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Astron. Astrophys. 358, 347-352 (2000)

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3. The Parker propagator

After having established the relevant parameters of transport for the three turbulence models, we now turn to the derivation of the propagator of Parker's Eq. (1).

Using Eq. (5) and the general Ansatz [FORMULA], Eq. (1) can be manipulated to obtain the form

[EQUATION]

where we have introduced the new variable (see also Jokipii 1967) [FORMULA] and the modulation parameter

[EQUATION]

Furthermore, we have used the abbreviations [FORMULA] and [FORMULA].

The general solution for the phase space distribution function [FORMULA] can be expressed in terms of the Green's function [FORMULA], i.e. Parker's propagator:

[EQUATION]

The Laplace-transformed Green's function

[EQUATION]

with [FORMULA] being the Laplace-Transformation, s the corresponding variable, [FORMULA] the Gamma function, and M and U denoting Kummer's functions, satisfies the ordinary inhomogeneous differential equation

[EQUATION]

resulting from Eq. (6). [FORMULA] is the delta function. The homogeneous part is the confluent hypergeometric differential equation, also called Kummer's equation.

Following the standard method of solving such equations and constructing the Green's functions one can derive, by applying an inverse Laplace transformation [FORMULA] (see formula 5.20.27 in Erdélyi et al. (1954), the exact solution for the differential intensity (Stawicki 1999):

[EQUATION]

Here, [FORMULA] and [FORMULA] are integration variables which result from the construction of the Laplace-transformed Green's function [FORMULA]. To simplify the notation we have introduced the functions

[EQUATION]

as well as the modulation parameter [FORMULA]. The quantity [FORMULA] is a modified Bessel function of the first kind.

This solution is valid for arbitrary source functions [FORMULA] with which super-alfvénic charged particles of momentum p are injected at position r. Notice that, in contrast to the solutions presented previously, no assumptions were made with regard to Kummer's functions or the source function.

For later application we observe two asymptotic representations of the general solution. First, for [FORMULA] one obtains with the corresponding asymptotic form of the Bessel function:

[EQUATION]

This limiting value of the differential intensity is finite and shows, that the particle spectrum depends on the source function not only close to the source but also far away from it.

Similarly, one can approximate Eq. (10) for low momenta, i.e. [FORMULA], and finds

[EQUATION]

Thus, for the case of sufficiently small p, we get [FORMULA]. In other words, the phase space distribution function [FORMULA] becomes constant as p approaches zero.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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