Free Access
Issue
A&A
Volume 519, September 2010
Article Number A9
Number of page(s) 11
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201014087
Published online 06 September 2010
A&A 519, A9 (2010)

Combined synchrotronand nonlinear synchrotron-self-Compton cooling of relativistic electrons

R. Schlickeiser1 - M. Böttcher2 - U. Menzler1

1 - Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität 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 high-frequency SSC component often dominates the low-frequency 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 self-Compton cooling for monoenergetic injection. We calculate the time-dependent fluxes and time-integrated fluences resulting from monoenergetic electrons cooling via synchrotron and SSC, and suggest this as a model for the broadband SED of Compton-dominated blazars.

Key words: radiation mechanisms: non-thermal - BL Lacertae objects: general - gamma rays: galaxies

1 Introduction

The new generation of air Cherenkov TeV $\gamma $-ray telescopes such as HESS, MAGIC, and VERITAS (for review see Hinton & Hofmann 2009), have so far detected 27 AGNs as powerful variable high-energy 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 Doppler-boosted 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 low-energy component whereas the high-energy component results from Compton interactions of the same relativistic electron population. A prime candidate for the source photons being Compton upscattered are the self-produced synchrotron photons, which defines the synchrotron self-Compton (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 self-consistently 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 high-frequency peaked BL Lac objects PKS 2155-304 (Aharonian et al. 2005), 1ES 1121-232 (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 $\gamma $-ray (Compton) component in the SED is usually even more pronounced in the lower-peaked blazars, in particular, flat spectrum radio quasars (FSRQs) like PKS 0528+134 (Sambruna et al. 1997) or low-frequency 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 $\gamma $-ray dominated SEDs including PKS 0227-369, PKS 0347-211, PKS 0454-234, PKS 1454-354, 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 power-law 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 Compton-dominated blazars are more plausibly represented by $\gamma $-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 power-law electron distribution, we here revisit this problem by investigating the radiative signatures of the injection and subsequent, self-consistent synchrotron and SSC cooling of monoenergetic electrons, including the case of SSC-dominated 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  $n(\gamma )$

\begin{displaymath}{L^*_{\rm SSC}\over L^*_{\rm sy}}={\int {\rm d}V\int_1^\infty...
... {\rm d}\gamma \; n(\gamma )\vert \dot{\gamma _{\rm S}}\vert }
\end{displaymath} (1)

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 first-order 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 Klein-Nishina limit with a much reduced scattering cross section; (ii) all SSC components including the first-order one operate in the extreme Klein-Nishina 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 first-order 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 SST-losses) the SST energy loss rate of a single electron

\begin{displaymath}\vert\dot{\gamma }\vert _{\rm SST}\simeq A_0\gamma ^2\int_0^\...
...) ,
\;\;
A_0={3\sigma _{\rm T}c_1P_0R\epsilon_0 ^2\over mc^2}
\end{displaymath} (2)

depends on the energy integral of the actual electron spectrum $n(\gamma ,t)$, 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

\begin{displaymath}\vert\dot{\gamma }\vert _{\rm S}=D_0\gamma ^2,\;\;\; D_0={4\o...
...{\rm T}\over mc^2}U_B=1.29\times 10^{-9}b^2\;~~ \hbox{s}^{-1},
\end{displaymath} (3)

which is solely determined by the magnetic field strength B=b Gauss and the electron Lorentz factor $\gamma $. The notation in Eqs. (2) and (3) is the same as in paper S: $P_0=3.2\times 10^{12}$ eV-1 s-1, $\epsilon_0 =1.16\times 10^{-8}b$ eV, R is the radius of the spherical source, c denotes the speed of light, $\sigma _{\rm T}=6.65\times 10^{-25}$ cm2 is the Thomson cross section, and c1=0.684.

The competition between the instantaneous injection of ultrarelativistic electrons ( $\gamma _0\gg 1$) at the rate $Q(\gamma ,t)=q_0\delta (\gamma -\gamma _0)\delta (t)$ at time t=0 and the electron radiative losses $\vert\dot{\gamma }\vert$ is described by the time-dependent kinetic equation for the volume-averaged relativistic electron population inside the radiating source (Kardashev 1962):

\begin{displaymath}{\partial n(\gamma ,t)\over \partial t} -
{\partial \over \pa...
...t n(\gamma,t)\right]=
q_0\delta (\gamma -\gamma _0)\delta (t),
\end{displaymath} (4)

where $n(\gamma ,t)$ denotes the volume-averaged 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 time-dependent optically thin synchrotronand SSC intensities, and time-integrated 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) fluence-energy 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.,

\begin{displaymath}\vert\dot{\gamma }\vert=\vert\dot{\gamma }\vert _{\rm S}+\ver...
...\rm d}\tilde\gamma ~ \tilde\gamma^2 n(\tilde\gamma ,t)\right].
\end{displaymath} (5)

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 $\gamma _0<1.94\times 10^4b^{-1/3}$, putting an upper limit on the maximum SSC photon energy (modulo beaming) of $\epsilon _{\rm SSC,max}=4\epsilon_0\gamma _0^4<6.6b^{-1/3}$ GeV. Our study therefore applies more to blazars observed in the Fermi survey, whose SEDs peak at sub-GeV energies, and not to TeV blazars. Throughout this work we therefore consider emission regions that have all of the first-order 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) reads

\begin{displaymath}{\partial n(\gamma ,t)\over \partial t}
-{\partial \over \pa...
...ilde\gamma ~
\tilde\gamma^2 n(\tilde\gamma ,t)\right)\right]=
\end{displaymath}


\begin{displaymath}\qquad \qquad q_0 \delta (\gamma -\gamma _0) \delta (t).
\end{displaymath} (6)

With the substitution y=A0t and $S (\gamma, t) = \gamma^2 n(\gamma, t)$, the kinetic Eq. (6) becomes

\begin{displaymath}{\partial S\over \partial y} -
\left[K_0 + \int_0^\infty {\r...
... \gamma } =
q_0\gamma ^2\delta (\gamma -\gamma _0)\delta (y),
\end{displaymath} (7)

where K0=D0/A0. Using $x=1/\gamma $ as independent variable, we obtain, with $x_0=1/\gamma _0$,

\begin{displaymath}{\partial S\over \partial y} +
{\partial S\over \partial x}\l...
...e x^{-2} S(\tilde x, y)\right]=
q_0\delta (x - x_0)\delta (y).
\end{displaymath} (8)

Now we define the implicit time variable T through

\begin{displaymath}{{\rm d}T\over {\rm d}y}=U(y)=K_0+\int_0^\infty {\rm d}x x^{-2}S(x, y).
\end{displaymath} (9)

Then Eq. (8) becomes

\begin{displaymath}{\partial S\over \partial T}+{\partial S\over \partial x}=q_0\delta (x-x_0)\delta (T),
\end{displaymath} (10)

which agrees exactly with Eq. (52) of Schlickeiser & Lerche (2007). According to their Eq. (54) the solution is

\begin{displaymath}S(x,T)=q_0\delta (x-T-x_0)\left(H[T]-H[T-x]\right).
\end{displaymath} (11)

