Spherically symmetric diffusion from a single source (Cosmic Gamma Radiation)

In the standard diffusion approximation (i.e. neglecting convection) the propagation of CRs is described by the familiar diffusion equation (Ginzburg and Syrovatskii, 1964) which in the spherically symmetric case reduces to the following form

Hereis the energy distribution function of particles at instant t and distance r from the source;is the continuous energy loss rate, and D denotes the energy-dependent diffusion coefficient D(y). It is assumed that D is independent of R , i.e. a homogeneous medium is supposed. For the J-function-type initial distribution of particles both in space and time, i.e. for the Green’s function with respect to R and t, Eq.(A.1) has a simple analytical solution for an arbitrary injection spectrum Q (Atoyan et al., 1995):

is the inverse function of #(y), defined

Here

is the inverse function of #(y), defined as

The variable Yt corresponds to the initial energy of particles which are cooled to a given Y during the time t, and

corresponds to the effective diffusion radius up to which the relativistic particles with energy 7 propagate during the time t after their injection from the source. The function A is determined from the following equation:

Eq.(A.2) was obtained without any specification of the initial spectrum, the energy losses P(7) , or the diffusion coefficient D(y). Since the function f in Eq.(A.2) depends on R in a simple exponential form, this solution is convenient for integration over spatially distributed sources, as well as over a finite particle injection time. For example, in the case of continuous injection of CRs from a stationary point source

In the case of stationary sources distributed uniformly in space at distances beyond some R0 from us, we can substitutebeing the specific injection rate per unit volume) and integrate over the region R > R0, which results in

whereis the error-function, and

Notice that, since erfc (0) = 1, in the limiting case Rq —0 this latter equation gives the familiar result for the distribution of particles which are injected stationarily and uniformly into interstellar space, and suffer continuous energy losses. In this case the dependence on D disappears completely.