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Astron. Astrophys. 330, 764-772 (1998)

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3. Hysteresis effect

The 11-year modulation of the cosmic-ray intensity shows some time lag behind the solar activity, in other words some kind of hysteresis effect against the activity (Moraal 1976; Mavromichalaki et al 1990 etc). In this paper, we show that there is a characteristic difference between even and odd solar cycles concerning the time lag between cosmic-ray intensity and the proxy indices of solar activity. A correlated analysis between the monthly values of the cosmic-ray intensity at Neutron Monitor Energies for the three solar cycles (20th, 21st, 22nd) and the solar activity is indicated by
the sunspot number Rz
the grouped solar flares Nf and
the Ap index
for the time period 1965-1995 as a function of the cosmic-ray intensity lag with respect to these parameters (Hatton, 1980; Mavromichalaki and Petropoulos,1987). The monthly values of the sunspot number, solar flares and Ap index during the period examined in this work are taken from the Solar Geophysical Data Reports. Monthly data from the two Neutron Monitor Stations, Climax (2.96 GV) and Inuvik (0.16GV) have been used. The pressure-corrected cosmic-ray data of each station for the period 1965-1995 were normalised with the intensity taken equal to 1.00 at solar minimum (May 1965) and equal to 0.00 at solar maximum (June 1991). The correlation coefficients for different time lags calculating over the three solar cycles are presented in Fig. 2. Each cycle has appeared separately in previous works (Mavromichalaki et al., 1984; Mavromichalaki et al., 1988; Marmatsouri et al., 1995). We can see that the cross-correlation coefficient for the sunspot number is at a maximum at a time lag of five months and for the grouped flares is at a maximum at a time lag of six months. The correlation coefficient of cosmic-ray intensity and geomagnetic activity expressed by Ap index does not show a pronounced maximum. One can distinguish two peaks: one at zero months and another at 14 months. This is consistent with the results of previous solar cycles (Balasubrahmanyan, 1969; Mavromichalaki and Petropoulos, 1984) where Bartels's Ap index correlates with the cosmic-ray intensity without pronounced phase lags or with the two maxima.

[FIGURE] Fig. 2. Correlation coefficients between monthly cosmic-ray intensity and sunspot number, grouped solar flares and Ap-index as functions of cosmic-ray intensity with respect to these indices for the time period 1965-1995. The statistical errors are also indicated.

The time lags of cosmic rays corresponding to the cross-correlation coefficient of each parameter for the cycles 20, 21 and 22 separately and for the three solar cycles are given in Table 2. According to Fisher's transformation of relation coefficient significance, the estimated correlation coefficients for our data series are at a 99%. It is noteworthy that the sunspot number phase lag for the 21st solar cycle is remarkably large (16 months), whereas it is small in the 20th and 22nd solar cycles (2 months and 4 months respectively), which are even cycles (Mavromichalaki et al., 1988). This gives us an evidence that there is a distinction between even and odd solar cycles concerning the hysteresis phenomenon. To clarify this distinction, we present the time lag of sunspot numbers with respect to cosmic-ray intensity for the last six solar cycles in Table 3. This result for the first three solar cycles has been adapted from Nagashima and Morishita (1980b), while the hysteresis for the last three cycles has been computed for the purposes of this paper. Inspecting the whole set of results, we can clearly distinguish between even and odd solar cycles concerning the sunspot number time lag. This is due to the 22-year variation in the time-lag already found by Nagashima and Morishita (1980b) and Otaola et al. (1985). Indeed particles reach the Earth more easily when their access route is by the heliospheric polar regions than when they gain access along the recurrent sheet. In this case, as the route of access becomes longer due to the waviness of the neutral sheet (Kota and Jokipii, 1991), the time-lag is also longer as one would expect from theoretical considerations. This model can't explain, however, the double-maximum structure of the even cycles.


Table 2. Cross-correlation coefficients and the corresponding time lags for each of the solar cycles 20, 21 and 22 separately and for the three cycles.


Table 3. Solar cycle dependence of time lag of the cosmic-ray intensity behind the sunspot number.

The Table 2 shows that during the 21st solar cycle the solar flares [FORMULA] affected mainly the cosmic-ray modulation and not the sunspot number, as in the 20th solar cycle. Storini (1995) underlined that the anticorrelation between cosmic-ray data and sunspot numbers is very high for all phases considered, excepting the declining ones of cycles 20 and 21, when high-speed solar wind streams coming from coronal holes affected the cosmic-ray propagation. The sunspot parameter is not the right index for solar-induced effects in the interplanetary medium. The anticorrelation during the 22nd cycle between cosmic-rays and solar magnetic field is very high (0.85) (Burlaga et al., 1993; Marmatsouri et al., 1995)

[FIGURE] Fig. 3. Schematic solar-cycle dependence of hysteresis curves of cosmic-ray intensity versus solar activity due to the polarity reversal of the solar polar magnetic field. (Nagashima and Morishita, 1980b)

If the effect of the polarity reversal is superposed on the hysteresis effect, the hysteresis curves split into two loops, as shown by the solid lines in Fig. 3a of Nagashima and Morishita (1980b). The upper and lower loops correspond respectively to parallel and antiparallel states of polarity to the galactic magnetic field. Practically, however, as the reversal is supposed to occur around every solar maximum (Babcock, 1961), the transition from the upper to the lower loop and vice versa can be expected to occur alternately every eleven years as shown by the dotted and chained lines in the figure. If we divide the hysteresis curve into two at solar minima, so that each curve corresponds to each period of solar cycle number, then the divided curves describe respectively the wider and narrower loops as shown in Fig. 3b. The upper panel of Fig. 4 shows the observed hysteresis curves for three solar cycles for the Inuvik cosmic-ray data. The reversals of the solar polar magnetic field are indicated. These curves clearly show respectively the above-mentioned patterns during solar cycles of the same type (even or odd). The curves during the even cycles are narrower than the curves of the odd cycles, while the existence of the secondary maximum is obvious and occurs after the polarity reversals.

[FIGURE] Fig. 4. Observed (upper panel) and calculated (lower panel) hysteresis curves for the three last solar cycles. Yearly mean intensities from Inuvik Neutron Monitor Station. The reversals from parallel to antiparallel state of the magnetic field and vice versa are indicated (P: parallel, A: antiparallel and R: Reversal).

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998