When Eq. 3.12.29 can not be satisfied, the terminal transition must be of a gradual type (without formation of terminal shock wave). For Pmojpou2 ~ 1/450 the value
1 _2pm2becomes equal to 0 at uju ~6.1. Therefore if uo/u^ > 6.1 we
obtain shock wave transition. The value uo/u^ is determined by Eq. 3.12.23. If we consider CR as a relativistic gas (yc = 4/3) then uo/u^ ~ 7 and we expect a shock wave transition. The problem is that non relativistic CR also contribute to CR pressure and stream instability and in this case the value uo/u^ becomes smaller and for final solution we need some additional analyses.
3.12.3. Main results for Heliosphere; possible nonlinear effects for stellar winds
It is shown that CR nonlinear effects (pressure and stream instability) considered here play a vital role in the Heliosphere: influence on solar wind propagation; a role in the formation of terminal shock wave and a boundary between solar plasma and the interstellar medium; in the generation of MHD waves and formation of CR diffusion coefficient and anisotropy; significant influence on CR propagation and modulation in interplanetary space, especially in the outer Heliosphere. It is expected that these nonlinear effects can play an important role in dynamics of other stellar winds. CR nonlinear effects are expected to be especially important for stellar winds from quickly rotated stars with frozen in big magnetic fields (in this case both effects of CR pressure and kinetic stream instabilities effects will be important for limiting and formation of the stellar-sphere).
3.13. CR nonlinear effects in the dynamic Galaxy
3.13.1. CR propagation in the dynamic model of the Galaxy
CR can give important information on galactic wind. The extended dynamical halo (galactic wind) was first taken into account in CR propagation in the convection-diffusion model by Bulanov et al. (1972). Dogiel et al. (1980) extended this model, taking into account the adiabatic losses. Bloemen et al. (1993) investigated in detail CR propagation in the Galaxy, taking into account the galactic wind (diffusion - convection processes and adiabatic energy losses in extended halo). It was assumed that the velocity of galactic wind u increased proportionally to the distance z from the equatorial plane of the disk: u = Voz. The problem is that we do not know exactly the coefficient Vo and effective diffusion coefficient in the halo Kh . In the framework of this model there was calculated the expected depth X of CR crossing of matter (in g/cm2 ) for the average time of CR living in the Galaxy, and relative contents f of radioactive isotope 10Be. Fig. 3.13.1 shows the results of calculations of Bloemen et al. (1993) of expected relations Vo - Kh for X = 6-8 g cm-2 (which fits experimental data on CR chemical composition) and for f = 0.2-0.3 (that fits experimental data on relative contents of radioactive isotope 10Be).
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