Sunspot Time Series: Passive and Active Intervals
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Abstract
Solar activity slowly and irregularly decreases from the first spotless day (FSD) in the declining phase of the old sunspot cycle and systematically, but also in an irregular way, increases to the new cycle maximum after the last spotless day (LSD). The time interval between the first and the last spotless day can be called the passive interval (PI), while the time interval from the last spotless day to the first one after the new cycle maximum is the related active interval (AI). Minima of solar cycles are inside PIs, while maxima are inside AIs. In this article, we study the properties of passive and active intervals to determine the relation between them. We have found that some properties of PIs, and related AIs, differ significantly between two group of solar cycles; this has allowed us to classify Cycles 8 – 15 as passive cycles, and Cycles 17 – 23 as active ones. We conclude that the solar activity in the PI declining phase (a descending phase of the previous cycle) determines the strength of the approaching maximum in the case of active cycles, while the activity of the PI rising phase (a phase of the ongoing cycle early growth) determines the strength of passive cycles. This can have implications for solar dynamo models. Our approach indicates the important role of solar activity during the declining and the rising phases of the solarcycle minimum.
Keywords
Spotless days Solar phase tracking Waldmeier effect Prediction1 Introduction
In a recently published paper (Zięba and Nieckarz 2012), we have shown that the relations based on the position of the longest spotless segment (LSS, the longest sequence of consecutive days when no spots were observed) with respect to locations of some characteristic extreme points in the daily sunspot numbers are statistically significant and have a predictive value. This indicates that the passive interval (PI) as well as the location and length of the LSS can be useful for studying of the physical processes responsible for the solar variability, especially since their values do not result from any smoothing procedure but are really observed.
In this article, we examine relationships between various parameters characterizing the passive and active intervals whose definition was given by Zięba et al. (2006). The active interval (AI) complements the passive interval in the sense that its covers also the time interval between the border spotless days but includes the cycle maximum. The proposed decomposition of sunspot time series into passive and active intervals is unambiguous. PI and related AI carry the same number, which is assigned from the number of the cycle of which minimum and maximum are inside of these intervals. Each passive interval and the active interval occurring after it form the ordered pair PI–AI. As the classically understood solar cycle and the pair PI–AI with the same number include the same maximum phase of solar cycle we use the name cycle also for the ordered pair PI–AI.
The basis for our research was and is the daily international sunspot number (ISN) series provided by the Solar Influences Data Center (SIDC, http://sidc.oma.be/ ) of the Royal Observatory of Belgium. Until 1980 the ISN, better known as the Wolf or Zürich number, was compiled by the Swiss Federal Observatory. In this work we use the daily ISN series covering the period between January 1830 and June 2012. This period includes the decline phase of Cycle 7, Cycles 8 – 23 and the initial rise of Cycle 24. Cycles 7 – 9 include the era of Schwabe’s records with a large number of days without observations (Wilson 1998). Cycles 10, 11, and the rise of Cycle 12 belong to Wolf’s era (years 1848 – 1882), while those of Cycles 13 – 21 belong to the Zürich era. Since 1981, when the International Astronomical Union World Data Center for sunspot numbers was transferred from the Zürich Observatory to Brussels, a new approach for calculation of the sunspot number has been established (Clette et al. 2007). The intrinsic nature and accuracy for four main eras of sunspot number observations are different in the ISN time series and this is the reason for treating the daily ISN series with care. However, we have shown in Zięba and Nieckarz (2012) that differences arising from heterogeneous ISN data among Cycles 8 – 23 can be neglected.
This article is organized as follows. In Section 2 we describe characteristic points of the daily ISN series and define various time intervals whose lengths and amplitudes are studied in Sections 3 and 4. Next, in Section 5 we discuss relations between lengths (periods) and amplitudes (strength) of the intervals considered, including the Waldmeier effect. Some predictions for Cycle 24 are given in Section 6, and Section 7 comprises a summary and a discussion.
2 Definition of Characteristic Points and Time Intervals
The passive interval (PI) is defined as the time interval (denoted ‘00’) from the first spotless day (FSD) after an old cycle maximum to the last spotless day (LSD) before the next new cycle maximum (Zięba et al. 2006; Zięba and Nieckarz 2012). All spotless days occur within the passive intervals. For each passive interval we have a minimum of activity (cycle minimum) and the occurrence of the longest spotless segment (LSS). We accept the center of LSS as the distinctive point of the PI.
The idea of using spotless days to find the minimum of activity was suggested by Waldmeier (1961) and McKinnon (1987). Wilson (1995) proposed to use the first spotless day as a predictor for the sunspot minimum. The possible connections of spotless days with the timing and length of the solar cycle were more accurately examined by Wilson and Hathaway (2005, 2006, 2007).
The time interval (denoted ‘xx’), between the last spotless day before the upcoming new cycle maximum and the first spotless day after it, includes all the days when sunspots are observed and we called it the active interval (AI). The day with the highest sunspot number (RP, the day with the maximum number of sunspots in a given cycle) is inside AI and we consider its position as an AI distinctive point.
Definitions of the intervals and the time distances used in this article.
Interval (distance)  Definition  Cycle  All cycle  

14  22  Mean  sem^{1}  
[days]  
00^{2}  Passive interval (PI), the time interval covering all the days from the first spotless day (FSD) after a cycle maximum to the last spotless day (LSD) before the next cycle maximum along with them  3548  1331  2334  216 
0s  Declining segment of the PI, the time interval from the FSD after a cycle maximum to the central day of the LSS  1983  774  1512  131 
s0  Rising segment of the PI, the time interval from the central day of the LSS to the LSD before the next cycle maximum  1565  557  822  100 
0m  The time distance from the FSD after a cycle maximum to the day of minimum given by the 810day Gaussian filter (Gaussian minimum, Gm)  2105  834  1404  138 
0p  The time distance from the FSD after a cycle maximum to the day with the maximal daily sunspot number (peak maximum, Px) inside the next cycle maximum  3653  2829  2919  164 
0g  The time distance from the FSD after a cycle maximum to the day of maximum given by the 810day Gaussian filter (Gaussian maximum, Gx) inside the next cycle maximum  3831  2327  3020  153 
sp  The time distance from the central day of the LSS to the successive Px  1671  2056  1407  74 
xx  Active interval (AI), the time interval covering all the days from the LSD after a cycle minimum to the FSD before the next cycle minimum without spotless days  445  2453  1638  173 
xp  The time distance from the LSD before a cycle maximum to the Px  106  1499  585  97 
px  The time distance from the Px to the FSD before the next cycle minimum  339  954  1052  109 
ss  The time distance between the central days of two successive LSS  4422  3927  4002  86 
Dates defining positions of characteristic points for solar cycles (N stands for the solarcycle number): the first spotless day (FSD), the central day of the longest spotless segment (LSS), the day of the Gaussian minimum (DGm), the last spotless day (LSD), the day of the peak maximum (DPx), the day of the Gaussian maximum (DGx). The numbers related to dates are enumerated from 1 January 1818. Other acronyms and abbreviations mean: nlss, the number of spotless days in the LSS, n00, the number of spotless days in the passive interval, RGm, the value of the Gaussian minimum, RPx, the maximal sunspot number of the indicated cycle (the peak maximum), RGx, the value of the Gaussian maximum, ‘00’, the length of the passive interval, ‘xx’, the length of the active interval, ‘0s’, the time distance from the FSD to the LSS, ‘0m’, the time distance from the FSD to the DGm, ‘0p’, the time distance from the FSD to the DPx, ‘0g’, the time distance from the FSD to the DGx.
