A&A 448, 861-871 (2006)
DOI: 10.1051/0004-6361:20053644
E. Massaro^{1} - A. Tramacere^{1} - M. Perri^{2} - P. Giommi^{2} - G. Tosti^{3}
1 - Dipartimento di Fisica, Università La Sapienza, Piazzale A. Moro 2,
00185 Roma, Italy
2 - ASI Science Data Center, ESRIN, 00044 Frascati, Italy
3 - Dipartimento di Fisica, Università di Perugia, via A. Pascoli,
Perugia, Italy
Received 16 June 2005 / Accepted 11 October 2005
Abstract
Curved broad-band spectral distributions of non-thermal sources like
blazars are described well by a log-parabolic law where the second
degree term measures the curvature.
Log-parabolic energy spectra can be obtained for relativistic electrons
by means of a statistical acceleration mechanism whose probability of
acceleration depends on energy.
In this paper we compute the spectra radiated by an electron population via synchrotron
and Synchro-Self Compton processes to derive the relations between
the log-parabolic parameters.
These spectra were obtained by means of an accurate numerical code
that takes the proper spectral distributions for single particle
emission into account.
We found that the ratio between the curvature parameters of the synchrotron spectrum
to that of the electrons is equal to 0.2 instead of 0.25, the value foreseen
in the -approximation.
Inverse Compton spectra are also intrinsically curved and can be approximated
by a log-parabola only in limited ranges.
The curvature parameter, estimated around the SED peak, may vary from a lower value than
that of the synchrotron spectrum up to that of emitting electrons depending on
whether the scattering is in the Thomson or in the Klein-Nishina regime.
We applied this analysis to computing the synchro-self Compton emission from the BL Lac
object Mkn 501 during the large flare of April 1997. We fit simultaneous BeppoSAX
and CAT data and reproduced intensities and spectral curvatures of both components
with good accuracy.
The large curvature observed in the TeV range was found to be mainly
intrinsic, and therefore did not require a large pair production absorption against the
extragalactic background. We regard this finding as an indication that the Universe is
more transparent at these energies than previously assumed by several models found
in the literature.
This conclusion is supported by recent detection of two relatively
high redshift blazars with HESS.
Key words: radiation mechanisms: non-thermal - galaxies: active - BL Lacertae objects: general - BL Lacertae objects: individual: Mkn 501
For the so-called Low-energy peaked BL Lac (LBL) objects, the frequency of the first peak is in the Infrared-Optical region, while it is in the UV-X ray range for the High-energy peaked BL Lac (HBL) (Padovani & Giommi 1995). The shape of these bumps is characterised in the Log vs. plot by a rather smooth curvature extending through several frequency decades. A very simple and successful analytical function that can model the shape of these broad peaks is a parabola in the logarithms of the variables (hereafter log-parabola). This function has only three spectral parameters, and was used by Landau et al. (1986) to fit the broad band spectra of some bright BL Lac objects from radio to UV. Sambruna et al. (1996) also used a log-parabola to fit blazars' SED to estimate peak frequencies and luminosities without, however, attributing a physical meaning to this shape. In two previous papers (Massaro et al. 2004a,b, hereafter Paper I and Paper II, respectively) we used the log-parabolic model to fit the BeppoSAX broad band X-ray spectra of the BL Lac objects Mkn 421 and Mkn 501 and studied the relations between the spectral parameters and the luminosity. We also showed that, under simple approximations, a log-parabolic synchrotron spectrum can be obtained by a relativistic electron population having a similar energy distribution, and then derived the main relations between their parameters. Furthermore, in Paper I we proposed a simple explanation for the log-parabolic energy distribution of the electrons as resulting from a statistical mechanism where the acceleration probability decreases with the particle energy.
Curved spectra in non-thermal sources have already been studied in the past and generally were related to the radiative ageing of the emitting electrons that have a single power law injection spectrum. The well-known solution by Kardashev (1962) predicts a change in the electron spectral index by a unity (0.5 for the synchrotron spectrum) around a break energy, whose value decreases with time. Probably for this reason, curved spectra have been modelled by means of rather complex functions like a double broken power law (Kataoka et al. 1999) or a continuous combination of two power laws after releasing the condition on the difference of spectral indices (see the discussion in Fossati et al. 2000). A similar spectrum is generally used in the spectral modelling of gamma-ray bursts (Band et al. 1993). On the same basis, Sohn et al. (2003) more recently introduced a spectral-curvature parameter, defined as the difference between two spectral indices in different frequency intervals, to study the evolution of synchrotron sources. As we will show later, this parameter is strongly dependent on the chosen frequency range and cannot be univocally related to the actual curved shapes.
In this paper we stress the advantages of using a log-parabolic law to model curved spectra over broad frequency ranges and present the results of an accurate study of the spectral distributions of the synchrotron radiation (hereafter SR) and of the inverse Compton (hereafter IC) scattered photons by a population of relativistic electrons with such energy spectrum. We study, in particular, the relations between the parameters of SR spectra and those of the electrons and show that they are generally approximated well by simple power behaviours and that they provide useful information on the electron spectrum. The relations with the IC spectrum are not so simple depending upon whether the majority of scatterings is in the Thomson or in the Klein-Nishina regime. In any case, our calculations indicate a good way to properly model the observed spectra. We also consider the case in which the probability of statistical acceleration is constant below a critical energy, thus producing a power law spectrum that turns into a log-parabola above this energy.
In the last part of the paper, we apply our model to the large flare from Mkn 501 observed in April 1997. This is a unique event in which the SR and IC components were observed simultaneously over energy bands that were broad enough to study the spectral curvature. We use the BeppoSAX data already analysed in Paper II together with those in the TeV band obtained by the CAT experiment (Djannati-Atai et al. 1999). Our approach of studying both the SR and IC spectral curvatures is useful for evaluating the relevance of the intergalactic absorption due to pair production from photon-photon interactions. In particular, we discuss the possibility that the observed TeV curvature is mainly intrinsic, as it implies rather low absorption and the possibility of detecting sources at larger distances than usually estimated.
The log-parabolic model is one of the simplest ways to represent curved
spectra when these show mild and nearly symmetric curvature around the
maximum instead of a sharp high-energy cut-off like that of an exponential.
In the following, we summarise the main properties of log-parabolic
spectra, while for details we refer to
Papers I and II. This law has only one more parameter than a simple power law
and can be written as:
(1) |
(2) |
(3) |
(4) |
(3') |
(5) |
One limit of the log-parabolic model, however, is that it can represent
only distributions symmetrically decreasing with respect to the
peak frequency.
However, it is not difficult to modify Eq. (1) to take into account a possible
asymmetry with respect to .
For instance, one can use two different values of b for energies lower
and higher than ,
where the two branches continuously match.
An interesting possibility, which can be simply explained in terms of
the energy dependence of the particle acceleration probability (see next
section), is that the low energy segment of the spectrum follows a single
power law with photon index
and that the log-parabolic
bending becomes apparent only above a critical value .
This behaviour can be described by the following model that takes into
account the continuity conditions on the flux and on
at :
F(E) | = | ||
F(E) | = | ||
(6) |
The energy spectrum of particles accelerated by some statistical mechanism,
e.g. a shock wave or a strong perturbation moving down a jet, is given by
a power law (Bell 1978; Blandford & Ostriker 1978; Michel 1981).
In Paper I we showed that a log-parabolic
energy spectrum is obtained when the condition that the acceleration probability
p is independent of energy is released and that its value
at the step i satisfies a power law relation as:
(7) |
(8) |
(9) |
(10) |
Shock wave acceleration is not the only statistical mechanism active in non-thermal sources. For instance, electron acceleration can occur in magnetohydrodynamical turbulence in which regions of magnetic field reconnection can develop in a very stochastic way. Recently, Nodes et al. (2004) presented the results of the numerical simulations of a relativistic particle acceleration in a three-dimensional turbulent electromagnetic field configuration, also taking their SR into account. These authors found energy spectral distributions that where significantly flatter than s=2 and, in a few cases, characterised by a steepening spectral index at high energies. We verified that over sufficiently wide energy ranges, the spectra given by Nodes et al. (2004) are represented well by a log-parabolic law or by a combination of a power law and a parabola. The resulting curvature parameters are generally small, but probably the curvature depends on the distribution and size of the acceleration regions, so one can expect that, under different assumptions, it could be higher. Nodes et al. (2004) computed the emerging SR spectra, which show an appreciable curvature over a few decade frequency range.
Statistical acceleration is not the only way to obtain curved electron spectra. Energy distributions showing a rather mild curvature have also been obtained in blazar physics. To model the SED of MeV blazars, Sikora et al. (2002) assumed that electrons are accelerated via a two-step process with a broken power-law energy distribution as injection. When the cooling effects are taken into account, the resulting electron spectrum (see, for instance, Fig. 6 in their paper) can be described well by a log-parabola over a range that is wider than three decades, as we verified (in this case we found r=0.44).
The assumption of Eq. (7) about the energy dependence of the
acceleration probability can be modified to take other
escape processes into account.
For instance, one can assume that the acceleration probability is constant
for low energies and that it begins to decrease above a critical Lorentz factor
(see, for instance Eq. (23) in Paper I).
An approximate expression for the energy distribution of accelerated
particles under this condition
for
should follow a power law with spectral
index
,
while for
it will approximate a log-parabolic spectrum like Eq. (8):
= | |||
= | (11) |
It is important to know the relations between the spectral parameters
a and b of the emitted radiation and those of the electron population,
namely s and r.
The spectral distribution
of the SR by relativistic electrons having a log-parabolic energy
distribution cannot be computed analitically.
For our purpose, however, the relations between the spectral parameters
can be derived under the usual -approximation and the
assumption that the electrons are isotropically distributed in a
homogeneous randomly oriented magnetic field with an average
intensity B
(12) |
(13) |
(14) |
(15) |
a | = | (s-1)/2 | |
b | = | r/4 . | (16) |
The relation between r and b given above (Eq. (16)) is not exact: one can expect that a precise calculation of the spectral curvature of the emitted radiation must give a b value smaller than in the approximation and that, for high values of r, b can be greater than unity (depending on the frequency interval used around the peak in which it is estimated) because of the exponential cutoff by the SR spectrum radiated by a single particle.
To compute the relations between the spectral parameters numerically
and to study the spectral evolution of the emitting particles,
as reported in the next subsection, we applied a time-dependent
numerical code (Tramacere & Tosti 2003; see
Tramacere 2002, for a detailed description of the code).
This code uses a numerical
integration routine that applies a modified Simpson rule ensuring
good convergence in short computing times.
In particular, this code includes an accurate evaluation of the synchrotron
frequency distribution (hereafter SFD):
(17) |
Figure 1: Synchrotron spectra emitted by an electron population with log-parabolic (dashed lines) and power-law log-parabolic energy distributions (Eq. (11)) (solid lines), computed for r values in the interval 0.50-1.20. Spectra were shifted on the vertical scale to avoidconfusion. | |
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We calculated the SR spectra and SEDs from Eq. (12) using a log-parabolic distribution without (LP) and with a low-energy power law branch (LPPL) (Eq. (11)) and the frequency distribution of Eq. (17). All the calculations in this and in the following subsections are performed in the co-moving frame, so relativistic beaming effects were not included.
Figure 1 shows some examples of the resulting SEDs. In these computations we assumed a uniform magnetic field B=0.14 G, and the spectral parameters for the electrons were , s=s_{0}=1.2 (Eq. (11)), while r values were increased from 0.5 to 1.20 by steps of 0.05. The electron density and the size of the emission region were taken to be equal to the typical values used in blazar modelling. In these conditions the synchrotron self-absorption is important at frequencies lower than 10^{10} Hz. Note in Fig. 1 that at low frequencies both spectral sets tend to a power law: in the LP case the spectral index corresponds to the SFD asymptotic behaviour (), while in the other case it has the energy spectral index . At high frequencies the log parabolic behaviour is very evident with a peak frequency decreasing for higher values of r, as expected from Eq. (3) since b is also increasing with r.
We first verified that the peak frequency in the SED of the SR component
is proportional to the Lorentz factor
at which
the electron distribution
has its maximum.
Applying the same formulae of Sect. 2, we have
(18) |
Figure 2: The relation between the peak frequencies of the synchrotron SED and those of the energy spectrum of the emitting electrons computed using the SFD (Eq. (17)) for the LP and LPPL cases (see text). The abscissa is the inverse of the electron spectrum curvature to show the linear trend expected from Eq. (18). The dashed line is the best fit computed for 1/r >0.33. Note the small deviations of LPPL points from the best fit line at low values of 1/r. | |
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Figure 3: The relation between the log-parabolic curvature parameters of the electron energy distribution r and those of the synchrotron radiation computed applying the approximation (crosses) and the SFD. The two LP point sets correspond to log-parabolic electron spectra with b evaluated by a best fit around the SED peak (filled circle) and on the high frequency branch (diamonds). LPPL points (open squares) correspond to a combined log-parabolic power law distribution. Dashed lines are linear best fits used for the calculation of the b/r ratio. | |
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We also used the spectra of Fig. 1 to study the relation between r and b under different conditions. The estimate of b is not simple because the radiation spectra do not have an exact log-parabolic shape. One can expect that this estimate is more accurate considering only the SED branches at frequencies higher than the peak: a quadratic best fit of the spectra of Fig. 1 confirms the linear relation between b and r, but the coefficient is equal to 0.22, slightly lower than the value found under the approximation (Eq. (16)). We also computed b considering a frequency interval centred at the peak frequency and spanning approximately two decades. The results are shown in Fig. 3. The relation between b and r was linear with very good accuracy, but the coefficient decreased to 0.18, as expected because of the smaller curvature of the low frequency branches. The last result does not change if the low energy electron spectrum follows a power law instead of a log-parabola (open squares in Fig. 3): the b/r ratio in this case was found equal to 0.19. We also verified our results by computing the spectra using the approximation and found b/r=0.2508, in very good agreement with the theory. An approximate rule derived from this analysis indicates that it is convenient to take the value of the ratio equal to 0.2, within an accuracy of 10%. The resulting r estimate can then be used as the input value for a more refined modelling of the SED.
Figure 4: Time evolution of the SR spectrum ( upper panel) and of the corresponding spectrum of electrons, multiplied by the square of the Lorentz, under SR losses. The injection spectrum was a log-parabola with s=1.2 and r=0.7, corresponding to a peak energy of . Spectra are plotted for equal time steps of 0.4 cooling times. Note the decrease in the peak energy and the deviation from a log-parabola due to difference in cooling time with . | |
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Figure 5: Time evolution of the log-parabolic curvature parameters r of the electron spectrum ( upper line) and b of the SR SED ( lower line) computed by means of a best fit around the peak. Solid lines are linear best fits to the computed values. | |
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We also studied the evolution of spectral curvature due only to synchrotron cooling. Of course, this study cannot be performed in general because the solution of the time-dependent continuity equation is a function of the assumed time evolution of the injection rate of emitting electrons and of several other parameters like the mean residence time of particles inside the emitting volume. We studied, therefore, the simplest case of a pure radiative SR cooling in a homogeneous and steady magnetic field without leakage and with a log-parabolic initial spectrum. The time evolution of SR and electron spectra are shown in the two panels of Fig. 4, where the decrease in both the peak frequency and intensity with time is very evident. Although radiation losses modify the spectral shape, it remains approximately log-parabolic, though with larger curvature parameters. These variations are shown in Fig. 5, where we plotted the r and b values, measured in energy/frequency intervals centered approximately at the distribution peaks, as a function of time, measured in units of the cooling time for the peak energy of the initial electron spectrum, taken as and corresponding to s with a magnetic field B=0.1 G. It is worth noting that the change in time of r and b is described well by linear relations. The observation of a decrease in associated with an increasing curvature during the dimming phase of an outburst is an indication of a radiative cooling evolution. Our calculations also show that the changes of SR peak frequency and b are correlated according to a power law as shown in the plot of Fig. 6.
