Formation of very hard electron and gamma-ray spectra of flat spectrum radio quasar in fast-cooling regime
aa r X i v : . [ a s t r o - ph . H E ] M a r MNRAS , 1–8 (2015) Preprint 24 October 2018 Compiled using MNRAS L A TEX style file v3.0
Formation of very hard electron and gamma-ray spectra offlat spectrum radio quasar in fast-cooling regime
Dahai Yan , ⋆ , Li Zhang † , Shuang-Nan Zhang ‡ Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Department of Astronomy, Key Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University, Kunming, 650091, China
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
In external Compton scenario, we investigate the formation of the very hard electronspectrum in the fast-cooling regime, using a time-dependent emission model. It isshown that a very hard electron distribution N ′ e ( γ ′ ) ∝ γ ′− p with the spectral index p ∼ . Fermi -LAT spectrum of the flat spectrum radio quasar 3C 279 during the extremegamma-ray flare in 2013 December.
Key words: radiation mechanisms: non-thermal — galaxies: jets — gamma rays:galaxies
Leptonic models have met with considerable successes inmodelling the broadband (from radio to γ -ray frequen-cies) spectral energy distribution (SED) of all classesof blazars (e.g., Ghisellini et al. 2010; Zhang et al. 2012;Ghisellini et al. 2014; Yan et al. 2016). In leptonic models,non-thermal emission is produced by synchrotron emissionof relativistic electrons in a comoving magnetic field and in-verse Compton (IC) emission of relativistic electrons on lowenergy photons. For IC scattering, low-energy seed photonsmay be provided by synchrotron radiation (synchrotron-selfCompton, SSC) and various external radiation fields (EC):(1) accretion disk radiation (EC-disk; Dermer & Schlickeiser1993, 2002), (2) broad-line region (BLR) radiation (EC-BLR; Sikora et al. 1994), and (3) dust IR radiation (EC-dust; B la˙zejowski et al. 2000). Time-dependent leptonicmodels have been developed to study the observed variabil-ity features (e.g., Mastichiadis & Kirk. 1997; Li & Kusunose2000; Kusunose et al. 2000; B¨ottcher & Chiang 2002;B¨ottcher & Dermer 2010; Chen et al. 2012; Saito et al.2015).Relativistic electron energy distribution (EED) is cru-cial for studying the non-thermal radiation from blazars.A static broken power-law EED (see Yan et al. 2013;Zhou et al. 2014, for constraining various EEDs with ⋆ E–mail: [email protected] † E–mail: [email protected] ‡ E–mail: [email protected] observations) is frequently used to model the SEDsof blazars (e.g., Tavecchio et al. 1998; Finke et al. 2008;Mankuzhiyil et al. 2011; Yan et al. 2012a, 2014; Zhang et al.2014; Kang et al. 2014). An initial single power-law electrondistribution can be deformed to become a broken power-law distribution due to radiative energy losses. In short, inthe slow-cooling regime (i.e.,the minimum energy of injectedEED γ ′ min less than the broken energy γ ′ b of cooled EED),the electron spectrum below γ ′ b has a spectral index p = s ,where s is the spectral index of the injected single power-law EED; above γ ′ b , the spectrum is softened by cooling,and has an index s + 1. In the fast-cooling regime (i.e., γ ′ b = γ ′ min ), p ∼
2, independent of s ; above γ ′ min , thespectrum also has an index s + 1 (e.g., Dermer & Menon2009; Finke 2013). Note that standard shock accelerationtheories predict s ∼
2. Modelling the SEDs of
Fermi -LATdetected blazars (Abdo et al. 2010) returns a standard elec-tron spectrum with p ∼ γ ′ b (e.g., Yan et al. 2012a,2014; Zhang et al. 2012; Kang et al. 2014).However, the standard picture mentioned above faceschallenges when trying to explain the very hard TeV emis-sion detected for several high-synchrotron-peaked BL Lacobjects (HSPs), e.g., 1ES 1101-232 and 1ES 0229+200. Var-ious approaches have been proposed to explain the veryhard TeV spectrum of HSPs: the leptonic models in ex-treme regime (e.g, Katarzy´nski et al. 2006; Tavecchio et al.2009), the modified leptonic models in normal regime (e.g.,Lefa et al. 2011; B¨ottcher et al. 2008; Yan et al. 2012b), andthe Ultra-high energy cosmic rays propagation models (e.g.,Essey et al. 2011; Murase et al. 2012; Yan et al. 2015). c (cid:13) D. H. Yan, L. Zhang, and S. N. Zhang
The recently observed very hard GeV spectrum of theflat spectrum radio quasar (FSRQ) 3C 279 during an ex-treme γ -ray flare (Hayashida et al. 2015) also challenges thestandard picture. Modeling by Hayashida et al. (2015) witha broken power-law EED showed that a very hard elec-tron spectrum with p ∼ γ ′ b ∼ Fermi -LAT spectrum. In the slow-cooling regime, such a hard emitting electron spectrum re-quires a very hard injection electron distribution. Severalmechanisms have been proposed to produce a very hardinjection electron distribution, for example, magnetic re-connection (e.g., Zenitani & Hoshino 2001; Guo et al. 2014,2015; Sironi & Spitkovsky 2014; Werner et al. 2016) and rel-ativistic shock (e.g., Stecker, Baring, & Summerlin 2007).Asano & Hayashida (2015) recently explained the very hardGeV spectrum in a stochastic acceleration model.Here we study the formation of the very hard electronspectrum of FSRQ in the fast-cooling regime. We show thata very hard electron spectrum with p ∼ . Fermi -LATspectrum of 3C 279. In Section 2, we describe our model;numerical results are showed in Section 3. In Section 4,we apply our approach to the very hard
Fermi -LAT spec-trum of 3C 279. In Section 5, we give summary and discus-sion. Throughout the paper, we use cosmology parameters H = 71 km s − Mpc − , Ω m = 0 .
27, Ω Λ = 0 . In a one-zone leptonic model, it is assumed that the emissionis produced by relativistic electrons injected in a homoge-neous blob of comoving radius R ′ . The emission blob moveswith relativistic speed (corresponding to the bulk Lorentzfactor Γ) towards us. Due to the beaming effect, the ob-served emission is strongly boosted. For a blazar, we assumethe Doppler factor δ D = Γ. Note that quantities in the framecomoving with the jet blob are primed.Relativistic electrons lose energy due to synchrotronand IC radiation. The kinetic equation governing the tem-poral evolution of the electrons distribution N ′ e ( γ ′ , t ′ ) is ∂N ′ e ( γ ′ , t ′ ) ∂t ′ = ∂∂γ ′ [ ˙ γ ′ N ′ e ( γ ′ , t ′ )] − N ′ e ( γ ′ , t ′ ) t ′ esc + Q ′ ( γ ′ , t ′ ) , (1)where N ′ e is the differential electron number and t ′ esc theescape timescale. ˙ γ ′ is the total cooling rate, and Q ′ ( γ ′ , t ′ )is the electron injection rate.We take into account radiative cooling due to syn-chrotron radiation, SSC and EC. Therefore, we have˙ γ ′ ( r ) = ˙ γ ′ syn + ˙ γ ′ SSC + ˙ γ ′ EC ( r ) . (2)The synchrotron cooling rate is given by˙ γ ′ syn = 4 cσ T m e c u B γ ′ , (3)where u B = B ′ π is the magnetic energy density and B ′ the comoving mag-netic field, c is the speed of light, m e is the electron mass,and σ T is the Thomson cross section.The SSC cooling rate using the full KN cross section is(e.g., Jones 1968; B¨ottcher et al. 1997; Finke et al. 2008)˙ γ ′ SSC = 3 σ T m e c Z ∞ dǫ ′ u ′ syn ( ǫ ′ ) ǫ ′ G ( γ ′ ǫ ′ ) (4)where G ( E ) = 83 E E (1 + 4 E ) − E E (cid:18)
