ISSN 0884-5913, Kinematics and Physics of Celestial Bodies, 2008, Vol. 24, No. 5, pp. 242–247. © Allerton Press, Inc., 2008. Original Ukrainian Text © M.I. Pishkalo, 2008, published in Kinematika i Fizika Nebesnykh Tel, 2008, Vol. 24, No. 5, pp. 370–378.
SOLAR PHYSICS
Preliminary Prediction of Solar Cycles 24 and 25 Based on the Correlation Between Cycle Parameters M. I. Pishkalo Astronomical Observatory, Taras Shevchenko Kiev National University, ul. Observatornaya 3, Kiev, 04053 Ukraine e-mail:
[email protected] Received December 13, 2007
Abstract—The correlation between various parameters of solar cycles 1–23 is investigated. The derived regressions are used to make predictions of solar cycles 24 and 25. It is expected that solar cycle 24 will reach its maximum amplitude of 110.2 ± 33.4 in April–June 2012 and the next minimum will occur in December 2018–January 2019. The duration of solar cycle 24 will be about 11.1 years. Solar cycle 25 will reach its maximum amplitude of 112.3 ± 33.4 approximately in April–June 2023. DOI: 10.3103/S0884591308050036
INTRODUCTION The solar activity varies with a period of about 11 years. The solar activity variations cause changes in interplanetary and near-Earth space. In turn, these affect the operation of spaceborne and ground- based technological systems (manned space flights, space navigation and aeronavigation, radars, high- frequency radio communication, ground power lines, etc.) and, in a certain way, the climate and living organisms on Earth. That is why it is important to know the level of solar activity in a solar cycle in advance. The most commonly used parameter of solar activity is the Wolf number, which characterizes the number of sunspots on the visible solar surface. Many methods for the prediction of solar activity based on analysis of periodicities in the hourly series of various observed phenomena on the Sun and correlations between them, on analysis of geophysical and climatological data, and on computer simulations of number series and artificial neuron networks have been suggested to date (see, e.g., [2, 20, 26] and reference therein). Most of the existing methods for the prediction of solar activity only predict the maximum Wolf number in a solar cycle; these predictions often differ significantly. In this short paper, we investigate the correlation coefficients between various parameters of solar cycles 1–23, predict the solar activity in cycles 24 and 25, and compare our prediction with published predictions. RESULTS AND DISCUSSION We used the following parameters of solar cycles 1–23 obtained from smoothed monthly values of the relative sunspot number: the epochs of solar minimum and maximum, the corresponding amplitudes at solar minimum and maximum, the cycle duration, and the durations of the cycle rise and fall phases. These data were taken at the site of the National Geophysical Data Center, USA ((ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SUNSPOT_NUMBERS/maxmin.new). For solar cycle 23, we refined the values based on the monthly Wolf numbers taken at the site of the Belgian Royal Observatory (http://sidc.oma.be). At present (late November 2007), the solar activity is at a very low level and, probably, at its minimum with an amplitude of 5.4 found by a double smoothing of the monthly Wolf numbers with a moving average over 13 points. In what follows, we assume that the amplitude of the relative sunspot number at the beginning of solar cycle 24 is 5.4 at epoch 2007.9. In the subsequent calculations, we will use this value, although, of course, it can be determined more accurately one year after the solar minimum. Therefore, the prediction made below is preliminary. First, we calculated in pairs the correlation coefficients between the Wolf numbers at solar minimum (Wmin) and maximum (Wmax), the cycle duration (T), and the durations of the rise (Trise) and fall (Tfall) phases using data for solar cycle 1–23. These are given in Table 1. The correlation coefficient between Wmax and Trise has the highest value, –0.82. Our analysis showed that the correlation coefficients exceeding 0.5 in absolute value are statistically significantly (p < 0.02–0.01). 242
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Table 1. Correlation coefficients between various parameters of solar cycles A\ B Wolf number at minimum Wolf number at maximum Duration of rise phase, years Duration of fall phase, years Cycle duration, years
Wolf number at minimum – (0.43)*
Wolf number Duration of rise Duration of fall Cycle duration, at maximum phase, years phase, years years 0.56 (0.50)* – (0.36)*
–0.55 (–0.33)* –0.82 (–0.38)* – (0.24)*
0.35 (–0.22)* –0.17 (–0.56)* 0.43 (–0.22)* –0.35 (–0.62)* –0.52 (0.31)* 0.41 (–0.19)* – (–0.03)* 0.55 (0.27)* – (0.10)*
* The correlation coefficient between parameter A in a certain cycle and parameter B in the previous cycle is given in parentheses.
