Explaining the Multiwavelength Emission of Hard-TeV BL Lac Objects Using a Truncated Conical Jet Model

Maichang Lei ,Yuan Zheng ,Jianfu Zhang ,and Jiancheng Wang

1 College of Physics and Engineering Technology,Xingyi Normal University for Nationalities,Xingyi 562400,China;maichanglei83@163.com

2 Department of Physics,Xiangtan University,Xiangtan 411105,China

3 Yunnan Observatories,Chinese Academy of Sciences,Kunming 650216,China

Abstract Hard-TeV BL Lac objects are newly identified populations of active galactic nuclei with the emitted γ-ray photons well above TeV energies.In this paper,we explain the multiwavelength emission of six Hard-TeV BL Lac objects by using a truncated conical emission region of the jet,where the electron distribution is obtained by numerically solving the evolution equation along the jet self-consistently.For comparison,we also apply the model to Mrk 421 and Mrk 501,which are the potential candidates for the hard TeV emissions.We demonstrate that the model can satisfactorily reproduce the spectral energy distributions of eight sources,particularly of six Hard-TeV sources,where no extreme minimum Lorentz factor of the electron population is required.In contrast with Mrk 421 and Mrk 501,six Hard-TeV sources have rather low magnetization in emitting regions and high cutoff energies of the electron distributions.

Key words: galaxies:active–(galaxies:) BL Lacertae objects:general–radiation mechanisms:non-thermal

1.Introduction

Blazars are jetted active galactic nuclei (AGNs) with bipolar relativistic plasma jet aligned closely with the line of sight(Urry&Padovani1995;Padovani2016).The nonthermal electromagnetic emission,covering a wider range from radio up to very high energy(VHE;Eγ?100 GeV)γ-rays,is generally attributed to the jet and strongly enhanced by Doppler boosting.The multifrequency emissions are highly variable,on timescales of order of minutes to years (Abdo et al.2010b).Blazars are subdivided,according to the broad emission line criterion(equivalent width ＞or ＜5 ?),into flat spectrum radio quasars(FSRQs)and BL Lacertae objects(BL Lacs)(Stickel et al.1991;Stocke et al.1991;Ghisellini et al.2009).Based on the synchrotron peak,,BL Lacs are further classified as lowpeaked BL Lacs (LBL,＜ 1014Hz),intermediate-peaked BL Lacs(IBL,1014≤≤1015Hz)as well as high-peaked BL Lacs (HBL,＞ 1015Hz) (Abdo et al.2010a).In contrast,BL Lacs commonly lack luminous external radiation fields,the accelerating electrons will suffer inefficient cooling and can be accelerated to higher energies within their jet.Accordingly,the peak energy of the radiated γ-ray photons will be up to several TeVs,this can be statistically depicted by “blazar sequence”(Fossati et al.1998;Ghisellini et al.1998,2017).

Hard-TeV BL Lac objects (Hard-TeV BL Lacs) are an emerging class,and belong to extreme high-peaked BL Lacs(EHBLs) with the very high frequencies of their two emission peaks (Costamante et al.2001;?entürk et al.2013;Foffano et al.2019;Biteau et al.2020).These objects have an important implication to explore the extragalactic background light (EBL),the intergalactic magnetic fields and the exotic physics at extreme energies inaccessible with human-made devices,such as Lorentz invariance violation and axion-like particles (Biteau et al.2020).A mini catalog of six Hard-TeV BL Lacs is presented by Costamante et al.(2018) (henceforth,PaperI),their characteristic Fermi-LAT spectra are well characterized by the hard spectral slope,typically,ΓLAT?1.6–1.9,and their Compton peaks in spectral energy distribution(SED) are above 2–10 TeV.Comparatively,their synchrotron peaks are located at medium or hard X-ray bands.These characters lead a challenge to standard one-zone synchrotron self-Compton (SSC) model,where the Klein–Nishina (K-N)effect takes action to make the TeV spectrum steepen.

