Synchrotron selfCompton flaring of TeV blazars
II. Linear and nonlinear electron cooling
C. Röken  R. Schlickeiser
Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und Astrophysik, RuhrUniversität Bochum, 44780 Bochum, Germany
Received 29 October 2008 / Accepted 9 May 2009
Abstract
A theoretical radiation model for the flaring of TeV blazars is discussed here for the case of a nonlinear electron synchrotron cooling in these sources. We compute analytically the optically thick and thin synchrotron radiation intensities and photon density distributions in the emission knot as functions of frequency and time followed by the synchrotron selfCompton intensity and fluence in the optically thin frequency range using the Thomson approximation of the inverse Compton cross section. At all times and frequencies, the optically thin part of the synchrotron radiation process is shown to provide the dominant contribution to the synchrotron selfCompton quantities, while the optically thick part is always negligible. Afterwards, we compare the linear to the nonlinear synchrotron radiation cooling model using the data record of PKS 2155304 on MJD 53944 favouring a linear cooling of the injected monoenergetic electrons. The good agreement of both the linear and the nonlinear cooling model with the data supports the relativistic pickup process operating in this source.
Additionally, we discuss the synchrotron selfCompton scattering, applying the full KleinNishina cross section to achieve the most accurate results for the synchrotron selfCompton intensity and fluence distributions.
Key words: radiation mechanisms: nonthermal  methods: analytical  galaxies: active
1 Introduction
A process describing the energy loss of ultrarelativistic electrons in cosmicray sources named inverse Compton scattering has been considered to be very likely responsible for the production of the highenergy radiation emitted by active galactic nuclei. In this process, lowenergy photons are scattered to higher energies by relativistic electrons within the jets of these sources. There are several possible external and internal generators of the lowenergy photon field, such as the accretion disc of the central black hole (Dermer et al. 1992; Dermer & Schlickeiser 1993), the broadline region (Sikora et al. 1994), dust surrounding the active galactic nucleus (Blazejowski et al. 2000; Arbeiter et al. 2002) or synchrotron radiation produced in the jet itself. In this work, the process of interest is the inverse Compton scattering of internal synchrotron radiation called synchrotron selfCompton scattering (Maraschi et al. 1992).
Numerical models were applied in most of the studies of this process, e.g. Mastichiadis & Kirk (1997), Dermer et al. (1997), Chiaberge & Ghisellini (1999), Sokolov et al. (2004) and Böttcher (2007). Here, we investigate analytically the influence of a nonlinear electron synchrotron cooling on the synchrotron selfCompton process following the analysis of the flaring of TeV blazars due to the synchrotron selfCompton process for a linear synchrotron radiation cooling behaviour of the injected electrons (Schlickeiser & Röken 2008), where a distribution approximation (Reynolds 1982; Dermer & Schlickeiser 1993) was used for the computation of the inverse Compton scattering rate . Most of the calculations of photon spectra modelling quantities have been based on such rather simple approximations. Therefore, we also determine the inverse Compton scattering rate, the synchrotron selfCompton intensities and fluences for the linear and nonlinear synchrotron photon densities and electron populations for the full KleinNishina cross section to obtain more general results (Appendix B).
We assume that a flare of the emission knot occurs at the time t = t_{0} due to a uniform instantaneous injection of monoenergetic ultrarelativistic electrons. The emission knot itself moves with a relativistic bulk speed V with respect to an external observer. We model the emission knot as a spherical magnetised, fully ionised plasma cloud of radius R consisting of cold electrons and protons with a uniform density distribution and a randomly oriented largescale timedependent magnetic field B(t) that adjusts itself to the actual kinetic energy density of the radiating electrons in these sources, yielding the different nonlinear synchrotron radiation cooling behaviour. This magnetic field is most likely generated from the interaction of the relativistically moving knot with the surrounding ambient intergalactic and interstellar medium that is also responsible for the injection of the ultrarelativistic particles by the relativistic pickup process (Pohl & Schlickeiser 2000; Gerbig & Schlickeiser 2007; Stockem et al. 2007). The existence of a magnetic field is mandatory for the generation of synchrotron radiation and hence, the synchrotron selfCompton process.