Now we calculate the time variable T explicitly as a function of y, using Eq. (11) in Eq. (9) to write

\begin{displaymath}{{\rm d}T\over {\rm d}y}=K_0+\int_0^\infty {\rm d}x x^{-2}S(x, y)=
\end{displaymath}

\begin{displaymath}K_0+q_0\int_0^\infty {\rm d}x x^{-2}\delta (x-T-x_0)\left(H[T]-H[T-x]\right)=
\end{displaymath}

\begin{displaymath}K_0+q_0H[T]H[x_0]{1\over (x_0+T)^2}=K_0+{q_0\over (x_0+T)^2}
\end{displaymath} (12)

for x0>0 and $T\ge 0$. With z(y)=x0+T(y), Eq. (12) becomes

\begin{displaymath}{{\rm d}z\over {\rm d}y}=K_0+{q_0\over z^2}={q_0+K_0z^2\over z^2},
\end{displaymath} (13)

which after separation of variables with the integration constant C1 leads to

\begin{displaymath}K_0y+C_1=z-\int {{\rm d}z \over 1+{K_0z^2\over q_0}}= z-\sqrt{q_0\over K_0}\arctan \left(\sqrt{K_0\over q_0}z\right),
\end{displaymath} (14)

or

\begin{displaymath}x_0+T(y)-\sqrt{q_0\over K_0}\arctan \left(\sqrt{K_0\over q_0}[x_0+T(y)]\right)
=K_0y+C_1.
\end{displaymath} (15)

The integration constant C1 is fixed by the condition that T=0 for y=0, yielding

\begin{displaymath}K_0y=T-\sqrt{q_0\over K_0}\Bigg[\arctan \left(\sqrt{K_0\over q_0}[x_0+T(y)]\right)
\end{displaymath}

\begin{displaymath}\qquad -\arctan \left(\sqrt{K_0\over q_0}x_0\right)\Bigg].
\end{displaymath} (16)

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 $\arctan $-function for small and large arguments compared to unity.

2.1 Injection parameter

The argument of the $\arctan $-function is always larger than $\alpha ^{-1}=x_0(K_0/q_0)^{1/2}$. Therefore, we consider the two cases $\alpha \gg 1$ and $\alpha \ll 1$, respectively.

The parameter $\alpha $ depends on the energy density of the initially injected relativistic electrons and can be written as

\begin{displaymath}\alpha ={q_0^{1/2}\over K_0^{1/2}x_0}={q_0^{1/2}\gamma _0\over K_0^{1/2}}={\gamma _0\over \gamma _B}
\end{displaymath} (17)

with the characteristic Lorentz factor

\begin{displaymath}\gamma _B={K_0^{1/2}\over q_0^{1/2}}={2\over 3}\sqrt{cU_B\over c_1P_0R\epsilon_0 ^2q_0}={106\over R_{15}^{1/2}q_5^{1/2}}
\end{displaymath} (18)

for standard blazar parameters q0=105q5 cm-3 and R=1015R15 cm. The $\gamma _B$ and $\alpha $ are independent of the magnetic field strength. Note that $\alpha ^2$ equals the ratio of initial SST to synchrotron losses.

In terms of the total number of instantaneously injected electrons $N=4\pi R^3q_0/3=10^{50}N_{50}$, the characteristic Lorentz factor $\gamma _B$ and the injection parameter $\alpha $ scale as

\begin{displaymath}\gamma _B={217R_{15}\over N_{50}^{1/2}},~\;\alpha =46{\gamma _4N_{50}^{1/2}\over R_{15}},
\end{displaymath} (19)

if we scale the electron injection Lorentz factor as $\gamma _0=10^4\gamma _4$. Obviously, the more compact the source, and the more electrons are injected, the smaller the characteristic Lorentz factor $\gamma _B$ is. If the injection Lorentz factor $\gamma _0$ is higher (smaller) than $\gamma _B$, the injection parameter $\alpha $ will be larger (smaller) than unity. For a compact source with a large number of injected relativistic electrons, the injection parameter $\alpha $ is much larger than unity.

In homogenous spherical sources the initial electron injection luminosity is

\begin{displaymath}L_{\rm e}=4\pi R^2c\gamma _0mc^2q_0=3c\gamma _0mc^2N/R
\end{displaymath}

\begin{displaymath}\qquad \qquad\qquad\qquad=7.4\times 10^{43}{\gamma _4N_{50}\over R_{15}}\;\; \hbox{erg/s},
\end{displaymath} (20)

from which one can determine the dimensionless injection compactness of the source

\begin{displaymath}l_{\rm e}={\sigma _{\rm T}L_{\rm e}\over 3\pi Rmc^2}={\sigma ...
...r \pi R^2}=6.35\times 10^9{\gamma _4N_{50}\over R_{15}^2}\cdot
\end{displaymath} (21)

The injection parameter (19) scales as $\alpha =5.8\times 10^{-4}\gamma _4^{1/2}l_{\rm e}^{1/2}$.

In Appendix A we demonstrate that for low values of the injection parameter $\alpha \ll 1$, corresponding to $\gamma _0\ll \gamma _{{B}}$, the time evolution of the electron distribution function is determined solely by the linear synchrotronlosses, and is given by

\begin{displaymath}n(\gamma ,\gamma _0,t)={q_0\over \gamma ^2}H[\gamma _0-\gamma ]
\delta \left(\gamma ^{-1}-\gamma _0^{-1}-D_0t\right)
\end{displaymath}


\begin{displaymath}\qquad\qquad =q_0H[\gamma _0-\gamma ]\delta \left(\gamma -{\gamma _0\over 1+D_0\gamma _0t}\right),
\end{displaymath} (22)

which agrees with the standard linear synchrotroncooling solution.

For large injection parameters $\alpha \gg 1$, corresponding to $\gamma _0\gg \gamma _{{B}}$, 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 $\gamma _0$ exceeds the characteristic value $\gamma _B$, which is determined by the number of injected electrons and the size of the source. According to Appendix A we obtain for early times

\begin{displaymath}n_1(\gamma ,\gamma _0,t<t_{\rm c})=
\end{displaymath}

\begin{displaymath}\qquad q_0H[\gamma _0-\gamma ]H[t_{\rm c}-t]\delta \left(\gamma -{\gamma _0\over (1+3\alpha ^2\gamma _0K_0A_0t)^{1/3}}\right)
\end{displaymath}

\begin{displaymath}\qquad =q_0H[\gamma _0-\gamma ]H[t_{\rm c}-t]\delta \left(\gamma -{\gamma _0\over (1+3q_0\gamma _0^3A_0t)^{1/3}}\right),
\end{displaymath} (23)

which agrees with the nonlinear SST solution of S, his Eq. (28). At late times

\begin{displaymath}n_2(\gamma ,\gamma _0,t\ge t_{\rm c})=
\end{displaymath}
\begin{displaymath}q_0H[\gamma _B-\gamma ]H[t-t_{\rm c}]\delta \left(\gamma -{\g...
...\over {1+2\alpha ^3\over 3\alpha ^3}+\gamma _BK_0A_0t}\right),
\end{displaymath} (24)

which is a modified linear cooling solution. Both solutions show that at the transition time

\begin{displaymath}t_{\rm c}={\alpha ^3-1\over 3\alpha ^3\gamma _BD_0}\simeq {1\over 3\gamma _BD_0}
\end{displaymath}
\begin{displaymath}\quad ={2.6\times 10^8\over \gamma _Bb^2}~ \hbox{s}={1.2\times 10^6N_{50}^{1/2}\over R_{15}b^2}~ \hbox{s}
\end{displaymath} (25)

the electrons have cooled to the characteristic Lorentz factor $\gamma _B$.

