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Register a new userAn assessment of the pulsar outer gap model. I: Assumptions, uncertainties, and implications for the gap size and the accelerating field
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Authors
Daniele Vigan`o
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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.
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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.
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.
Two of the dominant channels to produce the black-hole binary mergers observed by LIGO and Virgo are believed to be the isolated evolution of stellar binaries in the field and dynamical formation in star clusters. Their relative efficiency can be characterized by a mixing fraction. Pair instabilities prevent stellar collapse from generating black holes more massive than about $45 M_odot$. This mass gap only applies to the field formation scenario, and it can be filled by repeated mergers in clusters. A similar reasoning applies to the binarys effective spin. If black holes are born slowly rotating, the high-spin portion of the parameter space (the spin gap) can only be populated by black hole binaries that were assembled dynamically. Using a semianalytical cluster model, we show that future gravitational-wave events in either the mass gap, the spin gap, or both can be leveraged to infer the mixing fraction between the field and cluster formation channels.
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.
A two-dimensional electrodynamic model is used to study particle acceleration and non-thermal emission mechanisms in the pulsar magnetospheres. We solve distribution of the accelerating electric field with the emission process and the pair-creation process in meridional plane, which includes the rotational axis and the magnetic axis. By solving the evolutions of the Lorentz factor, and of the pitch angle, we calculate spectrum in optical through $gamma$-ray bands with the curvature radiation, synchrotron radiation, and inverse-Compton process not only for outgoing particles, but also for ingoing particles, which were ignored in previous studies. We apply the theory to the Vela pulsar. We find that the curvature radiation from the outgoing particles is the major emission process above 10 MeV bands. In soft $gamma$-ray to hard X-ray bands, the synchrotron radiation from the ingoing primary particles in the gap dominates in the spectrum. Below hard X-ray bands, the synchrotron emissions from both outgoing and ingoing particles contribute to the calculated spectrum. The calculated spectrum is consistent with the observed phase-averaged spectrum of the Vela pulsar. We show that the observed five-peak pulse profile in the X-ray bands of the Vela pulsar is reproduced by the inward and outward emissions, and the observed double-peak pulse profile in $gamma$-ray bands is explained by the outward emissions.
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