Anisotropic electron populations in BL Lac jets: consequences for the observed emission
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 12 November 2019 (MN L A TEX style file v2.2)
Anisotropic electron populations in BL Lac jets: consequences forthe observed emission
F. Tavecchio (cid:63) , E. Sobacchi , INAF – Osservatorio Astronomico di Brera, via E. Bianchi 46, I–23807 Merate, Italy Physics Department, Ben-Gurion University, P.O.B. 653, Beer-Sheva 84105, Israel Department of Natural Sciences, The Open University of Israel, P.O.B. 808, Raanana 4353701, Israel
12 November 2019
ABSTRACT
We investigate the impact on the properties of high-energy emitting BL Lac objects of apopulation of electrons with an anisotropic momentum distribution. We adopt a simple phe-nomenological description of the anisotropy, in which the most energetic electrons have asmall pitch angle and the least energetic electrons are isotropic, as proposed by Sobacchi &Lyubarsky (2019). We explore (i) a simple model that assumes a phenomenological shapefor the electron energy distribution, and (ii) a self-consistent scheme in which the electronsfollow a distribution which is the result of the balance between injection and radiative losses(we include the effects of the anisotropy on the synchrotron cooling rate). Considering theBL Lac object Mkn 421 as representative of the entire class, we show that in both cases theemission can be satisfactorily reproduced under equipartition between the magnetic field andthe relativistic electrons. This is in better agreement with the idea that jets are launched asPoynting dominated flows with respect to the standard isotropic scenario, which requires botha low magnetization and a low radiative efficiency to reproduce the observed emission. Thehard spectrum predicted for the inverse Compton continuum at TeV energies could be used asa potential test of the anisotropic model.
Key words:
BL Lac objects: general – radiation mechanisms: non-thermal – γ –rays: galaxies– galaxies: jets Blazars are Active Galactic Nuclei with a relativistic jet pointedtowards the Earth (e.g. Urry & Padovani 1995). Since the radiationfrom the jet is strongly beamed (and often completely outshinesthe other components, such as the accretion flow), blazars representideal natural laboratories to test our theoretical understanding of thephysical processes acting in relativistic jets.According to the standard view (e.g. Blandford & Znajek1977), extragalactic jets are powered by the rotational energy ofa supermassive black hole, which is transferred to the outflowingplasma via electromagnetic stresses. These jets start as highly mag-netized outflows accelerating through the conversion of the Poynt-ing flux into the kinetic flux under differential collimation. Thisscheme naturally predicts that the plasma remains highly magne-tized, or at most reaches equipartition between the magnetic andthe kinetic energy densities, until some dissipative process occurs(e.g. Komissarov et al. 2009; Lyubarsky 2010).The development of MHD instabilities may trigger the dissi-pation of the magnetic energy into the particle kinetic energy at the (cid:63)
E–mail: [email protected] emission sites, possibly through relativistic magnetic reconnection(e.g. Bromberg & Tchekhovskoy 2016; Barniol Duran et al. 2017).Kinetic simulations have shown that relativistic magnetic recon-nection is able to accelerate a population of non-thermal particles,whose energy is distributed according to a power law (e.g. Sironi& Spitkovsky 2014; Guo et al. 2014; Werner & Uzdensky 2017;Petropoulou et al. 2019). Interestingly, the power law index foundby these studies is not universal, being it dependent on the magne-tization of the plasma, on the strength of the guide field, and on thecomposition of the plasma, which one may expect to be different indifferent objects.The Spectral Energy Distribution (SED) of blazars is charac-terised by two broad non-thermal components, the first one peakingat IR-optical-UV frequencies, and the second one peaking in the γ rays. The SED follows a sequence (e.g. Fossati et al. 1998; Ghis-ellini et al. 2017): the brightest objects, which usually show strongemission lines and are therefore classified as Flat Spectrum RadioQuasars (FSRQ), peak at lower frequencies, while the less power-ful objects, which usually show weak or no emission lines and aretherefore classified as BL Lacs, peak at higher frequencies. The firstbump of the SED is due to the synchrotron radiation from a popu-lation of non-thermal electrons, while the second bump is usually c (cid:13) a r X i v : . [ a s t r o - ph . H E ] N ov Tavecchio & Sobacchi attributed to the Comptonization of either an external photon field(in the case of FSRQ), or of the synchrotron photons themselves(in the case of BL Lacs) (e.g. Sikora et al. 1994, 2009; Ghiselliniet al. 1998, 2010; Tavecchio et al. 1998).The classical one-zone Synchrotron-Self-Compton (SSC)model used to reproduce the SED of the BL Lacs challengesour theoretical understanding of relativistic jets in several aspects(Tavecchio & Ghisellini 2016; see also Inoue & Tanaka 2016; Nale-wajko & Gupta 2017; Costamante et al. 2018):(i) The magnetic energy density is much smaller than the kineticenergy density of the electrons. This is in contrast with the theoret-ical expectation that the emission sites are highly magnetized.(ii) The average radiative efficiency of the electrons (defined asthe ratio between the dynamical timescale and the radiative coolingtime) is low. However, fitting the SED requires the electrons belowa characteristic Lorentz factor γ b to be distributed according to auniversal power law N ( γ ) ∝ γ − (e.g. Tavecchio et al. 2010), whichis exactly what one would expect in the fast cooling regime. Sincereconnection is unlikely to produce such a universal energy distri-bution, the origin of the distribution remains unclear if the radiativelosses are inefficient.These conclusions stem from the fact that in the classical one-zone SSC model the maxima of the two bumps of the SED are pro-duced by the same electrons, whose energy can be robustly derived.A possible way-out to these conclusions is proposed by Sobacchi &Lyubarsky (2019), who have argued that (i) the non-thermal elec-trons may be accelerated in the direction of the magnetic field; (ii)the gyro-resonant pitch angle scattering makes the distribution ofthe electrons isotropic below a critical γ iso (of the order of the pro-ton to electron mass ratio), while the pitch angle of the most en-ergetic electrons remains small. If γ iso is smaller than the break γ b of the electron energy distribution, the two bumps of the SED areproduced by different populations of electrons, the synchrotron (in-verse Compton) bump being produced by the electrons at γ iso ( γ b ).As discussed by Sobacchi & Lyubarsky (2019), in such a scenariothe magnetization of the emission sites and the radiative efficiencyof the non-thermal electrons required to reproduce the SED may besignificantly higher with respect those inferred using the classicalone-zone SSC model .In this paper we develop a quantitative model for the SED ofblazars based on the anisotropy proposed by Sobacchi & Lyubarsky(2019). We apply the model to reproduce the well sampled SED ofthe prototypical BL Lac object Mkn 421 obtained during a low stateas reported in Abdo et al. (2011). The same SED has been modelledwith the standard one-zone SSC model by Tavecchio & Ghisellini(2016) and Tavecchio et al. (2019), who found a low magnetiza-tion and a low radiative efficiency in the emission sites. We showthat both these problems may be solved assuming an anisotropicelectron population.The paper is organized as follows. In Section 2 we develop a If the cooling is fast, the number of electrons at the energy γ , namely γ N ( γ ) where N ( γ ) is the number of electrons per unit energy (or Lorentzfactor), is proportional to the cooling time at that energy, t cool ( γ ) . Since t cool ∝ γ − for both the synchrotron and the inverse Compton processes,one eventually finds that N ( γ ) ∝ γ − . An alternative possibility is that the emission comes from one single re-connecting current sheet, which then fragments into plasmoids of differentsizes. Christie et al. (2019) suggested that the IC emission is boosted as aneffect of the relative velocity of different plasmoids, which alleviates thetension on the magnetisation required to reproduce the SED phenomenological model of blazar SED assuming an anisotropicelectron population. In Section 3 we include a self-consistent treat-ment of the particle cooling into the model. Finally, in Section4 we discuss our results and we conclude. Throughout the pa-per, the following cosmological parameters are assumed: H =