In our analysis we assume that the synchrotron radiation losses of relativistic electrons in a constant (linear case) or partition (nonlinear) magnetic field dominate over synchrotron selfCompton losses which also imply nonlinear electron energy losses (Schlickeiser 2009) because the energy density of the target synchrotron photons is given by an energy integral over the radiating electron distribution function. Therefore, our analysis applies to blazar sources where the observed lowenergy synchrotron component in the spectral energy distribution dominates over the highenergy synchrotron selfCompton component. Apart from the exceptional ray flare in July 2006 (Aharonian et al. 2009a) this applies in particular to the powerful blazar PKS 2155304 as the existing multifrequency campaigns of this source indicate (see Fig. 10 of Aharonian et al. 2005; Fig. 2 of Aharonian et al. 2009b), especially in low activity states. We therefore compare our results with the highenergy observations from this source.
Before starting the analysis we explain our assumption that magnetic field partition is instantaneously established as the electrons cool. Equipartition conditions are often invoked in astrophysical sources for convenience, as discussed e.g. in the review by Beck & Krause (2005). Observationally, for a variety of nonthermal sources the equipartition concept is supported by magnetic field estimates such as e.g. the Coma cluster of galaxies (Schlickeiser et al. 1987). From a theoretical point of view, there is no simple explanation of partition but we will outline the basic arguments here. An upper limit on the magnetic field strength can be derived by applying Chandrasekhar's (1961, p. 583) general result, derived from the virial theorem, that for the existence of a stable equilibrium in the radiating source it is necessary that the total magnetic field energy of the system does not exceed the system's gravitational potential energy. Such a magnetic field upper limit corresponds to lower limits on the system's parallel and perpendicular plasma betas, and , respectively, as biMaxwellian plasma distributions with different temperatures along and perpendicular to the magnetic field are the most likely distributions of cosmic plasmas. The solar wind plasma is the only cosmic plasma where detailed insitu satellite observations of plasma properties are available (Bale 2008). Ten years of Wind/SWE data (Kasper et al. 2002) have demonstrated that the proton and electron temperature anisotropies are bounded by ion cyclotron, mirror and firehose instabilities (Hellinger et al. 2002) at large values of the parallel plasma beta . In the parameter plane defined by the temperature anisotropy and the parallel plasma beta , stable plasma configurations are only possible within a rhomblike configuration around , whose limits are defined by the threshold conditions for these instabilities. If a plasma would start with parameter values outside this rhomblike configuration, it immediately would generate fluctuations via the instabilities, which quickly relax the plasma distribution into the stable regime within the rhombconfiguration. Similar anisotropic plasma distributions, such as relativistic kappadistributions, are to be expected during the pickup of interstellar particles by the interaction of the relativistic jet in the case of blazars with the surrounding ambient interstellar or intergalactic medium (Stockem et al. 2007). Such an interaction is a prominent example of the relativistic collision of plasma shells with different properties (temperature, density, composition etc.). Experimentally (Kapetanakos 1974; Tatarakis et al. 2003) and from numerous particleincell (PIC) simulations (e.g. Lee & Lampe 1973; Nishikawa et al. 2003; Silva et al. 2003; Frederiksen et al. 2004; Sakai et al. 2004; Jaroschek et al. 2005) such collisions of plasma shells lead to the onset of linear Weibeltype plasma instabilities perpendicular to the flow directions in both unmagnetised and slightly magnetised plasmas, and to the development of anisotropic relativistic plasma distributions. The PIC simulations of electronproton and electronpositron plasmas demonstrate that these instabilities generate magnetic fields in the form of aperiodic fluctuations at almost equipartition strength on the shortest plasma time scale. The aperiodic magnetic fluctuations will scatter the initially beamlike interstellar particles by rapid pitchangle scattering in the rest frame of the jet, leading to an efficient pickup of nearly monoenergetic relativistic electrons and protons (Schlickeiser et al. 2002). The superposition of the plasma shell particles and the scattered interstellar particles results in a total plasma distribution with finite anisotropy that then, as in the solar wind case, controls the magnetic field strength. Many details of the complicating plasma processes leading to partition still have to be worked out, starting with studies of the micro instabilities of relativistic anisotropic plasmas (e.g. Lazar et al. 2008).
