Blazars are a sub-category of radio-loud active galactic nuclei with relativistic jets pointing towards the observer. They exhibit non-thermal variable emission, which practically extends over the whole electromagnetic spectrum. Despite the plethora of multi-wavelength observations, the origin of the emission in blazar jets remains an open question. In this work, we construct a two-zone leptonic model: particles accelerate in a small region and lose energy through synchrotron radiation and inverse Compton Scattering. Consequently, the relativistic electrons escape to a larger area where the ambient photon field, which is related to Accretion Disk MHD Winds, could play a central role in the gamma-ray emission. This model explains the Blazar Sequence and the broader properties of blazars, as determined by Fermi observations, by varying only one parameter, the mass accretion rate onto the central black hole. Flat Spectrum Radio Quasars have a strong ambient photon field and their gamma-ray emission is dominated by the more extensive zone, while in the case of BL Lac objects, the negligible ambient photons make the smaller, i.e. acceleration, zone dominant.
The Fermi Gamma-Ray Space Telescope observations of blazars show a strong correlation between the spectral index of their gamma-ray spectra and their synchrotron peak frequency $ν_{\rm{pk}}^{\rm{syn}}$; additionally, the rate of Compton Dominance of these sources also seems to be a function of $ν_{\rm{pk}}^{\rm{syn}}$. In this work, we adopt the assumption that the nonthermal emission of blazars is primarily due to radiation by a population of Fermi-accelerated electrons in a relativistic outflow (jet) along the symmetry axis of the blazar's accretion disk. Furthermore, we assume that the Comp…
The blazars provide a considerable opportunity to peer into the workings within a few tens of parsecs of the central engine in AGN. This considerable opportunity involves significant challenges as different macroscopic dynamical processes and microscopic physical processes operating at different locations can be responsible for the observed emission. In this proceedings article I review recent theoretical and numerical results relevant to dynamics inside the blazar zone, review the particle acceleration processes capable of producing the high energy particles required by the observed emission,…
The author is developing a numerical code with thousands of emission zones to simulate the time-dependent multi-waveband emission from blazars. The code is based on a model in which turbulent plasma flowing at a relativistic speed down a jet crosses a standing conical collimation shock that accelerates electrons to maximum energies in the 5-100 GeV range. This paper reports early results produced by the model. The simulated light curves and time profiles of the degree and position angle of polarization have a number of features in common with the observational data of blazars. Maps of the pola…
A one-parameter two-zone leptonic model for the blazar sequence
INAF – Osservatorio Astronomico di Brera（意大利国家天体物理研究所 - 布里拉天文台）；Department of Physics, National and Kapodistrian University of Athens（雅典国立卡波季斯特里安大学物理系）；NASA Goddard Space Flight Center（美国国家航空航天局戈达德太空飞行中心）
A high-energy muon neutrino event, IceCube-170922A, was recently discovered in both spatial and temporal coincidence with a gamma-ray flare of the blazar TXS 0506+056. It has been shown, with standard one-zone models, that neutrinos can be produced in the blazar jet via hadronic interactions, but with a flux which is mostly limited by the X-ray data. In this work, we explore the neutrino production from TXS 0506+056 by invoking two physically distinct emission zones in the jet, separated by the broad line region (BLR). Using the Doppler-boosted radiation of the BLR as the target photon field, …
We present a newly developed time-dependent three-dimensional multi-zone hadronic blazar emission model. By coupling a Fokker-Planck based lepto-hadronic particle evolution code 3DHad with a polarization-dependent radiation transfer code, 3DPol, we are able to study the time-dependent radiation and polarization signatures of a hadronic blazar model for the first time. Our current code is limited to parameter regimes in which the hadronic $γ$-ray output is dominated by proton synchrotron emission, neglecting pion production. Our results demonstrate that the time-dependent flux and polarization …
We study the expected variability patterns of blazars within the two-zone acceleration model putting special emphasis on flare shapes and spectral lags. We solve semi-analytically the kinetic equations which describe the particle evolution in the acceleration and radiation zone. We then perturb the solutions by introducing Lorentzian variations in its key parameters and examine the flaring behavior of the system. We apply the above to the X-ray observations of blazar 1ES 1218+304 which exhibited a hard lag behavior during a flaring episode and discuss possibilities of producing it within the c…
Recently reported coincidences between high-energy neutrino events and major blazar outbursts reinforce the relevance of lepto-hadronic emission models for blazars. We study the influence of physical parameters on the neutrino output modeling blazar spectral energy distributions self-consistently assuming a relativistically propagating acceleration zone surrounded by a larger cooling zone. We find that the gross features of the spectral energy distribution can readily be explained with the model. A rigorous test requires time-resolved measurements of blazar spectral energy distributions during…
In this work, we investigate the 2014-2015 neutrino flare associated with the blazar TXS 0506+056 and a recently discovered muon neutrino event IceCube-200107A in spatial coincidence with the blazar 4FGL J0955.1+3551, under the framework of a two-zone radiation model of blazars where an inner/outer blob close to/far from the supermassive black hole are invoked. An interesting feature that the two sources share in common is that no evidence of GeV gamma-ray activity is found during the neutrino detection period, probably implying a large opacity for GeV gamma rays in the neutrino production reg…
Kink instabilities are likely to occur in the current-carrying magnetized plasma jets. Recent observations of the blazar radiation and polarization signatures suggest that the blazar emission region may be considerably magnetized. While the kink instability has been studied with first-principle magnetohydrodynamic (MHD) simulations, the corresponding time-dependent radiation and polarization signatures have not been investigated. In this paper, we perform comprehensive polarization-dependent radiation modeling of the kink instability in the blazar emission region based on relativistic MHD (RMH…
The high-energy-peaked blazar 1ES 1959+650 is a well-known and well studied nearby blazar that has undergone several episodes of multi-TeV flaring. In 2002 for the first time an orphan TeV flare was observed from this blazar. During a multiwavelength campaign between 29th April to 21st November 2016, MAGIC telescopes observed multi-TeV flarings during the nights of 13th, 14th June and 1st July 2016 when the position of the synchrotron peak was found to be above $10^{17}$ Hz. Also observed was that the second peak of the spectral energy distribution shifted towards higher energy, and exhibiting…
Mildly-relativistic shocks that are embedded in colliding magnetohydrodynamic flows are prime sites for relativistic particle acceleration and production of strongly variable, polarized multi-wavelength emission from relativistic jet sources such as blazars and gamma-ray bursts. The principal energization mechanisms at these shocks are diffusive shock acceleration and shock drift acceleration. In recent work, we had self-consistently coupled shock acceleration and radiation transfer simulations in blazar jets in a basic one-zone scenario. These one-zone models revealed that the observed spectr…
Context.Blazars, a sub-category of Active Galactic Nuclei, are characterized by their non-thermal variable emission. This emission extends over the whole electromagnetic spectrum and is a consequence of particle acceleration inside their relativistic jets. However, especially the relation of radio emission to that at higher frequencies remains an open question. Aims. Observations of blazar emission show that the location of radio might be very different from the one where the rest of the spectrum is produced and often requires separate modeling. We aim to produce both emissions within the cont…
Blazars radiate from relativistic plasma jets with bulk Lorentz factors Γ ~ 10, closely aligned along our line of sight. In a number of blazars of the Flat Spectrum Radio Quasar type such as 3C 454.3 and 3C 279 gamma-ray flares have recently been detected with very high luminosity and little or no counterparts in the optical and soft X-ray bands. They challenge the current one-zone leptonic models of emissions from within the broad line region. The latter envisage the optical/X-ray emissions to be produced as synchrotron radiation by the same population of highly relativistic electrons in the …
Polarization Dynamics of X-Ray Synchrotron Emission from a Multi-Zone Blazar Jet
Washington University in St Louis（圣路易斯华盛顿大学）；University of Maryland Baltimore County（马里兰大学巴尔的摩县分校）；NASA Goddard Space Flight Center（美国国家航空航天局戈达德太空飞行中心）
Particle acceleration signatures in the time-dependent one-zone synchrotron self-Compton model of blazar flares
LUX, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université（巴黎天文台，巴黎文理研究大学，法国国家科学研究中心，索邦大学）；Centre for Space Research, North-West University（西北大学空间研究中心）
