Interactions with Photon Fields (Cosmic Gamma Radiation) Part 1

The interaction of relativistic electrons with radiation fields through inverse Compton scattering provides one of the principal Y-ray 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 Y-ray domain. Since the Compton cooling time decreases linearly with energy, the process becomes especially effective at very high energies.

The electron-positron pair production in photon-photon collisions is tightly coupled with inverse Compton scattering. First of all, it is an absorption process that prevents the escape of energetic Y-rays from compact objects, and determines the "Y-ray horizon" of the Universe. At the same time, in an environment where the radiation pressure dominates over the magnetic field pressure, the photon-photon pair production and the inverse Compton scattering "work" together supporting the effective transport of high energy radiation via electromagnetic "Klein-Nishina" cascades.

Although the inverse Compton scattering of protons is suppressed by a factor oftmp16-173_thumbvery high energy protons effectively interact with the ambient photon fields through electron-positron pair-production and photomeson processes. While in the first process Y-rays are produced indirectly, via inverse Compton scattering of the secondary electrons, photo-meson reactions result in the direct production oftmp16-175_thumband their subsequent decay to Y-rays. Typically, at extremely high energies these interactions proceed effectively both in compact objects and large scale structures.

Inverse Compton scattering

The derivation of the cross-section for Compton scattering with given four-vector 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 angle-averaged total cross-section of inverse Compton scattering depends only on the product of the energies of the interacting electron e and photontmp16-176_thumb(where all energies are in units oftmp16-177_thumbIn the  nonrelativistic regimetmp16-178_thumbit approaches the classical (Thomson) cross- sectiontmp16-179_thumbwhile in the ultrarelativistic regimetmp16-180_thumb it decreases withtmp16-181_thumbWith an accuracy of better than 10 per cent in a very broad range of k0, the cross-section can be represented in the following simple form (Coppi and Blandford, 1990) tmp16-189_thumb

The total cross-section of Compton scattering as a function of k0 is shown in Fig. 3.4.

The energy distribution of up-scattered Y-rays is determined by the differential cross-section 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

tmp16-190_thumb

wheretmp16-191_thumbThe energy of the high energy Y-ray photon e7 varies in the limitstmp16-192_thumb tmp16-193_thumb

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) tmp16-197_thumb

The differential energy spectra of Y-rays for several fixed values of k0 are shown in Fig. 3.5. In the deep Klein-Nishina regimetmp16-198_thumbthe spectrum grows sharply towards the maximum attmp16-204_thumbThis 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).

Total cross-sections of inverse Compton scattering and photon-photon 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).

Fig. 3.4 Total cross-sections of inverse Compton scattering and photon-photon 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 energytmp16-201_thumb(curves 2 and 4).

In the Thomson regimetmp16-205_thumbthe average energy of the upscattered photon istmp16-206_thumbthus only a fractiontmp16-207_thumbof the primary electron energy is released in the upscattered photon.

For a power-law distribution of electrons,tmp16-208_thumbthe resulting Y-ray spectrum in the nonrelativistic regimetmp16-209_thumbhas a power- law form with photon indextmp16-226_thumb(Ginzburg and Syrovatskii, 1964). In the ultrarelativistictmp16-227_thumbregime the Y-ray spectrum is no ticeably steeper,tmp16-228_thumbwithtmp16-229_thumb(Blumenthal and Gould, 1970). Several useful analytical approximations for Y-ray spectra over a broad energy interval, including these two regimes and the Klein-Nishina transition regiontmp16-230_thumbcan be found in Atoyan  (1981c) and Coppi and Blandford (1990).

Differential spectra of Y-rays from inverse Compton scattering (upper panel) and electrons from photon-photon pair production (bottom panel) in an isotropic and mono-energetic photon field. The parametersare defined as

Fig. 3.5 Differential spectra of Y-rays from inverse Compton scattering (upper panel) and electrons from photon-photon pair production (bottom panel) in an isotropic and mono-energetic photon field. The parameterstmp16-217_thumbare defined as

tmp16-218_thumbThe same values of the parameterstmp16-219_thumbare indicated by the curves.

The spectral features of the inverse Compton radiation in power-law target photon fields have been studied by Zdziarski (1988)

Table 3.1 The mean fraction of primary energytmp16-231_thumbtransferred to the secondary photon in the inverse Compton scattering (ics) and the synchrotron radiation (syn) processes for different values of the parameterstmp16-232_thumbrespectively.

tmp16-242

0.01

0.1

1

10′2

104

106

tmp16-243

0.014

0.099

0.358

0.760

0.867

0.910

tmp16-244

0.44 • 10-2

0.033

0.118

0.241

0.250

0.250

The energy-loss rate of relativistic electrons in a monoenergetic field of photons with energy w0 and number density nph is given by the following equation.

tmp16-245_thumb

where

tmp16-246_thumb

In the Thomson and Klein-Nishina regimes Eq.(3.17) reduces to the well known expressions (e.g. Blumenthal and Gould, 1970)

tmp16-247_thumb

and

tmp16-248_thumb

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 Klein-Nishina regime it is almost energy independent. This implies that in the first case the steady-state electron spectrum becomes steeper, whereas the Compton losses in the Klein-Nishina 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 energy-loss 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

tmp16-249_thumb

where the radiation energy densitytmp16-250_thumband the electron energytmp16-251_thumbare expressed in units oftmp16-252_thumbrespectively. Note that the same equation describes the synchrotron energy losses if we replacetmp16-253_thumbby the magnetic field energy densitytmp16-254_thumb

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

tmp16-260_thumb

Photon-photon pair production

Photon-photon pair production is the inverse process to pair annihilation. Therefore the differential cross-section is identical to the pair annihilation cross-section, except for a different phase-space volume. In the relativistic regime this process is quite similar also to inverse Compton scattering. However, unlike the pair annihilation and Compton scattering, the photon-photon pair production has a strict kinematic threshold given by

tmp16-261_thumb

wheretmp16-262_thumbare the energies of two photons in units oftmp16-263_thumbcolliding at an angle 0 (in the laboratory frame).

The large cross-section makes the photon-photon 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 Y-rays 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 cross-section 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 cross-section in the monoenergetic isotropic photon field can be represented in the following analytical form

tmp16-266_thumb

The total cross-sections of inverse Compton scattering and pair production in an isotropic monoenergetic photon field of energytmp16-267_thumbare shown in Fig. 3.4 (curves 1 and 3, respectively). Both cross sections depend only on the product of the primarytmp16-268_thumband target photontmp16-269_thumbenergies,

tmp16-270_thumbWhile astmp16-271_thumbthe inverse Compton cross- section approaches the Thomson cross-section,tmp16-272_thumb the pair production cross-section approaches zero,tmp16-273_thumb

Fortmp16-274_thumbthe two cross-sections are quite similar and decrease with k0 andtmp16-275_thumbThe pair-

production cross-section has a maximum at the level oftmp16-276_thumb achieved attmp16-277_thumb

The parameter that characterises Y-ray absorption at photon-photon interactions in a source of size R is the so-called optical depth

 tmp16-289_thumb

wheretmp16-290_thumbdescribes 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 approximationtmp16-291_thumbwheretmp16-292_thumbis 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 sourcetmp16-293_thumbin fact non-negligible absorption can take place at low energies. For example, for a Planckian distribution of target photons, the optical depth t cannot be disregarded at energiestmp16-294_thumbbecause of interactions with photons from the Wien tail region (see Fig. 3.4). Very useful approximations for the optical depth of Y-rays in a Planckian photon gas can be found in Gould and Schreder (1967).

Because of narrowness of the pair-production cross-section, for a large class of broad band target photon energy distributionstmp16-300_thumbthe optical depth at given Y-ray energytmp16-301_thumbis essentially determined by a relatively narrow band of target photons with energy centered ontmp16-302_thumb(Her-terich, 1974). Therefore, the optical depth can be written in the form

tmp16-303_thumbwhere the normalization factor n depends on the spectral shape of the background radiation. For a power-law target photon spectrum,tmp16-304_thumbthe parameter n is calculated analytically,

tmp16-305_thumb(Svensson, 1987).

The energy spectrum of electrons produced at photon-photon pair production has been studied by Zdziarski and Light-man (1985), Coppi and Blandford (1990) and Bottcher and Schlickeiser (1997). For a low-energy monoenergetic photon fieldtmp16-306_thumb, and correspondinglytmp16-307_thumbthe spectrum of electron-positron pairs can be rep

resented, with an accuracy of better than a few per cent:

tmp16-316_thumb

The kinematic range of variation oftmp16-317_thumbis

 

tmp16-319_thumb

The differential energy spectra of Y-rays for several fixed values of the parameterstmp16-321_thumbare shown in Fig. 3.5. The spectra are symmetric around the pointtmp16-322_thumbAlthough the average energy of the secondary electrons istmp16-323_thumbfor very large s0 the interaction has a catastrophic character – the major fraction of the energy of the primary Y-ray 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 Y-rays with more realistic spectral distributions,tmp16-324_thumbone should integrate the product of spectrum given by Eq.(3.25) andtmp16-325_thumbover a broad primary

 

Y-ray energy interval. For a power-law spectrum of Y-rays,tmp16-331_thumb the spectrum of secondary pairs can be approximated, with an accuracy of better than 20 per cent:

tmp16-333_thumb

wheretmp16-334_thumband

tmp16-335_thumb(two electrons per interaction). Starting from the minimum (allowed by kinematics) energy attmp16-336_thumbthe electron spectrum sharply rises achieving its maximum attmp16-337_thumband then attmp16-338_thumbit behaves astmp16-339_thumb

Table 3.2 The mean fraction of energy of the primary 7-ray photon transferred to the leading secondary electron at electron-positron pair production in a mono-energetic radiation field (rad) and in the magnetic field (B) for different values of the parameterstmp16-346_thumbrespectively.

tmp16-349

1

3

10

W’2

104

10b

tmp16-350

0.500

0.701

0.797

0.891

0.948

0.966

tmp16-351

0.634

0.693

0.746

0.782

0.824

0.825

Two pairs of coupled processes – inverse Compton scattering and photon-photon pair production – determine the basic features of interactions of electrons and Y-rays 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 exceedtmp16-352_thumbthe processestmp16-353_thumb(Brown et al., 1973) andtmp16-354_thumb(Mastichiadis, 1991; Dermer and Schlickeiser, 1991) dominate over singletmp16-355_thumbpair 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 Y-rays around 1021 eV, and puts a robust limit on the mean free path of Y-rays of about 100 Mpc. Analogously, above 1020 eV the second process becomes more important than the conventional inverse Compton scattering. Because the tmp16-356_thumbchannels 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.

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