The paper is structured as follows: first we solve the partial differential equation for the timedependent evolution of the volumeaveraged relativistic electron population inside the radiation source assuming a nonlinear synchrotron radiation cooling of the electrons and provide the necessary intrinsic synchrotron radiation formulas (Sects. 2 and 3). Then we calculate the associated synchrotron radiation intensities (Sect. 4). In Sect. 5 the optically thin synchrotron selfCompton emission is determined using the Thomson limit of the KleinNishina cross section followed by the calculation of the nonlinear synchrotron selfCompton fluence distribution including a comparison of the linear to the nonlinear model by application to the data record of the PKS 2155304 flare on MJD 53944 in Sect. 6. In Appendix B we compute the linear and nonlinear synchrotron selfCompton intensities and fluences using the full KleinNishina cross section and compare the approximate to the exact results.
2 Nonlinear electron synchrotron cooling
To begin with, we choose the appropriate reference frame for the calculations of all physical quantities for the nonlinear cooling process of relativistic electrons due to synchrotron radiation to be a coordinate system comoving with the radiation source.
Once ultrarelativistic electrons ( ) enter the observed physical system with a largescale magnetic field at the rate and at the time t = t_{0}, here a jetplasmoid of an active galactic nucleus, they compete with electron synchrotron energy losses. The timedependent evolution of the competition process is mathematically described by a partial differential equation for the volumeaveraged relativistic eletron population inside a radiating source first derived by Kardashev (1962)
where
is the synchrotron energy loss rate depending on the magnetic field energy density U_{B}. The function denotes the volumeaveraged differential number density. Throughout this work we do not consider effects due to particle escape from the source (i.e. the emitting volume is a thick target for the radiating particles) and/or due to additional Coulomb/ionisation, bremsstrahlung and adiabatic expansion energy losses. A finite escape time T_{0}, independent of particle energy, can easily be handled yielding an additional exponential function in the solutions derived below. However, the high magnetic field strength in blazar sources leads to very short synchrotron radiation loss times, much shorter than T_{0} for electron Lorentz factors of interest generating synchrotron selfCompton photons. Also, because of the short synchrotron radiation loss times, the additional Coulomb/ionisation, bremsstrahlung and adiabatic expansion losses only affect the electron distribution function at very low energies where (Schlickeiser 2003) for standard blazar parameter values. Again these electron energies are not relevant in the production of highenergy radiation by the synchrotron selfCompton process.
In this work, we discuss as an illustrative, but physically justified example the case of one instantaneous monoenergetic injection of ultrarelativistic electrons
with the injection strength q_{0}. At all times we assume that the entering electrons are ultrarelativistic (
), resulting in the relation
,
implying
for the differential electron number densities. So we find for the energy integrated kinetic energy density of the relativistic electrons
In the case of nonlinear cooling under the fixed partition condition (Schlickeiser & Lerche 2007, 2008; Röken & Schlickeiser ) between the energy densities of the magnetic field U_{B}(t) and the relativistic electrons (3), the synchrotron energy loss rate (2) reads
(4) 
where . Then the solution of the kinetic Eq. (1) is (Schlickeiser & Lerche 2007)
with denoting the Heaviside step function and the characteristic nonlinear Lorentz factor
where is a normalised time scale for the nonlinear cooling.
3 Intrinsic synchrotron radiation formulas
An appropriate approximation for the pitchangle averaged synchrotron power of a single electron in vacuum (Crusius & Schlickeiser 1986, 1988) reads
where . The function yields approximately
with a_{0} = 1.151, exhibiting a similar asymptotic behaviour as the function (Schlickeiser & Lerche 2007) and, therefore, it appears appropriate to use it for the calculation of the intrinsic synchrotron radiation formulas.
In the nonlinear cooling case the gyrofrequency,
,
is a timedependent function due to the imposed partition condition between the energy densities of the magnetic field and the relativistic electrons leading to a timedependence of the magnetic field
(9) 
Hence, we obtain for the gyrofrequency, using (6),
(10) 
where . The synchrotron intensity from relativistic electrons expressed by the volumeaveraged differential density for a homogeneous source of radius R reads
depending on the spontaneous synchrotron emission coefficient
and the synchrotron optical depth, , where is the synchrotron absorption coefficient,
In a strict sense the approximations in (11) are valid only for the cases and . However, we use an analytic continuation of the approximated synchrotron intensity in order to cover the whole domain. This approximation is justified because the calculations shown in Sects. 5 and 6 indicate its accuracy.
