The dynamics of black hole magnetospheres critically depend on the black hole spin and on the structure of the accretion flow. In the limit of a Schwarzschild black hole immersed in a zero-net angular momentum flow, accretion is spherical. However, in the presence of a large-scale vertical magnetic field, the classical Bondi accretion model is significantly altered. The frozen-in field is stretched radially as the plasma is pulled inward by gravity. This continues until the restoring force from the magnetic tension suddenly expels the material and resets the field, allowing a new cycle to begin. Although this scenario has been well depicted in previous studies, it remains incomplete as the issues of dissipation and particle acceleration are not yet fully resolved. In this work, we aim to revisit these issues with a first-principles kinetic plasma model. We perform two-dimensional global general relativistic particle-in-cell simulations of magnetized spherical accretion onto a Schwarzschild black hole, for both pair and electron-ion plasmas. The simulations are evolved over long timescales to capture multiple flux eruption events and establish a quasi-steady state. For each accretion cycle, we find that the system goes through three main stages: (i) an ideal advection phase where magnetic flux through the horizon increases quasi-linearly with time; (ii) a reconnection-regulated phase where the net increase of the flux is slowed down by intermittent reconnection events near the horizon; and (iii) a flaring phase when a major, large-scale reconnection event expels the flux, leading to efficient particle acceleration. The emergence of large-amplitude quasi-periodic flux eruptions and concomitant particle acceleration is reminiscent of Sgr A* flaring activity. This phenomenon could also be applicable to quiescent black holes, especially isolated black holes accreting the interstellar medium.

Transitional millisecond pulsars constitute a peculiar subclass of neutron stars in which the pulsar alternates between accretion-powered and rotation-powered states, depending on the variations in the mass accretion flow coming from a low-mass companion star. A third intermediate state, referred to as “sub-luminous disk state”, has been identified. During this state, observations indicate the presence of a disk surrounding the pulsar, and the system exhibits intriguing features, such as broad optical and X-ray pulsations characterized by a high luminosity. To date, no ab initio model of a pulsar wind interacting with an accretion disk has been developed to address these observables. We perform three-dimensional particle-in-cell simulations of a pulsar magnetosphere surrounded by a perfectly conducting torus to model the interaction between the pulsar wind and the disk. We find that the presence of the disk induces a significant reconfiguration of the magnetosphere compared to the rotation-powered state, leading to enhanced plasma density at the inner disk boundary, increased magnetic field strength, and more efficient plasma isotropization and particle acceleration. As a result, the synchrotron radiation is substantially enhanced, and characterized by a strong continuous component and either one or two-peaked light curves, depending on the pulsar's magnetic obliquity. The polarization degree is reduced compared to isolated systems, and its energy dependence is explored. The rotation of the polarization angle can also be altered, depending on the observer's viewing angle. The model successfully reproduces some of the main features of the optical and X-ray pulsed emission originating from PSR J1023+0038, thereby corroborating the scenario in which these pulsations originate from synchrotron radiation generated as the pulsar wind interacts with the inner edge of the disk.

Rotating black holes are known to launch relativistic jets and accelerate particles if they accrete a magnetized plasma. It remains unclear, however, how the global magnetic field orientation affects the jet powering efficiency. We propose the first kinetic study of a collisionless plasma around a Kerr black hole embedded in a magnetic field that is inclined with respect to the black hole spin axis. Using three-dimensional general relativistic particle-in-cell simulations, we show that while oblique magnetic field configurations significantly reduce the jet power, particle acceleration still remains highly efficient. This suggests that black holes producing a weak jet might still be bright sources of nonthermal radiation and cosmic rays.

Rotation-powered pulsars represent the main class of identified gamma-ray sources in the Galaxy. The wealth of observational data collected by the AGILE and Fermi gamma-ray space telescopes in the GeV range, and by ground-based Cherenkov telescopes in the TeV band provide invaluable insights into how relativistic plasmas dissipate and accelerate particles. Decoding the information contained in the gamma-ray pulses profile is an important step to understand how pulsars work. In this study, we aim at putting an ab initio plasma model of pulsar magnetospheres to the test, in light of the most recent gamma-ray observations in the GeV and TeV bands. To this end, we present of a new series of global particle-in-cell simulations of an inclined pulsar magnetosphere. High-quality synthetic pulse profiles in the synchrotron and inverse Compton channels are reconstructed to study in greater details their morphology and their energy dependence. We also perform a fit of observed lightcurves with the model, using the third Fermi-LAT gamma-ray pulsar catalog. Reconnection in the wind current sheet powers synchrotron and inverse Compton emission. The modeled pulse profiles reproduce some of the salient features of observed gamma-ray pulsars, including the mysterious Vela-like lightcurves, such as: the generic double-peaked structure, the presence of a bridge or third peak in between the main pulses, the pulse narrowing with increasing energy. The bolometric synchrotron radiative efficiency is strictly limited by the reconnection rate. Our global kinetic simulations are able to match observed pulse profiles. Such direct comparisons will help drive and focus future simulation developments.

We present the first collisionless realization of two-dimensional axisymmetric black hole accretion consistent with a persistent magnetically arrested disk state. The accretion flow, consisting of an ion-electron disk plasma combined with magnetospheric pair creation effects, is simulated using first-principles general-relativistic particle-in-cell methods. The simulation is evolved over significant dynamical timescales during which a quasi-steady accretion state is reached with several magnetic flux eruption cycles. We include a realistic treatment of inverse Compton scattering and pair production, which allows for studying the interaction between the collisionless accretion flow and pair-loaded jet. Our findings indicate that magnetic flux eruptions associated with equatorial magnetic reconnection within the black hole magnetosphere and the formation of spark gaps are locations of maximal particle acceleration. Flux eruptions, starting near the central black hole, can trigger Kelvin-Helmholtz-like vortices at the jet-disk interface that facilitate efficient mixing between disk and jet plasma in this region. Transient periods of increased pair production following magnetic flux eruptions and reconnection events are responsible for most of the highly accelerated particles.

Simulations of relativistic plasmas traditionally focus on the dynamics of two-species mixtures of charged particles under the influence of external magnetic fields and those generated by particle currents. However, the extreme conditions of astrophysical plasmas near compact objects, such as black holes and neutron stars, are often characterized by mixtures of electrons, protons, and positrons, whose dynamics can differ significantly, because of the considerable mass contrast. We present the first two-dimensional particle-in-cell simulations of relativistic turbulence and magnetic reconnection in a three-species plasma, varying the relative abundances of electrons, protons, and positrons, while employing realistic mass ratios to achieve unprecedented accuracy. We find that turbulence leads to the formation of magnetic islands, current sheets, and plasmoids. Reconnection occurs between these structures, with plasma composition playing a key role in determining the number of reconnection sites and their energy conversion efficiency. In particular, as the proton fraction increases, very small-scale features of the turbulence are washed out, while global dissipative effects are amplified. Finally, using a novel generalization of Ohm’s law for a relativistic multispecies plasma, we find that the reconnection rate is primarily governed by the electric fields associated with the divergence of the positron and electron pressure tensors. These results provide new insights into dissipation and particle acceleration in turbulent relativistic plasmas, such as those near black holes and neutron stars, and can be used to interpret their high-energy emission and phenomenology.

Context. Relativistic magnetic reconnection is one of the most fundamental mechanisms that is considered responsible for the acceleration of relativistic particles in astrophysical jets and magnetospheres of compact objects. Understanding the properties of the dissipation of magnetic fields and the formation of non-ideal electric fields is of paramount importance to quantify the efficiency of reconnection at energizing charged particles. Aims. Recent results from particle-in-cell (PIC) simulations suggest that the fundamental properties of how magnetic fields dissipate in a current sheet might be captured by an "effective resistivity" formulation, which would locally enhance the amount of magnetic energy dissipated and favor the onset of fast reconnection. Our goal is to assess this ansatz quantitatively by comparing fluid models of magnetic reconnection with a non-constant magnetic diffusivity and fully kinetic models. Methods. We performed 2D resistive relativistic magnetohydrodynamic (ResRMHD) simulations of magnetic reconnection combined to PIC simulations using the same initial conditions (i.e., a Harris current sheet). We explored the impact of crucial parameters such as the plasma magnetization, its mass density, the grid resolution, and the characteristic plasma skin depth. Results. Our ResRMHD models with effective resistivity are able to quantitatively reproduce the dynamics of fully kinetic models of relativistic magnetic reconnection. In particular, they lead to reconnection rates consistent with PIC simulations, whereas for constant-resistivity fluid models, the reconnection dynamics is generally ten times slower. Even at modest resolutions, adopting an effective resistivity can qualitatively capture the properties of kinetic reconnection models and produce reconnection rates compatible with collisionless models (i.e., on the order of similar to 10(-1)).

Giant radio lobes forming at both ends of powerful extragalactic jets are often suspected to generate and host some of the highest-energy cosmic rays in the Universe. Using particle-in-cell simulations of a magnetized relativistic jet termination shock, we show that particles are efficiently accelerated up to the confinement limit of the system provided that the global transverse structure of the jet is taken into account. A strong tangential velocity discontinuity grows between the core and the edges of the jet leading to macroscopic shear-flow particle acceleration. Energetic particles accumulate in an over-pressured and under-dense bubble near the shock front before escaping in the downstream medium via a Von Karman vortex street. A possible observational signature of this mechanism would be the detection of an underluminous synchrotron cavity upstream of AGN jet hotspots. These results suggest that extragalactic jet termination shocks provide exquisite conditions to accelerate cosmic rays up to the highest energies.

Turbulence in classical fluids is characterized by persistent structures that emerge from the chaotic landscape. We investigate the analogous process in fully kinetic plasma turbulence by using high-resolution, direct numerical simulations in two spatial dimensions. We observe the formation of long-lived vortices with a profile typical of macroscopic, magnetically dominated force-free states. Inspired by the Harris pinch model for inhomogeneous equilibria, we describe these metastable solutions with a self-consistent kinetic model in a cylindrical coordinate system centered on a representative vortex, starting from an explicit form of the particle velocity distribution function. Such new equilibria can be simplified to a Gold–Hoyle solution of the modified force-free state. Turbulence is mediated by the long-lived structures, accompanied by transients in which such vortices merge and form self-similarly new metastable equilibria. This process can be relevant to the comprehension of various astrophysical phenomena, going from the formation of plasmoids in the vicinity of massive compact objects to the emergence of coherent structures in the heliosphere.

Context. Relativistic magnetic reconnection studies have so far focused on symmetric configurations, where the upstream plasma has identical properties on the two sides of the layer. Yet, just like nonrelativistic reconnection on the dayside of the Earth's magnetosphere, relativistic reconnection can also operate at the interface between highly asymmetric environments. The boundary layer between a relativistic jet and an accretion flow forming around a supermassive black hole can present asymmetric configurations in terms of plasma composition, bulk velocity, temperature, and magnetization. Aims. We conducted the first study of relativistic magnetic reconnection where the upstream plasma is composed of electron-positron pairs on one side, and electrons and ions on the other. We also investigated the impact of a relativistic symmetric shear flow applied along the reconnecting field lines. Methods. We simulated magnetic reconnection using 2D particle-in-cell simulations. The initial setup was adapted from a classic Harris layer without a guide field, modified to accommodate plasma-composition and shear asymmetries in the upstream medium. Results. For a composition-asymmetric setup, we find that the reconnection dynamics is driven by the electron-ion side, which is the plasma with the lowest magnetization. The energy partition favors accelerating ions at the expense of electrons even more than in a corresponding symmetric setup. With respect to shear, a super-Alfv & eacute;nic upstream decreases the laboratory-frame reconnection rate, but, unlike in nonrelativistic studies, does not shut off reconnection completely. Conclusions. The asymmetries examined in this work lower the overall efficiency of electron acceleration relative to corresponding symmetric configurations. In the context of a black hole jet-disk boundary, asymmetric reconnection alone is probably not efficient at accelerating electrons to very high energies, but it might facilitate plasma mixing and particle injection for other acceleration channels at the interface.

The particle-in-cell approach has proven effective at modeling neutron star and black hole magnetospheres from first principles, but global simulations are plagued with an unrealistically small separation between the scales where microphysics operates and the system-size scales due to limited numerical resources. A legitimate concern is whether the scale separation currently achieved is large enough, such that results can be safely extrapolated to realistic scales. In this work, our aim is to explore the effect of scaling physical parameters up, and to check whether salient features uncovered by pure kinetic models at smaller scales are still valid, with a special emphasis on particle acceleration and high-energy radiation emitted beyond the light cylinder. To reach this objective, we develop a new hybrid numerical scheme coupling the ideal force-free and the particle-in-cell methods, to optimize the numerical cost of global models. We propose a domain decomposition of the magnetosphere based on the magnetic field topology using the flux function. The force-free model is enforced along open field lines while the particle-in-cell model is restricted to the reconnecting field line region. As a proof of concept, this new hybrid model is applied to simulate a weak millisecond pulsar magnetosphere with realistic scales using high-resolution axisymmetric simulations. Magnetospheric features reported by previous kinetic models are recovered, and strong synchrotron radiation above 100MeV consistent with the Fermi-LAT gamma-ray pulsar population is successfully reproduced. This work further consolidates the shining reconnecting current sheet scenario as the origin of the gamma-ray emission in pulsars, as well as firmly establishes pulsar magnetospheres as at least TeV particle accelerators.

Black hole and neutron star environments often comprise collisionless plasmas immersed in strong magnetic fields and intense baths of low-frequency radiation. In such conditions, relativistic magnetic reconnection can tap the magnetic field energy, accelerating high-energy particles that rapidly cool by inverse Compton (IC) scattering the dense photon background. At the highest particle energies reached in bright gamma-ray sources, IC scattering can stray into the Klein-Nishina regime. Here, the Comptonized photons exceed pair-production threshold with the radiation background and may thus return their energy to the reconnecting plasma as fresh electron-positron pairs. To reliably characterize observable signatures of such Klein-Nishina reconnection, in this work, we present first-principles particle-in-cell simulations of pair-plasma relativistic reconnection coupled to Klein-Nishina and pair-production physics. The simulations show substantial differences between the observable signatures of Klein-Nishina reconnection and reconnection coupled only to low-energy Thomson IC cooling (without pair production). The latter regime exhibits strong harder-when-brighter behaviour; the former involves a stable spectral shape independent of overall brightness. This spectral stability is reminiscent of flat-spectrum radio quasar (FSRQ) GeV high states, furnishing evidence that Klein-Nishina radiative physics operates in FSRQs. The simulated Klein-Nishina reconnection pair yield spans from low to order-unity and follows an exponential scaling law in a single governing parameter. Pushing this parameter beyond its range studied here might give way to a copious pair-creation regime. Besides FSRQs, we discuss potential applications to accreting black hole X-ray binaries, the M87* magnetosphere, and gamma-ray binaries.

Magnetic reconnection is a ubiquitous phenomenon for magnetized plasmas and leads to the rapid reconfiguration of magnetic field lines. During reconnection events, plasma is heated and accelerated until the magnetic field lines enclose and capture the plasma within a circular configuration. These plasmoids could therefore observationally manifest themselves as hot spots that are associated with flaring behavior in supermassive black hole systems, such as Sagittarius A$^\ast$. We have developed a novel algorithm for identifying plasmoid structures, which incorporates watershed and custom closed contouring steps. From the identified plasmoids, we determine the plasma characteristics and energetics in magnetohydrodynamical simulations. The algorithm's performance is showcased for a high-resolution suite of axisymmetric ideal and resistive magnetohydrodynamical simulations of turbulent accretion discs surrounding a supermassive black hole. For validation purposes, we also evaluate several Harris current sheets that are well-investigated in the literature. Interestingly, we recover the characteristic power-law distribution of plasmoid sizes for both the black hole and Harris sheet simulations. This indicates that while the dynamics are vastly different, with different dominant plasma instabilities, the plasmoid creation behavior is similar. Plasmoid occurrence rates for resistive general relativistic magnetohydrodynamical simulations are significantly higher than for the ideal counterpart. Moreover, the largest identified plasmoids are consistent with sizes typically assumed for semi-analytical interpretation of observations. We recover a positive correlation between the plasmoid formation rate and a decrease in black-hole-horizon-penetrating magnetic flux. The developed algorithm has enabled an extensive quantitative analysis of plasmoid formation in black hole accretion simulations.

Context. Pulsar winds have been shown to be preferred sites of particle acceleration and high-energy radiation. Numerous studies have been conducted to better characterize the general structure of such relativistic plasmas in isolated systems. However, many pulsars are found in binary systems and there are currently no ab initio models available that would include both the pulsar magnetosphere and the wind of the pulsar in interaction with a spherical companion. Aims. We investigate the interaction between a pulsar wind and a companion to probe the rearrangement of the pulsar wind, assess whether it leads to an enhancement of particle acceleration, and predict the high-energy radiative signature that stems from this interaction. We consider the regime where the companion is small enough to hold between two successive stripes of the wind. Methods. We performed two-dimensional (2D) equatorial particle-in-cell simulations of an inclined pulsar surrounded by a spherical, unmagnetized, perfectly conducting companion settled in its wind. Different runs correspond to different distances and sizes of the companion. Results. We find that the presence of the companion significantly alters the structure of the wind. When the companion lies beyond the fast magnetosonic point, a shock is established and the perturbations are advected in a cone behind the companion. We observe an enhancement of particle acceleration due to forced reconnection as the current sheet reaches the companion surface. Hence, high-energy synchrotron radiation is also amplified. The orbital light curves display two broad peaks reaching up to 14 times the high-energy pulsed flux emitted by an isolated pulsar magnetosphere. These effects increase with the growth of the companion size and with the decrease of the pulsar-companion separation. Conclusions. The present study suggests that a pulsar wind interacting with a companion induces a significant enhancement of high-energy radiation that takes the form of an orbital-modulated hollow cone of emission, which should be detectable by galactic-plane surveys, possibly with long-period radio transient counterparts.

Particle acceleration in relativistic shocks is quenched in the presence of a transverse magnetic field, even for a moderately low upstream magnetization. Pulsar wind nebulae form downstream of an ultra-relativistic magnetized shock; yet these objects are one of the most efficient particle accelerators known in the Galaxy. We propose that the key to this striking discrepancy lies in the anisotropic nature of the magnetic field profile in the pulsar wind. Using particle-in-cell simulations, we show that it has a dramatic impact on the structure and evolution of the shock. The formation of a current sheet in the equatorial plane, combined with a large-scale velocity shear flow lead to strong plasma turbulence and efficient non-thermal particle acceleration near the Bohm limit. The interplay between these processes may power the bright synchrotron nebula surrounding pulsars and possibly the puzzling Crab gamma-ray flares. Another important feature of the predicted shock structure is the presence of hot macroscopic filaments whose formation is driven by reconnection along the equatorial plane. We argue that these compact plasma structures (giant plasmoids) may explain the mysterious knots contained within the Crab Nebula inner ring.

Extragalactic plasma jets are some of the few astrophysical environments able to confine ultra-high-energy cosmic rays, but whether they are capable of accelerating these particles is unknown. In this work, we revisit particle acceleration at relativistic magnetized shocks beyond the local uniform field approximation, by considering the global transverse structure of the jet. Using large two-dimensional particle-in-cell simulations of a relativistic electron-ion plasma jet, we show that the termination shock forming at the interface with the ambient medium accelerates particles up to the confinement limit. The radial structure of the jet magnetic field leads to a relativistic velocity shear that excites a von Kármán vortex street in the downstream medium trailing behind an over-pressured bubble filled with cosmic rays. Particles are efficiently accelerated at each crossing of the shear flow boundary layers. These findings support the idea that extragalactic plasma jets may be capable of producing ultra-high-energy cosmic rays. This extreme particle acceleration mechanism may also apply to microquasar jets.

Low-luminosity supermassive and stellar-mass black holes (BHs) may be embedded in a collisionless and highly magnetized plasma. They show non-thermal flares indicative of efficient dissipative processes in the vicinity of the BH. During NIR flares from the supermassive BH Sagittarius A* (Sgr A*), GRAVITY detected circular motion and polarization evolution which suggest the presence of transient synchrotron-emitting hot spots moving around the BH. We study 3D reconnecting current layers in the magnetosphere of spinning BHs to determine whether plasma-loaded flux ropes formed near the event horizon could reproduce the hot spot observations and help constraining the BH spin. We perform global 3D particle-in-cell simulations in Kerr spacetime of a pair plasma embedded in a strong large-scale magnetic field originating in a disk in prograde Keplerian rotation. A cone-shaped current layer develops which surrounds the twisted open magnetic field lines threading the event horizon. Magnetic field lines coupling the disk to the BH inflate and reconnect a few gravitational radii above the disk. Particles accelerate and accumulate in a few rotating macroscopic flux ropes. Once flux ropes detach, they propagate in the current layer following what appears as a rapidly opening spiral when seen face-on. A single flux rope carries enough relativistic particles to emit synchrotron radiation at levels suitable to reproduce the flares' peak-luminosity of Sgr A* but it quickly fades away as it flows away. Our kinematic analysis of flux ropes' motion favors a BH spin of 0.65 to 0.8 for Sgr A*. The flares' duration require that the underlying magnetic loop seeded in the disk mid-plane has a finite lifetime and azimuthal extent. In this scenario, the hot spot corresponds to a spinning arc along which multiple reconnection sites power the net emission as flux ropes episodically detach.

(Abridged) An issue of considerable interest in the theory of jet formation by the Blandford-Znajek mechanism, is how plasma is being continuously supplied to the magnetosphere to maintain it in a force-free state. Injection of electron-positron pairs via annihilation of MeV photons, emitted from a hot accretion flow, has been shown to be a viable possibility, but requires a high enough accretion rate. At lower accretion rates, and in the absence of any other form of plasma supply, the magnetosphere becomes charge starved, forming intermittent spark gaps that can induce intense pair cascades via interactions with soft disk radiation, enabling outflow formation. It is often speculated that enough plasma can penetrate the inner magnetosphere from the accretion flow through some rearrangement of magnetic field lines (e.g., interchange instability). However, the question arises whether such episodes of plasma intrusion can prevent the formation of spark gaps. To address this question we conducted a suite of numerical experiments, by means of radiative, 2D axisymmetric general relativistic particle-in-cell simulations, in which plasma is injected into specified regions at a prescribed rate. We find that when pair production is switched off, nearly complete screening is achieved when the plasma is injected within the outer light cylinder at a high enough rate. Injection beyond the outer light cylinder results in either, the formation of large vacuum gaps, or coherent, large-amplitude oscillations of the magnetosphere, depending on the injection rate. Within the allowed dynamic range of our simulations, we see no evidence for the system to approach a steady state as the injection rate is increased. Switching on pair production results in nearly complete screening of the entire magnetosphere in all cases, with some fraction of the maximum Blandford-Znajek power emitted as TeV gamma-rays.

Context: Accreting black holes (BHs) may be surrounded by a highly magnetized plasma threaded by a poloidal magnetic field. Non-thermal flares and high energy components could originate from a hot, collisionless and nearly force-free corona. The jets we often observe from these systems are believed to be rotation-powered and magnetically-driven. Aims: We study axisymmetric BH magnetospheres where some magnetic field lines anchored in a surrounding disk can connect to the event horizon of a rotating BH. We identify the sites of magnetic reconnection within 30 gravitational radii depending on the BH spin. Methods: With the fully general relativistic particle-in-cell code GRZeltron, we solve the time-dependent dynamics of the electron-positron pair plasma and of the electromagnetic fields around the BH. The disk is represented by a steady plasma in Keplerian rotation, threaded by a frozen dipolar field. Results: For prograde disks, twisted open magnetic field lines crossing the horizon power a Blandford-Znajek jet while beyond a critical distance, open field lines on the disk are open. In the innermost regions, coupling field lines ensure the transfer of significant amounts of angular momentum and energy between the BH and the disk. From the Y-point at the intersection, a current sheet forms where particle acceleration via magnetic reconnection takes place. We compute the synchrotron images of the current sheet emission. Conclusions: Our estimates for jet power and BH-disk exchanges match those derived from purely force-free models. Dissipation at the Y-point heats the corona and provides a physically motivated source of hard X-rays above the disk for reflection models. Episodic plasmoid ejection might explain millisecond flares observed in Cyg X-1. Particles flowing from the Y-point down to the disk could produce a hot spot at the footpoint of the outermost closed field line.

Accreting supermassive black holes can now be observed at the event-horizon scale at millimeter wavelengths. Current predictions for the image rely on hypotheses (fluid modeling, thermal electrons) which might not always hold in the vicinity of the black hole, so that a full kinetic treatment is in order. In this Letter, we describe the first 3D global general-relativistic particle-in-cell simulation of a black-hole magnetosphere. The system displays a persistent equatorial current sheet. Synthetic radio images are computed by ray-tracing synchrotron emission from nonthermal particles accelerated in this current sheet by magnetic reconnection. We identify several time-dependent features of the image at moderate viewing angles: a variable radius of the ring, and hot spots moving along it. In this regime, our model predicts that most of the flux of the image lies inside the critical curve. These results could help promote understanding of future observations of black-hole magnetospheres at improved temporal and spatial resolution.