A two-dimensional electrodynamical outer gap model for γ-ray pulsars: γ-ray spectrum

A two-dimensional electrodynamical model is used to study particle acceleration in the outer magnetosphere of a pulsar. The charge depletion from the Goldreich-Julian charge density causes a large electric field along the magnetic field lines. The charge particles are accelerated by the electric field and emit γ-rays via the curvature process. Some of the emitted γ-rays may collide with X-ray photons to make new pairs, which are accelerated again on the different field lines and emit γ-rays. We simulate the pair creation cascade in the meridional plane using the pair creation mean-free path, in which the X-ray photon number density is proportional to the inverse square of the radial distance. With the space charge density determined by the pair creation simulation, we solve the electric structure of the outer gap in the meridional plane and calculate the curvature spectrum. We investigate in detail the relation between the spectrum and total current, which is carried by the particles produced in the gap and/or injected at the boundaries of the gap. We demonstrate that the hardness of the spectrum is strongly controlled by the current carriers. Especially, the spectrum sharply softens if we assume a larger particle injection at the outer boundary of the outer gap. This is because the mean-free path of the pair creation of the inwardly propagating γ-ray photons is much shorter than the light radius, so many pairs are produced in the gap to quench the outer gap. Because the two-dimensional model can link both gap width along the magnetic field line and trans-field thickness with the spectral cut-off energy and flux, we can diagnose both the current through the gap and the inclination angle between the rotational and magnetic axes. We apply the theory to the Vela pulsar. By comparing the results with the Energetic Gamma Ray Experiment Telescope (EGRET) data, we rule out any cases that have a large particle injection at the outer boundary. We also suggest the inclination angle of αinc≥ 65°. The present model predicts the outer gap starting from near the conventional null charge surface for the Vela pulsar.