X-ray variability patterns in blazars
aa r X i v : . [ a s t r o - ph . H E ] O c t Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018November 13, 2018
X-ray variability patterns in blazars
K. Moraitis and A. Mastichiadis
Department of Physics, University of Athens, Panepistimiopolis, GR 15783 Zografou, GreeceReceived ... / Accepted ...
ABSTRACT
Aims.
We study the expected variability patterns of blazars within the two-zone acceleration model putting special emphasis on flareshapes and spectral lags.
Methods.
We solve semi-analytically the kinetic equations which describe the particle evolution in the acceleration and radiationzone. We then perturb the solutions by introducing Lorentzian variations in its key parameters and examine the flaring behavior of thesystem. We apply the above to the X-ray observations of blazar 1ES 1218 +
304 which exhibited a hard lag behavior during a flaringepisode and discuss possibilities of producing it within the context of our model.
Results.
The steady-state radio to X-rays emission of 1ES 1218 +
304 can be reproduced with parameters which lie well within theones generally accepted from blazar modeling. Additionally, we find that the best way to explain its flaring behavior is by varying therate of particles injected in the acceleration zone.
Key words.
Radiation mechanisms: non-thermal - Shock waves - Galaxies: active - BL Lacertae objects: individual: 1ES 1218 +
1. Introduction
Blazars, a subclass of Active Galactic Nuclei, show variabilitywhich is observed across the electro-magnetic spectrum and can,in some cases, be as fast as a few minutes (Aharonian et al. 2007;Albert et al. 2007). The working blazar scenario is that we ob-serve their radiation coming from a jet which is directed, within asmall angle, to our line-of-sight. The observed spectrum, whichis clearly non-thermal in origin, shows a characteristic double-peak in a ν F ν plot, and is produced from a population of rela-tivistic particles, presumably accelerated by shock waves withinthe jet. While there are still open questions concerning the originof the high energy gamma radiation, there is a consensus that theemission at lower frequencies, which usually extends from radioto X-rays, comes from electron synchrotron radiation. The ob-served variability can then be attributed directly to conditions inthe acceleration region (for a recent review of the various blazarmodels see B¨ottcher 2010).The detailed X-ray observations of many blazars reveal thatthese objects show a rich and complex structure in their flar-ing behavior. Especially interesting are the trends in the lagsbetween the soft and hard X-ray bands. The first observationswhich were detailed enough to show time structure in variousenergy bands (Takahashi et al. 1996) revealed the blazar Mrk421 having a soft lag flare – that is the soft X-rays lagged behindthe hard ones. This was explained as synchrotron cooling: lowerenergy electrons cool with a slower rate than higher energy onesand therefore soft X-rays appear after the harder ones, i.e. softlags hard.Subsequent observations, however, showed cases of the op-posite trend (Takahashi et al. 2000). This could not be explainedby synchrotron cooling, but it was attributed to particle accel-eration. As fresh particles are accelerated to high energies, theyradiate first soft and later hard photons, i.e. hard lags soft.Long, uninterrupted observations (Brinkmann et al. 2005)revealed that Mrk 421 shows both kinds of lags, with typicaltimescales of the order of 10 s. The same mixed behavior was also verified for the same source by Tramacere et al. (2009).Finally, blazar 1ES 1218 +
304 showed a hard lag during a giantflare (Sato et al. 2008).While several time-dependent models have been put for-ward to explain time variations (Mastichiadis & Kirk 1997; Li& Kusunose 2000; B¨ottcher & Chiang 2002), hard lag behav-ior needs a particle acceleration scheme to be taken explicitlyinto account (Kirk et al. 1998; Kusunose et al. 2000). Kirk et al.(1998, hereafter KRM) have shown that both soft and hard lagscan be obtained during a flare once a detailed scheme for particleacceleration is considered. Adopting a two-zone model in whichparticles are energized in an acceleration zone by the first-orderFermi process and then escape and radiate in a radiation zone,they argued that flares produced by an increase in the numberof the injected particles can produce soft lags if the radiatingelectrons are in the energy regime where cooling is faster thanacceleration and hard lags when the two timescales are compa-rable.More recently, two-zone models which include SSC losseshave been developed in Gra ff et al. (2008); Weidinger & Spanier(2010). In particular, Weidinger & Spanier include both first andsecond-order Fermi acceleration in their acceleration zone. Adetailed study of the two types of acceleration and of their ef-fect on SSC spectra, but in an one-zone model, can be found inKatarzy´nski et al. (2006).In the present paper we investigate further the expected vari-ability within the two-zone model by letting its key parametersvary in time. While we keep the same assumptions as in KRM,we add also to the photon spectrum radiation coming from theacceleration zone. In Sect. 2 we present the basic points of themodel and then examine some interesting cases in Sect. 3. InSect. 4 we apply our results to 1ES 1218 +
304 and discuss themin Sect. 5.
1. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars
2. The model
We employ the two-zone acceleration model as developed byKRM. According to this, electrons are accelerated by a non-relativistic shock wave which propagates along a cylindrical jetand then they escape in a wider region behind the shock wherethey radiate the bulk of their energy. Following the KRM nota-tion we shall refer to the region around the advancing shock asthe ’acceleration zone’ (AZ) and to the escape region as the ’ra-diation zone’ (RZ). The total size of the source is assumed tohave a finite extent.
The evolution of the electron distribution function (EDF) in theAZ, N ( γ, t ), is given by the continuity equation ∂ N ∂ t + ∂∂γ ( N ˙ γ ) + Nt esc = Q δ ( γ − γ inj ) . (1)The energy of each particle in this region changes with the rate˙ γ = γ t acc − αγ , (2)where the first term describes first order Fermi-type accelera-tion with the energy independent timescale t acc and the seconddescribes the energy losses due to synchrotron radiation in themagnetic field B with α = σ T B π m e c . (3)Particles are injected in the AZ at some low energy γ inj m e c withthe rate Q and escape from it to the RZ with the energy indepen-dent rate t − .In our model we assume that variability can be produced bythe change of one or more of the parameters t acc , t esc , Q and B in the AZ and thus treat them as time-dependent in Eq. (1).We define the dimensionless functions that include the time-dependency through the relations t acc ( t ) = t a , f a ( t ), t esc ( t ) = t e , f e ( t ), Q ( t ) = Q f q ( t ) and B ( t ) = B f B ( t ), where t a , , t e , , Q and B are the unperturbed values of the parameters.The solution of Eq. (1) can be obtained semi-analyticallywith the method of characteristics. This method requires the so-lution of the initial value problem d γ dt = ˙ γ ( γ, t ) , γ ( t ∗ ) = γ inj , (4)which can be written in the form ϕ a ( t ) γ − ψ ( t ) γ max = ϕ a ( t ∗ ) γ inj − ψ ( t ∗ ) γ max , (5)where γ max = ( α t a , ) − (6)and the functions ϕ a , ψ are given by the relations ϕ a ( t ) = exp Z t / t a , dt ′ f a ( t a , t ′ ) ! (7)and ψ ( t ) = Z t / t a , dt ′ f ( t a , t ′ ) ϕ a ( t a , t ′ ) . (8) The solution of Eq. (1) can then be written in the genericclosed form N ( γ, t ) = Q ( t ∗ ) γ γ ˙ γ ( γ inj , t ∗ ) ϕ a ( t ) ϕ a ( t ∗ ) ϕ e ( t ∗ ) ϕ e ( t ) ! s − S ( γ ; γ inj , γ ( t )) (9)where s = + t a , / t e , , S ( x ; a , b ) = a ≤ x ≤ b and zero oth-erwise, and the function ϕ e is given by Eq. (7) with f a replacedby f e . The quantity t ∗ = t ∗ ( γ, t ) in the case where both the ac-celeration timescale and the magnetic field are time independent( f a = f B =
1) has the analytic expression t ∗ = t − t a , ln γ max /γ inj − γ max /γ − ! (10)while in every other case it is obtained from the solution of theimplicit Eq. (5). The upper limit of the EDF is obtained fromEq. (5) for t ∗ = γ ( t ) = γ inj ϕ a ( t )1 + γ inj γ max ψ ( t ) . (11)In the case where all parameters are time-independent, theunperturbed distribution function of the AZ is N ( γ, t ) = Q t a , γ inj − γ max ! − s γ − s − γγ max ! s − S ( γ ; γ inj , γ ( t )) (12)and its upper limit is γ ( t ) = γ inj e t / t a , + γ inj γ max (e t / t a , − , (13)relation valid whenever f a = f B =