N  FSD  LSS  Date  nlss  n00  LSD  DPx  Date  RPx  00  0s  0p 

Date  DGm  Date  RGm  Date  DGx  Date  RGx  xx  0m  0g  
8  4 407  5 958  24.04.1834  34  397  6 379  6 992  21.02.1837  261  1 973  1 551  2 585 
24.01.1830  5 742  20.09.1833  14.0  19.06.1835  7 054  24.04.1837  127.2  1 791  1 335  2 647  
9  8 141  9 661  13.06.1844  21  402  10 777  10 820  16.08.1847  254  2 637  1 520  2 679 
15.04.1840  9 357  14.08.1843  16.7  04.07.1847  11 136  27.06.1848  113.2  678  1 216  2 995  
10  11 456  13 764  07.09.1855  49  655  14 705  15 520  28.06.1860  220  3 250  2 308  4 064 
13.05.1849  13 899  20.01.1856  7.9  05.04.1858  15 434  03.04.1860  90.7  1 281  2 443  3 978  
11  15 987  17 914  17.01.1867  38  406  18 823  19 231  26.08.1870  317  2 837  1 927  3 244 
08.10.1861  17 913  16.01.1867  13.8  14.07.1869  19 348  21.12.1870  119.2  1 404  1 926  3 361  
12  20 228  22 354  15.03.1879  54  1 028  24 009  24 116  10.01.1884  166  3 782  2 126  3 888 
19.05.1873  22 125  29.07.1878  6.3  25.09.1883  24 054  09.11.1883  63.9  473  1 897  3 826  
13  24 483  26 266  29.11.1889  26  736  27 014  27 616  10.08.1893  237  2 532  1 783  3 133 
11.01.1885  26 064  11.05.1889  6.6  17.12.1891  27 674  07.10.1893  80.5  1 423  1 581  3 191  
14  28 438  30 420  15.04.1901  69  934  31 985  32 091  11.11.1905  182  3 548  1 982  3 653 
10.11.1895  30 542  15.08.1901  5.0  28.07.1905  32 269  08.05.1906  59.2  445  2 104  3 831  
15  32 431  34 842  24.05.1913  92  1 023  36 069  36 378  07.08.1917  268  3 639  2 411  3 947 
17.10.1906  34 665  28.11.1912  3.2  02.10.1916  36 507  14.12.1917  86.8  1 284  2 234  4 076  
16  37 354  38 741  26.01.1924  39  534  39 645  39 847  05.02.1927  173  2 292  1 387  2 493 
09.04.1920  38 492  22.05.1923  10.0  18.07.1926  40 159  14.12.1927  71.1  1 522  1 138  2 805  
17  41 168  42 330  23.11.1933  36  538  42 943  43 494  30.01.1937  233  1 776  1 162  2 326 
18.09.1930  42 214  30.07.1933  8.0  29.07.1935  43 828  30.12.1937  107.1  2 292  1 046  2 660  
18  45 236  46 147  06.05.1944  36  269  46 644  47 261  25.05.1947  323  1 409  911  2 025 
07.11.1941  46 023  03.01.1944  15.0  15.09.1945  47 570  29.03.1948  140.5  1 921  787  2 334  
19  48 566  49 842  18.06.1954  30  446  50 329  51 127  24.12.1957  355  1 764  1 276  2 561 
20.12.1950  49 704  31.01.1954  12.8  18.10.1955  51 180  15.02.1958  185.8  2 205  1 138  2 614  
20  52 535  53 591  22.09.1964  15  227  54 278  55 207  24.02.1969  215  1 744  1 056  2 672 
01.11.1961  53 602  03.10.1964  16.2  10.08.1966  55 227  16.03.1969  106.0  2 538  1 067  2 692  
21  56 817  57 910  20.07.1976  24  272  58 273  59 118  10.11.1979  302  1 457  1 093  2 301 
23.07.1973  57 766  27.02.1976  16.8  18.07.1977  59 321  31.05.1980  150.4  2 317  949  2 504  
22  60 591  61 364  03.01.1986  21  273  61 921  63 420  21.08.1991  300  1 331  773  2 829 
22.11.1983  61 424  04.03.1986  17.5  14.07.1987  62 918  06.04.1990  148.3  2 453  833  2 327  
23  64 375  65 291  04.10.1996  42  309  65 753  66 675  19.07.2000  246  1 379  916  2 300 
02.04.1994  65 129  25.04.1996  13.4  09.01.1998  66 853  13.01.2001  112.4  2 208  754  2 478  
24  67 962  69 991  17.08.2009  32  817  70 718  ?  2 757  2 029  ?  
27.01.2004  69 656  16.09.2008  4.0  14.08.2011  ?  ?  1 694  ? 
The definition of PI and AI, as well as the location of the LSS and the peak maximum, is based on the precisely determined observables. For comparative purposes we also distinguish the days having the minimal (RGm) and maximal (RG) sunspot number after smoothing the ISN time series using an 810day Gaussian filter. The Gaussian filter, having a full width at half maximum (FWHM) equal to 810 days, effectively removes shortterm variations of solar activity on time scales of about two years that can produce double peaked maxima. The size of this filter is similar to the 24month Gaussian filter used when monthly averaged sunspot numbers are considered (Hathaway, Wilson, and Reichmann 1999; Hathaway 2010).
We suppose that the central day of the LSS represents better the time of real minimum, interpreted according to Cameron and Schüssler (2008) as the epoch when the sum of the activities of the old and the new cycle is minimal, than the Gaussian minimum. Various definitions of the cycle minimum were considered by Harvey and White (1999), who among others took into account the monthly averages of spotless days.
3 Lengths of Intervals – Two Types of Solar Cycles
Results of statistical tests calculated for various parameters of the passive and active intervals.
No.  Parameter  Cycles 9 – 15  Cycles 17 – 23  pvalue^{1}  

Mean  SD  Mean  SD  Test F  Test t  Test U  
1  Length of the PI ‘00’  3175.0  507.5  1551.4  200.1  0.039  0.00000  0.0006 
2  Length of the PI declining segment ‘0s’  2009.1  306.1  1027.7  171.1  0.183  0.00001  0.0006 
3  Length of the PI rising segment ‘s0’  1165.9  340.6  523.7  106.0  0.012  0.00046  0.0006 
4  Length of the AI ‘xx’  998.3  445.6  2276.3  199.0  0.071  0.00002  0.0006 
5  Length of the AI rising segment ‘xp’  341.4  287.8  880.1  308.9  0.868  0.00551  0.0070 
6  Length of the AI declining segment ‘px’  656.9  278.6  1396.1  253.7  0.826  0.00022  0.0023 
7  Ratio xx/00  0.330  0.171  1.485  0.209  0.637  0.00000  0.0006 
8  Time elapsed since the LSS to the day of the peak maximum ‘sp’  1507.3  236.9  1403.9  332.8  0.428  0.516  0.383 
9  Time distance between two subsequent LSS segments ‘ss’  4154.3  216.0  3951.6  422.0  0.128  0.280  0.209 
10  Mean sunspot number of the PI  30.86  6.82  22.07  4.95  0.455  0.01729  0.0379 
11  Mean sunspot number of ‘s0’ segment in the PI  30.39  11.29  18.13  2.87  0.004  0.01656  0.0262 
12  Mean sunspot number of the AI  85.64  23.20  106.15  20.77  0.795  0.10684  0.0973 
13  Mean sunspot number of ‘xp’ segment in the AI  82.37  14.26  105.48  27.56  0.134  0.07239  0.0728 
14  Number of spotless days in the PI  740.6  268.9  337.7  123.2  0.079  0.00362  0.0070 
15  Length of the LSS  49.86  24.84  29.14  9.60  0.036  0.062  0.0728 
16  Length of the longest segment of days with spot in a row in the PI  373.1  131.6  191.9  121.5  0.851  0.02011  0.0262 
 i)
Lengths of cycles measured from one LSS to the next one (see row 9) and the time distance from the LSS minimum to the peak maximum are the same for both types of cycles (row 8).
