1

where p and P are the mass density and thermal pressure of the solar wind gas including pickup ions, y = 5/3 is the polytropic index for the thermal gas,

Pc = (4n/3))p^vfdp is the CR pressure. In Eq. 3.10.4-3.10.6 the source and loss terms are determined by the production rates of pickup ions resulting from photoionisation Qph of and charge exchange Qce with, interstellar hydrogen.

Following Lee (1997), the terms were derived under the assumption that upstream of the heliospheric shock, the energy density loss of the solar wind, and Qce depend mainly on solar wind velocity u, but downstream of the heliospheric shock depends mainly on the root mean square velocity vms of solar wind protons. In Eq. 3.10.6 H (rsh - r) and H (r - rsh) are the Heaviside function with rsh the heliospheric shock radius. The coefficient api in Eq. 3.10.6 results from the transfer of pickup ions with p < mu across the threshold p = mu from the thermal to the supra-thermal population by adiabatic heating (according to Zank et al, 1993).

3.10.3. Using methods of numerical calculations

According to Le Roux and Fichtner (1997a,b), the parabolic transport Eq. 3.10.1 describes both anomalous CR resulting from the injection and diffusive shock acceleration of pickup ions at the heliospheric shock and galactic CR incorporated by prescribing an interstellar spectrum (see, e.g., McDonald et al., 1995) at the outer boundary at 120 AU. This equation was solved by using a combination of the implicit Crank-Nicholson method for spatial diffusion and the explicit monotonic transport scheme for convection and adiabatic energy changes. For the system of hyperbolic fluid Eq. 3.10.4-3.10.6 describing the thermal gas mixture, solved with a Riemann algorithm (LeVeque, 1994), standard solar wind conditions at the inner boundary r = 1 AU are used (u = 400 km s-1, p^ = 5 mp cm-3, T, = 105 K). At the outer boundary r0 = 120 AU, a constant downstream density (i.e. dpjdr r=r0 = 0), a mass flux decreasing proportional to 1/r2 (i.e. dmr2jdr r=r0 = 0 ), and a thermal pressure equal to the local interstellar pressure (i.e. p = PthLISM = 1 eV/cm3 ) are assumed.

3.10.4. Expected differential CR intensities on various heliocentric distances

For n, the injection efficiency of pickup protons into the process of diffusive acceleration at the heliospheric shock, Le Roux and Fichtner (1997a,b) found one high and one low value of the free parameter n resulting in CR flux levels consistent with Pi0neer and V0yager observations during 1987. Fig. 3.10.1 shows the differential intensity J(E^ ) = p2 f of combined pickup ions, anomalous and galactic CR as a function of kinetic energy Ek for various heliocentric distances for these solutions: (a) n = 0.0003 and (b) n = 0.9. Since the injection efficiency as defined above denotes only a fraction of those pickup ions with velocities w > u, the actual number of injected particles represents a smaller fraction of the total pickup ions population than is indicated by n. The percentage of pickup protons at the heliospheric shock having velocities greater than u as a consequence of adiabatic heating is found to be 98% and 92% for (a) and (b), respectively. For n = 0.0003 and n = 0.9, Le Roux and Fichtner (1997a,b) found that 0.03% and 83% of all pickup ions are diffusively accelerated for (a) and (b), respectively. From an analytical estimate employing an upstream pickup ions distribution derived by Vasyliunas and Siscoe (1976) in combination with the self-consistently determined solar wind deceleration, one obtains (a) 0.02% and (b) 15% for the actual injection rate. These numbers demonstrate not only that the values obtained numerically represent a tendency of the algorithm to accelerate particles too efficiently (Hawley et al., 1984), but also that, in order to reproduce the observed spectrum with a high-injection case, the actual injection fraction has to be increased. Such an increase could be achieved by the inclusion of a pre-acceleration mechanism for pickup ions (e.g., Chalov and Fahr, 1996; Fichtner et al., 1996; Lee et al., 1996; Zank et al., 1996).

Fig. 3.10.1. The combined pickup ions, anomalues and galactic CR differential intensities in particles m-2s-1 srad-1MeV-1 as a function of kinetic energy in GeV in the upwind direction. From bottom to top the spectra are shown at 2, 23, 42, 61, 72, and 75 AU, respectively, with 75 AU just downstream of the heliospheric shock. The top panel for n = 0.0003, the bottom panel for n = 0.9. The filled circles represent proton data in 1987 from Voyager 2 and Pioneer 10 at 23 and 42 AU, respectively (McDonald et al., 1996). According to Le Roux and Fichtner (1997a,b).

Fig. 3.10.1. The combined pickup ions, anomalues and galactic CR differential intensities in particles m-2s-1 srad-1MeV-1 as a function of kinetic energy in GeV in the upwind direction. From bottom to top the spectra are shown at 2, 23, 42, 61, 72, and 75 AU, respectively, with 75 AU just downstream of the heliospheric shock. The top panel for n = 0.0003, the bottom panel for n = 0.9. The filled circles represent proton data in 1987 from Voyager 2 and Pioneer 10 at 23 and 42 AU, respectively (McDonald et al., 1996). According to Le Roux and Fichtner (1997a,b).

3.10.5. Different cases of heliospheric shock structure and solar wind expansion

From Fig. 3.10.1 it can be seen that the modulated spectra for distances smaller than 60 AU are basically identical; it means that there are differences farther out owing to the different heliospheric shock structure. The parameters describing this structure are listed in Table 3.10.1 for both injection cases (a) and (b) along with those for three non-injection cases (1-3) serving as reference solutions.

Table 3.10.1. Heliospheric shock parameters for different cases. According to Le Roux and Fichtner (1997b)

case

energetic particles

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