Electrodynamics of an Outer-Gap Accelerator: Formation of a Soft Power-Law Gamma-ray Spectrum

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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 gamma-rays via synchro-curvature and inverse-Compton processes, some of which 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 distribution functions of particles and gamma-rays. By solving these Vlasov equations, we demonstrate that an external injection of charged particles at nearly Goldreich-Julian rate does not quench the outer gap, but shifts its position. For example, if the injection rate across the inner (or outer) boundary is close to the Goldreich-Julian value, the gap is located near to the light cylinder (or the star surface). The injected particles are accelerated in the gap and escape from it with large Lorentz factors. We show that such escaping particles emit significant curvature gamma-rays, which forms a soft power-law spectrum below GeV energies. We also discuss that the luminosity of the gamma-rays emitted outside of the gap in this way, is naturally proportional to the square root of the spin-down luminosity.

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