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

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5. Discussion and results

The solar modulation of galactic cosmic-rays describes the changes due to the solar influence on the isotropic and constant distribution of energetic particles from local interstellar space. The understanding of the modulation is still based on the standard model of diffusion, convection and adiabatic deceleration effects, where the path of individual particles through the heliosphere is determined by the interplanetary magnetic fieldlines including drift processes. This leads to characteristic differences between adjacent solar cycles due to the different polarity of the solar and large-scale interplanetary magnetic fields (Kunow, 1991). The polarity of the solar field reverses sign about every 11-years near the time of maximum solar activity or minimum cosmic ray intensity. Thus successive activity maxima are characterised by different solar field polarity. In this work searching the last four solar cycles we have noted systematic differences between successive 11-year cycles and similarities between alternate 11-year cycles which are consistent to the 22-year magnetic cycle. When the solar polar field points into the northern hemisphere, i.e. during the odd cycles, we observe a "pointed" type maximum, while during the even cycles, a "mesa"-type maximum appears. The recovery phase of the odd cycle is characterised by a relatively long-lasting (6-8 years) smoothed increase, while the even cycle is characterized by a relatively rapid (about 3 years) increase (Mavromichalaki et al., 1988: Ahluwalia, 1995). The existence of two maxima in solar indices during the even solar cycles is reported by many authors (Mavromichalaki et al., 1988). The mean time lag of the cosmic-ray intensity behind the solar activity is estimated at about 5 months for the period 1965-1994. Thus, considering as a mean solar wind speed 500 km/s, we estimate that the mean modulation barrier that is the limits the heliosphere is (70 AU (Simpson and Wang, 1967). This value is consistent with most estimates, which place the modulation barrier at [FORMULA] AU. These basic characteristics of a solar cycle and its declining phase, according to Lin et al. (1994), can be summarised as follows: One or two years after a sunspot maximum, a new solar cycle begins shortly after the polar field reversal of the Sun. The activity of the previous cycle moves towards the equator, leaving an empty space at high latitudes for the formation of polar coronal holes which begin to grow reaching their maximum extent shortly before the old cycle minimum. At solar maximum, the polar regions are occupied by an equal number of positive and negative magnetic elements. As the cycle progresses towards sunspot minimum, the magnetic field elements in each polar region change to predominantly one polarity, positive on one pole and negative on the other.

The diffusion-convection and adiabatic deceleration theory (Gleeson and Axford, 1967) of galactic cosmic rays into a spherically symmetric solar wind model would lead to a long-term variation. In the light of this model, the modulation is well explained by setting proper physical states in the modulating region, but it is not so clear how these states are related to solar activities. According to this theory several authors (Nagashima and Morishita, 1980a; Xanthakis et al., 1981; Mavromichalaki and Petropoulos, 1987) studing previous solar cycles have shown that the cosmic-ray modulation can be described by the following integral equation, derived from a generalisation of Simpson's coasting solar wind model (1963) as:


where I and I(t) are, respectively, the galactic (unmodulated) and modulated cosmic ray intensities, S(t-r) the source function representing some proper solar activity index at a time t-r (r(0), and f(r) the characteristic function that expresses the time dependence of solar disturbances represented by S(t-r).

In this work, it is pointed out that the modulation of cosmic rays during the last three solar cycles can be described on a monthly basis by the source function of Eq. (2) expressed by a linear combination of three indices: the sunspot number Rz, the solar flares of importance (1B Nf, and the geomagnetic index Ap. The characteristic function f(r) of all these indices has a constant value during a solar cycle, calculated by the RMS-minimization method. In this way, the modulated cosmic-ray intensity is equal to galactic cosmic-ray intensity (unmodulated) at a finite distance, corrected by properly selected parameters which cause the disturbances in interplanetary space and thus modulate the cosmic-ray intensity. This model reproduces to a certain degree the cosmic-ray modulation, which will be very useful to cosmic-ray research.

To estimate the degree of agreement between the observed cosmic-ray intensities and those calculated according to this model, we computed the residuals between the observed and calculated values, presented in Figs. 5 and 6. Our model describes the cosmic-ray long-term modulation very well for the time period 1965-1995, three consecutive solar cycles. Neverthless, we note a deviation from the observed values which becomes remarkable during the declining phase of the 21st solar cycle and also around the maximum of the 21st and 22nd solar cycles. Polarity reversals of the polar solar magnetic field occurred around these periods. These residual values may be explained by the following interpretation of Ahluwalia (1979). A secondary maximum of the even solar cycles is observed 1-2 years after reversal from negative north pole to positive north pole. This leads from a closed heliosphere magnetic topology to an open one. Particles of the interstellar medium get into the heliosphere by travelling through the polar field lines of the Sun. In the opposite case, the diffusion mechanism is the most prominent one (Smart and Shea, 1981). During the solar maximum of the 21st cycle, a reversal of the solar polarity from [FORMULA] to [FORMULA] occurred. The north pole of the Sun became negative, resulting in an inward magnetic field. A closed magnetic configuration of the heliosphere was formed. The result was weaker in cosmic-ray intensity than expected from the proposed model.

On the other hand, Popielawska (1995) noted that near maximum solar activity, a transient mode of modulation manifests itself as a distinct phenomenon distinguished by formation of so-called hysteresis "loops" on correlation plots for low- versus high-rigidity cosmic-ray intensity changes. These hysteresis loops close fully at high modulation levels. The solar field evolves from a roughly dipole like configuration at solar minimum to a complex state at solar maximum when the polar fields inverse. The duration of this situation is about four years for the 22-nd solar cycle (Hoeksema, 1991). Perhaps an improvement of our model using for example a new source function with the tilt of the heliospheric current sheet (for the onset and declining phase) and/or the transient phenomena (for the solar maximum) that are highly correlated with the cosmic-ray intensity will give a more appropriate description of galactic cosmic-ray intensity (unmodulated). This proposed model would give an integrated model for the cosmic-ray modulation for the coming solar cycles.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998