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

Abstract

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.

High-energy characteristics of the schizophrenic pulsar PSR J1846-0258 in Kes 75

“…Hirotani & Shibata (1999) demonstrated in a one-dimensional analysis of a pair-production cascade along the magnetic field line, that the outer gap does exist as solution of the Maxwell and Boltzmann equations. This model was further elaborated in a 2D scenario (Takata et al 2006;Hirotani 2006;) and finally the photon propagation and pair production is solved in the full 3D magnetosphere Hirotani 2008). For these self-consistent solutions the gap extends past the nullcharge surface toward the neutron star.…”

High-Energy Emission From Pulsar Magnetospheres

“…Extending the one-dimensional studies [92,93,94,95,96,86], Refs. [97,98] solved (a simpler form of) Eq. (12) to reveal that the gap inner boundary is located inside of the null surface owing to the pair creation within the gap, assuming that the particle motion immediately saturates in the balance between electric and radiation-reaction forces.…”

“…Extending Ref. [97,98] by solving the Lorentz-factor and pitch-angle evolution of individual e ± 's, H06 [78] solved the gap electrodynamics for the Crab pulsar. For h m > 0.047, the created current density j e becomes super Goldreich-Julian in the sense that ρ < ρ GJ < 0 holds at the inner boundary (left panel of fig.…”

The multicomponent model of the Crab Pulsar at energies above 25 GeV

“…A common feature of these models is that the broadband emission comes from components originating from different physical processes. Takata & Chang (2007) developed a 3D outer gap model based on the 2D analytical solution of the accelerating field and particle motion by Takata et al (2004Takata et al ( , 2006 and Hirotani (2006). In this model the X-ray emission of P1 is due to two separate components, with curved, roughly log-parabolic, spectra and originating from synchrotron emission of secondary pairs in different regions of the outer gap, i.e.…”