Issue 
A&A
Volume 519, September 2010



Article Number  A9  
Number of page(s)  11  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201014087  
Published online  06 September 2010 
Combined synchrotronand nonlinear synchrotronselfCompton cooling of relativistic electrons
R. Schlickeiser^{1}  M. Böttcher^{2}  U. Menzler^{1}
1  Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und Astrophysik, RuhrUniversität Bochum,
44780 Bochum, Germany
2 
Astrophysical Institute, Department of Physics and Astronomy, Clippinger 339, Ohio University, Athens, OH 45701, USA
Received 18 January 2010 / Accepted 29 April 2010
Abstract
The broadband SEDs of blazars exhibit two broad spectral components, which in leptonic emission models
are attributed to synchrotronradiation and SSC radiation of relativistic
electrons. During high state phases, the highfrequency SSC component often
dominates the lowfrequency synchrotron component, implying that the inverse Compton SSC losses of electrons
are at least equal to or greater than the synchrotronlosses of electrons. The linear synchrotroncooling, usually included
in radiation models of blazars, then has to be supplemented by the SSC cooling. Here, we
present an analytical solution to the kinetic equation of relativistic electrons subject to the
combined synchrotron and nonlinear synchrotron selfCompton
cooling for monoenergetic injection. We calculate the timedependent fluxes and timeintegrated fluences
resulting from monoenergetic electrons cooling via synchrotron and SSC, and suggest this as a model for
the broadband SED of Comptondominated blazars.
Key words: radiation mechanisms: nonthermal  BL Lacertae objects: general  gamma rays: galaxies
1 Introduction
The new generation of air Cherenkov TeV ray telescopes such as HESS, MAGIC, and VERITAS (for review see Hinton & Hofmann 2009), have so far detected 27 AGNs as powerful variable highenergy photon emitters. Apart from the three radio galaxies M 87, Cen A, and 3C 66B, the remaining 24 AGNs all belong to the blazar class that is characterized by rapid time variability at all wavelengths and that often have established superluminal motion components at radio and mm frequencies. The combination of high observed luminosities with the observed short time variability in blazar flares indicates that the photon emission in blazars originates in relativistic jet knots that are beamed and Dopplerboosted towards the observer (e.g. Schlickeiser 1996).The broadband continuum spectra of blazars are dominated by nonthermal emission and often consist of two distinct broad components. In leptonic emission models (for review see Böttcher 2007) synchrotronradiation from highly relativistic electrons generates the lowenergy component whereas the highenergy component results from Compton interactions of the same relativistic electron population. A prime candidate for the source photons being Compton upscattered are the selfproduced synchrotron photons, which defines the synchrotron selfCompton (SSC) process (e.g., Maraschi et al. 1992; Bloom & Marscher 1996). Because of the short radiative synchrotronand SSC cooling time scales it is necessary to calculate selfconsistently the time evolution of the energy spectrum of the radiating relativistic electrons when modeling the spectral energy distribution (SED) from these objects.
The SSC emission model has been applied very successfully to represent the SEDs of the highfrequency peaked BL Lac objects PKS 2155304 (Aharonian et al. 2005), 1ES 1121232 (Aharonian et al. 2007), and Mrk 421 (Fossati et al. 2008), for which during high state phases the SSC component in the SED dominates over the synchrotroncomponent. The dominance of the ray (Compton) component in the SED is usually even more pronounced in the lowerpeaked blazars, in particular, flat spectrum radio quasars (FSRQs) like PKS 0528+134 (Sambruna et al. 1997) or lowfrequency peaked BL Lac objects (LBLs) like BL Lacertae (e.g., Madejski et al. 1998). The recent Fermi survey of bright blazars (Abdo et al. 2010a) also revealed many sources with ray dominated SEDs including PKS 0227369, PKS 0347211, PKS 0454234, PKS 1454354, PKS 1502+106, PKS 2325+093, 3C 454.3 (Abdo et al. 2009) and 3C 279 (Abdo et al. 2010b).
In most modelling works reproducing the SEDs of blazars, a single or broken powerlaw distribution of electrons is assumed to produce both the synchrotron and Compton emission. Comparing the required parameters for modelling FSRQs and LBLs in such a framework, many authors have concluded that these Comptondominated blazars are more plausibly represented by ray emission being dominated by Comptonization of external radiation fields, e.g., from the accretion disk or the broad line region (e.g., Madejski et al. 1999; Böttcher & Bloom 2000). Since this conclusion depends on the assumption of an underlying powerlaw electron distribution, we here revisit this problem by investigating the radiative signatures of the injection and subsequent, selfconsistent synchrotron and SSC cooling of monoenergetic electrons, including the case of SSCdominated radiative output.
The dominance of the SSC component over the synchrotroncomponent in this case implies that the inverse
Compton SSC losses of electrons are at least equal to or greater than the synchrotronlosses of electrons,
even more when the intergalactic attenuation of the TeV emission from the cosmic infrared background
is accounted for. The ratio of the observed SSC to synchrotron photon luminosity from the same population of
electrons
directly reflects the ratio of the corresponding loss rates, because of the identical Doppler boosting factors (Dermer & Schlickeiser 2002) of synchrotronand SSC emission. That a distinct third broad emission component did not appear at high photon energies in blazar SEDs can be interpreted in two ways: (i) if the firstorder Compton scattering of synchrotron photons is in the Thomson regime, higher order SSC scattering components then do not operate in the Thomson regime but in the extreme KleinNishina limit with a much reduced scattering cross section; (ii) all SSC components including the firstorder one operate in the extreme KleinNishina limit so that the maximum photon energy is determined by the initial electron energy modulo beaming. Throughout this work we only consider the first case.
In the case of the dominance of the firstorder SSC component over the synchrotroncomponent, Schlickeiser (2009
 hereafter referred to as paper S) pointed out that the linear
synchrotroncooling rate, usually included in radiation models of
blazars, then has to be supplemented by the nonlinear SSC cooling rate.
In the Thomson limit (hereafter referred to as SSTlosses) the SST
energy loss rate of a single electron
depends on the energy integral of the actual electron spectrum , reflecting that the energy integral determines the number density of the target synchrotron photons in the source. The dependence on the energy integral is a collective effect completely different from the linear synchrotronenergy loss rate of a single electron in a constant magnetic field
which is solely determined by the magnetic field strength B=b Gauss and the electron Lorentz factor . The notation in Eqs. (2) and (3) is the same as in paper S: eV^{1} s^{1}, eV, R is the radius of the spherical source, c denotes the speed of light, cm^{2} is the Thomson cross section, and c_{1}=0.684.
The competition between the instantaneous injection of ultrarelativistic electrons
(
)
at the rate
at time t=0 and
the electron radiative losses
is described by the timedependent kinetic equation for the
volumeaveraged relativistic electron population inside the radiating source (Kardashev 1962):