Various theoretical scenarios have been proposed to account for the origin of hard-TeV spectrum.They include the finetuned electron distributions with extremely hard Maxwellian form (Saugé &Henri2004;Lefa et al.2011),very high lowenergy cutoff (Katarzyński et al.2006;Tavecchio et al.2009),large Doppler factor (Tavecchio et al.2009),extreme model parameters (Tavecchio et al.2010),and Compton upscattering of an external radiation field (Lefa et al.2011) or internal γ–γ absorption on a narrow-band radiation field (Aharonian et al.2008).Moreover,the X-ray and TeV emissions are assumed to arise from distinct emitting regions (B?ttcher et al.2008).On the other hand,the hard-TeV spectra are also explored via invoking hadronic processes,e.g.,as a secondary product of cascades induced by ultra-high-energy protons (Essey &Kusenko2010;Essey et al.2011;Prosekin et al.2012).For the archetypal Hard-TeV BL Lac 1ES 0229+200,its hard-TeV spectra are satisfactorily reproduced by intergalactic cascade scenario (Murase et al.2012),or by the secondary radiations from p–γ interaction (Cao &Wang2014),but the latter needs highly super-Eddington jet power,which is about six orders of magnitude higher than the Eddington luminosity and could be problematic under the canonical accretion paradigm (Zdziarski&Bottcher2015).Subsequently,Cerruti et al.(2015) applied the proton synchrotron and the p–γ-induced cascades to interpret the hard-TeV spectra whereas avoiding the extreme parameters encountered in pure SSC models and super-Eddington crisis mentioned above.The TeV flares are also studied based on the hadronic and leptohadronic emission models (Murase et al.2012;Cerruti et al.2015;MAGIC Collaboration et al.2019;Sahu et al.2019).Recently,the broadband SEDs of several EHBLs(including 1ES 0229+200)are modeled based on one-zone SSC,spine-layer and the proton synchrotron scenarios with substantially different parameters,especially for magnetization (Acciari et al.2020).

The extremely high-energy photons from the Hard-TeV BL Lacs indicate that some efficient energization processes of particles in the relativistic jet must be at work.At present,three types of acceleration mechanism are preferred:the first-Fermi(shock),second-Fermi (stochastic) accelerations from the interactions of shock waves and of random field of Alfvén waves and magnetic reconnection usually adopted to explain the most rapid flares (Baring et al.2017),which is notable incompatible with the modest variability shown by several Hard-TeV BL Lacs (Aliu et al.2014;Cologna et al.2015;Acciari et al.2020).Among them,the shock and turbulence accelerations are widely invoked to accelerate particles to higher energies(Lewis et al.2016,2018;Baring et al.2017).A hybrid acceleration process underwent by injected background particles,consisting of initial acceleration by turbulence followed by a second stage acceleration by shocks,could play an important role(Fan et al.2010;Petrosian2012;Kang2015;Baring et al.2017).The stochastic acceleration is also the promising candidate to produce the hard spectrum (Stawarz &Petrosian2008;Asano et al.2014).Most probably,the acceleration mechanisms operating during the flaring episodes could be their combination,with different dominated processes depending on the local conditions (Rieger et al.2007).Under the dramatically low magnetization and the requirement on rather high energy electrons,the shocks and some preheating mechanisms,e.g.,turbulent accelerations,could play a dominated role in the jet of the Hard-TeV BL Lacs (Zech &Lemoine2021).

In this paper,we focus on the broadband SED modeling of six Hard-TeV BL Lacs presented in PaperI,with an emphasis on the origin of the hard-TeV spectrum.The outline of the paper is as follows.In Section2,we outline the model framework.Section3presents the model application.Results are discussed in Section4and the conclusions are given in Section5.Throughout this paper,the following cosmological parameters are adopted:H0=70 km s-1Mpc-1,ΩM=0.3,ΩΛ=0.7.

2.Model Description

In leptonic framework,while both conical and spherical configurations have been widely adopted as the emitting region to insight into the multiwavelength emission of blazars(Ghisellini et al.1985;Moderski et al.2003;Potter &Cotter2012,2013;Lei et al.2018).Observationally,an axisymmetric jet flows,with a constant opening angle over much of the radical length,have been shown by Very Long Baseline Array (VLBA) images (Kovalev et al.2007).Meanwhile,Sokolovsky et al.(2011) invoked this structure and well explained the frequency-dependent core shifts obtained by them.Therefore,this paper adopts the conical structure of magnetized jet to explore the nature of multiwavelength emission from six Hard-TeV BL Lacs,where an assemble of isotropically non-thermal electrons are continually injected into a truncated conical region with radical length L′from the base,this truncated cone is called“emitting region”in the subsequent sections.It is emphasized that the primed quantities represent the ones measured in the comoving frame of the jet,whereas the quantities with subscript “obs” are measured in the observer’s frame.The emitting region is assumed to be filled with tangled magnetic field,the injected electrons are evolved following the evolution equation.Our model is the generalization of the one proposed by Potter &Cotter(2012)(henceforth,PaperII),this model mainly has two advantages,first,it can well reproduce the flat radio spectrum(Zheng&Yang2016;Lei et al.2018);second,compared with the models which involve to solve Fokker–Planck equation (Park &Petrosian1996),it is quite appropriate for studying the BL Lacs in which only the simply cooling process needs to be considered,because it cannot deal with the complex cooling processes selfconsistently.Thus,this only fits for Hard-TeV BL Lacs,a number of studies have shown that the γ-ray emitting region has rather weak magnetic field,the energy loss of the energetic electrons from inverse Compton cooling is very weak and can be ignored(Tavecchio et al.2009;Kaufmann et al.2011;Yan et al.2012;Cohen et al.2014;Costamante et al.2018).

