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In Fig. 3.11.2 the results of calculations are presented for y = 1.5, ui = 4 x10 cm/sec and N1 = 10 cm-3., and in Fig. 3.11.3 - the results are presented for y = 2.0,

Fig. 3.11.2. Expected u±(r) and x±(r)/r6>o for y = 1.5; U1 = 4x107 cm/sec, and N1 = 10 cm-3. Notations are the same as in Fig. 3.11.1.

A comparison of Fig. 3.11.1 with Fig. 3.11.2 shows that with growth of u1 and N1 the effect of non-linear interaction of CR with solar wind is substantially decreased. This results from a comparison of Fig. 3.11.1 and Fig. 3.11.3 that with increasing y the effect of non-linear interaction under consideration is increased.

Fig. 3.11.3. Expected u±(r) and x±(r)/r6>o for y = 2.0; U1 = 3 x107 cm/sec, and

N1 = 5 cm . Notations are the same as in Fig. 3.11.1.

Fig. 3.11.3. Expected u±(r) and x±(r)/r6>o for y = 2.0; U1 = 3 x107 cm/sec, and

N1 = 5 cm . Notations are the same as in Fig. 3.11.1.

The obtained results show that the effect of non-linear interaction of CR with solar wind in the direction normal to ecliptic plane, becomes substantial (i.e., x±(r)/[email protected] > 0.1-0.2) and it must be taken into account at the distances r > 15-20 AU with Ek min = 0.01 GeV and at r > 30-40 AU with Ek min = 0.1 GeV if y

=1.5, ui = 3x10 cm/sec, N = 5 cm . At the same time, if u\ = 4 x 10 cm/sec and N1 = 10 cm-3, the above distances are extended to 20-25 AU and 50-60 AU, respectively, for Ek ,min = 0.01 GeV and Ek ,min = 0.1 GeV. If, however, in the low energy range the power index in interstellar spectrum y = 2, then at u = 3 x10

cm/sec, N1 = 5 cm-3, the effect of non-linear transverse interaction becomes substantial at r > 7-10 AU if Ek,min = 0.01 GeV and at r > 25-30 AU if Ek,min =

0.1 GeV. In all of the mentioned cases, when non-linear effects of a transverse interaction of CR with solar wind become considerable, one must solve the self-consistent problem of CR modulation which was mention above when discussing Eq. 3.5.12.

3.11.3. The effect of the galactic CR gradients on propagation of solar wind in meridianal plane

Babayan and Dorman (1981) considered a self-consistent set of equations describing the hydrodynamic flow of solar wind in the medirional plane including the pressure of galactic CR. The hydrodynamic equations are linearized assuming a small difference of the solar wind parameters from the spherically symmetric case. The differential equations have been obtained which describe the variations of the solar wind parameters in the meridional plane depending on the galactic cosmic ray gradients. We shall treat the stationary one-fluid polytropic model wlthout magnetic fteld, i.e. we shall assume that the solar wind can be described by the following set of hydro-dynamic equations including equation of state:

and that the equation of state is

Here p, u, Pg are the density, velocity, and gas kinetic pressure of solar wind; r is the helio-centric distance; Pc is the pressure of galactic CR determined from the equation of anisotropic diffusion or by the Fokker-Planck equation including the diffusion, convection, drift, and energy change of CR particles. Gravitation is neglected since it is of significant importance only at the distances comparable with the distances at which the subsonic flow turns into supersonic flow. We are interested, however, in the distances much in excess of 1 AU.

Let the solution for the set determined by Eq. 3.11.10 be presented as

Pg = Pgo + Pg, p = p0 + p\ u = ua + u\ Pc = + Pc, (3.11.12)

where the parameters with a dash will be assumed to be small compared with the parameters labelled by the index 'o', i.e. with non-disturbed parameters. Treated as the zero approximation will be the solution of the spherically symmetric model of solar wind including the effect of the radial gradient of galactic CR and the charge exchange of the solar wind protons with interstellar neutral hydrogen (Babayan and Dorman, 1979a,b; see also Section 3.7), i.e. we shall assume that uo = {uo (r),0,0}, po (rpuo (r)r = const, Pg0p0 g = const,

1 oo

where n, p, v, and Ek are the differential energy spectrum, momentum, velocity, and kinetic energy of CR particles. It will be noted that, in turn, the factor n(Ek, r ) is determined from the condition of equality between the diffusive and convective fluxes of gaiactic CR is a function of uo .

After substituting Eq. 3.11.12 in Eq. 3.11.10 and linearizing the set of Eq. 3.11.10 for the solar wind propagation in medirional plane, the set of Eq. 3.11.10 will take the following forms: for the r component dr

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