Fig. 3.12.2. CR modulation in the outer Heliosphere at distances 40-60 AU for particles with kinetic energy 1 GeV, expected according to Dorman et al. (1990).
3.12.2. Self-consistent problem including effects of CR pressure and kinetic stream instability in the Heliosphere
In Section 3.12.1 it was assumed that the solar wind velocity is constant up to the boundary of the Heliosphere and we did not take into account the influence of CR pressure on the solar wind moving, although it was shown in Sections 3.5-3.8 that this influence is very important in the outer Heliosphere. To take into account both effects (stream instability and CR pressure) Zirakashvili et al. (1991) considered self-consistent problem on the basis of the following set of equations in the hydrodynamic approximation (Drury and Volk, 1981; McKenzie and Webb,
Here the value Pw = (1/2//W(m)dm is the pressure of MHD turbulence, and Pm is the magnetic pressure. We will consider an inner boundary problem at r = ro >> 1 AU, where Pw = 0 and the boundary of Heliosphere (the transition from supersonic to subsonic flow) will be determined on the basis of solution of the self-consistent problem described by the set of Eq. 3.12.15. To obtain some rough analytical solution of the problem Eq. 3.12.15 we assume that the dimension of layer between ro and the transition layer is small relative to the radial distance r
(i.e. r - ro << r), so we can consider it as one-dimensional problem. In this case in the set of Eq. 3.12.15 we can consider r2 = r^ + 2ro (r -ro ) + (r -ro / « rO (1 + 2 (r - ro )/ro )to be a slow function of r, and take it outside the differentiation d/dr and the set of Eq. 3.12.15 will transform to:
Because the magnetic field is frozen in moving plasma, we obtain from the first equation of the set described by Eq. 3.12.16 that (subscript 'o' means the values at r = ro ) for the magnetic field
for the magnetic pressure
u for the Alfven velocity
and for cosY = (u/uo)cosYo . (3.12.20) Then on the basis of the set described by Eq. 3.12.16 we obtain pu =PoUo, Pc = Pco + PoUo(uo -u) + Pmo(1 -(o/u(3.12.21)
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