1. The time it takes for steady-state to establish is a few tens t a , and then the EDF is a power-law of index s up to the maximum Lorentz factor γ max (Eq. 6)where acceleration balances energy losses. The escaping particles enter the radiation zone where they ra-diate a part of their energy in the constant magnetic field B .Assuming that synchrotron losses dominate, we can write anequation for the evolution of their di ff erential density n ( γ, x , t )in the RZ ∂ n ∂ t − ∂∂γ (cid:16) α γ n (cid:17) = N ( γ, t ) t esc δ ( x − x sh ( t )) , (14)where x sh ( t ) = u sh t is the position of the shock and u sh is theshock speed. The solution of Eq. (14) is easily found n ( γ, x , t ) = N ( γ ∗ , x / u sh ) u sh t esc ( x / u sh ) γ ∗ γ ! S ( x ; 0 , x sh ( t )) (15)with γ ∗ = γ − α ( t − x / u sh ) ! − . (16)Following KRM, we impose the additional restriction x > x sh ( t ) − L on the spatial limits of Eq. (15), in order to take into ac-count the finite size of the RZ, L . This is better expressed throughthe time the shock needs to travel the RZ t b = L / u sh . (17)
2. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars
Particles not fulfilling this restriction are assumed to escape theRZ in a region of practically zero magnetic field and thus donot contribute to the total radiation. The EDF in the RZ is thenobtained by spatially integrating the function n ( γ, x , t ).In the case where all parameters are time-independent, thesteady-state distribution in the RZ is a broken power-law hav-ing the same energy limits as the EDF in the AZ. The power-law has an index s for energies less than the breaking energy γ br ≃ γ max t a , / t b and ( s +
1) for γ > γ br . The break is formedbecause of the finite size of the source, since the particles with γ < γ br leave the source before they cool. Radiation from theseparticles is ignored because of the assumption that the magneticfield declines substantially outside the source. The time it needsthe EDF in the RZ to reach steady-state is of the order of t b . Knowing the electron distribution functions in both zones, it iseasy to compute the emitted synchrotron spectrum in the sourceframe through the relation I ν ( t ) = I AZ ν ( t ) + I RZ ν ( t ) = Z ∞ d γ N ( γ, t ) I γ ( ν ) + Z ∞ d γ I γ ( ν ) Z ∞ dxn ( γ, x , t ) (18)where I γ ( ν ) is the single particle emissivity for synchrotron radi-ation that is given by I γ ( ν ) = √ e Bm e c F ν γ ν ! , (19)with F ( x ) = x K / ( x ) K / ( x ) − x (cid:16) K / ( x ) − K / ( x ) (cid:17)! (20)(Crusius & Schlickeiser 1986) and the characteristic frequencyis ν = π eBm e c . (21)In Eq. (18) one can see the two di ff erences that our approachhas with the one adopted by KRM. The first is that we calculateradiation from both zones, while KRM ignore the contributionfrom the AZ. The second di ff erence is that we do not treat light-travel e ff ects inside the source. This assumption is equivalent totaking u sh ≪ c and in that case the shock speed is no longer aparameter of the problem. Our model thus considers the case ofnon-relativistic shocks. In the opposite case, light-travel e ff ectscan be important depending on the frequency (see the discus-sion in KRM) and then the intrinsic variations are smoothed outon the light-crossing timescale. A study of these phenomena inthe internal-shock model for blazars can be found in B¨ottcher &Dermer (2010).
3. Flare profiles
The model that we presented in the previous section can producea wide variety of flaring behaviors by varying one or more of itsbasic parameters. The temporal behavior of these is essentiallyanother free parameter as no theory can provide this information.While in many applications a short duration impulsive changeis assumed (Mastichiadis & Kirk 1997, KRM), here we adopt
13 14 15 16 17 18 1940.440.841.241.642.0 g fe d cba f l l og I [ e r g s - ] log [Hz] Fig. 1.