The current generation of Cherenkov telescopes, together with Fermi-LAT, have greatly increased our knowledge of $γ$-ray blazars. Among them, the high-frequency-peaked BL Lacertae object (HBL) PKS1424+240 (z $\simeq$ 0.6) is the farthest persistent emitter of very-high-energy (VHE; E $\geq$ 100 GeV) $γ$-ray photons. Current emission models can satisfactorily reproduce typical blazar emission assuming that the dominant emission process is synchrotron-self-Compton (SSC) in HBLs; and external-inverse-Compton (EIC) in LBLs and FSRQs. Hadronic models are also able to correctly reproduce the $γ$-ray…
The author presents a model for variability of the flux and polarization of blazars in which turbulent plasma flowing at a relativistic speed down a jet crosses a standing conical shock. The shock compresses the plasma and accelerates electrons to energies up to gamma(max) > 1E4 times their rest-mass energy, with the value of gamma(max) determined by the direction of the magnetic field relative to the shock front. The turbulence is approximated in a computer code as many cells, each with a uniform magnetic field whose direction is selected randomly. The density of high-energy electrons in the …
We present the results of observations of blazar PKS 1510-089 with the Herschel Space Observatory PACS and SPIRE instruments, together with multiwavelength data from Fermi/LAT, Swift, SMARTS and SMA. The source was found in a quiet state, and its far-infrared spectrum is consistent with a power-law with a spectral index of alpha ~ 0.7. Our Herschel observations were preceded by two 'orphan' gamma-ray flares. The near-infrared data reveal the high-energy cut-off in the main synchrotron component, which cannot be associated with the main gamma-ray component in a one-zone leptonic model. This is …
Various attempts have been made in the literature at describing the origin and the physical mechanisms behind flaring events in blazars with radiative emission models, but detailed properties of multi-wavelength (MWL) light curves still remain difficult to reproduce. We have developed a versatile radiative code, based on a time-dependent treatment of particle acceleration, escape and radiative cooling, allowing us to test different scenarios to connect the continuous low-state emission self-consistently with that during flaring states. We consider flares as weak perturbations of the quiescent …
Using blazar light curves from the optical All-Sky Automated Survey for Supernovae (ASAS-SN) and the $γ$-ray \textit{Fermi}-LAT telescope, we performed the most extensive statistical correlation study between both bands, using a sample of 1,180 blazars. This is almost an order of magnitude larger than other recent studies. Blazars represent more than 98\% of the AGNs detected by \textit{Fermi}-LAT and are the brightest $γ$-ray sources in the extragalactic sky. They are essential for studying the physical properties of astrophysical jets from central black holes. However, their $γ$-ray flare me…
Mildly relativistic, oblique shocks are frequently invoked as possible sites of relativistic particle acceleration and production of strongly variable, polarized multi-wavelength emission from relativistic jet sources such as blazars, via diffusive shock acceleration (DSA). In recent work, we had self-consistently coupled DSA and radiation transfer simulations in blazar jets. These one-zone models determined that the observed spectral energy distributions (SEDs) of blazars strongly constrain the nature of the hydromagnetic turbulence responsible for pitch-angle scattering. In this paper, we ex…
Diffusive shock acceleration (DSA) at relativistic shocks is widely thought to be an important acceleration mechanism in various astrophysical jet sources, including radio-loud active galactic nuclei such as blazars. Such acceleration can produce the non-thermal particles that emit the broadband continuum radiation that is detected from extragalactic jets. An important recent development for blazar science is the ability of Fermi-LAT spectroscopy to pin down the shape of the distribution of the underlying non-thermal particle population. This paper highlights how multi-wavelength spectra spann…
Blazars, a class of active galaxies whose jets are relativistic and collimated flows of plasma directed along the line of sight and are prone to a slew of magneto-hydrodynamic (MHD) instabilities. We aim to study the interplay of radiation and particle acceleration processes in regulating the multi-band emission and variability signatures from blazars. In particular, the goal is to decipher the impact of shocks arising due to MHD instabilities in driving the longterm variable emission signatures from blazars. In this regard, we have performed RMHD simulations of a representative section of bla…
The gamma-ray emission offers a powerful diagnostic tool to probe jets and their surroundings in flat spectrum radio quasars (FSRQ). In particular, sources emitting at high energies (>10 GeV) give us the strongest constraints. This motivates us to start a systematic study of flares with bright emission above 10 GeV, examining archival data of Fermi-LAT gamma-ray telescope. At the same time, we began to trigger Target of Opportunity observations to the Swift observatory at the occurrence of high-energy flares, obtaining a wide coverage of the spectral energy distributions for several FSRQs duri…