2.1.Diffusion Equation

The diffusion equation governing the evolution of the injected electron population along the jet is given by PaperIIas

the normalizationN0is related to the jet length and electron’s energy in such a way:

Subsequently,the synchrotron and SSC emissions can be calculated using the electron distribution given by

2.2.Jet Energetics

Combining these equations,N0andR0can be calculated as follows:

where the factorH is determined merely by three quantities,i.e.,the spectral index α,the minimum γ′minand the maximumof the injected electron distribution.Moreover,we assume that each segment of the jet will conserve the magnetic energy,the magnetic field will decline as the function of radius of the jet,that is,

2.3.Radiative Processes

The electrons are once injected into the emitting region and will diffuse along the jet,these electrons will inevitably produce emissions through synchrotron and SSC processes in the magnetic field.The synchrotron emissivity from electrons in a section of lengthdx′ is given as

We note that Equation(9)need to be corrected by the factorbecause of the synchrotron self-absorption,wherek∈′is the opacity.It is noted that the path length of a photon with energy ∈′escaped from a segment of widthdx′has approximately taken as R (x ′),the radius of the jet,where the photon is produced.Combining this correction factor and Equation (9),the energy density of the synchrotron radiation can be given as

For a photon survived after undergoing a path lengthdx′will suffer from the absorption of the remaining part of the emitting region,which will incorporate another correction factor,thus the total synchrotron emissivity emitted by the whole truncated cone is given by

where τtot(∈′ ,x′)is the opacity accounting for the absorption probability of a synchrotron photon moving from x′ to L′.

We calculate the inverse Compton emission in which the effect of the cross section reduction in the K-N regime has been considered.So,the SSC emissivity is given by

whereHis the Heaviside function,Fsscis the Compton scattering kernel for isotropic radiation fields of both photons and electrons,which has been given by Jones (1968),Blumenthal &Gould (1970),Finke et al.(2008),Dermer et al.(2009):

From Equations(12)and(13),the observed synchrotron and SSC fluxes are calculated according to

2.4.Numerical Implementation

For solving Equation (1) numerically,we need to discretize it as follows:

3.Applications

In the following,the model is applied to six Hard-TeV BL Lacs presented in PaperI,where preliminary SED modelings have been performed and the results revealed that the rather low magnetic field,of the order of mG,could be possible.Observationally,this extreme subclass of blazar usually showed mild flux variations,with the flux variation of a factor of two to three,over years at TeV energies (Aliu et al.2014;Cologna et al.2015;Acciari et al.2020).An exception is 1ES 1218+304,which presents a rapid flare over a few days(Acciari et al.2010b).Even though these sources showed weak flux variation at γ-rays,but at other frequencies,because of the large statistical uncertainties,the notable flux variation cannot be ruled out,such as X-ray.Interestingly,Mrk 501,a prototypical high-peaked BL Lac object,also showed EHBLlike behaviors during its flaring episodes (Ghisellini1999;Ahnen et al.2018),this indicates that the HBLs could be the candidate to produce the hard-TeV emission under flaring states.However,in this paper,we apply the model to Mrk 421 and Mrk 501 when they were in low γ-ray state,instead,there remains some flux change at other frequencies,this will help us to compare the properties of two extreme subclasses of BL Lacs in roughly same state of activity,and provide an explanation on why two HBLs cannot emit hard-TeV emission in non-flaring conditions,such a comparison will provide some clues on unveiling the origin of the Hard-TeV emission from the Hard-TeV BL Lacs.The SED data used for Mrk 421 and Mrk 501 are taken from 2008 August 5 to 2010 March 12 as well as 2009 March 15 to August 1,respectively (Abdo et al.2011a,2011b),where the quasi-simultaneous SED was reproduced using a one-zone leptonic SSC model and onezone hadronic one.Under the former,the electron distribution is typically parameterized with two breaks,which indicates the complex acceleration and cooling processes of electrons would occur within the emitting region.In contrast,we attempt to reproduce the spectral shape by using electron distribution with only one break.

3.1.To Six Hard-TeV BL Lac Objects

First,the model is used to reproduce the broadband SED of six Hard-TeV BL Lacs,i.e.,1ES 0229+200,1ES 0347-121,1ES 0414+009,RGB J0710+591,1ES 1101-232 and 1ES 1218+304.The multiwavelength data points are from PaperI,and are here presented in Figure1,in which the red filled circles represent the quasi-contemporaneous data,the data points taking close to the same epoch of the VHE ones are shown as blue filled circles,the archival data are plotted in gray triangles.The VHE data of RGB J0710+591 and 1ES 1218+304 are taken by VERITAS telescopes,while the corresponding data are from HESS telescopes for other sources,these data sets have been corrected for EBL absorption using the model of Franceschini et al.(2008).

Figure1shows the resulting SED modeling to six Hard-TeV BL Lacs.Overall,the model can well reproduce the multifrequency emission from radio to VHE γ-rays.An exception is 1ES 0414+009,its γ-ray emission cannot be well explained by our model,the dramatic incompatibility between flat Fermi-LAT spectrum and hard-TeV ones may originate from distinct SED emission component at different epochs,as stated in PaperI.Within this period,1ES 0414+009 has likely changed its emitting state and with different γ-ray SED properties,making the Compton peak shift to higher frequency.On the other hand,it is worth mentioning that the data points are at optical,where the predominant emission is contributed by the host galaxy and not considered in our SED modeling,these data points only act as an upper limit to the reproduced spectrum.The model parameters are presented in Table1,and the seven derived quantities are given in Table2.

From Table1,it is clear that the magnetic field strengthat the base of the emitting region are of the order of magnitude of mG,which are roughly consistent with the ones adopted by PaperI.A larger value ofis required for 1ES 0414+009,wheretakes 17 mG,which is about 18 times larger than that adopted in PaperI.The model requires an unusually low magnetic field to make the magnetic energy,,far below the electron energy,,by two to three orders of magnitude.On the other hand,because of low magnetic energy,the electrons can be easily accelerated to higher energy under weak radiative cooling from synchrotron process.As shown in Figure1,the good consistency of both flux and slope between the Fermi-LAT and the intrinsic VHE spectra suggest that the Compton peak would be located at the high-energy end or beyond,which is roughly related to the cutoff energy of electron distribution.Table1presents large value of,within the range of(1.0–8.8)×106,they are somewhat larger than the ones obtained by PaperI,where the maximum is 1.5×106for 1ES 0229+200.In our modeling,the larger value of 8.8×106is invoked for modeling the broadband SED of 1ES 0347-121,accordingly,this makes the Compton peak being at higher frequency.In comparison with 1ES 0347-121,the Compton peak of the other sources is nearly located at the tail of VERITAS or HESS observation.If this is true,the good energy coverage of the imaging atmospheric Cherenkov telescopes is vital to constrain on the high-energy peak of this unique subclass of blazar,and consequently to unveil the underlying acceleration processes.

Figure 1.Broadband SED modelings to six Hard-TeV BL Lacs.The heavy black line indicates the superposed intrinsic emission from synchrotron and SSC processes,the blue solid line represents the synchrotron emission and the red dotted-line is the EBL-corrected SSC spectrum using the EBL model proposed by Franceschini et al.(2008).

A crucial parameter,α,is closely related to the acceleration mechanisms of the particle,and the very hard spectral index is generally thought to be the typical property of the Hard-TeV BL Lacs at γ-rays.Our SED modeling requires α to be within the range from 1.63 to 2.1,where four sources,1ES 0229+200,1ES 0347-121,RGB J0710+591 and 1ES 1218+304,have harder spectra,the corresponding α are 1.63,1.75,1.7 and 1.8,respectively.In contrast,two sources,1ES 0414+009 and 1ES 1218+304,have softer spectra with the same spectral index of 2.1.These values are basically consistent withn1given in PaperI.In comparison with PaperI,1ES 0229+200 and 1ES 1218+304 have obvious differences.For 1ES 0229+200,our obtained α is 1.63 larger than their value of 1.4,while for 1ES 0229+200,the obtained value of 2.1 by us is obviously lower than their value of 2.85.

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Once the broadband SEDs are are well reproduced,the synchrotron and SSC peaks are shown in Table3,where for each source the synchrotron and the SSC peaks have nearly the same flux level.Our SED modeling shows that five of the Hard-TeV BL Lacs have SSC peak,measured in the observer’s frame,are well TeV energies,and the highest peak frequency of 1ES 0229+200 reaches 10.75 TeV.In contrast,1ES 0414+009 has low SSC peak,which is located at 0.23 TeV.According to the Doppler transformation,we can obtain the comoving frequency of peak.From Table3,we can see that the intrinsic SSC peaks of six Hard-TeV BL Lacs are lower than 1 TeV.

