## 4. Cosmic-ray simulationPrevious works (Xanthakis et al., 1981; Mavromichalaki et al., 1990; Marmatsouri et al., 1995) proposed an empirical model to describe the long-term cosmic-ray modulation during cycles 20, 21 and 22. We have now attempted to give a generalised model applied over the three solar cycles, as data are not available for the time before. This generalised model is derived by a generalisation of Simpson's solar wind model using the diffusion-convection-drift model (Nagashima and Morishita, 1980a). According to this, the modulated cosmic-ray intensity as it is measured at Neutron Monitor Stations can be computed by the difference between the galactic cosmic-ray intensity expressed by a constant C and the sum of some source functions appropriately selected from the solar and interplanetary indices that affect the cosmic-ray modulation (see discussion). The empirical relation is given by the following expression: where the constant C depends linearly on the cut-off rigidity of each station, Rz, Nf, Ap are the solar-terrestrial parameters incorporating the time-lag and i (i=1 to 3) are factors calculated by the RMS-minimisation (2.5, 1.8, and 0.5 respectively). The observed and calculated values of the cosmic-ray intensity for the Inuvik and Climax Neutron Monitor stations are presented in Fig. 5 and 6 respectively. The residuals are also indicated. The standard deviation between the observed and calculated values is about 10%, which suggests a very good approximation. It is noteworthy that this formula simulates fairly well the cosmic-ray intensity observed at the Earth during the onset and the declining phase of the solar cycles, whereas it is not so good during the maximum phase of solar activity. This is expected, because during the maximum phase the solar magnetic polarity usually changes configuration (Fig. 5 and 6). It is known that this change takes place over a period of several months. For example, in the last cycle it seems to have had a duration longer than one year (Webber and Lockwood, 1993). The hysteresis curves of the values calculated by Eq. (1) of the cosmic-ray intensity at Inuvik with the sunspot number are also presented in the lower panel of Fig. 4. We underline that these curves follow the hysteresis curves of the observed values of cosmic-ray intensity (upper panel). Loops for odd and even cycles correspond respectively to the left and right loops in Fig. 3b. The time lag of cosmic-ray intensity clearly shows a 22-year variation that is greater in odd number cycles than in even cycles. Comparing with the results of the correlation method (Table 2) we note that we have the same characteristics quantitatively in spite of their difference in the observation method. This fact also suggests that the 11-year modulation of the comic-ray intensity has been modulated by some disturbance with the 22-year periodicity through the three solar cycles. The sunspot number has no ability to produce such a 22-year variation (Nagashima and Morishita, 1980a).
© European Southern Observatory (ESO) 1998 Online publication: January 16, 1998 |