23 + 12 E + 18 E (cid:19) + ln(1 + 4 E ) (cid:18) E + 34 1 E + ln[1 + 4 E ]2 E − ln[4 E ] E (cid:19) −
52 1 E + 1 E ∞ X n =1 (1 + 4 E ) − n n − π E − , and u ′ syn ( ǫ ′ ) is the spectral energy density of synchrotronradiation.A fairly accurate approximation for the EC coolingrate, valid in the Thomson through Klein-Nishina regimes,is given by Moderski et al. (2005), and it is˙ γ ′ EC = 4 cσ T m e c u ′ γ ′ f KN (4 γ ′ ǫ ′ ) , (5)where u ′ and ǫ ′ are the energy density and dimensionlessphoton energy, respectively, of the external radiation field inthe comoving frame of blob. The correction function for KNeffect is given by f KN ( x ) = 1(1 + x ) . . (6)The external radiation includes emissions from broad-line region (BLR) and infrared dust torus. Their energy den-sities in the comoving frame as the functions of the dis-tance r from the black hole are given by (Sikora et al. 2009;Hayashida et al. 2012) u ′ BLR ( r ) = Γ τ BLR L disk πr c [1 + ( r/r BLR ) ] , (7) u ′ dust ( r ) = Γ τ dust L disk πr c [1 + ( r/r dust ) ] , (8)where τ BLR and τ dust are the fractions of the disk lumi-nosity reprocessed into BLR radiation and into dust ra-diation, respectively. The typical values of τ BLR ∼ . τ dust ∼ . L disk (Ghisellini & Tavecchio 2009;Ghisellini et al. 2014), i.e., r BLR = 10 ( L disk / erg s − ) / cm , (9) r dust = 10 ( L disk / erg s − ) / cm . (10)Then, we have u ′ BLR ( r ) ≃ . τ BLR r/r
BLR ) erg cm − , (11) MNRAS , 1–8 (2015) ery hard electron spectrum in fast-cooling regime
10 100 1000 10000 10000010001000010000010000001E71E81E9
EC, r=5r
BLR synchrotron EC, r=0.8r
BLR t ’ c oo l [ s ] ’ Figure 1.
Cooling time for synchrotron radiation and EC pro-cesses. B ′ = 1 G and Γ = 30 are used. u ′ dust ( r ) ≃ . τ dust r/r dust ) erg cm − . (12)Therefore, u ′ is also a function of r , i.e., u ′ ( r ) = u ′ dust ( r ) + u ′ BLR ( r ) . (13)BLR and IR dust radiation is assumed to be a dilutedblackbody radiation. Given that BLR radiation is dominatedby Ly α line photons, we adopt an effective temperature forthe BLR radiation of T BLR = 4 . × K, so that the energydensity of BLR radiation peaks at ≈ . k B T BLR /h ∼ = 2 . × Hz (corresponding to the mean dimensionless energy ǫ BLR = 2 × − ). We assume an effective temperature forthe IR dust radiation of T dust = 1000 K (Malmrose et al.2011), i.e., the mean dimensionless energy ǫ dust = 5 × − .Then we have ǫ ′ = Γ ǫ BLR for r ≤ r BLR , and ǫ ′ = Γ ǫ dust for r > r BLR .We neglect the electron energy loss due to adiabatic ex-pansion, because in FSRQs the adiabatic cooling with an ex-panding velocity ∼ . c is relevant only for very low energyelectrons which do not contribute to the radiative output.In Fig. 1, we show the cooling time ( γ ′ / ˙ γ ′ ) for syn-chrotron radiation and EC processes. We use B ′ = 1 G, Γ =30, τ BLR = 0 . τ dust = 0 .
3, and L disk = 1 . × erg s − .One can find that taking r = 0 . r BLR , significant KN correc-tion takes place at γ ′ & / ǫ BLR ∼ r = 5 r BLR ,the KN correction takes place at γ ′ > .We consider a constant injection during the injectiontime t ′ inj . The injection electron distribution is Q ′ ( γ ′ ) = Q ′ γ ′− s exp( − γ ′ /γ ′ cut ) H ( γ ′ ; γ ′ min ) , (14)where s is the spectral index, γ ′ min is the minimum injec-tion energy, γ ′ cut is the cut-off energy, and Q ′ [s − ] is thenormalization constant; H ( γ ′ ; γ ′ min ) = 1 for γ ′ > γ ′ min ,otherwise H ( γ ′ ; γ ′ min ) = 0. l og ’ N ( ’ ) log ’ L [ e r g s - ] Hz SSC
Figure 2.