70 km s − Mpc − , Ω M = . Ω Λ = . We implement a one-zone SSC model, including the anisotropy ofthe electron momentum distribution, modifying the model fully de-scribed in Maraschi & Tavecchio (2003). We assume that the num-ber of electron per unit energy follows a (widely adopted) smoothedbroken power law: N ( γ ) = K γ − n (cid:18) + γγ b (cid:19) n − n if γ min < γ < γ max . (1)The distribution extends from γ min to γ max , and has a break at γ b .The normalization factor K , which has units of cm − , controls thetotal number density of the electrons.As a phenomenological description of the anisotropy proposedby Sobacchi & Lyubarsky (2019), we assume that the diffusion ofthe particle momentum results in a maximum pitch angle that de-pends on the energy (or Lorentz factor) of the electrons as: θ max ( γ ) = π × (cid:40) γ min < γ < γ iso ( γ / γ iso ) − η if γ iso < γ < γ max (2)with η >
0. The velocity of most energetic electrons, with γ (cid:29) γ iso ,is therefore contained within a cone of semi-aperture θ max (cid:28) π / γ < γ iso havean isotropic distribution, namely θ max = π / locally (i.e. with respect to the local mag-netic field) anisotropic, but globally (i.e. on the scale of the emis-sion region) isotropic. Hence, the synchrotron emission (which de-pends on the electron’s pitch angle with respect to the local mag-netic field) is affected by the local anisotropy of the electron pop-ulation, while for the IC emission (which is only sensitive to theproperties of the electron distribution and the soft radiation field onthe scale of the emission region) we can apply the same treatmentof the isotropic model used by Maraschi & Tavecchio (2003). Alsonote that, as a consequence of the global isotropy of the electrondistribution, the non-thermal emission (i.e. synchrotron and IC) isisotropic in the frame of the emission region.It is well known (e.g. Rybicki & Lightman 1979) that if theelectron distribution is isotropic, θ max = π /
2, and follows a powerlaw, N ( γ ) ∝ γ − n , the synchrotron emissivity displays a power lawshape j ( ν ) ∝ ν − α , with α = ( n − ) /
2. As shown in Appendix A,for an anisotropic distribution following Eq. (2) the spectrum is de-scribed by a power with index α = ( n − + η ) / ( − η ) for frequen-cies above ν c ( γ iso ) , where ν c ( γ iso ) is the typical frequency of thephotons emitted by the electrons at γ iso . Therefore, if the electrondistribution follows an unbroken power law N ( γ ) ∝ γ − n , the syn-chrotron spectrum would show a break at the frequency ν ∼ ν c ( γ iso ) with ∆α = η ( n + ) / ( − η ) . Note that ∆α diverges when η → ν c becomes a decreasing function of γ c (cid:13) , 000–000 nisotropic electron populations in BL Lacs Figure 1.
Spectral energy distribution of Mkn 421 (black filled circles)obtained during the campaign reported in Abdo et al. (2011) reproducedwith the anisotropic electron distribution model assuming a phenomenolog-ical electron energy distribution (orange solid: model 1; blue long-dashed:model 2). The parameters are listed in Table 1. For comparison, the graydashed line shows the standard isotropic model reported in Tavecchio &Ghisellini (2016). when η > ν c ( γ iso ) when η >
2. In thefollowing we focus on the case η < j ( ν ) may becalculated integrating the standard pitch angle-dependent emittedpower P ( ν , γ , θ ) (e.g. Rybicki & Lightman 1979) over the (energy-dependent) range of pitch angles [ , θ max ( γ )] . We obtain: j ( ν ) = √ e m e c B (cid:90) γ max γ min N ( γ ) (cid:90) θ max ( γ ) sin θ F ( x ) d θ d γ , (3)where the argument of the F function (for its explicit form seeRybicki & Lightman 1979) is x = ν / ν c ( θ ) , with: ν c ( γ , θ ) = γ e π mc B sin θ . (4)In Fig. 1 we report two possible models consistent with theobserved SED of the BL Lac object Mkn 421. We assume thatthe phenomenological electron energy and pitch angle distributionsare described by Eqs. (1) and (2) respectively (the model parame-ters are listed in Table 1). As discussed by Sobacchi & Lyubarsky(2019), when γ iso < γ b the synchrotron peak of the SED is pro-duced by the electrons at γ iso due to the suppression of the syn-chrotron emissivity at small pitch angles, while the IC peak is stillproduced in the Klein-Nishina regime by the electrons at γ b . Weobtain a satisfactory fit using γ iso ∼ , consistent with the valuesuggested by Sobacchi & Lyubarsky (2019) for a sample of BLLacs. The magnetic field is around 1 G, much larger than whatrequired by the standard isotropic one-zone models, that providesvalues around 0 . Figure 2.
