As a consequence of both effects, case (b) gives flux levels similar to those of case (a) inside ~ 60 AU. At distances smaller than ~ 60 AU it is difficult to distinguish between the low- and the high-injection case (see Fig. 3.10.1—3.10.3). Thus, an observational discrimination between the two alternatives (before a spacecraft encounters the heliospheric shock) can only be made in that part of the precursor that is close to the heliospheric shock.

3.10.6. The summary of obtained results

From Fig. 3.10.1—3.10.3 and Table 3.10.1 it can be seen how the heliospheric shock is modified by the simultaneous presence of pickup ions, anomalous and galactic CR for the low (n = 0.0003) and high (n = 0.9) injection efficiency cases. The presence of pickup ions in the solar wind according to Le Roux and Fichtner (1997a,b) results in:

1. The solar wind decelerates upstream of the heliospheric shock by about 12.8%, because of the charge exchange between solar wind protons and interstellar hydrogen; 2. Consequently the solar wind ram pressure is lower and the heliospheric shock moves inward from 80.8 AU (the initial position without pickup ions or CR) to 73.7 AU;

3. The heliospheric shock compression ratio is reduced from 5 = 4 to 5 = 3.5, because the mixture of solar wind protons with the hot pickup ions decreased the upstream Mach number from = 14.1 to = 4.3.

According to Le Roux and Fichtner (1997a,b), for very small injection efficiency (n = 0.0003) the heliospheric shock moves outward from 73.7 to 74.3 AU. Because anomalous CR move the heliospheric shock outward and galactic CR move it inwards, it means that anomalous CR protons are more important than galactic CR protons in determining the heliospheric shock position. The outward movement of the heliospheric shock is caused by the loss of internal energy of the solar wind (including pickup ions) across the heliospheric shock owed to the transfer of pickup ions across the threshold to the anomalous CR population by adiabatic heating. The inward movement of the heliospheric shock by galactic CR is owed to the positive galactic CR gradient which decelerates the solar wind and reduces the ram pressure. The compression ratio is reduced slightly by the anomalous CR to 5 = 3.4. Mainly oing to galactic CR, a precursor to the heliospheric shock's subshock is formed with a scale length of = 34 AU which increases the total deceleration of the solar wind to = 19.8%. Despite the combined presence of both anomalues CR and galactic CR, the heliospheric shock position and compression ratio are still dominated by pickup ions.

In the case of a big injection efficiency (n = 0.9) the heliospheric shock according to Le Roux and Fichtner (1997a,b) also moves outward, but to a lesser degree. The heliospheric shock is strongly modified by mainly anomalous CR protons from 5 = 3.5 to 1.5. Largely owing to anomalous CR, a precursor to the heliospheric shock's subshock is formed with a shorter scale length of = 15 AU which increases the total solar wind deceleration dramatically to = 45%. In this case the heliospheric shock is predominantly modified by anomalous CR protons, while its position is still mainly determined by pickup ions. Galactic CR protons contribute the least in modifying the heliospheric shock.

In Fig. 3.10.1 the combined spectra of pickup ions, anomalous and galactic CR protons are shown as solid curves at different radial distances r. The pickup ions spectra are noticeable at kinetic energies Ek < 10-6 GeV, whilst for Ek > 10-6 GeV, anomalous CR proton intensities dominate the galactic CR proton intensities, except for r < 23 AU, and Ek > 200 MeV where the reverse is true. Let us consider two cases:

1. For n = 0.0003 the spectrum at the heliospheric shock is a power law except at the highest energies where it rolls over because k/u > r5h . This implies that the galactic CR induced heliospheric shock precursor is too small so that anomalous CR at all energies see the same effective 5 = 3.4 value. The dynamical influence of CR on the heliospheric shock is small, and a test particle approach would have given a similar result.

2. For n = 0.9 the spectral slope of anomalous CR at the heliospheric shock is steeper overall and has a clear energy dependence. At low energies the spectrum has a steep slope because the anomalous CR see only the subshock with 5 = 1.5, while undergoing diffusive shock acceleration. At higher energies the particles have a larger effective diffusion length and see additionally the CR induced precursor in crossing the heliospheric shock (effectively seeing a larger 5-value), leading to a decreasing slope. Despite these large differences in the spectral slopes of anomalues CR at the heliospheric shock, the roll over portion of the spectrum basically occurs at the same energies, because k/u is the same for both n-values.

Consequently from Fig. 3.10.2 can be seen that the modulated intensity levels for energies close to the roll over energy are similar for the two injection efficiencies n = 0.0003 and 0.9. They are also nearly the same at lower energies except close to the heliospheric shock beyond = 60 AU and below = 200 MeV, where the spectra have different radial gradients (this is owing to the Compton-Getting factor C because the CR radial gradient = Cu/k which is determined by the spectral slope at the heliospheric shock, see Eq. 3.10.7).

