Issue 
A&A
Volume 504, Number 3, September IV 2009



Page(s)  821  828  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/200912237  
Published online  16 July 2009 
Online Material
Appendix A: Logparabolic spectra
In this Appendix we show how electron populations with a logparabolic energy distribution of the form expressed by Eq. (3), that is,
emit logparabolic spectra via the SSC process. The related particle synchrotron emissivity
(A.2) 
is easily computed on using the close approximation to the single particle emission in the shape of a deltafunction (see Rybicki & Lightmann 1979), that is, with and (with B measured in Gauss) is the synchrotron critical frequency. This leads to (Massaro et al. 2004a) a logparabolic differential flux
(A.3) 
and to a SED again of logparabolic shape
(A.4) 
its slope at the synchrotron reference frequency is given in terms of s, by
(A.5) 
the spectral curvature by
and the peak value occurs at a frequency ( , n, B and R are defined in Sect. 2.1 of the main text).
For IC radiation in the Thomson regime we may write to a fair approximation
(see Rybicky & Lightmann 1979) where
is the power radiated by a singleparticle IC scattering in the Thomson regime, having denoted with
the synchrotron radiation density^{}
. We obtain once again a logparabolic SED
(A.7) 
where the slope at the IC reference frequency is given by
(A.8) 
and the spectral curvature reads
The peak value is attained at a frequency .
For the KleinNishina (KN) regime instead it necessary to consider the convolution
(A.10) 
where and are the electron frequencies before and after the scattering, respectively, is the number spectrum of seed photons, and is the full Compton kernel (Jones 1968). Only in the extreme KN regime one may again approximate
(A.11) 
on approximating with its mean value for a homogeneous, spherical optically thin source
(A.12) 
one obtains again a SED with a logparabolic shape
(A.13) 
Now the slope at
(A.14) 
is steeper, and the spectral curvature
is larger than in the Thomson regime. The peak value occurs at a frequency .
The transition between the two regimes occurs when
,
where
,
that is, when
(A.16) 
holds, or equivalently
(A.17) 
We close these calculations with two remarks. First, the (primed) quantities used here refer to the rest frame of the emitting region, while the observed (unprimed) quantities must be multiplied by powers of the beaming factor
(A.18) 
being the bulk Lorentz factor of the relativistic electron flow in the emitting region, the angle between its velocity and the line of sight, and ; so we obtain , , and , . Second, we note that the actual specific powers radiated by a single particle differ from deltafunction shape, slightly for the synchrotron emission and considerably for the IC radiation in the Thomson regime (see discussion in Rybicky & Lightmann 1979). However, the above spectral shapes still approximatively apply, as confirmed by numerical simulations (Massaro 2007); in detail, the convolution with a broader single particle power yields a less curved spectrum, so that Eqs. (A.6) and (A.9) become (Massaro et al. 2006) and , respectively.
Appendix B: Particle acceleration processes
Figure B.1: Example of time evolution of the distribution (with peak energy ) due to stochastic and systematic accelerations ( upper panel) with , from the initial value , and of the related evolution for the SSC SEDs ( lower panel). In terms of the stochastic acceleration time , the time interval between each pair of lines is , corresponding to an observed time interval of about two days. 

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Figure B.2: Example of time evolution of the distribution (with peak energy ) due to stochastic and systematic accelerations ( upper panel) with , from the initial value , and of the related evolution for the SSC SEDs ( lower panel). In terms of the stochastic acceleration time , the time interval between each pair of lines is , corresponding to an observed time interval of about two days. 

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Here we derive a logparabolic electron energy distribution
from a kinetic continuity equation of the FokkerPlanck type; following Kardashev (1962), in the jet rest frame this reads
where is the particle energy, t denotes time, and and describe systematic and stochastic acceleration rates, occurring on timescales and , respectively. For example, in the picture of Fermi accelerations (e.g., Vietri 2006), the accelerations rates and can be expressed in terms of physical quantities related to processes occurring in shocks; in this framework a plane shock front of thickness moves with speed and gas clouds of average size move downstream of the shock with speed V; it is found that and hold (see Kaplan 1956; Paggi 2007), . On the other hand, numerical values of and are directly derived from the emitted spectrum as shown later.
The FokkerPlanck Eq. (B.1) describes the evolution of the electron distribution function; with an initially monoenergetic distribution in the form of a deltafunction
(n is the initial particle number density) the solution at subsequent times t takes the form of the logparabolic energy distribution assumed in Eq. (3) of the main text (see also A.1), reading
Here the time depending slope at is given by
(B.3) 
meanwhile, the curvature of driven by the diffusive (stochastic) term in Eq. (B.1), irreversibly decreases in time after
(B.4) 
from the large initial values corresponding to the initially monoenergetic distribution. Correspondingly, the timedependent height at follows
(B.5) 
To wit, Eq. (B.2) describes the evolution of the electron distribution, growing broader and broader under the effect of stochastic acceleration, while its peak moves from to the current position
(B.6) 
under the contrasting actions of the systematic and stochastic accelerations. An important quantity to focus on for the emission properties is the rms energy
=  (B.7) 
which is also the position for the peak of the distribution .
We have already derived in Appendix A the shapes of the spectra (synchrotron, and IC in both the Thomson and KN regimes) emitted by the distribution given in Eq. (3); here we stress the time dependence of their main spectral features. We can write for the synchrotron emission^{}
for IC emission we have in the Thomson regime
(B.9) 
and in the extreme KN regime
Note that during flares, since , we expect the curvature to vary so little that we can therefore approximatively write (see main text).
Now we focus the above relations for the simple case of time independent
and ,
when
;
then we have
while for the rms energy we have
so for the peak frequency and the spectral curvature we have
where is a normalization frequency. It is seen that soon after the injection the curvature (proportional to r) drops rapidly, then progressively decreases more and more gently, while the peak frequency still increases.
The value of
can be evaluated from observing the synchrotron spectral curvatures b_{2} and b_{1} at two times t_{2} and t_{1}, respectively (recall that
); denoting with t_{2}  t_{1} this time interval, we have
(B.14) 
on the other hand, form observing the related synchrotron peaks and , the value of can be evaluated as
(B.15) 
For example, in the case of Mrk 501 in the states of 7 and 16 April 1997, we obtain (on assuming ) and , corresponding to acceleration times and . Note that with the current data the evaluations of and turn out to be affected by uncertainties considerably larger than the single curvatures and .
If the total energy available to the jet is limited (e.g., by the BZ limit, see text) we expect that and cannot grow indefinitely, but are to attain a limiting value. At low energies where holds, we have and as before; at higher energies when reaches its limit, holds, leading to and . Eventually also reaches its limit, and both the fluxes and the peak frequencies cannot grow any more.
Appendix C: Prefactors for Eqs. (4)(13)
The HSZ SSC model is, as stated before, characterized by five parameters: the rms particle energy
,
the particle density n, the magnetic field B, the size of the emitting region R and the beaming factor .
So the model may be constrained by five observables that we denote with
(C.1) 
(C.2) 
where the indexes i,j,k,h express the normalizations demonstrated below; in addition, we denote with the time in days for the source variations, and with D the distance of the source in Gpc.
In the Thomson regime we find
B  =  
(C.3)  
=  13.5  
(C.4)  
R  =  
(C.5)  
n  =  
(C.6)  
=  (C.7) 
For the extreme KN regime we obtain
B  =  
(C.8)  
=  
(C.9)  
R  =  
(C.10)  
n  =  
(C.11)  
=  
(C.12) 
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