where denotes the volumeaveraged differential number density. S solved this kinetic equation for the two cases, where the radiative loss rate is given either by the linear synchrotronloss rate (3) or the nonlinear SST loss rate (2), for the illustrative case of instantaneous injection of monoenergetic particles. From the respective relativistic electron number densities he then calculated timedependent optically thin synchrotronand SSC intensities, and timeintegrated total synchrotronand SSC fluence distributions, which can be compared with the observed SEDs of blazars. For the illustration case of instantaneous injection of monoenergetic particles, he found significant differences for both (i) the optically thin synchrotronand SSC radiation intensities and (ii) fluenceenergy spectra in the two different cooling cases.
It is purpose of the present analysis to solve the electron kinetic
equation for the more realistic case where the radiative loss rate is
the sum of the linear synchrotronand nonlinear SST cooling rate, i.e.,
In Sect. 2 we calculate the analytic solution of the electron kinetic equation under the combined synchrotronand SST losses (5) for the case of instantaneous injection of monoenergetic particles. The analytic solution is then used in Sects. 3 and 4 to calculate the optically thin synchrotronradiation intensity and total fluence distribution, respectively. In Sect. 5 we demonstrate that our analytical results agree well with synchrotronintensity and fluence resulting from the numerical radiation code of Böttcher et al. (1997) for this case. This code also provides the SSC intensity and fluence distributions.
In our analytical analysis we only take the synchrotronphotons into account as seed photons for electron cooling, and neglect any higher order SSC components. In contrast, the numerical radiation code of Böttcher et al. (1997) consistently accounts for all SSC radiation fields. The comparison of the analytical and numerical synchrotronradiation fluence distributions calculated below will therefore test the validity of the assumptions made in the analytical calculations.
Moreover, the nonlinear SSC electron loss rate (2) is approximated to operate in the Thomson limit. According to S, this limits the initial electron Lorentz factor to , putting an upper limit on the maximum SSC photon energy (modulo beaming) of GeV. Our study therefore applies more to blazars observed in the Fermi survey, whose SEDs peak at subGeV energies, and not to TeV blazars. Throughout this work we therefore consider emission regions that have all of the firstorder SSC emission initially in the Thomson regime. At later times after significant electron cooling, the Thomson approximation is fulfilled even better.
2 Solution of electron kinetic equation for combined synchrotron and SST cooling
For combined synchrotron and SST cooling (5) the kinetic equation of the electrons (4) readsWith the substitution y=A_{0}t and , the kinetic Eq. (6) becomes
where K_{0}=D_{0}/A_{0}. Using as independent variable, we obtain, with ,
Now we define the implicit time variable T through
Then Eq. (8) becomes
which agrees exactly with Eq. (52) of Schlickeiser & Lerche (2007). According to their Eq. (54) the solution is
Now we calculate the time variable T explicitly as a function of y, using Eq. (11) in Eq. (9) to write
for x_{0}>0 and . With z(y)=x_{0}+T(y), Eq. (12) becomes
which after separation of variables with the integration constant C_{1} leads to
or
The integration constant C_{1} is fixed by the condition that T=0 for y=0, yielding
Unfortunately, this dependence y(T) cannot be directly inverted to infer the general dependence T(y). However, an approximate, but reasonably accurate, inversion is possible by using the asymptotic expansions of the function for small and large arguments compared to unity.
2.1 Injection parameter
The argument of the function is always larger than . Therefore, we consider the two cases and , respectively.The parameter
depends on the energy density of the initially injected relativistic electrons and can be written as
with the characteristic Lorentz factor
for standard blazar parameters q_{0}=10^{5}q_{5} cm^{3} and R=10^{15}R_{15} cm. The and are independent of the magnetic field strength. Note that equals the ratio of initial SST to synchrotron losses.
In terms of the total number of instantaneously injected electrons
,
the characteristic Lorentz factor
and the
injection parameter
scale as
if we scale the electron injection Lorentz factor as . Obviously, the more compact the source, and the more electrons are injected, the smaller the characteristic Lorentz factor is. If the injection Lorentz factor is higher (smaller) than , the injection parameter will be larger (smaller) than unity. For a compact source with a large number of injected relativistic electrons, the injection parameter is much larger than unity.
In homogenous spherical sources the initial electron injection luminosity is
from which one can determine the dimensionless injection compactness of the source
The injection parameter (19) scales as .
In Appendix A we demonstrate that for low values of the injection parameter , corresponding to , the time evolution of the electron distribution function is determined solely by the linear synchrotronlosses, and is given by
which agrees with the standard linear synchrotroncooling solution.
For large injection parameters
,
corresponding to
,
nonlinear SST losses determine the electron distribution function at
early times. As a result, the time evolution of the electron
distribution function is affected by the nonlinear SST losses only if
the injection Lorentz factor
exceeds the characteristic value ,
which is determined by the number of injected
electrons and the size of the source. According to Appendix A we obtain for early times
which agrees with the nonlinear SST solution of S, his Eq. (28). At late times
which is a modified linear cooling solution. Both solutions show that at the transition time
the electrons have cooled to the characteristic Lorentz factor .
Summarizing this section, provided electrons are injected with Lorentz factors much higher than , given in Eq. (18), they initially cool down to the characteristic Lorentz factor by nonlinear SSTcooling until time . At later times they cool further to lower energies according to the modified cooling solution (24). If the electrons are injected with Lorentz factors much smaller than , they only undergo linear synchrotroncooling at all energies with no influence from the SST cooling. The characteristic Lorentz factor is only determined by the injection conditions, whereas the time scale also depends on the magnetic field strength.
This different cooling behavior for high and low injection energies affects the synchrotronand SSC intensities and fluences, which we investigate in the next sections.
3 Intrinsic optically thin synchrotron radiation intensities
In this section we analytically investigate the consequences of the combined synchrotronand SST cooling for the intensity spectra of optically thin synchrotronradiation. We closely follow the earlier analysis in S.The optically thin synchrotronintensity from relativistic electrons with the volumeaveraged differential density
is given by
where
denotes the synchrotronpower of a single electron (Crusius & Schlickeiser 1988) in a largescale random magnetic field of constant strength B.
3.1 High injection energy
Inserting the electron density (23) gives at early timeswhere we have introduced the initial characteristic synchrotronphoton energy
and the dimensionless time scale
with the linear synchrotroncooling time
Then
Likewise, inserting the late electron density (24) gives
in terms of the same dimensionless time (29).
The function CS(x) is approximated well by (Crusius & Schlickeiser 1988)
with a_{0}=1.151275 yielding
and
respectively, with the cutoff energies
With respect to photon energy , both synchrotronintensities exhibit the same increasing power law with exponential cutoff behavior; however, the cutoff energy differs for early and late times due to the different electron cooling behavior. Note that .
The cutoff energies (37) determine the time dependence of the peak energy of the synchrotronSED . At early and late times we obtain
and
respectively, which is illustrated in Fig. 1.
Figure 1: Time dependence of the peak energy of the synchrotronSED for high ( , lower curve) and low ( , upper curve) values of the injection parameter. 

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3.2 Low injection energy
For the low injection energy case we use the electron density (22), which in terms of the normalized time scale (29) reads asIn agreement with S we obtain for the synchrotronintensity at all times
here the synchrotronpeak energy
decreases from its intial maximum value (4E_{0}/3) proportional to t^{2} for . The upper curve in Fig. 1 refers to this case. For late times the high and low injection energy cases yield the same decrease of the peak energy.
3.3 Lightcurve peak times
Equations (38), (39), and (42) also provide the photon energy dependences of the intrinsic light curve peak time or the time of maximum intensity of the synchrotronflare . With Eqs. (31) the peak time is related to aswith the linear synchrotroncooling time (31).
For the low injection energy case (
), we reproduce the well known relation
for all energies below 3E_{0}.
For the high injection energy case ( ), we obtain
indicating a steeper powerlaw ( ) at photon energies above , whereas at lower energies the standard ( dependence results. This is also clearly visible in Fig. 2 where we compare the light curve peak times for low and high injection energy conditions. At high photon energies the high injection peak time is a factor shorter than the low injection peak time. This results from the faster additional SST cooling of relativistic electrons in the high injection case.
Figure 2: Photon energy ( ) dependence of the synchrotronlight curve peak time for high ( , full curve) and low ( , dashed curve) values of the injection parameter. 

