## Kyjrr0z2 1cosf22 21432

Then in the integral in Eq. 2.14.32 we make the substitution of variables according to the relations

2 (P1 + P2 ) = ® = r + ro ; VP1P2 = v = 2^ cos--. (2.14.33)

1 d and the known relation J0 (z)_

2.14.32 in the form

Using Eq. 2.14.33 and the known relation J0(z)_--J1(z) we write the Eq.

z dz

Comparing this expression with the integral representation of the product of Whittaker's functions (Eq. 2.14.24) we obtain the expression for the Green's function of the three-dimensional equation in the form which does not include the integration operation

The variables p1 and p2 are related with the variables r and ro by the expressions

P _ r + ro - |ro - r|, p _ r + ro + |ro - r| . (2.14.36)

Writing the differential operator in Eq. 2.14.35 in terms of these variables, we finally obtain

G(r,ro;®)_ if-|"WV2(yM",1/2(x) , (2.14.37)

4n|ro - r| ^ dx dy where x _ kp1, y _ kp2 . Let us write the solution of the transfer Eq. 2.14.1 through the Green's function described by Eq. 2.14.37. Taking into account Eq. 2.14.3, Eq. 2.14.6 and Eq. 2.14.9, we obtain for the particle density N(r,®):

As results from Eq. 2.14.38, the Green's function of Eq. 2.14.1 has the following form

2.14.4. Possible inclusion of the variations of particle energy

Now we show the way in which it is possible to take into account the process of particle energy variations in the framework of the considered formalism. In this case, instead of the equation Eq. 2.14.1, one should consider the equation

wherep is the particle's momentum. Using the Mellin transform n(r, 5, t )= J dpps-1n(r, p, t) (2.14.41)

we write the Eq. 2.14.40 in the form

<)n{r: ',t ) = (- u.V )(r, s, t ) + 2U°sn(r, s, t) + Qo (r, s, t ). (2.14.42)

The substitution of the unknown function according to the Eq. 2.14.7 transforms the Eq. 2.14.32 to the form

= AT(r,s,t)+ I 2-s I---2 T(r,s,t)+ Qo(r,s,t), (2.14.43)

3 ) r i.e. the Eq. 2.14.40 is actually transformed to Eq. 2.14.18 with inessential (in the framework of the considered method) variation of Coulomb's potential which is represented by the coefficient in the square brackets on the right-hand side of the Eq. 2.14.43. Thus, to determine Y(r, s, t), the developed above formalism can be completely applied (one should observe that making of the inverse Mailing transformation to determine the unknown Green's function may produce considerable mathematical difficulties).

2.14.5. The Green's function for the stationary isotropic diffusion in the case of power dependence of the diffusion coefficient on the distance

Basing on the non-stationary diffusive-convective transfer of CR in interplanetary space and taking into account adiabatic cooling of particles, Webb and Gleeson (1977) composed the equation with the source in the form of a five-dimensional 8-function (time, particle rigidity, and 3 spatial coordinates) to determine the Green's function. The further integration over five-dimensional volume made it possible to represent the Green's function in the form of solution of a multi-integral equation. In the special case of the stationary spherically symmetric model of isotropic diffusion the Green's function is written in analytical form through Bessel's function of the first kind. If the coefficient of isotropic diffusion k = k0 (p)rb, where p is a particle momentum, r is the distance from the Sun, Ko (p ) is an arbitrary scalar function of p, then the Green's function

2 y2

where u is the radial velocity of the solar wind, I|^ is the modified Bessel's function of the first kind. To abbreviate writing, the following notations are used in Eq. 2.14.44:

2.15. On a relation between the correlation function of particle velocities and pitch-angle and spatial coefficients of diffusion

### 2.15.1. Correlation function of particle velocities

Forman (1977a,b) developed the concept of the correlation function of particle velocities (v, (()■ ((')) which was proposed by Kubo (1957). If is the corresponding cosine of the angle between the velocity direction and the i-axis of coordinates, then v2 1 1 , > (vi((/vj(('/ = ~t I m^M IM'jdMjm'j;t-1''

where ,jj ;t - t^j is a number of particles between j and ju, + dju, in the time instant t which had in the instant t' the direction cosine j j . The function Q is a solution of the equation

V Xo y where D is the operator of the equation of a transfer in the pitch-angle space with the initial condition

and can be expanded in the series

«(j;t -1') = £Rk())(u')exp(- (t -1) )/ J(u)du. (2.15.4) k / -1

Here Rk(juj and Tk are the eigen-functions and eigen-values of the operator D (i.e. DRk = Rk¡Tk ). In a special case of isotropic scattering and injection the functions Rk (u) at u = 1 transform into Legendre's polynomials Pk (u) and Tk transforms into 2T1/k (k +1); in this case vt1 is the transport path of particles for scattering. The method developed of the correlation function of particle velocities makes it possible to apply the theory of CR diffusion to the actual cases when a scattering is not isotropic, but, for example, takes place mainly along the field.

2.15.2. Connection between the correlation function of particle velocities, pitch-angle and spatial coefficients of diffusion

In general form, a connection between the correlation function of particle velocities (v, (t)■ (t')) and the spatial coefficient of anisotropic diffusion Dj is determined by the relation