A&A 454, L1-L4 (2006)
LETTER TO THE EDITOR
L. Saugé1,2 - G. Henri2
1 - Institut de Physique Nucléaire de Lyon, UCBL/IN2P3-CNRS, 4 rue Enrico-Fermi, 69622 Villeurbanne Cedex, France
2 - Laboratoire d'Astrophysique de Grenoble, Université Joseph-Fourier, BP 53, 38041 Grenoble, France
Received 8 April 2006 / Accepted 7 June 2006
Aims. Most of current models of TeV blazars emission assume a Synchrotron Self-Compton mechanism where relativistic particles emit both synchrotron radiation and Inverse Compton photons. For sake of simplicity, these models usually consider only steady state emission. The spectral features are thus only related to the shape of the particle distribution, and do not depend on the timing of observations.
Methods. In this letter, we study the effect of, firstly, the lag between the beginning of the injection of the fresh particles and the trigger of the observation, and secondly, of a finite injection duration. We illustrate these effects considering an analytical time-dependent model of the synchrotron emission by a monoenergetic distribution of leptons.
Results. We point out that the spectral shape can be in fact very dependent on observational conditions if the particle injection term is time-dependent, particularly taking into account the effect of the time averaging procedure on the final shape of the SED. Consequences on the acceleration process are also discussed.
Key words: acceleration of particles - BL Lacertae objects: individual: Markarian 501 - galaxies: active - radiation mechanisms: nonthermal
Extended observations of TeV blazars like Mrk 501 ( , Quinn et al. 1996) show that synchrotron component can be very variable in the UV/X-ray range. The simplest approach is the well-known homogeneous or one-zone modeling. It considers the emission of a relativistic moving blob of plasma filled by a tangled magnetic field, where ultra-relativistic particle are injected and can cool freely (via both synchrotron and inverse Compton scattering process). Generally, in these models, the acceleration mechanism is not consistently taken into account and the resulting energy distribution function (EDF) of injected particles is prescribed. If we supposed that the acceleration zone could be dissociated from the radiative one, then we must distinguish two different kinds of EDF: the injected one and the cooled one. In the usual one-zone model, the first is directly the result of the acceleration mechanism. The second ensues from the radiative cooling of particles (and possible re-acceleration mechanism). It is mathematically speaking the solution of the standard kinetic equation which the injected EDF is the main source term. The exact shape of the SED could be only understood in the framework of time dependent modeling, depending on the detail of the cooling and injection processes (Chiaberge & Ghisellini 1999; Kirk et al. 1998). The majority of SSC models are steady-state: therefore the instantaneous SED is equal to the observed one. In this case, the authors use broken power-law EDF as the solution of the kinetic equation (characterized by the indices and ). Then the subsequent synchrotron spectra have also a broken power-law shape with indices over suitable frequency ranges (Blumenthal & Gould 1970). Spectral index variations can arise from the variation of i.e. by varying the physical conditions of the emission zone (see e.g. Kino & Takahara 2004).
We have recently investigated another solution, still in the framework of homogeneous modeling, but with a time dependent particle injection term (Saugé & Henri 2004, hereafter Paper I). The kinetic equation is solved numerically taking into account (i) the possible in situ pair reprocessing and (ii) both synchrotron and IC cooling term. The source term of the equation is chosen as a quasi-monenergetic distribution of particles (or "pileup''), injected during a finite time. Physically, the formation of such a distribution results from the combination of a stochastic heating via second order Fermi process and radiative cooling (Henri & Pelletier 1991; Schlickeiser 1985). Then the resulting cooled EDF is also partially a power-law, but with a constant index value over a dynamical range depending on the details of the cooling process. In this case, the instantaneous synchrotron spectrum is a power-law of index and no spectral index variation could be expected at this stage. However, such a variation can be obtained by averaging out the instantaneous SED over the time and considering that the observation starting time does not necessarily coincide with the beginning of injection phase.
In this letter we study in more detail this effect. First of all, we derive a basic model in Sect. 2 in the idealized case of monoenergetic injected EDF subject only to synchrotron cooling. Then we derive the global shape of the time average synchrotron part of the SED as a function of the observational parameters. Finally, we illustrate our approach on BeppoSAX archival data sets of the object Mrk 501 in Sect. 3, before concluding.
|Figure 1: Influence of the value of parameter on the time averaged SED (left panel) with and . Vertical dashed lines represent location of the frequency break calculated for each previous curve (see Eq. (6)). Resulting local spectral indices in normalized frequency range [-0.5,0] (upper solid curve), [-1,-0.5] (lower solid curve) and [-1,0] (dashed curve) are given on the right panel.|
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|Figure 2: Same as Fig. 1 for the influence of the parameter with . Considering these parameter values, we easily verify that spectral index above is going to be flat since .|
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Similarly, Fig. 2 shows the main effects of the increase of the parameter. In the case of , the spectral shape does not evolve. As expected, only the global flux level changes (decreasing linearly with ) just as the low-frequency cut-off (decreasing as maximum system time increases).
|Figure 3: Fits of Markarian 501 UV/X-ray synchrotron spectrum for three different periods.|
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Table 1: Parameters used in our simulations. All times are expressed in unit of
All parameter values needed for the synchrotron part of the SED fits and suitable estimations are given in Table 1. Again, the times are expressed in unit of . For each period, we represent a continuous set of curves as the function of ; also represented, our best estimation of .
First analysis done by Pian et al. (1998) using an one-zone model with a power-law
injection function show that the April 16 spectrum is correctly reproduced for
an injected EDF with an index s=1 over a dynamical range less than
10. In the usual context of shock acceleration process, the value of the index
s case, the index s directly depends on the physical conditions at the
shock (Blandford & Ostriker 1978; Protheroe & Stanev 1999),
In classical shock acceleration model, the difference between the 1997's high and medium state is explained by difference in the shock physical condition - e.g. via the compression ratio R - leading to a different index value s. In the light of our scenario and as already noted in Paper I, the difference in the shape of the spectra arises essentially from different observational conditions. The medium state corresponds to a previous injection observed in a later stage (with respect to beginning of the injection phase). This is corroborated by the flatness of the high energy part of the X-ray spectrum.
In the shock acceleration scenario the value of is related to svia the relation (Kino & Takahara 2004). Then in ourprecise case, it corresponds to an acceleration time quite larger than the escape one ( or, in term of compression ratio ). In our approach, this spectral shape is obtained considering an observation time larger than .
We have exposed a simple time dependent mechanism in order to explain the spectral shape of the synchrotron spectrum of blazar considering the injection of a pure monoenergetic distribution of ultra-relativistic particles over a finite time range. This latter EDF arises from stochastic acceleration mechanism (second order Fermi process). Spectral index and SED shape variations can be explained by the value of the starting observation time with respect to the beginning of the injection of fresh particles and from the variation of the acceleration condition even if the instantaneous synchrotron spectrum is universal with a constant spectral index equals to 1/2. Because blazars are extremely variable sources, there are often observed in "Target-of-Opportunity'' mode, where satellites observations are triggered by some other instruments after some delay leading naturally to . This mechanism leads to complex time dependent behaviors of the spectral shape; we show that we can reproduce various spectral shapes from the single to broken power-law shape, with strong or soft cut-off. This model differs from the usual one in the context of the shock acceleration model, where the spectral variability arises from the physical conditions at the shock.
Remarks of an anonymous referee helped to improve the content of this letter. LS particularly thanks Gaëlle Boudoul for her helpful comments and her interest. All members of the SHERPA team (Grenoble) and the IPNL team of the SNFactory collaboration (Lyon) are also warmly acknowledged. All computations and figures were performed with the free software R (http://www.r-project.org)