminosity.