Correlations between γ-ray luminosity and magnetization of the jet as well as relativistic electron injection power:cases for Mrk 421, 3C 454.3 and 3C 279
aa r X i v : . [ a s t r o - ph . H E ] F e b MNRAS , 1– ?? (0000) Preprint 16 February 2021 Compiled using MNRAS L A TEX style file v3.0
Correlations between γ -ray luminosity and magnetization of thejet as well as relativistic electron injection power: cases for Mrk421, 3C 454.3 and 3C 279 Wen Hu, Dahai Yan, ⋆ and Qianglin Hu Department of Physics, Jinggangshan University, Jiangxi Province, Ji’an 343009, People’s Republic of China Key Laboratory for the Structure and Evolution of Celestial Objects, Yunnan Observatories, Chinese Academy of Sciences,Kunming 650011, People’s Republic of China
Accepted 2021 February 12; Received 2021 February 11; in original form 2020 November 19
ABSTRACT
By fitting high-quality and simultaneous multi-wavelength (MWL) spectral energydistributions (SEDs) at multiple epochs with a one-zone leptonic jet model, we studyjet properties of the three famous blazars Mrk 421, 3C 454.3 and 3C 279. In the jetmodel, the emitting electron energy distributions (EEDs) are calculated by solving thekinetic equation of electron injection, escape, adiabatic and radiative energy losses.To explore multi-dimensional parameter space systematically, we employ a Markovchain Monte Carlo (MCMC) fitting technique. The properties of emission regions wederived here are consistent with those in previous studies, e.g., the particle-dominatedand low-magnetization jet. The new finding is that there is a tight correlation between γ -ray luminosity and electron injection power and an anti-correlation between γ -rayluminosity and jet magnetization parameter. The results suggest that same energy-dissipative mechanism (like a shock) could be operating in the jets of different typesof blazars, and the origin of γ -ray flares is associated with the particle accelerationprocess. Key words: radiation mechanisms: non-thermal — galaxies: active — galaxies: indi-vidual: Mrk 421, 3C 454.3 and 3C 279 — galaxies: jets.
Flat spectrum radio quasars (FSRQs) and BL Lacertae ob-jects (BL Lacs) constitute a subset of active galactic nu-clei (AGNs), which are called blazars. A blazar is charac-terized by flux variability at all wavelengths, high polariza-tion at optical and radio frequencies (e.g., D’arcangelo et al.2009; Marscher et al. 2010; Peceur et al. 2020), apparentlysuperluminal jet components (e.g., Jorstad et al. 2004, 2005;Homan et al. 2009), and non-thermal emission from a rel-ativistic jet pointed close to the observer’s line of sight(Urry & Padovani 1995)Compared with BL Lacs with featureless optical spec-tra, FSRQs display prominent emission lines with theequivalent width ≥ ⋆ E-mail: [email protected] sion of high-energy electrons in the jet, is generally locatedat infrared/optical bands. The second peak, which couldbe generated by inverse Compton (IC) scattering off lowenergy photons by high-energy electrons, is generally lo-cated at γ -ray energies. The low photon fields for the ICscattering can include the synchrotron emission from thehigh-energy electrons themselves (Synchrotron Self Comp-ton: SSC; Bloom & Marscher 1996; Finke et al. 2008) andexternal photon fields surrounding the jet (external Comp-ton scattering: EC). Depending on location of the emissionregion, the external photon field responsible for the EC emis-sion could be either from dusty torus (DT; Blazejowski et al.2000; Dermer et al. 2014; Yan et al. 2015; Hu et al. 2017;Wu et al. 2018), and/or from broad-line region (BLR;Sikora et al. 1994; Yan et al. 2012a; B¨ottcher et al. 2013;Hu et al. 2015).Electron acceleration in blazar jet is still an open ques-tion. In general, it can be achieved by the energy dissipatedin shocks and/or magnetic reconnections. If the jet is domi-nated by the kinetic energy flux at the dissipation distance,then shocks are natural candidates for powering the jet © Hu, Yan, and Hu emission and accelerating electrons to ultra-relativistic en-ergy (Marscher & Gear 1985; Kirk et al. 2000; Spada et al.2001; Sironi & Spitkovsky 2009; Summerlin & Baring 2012).The shock scenario is supported by multiple observation of γ − ray outbursts in coincidence with the emergence of ajet perturbation in or close to radio core of blazars (e.g.,Marscher et al. 2008; Jorstad et al. 2013; Abeysekara et al.2018). While, if the jet remains Poynting flux dominatedat the energy dissipation distance, shocks are generallyexpected to be weak, and magnetic reconnection is amore plausible candidate for acceleration of electrons (e.g.,Cerutti et al. 2012; Sironi et al. 2013; Sironi & Spitkovsky2014; Guo et al. 2014, 2015). The ratio between the Poynt-ing flux and kinetic energy flux, namely magnetization pa-rameter σ m ≡ B ′ / πρ ′ c , is a crucial quantity to discrimi-nate the two scenarios .Recently, particle-in-cell (PIC) simulations of relativis-tic magnetic reconnection have demonstrated that electronscan be efficiently accelerated to form a non-thermal distribu-tion for σ m & = 70 km s − Mpc − , Ω M = 0 .
3, and Ω Λ = 0 . We adopt a one-zone homogeneous leptonic jet model. It isassumed that emissions are produced in a spherical blob ofradius R ′ which moves relativistically with bulk Lorentz fac-tor Γ in the observer’s frame at an angle θ obs ∼ Γ − withrespect to our line of sight. The emission region is filledwith a uniform and tangled magnetic field of strength B ′ .The radius of the emission region is estimated from the mini-mum variability timescale t var , through R ′ = cδ D t var / (1+ z ),where z is the redshift of the source and Doppler factor δ D is Here, B ′ and ρ ′ are the magnetic field strength and the rest-mass density in the rest frame of the jet, respectively. approximated as Γ. Throughout this paper, all primed quan-tities refer to the comoving frame of the emission region andunprimed quantities denote the observer’s frame.It is assumed that a population of ultra-relativistic non-thermal electrons is continuously injected into the blob witha rate of Q ′ e ( γ ′ ) in units of s − . The injected electrons loseenergy through radiation and adiabatic processes. The ki-netic equation that governs the evolution of electrons canbe described by the time-dependent Fokker-Planck equation(Coppi et al. 1990; Chiaberge & Ghisellini 1999).With respect to our previous work, we further refine themodel, including a physically realistic, stratified BLR modelproposed by Finke (2016). A modification is including adi-abatic expansion, which may be important in modeling theSEDs of blazars (e.g., Lewis et al. 2016, 2018). The radia-tion coolings from both EC-BLR and EC-DT processes areconsidered to calculate the radiating electron distribution.We consider a single power-law (PL) of electron injec-tion, which is given by γ ′ Q ′ e ( γ ′ ) = Q γ ′ − n , γ ′ min ≤ γ ′ ≤ γ ′ max , (1)with Q ′ = L ′ inj m e c − nγ ′ max2 − n − γ ′ min2 − n ; n = 2 L ′ inj m e c ln ( γ ′ max /γ ′ min ) ; n = 2 , (2)where γ ′ min and γ ′ max are respectively the low- and high-energy cutoffs, L ′ inj is injection power of electrons in unitsof ergs/s, n is the power-law index , m e is the rest mass ofelectron, and c is the speed of light. The kinetic equation governing the evolution of the electronenergy distribution, N ′ e ( γ ′ ), is given by ∂N ′ e ( γ ′ ) ∂t ′ = − ∂∂γ ′ (cid:2) ˙ γ ′ N ′ e ( γ ′ ) (cid:3) − N ′ e ( γ ′ ) t ′ esc + Q ′ e ( γ ′ ) , (3)where t ′ esc is the escape timescale of electrons, and ˙ γ ′ is theenergy-loss rate of the electrons.In a blazar jet, the electrons lose energy through syn-chrotron and IC scattering on internal and external photonfields. The external radiation fields surrounding the jet in-clude BLR and DT photon fields. Therefore, the electronradiative cooling rate can be written as ˙ γ ′ rad = ˙ γ ′ syn + ˙ γ ′ ssc +˙ γ ′ BLR + ˙ γ ′ DT , where ˙ γ ′ syn , ˙ γ ′ ssc , ˙ γ ′ BLR and ˙ γ ′ DT are the cool-ing rate due to the synchrotron, SSC, EC-BLR and EC-DTradiation, respectively.The synchrotron energy-loss rate is given by − ˙ γ ′ syn = 4 σ T m e c U ′ B γ ′ , (4)where U ′ B = B ′ / π is the magnetic field energy density.The SSC energy-loss rate is given by − ˙ γ ′ ssc = 4 σ T m e c γ ′ Z ∞ dǫ ′ u ′ syn ( ǫ ′ ) f kn ( ǫ ′ , γ ′ ) , (5)where u ′ syn ( ǫ ′ ) is the spectral energy density of the syn-chrotron radiation, and f kn ( ǫ ′ , γ ′ ) = 916 Z γ ′ γ ′ l dγ ′′ F c ( x, q ) γ ′ − γ ′′ ǫ ′ γ ′ , (6) MNRAS , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj where the lower limit for the integration is γ ′ l ≃ γ ′ + ǫ ′ − γ ′ ǫ ′ γ ′ ǫ ′ , and the kernal function is F c ( x, q ) = h q ln q + q + 1 − q +( xq ) xq ) (1 − q ) i × H [ q ; 14 γ ′ , , (7)(Jones 1968; Blumenthal & Gould 1970). Here, H ( x ; a, b ) isthe Heaviside function defined as H = 1 if a ≤ x ≤ b andH = 0 otherwise. In the equation, x = 4 ǫ ′ γ ′ , q = ǫ ′ γ /γ ′ x (1 − ǫ ′ γ /γ ′ ) ,and the scattered photon energy is ǫ ′ γ = γ ′ + ǫ ′ − γ ′′ .The EC-DT energy-loss rate is given by − ˙ γ ′ ec = 4 σ T m e c γ Z ∞ dǫu ext ( ǫ ) f kn ( ǫ, γ ) , (8)where the quantities γ = δ D γ ′ and ǫ refer to the station-ary frame with respect to the black hole, and u ext ( ǫ ) is thespectral energy density of external photon field. Here, theDT radiation is assumed to be described as a dilute black-body spectrum with a peak frequency of ν DT = 3 × Hz in the lab frame (e.g., Tavecchio & Ghisellini 2008;Ghisellini & Tavecchio 2009), and the spectral energy den-sity is given by u ext ( ǫ ) = 15 U ( π Θ) ǫ exp ( ǫ/ Θ) − , (9)where Θ = h ν DT / . e c and U ≃ . × − erg / cm are the dimensionless temperature and energy density of theDT radiation field, respectively.The EC-BLR loss rate is given by˙ γ ′ BLR = − σ T m e c γ n =26 X i =1 I d Ω u BLR ( R loc , ǫ i , Ω) f kn ( ǫ i , γ ) , (10)where R loc is the location of the emission region, ǫ i is thedimensionless energy of the lines, n denotes the number ofdifferent lines in the BLR model, and f kn ( ǫ, γ ) = 932 Z γγ l dγ ′′ F c ( x, q ) γ − γ ′′ ǫ γ , (11)where γ l ≃ γ + ǫ − γǫ ǫ , and F c ( x, q ) = h y + y − − ǫ γ γǫy + (cid:18) ǫ γ γǫy (cid:19) i × H [ ǫ γ ; ǫ γ , γǫ ǫ ] , (12)where y = 1 − ( ǫ γ /γ ), ǫ = γǫ (1 − µµ obs ) and ǫ γ = γ + ǫ − γ ′′ (Dermer & Schlickeiser 1993; Dermer et al. 2009). In thecalculation, we adopt a stratified BLR model, with 26 dif-ferent lines emitting at different radii, in a spherical shellconfiguration (Finke 2016). In the model, the location andluminosities of the various lines are estimated by using em-pirical relations derived from reverberation mapping, whenthe disk luminosity L d has been specified.In addition to the radiative losses, electrons also lossenergy through adiabatic expansion of the outflowing plasmablob. The adiabatic losse is evaluated through − ˙ γ ′ adi = 3 cR ′ δ D γ ′ , (13) assuming a conical jet with opening angle θ op ∼ / Γ(B¨ottcher et al. 2013).Therefore, the total cooling rate for electrons is ˙ γ ′ =˙ γ ′ adi + ˙ γ ′ rad .We parameterize the escape timescale in terms of thelight crossing timescale as t ′ esc = ηR ′ /c with η >
1. Here, weadopted a typical value of η = 10 (B¨ottcher & Chiang 2002;Hu et al. 2020).With the above information, equation (3) is numeri-cally solved by using the full implicit scheme described byGraff et al. (2008) to calculate the steady-state EED. Sub-sequently, the observed SEDs of synchrotron and inverse-Compton (IC) emissions are calculated by using the formu-las in Finke et al. (2008) and Dermer et al. (2009). Here,the synchrotron self-absorption (SSA) process is considered(Rybicki & Lightman 1979; Crusius & Schlickeiser 1986).We also consider the contribution from an accretion diskfollowing Dermer & Schlickeiser (2002), which is assumed tobe a Shakura & Sunyaev (1973) disk.In summary, the model is characterized by eight param-eters, i.e., B ′ , n, t var , δ D , L ′ inj , γ ′ min , γ ′ max and R loc . By applying the model to the observed SED, the free pa-rameters and their uncertainties are estimated by perform-ing the MCMC fitting method, which is a powerful tool toexplore the multi-dimensional parameter space in blazar sci-ence (Yan et al. 2013, 2015). The details on MCMC tech-nique can be found in Lewis & Bridle (2002); Yuan et al.(2011); Liu et al. (2012).
We apply the model described in Section 2 to the well-sampled SEDs of three famous blazars 3C 454.3, Mrk 421and 3C 279. The results of the fits to the observed SEDsare shown in the Figures 1-3. The corner plots of the freeparameters are displayed in the left panels of Figures A1-A4in Appendix A. The fitted parameter values are tabulatedin Table 1. In the calculations, a relative systematic uncer-tainty of 5% was added in quadrature to the statistical er-ror of the IR-optical-UV and X-rays data (Poole et al. 2008;Abdo et al. 2011).
3C 454.3 is the brightest γ -ray FSRQ with a redshift z = 0 .
859 (Jackson & Browne 1991). In recent years, thesource attracts much attention, because of its remarkablyhigh activity over the entire electromagnetic spectrum(e.g.,Villata et al. 2006; Raiteri et al. 2008; Vercellone et al.2009, 2010, 2011; Abdo et al. 2011; Shah et al. 2017) andits broken PL γ -ray spectrum (Abdo et al. 2009). Here, wefocus on a simultaneous MWL campaign organized dur-ing 2009 November and December. The MWL SEDs arecollected from Bonnoli et al. (2011). For investigating thechange of the parameters in different active states, we also MNRAS , 1– ????
859 (Jackson & Browne 1991). In recent years, thesource attracts much attention, because of its remarkablyhigh activity over the entire electromagnetic spectrum(e.g.,Villata et al. 2006; Raiteri et al. 2008; Vercellone et al.2009, 2010, 2011; Abdo et al. 2011; Shah et al. 2017) andits broken PL γ -ray spectrum (Abdo et al. 2009). Here, wefocus on a simultaneous MWL campaign organized dur-ing 2009 November and December. The MWL SEDs arecollected from Bonnoli et al. (2011). For investigating thechange of the parameters in different active states, we also MNRAS , 1– ???? (0000) Hu, Yan, and Hu
Table 1.
Mean values and 1 σ errors of the parameters for 3C 454.3, Mrk 421 and 3C 279.state B ′ (G) δ D log L ′ inj log γ ′ min n log R loc χ r
3C 454.3Low 0 . ± .
22 30 . ± .
25 42 . ± .
09 2 . ± .
09 3 . ± . −− † . ± .
12 22 . ± .
69 43 . ± .
09 2 . ± .
12 2 . ± . −− . ± .
21 37 . ± .
68 42 . ± .
07 2 . ± .
06 3 . ± . −− . ± .
11 34 . ± .
70 43 . ± .
12 2 . ± .
12 2 . ± . −− . ± .
10 40 . ± .
12 43 . ± .
06 2 . ± .
07 3 . ± . −− . ± .
16 54 . ± .
52 43 . ± .
09 2 . ± .
14 3 . ± .
16 18 . ± .
06 1.3703/12 0 . ± .
07 42 . ± .
20 43 . ± .
06 2 . ± .
08 3 . ± .
10 18 . ± .
14 1.37Mrk 421Quiet 0 . ± .
01 59 . ± .
54 39 . ± .
01 2 . ± .
05 2 . ± .
04 5 . ± .
02 2.61Quiet † . ± .
003 23 . ± .
96 40 . ± .
03 3 . ± .
05 2 . ± .
04 6 . ± .
02 3.1255266 0 . ± .
01 73 . ± .
04 39 . ± .
02 2 . ± .
08 2 . ± .
05 6 . ± .
02 2.4755270 0 . ± .
02 54 . ± .
98 39 . ± .
04 2 . ± .
08 2 . ± .
07 5 . ± .
04 0.9655277 0 . ± .
06 48 . ± .
78 39 . ± .
10 2 . ± .
15 2 . ± .
14 5 . ± .
07 1.153C 279Period A 1 . ± .
09 37 . ± .
12 41 . ± .
03 2 . ± .
02 3 . ± . −− † . ± .
09 23 . ± .
79 42 . ± .
04 2 . ± .
04 3 . ± .
09 17 . ± .
10 0.80Period C 1 . ± .
09 39 . ± .
12 42 . ± .
03 2 . ± .
02 3 . ± . −− . ± .
06 42 . ± .
59 42 . ± .
06 2 . ± .
06 3 . ± . −− † denotes the results obtained with t var = 1 day.
11 13 15 17 19 21 23 25 log ν [Hz] −12−11−10−9−8 o g ν F ν [ e r g s / c m / s ] Low χ = 1.32 SYNSSCEC-DustEC-BLRDiskToT
11 13 15 17 19 21 23 25 log ν [Hz]6/11 χ = 1.03
11 13 15 17 19 21 23 25 og ν [Hz]27/11 χ = 1.02
11 13 15 17 19 21 23 25 og ν [Hz] −12−11−10−9−8 o g ν F ν [ e r g s / c m / s ] = 1.06
11 13 15 17 19 21 23 25 og ν [Hz]2/12 χ = 1.37
11 13 15 17 19 21 23 25 og ν [Hz]3/12 χ = 1.37 Figure 1.
Comparisons of the best-fitting SEDs with observed data of 3C 454.3. The different components are labelled in the legend. model the SED at the lowest γ -ray state since the beginningof Fermi/LAT observations.The bolometric luminosity of the accretion disc L d is3 × erg/s (Raiteri et al. 2007), and the mass of black holeM BH is 5 × M ⊙ (Bonnoli et al. 2011). To further reducethe number of model parameters, we take t var = 6 hours ac-cording to the analysis of γ -ray variability (Tavecchio et al.2010; Jorstad et al. 2013). Thus, there are six free param-eters in the model, i.e., B ′ , δ D , L ′ inj , γ ′ min , n and R loc . The best-fitting values of the parameters are summarized in Ta-ble 1.From Figure 1, one can see that the model providessatisfactory fits to six SEDs. In the states of 27/11, 2/12 and3/12, the γ -rays can be well interpreted as the superpositionof EC-BLR and EC-DT radiations; while in the other states,the γ -rays are attributed to the EC-DT emission. The X-ray emission in highest γ -ray flare is dominated by EC-DT MNRAS , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj component. In the other five states, the X-ray emission isalmost attributed to SSC component only. Mrk 421 ( z = 0 . t var = 1 hour is taken in the fittings (seeAleksi´c et al. 2015, for discussion).The best-fitting SEDs and observational data points areshown Figure 2 and the obtained parameters are reported inTable 1. EC processes are neglected in the source. The radiodata point from SMA reported in Abdo et al. (2011) is usedto constrain γ ′ min of the PL electron injection. From Figure2, we can see that the fitting to each SED is successful. For comparison, we also revisit the jet properties of thefamous FSRQ 3C 279 ( z =0.538), which was studied witha similar model in our previous work (Hu et al. 2020).Here, we focus on the results of a MWL observing cam-paign conducted during a phase of increased activity from2013 December to 2014 April. The MWL SEDs are takenfrom Hayashida et al. (2015). In the fitting, we adoptt var = 2 hours, which was obtained in the period D inHayashida et al. (2015). L d = 2 × ergs/s (Pian et al.1999) and M BH = 5 × M ⊙ (Gu et al. 2001; Woo & Urry2002; Nilsson et al. 2009) are adopted.Figure 3 shows the SED fitting results, with parame-ters summarized in Table 1. The SEDs can be fitted wellby the model. During the period A, we note that both theEC-DT and EC-BLR components are required to reproducethe γ -ray emission, while during period D, the γ -ray emis-sion is dominated by the EC-DT component. During theperiod C, an EC-BLR component is needed to account forthe γ -ray emission. For all three states, the X-ray spectrumis attributed to SSC emission. Evolution of model parameters could allow us to get a deepinsight into the cause of such a activity that may be associ-ated with changes in the physical conditions of jet, e.g., theinjection rate, Doppler factor, magnetic field strength (e.g.,B¨ottcher & Chiang 2002; Graff et al. 2008; Hu et al. 2015,2020), and/or change in the acceleration process (Yan et al.2013).
Our results show that the locations of emission regions R loc can be only constrained well for the data of 3C 454.3 on 02/12 and 03/12 (see Figures A1, A2 and A4 in AppendixA). It can be found from Table 1 that the marginalized95% CIs of R loc in logarithmic space are [17.88-18.12] cmand [17.95-18.52] cm, respectively. It agrees with the re-sult derived by Nalewajko et al. (2014) who used an inde-pendent method to constrain γ -ray emission site. For theother states of 3C 454.3 and 3C 279, a meaningful con-straint on R loc can not be obtained, and the 95% lowerlimits are reported in Table 1. Note that an upper limit, R loc ≤ R DT = 2 . × L / d, , can be imposed by theadopted DT geometry. Using the vales of L d obtained fromobservations, we have R DT ≃ . × cm for 3C 454.3. The magnetic field strength B ′ and the Doppler factor δ D are well constrained (see Figure A1-A4 in Appendix A).The value of B ′ is ∼ B ′ are more or less similarto that in the pc-scale jet as estimated from the core-shiftmeasurement (Pushkarev et al. 2012; Kutkin et al. 2014;Mohan et al. 2015). This implies that the γ -ray emission re-gions may be located at pc-scale. δ D is found to be larger than 30. The high δ D -valuesare roughly consistent with that estimated from the radiovariability time-scales (Hovatta et al. 2009). Moreover, it isalso supported by the studies of the kinematics of the jetof 3C 279 (Lister & Marscher 1997; Jorstad et al. 2004) and3C 454.3 (Lister et al. 2009; Jorstad et al. 2010, 2013). ForMrk 421, the high δ D -values are consistent with the resultsreported in Hervet et al. (2019). Note that δ D for Mrk 421is generally larger than that for 3C 454.3 and 3C 279. We find that the parameters of the injection electron spec-trum, i.e., L ′ inj , γ ′ min and n , are constrained very well for allSEDs (see Figure A1-A4 in Appendix A). From Table 1,it can be seen that the power-law indexes of injection elec-tron spectrum are restricted to relatively small ranges. It is n ≃ . − . n ≃ . − . . − . L ′ inj varies from ∼ × to 2 × ergs/s, and it is ∼ × ergs/s for Mrk 421. In L γ − L ′ inj plot, we find that the powers of injected electrons L ′ inj in-crease with L γ in 3C 454.3 and 3C 279. For the three sources,the best linear fit in log scale gives L ′ inj ∝ L . ± . γ witha Pearson correlation coefficient of r = 0 .
99 and a chanceprobability of p = 4 . × − .Our results show that γ ′ min is in the range of [340 , , ] for the twoFSRQs. Moreover, we note that γ ′ min increases with L γ in3C 279 (Hu et al. 2020), and there are no such a trend in3C 454.3 and Mrk 421. Notice that for the two FSRQs γ ′ min is in the fast-cooling regime, while for Mrk 421 it is in theslow-cooling regime. MNRAS , 1– ????
99 and a chanceprobability of p = 4 . × − .Our results show that γ ′ min is in the range of [340 , , ] for the twoFSRQs. Moreover, we note that γ ′ min increases with L γ in3C 279 (Hu et al. 2020), and there are no such a trend in3C 454.3 and Mrk 421. Notice that for the two FSRQs γ ′ min is in the fast-cooling regime, while for Mrk 421 it is in theslow-cooling regime. MNRAS , 1– ???? (0000) Hu, Yan, and Hu ν [Hz] −13 −12 −11 −10 −9 ν F ν [ e r g s / c m / s ] Quiet χ = 2.61 10 ν [Hz]MJD 55266 χ = 2.47 10 ν [Hz]MJD 55270 χ = 0.96 10 ν [Hz]MJD 55277 χ = 1.15 Figure 2.
Comparisons of the best-fitting SEDs with observed data of Mrk 421. The black and red dashed lines refer to the synchrotronand SSC emission, respectively. ν [Hz] −13 −12 −11 −10 −9 −8 ν F ν [ e r g s / c m / s ] Period A χ = 1.54 10 ν [Hz]Period C χ = 1.70 10 ν [Hz]Period D χ = 1.54 Figure 3.
Comparisons of the best-fitting SEDs with observed data of 3C 279.
Table 2.
Mean values and 1 σ errors of the derived parameters for 3C 454.3, Mrk 421 and 3C 279.state log P B log P e log P p log P r U ′ B /U ′ e (10 − ) σ m (10 − )3C 454.3Low 44 . ± .
27 45 . ± .
06 46 . ± .
06 44 . ± .
03 2 . ± .
84 2 . ± . † . ± .