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4 Fluences
To collect enough photons, intensities are often averaged or integrated over long enough time intervals. For rapidly varying photon intensities, this corresponds to fractional fluences given by the timeintegrated intensitieswith . The total fluence spectra result in the limit
4.1 Low injection energy
The synchrotronintensity (41) yields, after obvious substitutions, for the fractional fluencewith the constant
and . The fractional fluence (48) is identical to Eq. (S52) in S.
For the total fluence we obtain for low ( ) and high ( ) synchrotronphoton energies
where (see Appendix B)
Likewise, the fractional fractional fluence at high energies is
At small energies we find for small integration times
and for late times the timeindependent total fluence spectrum at small energies
4.2 Total fluence for high injection energy
Here the synchrotronintensities (28) and (33) yield, after obvious substitutions, for the total synchrotronfluencewith the constant
For high synchrotronphoton energies , we approximate
and
so that
For intermediate synchrotronphoton energies
,
we approximate
with the constant
and
so that
Finally, for low synchrotronphoton energies , we approximate
and
with the constant (51). In this case the fluence (55) is
For the high injection energy case, the total synchrotronfluence therefore varies as
4.3 Interlude
At high synchrotronphoton energies ( ) the total synchrotronfluences for low and high injection energy exhibit the same exponential cutoff. However, at low energies ( ) we find markedly different powerlaw behaviors for the two injection cases. In the low injection energy case ( ), the total synchrotronfluence exhibits the single powerlaw behavior . In the high injection energy case ( ), the total synchrotronfluence steepens from the powerlaw below to the powerlaw above .4.4 Total fluence synchrotron SED
For the total fluence SED we then find in the two cases of small (s) and high (h) injection energiesand
with the constant
and the characteristic break energy
In terms of the normalized synchrotronphoton energy and for the same , the two total SEDs (68) and (69) are
and
with and
Figure 3 shows the fluence SEDs N(x) for small ( ) and high ( ) injection conditions.
Figure 3: Total synchrotronfluence SED N(x) for high ( , full curve) and small ( , dashed curve) injection conditions calculated for . 

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whereas in the high injection case peaks at x=x_{B} with the peak value
The ratio of peak values is given by
For the case shown in Fig. 3, this ratio is . We summarize the differences for the total synchrotronfluence SEDs and light curves (numerical values hold for the cases shown in Figs. 2 and 3):
 D1) in the high injection case the synchrotronSED peaks at a photon energy, which is a factor less than the peak in the small injection case;
 D2) the high injection energy peak value decreases at early times more rapidly than the small injection energy peak value;
 D3) the high injection SED is a broken powerlaw with spectral indices +0.5 below and 0.5 above the peak energy , respectively, and it cuts off exponentially at photon energies x>1. Below the peak energy x_{B} the time of maximum synchrotronintensity decreases as , whereas above the peak energy x_{B} it decreases more rapidly as due to the severe additional SST losses;
 D4) the small injection SED is a single powerlaw with spectral index +0.5 below the peak energy , and it cuts off exponentially at photon energies x>1. Here, the time of maximum synchrotronintensity decreases as at all energies x<1 because in the small injection case the SSTlosses do not contribute.
5 Synchrotron and SSC fluence SEDs and light curves from numerical radiation code
In Fig. 4 we show the photon energy variation in the light curve peak time calculated with the numerical radiation code of Böttcher et al. (1997) using a magnetic field strength b=1 and an injection Lorentz factor for the high ( ), small ( ) and intermediate () injection cases. The numerical variations for and are in perfect agreement with the different powerlaw variations found analytically, which are included in Fig. 4 for orientation, and they confirm our earlier findings. For the intermediate case , for which no analytical solution is possible, one observes a peak time variation that lies between the variation and the variation of the high and small injection cases.Figure 4: Numerically calculated synchrotronlight curve peak times for small ( ), intermediate , and high ( ) injection conditions calculated for and b=1. 

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Figure 5: Numerically calculated fractional and total synchrotronand SSC fluence SEDs for high ( ) injection conditions calculated for . The full curves show the total fluence SEDs. The dashed and dotdashed lines show the asymptotic behavior of the total synchrotronfluence according to Eq. (67). The fractional fluences are shown at approximately logarithmically spaced times. The SSC emission has been artificially cut off at low frequencies as it would otherwise overwhelm the highenergy end of the synchrotron emission. 