In Figure2,we present the evolution of the electron population,under the well representation to the broadband SED in Figure1,along the jet for six hard-TeV BL Lacs.In our code,12 curves with the same logarithmic step are calculated and plotted,however,due to the weak cooling exists,only a few curves are shown.Because of distinct emitting conditions,the number of the curve is different for different sources.From right to left,a set of curves represent the electron distributions at the location from the base of the emitting region.It is clear that the spectral shape hardly keeps invariable for each location as long as the Lorentz factor of the electrons is below the cutoff,while above the cutoff the curve will bend down,this can be contributed to the exponential cutoff itself,the synchrotron cooling or K-N effect,or their combination.As for the K-N effect,the decline of the scattering cross-section will largely decrease the collisions between high-energy electrons and photons to low the flux at the high-energy end.Note.,Γjare the magnetic field strength at the base and the bulk Lorentz factor of the jet,respectively.α andare the spectral index and the cutoff energy for injected electron distribution,respectively.Aeqis the equipartition fraction between magnetic and electron’s energy measured in the comoving frame,is the injected energy.The last column represents the mass of the central BH,where the superscripts represent the corresponding references,such that:(a):Aharonian et al.(2007a),(b):Aharonian et al.(2007b),(c):Falomo et al.(2003),(d):Acciari et al.(2010a),(e):Aharonian et al.(2007c),(f):Acciari et al.(2009),(g):Albert et al.(2007a),(h):Albert et al.(2007b).As shown in Figure2,four sources show an excessive softening of the high-energy γ-ray spectrum,i.e.,1ES 0229+200,1ES 0414+009,1ES 1101-232 and 1ES 1218+304,the other two sources only show weak deviation from the injected spectrum.Taking both B′ andinto account,the radiative cooling will play an important role in shaping VHE spectrum,whereis the distance from the central black hole(BH) and is presented in Table2.It is calculated viais the viewing angle measured in the comoving frame,which is connected to θobsby tan() =Γjtan(θobs).

Figure 2.Electron number distribution vs.normalized energy γ′by mec2 for six Hard-TeV BL Lacs,which correspond to the SED modeling as shown in Figure 1.In each panel,the black line indicates the injected spectrum,whereas the remaining curves correspond to a series of grid points x′ at further distance.

Figure 3.Broadband SED modeling for two typical HBLs,Mrk 421 (left column) and Mrk 501 (right column).In SED modelings,the heavy black line is the superposed spectrum from synchrotron and the SSC processes,the red dotted line is the EBL-corrected SSC emission using the EBL model proposed by Franceschini et al.(2008).For each source,the bottom panel corresponds to energy distribution of electrons at different x′,where the black line is the spectrum at the injected point.The remaining curves,from right to left,represent the spectra at further locations,respectively.

Table 1 Input Parameters for the Model

Table 2 The Values of Several Derived Quantities

Table 3 Peaks and Peaking Fluxes for Synchrotron and SSC Humps

3.2.To HBLs Mrk 421 and Mrk 501

Figure3also shows the broadband SED modelings and the corresponding electron distributions,where Mrk 421 and Mrk 501 correspond to the left and right columns,respectively.The same as Figure1,the optical data are not considered and in principle can be explained by invoking the contributions from the host galaxy.For Mrk 501,the optical data accumulated to a mini hump with distinct difference to the remainder of the synchrotron hump in terms of spectrum and flux level.However the optical data of Mrk 421 has the similar spectral slope connecting the radio and X-ray data,but has slightly higher flux level.In order to well model the data of both optical and X-ray,the model curve will unavoidably exceed the radio data,unless a large minimum Lorentz factor is adopted in electron distribution.Generally,the radio emission is thought to be the superposition from the extended emission region,as shown in Lei et al.(2018).Therefore,the radio flux is commonly taken as an upper limit.In our SED modeling,the minimum Lorentz factor is fixed at 1.0,thus the model curve need to be below the optical data,this inversely implies that the optical data may be elevated by the emission from the host galaxy.As far as the spectral shape is considered,the spectrum of Mrk 421 is more steeper than that of Mrk 501 at low-energy frequencies before the synchrotron peak,this consequently requires a harder spectral index for Mrk 421 than for Mrk 501.