Summarizing this section, provided electrons are injected with Lorentz factors much higher than $\gamma _B$, given in Eq. (18), they initially cool down to the characteristic Lorentz factor $\gamma _B$ by nonlinear SST-cooling until time $t_{\rm c}$. 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 $\gamma _B$, they only undergo linear synchrotroncooling at all energies with no influence from the SST cooling. The characteristic Lorentz factor $\gamma _B$ is only determined by the injection conditions, whereas the time scale $t_{\rm c}$ 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 volume-averaged differential density  $n(\gamma ,t)$ is given by

\begin{displaymath}I(\epsilon,t)=Rj_{\rm S}(\epsilon,t)={R\over 4\pi }\int_0^\infty {\rm d}\gamma ~ n(\gamma ,t)p_{\rm S}(\epsilon,\gamma ),
\end{displaymath} (26)

where

\begin{displaymath}p_{\rm s}(\epsilon , \gamma )={P_0\epsilon \over \gamma ^2}CS \left({2\epsilon\over 3\epsilon_0 \gamma ^2}\right)
\end{displaymath} (27)

denotes the synchrotronpower of a single electron (Crusius & Schlickeiser 1988) in a large-scale random magnetic field of constant strength B.

3.1 High injection energy

Inserting the electron density (23) gives at early times $t<t_{\rm c}$

\begin{displaymath}I_1(\epsilon, \tau <\tau _{\rm c})=
{3RP_0q_0\epsilon_0 \epsi...
...ght]^{2/3}
CS \left({\epsilon[1+\tau ]^{2/3}\over E_0}\right),
\end{displaymath} (28)

where we have introduced the initial characteristic synchrotronphoton energy

\begin{displaymath}E_0={3\over 2}\epsilon_0 \gamma _0^2=1.74b\gamma _4^2\;\;\; \hbox{eV}
\end{displaymath} (29)

and the dimensionless time scale

\begin{displaymath}\tau =3A_0q_0\gamma _0^3t=3\alpha ^2D_0\gamma _0t=3\alpha ^2t/t_{\rm s},
\end{displaymath} (30)

with the linear synchrotroncooling time

\begin{displaymath}t_{\rm s}={7.75\times 10^4\over b^2\gamma _4}\;\; \hbox{s}.
\end{displaymath} (31)

Then

\begin{displaymath}\tau _{\rm c}=3A_0q_0\gamma _0^3t_{\rm c}={q_0\gamma _0^3\over \gamma _BK_0}{\alpha ^3-1\over \alpha ^3}=\alpha ^3-1.
\end{displaymath} (32)

Likewise, inserting the late electron density (24) gives

\begin{displaymath}I_2(\epsilon,\tau \ge \tau _{\rm c})={RP_0q_0\epsilon\over 4\...
...t(\gamma -{3\alpha ^3\gamma _B\over 1+2\alpha ^3+\tau }\right)
\end{displaymath}
\begin{displaymath}\qquad\qquad\qquad\times CS \left({2\epsilon\over 3\epsilon_0 \gamma ^2}\right)
\end{displaymath}
\begin{displaymath}\qquad \qquad\quad~~={RP_0q_0\epsilon_0 \epsilon\over 24\pi \alpha ^4E_0}\left(1+2\alpha ^3+\tau \right)^2
\end{displaymath}
\begin{displaymath}\qquad\qquad\qquad\times CS \left({\epsilon\over 9\alpha ^4E_0}\left[1+2\alpha ^3+\tau \right]^2\right)
\end{displaymath} (33)

in terms of the same dimensionless time (29).

The function CS(x) is approximated well by (Crusius & Schlickeiser 1988)

\begin{displaymath}CS(x)=a_0x^{-2/3}{\rm e}^{-x}
\end{displaymath} (34)

with a0=1.151275 yielding
$\displaystyle I_1(\epsilon,\tau <\tau _{\rm c})={3a_0\over 8\pi }RP_0q_0\epsilon_0 (\epsilon/E_0)^{1/3}\left[1+\tau \right]^{2/9}$      
$\displaystyle \times
\exp \left(-{\epsilon\over E_0}\left[1+\tau \right]^{2/3}\right)$      
$\displaystyle \propto (\epsilon/\epsilon_0 )^{1/3}\left[1+\tau \right]^{2/9}\exp \left( -{\epsilon\over \epsilon_1(\tau )}\right)$     (35)

and

\begin{displaymath}I_2(\epsilon,\tau \ge \tau _{\rm c})={3^{1/3}a_0RP_0q_0\epsil...
...4/3}}(\epsilon/E_0)^{1/3}\left[1+2\alpha ^3+\tau \right]^{2/3}
\end{displaymath}
\begin{displaymath}\qquad \qquad\qquad \times
\exp \left(-{\epsilon\over 9\alpha ^4E_0}\left[1+2\alpha ^3+\tau \right]^2\right)
\end{displaymath}
\begin{displaymath}\qquad \qquad\quad\propto (\epsilon/\epsilon_0 )^{1/3}\left[1...
...]^{2/3}\exp \left( -{\epsilon\over \epsilon_2(\tau )}\right)
,
\end{displaymath} (36)

respectively, with the cut-off energies

\begin{displaymath}\epsilon_1(\tau \le \tau _{\rm c})=E_0(1+\tau )^{-2/3},\;
\end{displaymath}

\begin{displaymath}\epsilon_2(\tau \ge \tau _{\rm c})=9\alpha ^4E_0\left[1+2\alpha ^3+\tau \right]^{-2}.
\end{displaymath} (37)

With respect to photon energy $\epsilon$, both synchrotronintensities exhibit the same increasing power law with exponential cut-off behavior; however, the cut-off energy differs for early and late times due to the different electron cooling behavior. Note that $\epsilon_1(\tau _{\rm c})=\epsilon_2(\tau _{\rm c})=E_0/\alpha ^2$.

The cut-off energies (37) determine the time dependence of the peak energy $\epsilon _{\rm p}(\tau )$ of the synchrotronSED $\epsilon I(\epsilon,\tau )$. At early and late times we obtain

\begin{displaymath}\epsilon_{\rm p}(\tau <\tau _{\rm c})={4\epsilon_1\over 3}={4...
...u )^{2/3}}={4E_0\over 3(1+3\alpha ^2{t\over t_{\rm s}})^{2/3}}
\end{displaymath} (38)

and

\begin{displaymath}\epsilon_{\rm p}(\tau \ge \tau _{\rm c})={4\epsilon_1\over 3}={12\alpha ^4E_0\over [1+2\alpha ^3+\tau ]^2}
\end{displaymath}
\begin{displaymath}\qquad\qquad ={12\alpha ^4E_0\over [1+2\alpha ^3+3\alpha ^2{t\over t_{\rm s}})]^2},
\end{displaymath} (39)

respectively, which is illustrated in Fig. 1.

\begin{figure}
\par\includegraphics[width=9cm]{14087fg1}
\end{figure} Figure 1:

Time dependence of the peak energy $\epsilon _{\rm p}(\tau )$ of the synchrotronSED for high ( $\alpha =10$, lower curve) and low ( $\alpha =0.1$, upper curve) values of the injection parameter.

Open with DEXTER
For the high injection case the synchrotronpeak energy decreases from its intial maximum value $E_{\rm p, max}=(4E_0/3)$ proportional to $(1+\tau )^{-2/3}$ for early times $\tau <\tau _{\rm c}=\alpha ^3-1$ to $E_{\rm p}=E_{\rm p, max}/\alpha ^2$. At later times $\tau \ge \tau _{\rm c}$ the peak energy decreases further proportional to $\tau ^{-2}$.