 ii)
There are significant differences between both types of cycles in the lengths of their segments, such as ‘0s’, ‘s0’, ‘xp’, and ‘px’ (rows 2, 3, 5, 6).
 iii)
The AI mean sunspot numbers of Cycles 17 – 23 are sightly higher than those of Cycles 9 – 15, but the differences are mainly in the ‘xp’ segments (rows 12, 13).
 iv)
The PI mean sunspot numbers of Cycles 9 – 15 are significantly larger than those in the other type of cycles; this is mainly because of the relatively large sunspot numbers in their rising segments ‘s0’, along which a high variability of the sunspot numbers is also observed (rows 10, 11).
 v)
The number of the spotless days, the longest spotless segments and longest segments of days with spots in a row are significantly larger in the PI of Cycles 9 – 15 (rows 14, 15, 16).
4 Amplitudes of Passive and Active Cycles
For the active interval, the day of the peak maximum is used as the split point, while for the passive interval, the central day of the LSS fulfills this role. The peak maxima in both segments of the passive interval are simple to define but for the active interval an additional rule is that the day corresponding to the maximum in each segment has to belong to a different solar rotation from the day of the split point.

All the three active interval peak amplitudes are apparently higher than peaks observed during passive intervals. Student’s ttest gives the pvalues (see the footnote in Table 3) smaller than 0.0006 for all the possible pairs of AI peak amplitudes versus PI peaks. Furthermore, we find that the nonparametric Wilcoxon T test (a signedrank paired difference test) confirms these differences by giving a pvalue of 0.018 for all the pairs.

In Figure 6, the peaks of the PI declining segments of active cycles (red circles) are typically higher than the peaks of the rising segments (red triangles). The nonparametric Wilcoxon T test gives a pvalue equal to 0.028.

The peaks of PI rising segments of active cycles are significantly lower than the same one in the passive cycles. Student’s ttest gives a pvalue equal to 0.000027 while pvalue of the Mann–Whitney U nonparametric test is 0.00058.
In Figure 7, the curves of the mean amplitudes support the same results we infer from the curves in Figure 6. The statistical significance of the differences between the mean amplitudes of the active and passive intervals for all the cycles is evident. Results of the comparison between the mean amplitudes of the passive and active cycles calculated separately for the passive and active intervals are given in Table 3 (row 10 and 12).
Amplitude  Symbol  a  T, the period  Φ  b  R ^{2} 

RPxx  FRPxx in Figure 6  41 ± 20  13.6 ± 2.7  1.8 ± 0.9  250 ± 14  0.27 
RMxx  FRMxx in Figure 7  25 ± 6  12.0 ± 1.2  1.0 ± 0.6  91 ± 5  0.57 
RGxx  FRGxx in Figure 8  40 ± 8  14.0 ± 1.3  1.9 ± 0.4  108 ± 6  0.65 
RGm00  FRGm23 in Figure 8  5.6 ± 0.9  13.9 ± 1.2  1.6 ± 0.4  11 ± 0.7  0.77 
RGm00  FRGm24 in Figure 8  5.8 ± 1.0  12.5 ± 0.9  1.2 ± 0.4  10 ± 0.7  0.71 
RGxx/RPxx  in Figure 8 in ZN^{1}  0.086 ± 0.008  14.0 ± 0.6  1.9 ± 0.2  0.42 ± 0.006  0.90 
All the bestfitted sinusoids indicate a period of about 14 solar 11year cycles, which can be associated with the Gleissbergcycle period (Gleissberg 1939; Hathaway, Wilson, and Reichmann 1999; Ogurtsov et al. 2002).
5 The Relations Between the Time Distances and the Amplitudes
5.1 The Relations Between the Amplitude and the Period of Activity
5.2 Waldmeier Effect
The Waldmeier Effect (Waldmeier 1935, 1939) is one of the widely quoted relationships in which the rise time elapsed between minimum and maximum of a cycle is inversely correlated with the cycle amplitude. The effect was analyzed using various monthly averages smoothed data sets such as: international (Wolf) and group sunspot numbers (Hathaway, Wilson, and Reichmann 2002), Wolf sunspot numbers and sunspot area (Dikpati, Gilman, and de Toma 2008), Wolf sunspot numbers, group sunspot numbers, sunspot area and 10.7 cm radio flux (Karak and Choudhuri 2011). Among the time distances defined using the LSS location, the time distances ‘s0’ and ‘sp’ are the closest to the most commonly used definition of the rise time when the cycle maximum has occurred. The correlation coefficients between ‘sp’ and the discussed amplitudes RPxx, RGxx and RIxx are not significant (they are about −0.23), while those between ‘s0’ and these amplitudes are −0.62, −0.76, and −0.79, respectively. All these values are significant with pvalues smaller than 0.011.
Beside the study of the classical Waldmeier effect, some authors have tried to find correlations showing that stronger cycles tend to rise faster (Lantos 2000; Cameron and Schüssler 2008; Kane 2008; Podladchikova, Lefebvre, and van der Linden 2008; Karak and Choudhuri 2011). There are no simple observables to measure the growth and decay rates of solar activity. Various authors proposed different approaches to define a parameter measuring a growth rate of solar activity. All these authors used monthly averages smoothed data, but took differently defined time distance values (Lantos 2000; Karak and Choudhuri 2011) or sunspot numbers difference (Cameron and Schüssler 2008) between two separated months selected during the cycle’s ascending phase for their growth rate definition. Because many of the abovementioned articles show that growth and decay rates can be treated as a precursor quantity, we examine for this purpose two values, RP0s/0s and RPs0/s0, formed by such observables as the segments ‘s0’ and ‘s0’ as well as the maximal sunspot number in these segments RP0s and RPs0. We call these ratios declining and rising indices.
The parameters of the multilinear regression models for the prediction of the maximal peak (RPxx) and Gaussian amplitude (RGxx) of the studied passive or active cycles using the following predictors (variables): RP0s/0s, the declining index, RPs0/s0, the rising index, nlss, the number of the spotless days in the LSS, n00, the number of the spotless days in the passive interval.
Passive cycle  

Model: RPxx=a+b1⋅RP0s/0s+b2⋅RPs0/s0+b3⋅nlss  Analysis of variance for ^{3}Regression  
Variable  Coefficient  Value  ^{1}Std. Error  ^{2} pvalue  
a  −50.306  24.9104  0.13672  SSM=15834.29 SSE=138.57 F=114.27 pvalue=0.00137 R ^{2}=0.9956 Adjusted R ^{2}=0.9826 see=6.8  
RP0s/0s  b1  729.391  254.7788  0.06443  
RPs0/s0  b2  1380.260  79.6949  0.00042  
nlss  b3  0.637  0.1316  0.01685 
Model: RGxx=a+b1⋅RP0s/0s+b2⋅RPs0/s0  Analysis of variance for Regression  

Variable  Coefficient  Value  Std. Error  pvalue  
a  −41.096  12.4554  0.02995  SSM=2993.77 SSE=89.04 F=67.24 pvalue=0.00083 R ^{2}=0.9711 Adjusted R ^{2}=0.95667 see=4.7  
RP0s/0s  b1  909.152  164.3010  0.00521  
RPs0/s0  b2  421.323  52.4448  0.0013 
Active cycle  

