In this paper we study the generation of high energy emission from normal pulsars. For this purpose we consider the particles accelerated in the outer magnetosphere sliding along the closed magnetic field lines. It has been shown that in due course of motion the initial small pitch angle increases and at a certain distance from the neutron star the synchrotron emission becomes significant. We found that a region covered from $150$ to $500$ stellar radii the emission pattern is characterised by energies in the interval $(0.1-10)$ MeV.
Surprisingly, the chronology of nearly 50 years of the pulsar magnetosphere and pulsar wind research is quite similar to the history of our civilization. Using this analogy, I have tried to outline the main results obtained in this field. In addition to my talk, the possibility of particle acceleration due to different processes in the pulsar magnetosphere is discussed in more detail.
With the neutron star rotating under a stationary magnetic field generating unipolar induction, charges are driven to the pulsar surface according to their signs, and are uploaded to the magnetosphere along the magnetic field lines. In the presence of an electron-positron plasma, the pulsar magnetosphere is represented by a unique magnetohydrodynamic (MHD) plasma. The pulsar equation of force-free equilibrium is solved analytically for two configurations. The first one is a light cylinder guided jet-like open magnetosphere, and the second one is a closed magnetosphere within the light cylinder…
We perform first-principles relativistic particle-in-cell simulations of aligned pulsar magnetosphere. We allow free escape of particles from the surface of a neutron star and continuously populate the magnetosphere with neutral pair plasma to imitate pair production. As pair plasma supply increases, we observe the transition from a charge-separated electrosphere solution with trapped plasma and no spin-down to a solution close to the ideal force-free magnetosphere with electromagnetically-dominated pulsar wind. We calculate the magnetospheric structure, current distribution and spin-down powe…
I give a brief overview of recent results from self-consistent modeling of electron-positrons cascades in pulsar polar caps. These results strongly suggest that the pulsar magnetosphere is a more complex system than was assumed before.
Recent advances in numerical algorithms and computational power have enabled first-principles simulations of pulsar magnetospheres using Particle-in-Cell (PIC) techniques. These ab-initio simulations seem to indicate that pair creation through photon-photon collision at the light cylinder is required to sustain the pulsar engine. However, for many rotation-powered pulsars pair creation operates effectively only near the stellar surface where magnetic field is high. How these "weak pulsars" fill their magnetospheres without efficient photon-photon pair conversion in the outer magnetosphere is s…
A new model of the pulsar force-free magnetosphere is suggested, which includes the presence of the polar, outer and slot gaps. It is based on a new exact solution of the pulsar equation in the form of an offset monopole and the resultant split-offset monopole scheme.
The magnetospheres around neutron stars should be very particular because of their strong magnetic field and rapid rotation. A study of the pulsar magnetospheres is of a crucial importance since it is the key issue to understand how energy inflow to the exterior is produced. In this paper, I discuss magnetohydrodynamic processes in the pulsar magnetosphere. I consider in detail the properties of magnetohydrodynamic waves that can exist in the magnetosphere and their instabilities.These instabilities lead to formation of magnetic structures and can be responsible for a short-term variability of…
Pulsar magnetospheres have strong magnetic fields and large amounts of plasma. The structures of these magnetospheres are studied using force-free electrodynamics. To understand pulsar magnetospheres, discussions must include their outer region. However, force-free electrodynamics is limited in it does not handle dissipation. Therefore, a resistive pulsar magnetic field model is needed. To break the ideal magnetohydrodynamic (MHD) condition $E \cdot B = 0$, Ohm's law is used. In this work, I introduce resistivity depending upon the distance from the star and obtain a self-consistent steady sta…
We argue that existing models of the ideal pulsar magnetosphere are incorrect because of the improper treatment of the singular current layer outside the light cylinder. We simulated the axisymmetric pulsar magnetosphere in the Force-Free limit of Strong-Field Electrodynamics. It turns out that even in the Force-Free limit, some field lines enter the singular current layer which lies beyond the light cylinder in the equatorial plane. As a result: (i) about 10% of the Poynting flux is dissipated between 1 and 1.5 light cylinder radii, (ii) there is no singular current layer within the light cyl…
We investigate the shapes of γ-ray pulsar light curves using 3D pulsar magnetosphere models of finite conductivity. These models, covering the entire spectrum of solutions between vacuum and force-free magnetospheres, for the first time afford mapping the GeV emission of more realistic, dissipative pulsar magnetospheres. To this end we generate model light curves following two different approaches: (a) We employ the emission patterns of the slot and outer gap models in the field geometries of magnetospheres with different conductivity σ. (b) We define realistic trajectories of radiating partic…
