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.

Received; accepted 1 takata@hku.hk 2 yuwang@hku.hk 3 hrspksc@hkucc.hku.hk -2 - ABSTRACTWe discuss outer gap closure mechanism in the trans-field direction with the magnetic pair-creation process near the stellar surface. The gap closure by the magnetic pair-creation is possible if some fraction of the pairs are produced with an outgoing momentum. By assuming that multiple magnetic field will affect the local field near the stellar surface, we show a specific magnetic field geometry near the stellar surface resulting in the outflow of the pairs. Together with the fact that the electric field is weak below null charge surface, the characteristic curvature photon energy emitted by incoming particles, which were accelerated in the outer gap, decreases drastically to ∼ 100MeV near the stellar surface. We estimate the height measured from the last-open field line, above which 100 MeV photons is converted into pairs by the magnetic pair-creation. We also show the resultant multiplicity due to the magnetic pair-creation process could acquire M e ± ∼ 10 4 − 10 5 . In this model the fractional outer gap size is proportional to P −1/2 . The predicted gamma-ray luminosity (L γ ) and the characteristic curvature photon energy (E c ) emitted from the outer gap are proportional to B 2 P −5/2 and B 3/4 P −1 respectively. This model also predicts that L γ and E c are related to the spin down power (L sd ) or the spin down age of pulsars (τ ) as L γ ∝ L 5/8 sd or L γ ∝ τ −5/4 , and E c ∝ L 1/4 sd or E c ∝ τ −1/2 respectively.

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We report on Chandra observations of the black widow pulsar, PSR B1957+20. Evidence for a binary-phase dependence of the X-ray emission from the pulsar is found with a deep observation. The binary-phase-resolved spectral analysis reveals non-thermal X-ray emission of PSR B1957+20, confirming the results of previous studies. This suggests that the X-rays are mostly due to intra-binary shock emission, which is strongest when the pulsar wind interacts with the ablated material from the companion star. The geometry of the peak emission is determined in our study. The marginal softening of the spectrum of the non-thermal X-ray tail may indicate that particles injected at the termination shock are dominated by synchrotron cooling.

We investigate the electrodynamics of an outer gap in the meridional plane of the aligned-rotator. The charge depletion from the Goldreich-Julian charge density causes a large electric field along the magnetic field line. The electrons or the positrons are accelerated by the field-aligned electric field and radiate the $\gamma$-rays tangentially to the local magnetic field line. Some of such $\gamma$-rays collide with $X$-rays to materialize as the electron-positron pairs on different field lines from the field line on which they were emitted. As a result, the electric field structure is expected to change across the field lines. Including these trans-field effects, we solve the formation of the electric field self-consistently with the curvature radiation and the pair creation processes. The $\gamma$-ray emission and the pair creation are treated by use of Monte Carlo technique. We demonstrate that the distribution of the electric field along the field lines is affected by both the gap geometry and the external currents coming into the gap through the boundaries. In the electrodynamical model, it has been known that the solution disappears if the current density carried by the electron-positron pairs produced in the gap exceeds a critical value. We show that the critical current density is significantly increased when the trans-field structure is taken into account. We also find that the location of the inner boundary of the gap shifts toward the stellar surface from the conventional null surface as the current density increases. The reason for the shift is derived from the stability condition of the inner boundary. We also argue that the ideal-MHD condition holds outside of the gap only when the low energy particles coexist with the high energy particles migrating from the gap.Comment: 28 pages, 8 figures. Accepted for publication in MNRA