After these considerations,the broadband SED modelings are carried out.As mentioned above,a relatively hard spectral index is required for Mrk 421,but we cannot provide a well fitting to the beginning data of the SSC hump if a rather hard value is adopted.Inversely,if we adopt a relatively soft spectral index,the modeling curve will be above the radio data.To give a better fitting to all the observational data,we use soft spectral index and increase Γjto make the multifrequency spectrum shifting toward the higher frequency.The modeling curve will pass through or low the radio data.On the other hand,we performed the SED modeling to Mrk 501 by taking a soft spectral index and a relatively low value of Γj.The resulting parameters are presented in Table1,it is clear that both sources have different parameters,where Mrk 421 has the harder spectral index of 1.81,the higher magnetic field of 1.0 G at the base of the emitting region and the higher Γjof 33.Due to larger magnetic field,the equilibrium between matter and magnetic energies is achieved,e.g.,Aeqequals to 1.4.However,Mrk 501 has the soft spectral index of 2.2,the lower magnetic field of 0.44 G and the normal Γjof 21.Both sources have also significant difference in,corresponding to 6.9×104and 4.5×105for Mrk 421 and Mrk 501,respectively.Moreover,the peaks and fluxes of synchrotron and SSC humps are also presented in Table3.Their synchrotron peaks have nearly the same value of frequency,but their SSC peaks are far below those of Hard-TeV BL Lacs.

In Figure3,the bottom two panels show the evolution of emitting electrons with the distance from the central BH for Mrk 421 and Mrk 501.In comparison with six Hard-TeV BL Lacs,their electron populations have the same trend of evolution.Once the Lorentz factor of the electrons are over the cutoff energy,the electron distribution drops exponentially.Since the stronger magnetic field in the emitting region with respect to Hard-TeV BL Lacs,the high-energy electron spectrum shows significantly change above,in which the spectrum becomes steeper when the grid approaches to the end of the emitting region.

3.3.Jet Power

In leptonic scenario,the multiwavelength emission originates predominantly from the nonthermal electrons and positrons (shorted as electrons) within the tangled magnetic field,whereas the jet dynamics is dominated by the cold protons.In Table2,we present the jet power loaded by electrons,magnetic field and the cold protons given by

The total jet power is given byIt is emphasized that the jet power related to the protons is calculated assuming one cold proton per electron.In Table2,we also show the ratio of the total jet power to the Eddington luminosity (Eddington ratio),the latter is given byLEdd?1.26×1038MBH/M⊙erg s-1,whereMBHandM⊙a(bǔ)re the central BH mass and the solar mass,respectively.We can see from Table2that half of the eight sources has lower Eddington ratio,including 1ES 0229+200,RGB J0710+591 and two HBLs,at the order of 10-3.Other three hard-TeV BL Lacs,1ES 0347-121,1ES 0414+009 and 1ES 1101-232 have larger jet power nearly at the factor 0.3 of the Eddington luminosity.The maximum jet power appears in 1ES 1218+304 with the factor 4.9 of the Eddington luminosity.

4.Discussion

We reproduce the broadband SED of six Hard-TeV BL Lacs by using the power-law electron distributions with the exponential cutoff,where the injected electron distribution spans a broader range with the Lorentz factor,γ′,from 1.0 to 5×107.These electrons are injected and evolved following the diffusion equation.To fit well the hard-TeV data,we require relatively hard spectral index for 1ES 0229+200,1ES 0347-121,RGB J0710+591 and 1ES 1101-232,while we adopt softer spectral index for 1ES 0414+009 and 1ES 1218+304.Their typical feature from the SED modeling is that the magnetic field strength at the injected point,as illustrated in Table1,is at the order of mG,consistent with ones obtained by previous studies (i.e.,PaperI).The weak magnetic field strength together with the short distance from the central BH could be problematic for powering the collimated outflow of the jet from sub-pc to kpc or even up to the spatial scale of the host galaxy,as illustrated by extensive radio observations.Actually,the collimation and acceleration take place up to hundreds of pc along the jet (Asada &Nakamura2012;Hada et al.2018),justified by the very long baseline interferometry (VLBI) by measuring kinematics of bright knots (Homan et al.2009,2015;Jorstad et al.2017).Generally,a low magnetization favors the shock accelerations,however,from the point of view of internal shock scenarios (Spada et al.2001),the particles will have low acceleration efficiency under the weak magnetization (Mimica &Aloy2012;Rueda-Becerril et al.2013,2014).Therefore,such weak magnetic fields required for reproducing the hard-TeV spectra could be “abnormal” to relativistic jet,where the distribution of magnetic field is homogeneous transversely.

On the other hand,it is well known that magnetohydrodynamic (MHD) turbulence is believed to accelerate the electrons of relativistic jet in AGN.In realistic situation,MHD turbulence may be accompanied with shocks and magnetic reconnections occurring inside the magnetized jet.MHD simulations also suggest that the Fermi-I acceleration will give rise to higher levels of stochastic turbulence (Inoue et al.2011),these turbulent processes will lead to the efficient acceleration of the injected particles.For inhomogeneous jet,the efficient re-acceleration processes will occur in the transition layer connecting the internal violently shocked spine and external stable sheath (Zech &Lemoine2021).Considering MHD turbulent acceleration and synchrotron and SSC radiative losses,Uzdensky (2018) derived a relationship between rms Lorentz factor of the electrons and the optical depth as.Thus,the higher the rms Lorentz factor is,the smaller the optical depth becomes.It is roughly consistent with our results of SED modelings,in which the larger scale of the emitting region is used,the low jet power is required.

The model is also applied to two typical HBLs Mrk 421 and Mrk 501,we obtain distinct spectral characteristics with respect to Hard-TeV BL Lacs,corresponding to the stronger magnetic field and the lower value of,corresponding to 1.0 G and 0.44 G,respectively.We note that a detailed SED modeling based on the χ2-minimization technique using the simple onezone syn+SSC leptonic model has been performed to investigate the radiation mechanisms and physical properties of the GeV–TeV BL Lacs,in which a slightly different values of parameters are given to low state (Zhang et al.2012),however,the obtained values ofremain higher than the ones of six Hard-TeV BL Lacs,such higher magnetic fields are also required to well interpret the multifrequency observations(Albert et al.2007b;Acciari et al.2011;Cao &Wang2013;Yan et al.2013;Zhang et al.2013;Chen2017;Zheng et al.2018).Thus,Mrk 421 and Mrk 501 have different properties from the Hard-TeV BL Lacs,this is mostly independent of the special model adopted.In terms of the acceleration mechanism,an equipartition of the matter and the magnetic energy as well as a hard spectral index for Mrk 421 support that the magnetic reconnection could play an important role to energize the background particles,while the weak magnetic field and the softer spectral index for Mrk 501 prefer to suggest that the shock acceleration will take action in accelerating the particles.

It is well known that blazars commonly show the spinesheath morphology in radio VLBI maps,i.e.,a limb-brightening component was interpreted as a slower external flow surrounding a fast spine (Ghisellini et al.2005),this morphology is also supported by the polarization VLBI observations (Zakamska et al.2008),a clear spine-sheath polarization structure was first observed from quasar 1055+018 on parsec scales (Attridge et al.1999),other strong observational supports for a spine-sheath structure to several TeV balzars were presented by Piner et al.(2009,2010) and MacDonald et al.(2015).Moreover,Kravchenko et al.(2017)performed polarimetry analysis on 20 AGNs jets using VLBA,and the observed variety of polarized signatures can be explained by a model of spine-sheath jet structure.Thus,the spine-sheath structure has been the common morphology of the relativistic jet in blazars.Theoretically,the various models based on spine-sheath structure have been proposed to explain the origin of the orphan γ-ray flares,where a blob of plasma moving relativistically along the spine of the jet inverse-Compton scatter the diffuse synchrotron photons emanating from a shocked sheath plasma (MacDonald et al.2015,2017).(iii).The broadband SED of the Hard-TeV HBL 2WHSP J073326.7+515354 can be better represented by a two-zone spine-layer model than the standard one-zone leptonic scenario(MAGIC Collaboration et al.2019).Such a structured jet is also used to explore the origin of the high-energy neutrinos detected by IceCube (Tavecchio et al.2014).

For spine-sheath structure,the higher magnetic field exists in the spine,while the magnetic field within the sheath is significantly weak,the speed of the internal spine flow is much larger than the external sheath flow.The high-energy electron populations originate presumably from the spine,where some efficient acceleration mechanisms will be at work,the highenergy electrons will be produced and reaccelerated continually along the spine.As the particle energy increases and the magnetic field declines,their Larmor radius becomes more larger.As a consequence,the high-energy electrons will go away from the spine,and naturally diffuse into the sheath.Hillas (1984) has shown that a particle with a larger Larmor radius will travel preferentially closer to the edge or sheath of the jet,where the apparent magnetic field strength may be much less than that of the spine.Due to lower synchrotron radiation field and weak magnetic field,these electrons will suffer from relatively low radiative loss.Also,the K-N turnover will also bring about inefficient energy losses at high-energy end.Maybe,at the interface between the spine and the sheath,the collisions of both shocks and surrounding matter induce outstanding turbulent processes,which will also act on the escaped high-energy electrons.Finally,a hard electron distribution appears in the sheath and subsequently produces synchrotron and SSC emissions,the latter could show a hard spectral shape,such that the hard-TeV spectrum occurs.Here,another point being emphasized is that the sheath is more beamed than the spine,e.g.,the viewing angle of the sheath is small relative to the spine,this ensures high-energy γ-ray emission from the sheath to be more beamed.In fact,such explanation on the origin of the hard-TeV spectrum in Hard-TeV BL Lacs has been proposed to explain the TeV emissions from radio galaxy M87,where the layer is responsible for the TeV photons,while the debeamed spine accounts for the emissions from radio to GeV energies(Tavecchio&Ghisellini2008).It is noted that such an idea is also used to clarify the origin of the high-energy particles generating during 2013 December 20 γ-ray flare from 3C 279,a typical FSRQ (Lewis et al.2019).