Since there is a significant correlation between some parameters of solar cycles, below we will attempt to use the value of Wmin that we found for the beginning of solar cycle 24 to calculate other parameters of this cycle. We will perform our calculation according to the following scheme: we will find Wmax and Trise from the known Wmin, then Tfall and, hence, T from the Trise found, and, subsequently, Wmin for the beginning of the next cycle. Obviously, the predicted parameters of solar cycle 24 obtained in this way will characterize it as some “average” cycle. This scheme was first used to make predictions of solar cycles 22 and 23 based on the known parameters of solar cycles 1–21 and 1–22, respectively. The predicted and actual parameters of these cycles are compared in Table 2; there is good agreement between the predicted and actual values for some of the parameters 22 23 23 23 ( W max , W max , W min , and T rise ) and satisfactory agreement for the remaining parameters. In Fig. 1, Wmax is plotted against Wmin in solar cycles 1–23 (the correlation coefficient (C. c.) is 0.56, p < 0.01). The solid line indicates a linear fit; the dashed line indicates how Wmax can be found for solar cycle 24 from the known Wmin by moving upward from the horizontal axis until the intersection with the straight line and then leftward until the intersection with the vertical axis. The solid line represents the regression equation W max = 77.99 ( ± 13.7 ) + 5.97 ( ± 1.93 )W min .
(1)
For the maximum of solar cycle 24, we obtain Wmax = 110.2 ± 33.4, where 33.4 is the root-mean-square (rms) deviation (the mean absolute deviation is 25.4). The predicted value of Wmax can also be found if the duration of the previous cycle T–1 is known (the correlation coefficient between these parameters is –0.62, p < 0.02). In this case, the regression equation is W max = 340.63 ( ± 63.50 ) – 20.71 ( ± 5.77 )T –1 . (2) For the maximum of solar cycle 24, we obtain Wmax = 102.5 ± 31.8 from Eq. (2), where 31.8 is the rms deviation (the mean absolute deviation is 25.3). We see that the values of Wmax in solar cycle 24 obtained from Eqs. (1) and (2) are close. The duration of the cycle rise phase Trise is plotted against the minimum Wolf number at the beginning of the cycle Wmin (the correlation coefficient is − 0.55, p < 0.01) in Fig. 2. The solid line indicates a linear fit; the dashed line indicates how the duration of the cycle rise phase can be found from Wmin in solar cycle 24. From the regression equation T rise = 5.34 ( ± 0.41 ) – 0.17 ( ± 0.06 )W min
Maximum sunspot number in cycle 250 C. c. = 0.56, p < 0.01
200 150 100 50
0
2
(3)
and Fig. 2, we obtain 4.42 ± 1.00, where 1.00 is the rms deviation (the mean absolute deviation is 0.76), for the duration of the rise phase of solar cycle 24. KINEMATICS AND PHYSICS OF CELESTIAL BODIES
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6 8 10 12 14 Minimum sunspot number in cycle
Fig. 1. Maximum Wolf number in the cycle versus minimum Wolf number for the beginning of the cycle. The solid line represents a linear fit; the dashed line indicates the value for solar cycle 24. Vol. 24
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Table 2. Comparison of the predicted and actual parameters for solar cycles 22 and 23 Parameter A found from known parameter B, A(B) Wmax(Wmin) Wmax(T–1) Trise(Wmin) Trise(Wmax) Tfall(Trise)
T = Trise + Tfall
Wmin(T–1)
Cycle 22
Cycle 23
Predicted value*
Actual value
Predicted value*
Actual value
a: 150.46 ± 35.03 b: 129.90 ± 31.58 c: 3.31 ± 0.71 a: 3.41 ± 1.04 b: 3.90 ± 1.04 a: 7.22 ± 1.10 b: 6.93 ± 1.10 c: 7.28 ± 1.10 a: 10.63 ± 1.10 b: 10.83 ± 1.10 c: 10.59 ± 1.10 6.77 ± 3.12
158.5 ″ 2.8 ″ ″ 6.8 ″ ″ 9.7 ″ ″ 12.3
a: 126.01 ± 34.22 b: 141.60 ± 32.23 c: 3.98 ± 1.02 a: 4.24 ± 0.70 b: 4.27 ± 0.70 a: 6.68 ± 1.09 b: 6.66 ± 1.09 c: 6.82 ± 1.09 a: 10.92 ± 1.09 b: 10.93 ± 1.09 c: 10.80 ± 1.09 8.17 ± 3.18
120.8 ″ 4.0 ″ ″ 7.5 ″ ″ 11.5 ″ ″ 8.0
* a, b, and c—the calculation begins with equations of type (1), (2), and (3), respectively.