Snapshots of the synchrotron spectrum in the source frame. Thecurve labeled ’a’ is the steady-state spectrum, while labels ’b’ to ’g’ arethe snapshots at t = t + kt a , for k = , ..., ν f , ν l ) for which the light-curves ofFig. 2 are calculated. the Lorentz profile for the changes which has a pulse shape andanalytic expression f L ( t ; t , w , n ) = + ( n − w t − t ) + w . (22)At t = t the pulse shows a maximum or a minimum, i.e. f L ( t ; t , w , n ) = n , depending on whether n > n < | t − t | ≫ w / f L =
1. The quantity w is thefull width at half maximum (or minimum, FWHM) of the pulse,since f L ( t ± w ; t , w , n ) = ( n + / A change in the injection rate can be attributed to the encounterof the shock with a higher or lower density area. The increase ofthe injection rate leads to an increase in the overall normalizationof the EDFs and thus to the production of a flare. The case of astep-function change was examined in KRM. Here we study itfor a Lorentz type change in more detail.A few snapshots of the synchrotron spectrum when the injec-tion rate varies according to f q ( t ) = f L ( t ; t , w , n ) with w = t a , , n = t ≫ t b – so that steady-state has been reached longbefore t = t – are shown in Fig. 1. For the steady-state spec-trum, the set of parameters γ inj = B = . t b = t a , , Q = s − and t a , , t e , such that γ max = and s = . ν max = γ ν , is then a few keV. The correspondinglight-curves are shown in Fig. 2.It is evident from Fig. 2 that the form of the flares is a func-tion of frequency. In order to quantify this, we define for eachflare the following parameters. If y ( t ) = I ν ( t ) / I ν (0) is the form ofthe light-curve normalized to the steady-state luminosity, I ν (0),then we define the time at which the flare peaks t pk and its peakvalue y pk ≡ y ( t pk ). We also define the flux doubling time duringthe rise of the flare, t r , and the flux halving time during the decay, t d , through the relations y ( t pk − t r ) = y ( t pk + t d ) = (1 + y pk ) /
2. Theratio t r / t d is then a measure of the time symmetry of the flare
3. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars -4 0 4 8 12 16 20 24 28 321.01.21.41.61.82.02.22.4 f l I( t ) / I( ) (t-t )/t a,0 Fig. 2.
Flares that are produced by a Lorentzian change of the injectionrate in the frequency range ν f = − ν max ≤ ν ≤ . ν max = ν l with step10 . ν max (from bottom to top, the first and the last curve are labeledwith their corresponding frequency). and w fl = t r + t d is the FWHM of the flare. We should note herethat we do not fit each flare with a specific function, but ratherwe calculate the above quantities independently from the exactform of the flares.In Fig. 3 we plot the quantities t pk , t r / t d , w fl and y pk as func-tions of frequency. In the first plot for ν . . ν max the timeat which the flare peaks decreases with frequency, while for ν & . ν max it increases. This means that for ν . . ν max the low-frequency flares precede the high-frequency ones, or, inother words, the flares show a ’soft-lag’. The opposite happensfor ν & . ν max where the flares show a ’hard-lag’.As noted in KRM, who also find this result, soft-lags canbe attributed to electrons for which the acceleration timescale ismuch faster than the cooling timescale, therefore for all prac-tical reasons these can be considered as pre-accelerated or in-jected at high energies with a ready power-law (see, for exampleMastichiadis & Kirk 1997). On the other hand, hard-lags are ob-served at higher frequencies where the two timescales are com-parable, therefore the observer sees the wave of freshly acceler-ated particles moving to high energies.Further insight can be gained by considering the exactchanges which the electron populations experience. The distri-bution function of the AZ can be written in this case as N ( γ, t ) = N ( γ, t ) f L ( t ∗ ; t , w , n ) (23)with t ∗ given by Eq. (10). This result indicates that the initialpulse propagates in the EDF from low to high energies, since itspeak is located at γ = γ ( t − t ), as can be easily found by setting t ∗ = t . Moreover, this moving pulse preserves the shape of theinitial pulse and thus it is symmetric around its center and hasan amplitude equal to n . So, if we consider radiation only fromthe AZ, we expect to get roughly symmetric flares of the sameamplitude which exhibit hard-lags.In the radiation zone the situation is more complicated, sincethe EDF there results from the integration of Eq. (15) along theextent of the source. The hard-lag behavior of the AZ is seen herealso, but only in high energies, where the contribution from theparticles residing just before the shock dominates in the integral. -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.01.21.62.02.4 log( / max ) y pk w f l / w t acc /t cool t r /t d hard-lagsoft-lag ( t pk - t ) /t a , Fig. 3.