The recent associations of neutrinos with blazars require the efficient interaction of relativistic protons with ambient soft photon fields. However, along side the neutrinos gamma-ray photons are produced which interact with the same soft photon fields producing electron-positron pairs. The strength of this cascade has significant consequences on the photon spectrum in various energy bands and puts severe constraints on the pion and neutrino production. In this study, we discuss the influence of the external thermal photon fields (accretion disk, broad-line region, and dusty torus) on the pro…
Electrons emitting non-thermal photons in blazars are possibly accelerated by turbulences developed in jets. In this paper, we consider the case so-called hard-sphere scattering as an interaction model between turbulences and electrons, in which the acceleration timescale is independent of the electron energy. We numerically simulate broad-band emission from blazar jets with a one-zone time-dependent code, taking into account the turbulence acceleration. Our model reproduces various blazar spectra with simple assumptions, such as constant particle injection rate, constant diffusion coefficient…
Aim : The nearby TeV blazar 1ES 1959+650 (z=0.047) was reported to be in flaring state during June - July 2016 by Fermi-LAT, FACT, MAGIC and VERITAS collaborations. We studied the spectral energy distributions (SEDs) in different states of the flare during MJD 57530 - 57589 using simultaneous multiwaveband data to understand the possible broadband emission scenario during the flare. Methods : The UV/optical and X-ray data from UVOT and XRT respectively on board Swift and high energy $γ$-ray data from Fermi-LAT are used to generate multiwaveband lightcurves as well as to obtain high flux states…
The jet composition and energy dissipation mechanism of Gamma-ray bursts (GRBs) and Blazars are fundamental questions which remain not fully understood. One plausible model is to interpret the $γ$-ray emission of GRBs and optical emission of blazars as synchrotron radiation of electrons accelerated from the collision-induced magnetic dissipation regions in Poynting-flux-dominated jets. The polarization observation is an important and independent information to test this model. Based on our recent 3D relativistic MHD simulations of collision-induced magnetic dissipation of magnetically dominate…
We present a new time-dependent leptonic code that we developed to model the varying multi-wavelength (MWL) emission during blazar flares. In our modeling, we assume that the blazar emission originates from a plasma blob located in the jet, and that relativistic electrons are injected into the blob and may undergo stochastic (Fermi II) or shock (Fermi I) acceleration. We numerically solve the kinetic equation for electron evolution in the blob, taking into account particle injection, escape, acceleration and radiative cooling. In order to calculate the spectral energy distribution (SED) of the…
H.E.S.S. detection and multi-wavelength study of the $z \sim$ 1 blazar PKS 0346$-$27
Dublin Institute for Advanced Studies, DIAS Dunsink Observatory（都柏林高级研究所，DIAS邓辛克天文台）；Max-Planck-Institut für Kernphysik（马克斯·普朗克核物理研究所）；University of Namibia（纳米比亚大学）；Centre for Space Research, North-West University（西北大学空间研究中心）；Universität Potsdam（波茨坦大学）；Université Paris Cité（巴黎西岱大学）；Linnaeus University（林奈大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步辐射中心）；Humboldt-Universität zu Berlin（柏林洪堡大学）；Universität Tübingen（蒂宾根大学）
The emission of the so-called extreme blazars challenges the particle acceleration models. The hardness of its spectrum, $<2$, demands extreme parameters using the standard one-zone SSC model in the high energy band. Some authors use both two-zone or hadronic/leptohadronic models to relax these extreme values. In this work, we present a leptohadronic two-zone model with external radiation fields to explain the broadband emission, where the contribution of two components forms the hard-spectrum in the $γ$-rays band. The first is produced by the photopion process, where accelerated protons in an…
Motivated by the observation of a $>290$ TeV muon neutrino by IceCube, coincident with a $\sim$6 month-long $γ$-ray flare of the blazar TXS 0506+056, and an archival search which revealed $13 \pm 5$ further, lower-energy neutrinos in the direction of the source in 2014-2015, we discuss the likely contribution of blazars to the diffuse high-energy neutrino intensity, the implications for neutrino emission from TXS 0506+056 based on multi-wavelength observations of the source, and a multi-zone model that allows for sufficient neutrino emission so as to reconcile the multi-wavelength cascade cons…
