Propagation of radio emission in a pulsar magnetosphere is reviewed. The effects of polarization transfer, induced scattering and reprocessing to high energies are analysed with a special emphasis on the implications for the gamma-ray pulsars. The possibilities of the pulsar plasma diagnostics based on the observed radio pulse characteristics are also outlined. As an example, the plasma number density profiles obtained from the polarization data for the Vela and the gamma-ray millisecond pulsars J1446-4701, J1939+2134 and J1744-1134 are presented. The number densities derived tend to be the highest/lowest when the radio pulse leads/lags the gamma-ray peak. In the PSR J1939+2134, the plasma density profiles for the main pulse and interpulse appear to fit exactly the same curve, testifying to the origin of both radio components above the same magnetic pole and their propagation through the same plasma flow in opposite directions. The millisecond radio pulse components exhibiting flat position angle curves are suggested to result from the induced scattering of the main pulse by the same particles that generate gamma rays. This is believed to underlie the wide-sense radio/gamma-ray correlation in the millisecond pulsars. The radio quietness of young gamma-ray pulsars is attributed to resonant absorption, whereas the radio loudness to the radio beam escape through the periphery of the open field line tube.

Pulsar electrodynamics is reviewed emphasizing the role of the inductive electric field in an oblique rotator and the incomplete screening of its parallel component by charges, leaving ‘gaps’ with $E_{\Vert }\neq 0$ . The response of the plasma leads to a self-consistent electric field that complements the inductive electric field with a potential field leading to an electric drift and a polarization current associated with the total field. The electrodynamic models determine the charge density, ${\it\rho}$ , and the current density, $\boldsymbol{J}$ ; charge starvation refers to situations where the plasma cannot supply ${\it\rho}$ , resulting in a gap and associated particle acceleration and pair creation. It is pointed out that a form of current starvation also occurs implying a new class of gaps. The properties of gaps are discussed, emphasizing that static models are unstable, the role of large-amplitude longitudinal waves and the azimuthal dependence that arises across a gap in an oblique rotator. Wave dispersion in a pulsar plasma is reviewed briefly, emphasizing its role in radio emission. Pulsar radio emission mechanisms are reviewed, and it is suggested that the most plausible is a form of plasma emission.

We review our high-time-resolution radio observations of the Crab pulsar and compare our data to a variety of models for the emission physics. The Main Pulse and the Low Frequency Interpulse come from regions somewhere in the high-altitude emission zones (caustics) that also produce pulsed X-ray and ${\it\gamma}$ -ray emission. Although no emission model can fully explain these two components, the most likely models suggest they arise from a combination of beam-driven instabilities, coherent charge bunching and strong electromagnetic turbulence. Because the radio power fluctuates on a wide range of time scales, we know the emission zones are patchy and dynamic. It is tempting to invoke unsteady pair creation in high-altitude gaps as the source of the variability, but current pair cascade models cannot explain the densities required by any of the likely models. It is harder to account for the mysterious High Frequency Interpulse. We understand neither its origin within the magnetosphere nor the striking emission bands in its dynamic spectrum. The most promising models are based on analogies with solar zebra bands, but they require unusual plasma structures which are not part of our standard picture of the magnetosphere. We argue that radio observations can reveal much about the upper magnetosphere, but work is required before the models can address all of the data.

We want to understand what drives magnetospheric dissipation in the equatorial current sheet. Numerical simulations have limitations and, unless we have a clear a priori understanding of the physical processes involved, their results can be misleading. We argue that the canonical pulsar magnetosphere is strongly dissipative and that a large fraction (up to 30–40 % in an aligned rotator) of the spindown luminosity is redirected towards the equator where it is dissipated into particle acceleration and emission of radiation. We show that this is due to the failure of the equatorial electric current to cross the Y-point at the tip of the corotating zone.

A critical revision of the essential principles of the physics of relativistic flows of cold plasma is given. We prove that the approximation of ideal magnetic hydrodynamics of the cold plasma is equivalent to the drift approximation of motion of charged particles in an electromagnetic field. The equations of magnetohydrodynamics are obtained from equations for the drift motion of the charged particles. The conditions of application of the equations of ideal magnetohydrodynamics are obtained. In the case of the Crab pulsar the violation of the frozen-in condition can happen at a distance that well exceeds the distance to the termination shock. One fluid MHD can be incorrect at the light cylinder provided that the Lorentz factor of the plasma exceeds $10^{4}$ and the curvature radius of the flow line is comparable with the light cylinder. It is shown that the electric currents in the cold plasma are the result of the inertial drift motion of the charged particles in the crossed electric and magnetic fields.

In the present work there is presented a model describing transfer of the Crab pulsar’s spin-down energy into the powerful synchrotron emission of the nebula. The process of the energy transfer consists of several consecutive stages. The physical processes underlying the theoretical model provide us with the synchrotron emission spectrum, which fits well with the observed one.