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In Fig. 7 we show the numerical synchrotronand SSC fluence SEDs using a magnetic field strength b=1 and an injection Lorentz factor for the intermediate () injection case. The spectral form of the synchrotronand SSC fluences are similar to the small injection case shown in Fig. 6, but the SSC peak fluence is only a factor 3 less than the synchrotronpeak fluence. Apparently, for no strong influence of nonlinear SST losses occurs for . For orientation, we have also plotted the analytic asymptotic behavior of the total synchrotronfluence of the small case according to Eq. (50) in this plot, which agrees well with the numerical spectral behavior.
Figure 6: Numerically calculated fractional and total synchrotronand SSC fluence SEDs for small ( ) injection conditions calculated for . The full curves show the total fluence SEDs. The dashed lines show the asymptotes according to Eq. (50). The fractional fluences are shown at approximately logarithmically spaced times. 

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Figure 7: Numerically calculated fractional and total synchrotronand SSC fluence SEDs for intermediate () injection conditions calculated for . The full curves show the total fluence SEDs. The dashed and dotdashed lines show the asymptotic behavior of the small case according to Eq. (50). The fractional fluences are shown at approximately logarithmically spaced times. 

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The radiation code also yields the SSC fluence SEDs. We note from Figs. 57 that for the high injection case, the SSC SED has a much higher amplitude than the synchrotronSED, whereas the opposite holds for the low and intermediate injection cases. Moreover, all SSC SEDs peak at the same photon energy, although the SSC peak value in the high injection case is a factor larger than in the small injection case.
6 Summary and conclusions
We have presented an analytical solution to the synchrotron and nonlinear synchrotronselfCompton cooling in the Thomson regime (SST) of monoenergetic electrons. Based on our analytical solution, we evaluated the timedependent synchrotron emissivity and timeintegrated fluences. We find qualitatively different results depending on whether electron cooling is initially Comptondominated (high injection energy parameter ), or it is always synchrotron dominated (low ). In the low case, the resulting fluence spectrum exhibits a simple exponentially cutoff powerlaw behavior, . In contrast, in the high case, we find a broken powerlaw with exponential cutoff, parametrized in the form . The analytically calculated synchrotronfluence SEDs and light curve peak times agree well with the corresponding numerically calculated quantities using the radiation code of Böttcher et al. (1997), which justifies the validity of the assumptions made in the analytical calculations.Based on our analysis we propose the following interpretation of multiwavelength blazar SEDs. Blazars, where the ray fluence is much stronger than the synchrotronfluence, are regarded as high injectionenergy sources. Here, the synchrotronfluence should exhibit the symmetric broken powerlaw behavior Eq. (67) around the synchrotronpeak energy that is a factor lower than the SSC peak energy. Below and above , the synchrotronlight curve peak times exhibit different frequency dependences and , respectively, resulting from the additional severe SSTlosses at .
Blazars, where the ray fluence is much less than the synchrotronfluence, are regarded as small injectionenergy sources. Here, the synchrotronfluence exhibits the single powerlaw behavior (D4) up to a higher synchrotronpeak energy that is a factor lower than the SSC peak energy. In this case, the synchrotronlight curve peak time exhibits the standard linear synchrotroncooling decrease at all frequencies.
If the injection Lorentz factor and the size of the source are the same, different values of the injection parameter result from different total numbers of instantaneously injected electrons, see Eq. (19); for example, the high injection case results for N_{50}=4.7, whereas the low injection case needs .
AcknowledgementsWe thank the referee for his/her constructive comments. R.S. acknowledges support from the German Ministry for Education and Research (BMBF) through Verbundforschung Astroteilchenphysik grant 05A08PC1 and the Deutsche Forschungsgemeinschaft through grants Schl 201/201 and Schl 201/231. M.B. acknowledges support from NASA through Fermi Guest Investigator grant NNX09AT82G and Astrophysics Theory Program Grant NNX10AC79G. This work was completed when M.B. was visiting guest scientist of the Research Department Plasmas with Complex Interactions at RuhrUniversity Bochum.
Appendix A: Determination of electron distribution functions for low and high injection energies
A.1 Low injection energy
In the case of low injection energies , the injection parameter is less than unity, so that the argument of the function in Eq. (15) is always greater than unity. For all values of T and y, Eq. (15) then simplifies toand with C_{0}=x_{0} to
In terms of y, the solution (12) then reads with x_{0}>0
yielding
which agrees with the standard linear synchrotroncooling solution.
Figure A.1: Comparison of exact solution y(T) (full curve) with asymptotic solution y_{1}(T) at early times (dashed curve) and asymptotic solution y_{2}(T) at late times (dotted curve) for the high injection energy case with q_{5}=1, x_{0}=10^{4} and K_{0}=1, corresponding to . 