We stress, however, that the estimate of b may not be simple when working with observational data because the position of the peak may be far from the centre of the detector frequency range, and the uncertainty on the interstellar and intrinsic absorption may be non-negligible, particularly in the soft X-ray band.
Figure 6: The relation between the peak frequency and the curvature parameter b of the SR spectra plotted in the upper panel of Fig. 4. The dashed line is the best fit to the computed values. | |
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We computed the spectral distribution of the SSC emission from a relativistic electron population with a log-parabolic energy spectrum in the framework of a homogeneous one-zone model in which the seed photons are isotropic in the bulk frame of the electrons. Our goal is to estimate the relations between the main spectral parameters: the peak frequencies of the SR and IC components and their curvatures indicated in the following by b_{S} and b_{C}, respectively.
Figure 7: The SR and SSC spectral energy distributions emitted by an electron population with power-law log-parabolic distributions (Eq. (11)), computed for r values in the interval 0.50-1.20. Upper panel: spectra for ; lower panel: spectra for . | |
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Figure 8: Dependence of the peak frequencies upon the particle characteristic energies in the SED of a single zone SSC model. Upper panel: frequency of SSC peak vs. electron Lorentz factors of the peaks for r values in the interval 0.50-1.90, the three curves correspond to (filled circles), (open cicrcles), 10^{4} (crosses). Lower panel: ratio between the peak frequencies of IC and SR components for the same values of the spectral parameters. | |
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The IC emissivity is given by (Jones 1968; Blumenthal & Gould 1970;
Band & Grindlay 1985)
(19) |
(20) |
(21) |
(22) |
The IC spectra show an evident curvature that is not univocally related to the r as in the SR case. Spectral curvature, in fact, depends on several parameters in a complex way. The spectral shape is generally different from a log-parabola, and it can be approximated by this function only in a limited frequency interval. Consequently, the estimate of the curvature parameter depends on the postion and amplitude of this interval. In particular, the curvature depends on the intrinsic electron spectral curvature, the energy of SR photons, and on the energy of the electrons that mostly contribute to the emission in the selected interval. These energies, in fact, determine if the scatterings happen in the Thomson limit or in the Klein-Nishina regime. We obtained spectra with a curvature less pronounced than SR only for (i.e. for dominant Thomson scattering) and b evaluated in an interval around the peak. In all the other cases, higher b values of IC spectrum resulted. Figure 9 shows some examples: the curvature parameter b of the IC spectrum, computed in a single zone SSC model, was evaluated in three adjacent frequency intervals, each having an amplitude of about a decade starting from the peak frequency. When the main contribution to IC emission comes from Thomson scatterings, the curvature is closer to that of SR (solid line), and it approaches that of the electron spectrum (dashed-dotted line) when the fraction of interactions in the Klein-Nishina regime increases. The two panels correspond to values differing by a factor of 20. In the upper panel, IC emission is dominated by Thomson scattering in the two first intervals, while in the lower panel the curvature approaches that of the electrons because the majority of interactions are in the Klein-Nishina regime. This property is really useful because a simultaneous measure of the curvature parameters of SR and IC emissions can help to discriminate between the two regimes of Compton scattering and to constrain the spectral parameters of emitting electrons.
Figure 9: The curvature parameter b of the IC spectrum for a single zone SSC model plotted against the electron spectral curvature r. Dashed-dotted lines are the curvature parameters for the electrons and solid lines for SR component. Dashed lines correspond to IC curvatures evaluated in three adjacent frequency intervals with an amplitude of about a decade selected starting from : lowest lines correspond to the intervals closest to the peaks and the highest ones to those at the highest frequencies. The upper panel shows b values computed for and lower panel those for . Note that when the largest contribution to IC emission is from Thomson scatterings ( upper panel), curvatures are closer to those of SR, while in the Klein-Nishina regime ( lower panel) b values result very close to r. | |
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Table 1: Model parameters for the 1997 April flare of Mkn 501.
In Sect. 4 we discussed the relations between the parameters of log-parabolic spectra for a SSC emission and showed that SSC emission model can give high frequency spectra with a b value ranging from that of the SR component to that of the electron spectrum depending on whether the scattering occurs in the Thomson or in the Klein-Nishina regime. From the calculations of Sect. 3, we know that an SR b value of about 0.15 corresponds to , and therefore a curvature of the IC component as that measured in the TeV range can be expected.
In Paper II we also presented a two component-model to describe the spectral evolution of the April 1997 X-ray outburst of Mkn 501. We assumed that the components had the same curvature (b=0.18) and differ for the peak energy and their relative intensity. In this way we explained the nearly constant flux observed at lower energies than 1 keV, while that around 100 keV changed by more than one order of magnitude, and the slight decrease of b in the higher states was interpreted as the effect of the sum of two log-parabolic distributions peaked at different energies.
An important effect to be considered when computing the spectral curvature in the TeV range is the absorption of the most energetic photons due to pair production against the infrared extragalactic background light (EBL). Dwek & Krennich (2005) have recently studied twelve EBL models and computed the corresponding intrinsic spectra of Mkn 501. These authors found that the majority of EBL models give intrinsic emitted spectra showing an unphysical rise at high TeV energies and that only the lowest intensity EBL models are acceptable.
We applied our radiation code to searching for the geometrical and physical parameters of the emitting region that can reproduce the observed intensities and spectral shapes, also taking EBL absorption into account. To compare the model with the data, we need to introduce a relativistic beaming factor , which does not affect the spectral shape, but which changes observed frequencies and fluxes by and , respectively, where z=0.034 is the source redshift. In the following we assume , in agreement with other models for Mkn 501 (see e.g. Kataoka et al. 1999).
We first considered a single-zone SSC model and tried to reproduce the two extreme states of the large outburst of April 1997, i.e. the low state on April 7 and the high state on April 16.
The spectrum of the emitting electrons was chosen with a curvature parameter r=b/ following the relation found in Sect. 4.1, and the parameter s was fixed to 1.2. Note that this value of r is consistent with the general behaviour presented in Fig. 9. The TeV curvature measured around the peak was about 0.4 for the low state and 0.45 for the high state (Djannati-Atai et al. 1999) and, for on the order of 10^{4} and even higher, we expect a value of r in the range 0.5-0.8. The values of the other parameters, given in Table 1, were chosen to achieve a satisfactory agreement between the data and the model. It is interesting to note the behaviour of the two correlated parameters (the peak frequency in the synchrotron SED) and (the value of of the electrons that contribute mostly to synchrotron peak emission). In fact the grows during the flare by a factor of 11 compared to the initial value while, according to Eq. (14), changes as the square root of growing rate. Spectral distributions of Fig. 10 show that the SR follows the X-ray data accurately in both the spectral shape and peak evolution. The TeV spectra in the two states were computed using the SSC code described in Sect. 4.2: in the upper panel of Fig. 10 we considered the EBL absorption for the LLL model by Dwek & Krennich (2005), while in the lower panel this absorption was neglected. In both cases TeV data were well fitted in both the peak position and spectral curvature. This suggests that this curvature could be partly intrinsic rather than produced entirely by EBL absorption. The LLL model could then be considered an upper limit to the extragalactic background.
Figure 10: Two spectral energy distributions of Mkn 501 during the low and high states observed on 7 and 16 April 1997, respectively. X-ray points are from Paper II, TeV points are simultaneous CAT data (Djannati-Atai et al. 1999) and soolid lines are the spectra computed in a 1-zone SSC model for the SR and IC components. In the upper panel IC, spectra have been absorbed (dashed lines) by interaction with infrared EBL photons according to the LLL model by Dwek & Krennich (2005). In the lower panel EBL absorption was neglected. | |
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Figure 11: Two spectral energy distributions of Mkn 501 during the high states observed on 7, 11 April 1997 ( upper panel) and 7, 16 April 1997 ( lower panel). X-ray points are from Paper II, and TeV points are simultaneous CAT data (Djannati-Atai et al. 1999). Thin solid lines are the spectra computed in a 2-zone SSC model for the SR and IC components, dashed lines are the spectra of the high-energy flaring component, and the thick solid line is that of a slowly evolving component. | |
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As proposed in Paper II we considered a two-zone SSC model. A first zone (Z1) is responsible for the "slowly'' variable emission, whereas the second zone (Z2) is the source of the high-energy flare: the observed fluxes are then the sum of these two contributions. We used the April 7 emission as representative of the "slowly'' variable state (Z1) and added the flaring component (Z2) to match the April 11 and 16 high states. In this way the stability of the flux at energies 1 keV is easily explained by the fact that the emission from Z2 is negligible when compared with that of Z1. The results of these calculations are represented in the SEDs of Fig. 11. We met some difficulty in obtaining a solution capable of modelling the observed spectral curvature mostly in the highest state. In fact, to deplete the SR from Z2 below 1 keV, we need an electron population with a very low emissivity in this band. This can be obtained only by introducing a low energy cut-off, with the consequence of a high curvature of the low energy portion of IC spectra at TeV energies. The values of parameters for this 2-zone SSC model are also given in Table 1.
These results can be used to obtain some information on the
acceleration process. According to Eq. (10) we can derive the
energy gain as a function of r:
(23) |
It is interesting to compare our SEDs of the 1997 flare of Mkn 501 with those calculated by other authors who adopted different physical models. Generally, there is no explicit evaluation of a spectral curvature parameter for both SR and IC components and consequently the electron energy distributions are assumed to be power laws with exponential cut-off (Konopelko et al. 2003; Krawczynski 2002) or a broken power law (Katarzynski 2001).
To reproduce the spectral curvature in IC spectra, these authors need to introduce a heavy contribution from EBL interaction with TeV photons. We stress that in our analysis the SR curvature is intrinsic, and we do not need break or cut-off to reproduce it. More interestingly, the IC curvature is also intrinsic and can be reproduced both using and neglecting the TeV photon absorption by EBL photons. In particular in the, in case of large flares where the peaks of SR and IC components increase their energy, models with break or cut-off do not predict intrinsically curved IC spectra, and so the spectral shape observed at TeV and the IC peak frequency are mainly modulated by what EBL model is chosen. Note also that EBL attenuation could consistently modulate the position of the IC peak because its optical depth varies with the energy and redshift. Observations of SR and IC components in objects with different redshifts and the study of the variations of their peak frequencies would be very useful for more accurate modelling of the EBL spectrum.
It is interesting to compare the values of the main model parameters. As a further point Konopelko et al. (2003) and Krawczynski et al. (2002) used values of 50 and 45, respectively, while Katarzynski et al. (2001) used 14, very close to our choice.
To find the exact relations between the spectral parameters of the energy distribution of the relativistic electrons and of their SR, we used a precise numerical code specifically developed for blazar applications. The main results can be summarised as follows:
The same conclusion has been reached very recently by Aharonian et al. (2005b), who report the detection of the blazar 1ES 1101-232 at z = 0.186 (by HESS in the TeV band). An important consequence of the intrinsic curvature of TeV spectra is that it can be variable in time and possibly related to the X-ray curvature. As a final remark, we stress the relevance of the simultaneous broad-band observations that are necessary to obtain a reliable estimate of curvature parameters for both components.
Acknowledgements
We acknowledge the helpful comments and suggestions of the referee, A. Djannati-Atai. This work was financially supported for the ASDC by the Italian Space Agency (ASI) and for the Physics Dept. by Università di Roma La Sapienza.