3.2 Low injection energy

For the low injection energy case $\alpha \ll 1$ we use the electron density (22), which in terms of the normalized time scale (29) reads as

\begin{displaymath}n(\gamma ,\gamma _0, \tau )=q_0H[\gamma _0-\gamma ]\delta \left(\gamma -{\gamma _0\over 1+{\tau \over 3\alpha ^2}}\right)\cdot
\end{displaymath} (40)

In agreement with S we obtain for the synchrotronintensity at all times

\begin{displaymath}I(\epsilon,\tau )={3RP_0q_0\epsilon_0 \epsilon\over 8\pi E_0}...
...ilon\over E_0}\left[1+{\tau \over 3\alpha ^2}\right]^2\right)
\end{displaymath}
\begin{displaymath}\qquad \quad\simeq{3a_0RP_0q_0\epsilon_0 \over 8\pi }\left({\...
... E_0}\right)^{1/3}\left(1+{\tau \over 3\alpha ^2}\right)^{2/3}
\end{displaymath}
\begin{displaymath}\qquad \qquad\times \exp \left(-{\epsilon\over E_0}\left[1+{\tau \over 3\alpha ^2}\right]^2\right)
\end{displaymath} (41)

here the synchrotronpeak energy

\begin{displaymath}\epsilon_{\rm p}={4E_0\over 3(1+{\tau \over 3\alpha ^2})^2}={4E_0\over 3(1+{t\over t_{\rm s}})^2}
\end{displaymath} (42)

decreases from its intial maximum value (4E0/3) proportional to t-2 for $t>t_{\rm s}$. The upper curve in Fig. 1 refers to this case. For late times the high and low injection energy cases yield the same $\propto
t^{-2}$-decrease of the peak energy.

3.3 Light-curve 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 $t_{\rm max}(\epsilon)$. With Eqs. (31) the peak time is related to $\tau _{\rm max}(\epsilon)$ as

\begin{displaymath}t_{\rm max}(\epsilon)=t_{\rm s}{\tau _{\rm max}(\epsilon)\over 3\alpha ^2}
\end{displaymath} (43)

with the linear synchrotroncooling time (31).

For the low injection energy case ( $\alpha \ll 1$), we reproduce the well known relation

\begin{displaymath}t_{\rm max}(\epsilon, \alpha \ll 1)=t_{\rm s}\left({E_0\over 3\epsilon}\right)^{1/2}-1
\end{displaymath} (44)

for all energies below 3E0.

For the high injection energy case ( $\alpha \gg 1$), we obtain

\begin{displaymath}t_{\rm max}(\epsilon, \alpha \gg 1)=t_{\rm s}
\cases{\left({E...
.../2}-1\right] & for $\epsilon\ge {E_0\over 3\alpha ^2}$ , \cr }
\end{displaymath} (45)

indicating a steeper power-law ( $t_{\rm max}\propto \epsilon^{-3/2}$) at photon energies above $E_0/3\alpha ^2$, whereas at lower energies the standard ($\propto $ $\epsilon^{-1/2})$ 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 $\epsilon>E_0/3\alpha ^2$ the high injection peak time is a factor $3\alpha ^ 2$ shorter than the low injection peak time. This results from the faster additional SST cooling of relativistic electrons in the high injection case.

\begin{figure}
\par\includegraphics[width=9cm]{14087fg2}
\end{figure} Figure 2:

Photon energy ( $x=\epsilon /E_0$) dependence of the synchrotronlight curve peak time for high ( $\alpha =100$, full curve) and low ( $\alpha =0.1$, 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 time-integrated intensities

\begin{displaymath}F_{\rm f}(\epsilon, t_{\rm f})=\int_0^{t_{\rm f}} {\rm d}t~ I...
...mma _0^3}\int_0^{\tau _{\rm f}}{\rm d}\tau ~ I(\epsilon,\tau )
\end{displaymath} (46)

with $\tau _{\rm f}=3A_0q_0\gamma _0^3t_{\rm f}$. The total fluence spectra result in the limit $t_{\rm f}\to \infty $

\begin{displaymath}F(\epsilon)=F_{\rm f}(\epsilon, t_{\rm f}=\infty )=\int_0^\infty {\rm d}t~ I(\epsilon,t)
\end{displaymath}
\begin{displaymath}\qquad \qquad\qquad\qquad={1\over 3A_0q_0\gamma _0^3}\int_0^\infty {\rm d}\tau ~ I(\epsilon,\tau ).
\end{displaymath} (47)

4.1 Low injection energy

The synchrotronintensity (41) yields, after obvious substitutions, for the fractional fluence

\begin{displaymath}F_{\rm f,s}(\epsilon, \tau _{\rm f})={RP_0\epsilon\over 12\pi A_0\gamma _0^5}
\end{displaymath}
\begin{displaymath}\qquad \qquad\quad\times \int_0^{\tau _{\rm f}}{\rm d}\tau ~ ...
...r\tau \right]^2CS \left({\epsilon\over E_0}[1+r\tau ]^2\right)
\end{displaymath}
\begin{displaymath}\qquad\qquad=F_{\rm0S}\left({E_0\over \epsilon}\right)^{1/2}
...
...lon(1+r\tau _{\rm f})^2/E_0}{\rm d}x~ x^{1/2}CS \left(x\right)
\end{displaymath} (48)

with the constant

\begin{displaymath}F_{\rm0S}={RP_0E_0\over 24\pi rA_0\gamma _0^5}={RP_0\epsilon_...
...a _0^3}={3\alpha ^2RP_0\epsilon_0 \over 16\pi A_0\gamma _0^3},
\end{displaymath} (49)

and $r=(3\alpha ^2)^{-1}$. The fractional fluence (48) is identical to Eq. (S-52) in S.

For the total fluence we obtain for low ( $\epsilon\ll E_0$) and high ( $\epsilon\gg E_0$) synchrotronphoton energies

\begin{displaymath}F_{\rm s}(\epsilon)\simeq F_{\rm0S}
\cases{c_0\left({E_0\over...
...exp \left(-\epsilon/E_0\right) & for $\epsilon\gg E_0$ , \cr }
\end{displaymath} (50)

where (see Appendix B)

\begin{displaymath}c_0=\int_0^\infty {\rm d}s~ s^{1/2}CS(s)=c(1/2)=0.95302.
\end{displaymath} (51)

Likewise, the fractional fractional fluence at high energies is

\begin{displaymath}F_{\rm f,s}(\epsilon\gg E_0, \tau _{\rm f})\simeq F_{\rm0S}\left({E_0\over \epsilon}\right)\exp \left(-\epsilon/E_0\right)
\end{displaymath}
\begin{displaymath}\qquad \qquad\times \left(1-(1+r\tau _{\rm f})^{-1}\exp \left[-{\epsilon\over E_0}[(1+r\tau _{\rm f})^2-1]\right]\right).
\end{displaymath} (52)

