## 4. Comparison to approximationsNow, we use the exact expression, given in Eq. (26) to specify the regimes of validity and the limitations of various approximations. The first detailed computation of the pair production spectrum was presented by Bonometto & Rees (1971). Based on the neglect of the energy input of the soft photon, they basically follow the same procedure as described above, but do not carry out the angle-integration (integration over in our formalism) analytically. In the case , it is in very good agreement with the exact result, but its evaluation is even more time-consuming than using the latter. For this reason, we will not consider it in detail, but concentrate on approximations which really yield simpler expressions than the exact one. ## 4.1. -function approximation for power-law spectraProbably the simplest expression for the pair spectrum injected by -rays interacting with a power-law of energy spectral index () is based on the assumption and on the well-known fact that photons of energy interact most efficiently with photons of energy which motivates the approximation for the - opacity of Gould & Schréder (1967) using a function approximation for the cross section, Since in this approach pair production takes place only near the pair production threshold (), the produced pairs have energies (Bonometto & Rees 1971). The resulting pair injection spectrum is therefore (e. g. Lightman & Zdziarski 1987) where (which, of course, reduces to for ) and is a numerical factor, depending only on the spectral index of the soft photon distribution. This approximation yields useful results, if the power-law photon spectra extend over a sufficiently wide range () and if for every high-energy photon of energy there is a soft photon of energy . Else, the injection spectrum calculated with Eq. (30) cuts off at the inverse of the respective cutoff of the soft photon spectrum, seriously underpredicting the injection of pairs of higher or lower energy, respectively, where the injection spectrum declines smoothly. Nevertheless, these pairs can still carry a significant fraction of the injected power. Eq. (30) fails also to describe the injection of low-energetic pairs in case of a high lower cut-off of the -ray spectrum even if soft photons of energy are present. For example, in the case of the interaction of a power-law spectrum extending from - with a soft power-law spectrum extending from - Eq. (30) overpredicts the injection of pairs slightly above by an order of magnitude and cuts off below this energy. A similar problem arises at the high-energy end of the injection spectrum. An example for this fact is shown in Fig. 2. In contrast, the approximation (30) can well be used to describe the injection of pairs at all energies if both photon fields extend up to (and down to, respectively) . For more general soft photon distributions which are different from a power-law (e. g. a thermal spectrum) the analogous -function approximation has first been introduced by Kazanas (1984).
## 4.2. -function in electron energyUsing the full cross section for - pair production as given by Jauch & Rohrlich (1959) instead of the -function approximation adopted in Eq. (30) does not reduce the limitations of this power-law approach significantly, but other soft photon distributions can be treated more successfully with this approximation which in the limit reads where and the limits are given by Eqs. (24) and (25). Here, we have assumed that the produced electron and positron have energy . This approach works equally well for power-law photon fields, but in contrast to Eq. (30), it tends to underpredict the injection of low-energetic pairs. The same is true for the interaction of -ray photon fields with thermal soft photon fields where the high-energy tail of the injection spectrum is described very accurately (a few % error) by Eq. (31). The accuracy of this approximation improves with decreasing lower cut-off of the -ray spectrum. E. g., the injection due to a power-law -ray spectrum from - interacting with a thermal spectrum of normalized termpature is described by Eq. (31) with a deviation of less than 30 % from the exact result down to . For , the deviation was much less than 10 %. We show an example for the latter situation in Fig. 3.
## 4.3. Approximation by Aharonian et al.A very useful approximation to the pair injection spectrum for all shapes of the soft photon spectrum under the condition has been found by Aharonian et al. (1983). They use a different representation of the pair production cross section and end up with a one-dimensional integral over which is equivalent to our integration in Eq. (21). They solve this integration analytically after simplifying the integrand and the integration limits according to the assumptions mentioned above. The resulting injection spectrum is It describes the power-law tail of the pair spectrum injected by power-law -ray photon fields perfectly and is much more accurate to the injection of low-energy pairs. Interaction with a power-law soft photon field is reproduced within errors of only a few %. Even if as well the -ray as the soft photon spectrum extend to , the error at increases only to 10 %. Problems with this approximation arise if the soft photon spectrum extends up to , but the -ray spectrum has a lower cut-off . In this case, the injection of low-energetic pairs is seriously overpredicted by Eq. (32). For power-law soft photon fields, the integration over in Eq. (32) can be carried out analytically, as was found by Svensson (1987). His Equation (B8) multiplied with the total absorption coefficient (by which the total injection rate had been normalized to unity) is in perfect agreement with the numerical results according to Eq. (32). The interaction of power-law -ray spectra with thermal soft photon fields is generally described within an error of a few % at all electron/positron energies if the soft photon temperature is , even if the -ray spectrum extends down to . Interestingly, even the interaction of a mildly relativistic thermal photon field () with itself (for which Aharonian's approximation was not designed) is reproduced reasonably well, but the result of Eq. (32) differs from the exact injection rate by a roughly constant factor. When artificially introducing a factor adjusting the high-energy tails of the injection spectra, Eq. (32) overpredicts the injection of low-energetic pairs by a factor of , but for there is very good agreement with the exact result. The deviation becomes more important with increasing photon temperature, and for the injection of cold pairs is already overpredicted by a factor of . Fig. 4 illustrates the accuracy of the various approximations for a compact thermal radiation of temperature .
We find that all the statements on soft photon or -ray power-law spectra made above are only very weakly dependent on the respective spectral index. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |