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).
where we have introduced the new variable (see also Jokipii 1967) and the modulation parameter
Furthermore, we have used the abbreviations and .
The general solution for the phase space distribution function can be expressed in terms of the Green's function , i.e. Parker's propagator:
The Laplace-transformed Green's function
resulting from Eq. (6). 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 (see formula 5.20.27 in Erdélyi et al. (1954), the exact solution for the differential intensity (Stawicki 1999):
Here, and are integration variables which result from the construction of the Laplace-transformed Green's function . To simplify the notation we have introduced the functions
as well as the modulation parameter . The quantity is a modified Bessel function of the first kind.
This solution is valid for arbitrary source functions 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.
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.
Thus, for the case of sufficiently small p, we get . In other words, the phase space distribution function becomes constant as p approaches zero.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000