At small energies $\epsilon\ll E_0$ we find for small integration times $t_{\rm f}\le (E_0/\epsilon)^{1/2}r^{-1}$

\begin{displaymath}F_{\rm f,s}\left(\epsilon\ll E_0, \tau _{\rm f}\le {E_0^{1/2}\over \epsilon^{1/2}r}\right)\simeq
\end{displaymath}
\begin{displaymath}\qquad \qquad\qquad{6F_{\rm0S}a_0\over 5}\left({\epsilon\over E_0}\right)^{1/3}\left[(1+r\tau _{\rm f})^{5/3}-1\right],
\end{displaymath} (53)

and for late times $t_{\rm f}\ge (E_0/\epsilon)^{1/2}r^{-1}$ the time-independent total fluence spectrum at small energies

\begin{displaymath}F_{\rm f,s}(\epsilon\ll E_0, \tau _{\rm f}\ge {E_0^{1/2}\over...
...\rm e}^{-1}\right]\left({\epsilon\over E_0}\right)^{-1/2}\cdot
\end{displaymath} (54)

4.2 Total fluence for high injection energy

Here the synchrotronintensities (28) and (33) yield, after obvious substitutions, for the total synchrotronfluence

\begin{displaymath}F_{\rm h}(\epsilon)={1\over 3A_0q_0\gamma _0^3}\left[\int_0^{...
...\tau _{\rm c}}^\infty {\rm d}\tau ~ I_2(\epsilon,\tau )\right]
\end{displaymath}
\begin{displaymath}\qquad\quad =F_{\rm0h}\left({E_0\over \epsilon}\right)^{3/2}
...
...E_0}^{\epsilon\alpha ^2/E_0}{\rm d}x~ x^{3/2}CS \left(x\right)
\end{displaymath}
\begin{displaymath}\qquad\qquad+{\epsilon\alpha ^2\over E_0}\int_{\epsilon\alpha ^2/E_0}^\infty {\rm d}x~ x^{1/2}CS \left(x\right)\Bigr]
\end{displaymath} (55)

with the constant

\begin{displaymath}F_{\rm0h}={RP_0E_0\over 8\pi A_0\gamma _0^5}={3RP_0\epsilon_0 \over 16\pi A_0\gamma _0^3}\cdot
\end{displaymath} (56)

For high synchrotronphoton energies $\epsilon\gg E_0$, we approximate

\begin{displaymath}\int _{\epsilon/E_0}^{\epsilon\alpha ^2/E_0} {\rm d}x~ x^{3/2...
...lon/E_0}^{\epsilon\alpha ^2/E_0} {\rm d}x~ x^{1/2}{\rm e}^{-x}
\end{displaymath}
\begin{displaymath}\qquad \qquad \simeq ({\epsilon/E_0})^{1/2}{\rm e}^{-\epsilon...
...{\epsilon\alpha ^2/E_0})^{1/2}{\rm e}^{-\epsilon\alpha ^2/E_0}
\end{displaymath} (57)

and

\begin{displaymath}\int _{\epsilon\alpha ^2/E_0}^\infty {\rm d}x~ x^{1/2}CS \lef...
..._{\epsilon\alpha ^2/E_0}^\infty {\rm d}x~ x^{-1/2}{\rm e}^{-x}
\end{displaymath}
\begin{displaymath}\qquad \qquad \qquad\qquad~~~\simeq ({\epsilon\alpha ^2/E_0})^{-1/2}{\rm e}^{-\epsilon\alpha ^2/E_0},
\end{displaymath} (58)

so that

\begin{displaymath}F_{\rm h}(\epsilon\gg E_0)\simeq F_{\rm0h}{E_0\over \epsilon}{\rm e}^{-\epsilon/E_0}.
\end{displaymath} (59)

For intermediate synchrotronphoton energies $(E_0/\alpha ^2)\ll \epsilon\ll E_0$, we approximate

\begin{displaymath}\int _{\epsilon/E_0}^{\epsilon\alpha ^2/E_0} {\rm d}x~ x^{3/2...
...(x\right)\simeq c_2-a_0\int _0^{\epsilon/E_0}{\rm d}x~ x^{5/6}
\end{displaymath}
\begin{displaymath}\qquad \qquad -\int_{\epsilon\alpha ^2/E_0}^\infty {\rm d}x~ ...
...x}
\simeq c_2-{6a_0\over 11}\left({\epsilon/E_0}\right)^{11/6}
\end{displaymath}
\begin{displaymath}\qquad \qquad-({\epsilon\alpha ^2/E_0})^{-1/2}{\rm e}^{-\epsilon\alpha ^2/E_0}
\simeq c_2
\end{displaymath} (60)

with the constant

\begin{displaymath}c_2=\int _0^\infty {\rm d}x~ x^{3/2}CS \left(x\right)=c(3/2)=0.67015,
\end{displaymath} (61)

and

\begin{displaymath}\int _{\epsilon\alpha ^2/E_0}^\infty {\rm d}x~ x^{1/2}CS \lef...
...epsilon\alpha ^2/E_0})^{-1/2}{\rm e}^{-\epsilon\alpha ^2/E_0},
\end{displaymath} (62)

so that

\begin{displaymath}F_{\rm h}\left({E_0\over \alpha ^2}\ll \epsilon\ll E_0\right)\simeq F_{\rm0h}c_2\left({E_0\over \epsilon}\right)^{3/2}\cdot
\end{displaymath} (63)

Finally, for low synchrotronphoton energies $\epsilon\ll E_0/\alpha ^2$, we approximate

\begin{displaymath}\int _{\epsilon/E_0}^{\epsilon\alpha ^2/E_0} {\rm d}x~ x^{3/2...
...\int _{\epsilon/E_0}^{\epsilon\alpha ^2/E_0} {\rm d}x~ x^{5/6}
\end{displaymath}
\begin{displaymath}\qquad\qquad\qquad\quad\qquad={6a_0\over 11}\left({\epsilon/E_0}\right)^{11/6}[\alpha ^{11/3}-1]
\end{displaymath} (64)

and

\begin{displaymath}\int _{\epsilon\alpha ^2/E_0}^\infty {\rm d}x~ x^{1/2}CS \lef...
...q
c_0-{6a_0\over 5}\left({\epsilon\alpha ^2/E_0}\right)^{5/6}
\end{displaymath} (65)

with the constant (51). In this case the fluence (55) is

\begin{displaymath}F_{\rm h}(\epsilon\ll {E_0\over \alpha ^2})\simeq F_{\rm0h}\a...
...6a_0\over 55}\left({\epsilon\alpha ^2/E_0}\right)^{5/6}\right]
\end{displaymath}

\begin{displaymath}\qquad \qquad\quad\simeq F_{\rm0h}\alpha ^2c_0\left({E_0\over \epsilon}\right)^{1/2}.
\end{displaymath} (66)

For the high injection energy case, the total synchrotronfluence therefore varies as

\begin{displaymath}F_{\rm h}(\epsilon)\simeq
F_{\rm0h}
\cases{c_0\alpha ^2\left...
...exp \left(-\epsilon/E_0\right) & for $\epsilon\gg E_0$ . \cr }
\end{displaymath} (67)

4.3 Interlude

At high synchrotronphoton energies ( $\epsilon\gg E_0$) the total synchrotronfluences for low and high injection energy exhibit the same exponential cut-off. However, at low energies ( $\epsilon\ll E_0$) we find markedly different power-law behaviors for the two injection cases. In the low injection energy case ( $\alpha \ll 1$), the total synchrotronfluence exhibits the single power-law behavior $\propto $ $ \epsilon^{-1/2}$. In the high injection energy case ( $\alpha \gg 1$), the total synchrotronfluence steepens from the power-law $\propto $ $ \epsilon^{-1/2}$ below $E_0/\alpha ^2$ to the power-law $\propto $ $\epsilon^{-3/2}$ above $E_0/\alpha ^2$.