4 Synchrotron radiation intensities
Inserting the nonlinear electron density distribution (5) into the synchrotron emissivities (12) and (13) and carrying out the integrations, we obtain in terms of the normalised frequency
and
where we used the characteristic frequency
for the characteristic values of the electron injection rate , the source radius and the initial Lorentz factor .
4.1 Synchrotron intensity variation
Following the discussion of the intensity variations for fixed frequencies as a function of time for optically thin emission frequencies
,
performed by Schlickeiser & Lerche (2007), we calculate the light curves at optically thick emission frequencies to
where . For times the optically thick intensity decreases , whereas at larger times it decreases . The time variation of the transition frequency along with the derivation of (17) can be found in Appendix A.
4.2 Synchrotron photon density distribution
With the synchrotron intensities (A.6), (A.7) and the definition
we obtain for the differential synchrotron photon number density
(18) 
for the case of optically thin energies
and for optically thick energies
where is the transition energy from the optically thick to the optically thin synchrotron emission regime. The time variation of the transition frequency can be found in Appendix A.
5 Synchrotron selfCompton emission
The differential number density of Compton scattered synchrotron photons with the normalised scattered photon energy is given by
using the distribution approximation for inverse Compton scattering introduced by Dermer & Schlickeiser (1993). The limit imposed on the integral restricts the scattering to the Thomson regime. Here, we neglect effects due to stimulated synchrotron selfCompton emission and absorption, as well as higher order synchrotron selfCompton scattering. Higher order contributions are expected to be small because they operate in the extreme KleinNishina limit where inverse Compton losses are much reduced (Schlickeiser 2009) and, therefore, are negligible compared to synchrotron and triplet pair production losses. This leads to the synchrotron selfCompton intensity
Inserting the relativistic electron distribution (5) into Eq. (21) we obtain
Equation (22) then yields
Making use of the synchrotron photon number densities (19) and (20) we find for the synchrotron selfCompton intensity
and
respectively.
We introduce a strictly decreasing function to simplify the following expressions and relations
(27) 
For times less and greater than (Appendix A) we obtain, analytically continuating the domain,
(28) 
and
(29) 
using Eqs. (A.2) and (A.5).
6 Synchrotron selfCompton fluences
We discuss the synchrotron selfCompton fluence distribution described by the timeintegrated synchrotron selfCompton intensity (22)
in two scattered photon energy ranges: for energies only the optically thin synchrotron photon distribution (26) contributes, whereas at lower energies, , both the optically thin and thick parts (25) and (26) of the synchrotron photon distribution have to be taken into account.
6.1 High scattered photon energies
For energies
only Eq. (26) contributes, so that
Substituting yields
where
We obtain with the new variable
(34) 
Defining s = y^{9/4} and the fluence reads
For high scattered photon energies, , the argument of the exponential function in the integral of Eq. (35) becomes very small for all values of s yielding approximately
=  
=  (36) 
For scattered photon energies we substitute to obtain for the fluence (35)
=  
=  (37) 
in terms of the generalised incomplete gamma function . The second argument of the generalised incomplete gamma function is much smaller than unity for all scattered photon energies , while the third argument is much larger than unity, so that the generalised incomplete gamma function can be asymptotically expanded (Abramowitz & Stegun 1972) leading to the approximated fluence distribution
(38) 
6.2 Low scattered photon energies
For low scattered photon energies, , Eqs. (25) and (26) contribute to the spectral fluence. Starting with the scattered photon range we find, substituting as before
(39) 
The integrand of the first integral can be approximated within the domain of integration by
(40) 
Using this approximation and the previous substitutions we obtain for the fluence
(41) 
The dominant contribution to the fluence again represents synchrotron photons from the optically thin part of the synchrotron spectrum
(42) 
For scattered photon energies the fluence reads
(43) 
Using the substitutions and approximations applied before we obtain
(44) 
Again the contribution representing the optically thick part of the spectrum is negligibly small compared to the contribution representing the optically thin part, demonstrating that the fluence distribution also holds in the scattered photon energy range .
Figure 1: Timeaveraged spectrum observed from PKS 2155304 on MJD 53944. The dashed line represents the fit for the linear cooling case (Schlickeiser & Röken 2008), whereas the solid curve illustrates the fit for the nonlinear cooling case. 

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6.3 AIC model selection test
Figure 1 shows the linear (dashed curve) and nonlinear (solid curve) fits to the timeaveraged spectrum observed from PKS 2155304 on MJD 53944 (Aharonian et al. 2007). For the generation of these fits we first had to transform the calculated fluence distributions from the comoving frame into the observer frame (asterisked quantities)
(45) 
followed by the construction of the linear model parametric function
(results from the linear synchrotron selfCompton fluence distribution in the distribution approximation (Schlickeiser & Röken 2008)) with the parameters , and and the nonlinear model parametric function
with , and from our calculated fluence distributions, which we fitted (3 parameters, 14 degrees of freedom) to the PKS 2155304 data. Finally we carried out a socalled AIC model selection test (Akaike 1974) that measures the quality of a fit of a statistical model, i.e. the precision and complexity of the model, where the preferred model is that with the smallest AIC value. We found the linear fit with an AIC value of 13.37 and a reduced value of 1.37 to be more consistent with the data than the nonlinear fit with an AIC value of 30.15 and a reduced value of 3.67. The parameters were determined to be , and , . The fits turn out to be independent of the parameters and , because both functions (46) and (47) converge as long as the relation is satisfied, leading always to the parameters listed above. Note that the test was performed for only one specific data record for which the linear fit was found to be the best. To transform this indication into a solid conclusion we have to perform tests on more statistical ensembles.
7 Summary and conclusions
Schlickeiser & Lerche (2007) developed a nonlinear model for the synchrotron radiation cooling of ultrarelativistic particles in powerful nonthermal radiation sources assuming a partition condition between the energy densities of the magnetic field and the relativistic electrons. Here, we used this model in order to calculate the synchrotron selfCompton process in flaring TeV blazars and compared it to the results obtained with the standard linear synchrotron cooling model. For simplicity we chose the case of instantaneously injected monoenergetic relativistic electrons as an example, although other injection scenarios, like the instantaneous injection of powerlaw distributed electrons (Schlickeiser & Lerche 2008), are also possible. After the nonlinear electron synchrotron cooling, the created synchrotron photons with nonrelativistic energies are multiple Thomson scattered off the cooled electrons in the source (synchrotron selfCompton process).
We calculated the optically thin and thick synchrotron radiation intensities as well as the synchrotron photon density distributions in the emission knot as functions of frequency and time. These synchrotron photons serve as target photons in the synchrotron selfCompton process. Using the Thomson approximation of the inverse Compton cross section, we determined the synchrotron selfCompton intensity and fluence for the nonlinear electron cooling. It is shown that the optically thick synchrotron radiation component provides only a negligible contribution to the synchrotron selfCompton quantities at all frequencies and times, as in the linear cooling case. In Appendix B, we extended our calculations to the full KleinNishina cross section and obtained additional positive and negative, nonvanishing fluence contributions only in the highenergy regime of the scattered photons, e.g. generalised incomplete functions or the new generalised dual hypergeometric functions. Surprisingly, for the special class of electron and synchrotron photon distributions used in this work, these contributions nearly cancel each other out on average, leaving only a small positive fluence contribution (see Figs. B.5 and B.6) which we expected to be due to the consideration of high energy photons in the scattering process. Because of the smallness of the new contribution it can be justified to model the photon spectra by applying the distribution approximation for the calculation of the differential inverse Compton scattering rate for electron and synchrotron densities of the form (B.15), (B.16) as well as (B.23) and (B.24).
Finally, we compared the linear to the nonlinear fluence distribution, fitting both to the observed TeV fluence spectrum of PKS 2155304 on MJD 53944 and performed a statistical qualityoffit test (AIC test). For this particular data record, we found the linear model to be more appropriate than the nonlinear. Actually, with the formalism we presented here, we cannot fit the entire spectral energy distribution because we neglect the synchrotron selfCompton component of energy losses and only discuss the synchrotron one. Therefore, Schlickeiser (2009) has shown that synchrotron selfCompton cooling is an alternative nonlinear cooling process that can be handled analytically as long as it operates in the Thomson limit. Nonetheless, the excellent agreement of both linear and nonlinear synchrotron selfCompton fluence spectra with the observation of the gammaray flare of PKS 2155304 supports the injection scenario of monoenergetic electrons by the relativistic pickup process.
Acknowledgements
This work was partially supported by the German Ministry for Education and Research (BMBF) through Verbundforschung Astroteilchenphysik grant 05 CH5PC1/6 and the Deutsche Forschungsgemeinschaft through grant Schl 201/162.
Appendix A: Synchrotron optical depth and photon spectra
A.1 Optical depth
The synchrotron emission of ultrarelativistic electrons is optically thin for frequencies and times satisfying the condition and optically thick for . The transition occurs at the frequency defined by . For (and its analytical continuation) the optical depth (15) reduces to
as long as . In this range the transition frequency reads
In the domain (and its analytical continuation) the optical depth simplifies to
Substituting and expanding asymptotically for z > 1 as well as for we find
with the proper transition frequency
In Fig. A.1, we present the timedependence of the transition frequency. For small times , the transition frequency is constant at the value . After this, it increases linearly to its maximum value at , followed by a decrease proportional to at times .
Figure A.1: Normalised synchrotron transition frequency as a function of time in the nonlinear cooling case plotted for . The instantaneous injection of monoenergetic ultrarelativistic electrons occurred at . 