Temporal evolution of EED (top panel) and SED (bot-tom panel) with r = 0 . r BLR . The lines from thin to heavycorrespond to t ′ = [0 . , , , × s, respectively. The SSCcomponent is at t ′ = 4 × s. We use γ ′ min = 10 , γ ′ cut = 10 , Q = 4 . × s − , s = 2 . B ′ = 1 G, δ D = 30, R ′ = 10 cm, τ BLR = 0 . τ dust = 0 .
3, and L disk = 1 . × erg s − . We naturally relate r to time by: r = r + ct ′ Γ , (15)where r is the distance where the injection starts. We numerically solve equation (1), adopting the numeri-cal method given by Chiaberge & Ghisellini (1999). In thecalculations, we use t ′ esc = t ′ inj = 10 s. We calculatethe synchrotron and IC spectra using the methods givenby Dermer & Menon (2009). Synchrotron-self absorption istaken into account. MNRAS , 1–8 (2015)
D. H. Yan, L. Zhang, and S. N. Zhang l og ’ N ( ’ ) log ’ L [ e r g s - ] Hz SSC
Figure 3.
Same as Fig. 2, but with r = r = 0 . r BLR . In Fig. 2, we show the temporal evolution of EED andSED in the case of the emission region initially locatedinside BLR. One can see that EED develops a very hard N ′ e ( γ ′ ) ∝ γ ′− . form below the minimum injection energy.This very hard spectrum is different from the standard shapeof N ′ ( γ ′ ) ∝ γ ′− expected in the case of Thomson or syn-chrotron cooling processes of the form ˙ γ ′ ∝ γ ′ . The hard-ening in the electron spectrum is mainly owing to KN en-ergy losses on the BLR radiation. The minimum energy γ ′ of the emitting electron distribution is affected by the evo-lution time t ′ when t ′ < t ′ esc , which can be evaluated bythe relation t ′ cool = t ′ . In the fast-cooling regime, we have γ ′ < γ ′ min . The γ -ray spectrum in Fig. 2 is the sum of EC-BLR and EC-dust components. Below ∼ Hz, we have avery hard γ -ray spectrum. When the emission region moves l og ’ N ( ’ ) log ’ EC-BLR L [ e r g s - ] HzEC-dust
Figure 4.
Temporal evolution of EED (top panel) and SED (bot-tom panel) with r = 5 r BLR . The lines from thin to heavy corre-spond to t ′ = [1 , , × s, respectively. The other parametersare same as those in Fig. 2. outside BLR, i.e., t ′ > s, a softening in EED and SEDoccurs.One can see that the variation in EED with time issignificant below γ ′ cut (see top panel in Fig. 2). The changein EED leads to the change in synchrotron spectrum below ∼ × B ′ γ ′ δ D Hz, and the change in EC spectrum below ∼ ǫ ′ δ D γ ′ Hz. Note that EC-BLR spectrum above ∼ ǫ ′ δ D γ ′ ∼ . × Hz is suppressed by the KN effect,where γ ′ KN ∼ / ǫ ′ . The variation of the X-ray spectrum isdue to the change of γ ′ .In above calculations, the energy density of BLR varieswith time because r ∼ r BLR . In Fig. 3, we show the re-sults for a constant energy density of BLR (correspondingto r ≪ r BLR ) by fixing r = r . In this case the electron dis-tribution above γ ′ min does not vary with time (see top panel MNRAS , 1–8 (2015) ery hard electron spectrum in fast-cooling regime l og ’ N ( ’ ) log ’ L [ e r g s - ] Hz SSC
Figure 5.
