Astron. Astrophys. 325, 866-870 (1997)

2. Kinematics

We consider an isotropic photon field where is the dimensionless photon energy in a rest frame which we call the laboratory frame. The Lorentz invariant scalar product of the four-momenta of two photons having energies colliding under an angle of cosine in the laboratory frame is then given by

Here, is the photon energy in the center-of-momentum frame. In order to allow for the possibility to create an electron-positron pair, conservation of energy implies , and the condition determines the pair-production threshold. is the Lorentz factor of the electron/positron in the cm frame where the produced electrons move with speed and . The definition of the angle variables needed in this calculation is illustrated in Fig. 1.

Fig. 1. Definition of the angles in cm and laboratory frame. denotes the direction of motion of an incoming photon, is the direction of motion of the produced electron and positron in the cm and the laboratory frame, and characterizes relative motion of the laboratory and the cm-frame, respectively.

The cm frame moves relative to the laboratory frame with velocity and Lorentz factor . The four velocity of the laboratory frame ( in the cm frame) is denoted by . The Lorentz factors of the produced pairs in the laboratory frame are related to the cm quantities by

Evaluating the Lorentz invariant scalar product

in the laboratory and the cm-frame, respectively, we find

and

Inserting Eq. (4) into Eq. (2) and using energy conservation () fixes the angle cosine u to

The differential cross section for - pair production (see Eq. [11]) depends on

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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