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Astron. Astrophys. 357, 301-307 (2000)

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2. Theory of relativistic inverse Compton scattering

For the sake of completeness we summarize in Sects. 2.1 and 2.2 the basic formulars describing the relativistic non-resonant and resonant inverse scattering processes.

2.1. Non-resonant inverse Compton scattering

Let us begin with Thomson case without a magnetic field 1. The average rate of photons scattered by relativistic electrons is given (e.g. Feenberg & Primakoff, 1948) by

[EQUATION]

where [FORMULA] denotes the energy of the infalling photons, [FORMULA] the angle between the moving direction of the electron in the laboratory frame and the direction of the infalling photon stream, [FORMULA] gives the number of infalling photons per unit volume dV, energy range [FORMULA], and solid angle element [FORMULA], and finally [FORMULA] represents the cross section. Stared letters denote quantities in the electron rest frame, unstared letters in the laboratory frame. Therefore, [FORMULA]. As we are not interested in the rate of scattered photons but in the amount of energy transfer from an electron to photons, we have to express [FORMULA] explicitly which results in

[EQUATION]

for the differential energy loss of the electron per moving unit [FORMULA]. The relation between the photon energies [FORMULA] and [FORMULA] before and after Thomson scattering can be found in text books (e.g. Jackson, 1998, p. 696):

[EQUATION]

with [FORMULA] the rest mass of the electron, [FORMULA] the angle between the direction of the scattered photon and the moving direction of the electron, and [FORMULA] the scattering angle, both angles in laboratory frame.

Taking a closer look at the cross section we have to distinguish between Thomson case and Klein-Nishina case i.e., between [FORMULA] and [FORMULA]. Following Feenberg and Primakoff (1948) we are now able to approximate the third integral in Eq. (2). With this and a change in the order of integration we come out with

[EQUATION]

Here, [FORMULA] is the Thomson cross section, [FORMULA] denotes the change from Thomson case to Klein-Nishina case, and the quantities

[EQUATION]

[EQUATION]

are approximately the average energy loss of the primary in a single Compton collision for [FORMULA] and [FORMULA], respectively.

2.2. Resonant inverse Compton scattering

In the presence of a (strong) magnetic field the cross section for inverse Compton scattering has a resonance when the Doppler shifted frequency of the infalling photon equals the gyro-resonance frequency of the electron in its rest frame. The later one can be written as

[EQUATION]

where [FORMULA] represents the strength of the magnetic field at the location of the electron in its rest frame. The Doppler shifted energy of the incoming photon is in resonance with the gyrating electron if

[EQUATION]

in the laboratory frame (here we made use of the fact, that electrons in pulsar magnetospheres travel along the magnetic field lines, which leads to [FORMULA]).

For a description of the resonance we use the Breit-Wigner formula. In general, the maximum cross section within the resonance can be expressed as

[EQUATION]

with the relaxation time

[EQUATION]

both formulated in the laboratory frame. 2 The width of the Breit-Wigner resonance is described by

[EQUATION]

We note, that the expression

[EQUATION]

is independent on the magnetic field strength.

To find a similar expression to Eq. (4) in the case of resonant scattering we approximate the integration over [FORMULA] by substituting it with a multiplication of the maximum resonant cross section [FORMULA] with the width [FORMULA] of the resonance. We find using Eq. (12):

[EQUATION]

Here, [FORMULA] has to be taken from Eq. (8), [FORMULA] is mentioned below Eq. (4), and [FORMULA] come out of Eqs. (5) and (6) with [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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