Same as Fig. 2, but with γ ′ min = 3 (corresponding tothe slow-cooling scenario) and Q = 10 s − . in Fig. 3). Below γ ′ min , the distribution obviously hardens at ∼ / ǫ BLR . When γ ′ ≪ / ǫ BLR , there is a clear shape N ′ e ( γ ′ ) ∝ γ ′− between γ ′ and γ ′ ∼ N ′ e ( γ ′ ) ∝ γ ′− isformed below γ ′ min . The γ -ray spectrum is also softer thanthat in Fig. 2. For comparison, we revisit the KN effect in the slow-coolingregion (e.g., Dermer & Atoyan 2002; Kusunose & Takahara2005; Georganopoulos et al. 2006). In Figs. 5 and 6, we showthe results in the slow-cooling regime; no very hard electron l og ’ N ( ’ ) log ’ L [ e r g s - ] HzEC-BLR EC-dust
Figure 6.
Same as Fig. 5, but with r = r = 0 . r BLR . spectrum with p < N ′ e ( γ ′ ) ∝ γ ′− ( s +1) ( s = 2 .
1) in the Thomson regime to N ′ e ( γ ′ ) ∝ γ ′− . in the KN regime. This produces a flat-tening in the EC spectrum (see bottom panel in Fig. 6).This situation is very similar to the flattening in EC spec-trum in FSRQs shown by Georganopoulos et al. (2006) andthe flattening in synchrotron X-ray spectrum in extended Chandra jet presented by Dermer & Atoyan (2002). γ -RAY SPECTRUM OF 3C 279 We apply our approach to the very hard spectrum with pho-ton spectral index Γ γ ≃ . γ -ray flare is estimated to be t var ≃ MNRAS , 1–8 (2015)
D. H. Yan, L. Zhang, and S. N. Zhang at t’=0.5 s at t’=2 s at t’=4 s l og ’ N ( ’ ) log ’ at t’=0.5 s at t’=2 s at t’=4 s F [ e r g c m - s - ] [Hz] SSC EC-dust
EC-BLR F [ e r g c m - s - ] [Hz] Figure 7.
Modeling the very hard γ -ray spectrum of 3C 279during 2013 December. The inset in bottom panel shows the de-tails of modelling at gamma-ray energies. The parameters are t ′ inj = 10 s, γ ′ min = 10 , γ ′ cut = 10 , B ′ = 0 . G , δ D = 42, R ′ = 7 . × cm, Q = 2 . × s − , s = 2 . , τ BLR = 0 . τ dust = 0 .
3, and r = 0 . r BLR . R ′ . ct var δ D / (1 + z ). We use L disk =1 . × erg s − (Yan, Zhang & Zhang 2015), then we have r BLR = 1 . × cm. We also assume an inefficient escape, t ′ esc = 10 s .A satisfactory modelling to the very hard γ -ray spec-trum as well as the optical data is seen in Fig. 7; modelparameters can be found in the caption of Fig. 7. No ex-treme parameter is required. The distance where the injec-tion starts is r = 0 . r BLR . The magnetic field is consis- We tested the EED with different t ′ esc and found that the escapeterm is negligible in our model. at t’=0.5 s at t’=0.8 s at t’= s l og ’ N ( ’ ) log ’ at t’=0.5 s at t’=0.8 s at t’= s F [ e r g c m - s - ] [Hz] Figure 8.
