RELATIONSHIP BETWEEN SOLAR MAXIMUM AMPLITUDE AND MAX MAX CYCLE LENGTH

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1 The Astronomical Journal, 132: , 2006 October # The American Astronomical Society. All rights reserved. Printed in U.S.A. RELATIONSHIP BETWEEN SOLAR MAXIMUM AMPLITUDE AND MAX MAX CYCLE LENGTH Z. L. Du National Astronomical Observatories, Chinese Academy of Sciences, Beijing , China; zldu@bao.ac.cn Received 2006 January 12; accepted 2006 June 4 ABSTRACT The maximum amplitude of a solar activity cycle is found to be inversely correlated (r ¼ 0:769) with the newly defined max max cycle length two cycles earlier in a 13 month mean of monthly sunspot numbers. Meanwhile, a 14 cycle periodicity is found in the fitted residuals. The max max cycle length can be used as one of the indicators to predict amplitudes. As a result, the amplitudes of cycles 24 and 25 are estimated to be 150:3 22:4 and 102:6 22:4, respectively, where the indicated error is the standard error. Key words: Sun: activity Sun: general sunspots 1. INTRODUCTION The most important index of solar activity is the Zürich or Wolf sunspot number (SN, now the international sunspot number; Hathaway et al. 2002). Since the discovery of the 11 yr periodicity in sunspots by Schwabe (1843), astronomers and astrophysicists have tried to explain the mechanism of this cycle ( Landscheidt 1999). Hale (1924) first noted that the 11 yr Schwabe cycle is actually caused by the 22 yr magnetic polarity cycle, in which every second (odd-numbered cycle) peak is reversed in sign ( Bracewell 1986; De Meyer 1998). In the dynamo model of Babcock (1961), the dynamics of the 22 yr magnetic cycle was related to the Sun s rotation on its axis, and the sunspot cycle is thought to be caused by the interaction of differential rotation, polar magnetic field, and convection in the solar plasma ( Landscheidt 1999). The Schwabe (1843) sunspot cycle is fundamental, but the length and amplitude are not constant. Studies of the relationship between amplitude and solar cycle length are important both for understanding the mechanism that drives the solar cycle ( Bracewell 1986; Dicke 1988) and for improving our knowledge of the connection between solar activity and space weather ( Eddy 1976; Javaraiah et al. 2005). There is evidence that the solar activity level may be one of the key factors determining the state of the lower atmosphere of the Earth (Ogurtsov 2005). Solar activity affects the solar-terrestrial system in various respects such as electric power transmission, radio communications, and satellite planning ( Hanslmeier et al. 1999; Hathaway et al. 1999; Kane 2001). Because the monthly SNs may have systematic uncertainties of about 25% ( Vitinský et al. 1986), accurate prediction of the level of solar activity using SNs is still a difficult task. Of the dozens of methods of prediction proposed in the past (Yule 1927; Waldmeier 1939a, 1939b; Kane 1978; Feynman 1982; Wilson 1990; Thompson 1993; Wang & Han 1997; Lantos & Richard 1998; Shastri 1998; Li et al. 2001a), the precursor methods (Gnevyshev & Ohl 1948; Ohl 1966; Ohl & Ohl 1979) are believed to be superior to nonprecursor methods (Kane 2001; Li et al. 2001b). Two important parameters in the solar activity of the 11 yr Schwabe (1843) cycle are the amplitude (R m ) and cycle length (period between two successive minima). A number of relationships between them have been found so far, e.g., sunspot cycles are asymmetric, with a fast rise and slow decline ( Waldmeier 1939a, 1939b); large amplitudes tend to take less time to reach their maxima than small ones ( Waldmeier 1939a, 1939b); the 1485 amplitude of a cycle is weakly correlated with the length of the previous cycle ( Waldmeier 1939a, 1939b; Hathaway et al ); the amplitude of an odd-numbered cycle is larger than that of the previous even-numbered one (the G-O rule; Gnevyshev & Ohl 1948); and the recently proposed three-cycle quasi-periodicity (Silverman 1992; Ahluwalia 1998; Du et al. 2006b; Du 2006a) and five-cycle quasi-periodicity ( Du 2006b). But the exact physics behind these relationships is not yet known (Solanki et al. 2002; Javaraiah et al. 2005). Recently, some other parameters have been studied, e.g., the temporal derivative ( Dmitrieva et al. 2000), the slope at the inflection point (Lantos 2000; Zhang & Wang 1999), the solar dynamo amplitude (Schatten & Pesnell 1993), and the ratio between the variations in the geomagnetic aa index and SN (Obridko 1995). The cycle length is conventionally defined as the time difference between two successive minima. As this length depends on how SNs are averaged or smoothed, Mursula & Ulich (1998) proposed the median cycle length, which is almost independent of the exact location of the minima, and Du et al. (2006a) proposed a cycle length in terms of weighted extremum epochs. The max max cycle length, the period between two successive maxima (P max ), is used in the present study in preference to min min cycle length because it is more strongly correlated with R m. In x 2 we briefly introduce the definition of the weighted average epochs of maxima. The relationship between R m and P max at various lags is discussed in x 3. We find that R m is significantly correlated with P max two cycles earlier. This correlation is tested for the past few cycles in x 4. The sizes of cycles 24 and 25 are estimated in x 5. Finally, our main conclusions are summarized in x WEIGHTED AVERAGE EPOCHS OF MAXIMA This paper employs the 13 month mean of monthly SNs (Gleissberg filter) of the more reliable data since cycle 8 ( Eddy 1976; Kane 1999), i.e., P max for cycles Monthly SNs are taken from the National Geophysical Data center Web site. 1 To study the long-term behavior of SNs, we make use of the weighted maximum epochs (Du et al. 2006a). Figure 1 shows the 13 month mean of monthly SNs from 1995 to 2005 including a maximum R m, where R 0 is the minimum from the beginning to the end of this cycle and ¼ 0:1(R m R 0 ). 1 See

2 1486 DU Vol. 132 Fig. 1. Thirteen-month mean of monthly SNs ( ) including a maximum R m,wherer 0 is the minimum from the beginning to end of this cycle, ¼ 0:1(R m R 0 ), E 1 and E n indicate the epochs at which R ¼ R m,ande m is the maximum epoch derived from eq. (1). Let E i (i ¼ 1; 2; :::;n) bethetime(ordate)ofr i,(r m R i ). The maximum epoch is thus defined as 1 X n E m ¼ P n i¼1 w E i w i ; ð1þ i i¼1 where w i ¼ 1/(R m R i ) is the weight of E i.ifr i ¼ R m, w i is taken as 3w 0,wherew 0 is the maximum weight for R i 6¼ R m (Du et al. 2006a, 2006b). The max max cycle length is defined as P max (i) ¼ E m (i) E m (i 1); where E m (i) ande m (i 1) are the maximum epochs of cycle i and (i 1), respectively. The parameters used in this paper are listed in Table 1. The official maximum epochs (E oa ) and the weighted ones (E m ) are listed in columns (2) and (3), respectively. The differences between them are usually 1 2 months but are half a year in cycles ð2þ Fig. 2. Correlation coefficients between R m and P max at lags from 0 to 6. The significant value is at lag 2, confidence level >99%. At lag 0 the correlation coefficient is 0.559, confidence level >95%. 18 and 23. The max max cycle lengths (P max ) and the amplitudes (R m ) are listed in columns (4) and (5), respectively. Other parameters are discussed below. 3. CORRELATION BETWEEN R m AND P max To study the relationship between R m and P max, we have calculated the correlation coefficients between R m and P max at lags from 0 to 6 (cycle), as shown in Figure 2. The correlation coefficient between R m and P max at lag 0 is with a confidence level >95%. A scatter plot is illustrated in Figure 3, and the regression equation is R F ¼ 346:3 1:73P max ; ¼ 31:8 26:4%; ð3þ where ¼ 31:8 is the regression standard deviation and 26:4% is its relative error. Although there is a weak correlation between R m and P max, we can employ it to predict neither of them, because if one is unknown, then so is the other. TABLE 1 Parameters and Results Cycle (1) E oa a (2) E m (3) P max (months) (4) R m b (5) R F (6) F (7) R 0 P (8) Feb 1848 Apr Feb 1860 Feb ( ) Aug 1870 Oct (+) 120.4(+) Dec 1883 Dec ( ) 95.3( ) Jan 1893 Dec (+) 130.4(+) Feb 1906 Apr ( ) 55.2( ) Aug 1917 Sep (+) 150.5(+) Apr 1928 Mar ( ) 80.3( ) ( ) Apr 1937 Jun (+) 107.9(+) (+) May 1948 Mar (+) 135.4(+) (+) Mar 1958 Feb (+) 173.0(+) (+) Nov 1969 Feb ( ) 127.9( ) ( ) Dec 1979 Dec (+) 153.0(+) (+) Jul 1989 Jul ( ) 120.4( ) ( ) May 2000 Sep ( ) 125.4(+) (+) a From Letfus (1994), except for cycle 23, which is from b The plus and minus signs are those of the differences R(i) R(i 1) between two successive cycles, indicating the varying trends.

3 No. 4, 2006 MAXIMUM AMPLITUDE AND MAX MAX CYCLE LENGTH 1487 Fig. 3. Scatter plot of R m vs. P max at lag 0. The regression equation is R m ¼ 346:3 1:73P max, ¼ 31:8 26:4%. Figure 2 clearly shows that the significant correlation coefficient is at a lag of 2 with a confidence level >99%, that is, R m correlates best with the P max value from two cycles earlier. Figure 4 shows the scatter plot of R m versus P max at lag 2. The linear regression equation is given by R F ¼ 451:2 2:51P max 2 ; ¼ 26:0 21:5%; ð4þ where P max 2 is the value of P max two cycles earlier. The fitted values, R F, are listed in column (6) of Table 1 and shown in Figure 5 (dashed line). It can be seen that in 12 out of 13 cases the varying trends (the signs in Table 1) of R F are the same as those of R m. The only exception is the even-odd pair of cycles 22 and 23. The residuals, F ¼ R F R m ; are listed in column (7) of Table 1 and shown in Figure 6 for clarity. It should be noted from Figures 5 and 6 that most fitted values for cycles are larger than the corresponding observed ones, whereas most fitted values for cycles are smaller than the corresponding observed ones. The reversal points are in cycles 16 and 23. If we use the values of P max in cycles 7 and 8 (162 and 88; Letfus 1994), then the extrapolated values from equation (4) are R F (9) ¼ 44:6 and R F (10) ¼ 230:3, and the fitted errors are F (9) ¼ 87:2 < 0 and F (10) ¼ 131:9 > 0, respectively. It suggests that another reversal point is likely in cycle 9. But these early data (P max before cycle 9) produce large ð5þ Fig. 5. Fitted (R F, dashed line) vs. observed (R m, solid line) values. regression errors, so we use the more reliable data after cycle 9 (Eddy 1976). The average residual for cycles is 1 ¼ 1 6 X 16 i¼11 F (i) ¼ 13:7; and the average residual for cycles is 2 ¼ 1 7 X 23 i¼17 F (i) ¼ 11:8: ð6þ ð7þ The absolute values of 1 and 2 are near to each other, and the mean is ¼ (j 1 jþj 2 j)=2 ¼ 12:8: The two residuals in equations (6) and (7) suggest a sign reversal of about cyclical drift 12.8 every other seven cycles. If this seven-cycle periodicity (23 16 ¼ 16 9) continues, the fitted values should be corrected by 8 >< ; 10 i 16; R 0 ¼ þ; 17 i 23; ð9þ >: ; 24 i 30: ð8þ Fig. 4. Scatter plot of R m vs. P max ; 2. The regression equation is R m ¼ 451:2 2:51P max 2, ¼ 26:0 21:5%. Fig. 6. Residual, F ¼ R F R m,where 1 (13:7) is the average for cycles 11 16, and 2 ( 11:8) is the average for cycles

4 1488 DU Vol. 132 Fig. 7. Predictions (RP 0, dashed line) ofthepasteightcyclesvs.theobserved values (R m, solid line). The mean arithmetic error is 25:4 18:5%. Fig. 8. Corrected predictions (R P, dashed line) from eq. (10) vs. the observed ones (R m, solid line). The mean arithmetic error is now 21:7 15:7%. After these corrections, the standard deviation of R F becomes ¼ 22:4, and its relative error is 18.5%. This phenomenon seems to imply that the amplitudes are likely modulated by a long-term trend of double 7 cycle periodicity or 14 cycle (154 yr) periodicity. Oliver et al. (1998) suggested that new magnetic flux was ejected to the dynamo region after cycle 16. Duhau (2003) pointed out that the sunspots before cycle 16 behaved differently from those after cycle 16. Wilson (1988) pointed out that the next local minimum of eightcycle modulation should be near cycle 22 (now the lower cycle 23). They all suggested that there may be a systematic drift every seven cycles, or there may be a long-term trend of about 14 cycles, as in the relationship between the amplitudes and periods (Du et al. 2006b). This long-term periodicity may be caused by the synthetic effect of the 179 yr periodicity related to the retrograde motion in the Sun s oscillation about the center of mass of the solar system (Jose 1965) and the lag of 2 (22 yr). 4. APPLICATION TO PAST CYCLES Now, let us test the effect of the correlation (at lag 2) in x 3 for the past eight cycles. To test the R m of cycle 23, we make use of the parameters for cycles 9 22 (values of P max are taken from cycles 9 20, and R m is taken from cycles 11 22). We recalculate the regression equation at lag 2, substitute the value P max ¼ 130 of cycle 21 (two cycles before 23) into this equation, and then obtain the predicted amplitude RP 0 (23) ¼ 125:9 of cycle 23. Repeating this process in the last eight cycles, we obtain eight predicted amplitudes, RP 0, in the same way. The results are listed in column (8) of Table 1 and displayed in Figure 7 (dashed line). The mean error of the predictions is 25.4, and its relative error is 18:5%. Considering the modulation of the long-term trend of about 14 cycles, the predictions should be corrected as R P ¼ R 0 P þ R 0: ð10þ The results (dashed line) and the observations (solid line) are displayed in Figure 8. The mean error of the predictions is now 21.7, and its relative error is 15:7%. The varying trends of R m (signs in Table 1) are all accurately predicted, except for the unusual even-odd pair of cycles 22 and 23 (although the predicted value, 125.9, is close to the observed one). In summary, this R m -P max 2 technique is effective for predicting the size of the solar cycle, at least for the past few cycles. 