It can be easily seen that at R >> X2Ho the asymptotic expression for function $((, R) will be ik+1 ik+1 ik+3 ik+3 tt2 -¡k+1 ik+1

so the asymptotic expression for Aeff, j will take the form

(e//,j )>>X2Ho njhiXf-2^^-1)(X^+1 -Xk+1) ' (1.9.32)

Analysis of the expressions presented above shows the following:

(1) If R/Ho >> X2 then at j = 1, Aeff ,1 — R for any k (the dependence is the same as in the case of magnetic clouds treated above). At j = 2, A eff 2 — R4; and at j = 3, Aeff,3 - R6.

(2) If R/Ho <<X1 then in all cases Aeff, j ^ i.e. the examined set of inhomogeneities proves, in contrast to magnetic clouds, to be transparent for low energy CR particles.

(3) A significant difference from the case of scattering by magnetic clouds will be also observed for the interval X1 < R/Ho < X2 when an appreciable scattering takes place only for a comparatively narrow interval of R near Ro which, for example for j = 1 and k = 1, is determined by the relation

Shown as an example in Fig. 1.9.3 is the dependence of Aeff1 on RHoX2 at a = 1, P = 0 and X1/X2 = 0.1.

It can be seen from Fig. 1.9.3 that Aeff,1 in the interval Xj < R/Ho < X2 varies comparatively little (by less than a factor of 2) and reaches its minimum (i.e. the scattering is most effective) at RHoX2 ~ 0.3 in accordance with Eq. 1.9.33. Beyond the above mentioned limits A eff1 increases rapidly, namely in proportion to R 2 at large R and even more rapidly (exponentially) at small R.

Fig. 1.9.3. Dependence of Aeff j on RH0X2; the quantity Aeffj/a (where

Fig. 1.9.3. Dependence of Aeff j on RH0X2; the quantity Aeffj/a (where a = 3.6e2 H2^3/)) i

) is counted along the ordinate axis.

The results of calculations of Aeff, j, made by Dorman and Sergeev (1975,

1976), according to Eq. 1.9.30 including Eq. 1.9.31-1.9.33, are presented in Fig. 1.9.4-1.9.8 for j = 1, 2, 3, and the values of parameters k = _3a + 2(l + j + 0)

2 = 10 , 10 , 10_J . Here the values Aeff j/Gj are put along the ordinate axis, where

Gj =n"V(2 j )2 ¡¡¿2 (1 -¿i/A tH0/h2 )2. (1.9.34)

Fig. 1.9.4. Dependence of Aeff,jfGj (where Gj is determined by Eq. 1.9.34) on R/H0A2 (or on R300H0^2 , if R in V, H0 in Gs, and ¿2 in cm) for a model of magnetic inhomogeneities of the type h = (0, h(x ),0) existing on a background of the general field Ho = (H0,0,0) for k = _1, 0, 1, 2, 3 and 4. Left panel for j = 1 and

Fig. 1.9.4. Dependence of Aeff,jfGj (where Gj is determined by Eq. 1.9.34) on R/H0A2 (or on R300H0^2 , if R in V, H0 in Gs, and ¿2 in cm) for a model of magnetic inhomogeneities of the type h = (0, h(x ),0) existing on a background of the general field Ho = (H0,0,0) for k = _1, 0, 1, 2, 3 and 4. Left panel for j = 1 and

¿l/ ¿2 = 10_V , right panel for j = 1 and ¿1/ ¿2 = 10_3 •

R/300HoX2

in Fig. 1.9.4, but for j = 2 and Xi/ X2 = 10 3 (left panel), and for j

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