4.4 Total fluence synchrotron SED

For the total fluence SED $S(\epsilon)=\epsilon F(\epsilon)$ we then find in the two cases of small (s) and high (h) injection energies

\begin{displaymath}S_{\rm s}(\epsilon)=S_0{\alpha ^2\over \gamma _0}\left({\epsilon\over E_0}\right)^{1/2}\exp \left(-\epsilon/E_0\right)
\end{displaymath} (68)

and

\begin{displaymath}S_{\rm h}(\epsilon)=S_0{\alpha ^2\over \gamma _0}\left({\epsi...
... _B\over \epsilon+\epsilon _B}\exp \left(-\epsilon/E_0\right),
\end{displaymath} (69)

with the constant

\begin{displaymath}S_0={3c_0mc^2\over 32c_1\sigma _{\rm T}}
\end{displaymath} (70)

and the characteristic break energy

\begin{displaymath}\epsilon _B={c_2E_0\over c_0\alpha ^2}=0.703{E_0\over \alpha ^2}\cdot
\end{displaymath} (71)

In terms of the normalized synchrotronphoton energy $x=\epsilon /E_0$ and for the same $\gamma _0$, the two total SEDs (68) and (69) are

\begin{displaymath}N_{\rm s}(x)=N_0\alpha _{\rm s}^2x^{1/2}{\rm e}^{-x}
\end{displaymath} (72)

and

\begin{displaymath}N_{\rm h}(x)=N_0\alpha _{\rm h}^2{x^{1/2}x_B\over x+x_B}{\rm e}^{-x}
\end{displaymath} (73)

with $N_0=S_0/\gamma _0$ and

\begin{displaymath}x_B={0.703\over \alpha _{\rm h}^2}\cdot
\end{displaymath} (74)

Figure 3 shows the fluence SEDs N(x) for small ( $\alpha _{\rm s}=0.1$) and high ( $\alpha _{\rm h}=100$) injection conditions.

\begin{figure}
\par\includegraphics[width=9cm]{14087fg3}
\end{figure} Figure 3:

Total synchrotronfluence SED N(x) for high ( $\alpha _{\rm h}=100$, full curve) and small ( $\alpha _{\rm s}=0.1$, dashed curve) injection conditions calculated for $\gamma _0=10^4$.

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In the small injection energy case $N_{\rm s}$ peaks at x=0.5 with the peak value

\begin{displaymath}N_{\rm s,peak}=N_{\rm s}(x=0.5)={\alpha _{\rm s}^2\over 2^{1/2}}N_0{\rm e}^{-1/2},
\end{displaymath} (75)

whereas in the high injection case $N_{\rm h}$ peaks at x=xB with the peak value

\begin{displaymath}N_{\rm h,peak}=N_{\rm h}(x=x_B)={\alpha _{\rm h}^2x_B^{1/2}\over 2}N_0{\rm e}^{-x_B}.
\end{displaymath} (76)

The ratio of peak values is given by

\begin{displaymath}{\cal R}={N_{\rm h,peak}\over N_{\rm s,peak}}={\alpha _{\rm h...
...^{1/2}\over 2^{1/2}\alpha _{\rm s}^2}{\rm e}^{{1\over 2}-x_B}=
\end{displaymath}
\begin{displaymath}0.593{\alpha _{\rm h}\over \alpha _{\rm s}^2}\exp \left[{1\ov...
...ight]
\simeq 0.97{\alpha _{\rm h}\over \alpha _{\rm s}^2}\cdot
\end{displaymath} (77)

For the case shown in Fig. 3, this ratio is ${\cal R}=9.7\times 10^3$. 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 $2x_B=1.4\alpha _{\rm h}^2=1.4\times 10^{-4}$ less than the peak in the small injection case;

D2) the high injection energy peak value decreases at early times $t<\alpha t_{\rm s}/3$ more rapidly than the small injection energy peak value;

D3) the high injection SED is a broken power-law with spectral indices +0.5 below and -0.5 above the peak energy $x_B\ll 1$, respectively, and it cuts off exponentially at photon energies x>1. Below the peak energy xB the time of maximum synchrotronintensity decreases as $t_{\rm max}\propto \epsilon^{-1/2}$, whereas above the peak energy xB it decreases more rapidly as $t_{\rm max}\propto \epsilon^{-1/2}$ due to the severe additional SST losses;

D4) the small injection SED is a single power-law with spectral index +0.5 below the peak energy $x_{\rm p}=0.5$, and it cuts off exponentially at photon energies x>1. Here, the time of maximum synchrotronintensity decreases as $t_{\rm max}\propto \epsilon^{-1/2}$ at all energies x<1 because in the small injection case the SST-losses do not contribute.
All four features are quantitatively visible in Figs. 1-3. These predicted differences for the total synchrotronfluence SED and the synchrotronlight curve behaviors provide a conclusive test for high or low injection energy conditions in blazars.

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 $\gamma _0=10^4$ for the high ( $\alpha _{\rm h}=100$), small ( $\alpha _{\rm s}=0.1$) and intermediate ($\alpha =1$) injection cases. The numerical variations for $\alpha _{\rm h}=100$ and $\alpha _{\rm s}=0.1$ are in perfect agreement with the different power-law variations found analytically, which are included in Fig. 4 for orientation, and they confirm our earlier findings. For the intermediate case $\alpha =1$, for which no analytical solution is possible, one observes a peak time variation $t_{\rm S}\propto \nu ^{-1}$ that lies between the $\nu ^{-3/2}$-variation and the $\nu ^{-1/2}$-variation of the high and small injection cases.

\begin{figure}
\par\includegraphics[width=9cm]{14087fg4}
\end{figure} Figure 4:

Numerically calculated synchrotronlight curve peak times for small ( $\alpha _{\rm s}=0.1$), intermediate $\alpha =1$, and high ( $\alpha _{\rm h}=100$) injection conditions calculated for $\gamma _0=10^4$ and b=1.

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In Figs. 5-7, we show the numerical synchrotronand SSC fluence SEDs using a magnetic field strength b=1 and an injection Lorentz factor $\gamma _0=10^4$ for the high ( $\alpha _{\rm h}=100$), small ( $\alpha _{\rm s}=0.1$) and intermediate ($\alpha =1$) injection cases. The first two synchrotronSEDs are in remarkable agreement with the analytical SEDs shown in Fig. 3. In particular, the numerical SEDs confirm all four predicted differences listed in the last section. For orientation, we have plotted in Figs. 5 and 6 the asymptotic synchrotronspectra (Eqs. (67) and (50)).

\begin{figure}
\par\includegraphics[width=9cm]{14087fg5}
\end{figure} Figure 5:

Numerically calculated fractional and total synchrotronand SSC fluence SEDs for high ( $\alpha _{\rm h}=100$) injection conditions calculated for $\gamma _0=10^4$. The full curves show the total fluence SEDs. The dashed and dot-dashed 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 high-energy end of the synchrotron emission.

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The good agreement of the numerically and analytically calculated synchrotronradiation fluence distributions confirms a posteriori the validity of the assumptions made in the analytical calculations, in particular the neglect of higher order SSC losses for the high injection case.

In Fig. 7 we show the numerical synchrotronand SSC fluence SEDs using a magnetic field strength b=1 and an injection Lorentz factor $\gamma _0=10^4$ for the intermediate ($\alpha =1$) 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 $\alpha =1$. For orientation, we have also plotted the analytic asymptotic behavior of the total synchrotronfluence of the small $\alpha $-case according to Eq. (50) in this plot, which agrees well with the numerical spectral behavior.

\begin{figure}
\par\includegraphics[width=9cm]{14087fg6}
\end{figure} Figure 6:

Numerically calculated fractional and total synchrotronand SSC fluence SEDs for small ( $\alpha _{\rm s}=0.1$) injection conditions calculated for $\gamma _0=10^4$. 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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\begin{figure}
\par\includegraphics[width=9cm]{14087fg7}
\end{figure} Figure 7:

Numerically calculated fractional and total synchrotronand SSC fluence SEDs for intermediate ($\alpha =1$) injection conditions calculated for $\gamma _0=10^4$. The full curves show the total fluence SEDs. The dashed and dot-dashed lines show the asymptotic behavior of the small $\alpha $-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. 5-7 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 $2\times 10^7$ larger than in the small injection case.

6 Summary and conclusions

We have presented an analytical solution to the synchrotron and nonlinear synchrotron-self-Compton cooling in the Thomson regime (SST) of monoenergetic electrons. Based on our analytical solution, we evaluated the time-dependent synchrotron emissivity and time-integrated fluences. We find qualitatively different results depending on whether electron cooling is initially Compton-dominated (high injection energy parameter $\alpha $), or it is always synchrotron dominated (low $\alpha $). In the low-$\alpha $ case, the resulting fluence spectrum exhibits a simple exponentially cut-off power-law behavior, $S_{\nu} \propto \nu^{1/2} {\rm e}^{-\nu/\nu_0}$. In contrast, in the high-$\alpha $ case, we find a broken power-law with exponential cutoff, parametrized in the form $S_{\nu} \propto \nu^{1/2} {\nu_B \over \nu + \nu_B} {\rm e}^{-\nu/\nu_0}$. 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 $\gamma $-ray fluence is much stronger than the synchrotronfluence, are regarded as high injection-energy sources. Here, the synchrotronfluence should exhibit the symmetric broken power-law behavior Eq. (67) around the synchrotronpeak energy that is a factor $(\alpha _{\rm h} \gamma _0)^2$ lower than the SSC peak energy. Below and above $\nu _B$, the synchrotronlight curve peak times exhibit different frequency dependences $t_{\rm max}(\nu <\nu _B)\propto \nu ^{-1/2}$ and $t_{\rm max}(\nu >\nu _b)\propto \nu ^{-3/2}$, respectively, resulting from the additional severe SST-losses at $\nu >\nu _B$.

Blazars, where the $\gamma $-ray fluence is much less than the synchrotronfluence, are regarded as small injection-energy sources. Here, the synchrotronfluence exhibits the single power-law behavior (D4) up to a higher synchrotronpeak energy that is a factor $\gamma _0^2$ lower than the SSC peak energy. In this case, the synchrotronlight curve peak time exhibits the standard linear synchrotroncooling decrease $t_{\rm max}(\nu )\propto \nu ^{-1/2}$ at all frequencies.

If the injection Lorentz factor $\gamma _0$ and the size of the source are the same, different values of the injection parameter $\alpha $ result from different total numbers of instantaneously injected electrons, see Eq. (19); for example, the high injection case $\alpha _{\rm h}=100$ results for N50=4.7, whereas the low injection case $\alpha _{\rm s}=0.1$ needs $N_{50}=4.7\times 10^{-6}$.

Acknowledgements
We 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 201/20-1 and Schl 201/23-1. 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 Ruhr-University Bochum.

Appendix A: Determination of electron distribution functions for low and high injection energies

A.1 Low injection energy $\gamma _0\ll \gamma _{{B}}$

In the case of low injection energies $\gamma _0\ll \gamma _{{B}}$, the injection parameter $\alpha \ll 1$ is less than unity, so that the argument of the $\arctan $-function in Eq. (15) is always greater than unity. For all values of T and y, Eq. (15) then simplifies to

\begin{displaymath}x_0+T(y)\simeq K_0y+C_0,
\end{displaymath} (A.1)

and with C0=x0 to

\begin{displaymath}T(y)\simeq K_0y.
\end{displaymath} (A.2)

In terms of y, the solution (12) then reads with x0>0

\begin{displaymath}S(x,x_0,y)=q_0H[x-x_0]\delta \left(x-x_0-K_0y\right),
\end{displaymath} (A.3)

yielding

\begin{displaymath}n(\gamma ,\gamma _0,t)={q_0\over \gamma ^2}H[\gamma _0-\gamma ]
\delta \left(\gamma ^{-1}-\gamma _0^{-1}-D_0t\right)
\end{displaymath}
\begin{displaymath}\qquad\qquad =q_0H[\gamma _0-\gamma ]\delta \left(\gamma -{\gamma _0\over 1+D_0\gamma _0t}\right),
\end{displaymath} (A.4)

which agrees with the standard linear synchrotroncooling solution.
\begin{figure}
\par\includegraphics[width=9cm]{14087fg8}
\end{figure} Figure A.1:

Comparison of exact solution y(T) (full curve) with asymptotic solution y1(T) at early times (dashed curve) and asymptotic solution y2(T) at late times (dotted curve) for the high injection energy case with q5=1, x0=10-4 and K0=1, corresponding to $\alpha =3\times 10^6$.

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A.2 High injection energy $\gamma _0\gg \gamma _{{B}}$

In the case of high injection energies $\gamma _0\gg \gamma _{{B}}$ the injection parameter $\alpha \gg 1$ is greater than unity. With the injection parameter (17) we rewrite Eq. (15) as

\begin{displaymath}K_0y+C_1=\alpha x_0\left[{1+{T\over x_0}\over \alpha }-\arctan \left({1+{T\over x_0}\over \alpha }\right)\right]
\end{displaymath} (A.5)

For times $0\le T\le T_{\rm c}$, where

\begin{displaymath}T_{\rm c}=(\alpha -1)x_0,
\end{displaymath} (A.6)

we expand $\arctan (x)\simeq x-(x^3/3)$ to third order in x to obtain

\begin{displaymath}K_0y_1+C_1\simeq {x_0\over 3\alpha ^2}\left(1+{T\over x_0}\right)^3
,
\end{displaymath} (A.7)

or

\begin{displaymath}y_1={(x_0+T)^3\over 3q_0}-C_2.
\end{displaymath} (A.8)

With T=0 for y=0, the integration constant C2=x03/3q0 is fixed so that

\begin{displaymath}y_1={(x_0+T)^3\over 3q_0}-{x_0^3\over 3q_0}\cdot
\end{displaymath} (A.9)

This solution is valid for $T\le T_{\rm c}$, corresponding with Eq. (A.6) to

\begin{displaymath}0\le y\le y_{\rm c}={x_0^3\over 3q_0}\left(\alpha ^3-1\right)={x_0\over 3\alpha ^2K_0}(\alpha ^3-1).
\end{displaymath} (A.10)

For times $T\ge T_{\rm c}$ or $y\ge y_{\rm c}$, the argument of the $\arctan $-function in Eq. (A.5) is greater than unity, yielding

\begin{displaymath}K_0y_2+C_3\simeq x_0+T,
\end{displaymath} (A.11)

or the linear relation

\begin{displaymath}y_2={x_0+T\over K_0}-C_4.
\end{displaymath} (A.12)