Zoom of Fig. 1 in the VHE range. The anisotropic scenario (bluelong-dashed and orange solid lines) predicts SSC spectra harder than thestandard model (gray dashed line). of magnitude), together with the concomitant reduction of the elec-tron density, allows to reach equipartition keeping all other param-eters within the standard range.In the standard isotropic one-zone model the observed slope ofthe optical-UV and X-ray continua constrain the index of the un-derlying electron energy distribution in the ranges n (cid:39) − . n (cid:39) − . n (cid:39)
3. This, togetherwith the larger radiative losses ensured by the higher magneticand radiative energy densities, allows one to adopt a scenario inwhich particle are continuously injected into the emission regionwith slope n inj ≈ γ inj ≈ γ b . Ra-diative losses increase the slope of the equilibrium distribution by ∆ n = γ > γ inj , and produce a γ − tail at γ < γ inj ,which results in a suitable energy distribution to reproduce the ob-served SED. While this scenario is attractive, it is not fully consis-tent, since it does not take into account that also the synchrotroncooling rate of the high-energy electrons is strongly reduced by thesmall pitch angles. A self-consistent picture is discussed in the nextsection.A last point concerns the IC spectrum. Since in the anisotropiccase the underlying electron distribution required to produce thehigh-energy tail of the synchrotron peak is harder than in theisotropic case, one expects a harder SSC spectrum at high-energy(i.e. after the SSC peak). Indeed, harder spectra are predicted in theVHE by the anisotropic model (Fig. 2). Quite interestingly, in thiscase, fluxes differences by a factor 2 − c (cid:13)000
3. This, togetherwith the larger radiative losses ensured by the higher magneticand radiative energy densities, allows one to adopt a scenario inwhich particle are continuously injected into the emission regionwith slope n inj ≈ γ inj ≈ γ b . Ra-diative losses increase the slope of the equilibrium distribution by ∆ n = γ > γ inj , and produce a γ − tail at γ < γ inj ,which results in a suitable energy distribution to reproduce the ob-served SED. While this scenario is attractive, it is not fully consis-tent, since it does not take into account that also the synchrotroncooling rate of the high-energy electrons is strongly reduced by thesmall pitch angles. A self-consistent picture is discussed in the nextsection.A last point concerns the IC spectrum. Since in the anisotropiccase the underlying electron distribution required to produce thehigh-energy tail of the synchrotron peak is harder than in theisotropic case, one expects a harder SSC spectrum at high-energy(i.e. after the SSC peak). Indeed, harder spectra are predicted in theVHE by the anisotropic model (Fig. 2). Quite interestingly, in thiscase, fluxes differences by a factor 2 − c (cid:13)000 , 000–000 Tavecchio & Sobacchi
Model γ min γ b γ max γ iso n n η B K R δ U B / U e [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] × × . × . . . .
25 7 × . .
352 500 7 × . × . .
25 1 . × .
65 15 1 . . × × − . . − .
06 3 . × . . Table 1.
Input model parameters for the models of Mkn 421 in Fig. 1 and derived magnetic to electron energy density ratio. For comparison we also reportthe parameters derived by Tavecchio & Ghisellini (2016) with the standard isotropic model (their model 2). [1]: model. [2], [3], and [4]: minimum, break andmaximum electron Lorentz factor. [5]: maximum Lorentz factor of electrons with isotropic pitch angle distribution. [6] and [7]: slope of the electron energydistribution below and above γ b . [8]: slope of the electron pitch angle distribution above γ iso . [9]: magnetic field in units of G. [10]: normalization of theelectron distribution in units of cm − . [11]: radius of the emission zone in units of 10 cm. [12]: Doppler factor. [13]: ratio between the magnetic and therelativistic electron energy densities. Telescope Array (we have checked that internal absorption is neg-ligible at these energies; see also Tavecchio et al. 2019).
From the standard synchrotron theory one derives that the coolingrate of electrons with pith angle θ is:˙ γ s = σ T m e c γ β sin θ U B . (5)For an anisotropic distribution one can calculate the average: (cid:104) ˙ γ s (cid:105) = σ T m e c γ β U B (cid:82) θ max sin θ d θ (cid:82) θ max sin θ d θ = A ( γ ) σ T m e c γ β U B , (6)where the function A ( γ ) is: A ( γ ) = − µ − µ µ ≡ cos θ max ( γ ) . The cooling timescale (taking into accountthe synchrotron losses only) can be defined as: t cool , s ( γ ) ≡ γ (cid:104) ˙ γ s (cid:105) = m e c A ( γ ) σ T γβ U B . (8)Note that when γ < γ iso (and therefore θ max = π /
2) the synchrotroncooling time is the same as in the standard isotropic case, whilewhen γ (cid:29) γ iso (and therefore θ max (cid:28) π /
2) the synchrotron coolingtime is a factor 1 / A ( γ ) ∼ ( / ) × ( / θ ) longer. If the pitch an-gle distribution is described by Eq. (2), one finds that t cool , s ∝ γ η − when γ (cid:29) γ iso .In Fig. 3 we show the synchrotron cooling time as a func-tion of the Lorentz factor for the isotropic case (cyan) and for theanisotropic case with η = . η = γ iso = .Clearly, the synchrotron cooling of the electrons is strongly sup-pressed above γ iso .An idea of the impact of the anisotropy on the emitted spec-trum can be grasped considering the simple case of the steady-statedistribution resulting from the balance between continuous injec-tion and radiative losses. In this case, the solution of the continuityequation is given by (e.g. Ghisellini 1989): N ( γ ) = γ tot (cid:90) ∞γ Q ( γ (cid:48) ) d γ (cid:48) (9)where Q ( γ ) is the injection term and ˙ γ tot is the total energy loss rate.In the following we assume an injected power law Q ( γ ) ∝ γ − n inj for γ inj < γ < γ max . For the SSC emission we have ˙ γ tot = ˙ γ s + ˙ γ IC .Note that ˙ γ tot ∝ γ when either (i) γ < γ iso , in which case bothboth ˙ γ s and ˙ γ IC are proportional to γ , or (ii) γ > γ iso , but the cooling Figure 3.
Synchrotron cooling time as a function of the Lorentz factor forelectrons with isotropic (long-dashed cyan) and anisotropic ( η = .
5, solidblue; η =
1, dashed red) momentum distribution. We assume γ iso = and B = is dominated by the inverse Compton (˙ γ IC (cid:29) ˙ γ s ) and the Klein-Nishina effects are not too strong. If ˙ γ tot ∝ γ at all γ , the solutionof Eq. (9) is N ( γ ) ∝ γ − at γ < γ inj and N ( γ ) ∝ γ − ( n inj + ) at γ > γ inj .In the most interesting case γ iso < γ inj , the synchrotron emissivity ischaracterised by j ( ν ) ∝ ν − α , where (i) α = / ν < ν c ( γ iso ) ; (ii) α = ( + η ) / ( − η ) if ν c ( γ iso ) < ν < ν c ( γ inj ) ; α = ( n inj + η ) / ( − η ) if ν > ν c ( γ inj ) . Therefore, the peak of the synchrotron luminosityis produced by the electrons at γ ∼ γ iso , as required by the model ofSobacchi & Lyubarsky (2019).If instead ˙ γ tot ∝ γ − η for γ > γ iso , which happens when thecooling is dominated by the synchrotron (˙ γ s (cid:29) ˙ γ IC ), the solution ofEq. (9) is N ( γ ) ∝ γ − at γ < γ iso , N ( γ ) ∝ γ − + η at γ iso < γ < γ inj ,and N ( γ ) ∝ γ − n inj − + η at γ > γ inj . In this case, the synchrotronemissivity is characterised by j ( ν ) ∝ ν − α , where (i) α = / ν < ν c ( γ iso ) ; (ii) α = ( − η ) / ( − η ) if ν c ( γ iso ) < ν < ν c ( γ inj ) ; α =( n inj − η ) / ( − η ) if ν > ν c ( γ inj ) . It is simple to realise that in thiscase the peak of the synchrotron luminosity would be produced bythe electrons at γ ∼ γ inj , and the mechanism invoked by Sobacchi & c (cid:13) , 000–000 nisotropic electron populations in BL Lacs Figure 4.
Upper panel: electron energy distribution at different times as-suming constant injection Q ( γ ) ∝ γ − . with γ inj =
10 and γ max = andthe pitch angle distribution Eq. 2 with η = .