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A.2 Synchrotron spectra
According to Eq. (11), the synchrotron intensity in the optically thick frequency domain is given by
where . In the optically thin frequency interval, , we obtain
In Fig. A.2, we show the intensity distribution as a function of the frequency at various times and 10^{11}. Consequently, the transition intensity reads
Applying the transition frequencies (A.2) and (A.5) we obtain for the transition intensity at normalised times
whereas at times the result is
(A.10) 
At small times the transition intensity turns out to be
(A.11) 
The timedependence of the transition intensity is presented in Fig. A.3.
Figure A.2: Synchrotron intensity distribution as a function of frequency at different times (solid curve), (dashdotted curve), (dotted curve) and (dashed curve). The injection occurred at with the parameter . 

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Figure A.3: Synchrotron transition intensity as a function of time plotted for at the injection time . 

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Appendix B: Synchrotron selfCompton scattering with the full KleinNishina cross section
B.1 The process of synchrotron selfCompton emission
The differential inverse Compton scattering rate
in a coordinate system comoving with the radiation source (unprimed quantities) is given by (Dermer et al. 1992)
where is the normalised energy of a single electron, the normalised energy of a scattered photon, N the scattered photon number, the differential electron density and and are the solid angles of the scattered photons and electrons, respectively. The single electron differential scattering rate reads
(B.2) 
with the normalised energy of an incoming photon , the differential synchrotron photon density n and . is the velocity of the electron and is the wave vector of the incoming photon. The differential KleinNishina cross section has the most compact form in the electron rest frame (primed quantites) (Jauch & Rohrlich 1976)
(B.3) 
where
(B.4) 
and with the wave vector of the scattered photon. We assume isotropic electron and photon distributions in the comoving frame. Hence, the differential Compton scattering rate does not depend on the direction of the outgoing photons, so (B.1) can be reduced to
(B.5) 
Following the paper of Arbeiter et al. (2005) we can write
(B.6) 
where denotes the Heaviside step function. Using the socalled headon approximation (Arbeiter 2005) and we obtain for the highenergy regime (Jones 1967; Blumenthal & Gould 1970)
where , and for the lowenergy regime
In the following calculations of the synchrotron selfCompton intensities and fluences we disregard (B.8) and only use the dominant contribution (B.7) of the differential scattering rate for a single electron.
B.2 Linear and nonlinear synchrotron selfCompton intensities
Neglecting effects due to stimulated synchrotron selfCompton emission and absorption the synchrotron selfCompton intensity reads
where is the spontaneous synchrotron emission coefficient and R the radius of the source. For the differential scattering rate we use the dominant highenergy contribution (B.7) yielding
Consequently, we have to distinguish between the three cases , and , where is the transition energy from the optically thin to the optically thick synchrotron emission regime. For the linear cooling case the transition energy reads (Schlickeiser & Röken 2008)
(B.11) 
and for the nonlinear electron cooling case
(B.12) 
Here, we discuss only the case as explained below in Appendix B.3. Therefore, we define the functions
(B.13) 
and
(B.14) 
with , and .
B.2.1 Linear electron cooling
To compute the linear differential scattering rate we insert the linear differential electron density (Schlickeiser & Lerche 2007)
and the linear synchrotron photon density
into Eq. (B.10). Then we obtain
(B.17) 
with the constant and the functions , and . Transforming into we obtain
where . As shown in Appendix B.3 we expect a change of the intensity and fluence behaviours to occur only at high energies, so that (B.18) is sufficiently well approximated by
(B.19) 
Substituting we obtain
=  
(B.20) 
which is solved by generalised incomplete Gamma functions
=  
(B.21) 
We, thus, find the linear synchrotron selfCompton intensity to be
where . In Figs. B.1 and B.2 we show the linear synchrotron intensity (B.22) as a function of the energy at four different times ( ) and as a function of the time at four different energies ( ).
Figure B.1: Normalised linear synchrotron intensity as a function of energy at four different times (solid curve), 10 (dotted curve), 10^{2} (dashed curve) and 10^{3} (dashdotted curve). 

