## 3. The Parker propagatorAfter 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 , Eq. (1) can be manipulated to obtain the form 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 with being the
Laplace-Transformation, 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 For later application we observe two asymptotic representations of the general solution. First, for one obtains with the corresponding asymptotic form of the Bessel 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. Similarly, one can approximate Eq. (10) for low momenta, i.e. , and finds Thus, for the case of sufficiently small © European Southern Observatory (ESO) 2000 Online publication: June 26, 2000 |