Characteristics of the flares shown in Fig. 2.
In lower energies the perturbation propagates towards the oppo-site direction, as the intrinsic variations of N ( γ, t ) are smoothedout by the spatial integration and only the e ff ect of the fadingpulse in the AZ stands out.In the second and third plot of Fig. 3 one sees that the widthof the flares decreases with frequency, while the ratio t r / t d in-creases. This means that the flares become narrower and moresymmetric as the frequency increases. If the rise time of theflares was connected with the acceleration timescale and the de-cay with the cooling timescale, then the ratio t r / t d will be pro-portional to t acc t cool = t a , ( α γ ) − = γγ max ≃ νν max ! / . (24)The plot of this curve in the t r / t d diagram reveals that this isnot the case and thus the relation of t r , t d to the acceleration andcooling timescales is more complicated.Finally, in the last plot of Fig. 3 one sees that the amplitude ofthe flares increases with frequency until it saturates to a constantvalue which is somewhat lower than the amplitude of the elec-tron variation. The level of saturation depends on the length ofthe pulse, because the longer the duration of the burst becomes,the closer the EDF gets to a steady state. As a next case we examine a perturbation in the escapetimescale. More precisely we assume that some physical pro-cess impedes momentarily particle escape, i.e. decreases the rateat which particles escape from the acceleration region. An in-spection of the EDF (Eq. 12) shows that an increase in the escape
4. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars -12 -8 -4 0 4 8 12 16 20 24 28 321.01.21.41.61.82.02.22.42.6 lf I( t ) / I( ) (t-t )/t a,0 Fig. 4.
Flares that are produced by a Lorentzian change of the escapetimescale in the frequency range ν f = − ν max ≤ ν ≤ . ν max = ν l with step 10 . ν max (from bottom to top, the first and the last curve arelabeled with their corresponding frequency). timescale leads to a decrease in the electron index. This will pro-duce a flare, the form of which, at various frequencies, is shownin Fig. 4. We assume that the change of the escape timescale fol-lows again a Lorentzian variation, i.e. f e ( t ) = f L ( t ; t , w , n ) with w = t a , , n = / t ≫ t b , while the parameters of thesteady-state spectrum are the same as before.The specific features of the flares are shown in Fig. 5. As be-fore the flares exhibit a soft-lag in low frequencies and a hard-lagclose to the maximum frequency. The frequency range in whichhard-lags are observed is much smaller than in the previous caseand also the flares now peak closer to the peak of the perturba-tion pulse, i.e. closer to t = t .The shape of the flares is asymmetric in time at low fre-quencies, having a rise time which is shorter than the decay one(second plot of Fig. 5). This however becomes symmetric, i.e. t r ≃ t d , at higher frequencies. The rising part of the curve indi-cates that t r / t d is related to t acc / t cool and thus the rise of the flaresis controlled by the acceleration and the decay by the coolingtimescale.The width of the low-frequency flares decreases as in theprevious case but in higher frequencies it increases. We note,however, that w fl ≫ w , i.e. even a narrow pulse in the escapetimescale produces a rather broad X-ray flare. The amplitude ofthe flares increases with frequency as in the previous case, butnow it does not saturate to a maximum value.The EDF of the AZ, Eq. (9), reads in this case N ( γ, t ) = N ( γ, t ) exp " w ( n − s − √ n × tan − t − t w √ n ! − tan − t ∗ − t w √ n !! (25)where t ∗ is given by Eq. (10). The perturbation in the EDF prop-agates again from low to high energies but is no longer pulse-shaped and has a much broader extent. Moreover, the perturba-tion moves twice as fast now, since the location of its peak isgiven by γ = γ (2( t − t )). One notices that, qualitatively speak-ing, the trends in the present case are similar to the previous one. -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.01.21.62.02.4 log( / max ) y pk w f l / w t acc /t cool t r /t d ( t pk - t ) /t a , Fig. 5.
Characteristics of the flares shown in Fig. 4.
