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Fig. 3.7.7. The character of a decrease of solar wind velocity u with a distance from the Sun

r including the process of re-charging at u = 3x10 cm/sec, N1 = 5 cm , ß = 0,

NHo = 0.1cm (a), NHo = 0.5 cm (b), and NHo = 1.0 cm (c); curves: 1 -

no x Ek2'5, Wc = 1 eV/cm3 ; 2 - no E-2-5, Wc = 2.67 eV/cm3 ; 3 - no ~ E-3 0,

Wc = 1 eV/cm3 ; 4 - no ~ E-3 0, Wc = 7.11 eV/cm3 ; 5 - no R"2 5, Wc = 1 e^cm3 .

The calculations for Fig. 3.7.7 were made for various values of the interstellar density of energy of CR: curves 1, 3 and 5 correspond to Wc = 1 eV/cm-3, and curves 2 and 4 are related to the values Wc = 2.67 and 7.11 eV/cm-3. A character of dependence of u/u on r is substantially dependent on Wc, Nho and on a form of interstellar energy spectrum. In Fig. 3.7.8 the dependences of u/u on r are presented for Nho = 0.5 cm-3 at Wc = 1 eV/cm-3 and the spectrum form ^ E- for ß= 0, 0.5 and 1. Fig.3.7.8. The character of a decrease of solar wind velocity u with distance from the Sun r, including re-charging process depending on parameters ¡3 at u = 3 x10 cm/sec, N1 = 5 cm-3 , NHo = 0.5 cm-3 , n0 ~ £-3 0 , Wc = 1 e^cm3 . Curves 1, 2 and 3 correspond to ¡3= 0, 0.5 and 1.

Fig.3.7.8. The character of a decrease of solar wind velocity u with distance from the Sun r, including re-charging process depending on parameters ¡3 at u = 3 x10 cm/sec, N1 = 5 cm-3 , NHo = 0.5 cm-3 , n0 ~ £-3 0 , Wc = 1 e^cm3 . Curves 1, 2 and 3 correspond to ¡3= 0, 0.5 and 1.

It is seen from Fig. 3.7.8 that with a growth of parameter 3 in Eq. 3.7.8, the dimension of the region occupied by solar wind should be somehow increased: the distance where u/u is decreased by an order, equals 72, 100, and 112 AU, respectively, at 3 = 0, 0.5, and 1.

The character of CR modulation at Nh0 = 0.5 cm-3 is seen from Fig. 3.7.9 for particles with the rigidity R from 0.5 to 100 GV.

A comparison with the results of Section 3.7.2 (where it was assumed Nh0 = 0) shows that including of the process of re-charging results in some weakening of the modulation depth; it is caused by more rapid deceleration of solar wind with a growth of Nh0 . Fig. 3.7.9. Modulation depth of CR n/no depending on a distance to the Sun r for particles with the rigidity R = 0.5, 1, 5, 10, and 100 GV (Curves 1-5, respectively) at Nh = 0.5 cm-3 , 0= 0, u = 3 x107 cm/sec, N1 = 5 cm-3 , no - E-2'5 , Wc = 1 eV/cm3 .

Fig. 3.7.9. Modulation depth of CR n/no depending on a distance to the Sun r for particles with the rigidity R = 0.5, 1, 5, 10, and 100 GV (Curves 1-5, respectively) at Nh = 0.5 cm-3 , 0= 0, u = 3 x107 cm/sec, N1 = 5 cm-3 , no - E-2'5 , Wc = 1 eV/cm3 .

3.8. Expected change of solar wind Mach number accounting the effects of radial CR pressure and re-charging with neutral interstellar atoms

If we introduce the Mach number M = u/Vs , where vs AyPglp}11 (3-8.!)

is the sound velocity, then on the basis of Eq. 3.7.13, taking into account the analytical solution of Eq. 3.7.11 for CR isotropic diffusion, we obtain: 