Qor cose sine u

2vpcr

p is the momentum of CR particle, v is its speed, and q its charge; usw is the velocity of the solar wind, Qo is the angular velocity of the Sun; r, 6, and 0 are the usual spherical polar coordinate centered in the Sun. To simplify the calculations Jokipii and Levy (1977) assume that Krr, Kqq, Vdr are independent of energy.

They ran simulations using 60,000 to 200,000 particles with: usw = 4 X10 cm s ,

Krr = 5 X10 cm2s 3, Kqq = 0.1Krr, A = ±4.7 X10 cm2Gs (corresponding to

|B| = 5 x10-5 Gs at 1 AU), pc = 1 GeV, ra = 1012 cm The solution is symmetric about Q = n/2, so it was considered the range

2.35.3. The illustration results on the nature of CR drift modulation

Results shown in Fig. 2.35.1 and Fig. 2.35.2 are histograms of the number density of CR particles within 27° of the equatorial plane, averaged over 0.5 AU, with the density normalized to 1 at the outer boundary ro = 10 AU (really, as it is considered in Chapter 3, ro ~ 100 AU, but results shown in Fig. 2.35.1 and Fig. 2.35.2 are interested as illustration of drifts influence on CR propagation and modulation). In Fig. 2.35.1, A is positive, and in Fig. 2.35.2, A is negative. The dashed lines in each case correspond to the modulation solution with the drifts set equal to zero.

Fig. 2.35.1. Histograms of the number of CR particles within ± 27° of the solar equatorial plane, as a function of heliocentric radius r, for positive A. The solid line is the solution with drifts; the dashed line is the solution in the absence of drifts. The statistical uncertainties scale as r— and are about ± 15% for the innermost bin shown. From Jokipii and Levy (1977).

Fig. 2.35.1. Histograms of the number of CR particles within ± 27° of the solar equatorial plane, as a function of heliocentric radius r, for positive A. The solid line is the solution with drifts; the dashed line is the solution in the absence of drifts. The statistical uncertainties scale as r— and are about ± 15% for the innermost bin shown. From Jokipii and Levy (1977).

Fig. 2.35.2. Same as in Fig. 2.35.1, but for A negative. The bump at ~ 3 AU is a statistical deviation. From Jokipii and Levy (1977).

From Fig. 2.35.1 and Fig. 2.35.2 may be clear seen that for both positive and negative A, the drifts considerably change the modulated CR density. An item of major interest is that the CR radial gradient may be substantially reduced by the inclusion of realistic drifts. According to Jokipii et al. (1977), since the divergence of the average drift velocity is zero, the drift by itself cannot cause CR modulation; however, in the presence of a CR particle gradient produced by the usual convection-diffusion modulation, the drifts can have a substantial effect.

2.36. Drifts, perpendicular diffusion, and rigidity dependence of near-Earth latitudinal proton density gradients

Burger et al. (1999) note that from September 1994 to July 1995, the Ulysses spacecraft executed a fast latitude scan by moving from 80° South to 80° North at solar distances between 1.3 and 2.2 AU. During this first comprehensive exploration of the latitudinal dependence of modulation, a number of discoveries were made (see Simpson, 1998 and McKibben, 1998 for recent overviews). It was observed the unexpected small latitudinal CR proton density gradients, and its rigidity dependence (Heber et al., 1996.) These authors also attempted to model the observed gradients. They found that the discrepancy between measurements and model results increased as rigidity is decreased. The magnitude problem was subsequently solved (Potgieter et al, 1997, 1999; Hattingh et al., 1997) by using anisotropic perpendicular diffusion (Jokipii and Kota, 1995). Burger et al. (1999) show that it is the rigidity dependence of the perpendicular diffusion coefficient in the polar direction that controls that of the latitudinal gradient, and that this coefficient's rigidity dependence cannot be the same as that of parallel diffusion.

2.36.2 The propagation and modulation model, and diffusion tensor

According to Burger et al. (1999) the modulation of galactic CR is described by Parker's transport equation (Parker, 1965) for the omni-directional distribution function fo (r, p ) for particles with rigidity p at position r, which can be written in the steady state as i 1 ))dfo + i 1 d , )dfo d2,fo + Kqq d2fo

Here r and 6 are heliocentric radial distance and colatitude (polar angle) respectively, usw is the solar wind speed, and vdr is the drift velocity. The coefficient kqq describes diffusion perpendicular to the mean magnetic field in the polar direction, while the radial coefficient is

is where K// is the diffusion coefficient parallel to the mean magnetic field, K the diffusion coefficient perpendicular to the field in the radial/azimuthal direction and ^ is the spiral angle. In the two-dimensional model this coefficient acts only in the radial direction. In Eq. 2.36.1 the first 4 terms described diffusion, 5th and 6th -drifts, 7th - convection, and the last, 8th - adiabatic energy loss. Burger et al. (1999) used a steady-state two-dimensional model that simulate the effect of a wavy current sheet (Burger and Hattingh, 1995, Hattingh and Burger 1995) by using for the three-dimensional drift pattern in the region swept out by the wavy current sheet, an averaged field with only an r-and a 9-component. The Heliospheric boundary is assumed at 100 AU while the solar wind speed is 400 km/s within ~30° of the ecliptic plane and increases within ~10° to 800 km/s in the polar regions. A modified Heliospheric magnetic field (HMF) is used (Jokipii and Kota 1989). The tilt angle of the wavy current sheet is 15°. The diffusion tensor on which the current one is based, is described in detail in Burger and Hattingh (1998). For diffusion parallel to the magnetic field, Burger et al. (1999) used

9vB5J3l5J3 f R Y/3

28n sCs V c

if the quantity D = (c/R )Bals is greater than 1, while if it is less than one

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