Up to now,an immediate question is that why the hard-TeV spectra only present in Hard-TeV BL Lacs and not in other typical HBLs,such as Mrk 421 and Mrk 501.The main reason could be the magnetic field in the jet.Actually,as shown in Table1,we can see that the magnetic field strength,,of Mrk 421 and Mrk 501 are far larger than that of the Hard-TeV BL Lacs.The numerical simulations have shown that the development of sheath flow around a relativistic jet spine will help to clarify the partial stabilization of the jets,meanwhile the stabilization of spine and sheath as well as the velocity discrepancy of both will mainly depend on the magnetization(Hardee&Hughes2003;Mizuno et al.2007).In other words,as the magnetic field increases,the jet will become homogeneous,both spine and sheath tend to have the nearly same velocity and properties,the spine-sheath structure will disappear.The conditions for generating the hard electron spectrum cannot be achieved,and the hard-TeV spectrum will be hardly produced.

5.Conclusion

Because the observation at energies above 580 GeV from 1ES 0229+200(Aharonian et al.2007a),the origin of the hard-TeV spectrum has been attracted more attentions,many attempts are made to explore the nature of the underlying particle acceleration mechanisms both theoretically and observationally.Up to 2018,the number of the Hard-TeV BL Lacs increases up to six,it is noted that the sample has been further enriched by recent observations (Acciari et al.2020;Biteau et al.2020),this thus makes this class of sources be emerging subclass of BL Lacs,where a prior efficient acceleration of the particle and an accompanying inefficient cooling must be matched each other.In this paper,we propose a leptonic one-zone model with a truncated conical structure to explore the origin of the hard-TeV spectra of six Hard-TeV BL Lacs,in which the electron population is injected at the base of the emitting region.During the evolution of the electrons along the jet,we merely consider the synchrotron radiative loss selfconsistently.For comparison with six Hard-TeV BL Lacs,the model is also applied to two typical HBLs,Mrk 421 and Mrk 501.Our main results are summarized as follows:

(1)By fixing a broader electron energy distribution with the Lorentz factor from 1.0 to 5.0×105,our model can well reproduce the broadband SEDs of six Hard-TeV BL Lacs and two HBLs.Compared with two HBLs,the Hard-TeV BL Lacs require the higher cutoff energies to the electron distributions,at the order of 106,larger than that of two HBLs by nearly one order of magnitude.

(2) Our SED modelings show that four Hard-TeV BL Lacs,1ES 0229+200,1ES 0347-121,RGB J0710+591 and 1ES 1101-232,have hard spectral index of 1.63,1.75,1.7 and 1.8,respectively,while 1ES 0414+009 and 1ES 1218+304 have softer spectral index of 2.1.On the other hand,in terms of SSC peak,five out of six have SSC peaks well above 1 TeV.

(3) The SED modelings to Hard-TeV BL Lacs require a lower magnetic field,at the order of mG,thus the jets are matter-dominated.While for Mrk 421 and Mrk 501,the required magnetic field are 1.0 G and 0.44 G,respectively.For Mrk 421,the relatively high magnetic field makes that the matter and the magnetic field energies are in equipartition.In contrast,the SED modelings of Mrk 501 and six Hard-TeV BL Lacs require the magnetic energy densities to be well below equipartition.

(4)As far as the jet power is concerned,only 1ES 1218+304 has large jet power more than the Eddington luminosity by a factor of 4.9,the other sources have a lower ratio of Eddington luminosity.

Acknowledgments

We acknowledge the anonymous referee for insightful comments and valuable suggestions that have helped us improve the presentation.We also sincerely thank Dr.Luigi Costamante and David Paneque for sending us the multiwavelength data sets.We acknowledge the financial supports from the growth project of young scientific and technological talents in colleges and universities in Guizhou Province(Qianjiaohe -KY-Zi[2020]221),and Scientific Research Foundation for Doctoral Program of Xingyi Normal University for Nationalities (20XYBS16).