There is a simple explanation for this. The perturbation intro-duced causes the escape timescale to increase first and then de-crease back to its original value. This in turn causes the electronspectral index to flatten and then steepen again. Therefore, thenumber of accelerated particles increases and then decreases toits unperturbed value. Thus this case can be considered as equiv-alent to the previous one which treated variations in the injectionof particles at low energies. Note that, in the present case, theflattening of the EDF is partly compensated from the fact that, atthe same time, the particles decrease in the RZ (see Eq. 15).
As a next example we investigate changes in the accelerationtimescale and more specifically the variations induced when thisdecreases. This change has two e ff ects on the EDFs and, there-fore, to the produced spectrum. First, an inspection of the EDF(Eq. 12) reveals that if the acceleration timescale decreases, thenthe electron index becomes flatter. Second, the maximum elec-tron energy will increase – see Eq. (6). The flares which are pro-duced for a Lorentzian change with w = t a , and n = / γ max in the EDFs leads to the simultaneous increase of the maximumemitted frequency in the spectrum. This produces a large flare athigh frequencies, since during pre-flare this part of the spectrumwas in the exponentially decaying synchrotron regime.
5. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars -10 -5 0 5 10 15 20123456 l f I( t ) / I( ) (t-t )/t a,0 Fig. 6.
Flares that are produced by a Lorentzian change of the accelera-tion timescale in the frequency range ν f = − . ν max ≤ ν ≤ . ν max = ν l with step 10 . ν max (from bottom to top, the first and the last curve arelabeled with their corresponding frequency). As a last case we examine variations of the magnetic fieldstrength which are assumed to occur in the AZ only. A decreasein the magnetic field leads to an increase of the maximum elec-tron energy. Note that the maximum photon frequency increasesalso, even if B decreases and this results to a production of aflare. The form of the flares which are produced for a Lorentzianchange of the magnetic field with w = t a , and n = . γ max which compensates for this reduction.The flares exhibit only soft-lags for a similar reason to thatof the previous case. The flares also become narrower and moresymmetric with frequency. In total, this case has many similari-ties with the change of the acceleration timescale, since in bothcases the maximum electron energy follows the same temporalbehavior.
4. Application to 1ES 1218+304 +
304 is a high-frequency peaked BL Lac object at aredshift z = . +
304 is shown inFig. 8 together with the steady-state spectrum of our model asthis is given in Eq. (18). The transformation to the observer’s -10 -5 0 5 10 151.01.21.41.61.82.0 l f I( t ) / I( ) (t-t )/t a,0 Fig. 7.
Flares that are produced by a Lorentzian change of the magneticfield in the frequency range ν f = − . ν max ≤ ν ≤ . ν max = ν l withstep 10 . ν max (from bottom to top, the first and the last curve are labeledwith their corresponding frequency). frame is made through the relations F obs ν = δ (1 + z ) I ν π d (26) ν obs = δ + z ν (27)where δ is the Doppler factor and d L =
880 Mpc is the lu-minosity distance of 1ES 1218 + Ω Λ = . Ω m = . H =
70 km s − Mpc − .The maximum photon frequency in the spectrum is ν obsmax ≃ . Hz, leading to the relation δ B γ ≃ G . (28)We choose a random combination of these parameters so thatEq. (28) is satisfied and also take γ inj = t b = t a , and s = .
8. The value of t b is dictated by the frequency where thespectrum appears to break, ν obsbr ≃ Hz, while the value ofthe electron index is chosen so that the photon index in radio fre-quencies is α r ≃ .
6. The parameter γ inj has a small e ff ect in thenormalization of the spectrum and thus its value is unimportant.The remaining parameters can be further constrained by thevariability timescale of 1ES 1218 + t pk ( ε )given in Sato et al. for the time at which the flares peak, wededuce that the variation responsible for producing these flaresis the change in the injection rate. This is because the energyrange in which t pk is rising is quite broad and rules out the caseof the escape timescale change. As in Sato et al. we define theamount of hard-lag simply by the di ff erence of the peak times ofthe flares in the various energy bands from the peak time of theflare in the highest energy band, namely τ hard ( ε ) = t pk ( ε max ) − t pk ( ε ) , (29)where ε max = √
50 keV is the logarithmic mean energy ofthe 5 −
10 keV energy band. The theoretical curve for τ hard ( ε )for the Lorentzian change of the injection rate with parameters w = . t a , and n = .