The large number of ${\it\gamma}$ -ray pulsars discovered by the Fermi Gamma-Ray Space Telescope since its launch in 2008 dwarfs the handful that were previously known. The variety of observed light curves makes possible a tomography of both the ensemble-averaged field structure and the high-energy emission regions of a pulsar magnetosphere. Fitting the ${\it\gamma}$ -ray pulsar light curves with model magnetospheres and emission models has revealed that most of the high-energy emission, and the particles acceleration, takes place near or beyond the light cylinder, near the current sheet. As pulsar magnetosphere models become more sophisticated, it is possible to probe magnetic field structure and emission that are self-consistently determined. Light curve modelling will continue to be a powerful tool for constraining the pulsar magnetosphere physics.

Successful phenomenological models of pulsar wind nebulae assume efficient dissipation of the Poynting flux of the magnetized electron–positron wind as well as efficient acceleration of the pairs in the vicinity of the termination shock, but how this is realized is not yet well understood. This paper suggests that the corrugation of the termination shock, at the onset of nonlinearity, may lead towards the desired phenomenology. Nonlinear corrugation of the termination shock would convert a fraction of order unity of the incoming ordered magnetic field into downstream turbulence, slowing down the flow to sub-relativistic velocities. The dissipation of turbulence would further preheat the pair population on short length scales, close to equipartition with the magnetic field, thereby reducing the initial high magnetization to values of order unity. Furthermore, it is speculated that the turbulence generated by the corrugation pattern may sustain a relativistic Fermi process, accelerating particles close to the radiation reaction limit, as observed in the Crab nebula. The required corrugation could be induced by the fast magnetosonic modes of downstream nebular turbulence; but it could also be produced by upstream turbulence, either carried by the wind or seeded in the precursor by the accelerated particles themselves.

Pulsar-wind nebulae emit an extremely broad spectrum of continuum radiation, from low radio frequencies to TeV gamma rays. The part of the spectral energy distribution (SED) from radio through MeV gamma rays is due to synchrotron emission from a distribution of relativistic electrons (or pairs) which can be described by one or more power laws. This spectrum exhibits that particle energy distribution, responsible also for the higher-energy (GeV–TeV) part of the SED, due to inverse-Compton upscattering of one of three photon fields: the synchrotron spectrum, the cosmic microwave background, or ambient optical/infrared photons. However, in a few sources, primary hadrons may produce GeV–TeV gamma rays through the decay of neutral pions produced in inelastic cosmic-ray collisions with thermal gas. The higher-energy end of the particle spectrum, producing synchrotron photons above approximately 10 keV, holds clues to the particle acceleration process. However, its detailed study requires imaging spectroscopy in this energy range, not available until the NuSTAR mission beginning in 2012, which performs true imaging between 3 and 78 keV with ${\sim}1^{\prime }$ angular resolution. I review NuSTAR observations of the first three pulsar-wind nebulae (PWNe) to be examined in this way: the Crab Nebula, G21.5–0.9 and MSH 15–52. All three show spectral structure not previously known: spectral steepening in certain locations and overall source shrinkage with increasing photon energy. The Crab Nebula has different shrinkage rates along the torus and along the northwest counter-jet. The latter rate is similar to that for both the other sources (FWHM $\propto E^{m}$ with $m\sim -0.2$ ). I discuss implications of these results for models of particle transport in PWNe.

Neutron stars are fascinating astrophysical objects immersed in strong gravitational and electromagnetic fields, at the edge of our current theories. These stars manifest themselves mostly as pulsars, emitting a timely very stable and regular electromagnetic signal. Even though discovered almost fifty years ago, they still remain mysterious compact stellar objects. In this review, we summarize the most fundamental theoretical aspects of neutron star magnetospheres and winds. The main competing models explaining their radiative properties like multi-wavelength pulse shapes and spectra and the underlying physical processes such as pair creation and radiation mechanisms are scrutinized. A global but still rather qualitative picture slowly emerges thanks to recent advances in numerical simulations on the largest scales. However considerations about pulsar magnetospheres remain speculative. For instance, the exact composition of the magnetospheric plasma is not yet known. Is it solely filled with a mixture of $e^{\pm }$  leptons or does it contain a non-negligible fraction of protons and/or ions? Is it almost entirely filled or mostly empty except for some small anecdotal plasma filled regions? Answers to these questions will strongly direct the description of the magnetosphere to seemingly contradictory results leading sometimes to inconsistencies. Nevertheless, accounts are given as to the latest developments in the theory of pulsar magnetospheres and winds, the existence of a possible electrosphere and physical insight obtained from related observational signatures of multi-wavelength pulsed emission.

In the last decade, the relativistic magnetohydrodynamic (MHD) modelling of pulsar wind nebulae, and of the Crab nebula in particular, has been highly successful, with many of the observed dynamical and emission properties reproduced down to the finest detail. Here, we critically discuss the results of some of the most recent studies: namely the investigation of the origin of the radio emitting particles and the quest for the acceleration sites of particles of different energies along the termination shock, by using wisp motions as a diagnostic tool; the study of the magnetic dissipation process in high magnetization nebulae by means of new long-term three-dimensional simulations of the pulsar wind nebula evolution; the investigation of the relativistic tearing instability in thinning current sheets, leading to fast reconnection events that might be at the origin of the Crab nebula gamma-ray flares.