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Figure B.2: Normalised linear synchrotron light curve as a function of time at four different energies (solid curve), 10^{4} (dotted curve), 10^{5} (dashed curve) and 10^{6} (dashdotted curve). 

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B.2.2 Nonlinear electron cooling
With the nonlinear differential electron density (Schlickeiser & Lerche 2007)
and the nonlinear synchrotron photon density
the differential scattering rate reads
(B.25) 
where , and , applying the same substitutions and approximations as for the linear case. We obtain for the intensity
In Figs. B.3 and B.4 we present the nonlinear synchrotron intensity (B.26) as a function of the energy at five different times ( ) and as a function of the time at four different energies ( ). The different electron synchrotron cooling behaviours can be well observed by comparing for example Figs. B.1 and B.3. We can see that the functional behaviour of the linear intensity distribution at fits the behaviour of the nonlinear intensity distribution at later times due to a faster linear cooling process.
Figure B.3: Normalised nonlinear synchrotron intensity as a function of energy at five different times (solid curve), 10 (dotted curve), 10^{2} (dashed curve), 10^{3} (dashdotted curve) and 10^{5} (longdashed curve). 

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Figure B.4: Normalised nonlinear synchrotron light curve as a function of time at four different energies (solid curve), 10^{4} (dotted curve), 10^{5} (dashed curve) and 10^{6} (dashdotted curve). 

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B.3 Linear and nonlinear synchrotron selfCompton fluences
By the timeintegration of the synchrotron selfCompton intensity (B.9) we obtain the synchrotron selfCompton fluence distribution
Here, we only have to examine the range for high scattered photon energies because the action of the full KleinNishina cross section becomes apparent in the highenergy scattered photon regime for both the linear and nonlinear case, as shown in the following subsections. For energies much lower than the characteristic energy , we find the same solutions as in Schlickeiser & Röken (2008) and as in Sect. 6, leading to the conclusion that the fluence behaviour for is identical to the distributionapproximated Thomson regime fluence behaviour. Thus, we need only the optically thin synchrotron photon distributions (B.16) and (B.24) for the calculation of the synchrotron selfCompton fluence.
B.3.1 Linear electron cooling
With the linear synchrotron selfCompton intensity distribution (B.22) the synchrotron selfCompton fluence (B.27) reads for
=  
(B.28) 
where . We start with the computation of the fluence contribution for the first summand of function . Substituting a new timevariable, , we obtain
=  
(B.29) 
Transforming and defining , the integral yields
For scattered photon energies the integrands contribution for is dominant, so that (B.30) can be reduced to
=  
(B.31) 
Partial integration leads to the solution
consisting of incomplete and generalised incomplete functions multiplied by simple power functions. The argument is much larger than unity while the argument is much smaller than unity. Hence, the fluence (B.32) can be approximated by
(B.33) 
For high scattered photon energies, , we split expression (B.30) for the fluence into two integrals, and after partial integration we obtain for the first integral
Because all integrals in the highenergy scattered photon range are finally of the form of the integral on the righthandside of (B.34), we solve this in detail to demonstrate the analytical methods used. For this purpose, the exponential function can be Taylorexpanded leading to
(B.35) 
The denominator of the integrand can be written as a generalised geometric series
(B.36) 
where is the Pochhammer symbol. Thus, the integral is simply solved by a sum of hypergeometric functions
(B.37) 
The dominating contribution of the sum is of zeroth order, so that we obtain approximately
(B.38) 
The whole solution of the integral (B.34) reads
(B.39) 
The second integral of (B.30),
(B.40) 
yields the solution
For the first time, defining generalised dual hypergeometric functions
(B.42) 
the solution (B.41) can be written in a more compact form
(B.43) 
The whole first fluence contribution (B.30) then reads
We obtain for the second term of for a negligibly small fluence contribution
(B.45) 
and for
where the total fluence is finally .
B.3.2 Nonlinear electron cooling
Figure B.5: Normalised linear fluence distributions for the full KleinNishina cross section (solid curve) and for the distributionapproximated cross section (dashed curve) for with . 

