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


Astron. Astrophys. 327, 432-440 (1997)

Previous Section Next Section Title Page Table of Contents

Appendix A: derivation of the propagators

The propagators are evaluated by inserting the expansions (13) into Eq. (10), inverting the Fourier transformation analytically (since we fix [FORMULA]) and using the method of steepest descents to find the asymptotic dependence of the [FORMULA] at large [FORMULA]. For the one-dimensional case, a standard integral leads to

[EQUATION]

with [FORMULA], from which, by the method of steepest descents (e.g., Mathews & Walker 1970 ) one finds

[EQUATION]

In the case of diffusion ([FORMULA]), this result is correctly normalised, i.e., [FORMULA]. However, in general the normalisation is lost in this approach (cf. Rax & White 1992 ). Correcting for this, we find for the constants used in Eq. (17) the expressions

[EQUATION]

Appendix B: the synchrotron brightness distribution

The expression for the density (26) can be directly integrated numerically. Alternatively, it is easily computed by expanding the factor [FORMULA] and integrating over [FORMULA], which yields

[EQUATION]

where

[EQUATION]

and

[EQUATION]

Note that for p integer and larger than 2, the series reduces to [FORMULA] terms.

This expression recovers, for 3-dimensional diffusion ([FORMULA], [FORMULA]), the formula derived by Wilson (1975 ), and used by Valtaoja (1984 )

[EQUATION]

where U is the confluent hypergeometric function of the second kind (Abramowitz and Stegun 1972 ). This can be seen either directly from Eq. (26) or, in the special case [FORMULA], by noting that

[EQUATION]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
helpdesk.link@springer.de