where j was determined by Eq. 3.9.1, and 3 - by Eq. 3.7.14, Lex = QexNHo, and L is the distance at which Nh (r) decreases by a factor of e. The behavior of integral curves of Eq. 3.9.2 near the critical point r = rc will be determined by the characteristic equation
B = -3'Kr3j + 3(rcL^ -LL-\ -20]; C = ^(^(3 + j) + 3l-1 ). (3.9.4)
The position of the critical point r = rc is determined by the equation which follows from Eq. 3.9.2:
According to the characteristic Eq. 3.9.3, the main Eq. 3.9.2 has critical point r = rc
(which is determined by Eq. 3.9.5) of the type knot (gradual transition) if
2 2 B - 4C > 0 and of type focus (shock wave transition) if B - 4C < 0 . It means that if j < jc we obtain near the critical point r = rc gradual transition from supersonic flow to subsonic flow, but if j > jc we expect near the critical point r = rc shock wave transition from supersonic flow to subsonic flow. The value of jc is determined from the condition B - 4C = 0 :
212 rcLex + LLex I
According to Semar (1970) and McDonough and Brice (1971) the probable value of L = 3 ■ 5 AU = (4.5 ■ 7.5)x 1013 cm and Lex = 5x 1015cm (at NHo = 0.1cm-3), so 2LL-X =(1.8 ■ 3.0)x 10-2. We expect that rc = 100 AU, so rcL-l = 0.3, and jc =- (2.29-2.30). The value jc depends very weakly on parameters L, Lex and rc . For example, if rc = 150 AU, then jc = - (2.32-2.33). From data on the radial
CR gradient obtained for the inner Heliosphere by space probes Pioneer, Voyager, and others, it is expected that the value j > -2 (therefore we expect a shock wave transition), but what will be the situation near the critical point r = rc , is not exactly clear. It needs a special investigation, including consideration of kinetic stream instability in the outer Heliosphere, which can change the diffusion coefficient and therefore the dependence determined by Eq. 3.9.1.
3.10. Non-linear influence of pickup ions, anomalous and galactic CR on the Heliosphere's termination shock structure
3.10.1. Why are investigations of the Heliosphere's termination shock important?
According to Le Roux and Fichtner (1997a,b) the shock transition terminating the supersonic solar wind, the so-called heliospheric shock, has received increasing attention for several reasons:
First, the deep space probes Pioneer and Voyager are entering the outer heliospheric region where the heliospheric shock is supposedly located, and it is of importance to have some expectation of how it might show up in the data (e.g., Barnes, 1993; Suess, 1993; Paularena et al., 1996). An indication that Pioneer 10 and Voyager 1, both located beyond a heliocentric distance of ~ 60 AU, might, in fact, be relatively close to the heliospheric shock is given by the detection of anomalous hydrogen by these spacecrafts (Christian et al., 1995; McDonald et al., 1995; Stone et al., 1996).
Second, the heliospheric shock is a key element in structuring the global Heliosphere, which is currently the subject of extensive numerical modeling (e.g., Baranov and Malama 1993; Karmesin et al., 1995; Linde et al., 1996; Pauls and Zank, 1996; Ratkiewicz et al., 1996).
Third, the notion that beyond 120 AU the pressure of pickup ions might be much larger than the thermal pressure of the solar wind (Isenberg, 1986) or than the magnetic field pressure (Whang et al., 1995) drives some interest in its influence on the dynamics of the outer Heliosphere (e.g., Fahr and Fichtner, 1995) concerning the location and modification of the heliospheric shock (Zank et al., 1993; Lee, 1997).
Fourth, the properties of anomalous CR, probably produced at the heliospheric shock, give rise to the question of how the heliospheric shock structure, determining the diffusive shock acceleration process, looks in detail (e.g., Lee, 1997; Le Roux et al., 1996).
Previously the influence of pickup ions, anomalous CR, and galactic CR on the structure of the heliospheric shock have been studied separately (e.g., Ko et al., 1988; Lee and Axford 1988; le Roux and Ptuskin 1995a,b) or, in a simplified approach, for combinations of some or all of the energetic particle populations (Zank et al., 1993; Lee et al., 1996; Le Roux et al., 1996). Le Roux and Fichtner (1997a,b) studied the simultaneous influence of all energetic particle populations on the structure of the heliospheric shock and developed a self-consistent time-dependent model on the non-linear influence of pickup ions, anomalous and galactic CR on the Heliosphere's termination shock structure. They demonstrate that on the basis of the currently available data the heliospheric shock structure cannot be clarified unambiguously, but that there are at least two different alternatives consistent with observations obtained so far.
3.10.2. Description of the self-consistent model and main equations
Le Roux and Fichtner (1997a,b) developed a self-consistent time-dependent model of the non-linear influence of pickup ions, anomalous and galactic CR on the Heliosphere's termination shock structure. This model generalized earlier approaches of Ko et al. (1988) and Donohue and Zank (1993) by taking into account the self-consistent interaction of the thermal plasma of solar wind (including pickup ions) with anomalous and galactic CR which propagation is described by the transport equation (Parker, 1965):
In the transport Eq. 3.10.1 u(r,t) is the solar wind velocity, f (r,p,t) is the omnidirectional CR distribution function, and
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