3.11. Expected CR pressure effects in transverse directions in Heliosphere

3.11.1. CR transverse gradients in the Heliosphere and its possible influence on solar wind moving

The data on annual and semi-annual variations of CR show a presence of transverse CR gradients in the Heliosphere. A presence of these gradients results also from an assumption that a part of the observed hysteresis of CR (Dorman and Dorman, 1967a,b,c) is caused by a shift of a zone of solar activity toward low latitudes during a cycle of solar activity (Stozhkov and Charakchyan, 1969). The existing of CR transverse gradients follows also from the analysis of NM data and muon telescopes underground data for about 25 years on CR drift and convection diffusion anisotropies (Ahluwalia and Dorman, 1995a,b; Dorman and Ahluwalia, 1995; see also Chapter 16). Whereas solar activity is concentrated generally in the latitude band ± 30°, one should expect that CR density be to decrease with approaching helio-equator. Therefore a compression of solar wind will take place. A focusing action of CR over helio-latitude is produced in this way. Approximate estimation of this focusing action was performed by Dorman and Dorman (1969) with assumption that CR density in interplanetary space is a power function depending on a distance r to the Sun. Babayan and Dorman (1977b) carried out more accurate calculations of a transverse interaction of CR with solar wind including a solution of the problem of non-linear interaction of CR with solar wind in the radial direction (Babayan and Dorman, 1977a) which made it possible to find actually the dependence of density and pressure of CR on r.

3.11.2. The simple model for estimation of upper limit of CR transverse effects on solar wind

Let us consider the following idealized simple model. Let solar wind be distributed uniformly over longitude and concentrated on the both sides from the helio-equator in the helio-latitude zone -do <d< +do. Outside this helio-latitude band let the density of CR be equal to its interstellar value no (Ek) and the pressure of CR on the latitude boundaries

(here p and v are a momentum and velocity of particles). To simplify the problem we shall consider that a distribution of CR density n(r,Ek) in the helio-equatorial plane is determined by a solution of the problem of non-linear modulation in a spherically symmetric case (see Sections 3.7-3.9). Respectively, the pressure of CR in the plane of the helio-equator will be

Let the CR pressure at 6 = ±6o be equal to the pressure in the interstellar space Pco . In this case the average value of the gradient of CR across the plane of the helio-equator at a distance r from the Sun will be

The energy spectrum of CR in the plane of the helio-equator at a distance r from the Sun, according to diffusion-convection theory, is determined by the expression i{r,Ek ) = n0(Ek )exp - J

where no (Ek) is the spectrum outside of solar wind, A(r) is transport path for scattering of particles, u(r) is the radial velocity of solar wind, determined including an inverse action of CR onto solar wind, respective to Section 3.7.2.

Whereas the problem is symmetric relative to the plane of the helio-equator, consider then a motion of solar plasma above the helio-equator. For a motion of solar plasma under the action of CR across the plane of the helio-equator, we have the equation p(uV)U± (r)« (PCo - Pc (r))) , (3.11.5)

where p is solar wind density and u±(r) is the velocity of solar wind plasma in the transverse direction. In Eq. 3.11.5 the terms are absent which take into account the gas kinetic pressure and the pressure of magnetic field, because we can assume for the first approximation that the transverse gradients of these pressures are equal to zero. It should be noted that with a strong compression of solar wind by CR there must arise a difference in gas kinetic and magnetic pressures inside and outside the modulation zone of CR; in this case the respective terms will appear in Eq. 3.11.5, which will result in a decrease of the compression effect. Moreover, in the case of strong compression of the solar wind one must take into account that there will occur a rapprochement of magnetic inhomogeneities, and therefore a decrease of transport path for scattering A, i.e. intensification of CR modulation which, in its turn, should result in an increase of the transverse gradient of CR pressure and, finally, in an increase of the effect of solar wind compression. A variation of CR modulation will result in a respective variation of the dependence u(r). Therefore in the general case, we have a considerably complicated self-consistent non-linear problem, the solving of which is difficult. Therefore Babayan and Dorman (1977b) presented only estimates obtained on the basis of Eq. 3.7.12 including Eq. 3.7.8, Eq. 3.7.9 and Eq. 3.7.11 of a solution for u(r ) which was presented in Section 3.7.2, to determine what is the expected effect of solar wind compression by CR and when the effect of non-linear interaction of CR with solar wind in the transverse direction is substantial. Let us add to Eq. 3.7.12 the equation of continuity for the solar wind p(r )r2u(r ) = P1r2u1, (3.11.6)

where p1 and u1 are, respectively, a density and velocity of the wind at the distance of the radius of the Earth's orbit from the Sun r1 = 1 a.u. Substituting Eq. 3.11.6 in Eq. 3.11.5 we obtain du±(r) = r(Pco -Pc(r)) (3117)

Let us follow an element of solar wind that had, in the instant of ejection from the Sun (r = 0) only the radial component of its velocity, i.e., u^(r = 0) = 0, and find a transverse velocity at a distance r. Eq. 3.11.7 results in u±(r) = (/W126>0 j-1 J(Pc0 — Pc(r))rdr . (3.11.8)

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