velocity near the Earth's orbit in 1965-1990: u = 4.41x10 cm/s = 7.73 AU/(average month), the estimated dimension of modulation region in cycle 22 will be ~ 100 AU

for CR with rigidity of 10-15 GV and about 80 AU for CR with rigidity of 35-40 GV. It means that at distances more than 80 AU the magnetic fields in solar wind and in inhomogeneities are too weak to influence intensity of 35-40 GV particles.

2.46. The inverse problem for small energy galactic CR propagation and modulation in the Heliosphere on the basis of satellite data

2.46.1. Diffusion time lag for small energy particles

As it was shown by Dorman and Dorman (1965), the time of diffusion propagation through the Heliosphere of particles with rigidity greater than 10 GV (to whom NM are sensitive) should be shorter than one month. This time is at least one order of magnitude smaller than the observed time-lag in the hysteresis phenomenon. It means that the CR long-term variation on the basis of NM data can be considered as a quasi-stationary problem with parameters of CR propagation changing with time. In this case, according to Parker (1958), Dorman (1959)

where n(R, r0bs, t) is the measured differential rigidity CR density at the time t, at the distance r0bs from the Sun; n0 (R) is the differential rigidity density spectrum in the local interstellar medium out of the Heliosphere; a ~ 1.5; u(r, t) is the effective solar wind velocity (taking into account also shock waves and high speed solar wind streams); and Dr(R, r, t) is the radial diffusion coefficient, in dependence of the distance r from the Sun, of particles with rigidity R at the same time t of observations (if we neglect the time of diffusion through the Heliosphere). In Dorman (2003a,b) it was taken into account the time lag of processes in the interplanetary space relative to processes on the Sun, determined by the value r/u .

For small energy particles measured on satellites and balloons, it is necessary to take into account the additional time-lag Tdif ((, r0bs, r, r0) caused by the particle diffusion through the Heliosphere from r to r0bs . This diffusion time-lag can be approximately estimated. In Dorman et al. (1997) it was shown that in a first approximation the value u/Dr in Eq. 2.46.1 can be considered as not dependent from r, and some effective values of solar wind speed uef (t) and of diffusion coefficient Dref (R, t) can be used. In this case, instead of Eq. 2.46.1, we obtain n(R robs,t )lno (r ):

The diffusion propagation time of CR particles with rigidity R from the distance r to the distance of observations robs can be approximately estimated as

Tdlf ((, t, robs, r, ro) _(o ._ r ^ « C (R, t , (246 3)


6a and Dr ef (R, t) was determined by Eq. 2.46.2. Instead of the distances from the Sun it is possible to introduce the variables used by Dorman (2003a,b):

these variables and Tdif are in units of av. month = (365.25/12) days =30.44 days = 2.628 x106 sec. By combining Eq. 2.46.5 and Eq. 2.46.3 we obtain

From Eq. (6) it follows that Tdif (, t, Xobs, X,Xo) reaches the maximum value at

X = Xo, and the coefficient C(R, t) reaches the maximum value, according to Eq. 2.46.4, at the minimum of CR intensity (near the maximum of solar activity; then

For high and middle latitude NM data (effective particle rigidity 10-15 GV) the amplitude of 11-year modulation is about 25% and according to Eq. 2.46.4 we obtain for solar maximum C(R, t)=0.028. It means that Tdf (R,t,Xobs,X,Xo))(( _Xobs)< 0.028 according to Eq. 2.46.7, i.e. the diffusion time-lag is negligible in comparison with the time propagation of solar wind from the Earth's orbit to the boundary of Heliosphere. On the basis of Burger and Potgieter (1999) we estimate C(R, t) for smaller rigidities, observed on satellites. Results are shown in Table 2.46.1.

Table 2.46.1. Coefficient C (R, t ) for different rigidities, for periods of maximum and minimum solar activity

particle rigidity and kinetic energy

solar activity

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