del ds

Po ds ds

Po ds

where

K1cl

ds Po J

3.12. Effects of CR kinetic stream instability in the Heliosphere

3.12.1. Rough estimation of stream instability effect at constant solar wind speed

First let us consider (Dorman et al., 1990) only the effects of CR kinetic stream instability in the outer Heliosphere without taking into account effects of CR pressure on the movement of the solar wind. Therefore we assume that the velocity of the solar wind u = 400 k^sec = const, the radius of the Heliosphere ro = 60 AU, the spiral interplanetary magnetic field has components

Hr = [r/rs)~2Hs, Hp = [ur/riQsl^s smtf, He = 0, (3.12.1)

where rs = 7 x 1010cm is the radius of the Sun, Hs ~ 2 Gs is the strength of the Sun's general magnetic field on the surface, Qs = 27 x 10-6sec-1 is the angular velocity of solar rotation, 9 is the polar angle ( d = n/2 is the solar equator). At large distances from the Sun the full spiral interplanetary magnetic field H and the angle Y between magnetic field and radial direction will be

where r = 1 AU is the radius of the Earth's orbit, and H\ = H(r) = 5x 10 5 Gs is the strength of IMF at the distance of 1 AU from the Sun. The galactic CR

anisotropy generated in the interplanetary space is determined by the spiral magnetic field, particle scattering, and CR gradients and convection. Near the Earth's orbit the average anisotropy has an amplitude of about 0.5 % and is perpendicular to the radial direction. In the first approximation the amplitude of average anisotropy must be x (cos ¥)-1 rc r sin#, and at large distances from the Sun we expect a large amplitude of CR anisotropy. Therefore we expect that the effects of stream instability in the outer Heliosphere must be very important.

According to the Section 3.3 we consider the generation by CR stream instability of MHD waves propagated along the magnetic field (axis Z). the growth rate r(kz) determines the evolution of spectral energy density of MHD waves with wave number kz in Z - direction and with accidental phases W(kz) will be determined by:

Resonance interaction of particles with momentum p and cosines of pitch angle ¡u = (pH)/(pH) will be with waves numbers kz = ± ZeH/(pc¡¡). We consider here only particles with velocity v ~ c >> Va = h/(4.p)0'5 = 5 X106 cm/sec for conditions in the interplanetary space. On the basis of Eq. 3.3.1 we obtain:

2 c2

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