5. The dashed line show the dis-tribution in the standard isotropic case. The calculation assumes γ iso = and B = t (cid:48) = × s (red), t (cid:48) = × s (green), t (cid:48) = s (blue) and t (cid:48) = × s (cyan). The dotted line show the distribu-tions taking into account inverse Compton cooling off photons with a totalenergy density of U rad = U B and with the spectral shape of the synchroroncomponent in Fig.1. Lower panel: as for the upper panel but with injectionbetween γ inj = × and γ max = × . Lyubarsky (2019) would not operate irrespective of the pitch angledistribution.To explore the general case of a time-dependent electron en-ergy distribution it is convenient to resort to numerical calculations.We implemented the method described in Chiaberge & Ghisellini(1999), who adopted the numerical scheme of Chang & Cooper(1970) to solve the time-dependent continuity equation includinga source term and radiative and adiabatic losses (for simplicity thelatter are not considered here). We have used Eq. (6) to describe thesynchrotron losses for the anisotropic distribution.As an example of the behavior of the electron distribution,in Fig. 4 we report the evolution at different times of the electronenergy distribution assuming a constant injection Q ( γ ) ∝ γ − n inj with n inj = .
5, and with γ iso = and B = γ inj (cid:28) γ iso , while in the lower panel we reportthe evolution for a case with γ inj > γ iso . The dashed lines show thestandard evolution for the isotropic case.For the case γ inj (cid:28) γ iso at different times t (cid:48) the distributionsfollow the injected spectra up to a Lorentz factor γ c for which t cool ( γ c ) = t (cid:48) . Above this energy the distributions relax and followthe cooled distribution with slope n inj +
1. For anisotropic distribu-tions, the inefficient cooling caused by the small pitch angles above γ iso determines the presence of a hard tail, asymptotically follow- Figure 5.
SED of Mkn 421 reproduced with the anisotropic model assuminga self-consistent electron distribution. See text for details. ing the uncooled slope n inj . For times at which γ c (cid:28) γ iso , above γ c the distribution follows the standard one and then rapidly raisesto reach the uncooled distribution at γ (cid:29) γ iso . Of course, when thesynchrotron cooling becomes very small, the IC losses can be rel-evant, even if scatterings occur deeply in the KN regime. This isshown by the dotted curves which include both synchrotron and IClosses (note that the IC scattering is not efficient to produce pitchangle diffusion, see the discussion in Appendix B). The IC coolingis calculated assuming the photon energy density associated to thesynchrotron component in Fig. 1.For the case γ inj > γ iso (lower panel), below γ inj the coolingproduces a tail extending to the cooling Lorentz factor defined by γ c for which t cool ( γ c ) = t (cid:48) . When the IC losses are considered, theslope of this cooling tail follows the standard γ − slope. However,if IC losses are negligible (solid lines), the suppression of the syn-chrotron cooling determines a slope in the interval γ iso < γ < γ inj harder than − t (cid:48) inj = R / c . The injection term isdescribed by a power law between γ inj = × and γ max = × with slope n inj = .
75. Below γ inj a cooled tail ∝ γ − develops downto a Lorentz factor for which the cooling time equal the injectiontime. A good agreement with the data can be found if we furtherassume B = . δ = . R = × cm, γ iso = . × , η = .
5. The result is shown in Fig. 5. It should be remarked that inthis case the IC losses are relevant for energies around γ iso and thisallows the formation of the standard γ − tail. In this situation thepitch angle suppression of the emission can operate and the SEDcan be reproduced in equipartition condition (we derive U B / U e = . c (cid:13)000
5. The result is shown in Fig. 5. It should be remarked that inthis case the IC losses are relevant for energies around γ iso and thisallows the formation of the standard γ − tail. In this situation thepitch angle suppression of the emission can operate and the SEDcan be reproduced in equipartition condition (we derive U B / U e = . c (cid:13)000 , 000–000 Tavecchio & Sobacchi
