When a photon is scattered by an electron, the energy and direction of motion of both the photon and the electron are usually altered. The change in properties of the photon is described by the usual Compton scattering formula
where the electron is assumed to be at rest before the interaction, and ' are the photon energies before and after the interaction, and _{12} is the angle by which the photon in deflected in the encounter (see Fig. 3).
For low-energy photons and mildly relativistic or non-relativistic electrons, << m_{e} c^{2} and the scattering is almost elastic (' = ). This limit is appropriate for the scatterings in clusters of galaxies that cause the Sunyaev-Zel'dovich effect, and causes a considerable simplification in the physics. Although the scatterings are usually still referred to as inverse-Compton processes, they might better be described as Thomson scatterings in this limit.
Scatterings of this type will also cause Sunyaev-Zel'dovich effects from the relativistic plasma of radio galaxies. The lobes of radio galaxies emit strong synchrotron radiation, and must contain electrons with Lorentz factors _{e} 10^{8}. In the rest frames of such electrons the microwave background radiation appears to have a peak at photon energies ~ 0.1 m_{e} c^{2}, and the assumption of elastic scattering will be inappropriate. Little theoretical work has been done on the spectrum of the scattered radiation in this limit, but see Section 5.
In this thermal scattering limit, the interaction cross-section for a microwave background photon with an electron need not be described using the Klein-Nishina formula,
but rather the classical Thomson cross-section formula which results in the limit ' -> . Then if the geometry of the collision process in the electron rest frame is as shown in Fig. 3, the probability of a scattering with angle is
where the electron velocity v_{e} = c, and µ = cos. The probability of a scattering to angle ' is
(Chandrasekhar 1950; Wright 1979), and the change of photon direction causes the scattered photon to appear at frequency
with µ' = cos'.
It is conventional (Wright 1979; Sunyaev 1980; Rephaeli 1995b) to express the resulting scattering in terms of the logarithmic frequency shift caused by a scattering, s (Sunyaev uses u for a related quantity),
when the probability that a single scattering of the photon causes a frequency shift s from an electron with speed c is
Using (23-25), this becomes
where µ' can be expressed in terms of µ and s as
(from equations 25 and 26), and the integral is performed only over real angles, so that
in (28). The integration can be done easily, and Fig. 4 shows the resulting function for several values of . The increasing asymmetry of P (s; ) as increases is caused by relativistic beaming, and the width of the function to zero intensity in s,
increases because increasing causes the frequency shift related to a given photon angular deflection to increase.
Figure 4. The scattering probability function P (s; ), for = 0.01, 0.02, 0.05, 0.10, 0.20, and 0.50. The function becomes increasingly asymmetric and broader as increases. |