22 45 . ± .
02 46 . ± .
07 45 . ± .
03 4 . ± .
46 3 . ± . . ± .
26 45 . ± .
05 47 . ± .
06 45 . ± .
03 2 . ± .
55 1 . ± . . ± .
26 45 . ± .
04 47 . ± .
07 45 . ± .
06 0 . ± .
32 0 . ± . . ± .
18 45 . ± .
03 47 . ± .
05 45 . ± .
03 0 . ± .
49 0 . ± . . ± .
25 46 . ± .
07 48 . ± .
13 45 . ± .
03 4 . ± .
52 0 . ± . . ± .
16 46 . ± .
02 47 . ± .
06 45 . ± .
02 0 . ± .
33 0 . ± . . ± .
03 43 . ± .
04 43 . ± .
04 41 . ± .
04 0 . ± .
04 4 . ± . † . ± .
03 43 . ± .
04 43 . ± .
04 42 . ± .
03 0 . ± .
13 28 . ± . . ± .
08 44 . ± .
09 43 . ± .
06 41 . ± .
08 0 . ± .
05 2 . ± . . ± .
08 43 . ± .
09 43 . ± .
06 41 . ± .
08 0 . ± .
10 4 . ± . . ± .
14 43 . ± .
17 43 . ± .
09 41 . ± .
17 0 . ± .
53 11 . ± . . ± .
06 44 . ± .
02 46 . ± .
03 44 . ± .
03 2 . ± .
39 2 . ± . † . ± .
11 45 . ± .
02 46 . ± .
05 44 . ± .
02 23 . ± .
19 10 . ± . . ± .
05 45 . ± .
02 46 . ± .
03 44 . ± .
03 1 . ± .
23 1 . ± . . ± .
14 45 . ± .
04 46 . ± .
05 45 . ± .
04 0 . ± .
11 0 . ± . , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj −2 −1 B ′ [ G ] δ D l o g L ′ i n j [ e r g s / s ] L γ [ergs/s]2.02.53.0 l o g γ ′ m i n Figure 4.
Evolutions of the model parameters as a function ofthe observed γ -rays luminosity. The red, blue and green squaresare the results of Mrk 421, 3C 454.3 and 3C 279, respectively.The open stars denote the results obtained with t var = 1 dayfor the quiescent states of the three sources. For 3C 279, themodel parameters reported in Hu et al. (2020) are also shown inthe each panel, and are denoted by the gray open circles. In the L γ − L ′ inj plot, the solid and dashed lines denote the best linearfits to the results with a short and a long variability timescale forthe quiescent states, respectively. We evaluate the powers of the relativistic jet from our spec-tral fits. To consider the uncertainties on the transport pa-rameters, we obtain the values of the derived parameters byusing the MCMC code adopted. In the right panels of Fig-ures A1-A4 in Appendix A, we display the corner plots ofthe derived parameters, and the mean values and 1 σ uncer-tainties are listed in Table 2.Assuming that there is one proton per radiating elec-tron, the jet powers carried by the magnetic field P B , rel-ativistic electrons P e and protons P p , as well as radia-tion P r , are evaluated through the method implemented byCelotti & Fabian (1993).The jet powers of FSRQs 3C 454.3 and 3C 279 aresignificantly larger than that of HBL Mrk 421. Further,one can find P p ≃ P e > P B > P r for Mrk 421, and P p > P e > P r & P B for 3C 454.3 and 3C 279. It generallyagrees with previous works for BL Lacs (e.g., Zhang et al. 2012; Yan et al. 2014) and FSRQs (e.g., Celotti & Ghisellini2008; Ghisellini et al. 2010; Ghisellini & Tavecchio 2015). Using our modeling results, we can obtain the equiparti-tion parameter U ′ B /U ′ e , where U ′ B and U ′ e are respectivelythe magnetic field and relativistic electrons energy densitiesin the rest frame of the jet. In addition, we calculate mag-netization parameter σ m ≡ ( υ A /c ) (Cerutti et al. 2012;Sironi et al. 2013; Sironi & Spitkovsky 2014), where υ A isequal to B ′ / √ πn p m p , with n p and m p denoting the ther-mal proton number density and rest mass, respectively.The variations of the equipartition ( U ′ B /U ′ e ) and mag-netization ( σ m ) with activities are shown in Figure 5 withthe values summarized in Table 2. U ′ B /U ′ e ranges from 0.01to 0.3, deviating from equipartition. No difference is foundbetween the values for HBL and FSRQ. U ′ B /U ′ e is not cor-related with L γ .With a large sample of γ -ray blazars, Chen (2018) foundthat the equipartition parameter of blazars have a largewidth in its distribution, and BL Lacs have much smaller U ′ B /U ′ e , comparing with FSRQs. σ m is in the range of 0.002 and 0.1. Mrk 421 haslarger σ m than that of 3C 454.3 and 3C 279. σ m variesfrom 0.03 to 0.2 for Mrk 421, which is in good agreementwith the results of Lewis et al. (2016) obtained by compar-ing the theoretical model with the X-ray time lags dur-ing the 1998 April 21 flare. In particular, using a relativis-tic oblique shock acceleration + radiation-transfer model,B¨ottcher & Baring (2019) successfully explained the SEDsand variabilities of 3C 279 during the flaring activity con-sidered in the work. They obtained the non-relativistic mag-netization σ = 3 .
42 in period A and 1.84 in period C. Usingthe relation σ m = σ Γ sh / ( m p /m e ) and Γ sh = δ D , we obtain σ m = 6 . × − and 3 . × − , respectively. They are alsoin good agreement with our results.There is a good correlation between σ m and L γ . It yields σ m ∝ L − . ± . γ with a Pearson correlation coefficient of r = − .
86 and a chance of probability of p = 1 . × − . Using a single-zone leptonic model and the MCMC fit-ting technique, we model the high-quality and simultane-ous MWL SEDs at multiple epochs of three typical blazars(3C 454.3, 3C 279 and Mrk 421). For 3C 454.3, the SEDsin a low γ -ray state and five highest γ -ray flaring states(Bonnoli et al. 2011) are considered. For Mrk 421, we con-sider the SED in a γ -ray quiescent state (Abdo et al. 2011)and the SEDs in three highest γ -ray flaring states reportedin Aleksi´c et al. (2015). For 3C 279, we consider three SEDsin Hayashida et al. (2015), which are the representatives for Notice that the definition of σ m can be related to the non-relativistic magnetization defined by Baring et al. (2017), throughthe relation σ m = σ Γ sh / ( m p /m e ). On the other hand, the defi-nition of σ m may be equivalent to the definition of σ ≡ P B /P m ,where P m = P e + P p is the kinetic power of matter. For Mrk 421,one can obtain σ m ≃ P B /P p ≃ P B /P e , since P e ∼ P p . For 3C454.3 and 3C 279, one can obtain σ m ≃ P B /P p , since P p ≫ P e .MNRAS , 1– ????
86 and a chance of probability of p = 1 . × − . Using a single-zone leptonic model and the MCMC fit-ting technique, we model the high-quality and simultane-ous MWL SEDs at multiple epochs of three typical blazars(3C 454.3, 3C 279 and Mrk 421). For 3C 454.3, the SEDsin a low γ -ray state and five highest γ -ray flaring states(Bonnoli et al. 2011) are considered. For Mrk 421, we con-sider the SED in a γ -ray quiescent state (Abdo et al. 2011)and the SEDs in three highest γ -ray flaring states reportedin Aleksi´c et al. (2015). For 3C 279, we consider three SEDsin Hayashida et al. (2015), which are the representatives for Notice that the definition of σ m can be related to the non-relativistic magnetization defined by Baring et al. (2017), throughthe relation σ m = σ Γ sh / ( m p /m e ). On the other hand, the defi-nition of σ m may be equivalent to the definition of σ ≡ P B /P m ,where P m = P e + P p is the kinetic power of matter. For Mrk 421,one can obtain σ m ≃ P B /P p ≃ P B /P e , since P e ∼ P p . For 3C454.3 and 3C 279, one can obtain σ m ≃ P B /P p , since P p ≫ P e .MNRAS , 1– ???? (0000) Hu, Yan, and Hu −2 −1 U ′ B / U ′ e L γ [ergs/s]10 −3 −2 −1 σ m Mrk 4213C 454.3
3C 279
Figure 5.
Evolutions of the equipartition (upper) and magnetiza-tion (lower) parameters as a function of the observed γ -rays lu-minosity. The open stars denote the results derived with t var = 1day for the quiescent states of the three sources. In the L γ − σ m plot, the solid and dashed lines denote the best linear fits to theresults with a short and a long variability timescale for the qui-escent states, respectively. the 14 SEDs studied in Hu et al. (2020). The jet model usedhere is a refined one of Hu et al. (2020).We find that the model can explain the SEDs well.The properties of the emission regions obtained by usingour model are consistent with previous studies. Mrk 421 hassmaller magnetic field strength and larger Doppler factorthan 3C 454.3 and 3C 279. The electron injection spectrumof 3C 279 and 3C 454.3 is steeper than that of Mrk 421. Mrk421 has smaller electron injection power and larger magne-tization than 3C 454.3 and 3C 279. The three jets are foundto be low-magnetization and non-equipartition, suggestingthat electrons may be accelerated by shock in the jet.The new findings are the tight correlations between L γ and L ′ inj , σ m . The correlation between L γ and L ′ inj has beenfound in modeling the 14 SEDs of 3C 279 (Hu et al. 2020).Here, after including the HBL Mrk 421 and 3C 454.3, thisrelation still holds. L ′ inj can be associated with accelerationin the jet. The correlation between L γ and L ′ inj suggeststhat particle acceleration process play an important role indriving γ -ray flares of blazars.As mentioned above, the low magnetization ( σ m ≤ . L γ and σ m is firstly found inmodeling blazars SEDs. This suggests that the same energy- dissipative mechanism behind the jet emission could be atwork in different types of blazars.In the above modelings, we used the minimum vari-ability timescale measured from the γ -ray flares for eachsource. The same variability timescale is adopted for theSED in the quiescent state. This implies that the gamma-rays in quiescent states and flaring states are producedin the same region. Alternatively, gamma-rays in differ-ent states may be produced in different regions (e.g.,Acharyya, Chadwick, & Brown 2021). In this case, a longervariability timescale should be used for the quiescent state.In order to investigate this scenario, we also performed thefitting to the SED at quiescent state for each source using t var = 1 day. The modeling results are shown in Figure B1,and the corresponding corner plots of the input and outputparameters are shown in Figure B2 with the parameters val-ues reported in Tables 1 and 2. We find that the increasingof t var could primarily lead to decrease in δ D , and increasein L ′ inj . B ′ , γ ′ min , n as well as R loc are found to changeslightly. For comparison, the parameters values are plottedin the Figures 4 and 5 (open stars). Interestingly, it is foundthat the correlation between L ′ inj and L γ is still significant,with r = 0 .
91 and p = 1 . × − . In the σ m − L γ panel ofFigure 5, the Pearson test also gives a significant correlationwith r = − .
87 and p = 1 . × − . The results may indi-cate that the correlations are independent of the location ofthe gamma-ray emission region.The variability may be caused by changes in physicalcondition of the jet, e.g., injection, magnetic field, Dopplerfactor (Diltz & B¨ottcher 2014), with possible intervention ofshock waves or turbulence (e.g., Sikora et al. 2001; Marscher2014). In the frame of the one-zone leptonic model, our re-sults indicate that the electron injection may be the maindriver of the variability. In other words, the acceleration ofelectrons causes the γ -ray variability.In addition to the intrinsic origin of the variability, itcould also be attributed to a geometrical effect of changingin the viewing angle (Raiteri et al. 2009, 2017; Villata et al.2007; Liodakis et al. 2020). It is interesting to make a dis-tinction between two explanations of variability. However,it is difficult to do such a distinction based on the currentobservations.It should be pointed out that the collimation parame-ter of the emitting region Γ θ obs = 1 is fixed in the fitting.This assumption is frequently adopted in blazar SED model-ing (e.g., Zhang et al. 2012; B¨ottcher et al. 2013; Yan et al.2014, 2015; Nalewajko et al. 2014). However, the VLBI ob-servations indicate that Γ θ obs ≪ δ D , which can keep constant by decreasing Γ and θ obs 3 . However, we would like to stress that the decreasing ofΓ θ obs will reduce the intrinsic jet power. Therefore, it shouldbe considered with caution when the jet powers from thespectral fitting are compared to the powers predicted by the In the limit Γ ≫ θ obs ≪
1, the Doppler factor δ D canbe related to Γ through the relationship δ D = 2Γ / (1 + Γ θ obs )(Dermer et al. 2014). MNRAS , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj Blanford-Payne (BP; Blandford & Payne 1982) or Blanford-Znajek (BZ; Blandford & Payne 1982) mechanisms.From Table 2, one can find that the averaged jet powersestimated from the SED fittings are 4 . × , . × and 1 . × erg/s for 3C 454.3, 3C 279 and Mrk 421,respectively. On the other hand, the jet powers can also beestimated by modeling the observed SEDs with hadronicmodels (e.g., B¨ottcher et al. 2013; Petropoulou & Dermer2016; Barkov, Aharonian & Bosch-Ramon 2010). In partic-ular, the hadronic models received a wide attention sincethe detection of coincident neutrinos and γ -rays from blazarTXS 0506+056 (Aartsen et al. 2018a,b). The required powerin relativistic protons L p is of the order of magnitude of 10 erg/s for 3C 454.3 and 10 erg/s for 3C 279 (B¨ottcher et al.2013), while for Mrk 421 L p ranges from 2 . × to7 . × erg/s (Petropoulou, Coenders & Dimitrakoudis2016). It can be seen that the jet powers estimated fromhadronic models are much greater than that estimated fromleptonic model.We can compare the jet power against the powerof the BZ process (Blandford & Znajek 1977), which isbelieved to be the plausible explanation for the jetlaunch. Generally, the predicted jet power can be writ-ten as P BZ = η j P acc , where η j is the jet produc-tion efficiency, and the accretion power is P acc = ˙ M c with ˙ M denoting the mass accretion rate. As it hasbeen theoretically estimated and numerically confirmed, η j in a magnetically choked accretion flow scenario canexceed unity (Tchekhovskoy, Narayan & McKinney 2010;McKinney, Tchekhovskoy & Blandford 2012), and reaches ∼ . P acc = L d /η d , where η d is the accretion disk radiative efficiency.With assumption of η d = 0 . . P acc ≃ (1 − × erg/s for 3C 454.3, and P acc ≃ (0 . − × erg/s for 3C 279. Compared withFSRQs, BL Lacs are believed to have low accretion rates(e.g., Wang, Staubert & Ho 2002; Xu, Cao & Wu 2009). ForMrk 421, we estimate the accretion power through the re-lation P acc = ˙ mL Edd ≃ . × ˙ mM ergs/s, where L Edd is the Eddington luminosity, ˙ m is the mass accre-tion rate in units of ˙ M Edd = L Edd /c , and M is the BHmass in units of 10 M ⊙ ( M ⊙ denotes the solar mass.). As-suming ˙ m ∼ (3 − × − (Ghisellini & Tavecchio 2008;Meyer et al. 2011), we obtain P acc ≃ (0 . − . × erg/sfor Mrk 421, when we take log M BH /M ⊙ = 8 .
28 derived fromthe measurement of stellar velocity dispersion (Woo & Urry2002). Therefore, it seems that the relativistic jets in thethree sources may be governed by the BZ process.Our results show that the energy density ratio of mag-netic field and radiating electrons U ′ B /U ′ e varies from ∼ . ∼ .
5. This is consistent with results from modeling theSEDs of the large sample blazars (e.g., Celotti & Ghisellini2008; Ghisellini et al. 2014; Chen 2018) or individual sources(e.g., Yan et al. 2013; Dermer et al. 2014; Hu et al. 2015).Interestingly, we note that in the quiescent states of the threesources U ′ B /U ′ e is closer to equipartition compared to theflaring states. In fact, the results may be supported by theradio observations. The studies of the core-shift-effect haveshown that the distance of the core from the jet base r c , thecore size W and the light-curve time lag ∆ t all depend onthe observation frequency ν as r c ∝ W ∝ ∆ t ∝ ν − /k with k ≃ r c ∝ ν − is disrupted during flares. The au-thors concluded that the observed flux density variabilityand the variations of the core position in a flaring jet aremainly caused by significant increase in emitting electrondensity and slight decrease in the magnetic field. This indi-cates that the electron energy density dominates over mag-netic field energy density during a flaring activity. Moreover,the extreme brightness temperatures ≥ K observedby RadioAstron also support that equipartition may be vi-olated during flares (e.g., G´omez et al. 2016; Bruni et al.2017; Pilipenko et al. 2018; Kutkin et al. 2018).
ACKNOWLEDGEMENTS
We thank the reviewer for constructive suggestions and com-ments. We acknowledge the National Natural Science Foun-dation of China (NSFC-11803081, NSFC-12065011,NSFC-U1831124) and the joint foundation of Department of Sci-ence and Technology of Yunnan Province and Yunnan Uni-versity [2018FY001(-003)]. The work of D. H. Yan is alsosupported by the CAS Youth Innovation Promotion As-sociation and Basic research Program of Yunnan Province(202001AW070013).
DATA AVAILABILITY
No new data were generated or analysed in support of thisresearch.
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APPENDIX A: CORNER PLOTS OF MODELPARAMETERSAPPENDIX B: SED FITTING WITH THEVARIABILITY TIMESCALE OF ONE DAY FORQUIESCENT STATES
Figure B1 shows the best-fitting SEDs to the three sourcesat the quiescent states with t var = 1 day, and Figure B2shows the corner plots of the model parameters. The resultsof this analysis are discussed in detail in Section 4, and thevalues are shown in Tables 1 and 2. MNRAS , 1– ????
Figure B1 shows the best-fitting SEDs to the three sourcesat the quiescent states with t var = 1 day, and Figure B2shows the corner plots of the model parameters. The resultsof this analysis are discussed in detail in Section 4, and thevalues are shown in Tables 1 and 2. MNRAS , 1– ???? (0000) Hu, Yan, and Hu P r obab ili t y log (R loc ) . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . . l og ( R l o c ) δ D log (L inj ′ ) . . . log ( γ min ′ ) . . n . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) (P e ) . . . log (P p ) . . . log (P r ) . U B ′ /U e ′ (0.1) P r obab ili t y log (R loc ) . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . . . . l og ( R l o c ) δ D log (L inj ′ ) . log ( γ min ′ ) . . n . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . .
24 log (P e ) . . log (P p ) . . log (P r ) . . . . U B ′ /U e ′ (0.1) P r obab ili t y log (R loc ) . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n
23 B ′ . l og ( R l o c ) δ D log (L inj ′ ) . log ( γ min ′ ) . . . n P r obab ili t y σ m (0.01) . . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . )
012 log (P B ) σ m ( . ) (P e ) . . log (P p ) . . log (P r ) . . . . U B ′ /U e ′ (0.1) Figure A1.