We have developed a model for the SED of the high-energy emit-ting BL Lacs, implementing the idea proposed by Sobacchi &Lyubarsky (2019) that the momentum distribution of the most en-ergetic electrons is anisotropic. The reduced pitch angle of the elec-trons above a certain Lorentz factor γ iso implies a severe suppres-sion of their synchrotron emission. If γ iso < γ b , where γ b is theLorentz factor of the electrons emitting at the synchrotron and theIC peaks of the SED in the standard SSC model, the anisotropy hasan important effect on the modelling of the observed emission. Wehave shown that the representative SED of the BL Lac object Mkn421 can be reproduced under equipartition between the magneticfield and the relativistic electrons, contrary to what happens withthe commonly adopted isotropic scenario. We have also exploreda self-consistent scheme in which the electrons follow an energydistribution resulting from the balance between injection and cool-ing (synchrotron and IC). In this case we have remarked that themechanism proposed by Sobacchi & Lyubarsky (2019) can operateonly in the presence of an effective IC cooling of the electrons with γ (cid:38) γ iso .Apart from the stronger magnetic field, the values of the pa-rameters derived for the jet of Mkn 421 in the anisotropic scenarioare similar the usual ones (e.g Tavecchio et al. 2010). It is how-ever worth noting that the Doppler factor can be kept to somewhatsmaller values than those required by the standard model (see e.g.Tavecchio & Ghisellini 2016). This implies a significantly largerradiation energy density in the emission sites, with consequencesfor the transparency of the source at TeV energies and a poten-tial impact on the multimessenger role of highly-peaked BL Lacs(Tavecchio et al. 2019).From the physical point of view, we remark that the scenariosketched in this paper is not fully self-consistent. In particular, ourchoice for the pitch angle distribution is purely phenomenological.The detailed description of the anisotropy as a function of the en-ergy has a strong impact on the derived spectra and it would there-fore be important to have a physically-motivated model for it. Themechanism proposed by Sobacchi & Lyubarsky (2019) requires (i)the electrons to be accelerated in the direction of the magnetic field;(ii) a significant proton component to be present, thus providingthe isotropization of the least energetic electrons. Regarding theitem (i), promising candidates are reconnection with a strong guidefield and/or dissipation of relativistic turbulence. Interestingly, re-cent kinetic simulations of decaying turbulence have suggested thatthe particles at γ ∼ γ inj are primarily accelerated by an electric fieldaligned with the local magnetic field (e.g. Comisso & Sironi 2018).Unfortunately, these simulations did not model radiative coolingand were limited to the case of a pair plasma. Regarding the item(ii), there are compelling arguments supporting the presence of a(cold) proton component in FSRQ (e.g. Sikora & Madejski 2000;Ghisellini & Tavecchio 2010; Ghisellini et al. 2014), but for BLLacs the situation is less clear. The potential association of the BLLac object TXS 0506+056 with high-energy neutrinos detected byIceCube (IceCube Collaboration et al. 2018) (thought to be pro-duced through photomeson interactions involving relativistic pro-tons and radiation) supports the existence of an important barioniccomponent also in BL Lac jets. It is worth remarking that the radiation energy density is proportional to δ − , where δ is the Doppler factor. ACKNOWLEDGMENTS
We thank the referee for useful comments. We thank G. Ghiselliniand Y. Lyubarsky for useful discussions. FT acknowledges con-tribution from the grant INAF CTA–SKA “Probing particle accel-eration and γ -ray propagation with CTA and its precursors” andthe INAF Main Stream project “High-energy extragalactic astro-physics: toward the Cherenkov Telescope Array”. ES acknowl-edges contribution from the Israeli Science Foundation (grant719/14) and from the German Israeli Foundation for Scientific Re-search and Development (grant I-1362-303.7/2016). REFERENCES
Abdo A. A. et al., 2011, ApJ, 736, 131Barniol Duran R., Tchekhovskoy A., Giannios D., 2017, MNRAS,469, 4957Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433Bromberg O., Tchekhovskoy A., 2016, MNRAS, 456, 1739Chang J. S., Cooper G., 1970, Journal of Computational Physics,6, 1Chiaberge M., Ghisellini G., 1999, MNRAS, 306, 551Christie I. M., Petropoulou M., Sironi L., Giannios D., 2019,arXiv e-prints, arXiv:1908.02764Comisso L., Sironi L., 2018, Physical Review Letters, 121,255101Costamante L., Bonnoli G., Tavecchio F., Ghisellini G., Taglia-ferri G., Khangulyan D., 2018, MNRAS, 477, 4257Fossati G., Maraschi L., Celotti A., Comastri A., Ghisellini G.,1998, MNRAS, 299, 433Ghisellini G., 1989, MNRAS, 236, 341Ghisellini G., Celotti A., Fossati G., Maraschi L., Comastri A.,1998, MNRAS, 301, 451Ghisellini G., Righi C., Costamante L., Tavecchio F., 2017, MN-RAS, 469, 255Ghisellini G., Tavecchio F., 2010, MNRAS, 409, L79Ghisellini G., Tavecchio F., Foschini L., Ghirlanda G., MaraschiL., Celotti A., 2010, MNRAS, 402, 497Ghisellini G., Tavecchio F., Maraschi L., Celotti A., Sbarrato T.,2014, Nature, 515, 376Guo F., Li H., Daughton W., Liu Y.-H., 2014, Physical ReviewLetters, 113, 155005IceCube Collaboration et al., 2018, Science, 361, eaat1378Inoue Y., Tanaka Y. T., 2016, ApJ, 828, 13Komissarov S. S., Vlahakis N., Königl A., Barkov M. V., 2009,MNRAS, 394, 1182Lyubarsky Y. E., 2010, MNRAS, 402, 353Maraschi L., Tavecchio F., 2003, ApJ, 593, 667Nalewajko K., Gupta M., 2017, A&A, 606, A44Petropoulou M., Sironi L., Spitkovsky A., Giannios D., 2019,ApJ, 880, 37Rybicki G. B., Lightman A. P., 1979, Radiative processes in as-trophysics. Wiley, New YorkSikora M., Begelman M. C., Rees M. J., 1994, ApJ, 421, 153Sikora M., Madejski G. M., 2000, ApJ, 534, 109Sikora M., Stawarz Ł., Moderski R., Nalewajko K., MadejskiG. M., 2009, ApJ, 704, 38Sironi L., Spitkovsky A., 2014, ApJ, 783, L21Sobacchi E., Lyubarsky Y. E., 2019, MNRAS, 484, 1192Tavecchio F., Ghisellini G., 2016, MNRAS, 456, 2374Tavecchio F., Ghisellini G., Ghirlanda G., Foschini L., MaraschiL., 2010, MNRAS, 401, 1570 c (cid:13) , 000–000 nisotropic electron populations in BL Lacs Tavecchio F., Maraschi L., Ghisellini G., 1998, ApJ, 509, 608Tavecchio F., Oikonomou F., Righi C., 2019, MNRAS, 1890Urry C. M., Padovani P., 1995, PASP, 107, 803Werner G. R., Uzdensky D. A., 2017, ApJ, 843, L27
APPENDIX A: SYNCHROTRON SPECTRUM FROMANISOTROPIC ELECTRONS
We calculate the synchrotron emissivity for frequencies above thatemitted by the electrons at γ iso . From Eq. (4) one may calculate thetypical synchrotron frequency emitted by the electrons at γ as ν ∼ ν c ( θ max ) ∼ γ e π mc B θ max ∝ γ − η , (A1)where we have used our assumption that θ max ∝ γ − η , and the factthat θ max (cid:28) π / γ (cid:29) γ iso (see Eq. 2). Note that ν c becomesa decreasing function of γ when η > ν j ( ν ) emitted at the fre-quency ν on the Lorentz factor γ can be estimated as ν j ( ν ) ∝ γ N ( γ ) B sin θ max ∝ γ − n − η , (A2)where we have assumed that the electron distribution follows apower law N ( γ ) ∝ γ − n .Inverting Eq. (A1) we find that γ ∝ ν / ( − η ) . Substituting intoEq. (A2), we finally get ν j ( ν ) ∝ ν ( − n − η ) / ( − η ) , (A3)or equivalently j ( ν ) ∝ ν ( − n − η ) / ( − η ) . (A4)Note that if the pitch angle does not depend on the energy ( η = j ( ν ) ∝ ν ( − n ) / . APPENDIX B: PITCH ANGLE DIFFUSION DUE TOINVERSE COMPTON SCATTERING
Let us consider a particle of Lorentz factor γ (cid:29) N photons of energy ε per unit time. After eachscattering, a photon of energy γ ε is emitted within an angle of1 / γ with respect of the direction of the particle motion. Hence, theparticle looses an amount of energy ∆ E ∼ γ ε , and the cooling time(due to IC scattering only) can be calculated as t cool , IC ∼ N mc γε . (B1)The particle pitch angle instead varies in a diffusive way. At eachscattering, the momentum transverse to the direction of motion thatis transferred to the particle is ∆ p ∼ ( / γ ) × ( γ ε / c ) ∼ γε / c . Hence,the pitch angle varies by ∆θ ∼ ∆ p / γ mc ∼ ε / mc . Over a coolingtime, the pitch angle diffuses by θ ∼ (cid:113) ˙ Nt cool , IC ( ∆θ ) ∼ (cid:114) εγ mc . (B2)Since the IC scattering is efficient only if ε (cid:46) mc / γ (otherwisethe scattering occurs in the Klein-Nishina regime, which stronglysuppresses the cross section), one finds that θ (cid:46) / γ . Hence, thepitch angle diffusion due to IC scattering may be neglected. c (cid:13)000