SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

2. Kinematics

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

[EQUATION]

Here, [FORMULA] 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 [FORMULA], and the condition [FORMULA] determines the pair-production threshold. [FORMULA] is the Lorentz factor of the electron/positron in the cm frame where the produced electrons move with speed [FORMULA] and [FORMULA]. The definition of the angle variables needed in this calculation is illustrated in Fig. 1.

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

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

[EQUATION]

Evaluating the Lorentz invariant scalar product

[EQUATION]

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

[EQUATION]

and

[EQUATION]

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

[EQUATION]

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

[EQUATION]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

helpdesk.link@springer.de