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Register a new userAn assessment of the pulsar outer gap model. II: Implications for the predicted $gamma$-ray spectra
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Authors
Daniele Vigan`o
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One of the most important predictions of any gap model for pulsar magnetospheres is the predicted $gamma$-ray spectra. In the outer gap model, the properties of the synchro-curvature radiation are sensitive to many parameters, whose realistic ranges have been studied in detail in an accompanying paper. There we demonstrated that the uncertainty in the radius of curvature, the magnetic field geometry, and the X-ray surface flux may affect by orders of magnitude the predicted flux and spectral peak in the $gamma$-ray regime. Here, we present a systematic, numerical study of the impact of the different parameters on the particle dynamics along the gap and calculate the emitted synchro-curvature radiation along the trajectory. By integrating the emitted radiation along the gap and convolving it with a parametrized particle distribution, we discuss how the comparison with the wealth of {em Fermi}-LAT data can be used to constrain the applicability of the model. The resulting spectra show very different energy peaks, fluxes and shapes, qualitatively matching the great variety of the observed {em Fermi}-LAT pulsars. In particular, if we see a large fraction of photons emitted from the initial part of the trajectory, we show that the spectra will be flatter at the low-energy {it Fermi}-LAT regime (100 MeV -- 1 GeV). This provides a solution for such observed flat spectra, while still maintain synchro-curvature radiation as the origin of these photons.
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The popular outer gap model of magnetospheric emission from pulsars has been widely applied to explain the properties observed in $gamma$-rays. However, its quantitative predictions rely on a number of approximations and assumptions that are usually overlooked. Here we examine them, reviewing the main ingredients entering in the model, evaluating their range of uncertainties. Usually, in the quantitative applications of the model, key parameters like the radius of curvature and the energies of the interacting photons are taken to be a fixed, single value. Instead, here we explore their realistic ranges, and the impact of these on the consistency of the model itself. We conclude that the popular evaluation of the trans-field size of the gap as a function of period and period derivative, is unreliable and affected by a huge dispersion. Last, the exploration of the possible values for the radius of curvature, the local magnetic field and other quantities deserve more attention for quantitative applications of the outer gap model, like the calculation of $gamma$-ray spectra, which is the subject of an accompanying paper.
We develop a model for gamma-ray emission from the outer magnetosphere of pulsars (the outer-gap model). The charge depletion causes a large electric field which accelerates electrons and positrons. We solve the electric field with radiation and pair creation processes self-consistently, and calculate curvature spectrum and Inverse-Compton (IC) spectrum. We apply this theory to PSR B0833-45 (Vela) and B1706-44 for which their surface magnetic fields, observed thermal X-rays are similar to each other. We find that each observed cut-off energies of the gamma-rays are well explained. By inclusion of emission outside the gap, the spectrum is in better agreement with the observations than the spectrum arising only from the inside of the gap. The expected TeV fluxes are much smaller than that observed by CANGAROO group in the direction of B1706-44.
With the large sample of young gamma-ray pulsars discovered by the Fermi Large Area Telescope (LAT), population synthesis has become a powerful tool for comparing their collective properties with model predictions. We synthesised a pulsar population based on a radio emission model and four gamma-ray gap models (Polar Cap, Slot Gap, Outer Gap, and One Pole Caustic) normalizing to the number of detected radio pulsars in select group of surveys. The luminosity and the wide beams from the outer gaps can easily account for the number of Fermi detections in 2 years of observations. The wide slot-gap beams requires an increase by a factor of ~10 of the predicted luminosity to produce a reasonable number of gamma-ray pulsars. Such large increases in the luminosity may be accommodated by implementing offset polar caps. The narrow polar-cap beams contribute at most only a handful of LAT pulsars. Standard distributions in birth location and pulsar spin-down power (Edot) fail to reproduce the LAT findings: all models under-predict the number of LAT pulsars with high Edot, and they cannot explain the high probability of detecting both the radio and gamma-ray beams at high Edot. The beaming factor remains close to 1 over 4 decades in Edot evolution for the slot gap whereas it significantly decreases with increasing age for the outer gaps. The evolution of the slot-gap luminosity with Edot is compatible with the large dispersion of gamma-ray luminosity seen in the LAT data. The stronger evolution predicted for the outer gap, which is linked to the polar cap heating by the return current, is apparently not supported by the LAT data. The LAT sample of gamma-ray pulsars therefore provides a fresh perspective on the early evolution of the luminosity and beam width of the gamma-ray emission from young pulsars, calling for thin and more luminous gaps.
We explore a non-stationary outer gap scenario for gamma-ray emission process in pulsar magnetosphere. Electrons/positrons that migrate along the magnetic field line and enter the outer gap from the outer/inner boundaries activate the pair-creation cascade and high-energy emission process. In our model, the rate of the particle injection at the gap boundaries is key physical quantity to control the gap structure and properties of the gamma-ray spectrum. Our model assumes that the injection rate is time variable and the observed gamma-ray spectrum are superposition of the emissions from different gap structures with different injection rates at the gap boundaries. The calculated spectrum superposed by assuming power law distribution of the particle injection rate can reproduce sub-exponential cut-off feature in the gamma-ray spectrum observed by Fermi-LAT. We fit the phase-averaged spectra for 43 young/middle-age pulsars and 14 millisecond pulsars with the model. Our results imply that (1) a larger particle injection at the gap boundaries is more frequent for the pulsar with a larger spin down power and (2) outer gap with an injection rate much smaller than the Goldreich-Julian value produces observe $>10$GeV emissions. Fermi-LAT gamma-ray pulsars show that (i) the observed gamma-ray spectrum below cut-off energy tends to be softer for the pulsar with a higher spin down rate and (ii) the second peak is more prominent in higher energy bands. Based on the results of the fitting, we describe possible theoretical interpretations for these observational properties. We also briefly discuss Crab-like millisecond pulsars that show phase-aligned radio and gamma-ray pulses.
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 $gamma$-rays via the curvature process. Some of the emitted $gamma$-rays may collide with $X$-ray photons to make new pairs, which are accelerated again on the different field lines in the gap and proceed similar processes. 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 inverse square of 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. 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 inclination angle between the rotational and magnetic axes. We apply the theory to the Vela pulsar. By comparing the results with the $EGRET$ data, we rule out any cases that have a large particle injection at the outer boundary. We also suggest the inclination angle of $alpha_{inc}geq65^{circ}$. The present model predicts the outer gap starting from near the conventional null charge surface for the Vela pulsar.
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