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

3. The pair yield

The differential yield of produced pairs is calculated as

where

The differential cross section has been evaluated by Jauch & Rohrlich (1959):

where

We may express the solid angle element . Using Eq. (2), we find

This enables us to carry out the u -integration in Eq. (9) immediately. If we write the denominators in Eq. (11) as

with

we find

where

with

and we used the integrals

and the identity

which follows from Eqs. (5), (6) and (14). Now, inserting Eq. (15) into Eq. (8) yields the exact expression for the differential pair injection rate. Using Eq. (1) we transform the µ integration into an integration over . This leads us to

which can be calculated analytically. The integration limits follow from and the condition which yields

where

Using the integrals 2.271.4, 2.271.5, 2.272.3, 2.272.4, and 2.275.9, of Gradshteyn & Ryzhik (1980), we find as final result for the differential pair yield

where for we have

and

For we find

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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