The constant C4 is determined by the equality of the two solutions $y_1(T_{\rm c})=y_2(T_{\rm c})=y_{\rm c}$ at $T_{\rm c}$ providing

\begin{displaymath}C_4={\alpha x_0\over K_0}+{x_0^3\over 3q_0}(1-\alpha ^3)={x_0^3\over 3q_0}\left[1+2\alpha ^3\right],
\end{displaymath} (A.13)

so that

\begin{displaymath}y_2={x_0+T\over K_0}-{2q_0^{1/2}\over 3K_0^{3/2}}-{x_0^3\over 3q_0}\cdot
\end{displaymath} (A.14)

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

\begin{displaymath}T_1(y<y_{\rm c})=\left[3q_0y+x_0^3\right]^{1/3}-x_0
\end{displaymath}
\begin{displaymath}\qquad \qquad=x_0\left[\left(1+{3\alpha ^2K_0y\over x_0}\right)^{1/3}-1\right]
\end{displaymath} (A.15)

and

\begin{displaymath}T_2(y\ge y_{\rm c})=
x_0\left[{1\over 3\alpha ^2}\left({3\alpha ^2K_0y\over x_0}+1+2\alpha ^3\right)-1\right].
\end{displaymath} (A.16)

In terms of y we obtain from Eq. (11) the two solutions

\begin{displaymath}S_1(x,x_0,y<y_{\rm c})=
\end{displaymath}
\begin{displaymath}q_0H[x-x_0]H[y_{\rm c}-y]\delta \left(x-x_0\left(1+{3\alpha ^2K_0y\over x_0}\right)^{1/3}\right)
\end{displaymath} (A.17)

and
        $\displaystyle S_2(x,x_0,y >y_{\rm c})$ = $\displaystyle q_0H[x-\alpha x_0]H[y-y_{\rm c}]\delta$  
  $\textstyle \times$ $\displaystyle \left(x-\left[K_0y+ {1+2\alpha ^3\over 3\alpha ^2}x_0\right]\right).$ (A.18)

We then find for early times

\begin{displaymath}n_1(\gamma ,\gamma _0,t<t_{\rm c})=
\end{displaymath}
\begin{displaymath}q_0H[\gamma _0-\gamma ]H[t_{\rm c}-t]\delta \left(\gamma -{\gamma _0\over (1+3\alpha ^2\gamma _0K_0A_0t)^{1/3}}\right)=
\end{displaymath}
\begin{displaymath}\qquad q_0H[\gamma _0-\gamma ]H[t_{\rm c}-t]\delta \left(\gamma -{\gamma _0\over (1+3q_0\gamma _0^3A_0t)^{1/3}}\right),
\end{displaymath} (A.19)

which agrees with the nonlinear SST solution of S, his Eq. (28). At late times,

\begin{displaymath}n_2(\gamma ,\gamma _0,t\ge t_{\rm c})=
\end{displaymath}
\begin{displaymath}q_0H[\gamma _B-\gamma ]H[t-t_{\rm c}]\delta \left(\gamma -{\g...
...\over {1+2\alpha ^3\over 3\alpha ^3}+\gamma _BK_0A_0t}\right),
\end{displaymath} (A.20)

which is a modified linear cooling solution. Both solutions show that, at time

\begin{displaymath}t_{\rm c}={y_{\rm c}\over A_0}={\alpha ^3-1\over 3\alpha ^3\gamma _BD_0}\simeq {1\over 3\gamma _BD_0}
\end{displaymath}
\begin{displaymath}\quad\qquad~ ={2.6\times 10^8\over \gamma _Bb^2}~ \hbox{s}={1.2\times 10^6N_{50}^{1/2}\over R_{15}b^2}~ \hbox{s},
\end{displaymath} (A.21)

the electrons have cooled to the characteristic Lorentz factor $\gamma _B$.

Appendix B: Calculation of synchrotron radiation constants

Here we determine the constants

\begin{displaymath}c(k)\equiv \int_0^\infty {\rm d}x~ x^k CS(x)
\end{displaymath} (B.1)

for 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

\begin{displaymath}c(k)={2\over \pi }G^{'}(2k+1)={2^k(3k+5)(3k-1)\over 18\pi ^{1/2}k(k+2)}
\end{displaymath}
\begin{displaymath}\qquad \times {\Gamma \left[{k+1\over 2}\right]\Gamma \left[{...
...left[{3k-1\over 6}\right]\over \Gamma \left[{k\over 2}\right]}
\end{displaymath} (B.2)

with the triplication formula of gamma functions for z=(3k-1)/6, Eq. (B.2) simplifies to

\begin{displaymath}c(k)={2^k\pi ^{1/2}(3k+5)(3k-1)\over 3^{3k+2\over 2}k(k+2)}{\...
...{3k-1\over 2}\right]\over \Gamma \left[{k\over 2}\right]}\cdot
\end{displaymath} (B.3)

We obtain c(1/2)=0.95302 and c(3/2)=0.67015.

References

All Figures

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg1}
\end{figure} Figure 1:

Time dependence of the peak energy $\epsilon _{\rm p}(\tau )$ of the synchrotronSED for high ( $\alpha =10$, lower curve) and low ( $\alpha =0.1$, upper curve) values of the injection parameter.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg2}
\end{figure} Figure 2:

Photon energy ( $x=\epsilon /E_0$) dependence of the synchrotronlight curve peak time for high ( $\alpha =100$, full curve) and low ( $\alpha =0.1$, dashed curve) values of the injection parameter.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg3}
\end{figure} Figure 3:

Total synchrotronfluence SED N(x) for high ( $\alpha _{\rm h}=100$, full curve) and small ( $\alpha _{\rm s}=0.1$, dashed curve) injection conditions calculated for $\gamma _0=10^4$.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg4}
\end{figure} Figure 4:

Numerically calculated synchrotronlight curve peak times for small ( $\alpha _{\rm s}=0.1$), intermediate $\alpha =1$, and high ( $\alpha _{\rm h}=100$) injection conditions calculated for $\gamma _0=10^4$ and b=1.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg5}
\end{figure} Figure 5:

Numerically calculated fractional and total synchrotronand SSC fluence SEDs for high ( $\alpha _{\rm h}=100$) injection conditions calculated for $\gamma _0=10^4$. The full curves show the total fluence SEDs. The dashed and dot-dashed 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 high-energy end of the synchrotron emission.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg6}
\end{figure} Figure 6:

Numerically calculated fractional and total synchrotronand SSC fluence SEDs for small ( $\alpha _{\rm s}=0.1$) injection conditions calculated for $\gamma _0=10^4$. 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.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg7}
\end{figure} Figure 7:

Numerically calculated fractional and total synchrotronand SSC fluence SEDs for intermediate ($\alpha =1$) injection conditions calculated for $\gamma _0=10^4$. The full curves show the total fluence SEDs. The dashed and dot-dashed lines show the asymptotic behavior of the small $\alpha $-case according to Eq. (50). The fractional fluences are shown at approximately logarithmically spaced times.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=9cm]{14087fg8}
\end{figure} Figure A.1:

Comparison of exact solution y(T) (full curve) with asymptotic solution y1(T) at early times (dashed curve) and asymptotic solution y2(T) at late times (dotted curve) for the high injection energy case with q5=1, x0=10-4 and K0=1, corresponding to $\alpha =3\times 10^6$.

Open with DEXTER
In the text


Copyright ESO 2010

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