The blazar zone in quasars is commonly assumed to be located inside the broad-line region at some hundreds of Schwartzschild radii from the central black hole. Now, the simultaneous Fermi/LAT and MAGIC observations of a strong flare in the FSRQ PKS 1222+21 (4C 21.35, z=0.432) on 2010 June 17 challenge this picture. The spectrum can be described by a single power law with photon index 2.72+/-0.34 between 3 GeV and 400 GeV, and this is consistent with emission from a single component in the jet. The absence of a spectral cutoff constrains the gamma-ray emission region to lie outside of the broad…
One of the most important questions in blazar physics is the origin of broadband emission and fast-flux variation. In this work, we studied the broadband temporal and spectral properties of a TeV blazar 1ES 1727+502 and explore the one-zone synchrotron-self Compton (SSC) model to fit the broadband spectral energy distribution (SED). We collected the long-term (2014-2021) multiband data which includes both the low and high flux states of the source. The entire light curve is divided into three segments of different flux states and the best-fit parameters obtained by broadband SED modeling corre…
The multi-wavelength emission from a newly identified population of `extreme-TeV' blazars, with Compton peak frequencies around 1 TeV, is difficult to interpret with standard one-zone emission models. Large values of the minimum electron Lorentz factor and quite low magnetisation values seem to be required. We propose a scenario where protons and electrons are co-accelerated on internal or recollimation shocks inside the relativistic jet. In this situation, energy is transferred from the protons to the electrons in the shock transition layer, leading naturally to a high minimum Lorentz factor …
The matter content of extragalactic relativistic jets is still an unsolved issue. There are strong arguments against pure electron-positron pair jets, but pairs could outnumber the electrons associated with protons by a factor 10-20. This impacts on the estimate of the jet kinetic power, by reducing it by the same factor, and on the total energy delivered to leptons by the particle acceleration mechanism. Pairs cannot be created in the same jet-zone responsible for the high energy gamma-ray emission we see in blazars, because the reprocessing of the created pairs would overproduce the X-ray fl…
Detection of the IceCube-170922A neutrino coincident with the flaring blazar TXS 0506+056, the first and only 3-sigma high-energy neutrino source association to date, offers a potential breakthrough in our understanding of high-energy cosmic particles and blazar physics. We present a comprehensive analysis of TXS 0506+056 during its flaring state, using newly collected Swift, NuSTAR, and X-shooter data with Fermi observations and numerical models to constrain the blazar's particle acceleration processes and multimessenger (electromagnetic and high-energy neutrino) emissions. Accounting properl…
In this work we investigate the nature of multi-wavelength variability of blazars from a purely numerical approach. We use a time-dependent one-zone leptonic blazar emission model to simulate multi-wavelength variability by introducing stochastic parameter variations in the emission region. These stochastic parameter variations are generated by Monte Carlo methods and have a characteristic power law index of $α=-2$ in their power spectral densities. We include representative blazar test cases for a flat spectrum radio quasar and a high synchrotron peaked BL Lacertae object for which the high e…
A very bright and fast varying gamma-ray flare has been detected from the blazar 3C 279 on June 16, 2015. We have studied the broadband spectral energy distribution of the source during the flaring episode and in the low activity state using a simple one zone leptonic model. We find that an electron energy distribution described by a broken power law can be used to reproduce the broadband emissions during the high and low activity states. The flux measurements at radio, infrared and optical frequencies are reproduced by the synchrotron emission resulting from the relativistic electrons in a je…
PKS 1424+240 is a distant very high energy gamma-ray BL Lac object with redshift $z=0.601$. It was found that pure synchrotron self-Compton (SSC) process normally need extreme input parameters (e.g., very low magnetic field intensity and extraordinarily large Doppler factor) to explain its multi-wavelength spectral energy distributions (SEDs). To avoid the extreme model parameters, different models have been proposed (e.g., two-zone SSC model or lepto-hadronic model). In this work, we employ the traditional one-zone leptonic model after including a weak external Compton component to re-explore…
Luminosities of many powerful blazars are strongly dominated by $γ$-rays which most likely result from Comptonization of radiation produced outside a jet. This observation sets certain constraints on composition and energetics of the jet as well as the surrounding quasar environment. We study the dependence of a Compton dominance on a jet magnetization (the magnetic-to-matter energy flux) and on the location of the 'blazar zone'. Calculations are performed for two geometries of broad-emission-line and hot-dust regions: spherical and planar. The jet magnetization corresponding to the large obse…
Blazars are a subclass of active galactic nuclei (AGN) that have a relativistic jet with a small viewing angle towards the observer. Recent results based on hadronic scenarios have motivated an ongoing discussion of how a blazar can produce high energy neutrinos during a flaring state and which scenario can successfully describe the observed gamma-ray behaviour. Markarian 421 is one of the closest and brightest objects in the extragalactic gamma-ray sky and showed flaring activity over a 14-day period in March 2010. In this work, we describe the performed analysis of \textit{Fermi}-LAT data fr…
In the third catalog of active galactic nuclei detected by the $Fermi$ Large Area Telescope Clean (3LAC) sample, there are 402 blazars candidates of uncertain type (BCU). The proposed analysis will help to evaluate the potential optical classification flat spectrum radio quasars (FSRQs) versus BL Lacertae (BL Lacs) objects of BCUs, which can help to understand which is the most elusive class of blazar hidden in the Fermi sample. By studying the 3LAC sample, we found some critical values of $γ$-ray photon spectral index ($Γ_{\rm ph}$), variability index (VI) and radio flux (${\rm F_R}$) of the …
High energy photons from blazars interact within tens of kpc with the extragalactic photon background, initiating electromagnetic pair cascades. The charged component of such cascades is deflected by extragalactic magnetic fields (EGMF), leading to halos even around initially point-like sources. We calculate the intensity profile of the resulting secondary high-energy photons for different assumptions on the initial source spectrum and the strength of the EGMF, employing also fields found earlier in a constrained simulation of structure formation including MHD processes. We find that the obser…
We observed 5 gamma-ray loud blazars at redshift greater than 2 with the X-Ray Telescope (XRT) and the UltraViolet and Optical Telescope (UVOT) onboard the Swift satellite, and the Gamma-Ray burst Optical Near-Infrared Detector (GROND) instrument. These observations were quasi simultaneous, usually within a few hours. For 4 of these blazars the near-IR to UV data show the presence of an accretion disc, and we could reliably estimate its accretion rate and black hole mass. One of them, PKS 1348+007, was found in an extraordinarily high IR-optical state, almost two orders of magnitude brighter t…
Blazar flares from plasma blobs crossing the broad-line region
Université Paris Cité, CNRS, Astroparticule et Cosmologie（巴黎西岱大学）；Centre for Space Research, North-West University（西北大学空间研究中心）；LUX, Observatoire de Paris, Université PSL, Sorbonne Université, CNRS（巴黎天文台）
Diffusive shock acceleration (DSA) at relativistic shocks is likely to be an important acceleration mechanism in various astrophysical jet sources, including radio-loud AGN. An important recent development for blazar science is the ability of Fermi-LAT data to pin down the power-law index of the high energy portion of emission in these sources, and therefore also the index of the underlying non-thermal particle population. This diagnostic potential was not possible prior to Fermi launch, when gamma-ray information was dominated by the highly-absorbed TeV band. This paper highlights how multiwa…
Low ($\lesssim 1\%$) levels of circular polarization (CP) detected at radio frequencies in the relativistic jets of some blazars can provide insight into the underlying nature of the jet plasma. CP can be produced through linear birefringence, in which initially linearly polarized emission produced in one region of the jet is altered by Faraday rotation as it propagates through other regions of the jet with varying magnetic field orientation. Marscher has begun a study of jets with such magnetic geometries using the Turbulent Extreme Multi-Zone (TEMZ) model, in which turbulent plasma crossing …
We performed 3D MHD calculations of stream accretion in cataclysmic variable stars for which the white dwarf primary star possesses a strong and complex magnetic field. These calculations are motivated by observations of polars; cataclysmic variables containing white dwarfs with magnetic fields sufficiently strong to prevent the formation of an accretion disk. So an accretion stream flows from the L1 point and impacts directly onto one or more spots on the surface of the white dwarf. Observations indicate that the white dwarf, in some binaries, possesses a complex (non-dipolar) magnetic field.…
