## 2. Theory of relativistic inverse Compton scatteringFor 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 scatteringLet us begin with Thomson case without a magnetic
field where denotes the energy of the
infalling photons, the angle between
the moving direction of the electron in the laboratory frame and the
direction of the infalling photon stream,
gives the number of infalling
photons per unit volume for the differential energy loss of the electron per moving unit . The relation between the photon energies and before and after Thomson scattering can be found in text books (e.g. Jackson, 1998, p. 696): with the rest mass of the electron, the angle between the direction of the scattered photon and the moving direction of the electron, and 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 and . 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 Here, is the Thomson cross section, denotes the change from Thomson case to Klein-Nishina case, and the quantities are approximately the average energy loss of the primary in a single Compton collision for and , respectively. ## 2.2. Resonant inverse Compton scatteringIn 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 where 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 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 ). 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 both formulated in the laboratory
frame. 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 by substituting it with a multiplication of the maximum resonant cross section with the width of the resonance. We find using Eq. (12): Here, has to be taken from Eq. (8), is mentioned below Eq. (4), and come out of Eqs. (5) and (6) with . © European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |