F f r W V33 f t Wmax

Ai (w, r, t) = A i (Wmax ) Wt --j jWmax 1 ^ V "" J , (1.14.7)

where Wmax = 200 and Ai (Wnax)012 cm.

In the some rough approximation the convection-diffusion global modulation described by Eq. 1.14.1 can be determined as

where ro is the size of the modulation region, the parameter y determines the dependence a(r )<*

is the particle rigidity (in GV), and

¡3 = v/c = (e2 + 2Ekmpc2 )/2/(Ek + mpc2) (1.14.10)

is the particle velocity for protons in units of light speed c, and B(t) is the parameter of modulation. According to Dorman and Dorman (1967a,b, 1968), Dorman (M1975b), Zusmanovich (M1986), Dorman, Villoresi et al. (1997a,b), the parameter B(t) changes with solar activity in the first approximation inverse proportional to Aj . Near the minimum of solar activity Bmin = (0.3 + 0.4) GV . In the maximum of solar activity the modulation became higher and Bmax = (1.6 - 2.5) GV for different solar cycles in dependence of direction of solar general magnetic field and sign of CR particles charge (taking into account drift effects). For a-particles in galactic CR the space distribution of the modulated spectrum will be

j where B((), R, y, and ¡3 are the same as in Eq. 1.14.8 for protons, and Nga(Ek) is the a - particle spectrum outside of the Heliosphere, which according to Simpson (1983)is

Nga(Ek ) = 0.07^ + mpc2) particle.sr-1.cm -2.s-1 (GeV/nucleon)-1,

and Ek is the kinetic energy of a-particles per nucleon. For heavier particles with ^~2Z we have an equation, similar to Eq. 1.14.11, but

Described above is the modulation of galactic CR caused by convection-diffusion processes. To this modulation it is necessary to add modulation caused by drift effects which can be determined mainly by the value of the tilt angle between the neutral current sheet and the solar equatorial plane (Burger and Potgieter, 1999; Dorman, 2001c; Dorman, Dorman et al., 2001). This type of modulation changes the sign in periods of solar magnetic field reversal (near maxima of solar activity). The amplitude of drift modulation as well as dimension of the modulation region ro where determined in Dorman (2001c) as average for even and odd cycles on the basis of data for four solar cycles 19-22 (see Fig. 1.14.1).

Fig. 1.14.1. Observed long term modulation of galactic CR in 1953-2000 according to Climax NM data (effective rigidity Ref ~ 10GV, curve LN(CL11M)) in comparison with that expected (curve EXPTOT14) at ro = 14av.monthXuav ~ 108 AU (for this period uav ~ 7.7 AU/av.month). Convection-diffusion modulation (curve ECDTOT14) and drift modulation (curve DRIFT) are also shown. Left, Y-scale for natural logarithm of Climax NM counting rate; right, Y-scale for drift modulation. Interval between two horizontal lines corresponds to 5% variation. According to Dorman (2001c).

Fig. 1.14.1. Observed long term modulation of galactic CR in 1953-2000 according to Climax NM data (effective rigidity Ref ~ 10GV, curve LN(CL11M)) in comparison with that expected (curve EXPTOT14) at ro = 14av.monthXuav ~ 108 AU (for this period uav ~ 7.7 AU/av.month). Convection-diffusion modulation (curve ECDTOT14) and drift modulation (curve DRIFT) are also shown. Left, Y-scale for natural logarithm of Climax NM counting rate; right, Y-scale for drift modulation. Interval between two horizontal lines corresponds to 5% variation. According to Dorman (2001c).

In Dorman, Dorman et al. (2001) the drift modulation was determined for cycle 22 as a dependence on effective particle rigidity Ref. For galactic electrons the modulation in the Heliosphere will be determined also by Eq. 1.14.1 or Eq. 1.14.8, but the drift effects will be opposite in comparison with protons and a - particles .

1.14.3. The 2nd factor: space-time distribution of solar wind matter

The detail information on this factor we considered above, in Section 1.13.3 (see Eq. 1.13.14 and Eq. 1.13.15).

1.14.4. The 3rd factor: gamma ray generation by galactic CR in the Heliosphere

According to Stecker (M1971), Dermer (1986a,b) the neutral pion generation by nuclear interactions of energetic protons with hydrogen atoms (reactions p + p ^no + anything) will be determined by

( r,B,t)= Arn{r,0, t) °¡dEkNp ^, r, ^o^E ^E, e))^!.!4.!4)

where n(r,d,t) is determined by Eq. 1.13.14, EkminEn) is the threshold energy for pion generation, Np(Ek,r,t) is determined by Eq. 1.14.1, (gcrn(Ek)) is the inclusive cross section for reactions p + p + anything (see Section 1.12.2), and

Gamma ray emissivity caused by nuclear interactions of galactic CR protons with solar wind matter will be determined according to Stecker (M1971), Dermer (1986a,b) by

FPH ((,r,0, t)= 2 JdEAEl - m2nc4) F^h (e„,r,0, t), (1.14.15)

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