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

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3. The pair yield

The differential yield of produced pairs is calculated as

[EQUATION]

[EQUATION]

where

[EQUATION]

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

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

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

[EQUATION]

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

[EQUATION]

with

[EQUATION]

we find

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

with

[EQUATION]

[EQUATION]

and we used the integrals

[EQUATION]

[EQUATION]

and the identity

[EQUATION]

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 [FORMULA]. This leads us to

[EQUATION]

[EQUATION]

[EQUATION]

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

[EQUATION]

where

[EQUATION]

[EQUATION]

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

[EQUATION]

[EQUATION]

where for [FORMULA] we have

[EQUATION]

[EQUATION]

[EQUATION]

and

[EQUATION]

[EQUATION]

For [FORMULA] we find

[EQUATION]

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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