We demonstrate a method for quantitatively comparing gamma-ray pulsar light curves with magnetosphere beaming models. With the Fermi LAT providing many pulsar discoveries and high quality pulsar light curves for the brighter objects, such comparison allows greatly improved constraints on the emission zone geometry and the magnetospheric physics. Here we apply the method to Fermi LAT light curves of a set of bright pulsars known since EGRET or before. We test three approximate models for the magnetosphere structure and two popular schemes for the location of the emission zone, the Two Pole Caus…
We have examined the cyclotron resonance instability developing in the relativistic out-flowing plasma in the pulsar magnetosphere. The instability condition leads to radio emission in the sub-GHz frequency regime which is likely to be seen as off-pulse emission. Recent studies have shown the presence of off-pulse emission in long period pulsars, and we demonstrate this plasma process to be an energetically viable mechanism.
Thermal effects on the properties of four electromagnetic waves propagating in the pulsar magnetosphere are analyzed. It is shown that thermal effects change only quantitatively the dispersion properties of superluminal ordinary O-mode freely escaping the pulsar magnetosphere; properties of the extraordinary X-mode remain unchanged. As for two subluminal waves propagating along magnetic field lines, for them thermal effects result in essential absorption. However, this attenuation occurs at considerable distances from the neutron star, so there is no doubt in their existence.
Magnetohydrodynamic (MHD) instabilities can play an important role in the structure and dynamics of the pulsar magnetosphere. We consider the instabilitycaused by differential rotation that is suggested by many theoretical models. Stability is considered by means of a linear analysis within the frane of the force-free MHD. We argue that differentially rotating magnetospheres are unstable for any particular geometry of the magnetic field and rotation law. The characteristic growth time of instability is of the order of the rotation period. The instability can lead to fluctuations of the emissio…
Following the basic principles of a charge separated pulsar magnetosphere \citep{goldreich1969}, we consider the magnetosphere be stationary in space, instead of corotating, and the electric field be uploaded from the potential distribution on the pulsar surface, set up by the unipolar induction. Consequently, the plasma of the magnetosphere undergoes guiding center drifts of the gyro motion due to the transverse forces to the magnetic field. These forces are the electric force, magnetic gradient force, and field line curvature force. Since these plasma velocities are of drift nature, there is…
Stationary pulsar magnetospheres in the force-free system are governed by the pulsar equation. In 1999, Contopoulos, Kazanas, and Fendt (hereafter CKF) numerically solved the pulsar equation and obtained a pulsar magnetosphere model called the CKF solution that has both closed and open magnetic field lines. The CKF solution is a successful solution, but it contains a poloidal current sheet that flows along the last open field line. This current sheet is artificially added to make the current system closed. In this paper, we suggest an alternative method to solve the pulsar equation and constru…
Eight pulsars have low braking indices, which challenge the magnetic dipole braking of pulsars. 222 pulsars and 15 magnetars have abnormal distribution of frequency second derivatives, which also make contradiction with classical understanding. How neutron star magnetospheric activities affect these two phenomena are investigated by using the wind braking model of pulsars. It is based on the observational evidence that pulsar timing is correlated with emission and both aspects reflect the magnetospheric activities. Fluctuations are unavoidable for a physical neutron star magnetosphere. Young p…
The pulsar light curves and energy spectra are explored in dissipative pulsar magnetospheres with the Aristotelian electrodynamics (AE), where particle acceleration is fully balanced with radiation reaction. The AE magnetospheres with non-zero pair multiplicity are computed by a pseudo-spectral method in the co-moving frame. The dissipative region near the current sheet outside the LC is accurately captured by the high-resolution simulation. The pulsar light curves and spectra are computed by the test particle trajectory method including the influence of both the consistent accelerating electr…
The dipolar magnetic field structure of the neutron star is modified by the plasma originating in the pulsar magnetopshere. In the simplest case of a stationary axisymmetric force-free magnetosphere, a self-consistent description of the fields and currents is given by the well-known pulsar equation. Here we revise the commonly used boundary conditions of the problem in order to incorporate the plasma-producing gaps and to provide a framework for a truly self-consistent treatment of the pulsar magnetosphere. The generalized multipolar solution of the pulsar equation is found, which, as compared…
It has recently been demonstrated that self-consistent particle-in-cell simulations of low-obliquity pulsar magnetospheres in flat spacetime show weak particle acceleration and no pair production near the poles. We investigate the validity of this conclusion in a more realistic spacetime geometry via general-relativistic particle-in-cell simulations of the aligned pulsar magnetospheres with pair formation. We find that the addition of frame-dragging effect makes local current density along the magnetic field larger than the Goldreich-Julian value, which leads to unscreened parallel electric fi…
In this paper we study the possibility of efficient pair production in a pulsar's magnetosphere. It has been shown that by means of the relativistic centrifugal force the electrostatic field exponentially amplifies. As a result the field approaches the Schwinger limit leading to pair creation process in the light cylinder area where the effects of rotation are very efficient. Analysing the parameters of the normal period ($\sim 1$ sec) pulsars we have found that the process is so efficient that the number density of electron-positron pairs exceeds the Goldreich-Julian density by five orders of…
The current state of the art in the modeling of pulsar magnetospheres invokes either the vacuum or force-free limits for the magnetospheric plasma. Neither of these limits can simultaneously account for both the plasma currents and the accelerating electric fields that are needed to explain the morphology and spectra of high-energy emission from pulsars. To better understand the structure of such magnetospheres, we combine accelerating fields and force-free solutions by considering models of magnetospheres filled with resistive plasma. We formulate Ohm's Law in the minimal velocity fluid frame…
It is proposed that there exists a class of pulsars, called weak pulsars, for which the large-scale magnetosphere, and hence the gamma-ray emission, are independent of the detailed pattern of plasma production. The weak pulsar magnetosphere and its gamma-ray emission are uniquely determined by just three parameters: spin, dipole, and the spin-dipole angle. We calculate this supposedly unique pulsar magnetosphere in the axisymmetric case. The magnetosphere is found to be very close to (although interestingly not fully identical with) the magnetosphere we have previously calculated, explaining t…
We examine if an efficient particle acceleration takes place by a magnetic-field-aligned electric field near the light cylinder in a rotating neutron star magnetosphere. Constructing the electric current density with the actual motion of collision-less plasmas, we express the rotationally induced, Goldreich-Julian charge density as a function of position. It is demonstrated that the 'light cylinder gap', which emits very high energy photons via curvature process by virtue of a strong magnetic-field-aligned electric field very close to the light cylinder, will not arise in an actual pulsar magn…
Pulsars are famous for their rotational stability. Most of them steadily spin down and display a highly repetitive pulse shape. But some pulsars experience timing irregularities such as nulling, intermittency, mode changing and timing noise. As changes in the pulse shape are often correlated with timing irregularities, precession is a possible cause of these phenomena. Whereas pulsar magnetospheres are filled with plasma, most pulsar precession studies were carried out within the vacuum approximation and neglected the effects of magnetospheric currents and charges. Recent numerical simulations…
We apply a computationally efficient technique to validate the global structure of the pulsar magnetosphere. In this first of a series of studies, a 3D, computationally intensive, implicit Crank-Nicolson finite-difference scheme is developed. The region of magnetic influence is evolved under the approximation of force-free electrodynamics. The main objective of this paper is to present our code and use it to demonstrate and verify the now widely - accepted global features of a pulsar magnetosphere. Our results qualitatively agree with previously developed time-dependent models for an oblique r…
A good compromise between the resistive model and the PIC model is Aristotelian electrodynamics, which can include the back-reaction of the radiative photons onto particle motion and allow for a local dissipation where the force-free condition is violated. We study the dissipative pulsar magnetosphere with Aristotelian electrodynamics where particle acceleration is fully balanced by radiation. The expression for the current density is defined by introducing a pair multiplicity. The 3D structure of the pulsar magnetosphere is then presented by solving the time-dependent Maxwell equations using …
The key properties of the wave propagation theory in the magntosphere of radio pulsars based on the Kravtsov-Orlov equation are presented. It is shown that for radio pulsars with known circular polarization and the swing of the linear polarization position angle one can determine which mode, ordinary or extraordinary one, forms mainly the mean profile of the radio emission. The comparison of the observational data with the theory predictions demonstrates their good agreement.
The equatorial current sheets outside the light cylinder(LC) are thought as the promising site of the high energy emission based on the results of the recent numerical simulations. We explore the pulsar light curves and energy spectra by computing the curvature radiation based on the FIDO magnetospheres. The FIDO magnetospheres with a near force-free regime inside the LC and a finite but high conductivity outside the LC are constructed by a spectral algorithm. The pulsar high energy emission properties are explored by integrating the trajectories of the test particles under the influence of bo…
The stationary axisymmetric force-free magnetosphere of a pulsar is considered. We present an exact dipolar solution of the pulsar equation, construct the magnetospheric model on its basis and examine its observational support. The new model has toroidal rather than common cylindrical geometry, in line with that of the plasma outflow observed directly as the pulsar wind nebula at much larger spatial scale. In its new configuration, the axisymmetric magnetosphere consumes the neutron star rotational energy much more efficiently, implying re-estimation of the stellar magnetic field, $B_{\mathrm …
An oblique pulsar magnetosphere with a plasma conductivity is studied by using a pseudo-spectral method. In the pseudo-spectral method, the time-dependent Maxwell equations are solved, both electric and magnetic fields are expanded in terms of the vector spherical harmonic (VSH) functions in spherical geometry and the divergencelessness of magnetic field is analytically enforced by a projection method. The pulsar magnetospheres in infinite (i. e., force-free approximation) and finite conductivities are simulated and a family of solutions that smoothly transition from the Deutsch vacuum solutio…
We present the structure of the 3D ideal MHD pulsar magnetosphere to a radius ten times that of the light cylinder, a distance about an order of magnitude larger than any previous such numerical treatment. Its overall structure exhibits a stable, smooth, well-defined undulating current sheet which approaches the kinematic split monopole solution of Bogovalov 1999 only after a careful introduction of diffusivity even in the highest resolution simulations. It also exhibits an intriguing spiral region at the crossing of two zero charge surfaces on the current sheet, which shows a destabilizing be…
The detection of the pulsed $\sim 1 $~TeV gamma-ray emission from the Crab pulsar reported by MAGIC and VERITAS collaborations demands a substantial revision of existing models of particle acceleration in the pulsar magnetosphere. In this regard model independent restrictions on the possible production site of the VHE photons become an important issue. In this paper, we consider limitations imposed by the process of conversion of VHE gamma rays into $e^{\pm}$ pairs in the magnetic field of the pulsar magnetosphere. Photons with energies exceeding 1~TeV are effectively absorbed even at large di…
Acceleration of bulk motion of plasma due to magneto centrifugal mechanism is investigated for different shapes of the field lines of 3-D magnetic field. It is shown that this mechanism can be efficient provided that the field line is twisted into direction of rotation. According to the last results of the numerical modelling a fraction of the field lines with such geometrical shape apparently exist in the pulsar magnetosphere close to the last closed field line. In this case the Lorentz factor $γ$ increases with the radius $r$ along these field lines as $γ\sim (1-(rΩ/c)^2)^{-1}$, where $Ω$ - …
The SKA will discover tens of thousands of pulsars and provide unprecedented data quality on these, as well as the currently known population, due to its unrivalled sensitivity. Here, we outline the state of the art of our understanding of magnetospheric radio emission from pulsars and how we will use the SKA to solve the open problems in pulsar magnetospheric physics.
We propose that the magnetosphere reconfiguration induced by magnetic twists in the closed field line region can account for the mode-switching of intermittent pulsars. We carefully investigate the properties of axisymmetric force-free pulsar magnetospheres with magnetic twists in closed field line region around the polar caps. The magnetosphere with twisted closed lines leads to enhanced spin-down rates. The enhancement in spin-down rate depends on the size of region with twisted closed lines. Typically, it is increased by a factor of $\sim2$, which is consistent with the intermittent pulsars…
Pulsars emit low-frequency radio waves through to high-energy gamma-rays that are generated anywhere from the surface out to the edges of the magnetosphere. Detecting correlated mode changes in the multi-wavelength emission is therefore key to understanding the physical relationship between these emission sites. Through simultaneous observations, we have detected synchronous switching in the radio and X-ray emission properties of PSR B0943+10. When the pulsar is in a sustained radio 'bright' mode, the X-rays show only an un-pulsed, non-thermal component. Conversely, when the pulsar is in a rad…
We present solutions of force-free pulsar magnetospheres coupled with a uniform external magnetic field aligned with the dipole magnetic moment of the pulsar. The inclusion of the uniform magnetic field has the following consequences: The equatorial current sheet is truncated to a finite length, the fraction of field lines that are open increases, and the open field lines are confined within a cylindrical surface instead of becoming radial. A strong external magnetic field antiparallel to the dipole allows for solutions where the pulsar magnetic field is fully enclosed within an ellipsoidal su…
In view of recent efforts to probe the physical conditions in the pulsar current sheet, we revisit the standard solution that describes the main elements of the ideal force-free pulsar magnetosphere. The simple physical requirement that the electric current contained in the current layer consists of the local electric charge moving outward at close to the speed of light, yields a new standard solution for the pulsar magnetosphere everywhere ideal force-free except in the current layer. The main elements of the new solution are a) the pulsar spindown rate of the aligned rotator is 23% times lar…
We consider magnetospheric structure of rotating neutron stars with internally twisted axisymmetric magnetic fields. The twist-induced and rotation-induced toroidal magnetic fields align/counter-align in different hemispheres. Using analytical and numerical calculations (with PHAEDRA code) we show that as a result the North-South symmetry is broken: the magnetosphere and the wind become "angled", of conical shape. Angling of the magnetosphere affects the spindown (making it smaller for mild twists), makes the return current split unequally at the Y-point, produces anisotropic wind and linear a…
It is shown that the ion-proton magnetosphere is unstable in a limited area of the P - Pdot plane against transitions to a self-sustaining inverse Compton scattering mode in which the particles accelerated are mainly protons with a small component of positrons. It is argued that this mode cannot be absolutely stable. The number density of any outward-moving pair plasma is small and electron and positron Lorentz factors too high to support growth of any collective mode capable of exciting normal pulsar coherent radio emission. Particle fluxes and the position at which they pass through the ligh…
University of Southampton（南安普顿大学）；University of Oxford（牛津大学）；The University of Manchester（曼彻斯特大学）；Indian Institute of Technology Indore（印度理工学院印多尔分校）；National Centre for Radio Astrophysics (TIFR)（国家射电天文中心（泰米尔纳德邦印度理工学院））；Indian Institute of Technology, Roorkee（印度理工学院鲁尔基分校）；State University of New York at Oswego（纽约州立大学奥斯威戈分校）；Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）；Max-Planck-Institut für Radioastronomie（马克斯·普朗克射电天文研究所）；Swinburne University of Technology（斯威本科技大学）
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 curren…
Based on the exact dipolar solution of the pulsar equation the self-consistent two-fluid model of the pulsar magnetosphere is developed. We concentrate on the low-mass limit of the model, taking into account the radiation damping. As a result, we obtain the particle distributions sustaining the dipolar force-free configuration of the pulsar magnetosphere in case of a slight velocity shear of the electron and positron components. Over most part of the force-free region, the particles follow the poloidal magnetic field lines, with the azimuthal velocities being small. Close to the Y-point, howev…
We present the first self-consistent global simulations of pulsar magnetospheres with operating $e^\pm$ discharge. We focus on the simple configuration of an aligned or anti-aligned rotator. The star is spun up from zero (vacuum) state to a high angular velocity, and we follow the coupled evolution of its external electromagnetic field and plasma particles using the "particle-in-cell" method. A plasma magnetosphere begins to form through the extraction of particles from the star; these particles are accelerated by the rotation-induced electric field, producing curvature radiation and igniting …
