The interaction of relativistic electrons with radiation fields through inverse Compton scattering provides one of the principal Yray production processes in astrophysics. It works effectively almost everywhere, from compact objects like pulsars and AGN to extended sources like supernova remnants and clusters of galaxies. Because of the universal presence of the 2.7 K CMBR, as well as low gas densities and low magnetic fields, inverse Compton scattering proceeds with very high efficiency in the intergalactic medium over the entire Yray domain. Since the Compton cooling time decreases linearly with energy, the process becomes especially effective at very high energies.
The electronpositron pair production in photonphoton collisions is tightly coupled with inverse Compton scattering. First of all, it is an absorption process that prevents the escape of energetic Yrays from compact objects, and determines the "Yray horizon" of the Universe. At the same time, in an environment where the radiation pressure dominates over the magnetic field pressure, the photonphoton pair production and the inverse Compton scattering "work" together supporting the effective transport of high energy radiation via electromagnetic "KleinNishina" cascades.
Although the inverse Compton scattering of protons is suppressed by a factor ofvery high energy protons effectively interact with the ambient photon fields through electronpositron pairproduction and photomeson processes. While in the first process Yrays are produced indirectly, via inverse Compton scattering of the secondary electrons, photomeson reactions result in the direct production ofand their subsequent decay to Yrays. Typically, at extremely high energies these interactions proceed effectively both in compact objects and large scale structures.
Inverse Compton scattering
The derivation of the crosssection for Compton scattering with given fourvector momenta of the electron and photon can be found in Akhiezer and Berestetskii (1965). The basic expressions of the inverse Compton scattering (i.e. when the energy of the electron significantly exceeds the energy of the target photon) have been comprehensively analysed by Jones (1968), Blumenthal and Gould (1970) and Coppi and Blandford (1990) for the case of isotropically distributed photons and electrons.
The angleaveraged total crosssection of inverse Compton scattering depends only on the product of the energies of the interacting electron e and photon(where all energies are in units ofIn the nonrelativistic regimeit approaches the classical (Thomson) cross sectionwhile in the ultrarelativistic regime it decreases withWith an accuracy of better than 10 per cent in a very broad range of k0, the crosssection can be represented in the following simple form (Coppi and Blandford, 1990)
The total crosssection of Compton scattering as a function of k0 is shown in Fig. 3.4.
The energy distribution of upscattered Yrays is determined by the differential crosssection of the process. Assuming that a monoenergetic beam of low energy photons w0 penetrates an isotropic and homogeneous region filled with relativistic electrons of energy ee, the spectrum of radiation scattered at the angle 0 relative to the initial photon beam is written as
whereThe energy of the high energy Yray photon e7 varies in the limits
In the case of isotropically distributed electrons and photons, the integration of Eq.(3.15) over the angle 0 gives (Jones, 1968, Blumenthal and Gould, 1970)
The differential energy spectra of Yrays for several fixed values of k0 are shown in Fig. 3.5. In the deep KleinNishina regimethe spectrum grows sharply towards the maximum atThis implies that in this regime just one interaction is sufficient to transfer a substantial fraction of the electron energy to the upscattered photon (see also Table 3.1).
Fig. 3.4 Total crosssections of inverse Compton scattering and photonphoton pair production in isotropic radiation fields. Two spectral distributions for the ambient photon gas are assumed: (i) monoenergetic with energy wo (curves 1 and 3), and (ii) Planckian with the same mean photon energy(curves 2 and 4).
In the Thomson regimethe average energy of the upscattered photon isthus only a fractionof the primary electron energy is released in the upscattered photon.
For a powerlaw distribution of electrons,the resulting Yray spectrum in the nonrelativistic regimehas a power law form with photon index(Ginzburg and Syrovatskii, 1964). In the ultrarelativisticregime the Yray spectrum is no ticeably steeper,with(Blumenthal and Gould, 1970). Several useful analytical approximations for Yray spectra over a broad energy interval, including these two regimes and the KleinNishina transition regioncan be found in Atoyan (1981c) and Coppi and Blandford (1990).
Fig. 3.5 Differential spectra of Yrays from inverse Compton scattering (upper panel) and electrons from photonphoton pair production (bottom panel) in an isotropic and monoenergetic photon field. The parametersare defined as
The same values of the parametersare indicated by the curves.
The spectral features of the inverse Compton radiation in powerlaw target photon fields have been studied by Zdziarski (1988)
Table 3.1 The mean fraction of primary energytransferred to the secondary photon in the inverse Compton scattering (ics) and the synchrotron radiation (syn) processes for different values of the parametersrespectively.
0.01 
0.1 
1 
10′^{2} 
10^{4} 
10^{6} 

0.014 
0.099 
0.358 
0.760 
0.867 
0.910 

0.44 • 10^{2} 
0.033 
0.118 
0.241 
0.250 
0.250 
The energyloss rate of relativistic electrons in a monoenergetic field of photons with energy w0 and number density nph is given by the following equation.
where
In the Thomson and KleinNishina regimes Eq.(3.17) reduces to the well known expressions (e.g. Blumenthal and Gould, 1970)
and
The energy losses in these two regimes have quite a different dependence on the electron energy. While in the Thomson regime the loss rate is proportional to e^, in the KleinNishina regime it is almost energy independent. This implies that in the first case the steadystate electron spectrum becomes steeper, whereas the Compton losses in the KleinNishina regime make the electron spectrum harder.
For calculations of electron energy losses in radiation fields with more realistic distributions one should integrate Eqs.(3.17)(3.19) over w0. It is interesting to note that in the Thomson regime the energyloss rate does not depend on the spectral distribution of target photons, but depends only on the total energy density of radiation ur. Correspondingly, the cooling time of electrons due to Thomson scattering is given by
where the radiation energy densityand the electron energyare expressed in units ofrespectively. Note that the same equation describes the synchrotron energy losses if we replaceby the magnetic field energy density
The comparison of Eq.(3.20) with the bremsstrahlung cooling time given by Eq.(3.4) shows that Compton and synchrotron losses dominate over the bremsstrahlung cooling, if
Photonphoton pair production
Photonphoton pair production is the inverse process to pair annihilation. Therefore the differential crosssection is identical to the pair annihilation crosssection, except for a different phasespace volume. In the relativistic regime this process is quite similar also to inverse Compton scattering. However, unlike the pair annihilation and Compton scattering, the photonphoton pair production has a strict kinematic threshold given by
whereare the energies of two photons in units ofcolliding at an angle 0 (in the laboratory frame).
The large crosssection makes the photonphoton pair production one of the most relevant elementary processes in high energy astrophysics. The role of this process in the context of intergalactic absorption of Yrays was first pointed out by Nikishov (1962). Bonometto and Rees (1971) were first who emphasised the importance of this process in dense radiation fields of compact objects.
Several convenient approximations for the total crosssection of this process in the isotropic radiation field have been proposed by Gould and Schreder (1967), Coppi and Blandford (1990). With an accuracy of better than 3 per cent, the total crosssection in the monoenergetic isotropic photon field can be represented in the following analytical form
The total crosssections of inverse Compton scattering and pair production in an isotropic monoenergetic photon field of energyare shown in Fig. 3.4 (curves 1 and 3, respectively). Both cross sections depend only on the product of the primaryand target photonenergies,
While asthe inverse Compton cross section approaches the Thomson crosssection, the pair production crosssection approaches zero,
Forthe two crosssections are quite similar and decrease with k0 andThe pair
production crosssection has a maximum at the level of achieved at
The parameter that characterises Yray absorption at photonphoton interactions in a source of size R is the socalled optical depth
wheredescribes the spectral and spatial distribution of the target photon field in the source. For a homogeneous source with a narrow spectral distribution of photons, for order of magnitude estimates one can use the approximationwhereis the average target photon energy. However, generally one has to be careful with this type of estimate, especially at low energies; while this approximation implies a completely transparent sourcein fact nonnegligible absorption can take place at low energies. For example, for a Planckian distribution of target photons, the optical depth t cannot be disregarded at energiesbecause of interactions with photons from the Wien tail region (see Fig. 3.4). Very useful approximations for the optical depth of Yrays in a Planckian photon gas can be found in Gould and Schreder (1967).
Because of narrowness of the pairproduction crosssection, for a large class of broad band target photon energy distributionsthe optical depth at given Yray energyis essentially determined by a relatively narrow band of target photons with energy centered on(Herterich, 1974). Therefore, the optical depth can be written in the form
where the normalization factor n depends on the spectral shape of the background radiation. For a powerlaw target photon spectrum,the parameter n is calculated analytically,
The energy spectrum of electrons produced at photonphoton pair production has been studied by Zdziarski and Lightman (1985), Coppi and Blandford (1990) and Bottcher and Schlickeiser (1997). For a lowenergy monoenergetic photon field, and correspondinglythe spectrum of electronpositron pairs can be rep
resented, with an accuracy of better than a few per cent:
The kinematic range of variation ofis
The differential energy spectra of Yrays for several fixed values of the parametersare shown in Fig. 3.5. The spectra are symmetric around the pointAlthough the average energy of the secondary electrons isfor very large s0 the interaction has a catastrophic character – the major fraction of the energy of the primary Yray photon is transferred to the leading electron. This fraction exceeds 0.5 and asymptotically approaches 1 (see Table 3.2).
For calculation of electron spectra produced in radiation fields by Yrays with more realistic spectral distributions,one should integrate the product of spectrum given by Eq.(3.25) andover a broad primary
Yray energy interval. For a powerlaw spectrum of Yrays, the spectrum of secondary pairs can be approximated, with an accuracy of better than 20 per cent:
(two electrons per interaction). Starting from the minimum (allowed by kinematics) energy atthe electron spectrum sharply rises achieving its maximum atand then atit behaves as
Table 3.2 The mean fraction of energy of the primary 7ray photon transferred to the leading secondary electron at electronpositron pair production in a monoenergetic radiation field (rad) and in the magnetic field (B) for different values of the parametersrespectively.
1 
3 
10 
W’^{2} 
10^{4} 
10^{b} 

0.500 
0.701 
0.797 
0.891 
0.948 
0.966 

0.634 
0.693 
0.746 
0.782 
0.824 
0.825 
Two pairs of coupled processes – inverse Compton scattering and photonphoton pair production – determine the basic features of interactions of electrons and Yrays in the radiation dominated environments. At extremely high energies higher order QED processes may compete with these basic channels. Namely, when the product of the energies of colliding cascade particles (electrons or photons) E and the background photons w significantly exceedthe processes(Brown et al., 1973) and(Mastichiadis, 1991; Dermer and Schlickeiser, 1991) dominate over singlepair production and Compton scattering, respectively. For example, in the 2.7 K CMBR the first process stops the linear increase of the mean free path of the highest energy Yrays around 1021 eV, and puts a robust limit on the mean free path of Yrays of about 100 Mpc. Analogously, above 1020 eV the second process becomes more important than the conventional inverse Compton scattering. Because the channels result in production of 2 additional electrons, they substantially change the character of interactions. Note, however, that an effective realization of these processes is possible only under very specific conditions with an extremely low magnetic field and narrow energy distribution of the background photons.