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Astron. Astrophys. 325, 401-413 (1997)

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6. Compton scattering

The presence of photons in a thermal plasma leads to essential energy losses due to Compton scattering. Thomson limit remains a good approximation while the photon energy is [FORMULA] (the rest mass of the electron) and the electron Lorentz factor is not too high. As the photon energy reaches [FORMULA] the difference from the classical limit becomes large, the principal effect is to reduce the cross section from its classical value. Numerous X-ray experiments show that the actual temperature of plasmas in astrophysical sources (far) exceeds 0.05 and the particle Lorentz factor exceeds often few units, that is why we consider the Klein-Nishina cross section.

The particle energy loss rate due to Compton scattering is given by

[EQUATION]

where [FORMULA], [FORMULA] are the LS particle Lorentz factor and speed prior to scattering, [FORMULA] is the initial photon energy in the LS, the background photon distribution [FORMULA] is normalized on the photon number density [FORMULA] or on the energy density as [FORMULA], [FORMULA], and an average particle energy change due to the scattering is

[EQUATION]

The Klein-Nishina differential cross section in the positron-rest-system (PRS) is expressed in terms of initial k and final [FORMULA] photon energies (Jauch & Rohrlich 1976),

[EQUATION]

[EQUATION]

where [FORMULA] is the photon scattering angle in this system. The particle energy change in the LS due to the recoil effect is

[EQUATION]

where [FORMULA] is the angle between the incoming photon and positron velocity vectors in the PRS, [FORMULA].

After the integration one can obtain

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and [FORMULA] is the dilogarithm

[EQUATION]

[EQUATION]

Formulas (38)-(39) give exactly the same result as Jones' (1965) Eq. (13). The delta-function approximation of the photon distribution [FORMULA] can sometimes be used for evaluation of the integral (38). We have found that in some cases it shows a good agreement with exact calculations, e.g. for the Planck's distribution with [FORMULA] (see Fig. 2).

The Thomson limit of the Compton scattering can be obtained similarly by equating [FORMULA] in Eq. (36)

[EQUATION]

For the energy dispersion rate one can get

[EQUATION]

where [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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