We review multiwavelength properties of pulsar wind nebulae created by supersonically moving pulsars and the effects of pulsar motion on the pulsar wind nebulae morphologies and the ambient medium. Supersonic pulsar wind nebulae are characterized by bow-shaped shocks around the pulsar and/or cometary tails filled with the shocked pulsar wind. In the past several years significant advances in supersonic pulsar wind nebula studies have been made in deep observations with the Chandra and XMM-Newton X-ray observatories and the Hubble Space Telescope. In particular, these observations have revealed very diverse supersonic pulsar wind nebula morphologies in the pulsar vicinity, different spectral behaviours of long pulsar tails, the presence of puzzling outflows misaligned with the pulsar velocity and far-UV bow shocks. Here we review the current observational status focusing on recent developments and their implications.

The extreme properties of the gamma-ray flares in the Crab nebula present a clear challenge to our ideas on the nature of particle acceleration in relativistic astrophysical plasma. It seems highly unlikely that standard mechanisms of stochastic type are at work here and hence the attention of theorists has switched to linear acceleration in magnetic reconnection events. In this series of papers, we attempt to develop a theory of explosive magnetic reconnection in highly magnetized relativistic plasma which can explain the extreme parameters of the Crab flares. In the first paper, we focus on the properties of the X-point collapse. Using analytical and numerical methods (fluid and particle-in-cell simulations) we extend Syrovatsky’s classical model of such collapse to the relativistic regime. We find that the collapse can lead to the reconnection rate approaching the speed of light on macroscopic scales. During the collapse, the plasma particles are accelerated by charge-starved electric fields, which can reach (and even exceed) values of the local magnetic field. The explosive stage of reconnection produces non-thermal power-law tails with slopes that depend on the average magnetization $\unicode[STIX]{x1D70E}$ . For sufficiently high magnetizations and vanishing guide field, the non-thermal particle spectrum consists of two components: a low-energy population with soft spectrum that dominates the number census; and a high-energy population with hard spectrum that possesses all the properties needed to explain the Crab flares.

Using analytical and numerical methods (fluid and particle-in-cell simulations) we study a number of model problems involving merger of magnetic flux tubes in relativistic magnetically dominated plasma. Mergers of current-carrying flux tubes (exemplified by the two-dimensional ‘ABC’ structures) and zero-total-current magnetic flux tubes are considered. In all cases regimes of spontaneous and driven evolution are investigated. We identify two stages of particle acceleration during flux mergers: (i) fast explosive prompt X-point collapse and (ii) ensuing island merger. The fastest acceleration occurs during the initial catastrophic X-point collapse, with the reconnection electric field of the order of the magnetic field. During the X-point collapse, particles are accelerated by charge-starved electric fields, which can reach (and even exceed) values of the local magnetic field. The explosive stage of reconnection produces non-thermal power-law tails with slopes that depend on the average magnetization $\unicode[STIX]{x1D70E}$ . For plasma magnetization $\unicode[STIX]{x1D70E}\leqslant 10^{2}$ the spectrum power-law index is $p>2$ ; in this case the maximal energy depends linearly on the size of the reconnecting islands. For higher magnetization, $\unicode[STIX]{x1D70E}\geqslant 10^{2}$ , the spectra are hard, $p<2$ , yet the maximal energy $\unicode[STIX]{x1D6FE}_{\text{max}}$ can still exceed the average magnetic energy per particle, ${\sim}\unicode[STIX]{x1D70E}$ , by orders of magnitude (if $p$ is not too close to unity). The X-point collapse stage is followed by magnetic island merger that dissipates a large fraction of the initial magnetic energy in a regime of forced magnetic reconnection, further accelerating the particles, but proceeds at a slower reconnection rate.

We develop a model of gamma-ray flares of the Crab Nebula resulting from the magnetic reconnection events in a highly magnetised relativistic plasma. We first discuss physical parameters of the Crab Nebula and review the theory of pulsar winds and termination shocks. We also review the principle points of particle acceleration in explosive reconnection events [Lyutikov et al., J. Plasma Phys., vol. 83(6), p. 635830601 (2017a); J. Plasma Phys., vol. 83(6), p. 635830602 (2017b)]. It is required that particles producing flares are accelerated in highly magnetised regions of the nebula. Flares originate from the poleward regions at the base of the Crab’s polar outflow, where both the magnetisation and the magnetic field strength are sufficiently high. The post-termination shock flow develops macroscopic (not related to the plasma properties on the skin-depth scale) kink-type instabilities. The resulting large-scale magnetic stresses drive explosive reconnection events on the light-crossing time of the reconnection region. Flares are produced at the initial stage of the current sheet development, during the X-point collapse. The model has all the ingredients needed for Crab flares: natural formation of highly magnetised regions, explosive dynamics on the light travel time, development of high electric fields on macroscopic scales and acceleration of particles to energies well exceeding the average magnetic energy per particle.