5. PREDICTIONS ON CYCLES 24 and 25 To estimate the amplitudes in cycles 24 and 25, we combine equations (4), (8), (9), and (10), R P ¼ 451:2 2:51P max 2 12:8: ð11þ Substituting P max ¼ 115 and 134 of cycles 22 and 23 into equation (11), we can estimate the amplitudes of cycles 24 and 25, R P ¼ 150:3 22:4 and102:6 22:4, respectively. It should be noted that the estimated amplitude, 150.3, of cycle 24 is greater than that found by some authors (see Table 2). The two predictions suggest that cycle 24 should be stronger than cycle 23 and that the even-odd pair of cycles 24 and 25 should violate the G-O rule as predicted by Duhau (2003), Hathaway & Wilson (2004), and Du et al. (2006b). But the predictions on cycles 24 and 25 by Solanki et al. (2002) should obey the G-O rule. 6. DISCUSSIONS AND CONCLUSIONS The newly defined max max cycle length is found to be inversely correlated (r ¼ 0:769) with the amplitude two cycles later. Meanwhile, a weak correlation (r ¼ 0:559) exists between the amplitude and max max cycle length of the same cycle. Considering that the background for the 11 yr cycle is the 22 yr Hale magnetic polarity cycle, in which every second (oddnumbered cycle) peak is reversed in sign ( Hale 1924; Bracewell TABLE 2 Predictions on Cycles 24 and 25 Reference Cycle 24 Cycle 25 De Meyer (1998) Kane (1999) Solanki et al. (2002) Wang et al. (2002) Duhau (2003) <87 Hathaway & Wilson (2004) Du et al. (2006b) This paper

5 No. 4, 2006 MAXIMUM AMPLITUDE AND MAX MAX CYCLE LENGTH ), the max max cycle length is in fact the duration from one polarity to the following reversed one ( ascending time in the Hale cycle). Thus, the correlation at lag 0 can be expressed as large amplitudes tend to take less time to reach their maxima than small ones in the Hale cycle. This is similar to the case of the Schwabe cycle ( Waldmeier 1939a, 1939b), but the exact physics behind this relationship is not yet known (Solanki et al. 2002; Javaraiah et al. 2005). It may suggest that a physical quantity, e.g., magnetic energy in a (Hale) cycle, has a tendency of stability: if it has a higher rise rate, then it needs less time to reach larger amplitude, and vice versa. But solar cycles are modulated by long-term cycles, e.g., the 80 yr Gleissberg (1971) cycle and 179 yr periodicity (Jose 1965; Landscheidt 1999). The correlation at lag 2 may result from the modulation of these long-term cycles. Regression analysis shows that most fitted values before cycle 17 are larger than the corresponding observed ones and that most fitted values from cycle 17 to cycle 23 are smaller than the corresponding observed ones. They both suggest that the amplitudes may be modulated by a long-term trend of about 14 cycles (154 yr). After the corrections of equation (9), the regression standard deviation becomes ¼ 22:4, and the amplitudes of cycles 24 and 25 are estimated to be 150:3 22:4 and 102:6 22:4, respectively. The value satisfies the condition R m (even) 125 for violations of the G-O rule suggested by Komitov & Bonev (2001). It further suggests that the pair of cycles 24 and 25 should violate the G-O rule. The past predictions on amplitude with statistical methods tend to be lower than the observations (Li et al. 2001b; Kane 2001). Considering the long-term trend of about 154 yr, predictions with statistical methods would be larger than observations from cycle 24 onward. According to the analysis above, our main conclusions are as follows: 1. The maximum amplitude of a solar activity cycle is inversely correlated ( 0.769) with the newly defined max max cycle length two cycles earlier. 2. The negative correlation coefficient between the amplitude and ascending time (either in the Schwabe or the Hale cycle) suggests that the energy in a solar activity cycle has a tendency of stability: if it has a higher rise rate, then it needs less time to reach larger amplitude, and vice versa. 3. Linear regression analysis shows that there exists a 14 cycle (154 yr) periodicity in solar activity. 4. The amplitudes of cycles 24 and 25 are estimated to be 150:3 22:4 and 102:6 22:4, respectively. This even-odd pair should violate the G-O rule. The authors are grateful to the anonymous referee for useful comments. 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