Solar cycles: the past evolution influence
aa r X i v : . [ a s t r o - ph . S R ] O c t Solar and Stellar Variability Impact on Earth and PlanetsProceedings IAU Symposium No. 264, 2009A. H. Andrei, A. Kosovichev, J.-P. Rozelot, eds. c (cid:13) Solar cycles : the past evolution influence
Alexis KLUTSCH , and Rubens FREIRE FERRERO Universidad Complutense de Madrid, Departamento de Astrof´ısica, Facultad C.C. F´ısicas,28040 Madrid, Spain, email: [email protected] Observatoire Astronomique, Universit´e de Strasbourg & CNRS, UMR 7550, 11 rue del’Universit´e, 67000 Strasbourg, France, email: [email protected]
Abstract.
The so-called solar cycle is generally characterized by the quasi-periodic oscillatoryevolution of the photospheric spots number. This quasi-periodic pattern has always been anintriguing question. Several physical models were proposed to explain this evolution and manymathematical data analysis were employed to determine the principal frequencies noticeable inthe measured data. Both approaches try to predict the future evolution of the solar activity andto understand the physical phenomena producing these cycles. Here we present the analysis ofthe sunspots number evolution using the time-delay approach. Our results show than the solarcycle can also be characterized by this behavior implying the influence of the past evolution overthe present one, suggesting an histeresis mechanism, linked probably with magnetic activity.
Keywords.
Sun: sunspots, Sun: activity, Sun: evolution
1. Introduction
Solar activity can be seen through the evolution of sunspots number in quasi-periodicoscillatory series with periods going from 8 to 15 years and with a mean period of 11 years.Due to the change of magnetic field polarity in solar hemispheres alternatively each cycle,the period is rather 22 years. The quasi-periodic evolution of many activity phenomenais still a unsolved key problem in solar physics (along with, e.g., heating of the solarchromosphere and corona, and solar flares). An important issue of this understanding isdue to the influence of solar activity over the terrestrial climate (Archibald 2006).Many physical models were proposed to understand the basic mechanisms (e.g., Benev-olenskaya 1998) producing solar/stellar activity and its quasi-cyclic evolution. Fromsunspots time series, mathematical approaches have highlighted several other hiddenperiods other that the well-apparent 11-years period. These works contributed to pro-vide some observed parameters to constraint theoretical models, and to predict the futureevolution of the sunspots number (e.g., Clilverd et al.
2. The solar cycle as a temporal delay phenomena
We consider a temporal delay behavior (Eq. 2.1) to rely the present evolution of aphenomena with some particular events in its past evolution. The variation of N in timeis not only related with its current value N ( t ) but also with its past values N ( t − T ),1 Alexis Klutsch & Rubens Freire Ferrero Figure 1.
Left panel:
Correlationfactor obtained by chi-square mini-mization versus the temporal delay.
Right panel:
Sunspots number ratioversus sunspots number with a 8-yearsdelay. The blue line shows a linear fit. where T is our temporal delay. The simplest way is to assume only one past temporalinfluence and some proportionality between the variables (e.g., Murray 1993), as follows: d N ( t ) d t = a N ( t ) { − b N ( t − T ) } → ∆ N i ∆ t = a N i { − b N i − J } t is equal to 1. The present and past values of N arealready known and we assumed some values for the parameter T ( T = 0 , , , . . . ; Fig. 1,left panel). So we can determine constants a and b (Eq. 2.2) from the correlation betweenthe sunspots number ratio and the past sunspots number (Fig. 1 right panel). N i + ∆ N i = a N i { − b N i − J } + N i → N i +1 N i = { a + 1 } − ab N i − J (2.2)
3. Results and conclusions
The T values which minimize the correlation between the sunspots number ratio andthe past spots number, are 7 or 8 years (Fig. 1). To test the accuracy of our method,we applied it on the solar cycle 23 (Table 1 and Fig. 2, left panel). Our predictions areconsistent with the observations, except for the unusually long period of low activity atthe end of this cycle. We also characterized the next solar cycle 24 (Table 2 and Fig. 2)using our modelling of the solar cycle 23 as well as the spots number observed until2006 . .
5. The next maximum sunspots number would take place between 2011and 2012 and should be close to 60 (Fig. 2, right panel) as for the solar cycle 14. Usingthe solar minimum occurred in 2008 . R min = 2 . et al. (2009). Moreover thenext solar minimum should not occur before 2019 or even 2020. Table 1.
Observed and predicted values for the cycle 23. Results showed in Fig. 2 (left panel).Observed parameters Epoch of solar Maximum sunspot Epoch of the endof solar minimum maximum number of cycleEpoch Number Observed Predicted Observed Predicted Observed Predicted1996 . . . . . . . ? 2006 . These preliminary results are encouraging because we find a similar delay as thatobserved between the geomagnetic activity and solar cycle peaks (Hathaway & Wilson2006) linked probably with magnetic activity by some kind of histeresis mechanism. Figure 2.
The observed ( plus symbols and dotted line ) and selected ( asterisks and blue line ) yearly valuesof the relative sunspots number. Our predictions using the temporal delay method are also marked ( diamondsand red line ). Left panel:
Prediction for the solar cycle 23 and 24.
Middle and Right panels:
Prediction of thesolar cycle 24 using the sunspots number observed until 2006.5 and 2007.5 as input.
Table 2.
Predicted values for the solar cycle 24 using various data set in input.Panel of Input parameters of Predicted parameters Predicted parameters ofFig. 2 solar minimum of solar maximum next solar minimumEpoch Number Epoch Number Epoch NumberLeft 2006 . . . . . . . . . . . . . . . . . . Figure 3.
Maximum ( R max ) versus minimum ( R min ) sunspots num-ber of a given solar cycle ( dots ). The cycle number is also marked. Blue,green and red diamonds show the locus of our predicted maxima usingour modelling of the solar cycle 23 as well as the spots number observeduntil 2006 . .
5, respectively (Table 2). The predicted range ofBrajsa et al. (2009) is plotted as well ( vertical line ). The blue and redlines show linear fits using data of both axis and the linear least-squarefit like one given by Brajsa et al. (2009, Fig. 6), respectively.
At present, we can successfully reproduce some previous solar cycles (e.g., epoch andsolar maximum). We plan to improve our predictive method including the influence ofdelays for which there is a good correlation. This more detailed description could allowto solve the overestimation of the next solar minimum and to reproduce the asymmetriesobserved during the previous cycles. Our final aim is to obtain a reliable prediction of thewhole solar activity by identifying the fundamental lower and high activity precursors ofits present and future evolution. All these improvements are still needed to be able topredict future low solar activity periods (e.g., Maunder/Dalton Minimum).
References
Archibald, D. 2006, Energy & Environment, 17, 29Benevolenskaya, E. E. 1998,
ApJ , 509, L49Brajˇsa, R., W¨ohl, H., Hanslmeier, A., et al. 2009,
A&A , 496, 855Clilverd, M. A, Clarke, E., Rishbeth, H., et al. 2003,
Astronomy and Geophysics , 44, 5.20Hathaway, D. H. & Wilson, R. M. 2006, Geophys. Res. Lett., 33, L18101Murray, J. D. 1993,
Mathematical Biology, 2nd corr. ed. (Springer)Sello, S. 2003,