4, is shown in the upper panel of Fig. 9
6. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars
10 12 14 16 18 20-15-14-13-12-11 l og F [ e r g c m - s - ] log [Hz] Fig. 8.
Radio to X-rays spectrum of 1ES 1218 +
304 (starred symbols,data taken from Sato et al. 2008) together with the steady-state theoret-ical spectrum (solid line) for the parameters given in text. The contri-butions from the acceleration zone (dashed line) and from the radiationzone (dotted line) are also shown. The optical data can be attributed tothe host galaxy of 1ES 1218 +
304 (R¨uger et al. 2010) and are thus notincluded in the fit. together with the observational data. From the fit we estimatethe acceleration timescale t a , = . δ s, where δ = δ/ +
304 as B = . δ − / G and γ max = . δ − / .Since the Doppler factor cannot be determined, we choose δ = Q = . s − . With these parameters we can estimate thesize of the source L = . ( u sh / . c ) cm and the energycontent of the source in electrons as E e = erg.In the lower panel of Fig. 9 we plot the theoretical curve ofthe asymmetry of the flares, t r / t d , together with the observationaldata. Our definition of t r and t d di ff ers from the one in Sato et al.but the ratio of the two quantities is the same in both cases. Onesees that we reproduce the general trend of the data quite well butwe overestimate the ratio t r / t d in the energy interval 1 −
5. Conclusions
In the present paper we have examined the expected variabilityof blazars in the context of the two-zone acceleration model. Byvarying four of the key parameters of the model, we found that inall cases flaring can be induced. In all cases the simulated flaresexhibit soft lags – only in two cases, i.e. when varying the injec-tion rate of particles or their escape rate, one can find frequencyregimes where hard lags appear. Thus, we find that this schemecannot produce hard lags at frequencies much lower than themaximum synchrotron frequency. Another conclusion is that theflares tend to mimic the electron variations at high frequencies.As a rule at low frequencies flares tend to be asymmetric withrise timescales much faster than decay ones.We were able also to make an acceptable fit to the hard lagflare of 1ES 1218 +
304 (Sato et al. 2008) by varying the rate ofparticles injected into the acceleration process. We note that thefit can be improved if we were to vary two parameters instead ofone but this lies outside of the aims of the present paper. The fit- -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.20.30.40.50.60.70.80.91.0 log obs [keV] t r /t d l og h a r d [ s ] Fig. 9.
Amount of hard-lag and the ratio t r / t d of 1ES 1218 +
304 in thevarious X-ray energy bands together with the corresponding theoreticalcurves. ting we obtained gives parameters well within the ones acceptedfrom blazar modeling. Note that the acceleration timescale de-duced ( t a , ≃ δ s) might be long for first order Fermi accel-eration (Tammi & Du ff y 2009), however we have to emphasizethat from the setup of the model, both acceleration and escapetimescales were assumed to be independent of the energy γ andthe magnetic field strength B , therefore we do not try to connectthem directly with theories of particle acceleration (Drury 1983;Blandford & Eichler 1987).We have restricted our analysis to the cases where syn-chrotron losses dominate and did not consider the inverseCompton scattering as a possible electron energy loss mecha-nism. This can be justified for sources where the energy den-sity due to magnetic fields exceeds the one due to photons. Inthis case the variability patterns due to inverse Compton scatter-ing will, in general, follow the ones due to synchrotron (Kirk& Mastichiadis 1999). On the other hand, an example of amodel dealing with X- and gamma-ray variability where in-verse Compton losses might be of importance can be found inMastichiadis & Moraitis (2008) – note, however, that this is anone-zone acceleration model.Our model, despite its limitations, is capable of producinga wide variety of flaring behaviors that could, in principle, betested against the growing X-ray data on blazar variability. Wenote that recently, Garson et al. (2010) have also attributed the
7. Moraitis and A. Mastichiadis: X-ray variability patterns in blazars properties of X-ray flares on intrinsic changes of the accelerationprocess.
Acknowledgements.
The authors would like to thank the anonymous referee forhis / her comments that helped improve the manuscript. KM acknowledges finan-cial support from the Greek State Scholarships Foundation (IKY). References
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