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Using the nonlinear synchrotron selfCompton intensity (B.26), the nonlinear synchrotron selfCompton fluence reads for
=  
(B.47) 
where . Defining and using the same methods as in Appendix B.3.1 we obtain for
(B.48) 
and
(B.49) 
as well as for
and
In Figs. B.5 and B.6, we show the linear,
Figure B.6: Normalised nonlinear fluence distributions for the full KleinNishina cross section (solid curve) and for the distributionapproximated cross section (dashed curve) for with . 

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(B.52) 
and
(B.53) 
Despite the different mathematical form of the full KleinNishina fluence distributions and the approximated fluence distributions in the highenergy regime, in both electron synchrotron cooling cases the plots look nearly identical, except for the expected, but small positive fluence contribution in the KleinNishina plots due to the consideration of high energy photons. This is caused by the almost cancellation of the significant positive and negative additional contributions of the full KleinNishina fluence distribution between each other.
References
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All Figures
Figure 1: Timeaveraged spectrum observed from PKS 2155304 on MJD 53944. The dashed line represents the fit for the linear cooling case (Schlickeiser & Röken 2008), whereas the solid curve illustrates the fit for the nonlinear cooling case. 

Open with DEXTER  
In the text 
Figure A.1: Normalised synchrotron transition frequency as a function of time in the nonlinear cooling case plotted for . The instantaneous injection of monoenergetic ultrarelativistic electrons occurred at . 

Open with DEXTER  
In the text 
Figure A.2: Synchrotron intensity distribution as a function of frequency at different times (solid curve), (dashdotted curve), (dotted curve) and (dashed curve). The injection occurred at with the parameter . 

Open with DEXTER  
In the text 
Figure A.3: Synchrotron transition intensity as a function of time plotted for at the injection time . 

Open with DEXTER  
In the text 
Figure B.1: Normalised linear synchrotron intensity as a function of energy at four different times (solid curve), 10 (dotted curve), 10^{2} (dashed curve) and 10^{3} (dashdotted curve). 

Open with DEXTER  
In the text 
Figure B.2: Normalised linear synchrotron light curve as a function of time at four different energies (solid curve), 10^{4} (dotted curve), 10^{5} (dashed curve) and 10^{6} (dashdotted curve). 

Open with DEXTER  
In the text 
Figure B.3: Normalised nonlinear synchrotron intensity as a function of energy at five different times (solid curve), 10 (dotted curve), 10^{2} (dashed curve), 10^{3} (dashdotted curve) and 10^{5} (longdashed curve). 

Open with DEXTER  
In the text 
Figure B.4: Normalised nonlinear synchrotron light curve as a function of time at four different energies (solid curve), 10^{4} (dotted curve), 10^{5} (dashed curve) and 10^{6} (dashdotted curve). 

Open with DEXTER  
In the text 
Figure B.5: Normalised linear fluence distributions for the full KleinNishina cross section (solid curve) and for the distributionapproximated cross section (dashed curve) for with . 

Open with DEXTER  
In the text 
Figure B.6: Normalised nonlinear fluence distributions for the full KleinNishina cross section (solid curve) and for the distributionapproximated cross section (dashed curve) for with . 

Open with DEXTER  
In the text 
Copyright ESO 2009