The predicted Trise can also be determined if Wmax is known. The correlation coefficient between these parameters is –0.82 (p < 0.01) and the regression equation is T rise = 7.07 ( ± 0.44 ) – 0.024 ( ± 0.004 )W max .
(4)
Using Wmax = 110.2 and 102.5, we obtain 4.42 ± 0.68 and 4.61 ± 0.68, respectively, for the duration of the rise phase of solar cycle 24 from Eq. (4). Here, 0.68 is the rms deviation (the mean absolute deviation is 0.53). We see that the values of Trise for solar cycle 24 obtained from Eqs. (3) and (4) are close. In Fig. 3, the duration of the cycle fall phase Tfall is plotted against the duration of the rise phase Trise (the correlation coefficient is –0.52, p < 0.02). The solid line represents a linear fit; the dashed line shows how Tfall can be found from Trise for solar cycle 24. In this case, the regression equation is T fall = 9.07 ( ± 0.88 ) – 0.55 ( ± 0.20 )T rise .
(5)
Duration of cycle fall, years 12
Duration of cycle rise, years 8 C. c. = –0.55, p < 0.01
6
C. c. = –0.52, p < 0.02
10 8 6
4
4 2
0
2
2
4
6 8 10 12 14 Minimum sunspot number in cycle
Fig. 2. Duration of the cycle rise phase versus minimum Wolf number for the beginning of the cycle. The solid line represents a linear fit; the dashed line indicates the value for solar cycle 24.
0 2
3
4
5 6 7 Duration of cycle rise, years
Fig. 3. Duration of the cycle fall phase versus duration of the cycle rise phase. The solid line represents a linear fit; the dashed line indicates how Tfall can be found from Trise for solar cycle 24.
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Table 3. Prediction of solar cycles 24 and 25
Author
Wolf Duration Duration Cycle Year number Year Wolf number of rise of fall duration, of minimum at mini- of maximum at maximum phase, phase, years mum years years Cycle 24
This paper
2007.9
Chistyakov, 1983 [5]
2011.4
5.4
2012.5 ± 0.7 110.2 ± 33.4 4.5 ± 0.7 6.6 ± 1.1 11.1 ± 1.1 2016.0
Kontor et al., 1983 [3] Tsirulnik et al., 1997 [29] Badalyan et al., 2001 [6]
2013
113
2014
180
2010–2011
50
2008 2006–2007
88
Khramova et al., 2001 [4]
2011.3
127 ± 30 101.3 ± 18.1
Wang et al., 2002 [30]
87.5 ± 23.5
Duhau, 2003 [12] Sello, 2003 [24]
2011
115 ± 21
Hathaway, Wilson, 2004 [13]
2010
145 ± 30
Svalgaard et al., 2004 [25]
2011
75 ± 8
Li et al., 2005 [21]
2006.9
2011.3
189.9 ± 15.5
2008.5
2013.2
137(80) 80 ± 30
Schatten, 2005 [23]
42 ± 34
Clilverd et al., 2006 [7] Du, 2006 [9]
150.3 ± 22.4
Du, Du, 2006 [10]
114.8 ± 17.4
Du et al., 2006 [11]
149.5 ± 27.6 160 ± 25
Hathaway, Wilson, 2006 [14] Lantos, 2006 [19] Maris, Oncica, 2006 [22]
2011 2006.5
18
2009.9
108 ± 38 145 70 ± 10
Abdusamatov, 2007 [1] Hiremath, 2007 [15]
10.8 ± 0.7
110 ± 11
2007.73
9.34
74 ± 10
Javaraiah, 2007 [16] Kane, 2007 [17]
2011–2012
142 ± 24 129.7 ± 16.3
Kane, 2007 [18] 2011.3 ± 0.7
Tlatov, 2007 [27]
110 ± 27 135 ± 12
Tlatov, 2007 [28] Cycle 25 This paper Chistyakov, 1983 [5]
2019.0 ± 1.1 5.8 ± 3.1 2023.4 ± 0.7 112.3 ± 33.4 4.4 ± 0.7 6.6 ± 1.1 11.0 ± 1.1 2024.6
2028.5
121
Kontor et al., 1983 [3]
2024
117
Hathaway, Wilson, 2004 [13]
2023
70 ± 30
Du, 2006 [9]
102.6 ± 22.4
Du, Du, 2006 [10]
111.6 ± 17.4
Du et al., 2006 [11]
144.3 ± 27.6 50 ± 15
Abdusamatov, 2007 [1] Hiremath, 2007 [15]
110 ± 11
2017.07
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Minimum sunspot number in cycle 14 C. c. = –0.56, p < 0.01
12
Adding the values of Trise and Tfall obtained, we find the cycle duration T. It is about 11.1 years, i.e., solar cycle 24 will reach its maximum of 110.2 in April–June 2012 and the next solar minimum will be reached in December 2018–January 2019.
10 8 6 4 2 0 9
We obtain Tfall = 6.64 ± 1.07, where 1.07 is the rms deviation (the mean absolute deviation is 0.86), for the duration of the fall phase of solar cycle 24 from the value of Trise = 4.42 found above.
10
11 12 13 14 Duration of previous cycle, years
Fig. 4. Minimum Wolf number in the cycle versus duration of the previous cycle. The solid line represents a linear fit; the dashed and dotted lines indicate the minimum Wolf number for solar cycles 24 and 25, respectively.
Table 3 gives our predicted parameters for solar cycles 24 and 25 and published predictions of other authors in chronologically alphabetic order. Among the papers that were not included in Table 3, we will note [8]. It was found in [8] that solar cycle 24 would be more intense than the previous solar cycle 23 by 30–50%, i.e., its amplitude would be 155–180.
Our predicted Wolf number at the maximum of solar cycle 24 (110.2 ± 33.4) agrees well with the predictions in [3, 15, 19, 27, 30] and is considerably smaller than that given by the predictions based on geomagnetic data in [14, 17]. At the same time, considerably lower predicted amplitudes were obtained for solar cycle 24 in several papers [1, 6, 7, 16, 25]. In Fig. 4, the minimum Wolf number at the beginning of the cycle Wmin is plotted against the duration of the previous cycle T–1 (the correlation coefficient is –0.56, p < 0.01). The solid line indicates the linear fit represented by the regression equation W min = 25.47 ( ± 6.24 ) – 1.77 ( ± 0.55 )T –1 ;
(6)
the dashed line shows how the Wolf number at the minimum of solar cycle 24 can be found from the duration of solar cycle 23; the dotted line indicates how the Wolf number at the minimum of solar cycle 25 can be found from the duration of solar cycle 24. For the minimum Wolf number in solar cycles 24 and 25, we obtain, respectively, 5.1 ± 3.1 and 5.8 ± 3.1 from the cycle durations, 11.5 (cycle 23) and 11.1 (cycle 24) years. Here, 3.1 is the rms deviation (the mean absolute deviation is 2.4). The predicted value of the minimum for the beginning of solar cycle 24 agrees well with the minimum Wolf number obtained above. If our value of Wmin for the beginning of solar cycle 25 (5.8) is used to determine the parameters of solar cycle 25 following the scheme described above, then we will find that the maximum smoothed monthly Wolf number in solar cycle 25 will be 112.3 at epoch 2023.4 and the minimum of solar cycle 26 will occur at epoch 2030.0. In this case, the duration of solar cycle 25 will be 11.0 years, the duration of the cycle rise phase will be 4.4 years, and the duration of the fall phase will be 6.6 years. The result does not depend on whether 23 or 24 previous cycles have been chosen to derive regressions (1), (3), and (5). Our prediction of solar cycle 25 and published predictions of other authors are given in Table 3. CONCLUSIONS Based on the derived regressions between various parameters of solar cycles 1–23, we made predictions of solar cycles 24 and 25. The smoothed monthly Wolf number in solar cycle 24 will reach its maximum value of 110.2 ± 33.4 in April–June 2012 and the next minimum with an amplitude of 5.8 ± 3.1 is expected in December 2018–January 2019. The duration of solar cycle 24 will be about 11.1 years, the duration of the rise phase will be 4.5 years, and the duration of the fall phase will be 6.6 years. The maximum amplitude in solar cycle 25 will be 112.3 ± 33.4 in April–June 2023; the minimum of the next solar cycle 26 will occur in late 2029–early 2030. The duration of solar cycle 25 will be about 11.0 years, the duration of the rise phase will be 4.4 years, and the duration of the fall phase will be 6.6 years. KINEMATICS AND PHYSICS OF CELESTIAL BODIES
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ACKNOWLEDGMENTS This work was supported in part by the State Foundation for Basic Research of Ukraine (project no. F25.2/094). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
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