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 susceptible to explain their radiative properties like multi-wavelength pulse shapes and spect…
The self-consistent two-fluid model of the pulsar magnetosphere is considered. We concentrate on the case of vanishingly small inertia of the particles. Our approach allows to obtain the realistic particle distributions sustaining the force-free magnetic field configuration of a monopolar structure. The result differs substantially from the customary picture of the radial speed-of-light motion of massless particles. In our case, the electron and positron constituents follow slightly curved trajectories and are characterized by definite number densities and distinct velocities. The velocity shi…
We explore the emission properties of a dissipative pulsar magnetosphere model introduced by Kalapotharakos et al. (2014), comparing its high energy light curves and spectra, due to curvature radiation, with data collected by the Fermi LAT. The magnetosphere structure is assumed to be near the force-free solution. The accelerating electric field, inside the light-cylinder, is assumed to be negligible, while outside the light-cylinder it rescales with a finite conductivity (σ). In our approach we calculate the corresponding high energy emission by integrating the trajectories of test particles …
The radio emission anomaly coincident with the 2016 glitch of the Vela pulsar may be caused by a star quake that launches Alfvén waves into the magnetosphere, disturbing the original radio emitting region. To quantify the lifetime of the Alfvén waves, we investigate a possible energy loss mechanism, the conversion of Alfvénwaves into fast magnetosonic waves. Using axisymmetric force-free simulations, we follow the propagation of Alfvén waves launched from the stellar surface with small amplitude into the closed zone of a force-free dipolar pulsar magnetosphere. We observe mode conversion happe…
We present a simple numerical model of the plasma flow within the open field line tube in the pulsar magnetosphere. We study how the plasma screens the rotationally induced electric field and maintains the electric current demanded by the global structure of the magnetosphere. We show that even though bulk of the plasma moves outwards with relativistic velocities, a small fraction of particles is continuously redirected back forming reverse plasma flows. The density and composition (positrons or electrons, or both) of these reverse flows are determined by the distribution of the Goldreich-Juli…
To explain the pulsed emission of the rotation powered pulsars from radio to gamma-ray, the polar cap models, the slot gap models, and the outer gap models are proposed. The recent observations suggest that these models are likely to co-exist in the same magnetosphere. If so, their mutual relation is known to be troublesome (Harding 2009) due to the boundary conditions and the direction of the current which are properly assumed in each acceleration models. We performed a particle simulation for the global magnetospheric structure. Based on the simulation, we present a new picture of the global…
A pseudo-spectral method with an absorbing outer boundary is used to solve a set of the time-dependent force-free equations. In the method, both electric and magnetic fields are expanded in terms of the vector spherical harmonic (VSH) functions in spherical geometry and the divergencelessness of magnetic field is analytically enforced by a projection method. Our simulations show that the Deutsch vacuum solution and the Michel monopole solution can be well reproduced by our pseudo-spectral code. Further the method is used to present the time-dependent simulation of the force-free pulsar magneto…
We present first-principles relativistic particle-in-cell simulations of the oblique pulsar magnetosphere with pair formation. The magnetosphere starts to form with particles extracted from the surface of the neutron star. These particles are accelerated by surface electric fields and emit photons capable of producing electron-positron pairs. We inject secondary pairs at locations of primary energetic particles, whose energy exceeds the threshold for pair formation. We find solutions that are close to the ideal force-free magnetosphere, with the Y-point and current sheet. Solutions with obliqu…
Many millisecond pulsars are thought to be old neutron stars spun up (`recycled') during an earlier accretion phase. They typically have relatively weak ($\lesssim 10^{9} \text{ G}$) dipole field strengths, consistent with accretion-induced magnetic burial. Recent data from the Neutron Star Interior Composition Explorer indicate that hot spots atop the recycled pulsar PSR J0030$+$0451 are not antipodal, so that the magnetic field cannot be that of a centered dipole. In this paper it is shown that multipolarity is naturally expected in the burial scenario because of equatorial field line compre…
