The expressions for similar quantities B^, D^, n^, follow from the above expressions by substituting /n^A,«o , Fedorov et al.

(2002) note that both terms Gus(y,r) and Go(y,T vanish in the diffusive limit owed to the remaining factor of exp(-vor); so only Gds (y,r) gives the main contribution in this limit.

2.29.6. Expected temporal profiles for neutron monitors and comparison with observations

According to Fedorov et al. (2002), the main peculiarity of the solar CR events is connected with some neutron monitors (Hobart - HO, Mt. Wellington - WE, Lomnicky Stit - LS) having the narrow peak of the anisotropic stream of the first fast particles, other neutron monitors (Oulu - OU, Apatity - AP, Thule - TH, Durham - DU, Mt. Washington - WA) show a diffusive tail with a wide maximum at a later time, or, show both - the first narrow peak with a second diffusion maximum (South Pole - SP). For example, some selected NM data for the 24 May 1990 are demonstrated in Fig. 2.29.1. The time (in min) is measured from the onset of particle injection taken as 20.50 UT of May 24, 1990.

Fig. 2.29.1. Two groups of NM records of the event on 24 May 1990. Left - have the narrow peak of the anisotropic stream of the first fast particles (HO - Hobart, WE- Mt.Wellington, LS - Lomnicky Stit); right - show a diffusive tail with a wide maximum at a later time (OU - Oulu, DU - Durham, WA- Mt.Washington). According to Fedorov et al. (2002).

Fig. 2.29.1. Two groups of NM records of the event on 24 May 1990. Left - have the narrow peak of the anisotropic stream of the first fast particles (HO - Hobart, WE- Mt.Wellington, LS - Lomnicky Stit); right - show a diffusive tail with a wide maximum at a later time (OU - Oulu, DU - Durham, WA- Mt.Washington). According to Fedorov et al. (2002).

Comparison of the NM data with the theoretical prediction based on the kinetic equation solution requires a choice of a 'normalizing NM station' and consequent rigidity-dependent re-calculation of input parameters entering into the theoretical profile calculation. The NM station HO was chosen to be that station because it allows to determine the starting parameters at its mean rigidity R (HO) = 2.3 GV. This value, as well as the others, have been calculated by assumption of a particle rigidity spectrum roughly « R- in the initial phase. For other NMs was taken the following calculated values of the mean rigidity: RWE = 2.3 GV, RLS = 2.3 GV, ROU = 1.0 GV, Rdu = 2.0 GV, Rwa = 1.8 GV, and RSP = 0.8 GV. The mean rigidity R was obtained from trajectory computations for R < 10 GV with a step of 0.01 GV by a technique owed to Kassovicova and Kudela (1998). Each allowed trajectory was assigned by the weight corresponding to the solar proton spectra ^ R— and the coupling function according to Dorman (M1975a). The geomagnetic field model for trajectory calculations included the IGRF plus the Tsyganenko 89 model (Tsyganenko, 1989) for Kp > 5.

The asymptotic directions Ao for NMs have been obtained by numerical integration of particle motion in the geomagnetic field (by the method described in Kassovicova and Kudela, 1998; see in details in the Chapter 3 of Dorman, M2006) for the given epoch at 21:00 hours, and then they were averaged over both the rigidity-dependent response function of NM and the particle rigidity spectrum. For each allowed trajectory the pitch-angle was assigned and the mean value Ao as well as dispersion A^ were obtained from the histogram of the expected pitch-angle distribution (for the computations are used vertically incident particles). This simplification is used because:

(a) the contribution to NM count rate is in the geometric approach inversely proportional to cosine of zenith angle,

(b) the main limitation for the trajectory computations is the magnetic field model (Smart et al., 2000),

(c) these computed results (Kassovicova and Kudela, 1998) can be compared with the vertical cutoff rigidities obtained by other methods (Shea and Smart, 2001).

The mean transport path A in IMF is supposed to be independent of rigidity for the considered interval of NM sensitivity rigidity. Elementary calculation shows that

where z and y = z/A is the distance of the detector (the Earth) from the source (the Sun) and the dimensionless one, respectively. Therefore the fit of the theoretical curve to experimental data of HO determines vo =vo (tm) for given yHO . The best fit gives tm = 12.4 min for yHO = 0.6. Values of vo for the other NMs are calculated assuming the rigidity dependence of tm x R-& using the given value yHO . The spectrum index characterizes shape of a low energy particle delay in the corona. Fedorov et al. (2002) have used the value of = 1.

For comparison of theoretical dependences with the experimental data the dependence of GCR intensity on particle rigidity also has been taken into consideration. Let this dependence be Ig xR Yg, where yg is GCR spectrum index, and the SCR rigidity dependence at the instant of its injection into IMF is Is x R~Ys . All theoretical curves are standardized to a maximum relative to the mean rigidity of HO, i.e., the curve HO has the value 1 in maximum. The multipliers (RHO/Ri) Yg (where i = OU, WE, WA, etc.) which take into account the rigidity spectra of GCR and SCR, must be used in calculation of the rigidity dependence of the i - th NM. The values of the maxima of the theoretical curves of the i - th NM are conditioned by the difference of Ay = ys -yg .

The experimental records of temporal profiles can be divided roughly into two groups, one of NMs which in the initial phase (about the first hour after the particle onset) have asymptotic direction near the regular IMF direction (Fig. 2.29.1, left panel), and the others whose the asymptotic direction differs from it (Fig. 2.29.1, right panel). The theoretically predicted temporal profiles for the selected NMs in the model described in Section 2.29.5 are demonstrated in Fig.

2.29.2, left and right panels, respectively, using the calculated asymptotic direction for each NM station. This calculation shows that HO and WE have very similar characteristics, Ao is 0.9 and 0.86, respectively, with AA = -0.26. Station LS has Ao = 0.34, AA = 0.4. In the second group of NM's, OU, DU, and WA have Ao = -0.94,-0.9, -0.85 and AA = 0.06, 0.1, 0.3, respectively. Note that Oulu and Apatity give absolutely the same theoretical curves resulting from their similar characteristics and very similar temporal profiles of the event. The last two NMs (DU and WA) experienced small increases at 1-2 hours after onset, as the theory predicts, see Fig. 2.29.2 (right panel), owing to smaller Ao and larger AA and larger mean R .

1.0 —ji—i-p-i—i-i—i-1-i—i-p— 0.5 —i-.-1—.-p-1-1-1-1—i-p—

QUU.-I-.-I-U—I--.--.--.--.—j--o1—'^-'-1-1-1-1-'-1-1-1-

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