Same as Fig. 7, but using t ′ inj = 0 . × s. tent with that in other activities derived by Dermer et al.(2014) and Yan, Zhang & Zhang (2015); Yan et al. (2016)who used a static log-parabola EED to model the SEDsof 3C 279. The comoving frame electron injection power is L ′ inj ≈ × erg s − . The ratio of emitting electron energydensity (at t ′ = 0 . × s) to magnetic energy density is ∼
4. The EED develops a clear form of N ′ e ( γ ′ ) ∝ γ ′− . below γ ′ min when t ′ < s; as the blob moves outside the BLR( t ′ > s) the cooling due to EC-dust becomes relevant,and consequently a softer EED occurs.In Fig. 8, we show the temporal evolutions of the SEDand EED when t ′ > t ′ inj = 0 . × s. It is noted that theoptical emission soon becomes undetectable after stoppingthe injection (see Fig. 8). We note that the EED is quicklynarrowed after stopping the injection. MNRAS , 1–8 (2015) ery hard electron spectrum in fast-cooling regime We have fully investigated the evolution of the EED in thejet of FSRQ using a time-dependent model. We found thata very hard electron spectrum with p ∼ . ∼ γ -ray spectrum of 3C 279 observed in the extreme flare during2013 December. The satisfactory modelling is sensitive tothe γ -ray emission site, and requires the γ -ray emission tak-ing place inside the BLR. External γγ absorption effects areunimportant in the Fermi -LAT spectrum of 3C 279, mea-sured below ≈
10 GeV. Absorption on BLR photons can beimportant above ∼
25 GeV (e.g., Dermer et al. 2014).Our model expects that X-rays lag optical and γ -rayemission, which can be tested by future observations. A morecomplicated injection rate might be needed to reproduce the γ -ray light-curve profile. During the extreme γ -ray flare, theoptical emission showed weak variability (Hayashida et al.2015). The problem of lack of simultaneous optical variabil-ity might be resolved in the fast-cooling regime where theelectrons making optical emission by synchrotron radiationdo not make a substantial contribution to the LAT spec-trum.The impacts of IC scattering in the KN regimeon EED have been investigated by previous studies inthe slow-cooling scenario (e.g., Dermer & Atoyan 2002;Kusunose & Takahara 2005; Georganopoulos et al. 2006;Sikora et al. 2009). For comparison, we also revisited thisscenario. We showed that, in the slow-cooling scenario, theelectron distribution becomes harder at ∼ γ ′ KN with thespectral index from ∼ s + 1 ( s = 2 . ∼ .
5. This moderate hardening in EEDresults in a flat EC-BLR/dust component, which is similarto the finding in Georganopoulos et al. (2006), also see theresults in Dermer & Atoyan (2002). We do not see a dip in γ ′ N ′ e ( γ ′ ) distribution presented by Kusunose & Takahara(2005). We note that Kusunose & Takahara (2005) also ob-tained a similar flat EC component. However, the formationmechanisms for such flat spectrum in the Fermi -LAT energyrange are not unique.The difference between the cooling behaviours in theThomson and KN regimes not only affects the EED/ γ -rayspectrum, but also affects the decay of γ -ray light curve. Inthe KN regime, cooling time is energy-independent, while inthe Thomson regime cooling time is energy-dependent. Thisdifference has been proposed to constrain the γ -ray emissionsite in FSRQs (Dotson et al. 2012, 2015).It is interesting to note that the mean Γ γ for the Fermi -LAT detected FSRQs is 2 . ± .
20 (Ackermann et al. 2015).Analyses of the γ -ray spectra with Γ γ > γ -ray emissionsof 3C 279 take place outside the BLR (Dermer et al. 2014;Paliya 2015; Yan, Zhang & Zhang 2015; Yan et al. 2016).As a last remark, we note that very recentlyUhm & Zhang (2014) and Zhao et al. (2014) showed thata very hard electron spectrum with p ∼ γ - ray burst (GRB) prompt emission spectra whose low-energyphoton spectral index has a value ∼ ACKNOWLEDGMENTS
We thank the referee for a very helpful report. We are grate-ful to Krzysztof Nalewajko for providing us the data of 3C279. We thank Xiang-Yu Wang and Zhuo Li for bringingthe KN effect in GRB study (e.g., Wang et al. 2009) to ourattention when this work is presented at the 8th black holeconference, held in Kunming 14 - 16th October 2015. Thiswork is partially supported by the National Natural ScienceFoundation of China (NSFC 11433004) and Top TalentsProgram of Yunnan Province, China. DHY acknowledgesfunding support by China Postdoctoral Science Foundationunder grant No. 2015M570152, and by the National Nat-ural Science Foundation of China (NSFC) under grant No.11573026, and by Key Laboratory of Astroparticle Physics ofYunnan Province (No. 2015DG035). SNZ acknowledges par-tial funding support by 973 Program of China under grant2014CB845802, by the National Natural Science Foundationof China (NSFC) under grant Nos. 11133002 and 11373036,by the Qianren start-up grant 292012312D1117210, and bythe Strategic Priority Research Program “The Emergenceof Cosmological Structures” of the Chinese Academy of Sci-ences (CAS) under grant No. XDB09000000.
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