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A.2 High injection energy
In the case of high injection energies the injection parameter is greater than unity. With the injection parameter (17) we rewrite Eq. (15) asFor times , where
we expand to third order in x to obtain
or
With T=0 for y=0, the integration constant C_{2}=x_{0}^{3}/3q_{0} is fixed so that
This solution is valid for , corresponding with Eq. (A.6) to
For times or , the argument of the function in Eq. (A.5) is greater than unity, yielding
or the linear relation
The constant C_{4} is determined by the equality of the two solutions at providing
so that
In Fig. A.1 we compare the two approximate solutions (A.9) and (A.14) with the exact solution (16). The agreement is reasonably good.
Both approximate solutions (A.9) and (A.14) can be inverted to yield
and
In terms of y we obtain from Eq. (11) the two solutions
and
We then find for early times
which agrees with the nonlinear SST solution of S, his Eq. (28). At late times,
which is a modified linear cooling solution. Both solutions show that, at time
the electrons have cooled to the characteristic Lorentz factor .
Appendix B: Calculation of synchrotron radiation constants
Here we determine the constantsfor general values of k>1/3. The integral (B.1) is directly related to the integral G^{'}(s) computed in Eq. (29) of Crusius & Schlickeiser (1986) if s=2k+1. This yields
with the triplication formula of gamma functions for z=(3k1)/6, Eq. (B.2) simplifies to
We obtain c(1/2)=0.95302 and c(3/2)=0.67015.
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All Figures
Figure 1: Time dependence of the peak energy of the synchrotronSED for high ( , lower curve) and low ( , upper curve) values of the injection parameter. 

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In the text 
Figure 2: Photon energy ( ) dependence of the synchrotronlight curve peak time for high ( , full curve) and low ( , dashed curve) values of the injection parameter. 

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In the text 
Figure 3: Total synchrotronfluence SED N(x) for high ( , full curve) and small ( , dashed curve) injection conditions calculated for . 

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In the text 
Figure 4: Numerically calculated synchrotronlight curve peak times for small ( ), intermediate , and high ( ) injection conditions calculated for and b=1. 

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In the text 
Figure 5: Numerically calculated fractional and total synchrotronand SSC fluence SEDs for high ( ) injection conditions calculated for . The full curves show the total fluence SEDs. The dashed and dotdashed lines show the asymptotic behavior of the total synchrotronfluence according to Eq. (67). The fractional fluences are shown at approximately logarithmically spaced times. The SSC emission has been artificially cut off at low frequencies as it would otherwise overwhelm the highenergy end of the synchrotron emission. 

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In the text 
Figure 6: Numerically calculated fractional and total synchrotronand SSC fluence SEDs for small ( ) injection conditions calculated for . The full curves show the total fluence SEDs. The dashed lines show the asymptotes according to Eq. (50). The fractional fluences are shown at approximately logarithmically spaced times. 

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In the text 
Figure 7: Numerically calculated fractional and total synchrotronand SSC fluence SEDs for intermediate () injection conditions calculated for . The full curves show the total fluence SEDs. The dashed and dotdashed lines show the asymptotic behavior of the small case according to Eq. (50). The fractional fluences are shown at approximately logarithmically spaced times. 

Open with DEXTER  
In the text 
Figure A.1: Comparison of exact solution y(T) (full curve) with asymptotic solution y_{1}(T) at early times (dashed curve) and asymptotic solution y_{2}(T) at late times (dotted curve) for the high injection energy case with q_{5}=1, x_{0}=10^{4} and K_{0}=1, corresponding to . 

Open with DEXTER  
In the text 
Copyright ESO 2010
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