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Astron. Astrophys. 324, 395-409 (1997)

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6. Photon spectra

From the observational point of view, it is most interesting to know both the evolution of the radiation spectra with time and the time-integrated photon spectra, which are emanating from the pair plasmas treated in the previous sections. We first evaluate all the contributions to the photon spectra in the frame of reference of the jet (blob frame) and then transform to the observer's frame using

[EQUATION]

where the asterisk denotes quantities measured in the observer's frame (which we idenfity with the accretion-disk frame), [FORMULA] is the spectral photon production rate in the energy interval [FORMULA] ([FORMULA]) in the solid angle interval [FORMULA], integrated over the whole blob volume [FORMULA], [FORMULA] is the Doppler factor, [FORMULA] is the cosine of the observing angle in the observer's frame and [FORMULA] is the observing angle cosine in the blob frame. A time interval [FORMULA] measured in the blob frame is related to a light reception time interval [FORMULA] by [FORMULA] and to an accretion disk frame time interval [FORMULA] by [FORMULA].

The most important contributions to the [FORMULA] -ray spectra in the MeV - TeV regime come from inverse-Compton interactions. The photon spectrum resulting from inverse-Compton scattering of accretion disk photons is calculated as

[EQUATION]

[EQUATION]

where [FORMULA] is the normalized energy and [FORMULA] the solid angle of the motion of the scattered photons, [FORMULA] is the differential number density of accretion disk photons at the location of the scattering event as discussed in Sect. 2.1 and the angle variables are illustrated in Fig. 2 and Eqs. (6) and (7).

It has turned out to be most convenient for this calculation (using the full Klein-Nishina cross section) to use the differential Compton scattering cross section in the blob frame:

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

(Jauch & Rohrlich 1976). (Of course, using the KN cross section in the electron system and transforming the final photon state back to the blob frame leads to the same final expression for the spectrum of scattered photons. It is more straightforward to use directly the cross section [50].)

With the assumptions we made above, the SSC spectrum is isotropic in the blob frame, and

[EQUATION]

[EQUATION]

Technically, we include higher order SSC scattering by using in each time step the SSC photon density [FORMULA] of the foregoing time step together with the synchrotron photon density as seed photon field for SSC scattering.

Using the cross section (50), Eq. (52) can be rewritten as

[EQUATION]

[EQUATION]

where [FORMULA] is the azimuthal angle between photon and electron direction of motion around the direction of motion of the scattered photon, the other angle variables are the same as in Sect. 2.1, [FORMULA] is given by Eq. (52), [FORMULA] denotes the sum of synchrotron and previously produced SSC photons, and

[EQUATION]

[EQUATION]

[EQUATION]

If pair bremsstrahlung can contribute significantly to the time-integrated photon spectra (which we find to be the case if [FORMULA] cm-3) we evaluate the rate of production of bremsstrahlung photons per unit volume, [FORMULA], in the comoving fluid frame as

[EQUATION]

[EQUATION]

where we use the differential cross section

[EQUATION]

[EQUATION]

(Alexanian 1968) in the center-of-momentum frame of the scattering particles, [FORMULA] is the hyperfinestructure constant, and the integration limits are given by

[EQUATION]

The calculation of the synchrotron spectrum has been described in Sect. 2.2. Pair annihilation radiation is negligible for an ultrarelativistic pair plasma.

After having calculated the resulting synchrotron and [FORMULA] -ray spectra, we have to check whether [FORMULA] -rays of energies [FORMULA] TeV can escape the emitting region. For this purpose we calculate the optical depth due to [FORMULA] - [FORMULA] pair production of high-energy [FORMULA] -rays interacting with the radiation produced in the blob. This calculation is carried out in the blob frame where we assume all contributions to be isotropic (which is not the case for the external IC component; but for a [FORMULA] -ray traveling with small angle with respect to the jet axis, the assumption of isotropy even overestimates the pair production optical depth). The results show that even in the first time step (which, of course, is the most critical one) the [FORMULA] - [FORMULA] optical depth does not exceed one for parameters suitable to fit the broadband spectrum of Mrk 421 (see Sect. 8). Nevertheless, we include the small effect of [FORMULA] - [FORMULA] absorption when calculating the emanating photon spectra which is the reason for the total emission (corrected for [FORMULA] - [FORMULA] absorption) being slightly lower than the EIC component in Fig. 7 a. The injection of pairs due to [FORMULA] - [FORMULA] pair production (for details see Böttcher & Schlickeiser 1997) is negligible even in the case of relatively high density as assumed for Figs. 7 and 8.

[FIGURE] Fig. 7. Instantaneous broadband spectra from the blob of simulation 1. Parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA] cm, [FORMULA] cm-3, [FORMULA] G, [FORMULA], [FORMULA] [FORMULA] pc, [FORMULA], [FORMULA]. Total emission (corrected for [FORMULA] - [FORMULA] absorption): solid; IC scattering of accretion disk radiation: dashed; SSC radiation: dot-dashed; synchrotron radiation: long dashed. (all quantities in the observer's frame)
[FIGURE] Fig. 8. Time-integrated broadband spectrum from the evolving blob illustrated in Fig. 7 (total [FORMULA] -ray spectrum: solid; EIC: dashed; SSC: dot-dashed; bremsstrahlung: dotted; synchrotron: long dashed)
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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