Corner plots of the free model parameters (left) and derived parameters (right) for FSRQs 3C 454.3. For the two-dimensionalconfidence contours of the parameters, the inner and outer contours denote the 68 and 95 per cent confidence intervals, respectively. Forthe one-dimensional probability distributions of the parameters, the dashed lines show the maximum likelihood distributions and solidlines show the marginalized probability distributions. From top to bottom, the plots are the results obtained from fitting SEDs at Low γ -ray state, 11/6 and 27/11 reported in Bonnoli et al. (2011), respectively. MNRAS , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj P r obab ili t y log (R loc ) . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . l og ( R l o c ) δ D log (L inj ′ ) . . . log ( γ min ′ ) . . n . . . . . . P r obab ili t y σ m (0.01) . . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . )
123 log (P B ) σ m ( . ) . . . . (P e ) . . . log (P p ) . . . log (P r ) . . . U B ′ /U e ′ (0.1) P r obab ili t y log (R loc ) . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . l og ( R l o c ) δ D log (L inj ′ ) . log ( γ min ′ ) . . . n . P r obab ili t y σ m (0.01) . . . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . (P e ) . log (P p ) . . log (P r ) . . U B ′ /U e ′ (0.1) P r obab ili t y log (R loc ) . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . l og ( R l o c ) δ D log (L inj ′ ) . . log ( γ min ′ ) . . n . . P r obab ili t y σ m (0.01) . . . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . . . . (P e ) . . log (P p ) . . log (P r ) . . U B ′ /U e ′ (0.1) . . Figure A2.
Same as Figure A1, but for the SEDs on 01/12 (top), 02/12 (middle) and 03/12 (bottom).MNRAS , 1– ????
Same as Figure A1, but for the SEDs on 01/12 (top), 02/12 (middle) and 03/12 (bottom).MNRAS , 1– ???? (0000) Hu, Yan, and Hu P r obab ili t y log ( γ max ′ ) . . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . l og ( γ m a x ′ ) δ D log (L inj ′ ) . . log ( γ min ′ ) . . n . . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . . . .
46 log (P e ) . . log (P p ) . . log (P r ) . . . U B ′ /U e ′ (0.1) . . . . . P r obab ili t y log ( γ max ′ ) . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . l og ( γ m a x ′ ) δ D log (L inj ′ ) . . . log ( γ min ′ ) . . n . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) .
246 log (P e ) . . log (P p ) . log (P r ) . . U B ′ /U e ′ (0.1) . . . P r obab ili t y log ( γ max ′ ) . . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . . . l og ( γ m a x ′ ) δ D log (L inj ′ ) . .
73 9 . . . log ( γ min ′ ) . . n . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . .
510 log (P e ) . . log (P p ) . log (P r ) . . U B ′ /U e ′ (0.1) . . . P r obab ili t y log ( γ max ′ ) . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . l og ( γ m a x ′ ) δ D log (L inj ′ ) . log ( γ min ′ ) . n . . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . )
012 log (P B ) σ m ( . ) . . . (P e ) . log (P p ) . . log (P r ) . U B ′ /U e ′ (0.1) Figure A3.
Corner plots of the input (left) and output parameters (right) for HBL Mrk 421. From top to bottom, the plots are arrangedin the following order quiescent state, 55266, 55270 and 55277. MNRAS , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj P r obab ili t y log (R loc ) δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . . . l og ( R l o c ) δ D log (L inj ′ ) . . . log ( γ min ′ ) . . n . . . P r obab ili t y σ m (0.01) . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . )
234 log (P B ) σ m ( . ) . . . (P e ) . . log (P p ) . . . . log (P r ) . . . U B ′ /U e ′ (0.1) P r obab ili t y log (R loc ) δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . l og ( R l o c ) δ D log (L inj ′ ) . log ( γ min ′ ) . . . n . . . P r obab ili t y σ m (0.01) . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . . . (P e ) . . . log (P p ) . . . log (P r ) .
54 4 . . . U B ′ /U e ′ (0.1) . P r obab ili t y log (R loc ) δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . . l og ( R l o c ) δ D log (L inj ′ ) . . log ( γ min ′ ) . n . P r obab ili t y σ m (0.01) . . l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . (P e ) . .
44 5 . .
54 5 . log (P p ) . .
74 6 . .
84 6 . log (P r ) . . . U B ′ /U e ′ (0.1) . . . . Figure A4.
Corner plots of the free model parameters (left) and derived parameters (right) for FSRQs 3C 279. From top to bottom, theplots are the results obtained from fitting SEDs in Periods A, C, D reported in Hayashida et al. (2015), respectively.MNRAS , 1– ????
Corner plots of the free model parameters (left) and derived parameters (right) for FSRQs 3C 279. From top to bottom, theplots are the results obtained from fitting SEDs in Periods A, C, D reported in Hayashida et al. (2015), respectively.MNRAS , 1– ???? (0000) Hu, Yan, and Hu
10 15 20 25 log ν [Hz] −13.0−12.5−12.0−11.5−11.0−10.5−10.0−9.5−9.0 l o g ν F ν [ e r g s / c m / s ]
3C 454.3 χ = 1.44 ν [Hz] −13 −12 −11 −10 −9 −8 ν F ν [ e r g s / c m / s ] Mrk 421 χ = 3.12 ν [Hz] −13 −12 −11 −10 −9 ν F ν [ e r g s / c m / s ]
3C 279 χ = 0.8 Figure B1.
Comparisons of the best-fitting SEDs with observed data of 3C 454.3, Mrk 421 and 3C 279 at the quiescent states with t var = 1 day. MNRAS , 1– ?? (0000) orrelations between L γ and σ m , L ′ inj P r obab ili t y log (R loc ) . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n
34 B ′ . l og ( R l o c ) δ D log (L inj ′ ) . . log ( γ min ′ ) . n P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . . (P e ) . . . log (P p ) . . log (P r ) . . . . U B ′ /U e ′ (0.1) P r obab ili t y log ( γ max ′ ) . . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . l og ( γ m a x ′ ) δ D log (L inj ′ ) . . log ( γ min ′ ) . . . n . . . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . . (P e ) . . log (P p ) . . . log (P r ) . . . U B ′ /U e ′ (0.1) . . . P r obab ili t y log (R loc ) . . . δ D l og ( L i n j ′ ) l og ( γ m i n ′ ) n ′ . . l og ( R l o c ) δ D log (L inj ′ ) . .
84 2 . .
94 2 . log ( γ min ′ ) . . . n . . . P r obab ili t y σ m (0.01) l og ( P e ) l og ( P p ) l og ( P r ) U B ′ / U e ′ ( . ) (P B ) σ m ( . ) . . . . (P e ) . . .
14 5 . . . . log (P p ) . . . log (P r ) . . . . .
94 4 . . U B ′ /U e ′ (0.1) Figure B2.
Corner plots of the free model parameters (left) and derived parameters (right) for the quiescent states with t var = 1 day.From top to bottom, the plots are the results obtained from fitting SEDs for 3C 454.3, Mrk 421 and 3C 279, respectively.MNRAS , 1– ????