Model: RPxx=a+b1⋅RP0s/0s+b2⋅RPs0/s0+b3⋅n00  Analysis of variance for Regression  
Variable  Coefficient  Value  Std. Error  pvalue  
a  −22.716  27.4782  0.46902  SSM=15634.5 SSE=285.5 F=54.75 pvalue=0.004056 R ^{2}=0.9910 Adjusted R ^{2}=0.9641 see=9.8  
RP0s/0s  b1  1074.876  108.6869  0.00220  
RPs0/s0  b2  771.194  103.9258  0.00506  
n00  b3  0.116  0.0340  0.04216 
Model: RGxx=a+b1⋅RP0s/0s+b2⋅RPs0/s0+b3⋅n00  Analysis of variance for Regression  

Variable  Coefficient  Value  Std. Error  pvalue  
a  −25.040  30.1489  0.46714  SSM=4807.0 SSE=343.7 F=13.98 pvalue=0.02868 R ^{2}=0.9333 Adjusted R ^{2}=0.8665 see=10.7  
RP0s/0s  b1  669.175  119.2504  0.01119  
RPs0/s0  b2  321.174  114.0266  0.06692  
n00  b3  0.0662  0.0373  0.17409 
6 Predictions for Cycle 24
In this section we address the prediction of the timings and the amplitude of the ongoing sunspot active interval for Cycle 24, using relations based on the description of the sunspot time series as a sequence of nonoverlapping segments called passive and active intervals. Some relations inferred from the location of LSS with respect to the occurrences of the FSD and the former sunspot maximum were discussed in Zięba and Nieckarz (2012) for a prediction of the peak and the Gaussiansmoothed maximum sunspot number for the present cycle.
To assess the length of the ongoing active interval we have examined how the PI lengths ‘00’, ‘0s’ and ‘s0’ affected the length ‘xx’ of the subsequent active interval. The highest correlation coefficient was found between ‘xx’ and ‘s0’. Figure 16 presents this relation and indicates a clear concentration of active cycles near the coordinates (524, 2276). It is evident, from the curves for the passive and active cycles in Figure 17, that the lengths of the active cycles do not correlate with the ‘s0’ lengths, while the lengths of the passive cycles do.
The position of the peak maximum can also be predicted from the relations between the time distance ‘0p’ and the PI lengths ‘00’ as well as ‘0s’ (Figure 18). The correlations of these parameters with the position of the Gaussian maximum ‘0g’ are higher than those with ‘0p’ and are equal to 0.97 and 0.96, respectively.
Some predictions for the AI in Cycle 24 derived from the relations found from previous cycles.
Variables  r  Fig.  Equation y=a+bx  See  y  Prediction for the AI of Cycle 24  

x  y  a  b  
s0=727  xx  −0.89  2321  −1.13  243  1499  ^{1} FSD 72217 Sep 2015  
00=2757  0p  0.91  18a  1310  0.69  248  3212  ^{2} DPx 71174 Nov 2012 
0g  0.97  –  1414  0.69  148  3316  ^{2} DGx 71278 Feb 2013  
0s=2029  0p  0.90  18b  1218  1.12  297  3490  DPx 71452 Aug 2013 
0g  0.96  –  1327  1.12  170  3599  DGx 71561 Dec 2013  
RP0s/0s=0.0527  RPxx  R ^{2} 0.983  See Table 5 for the parameters of the linear regression.  7  198  95 % CI 175 – 222  
RPs0/s0=0.1375 nlss=32  RGxx  R ^{2} 0.956  5  64  54 – 76 
The number of days, which elapsed from the last spotless day of the PI of Cycle 24, is 809 days counted from 31 October 2013 (the day on which this article was submitted). According to the relation presented in Figure 17, we assume that the end of the maximum phase of Cycle 24 (i.e. the active interval) can occur in the second half of 2015. The daily maximal ISN was 136 on 21 October 2011, but on 16 May 2013 this number was 135. Our predictions for the day of the peak maximum do not exclude that it can still happen, but the maximum value of the sunspot number is smaller than the one predicted by us (198) up to now. Our prediction for the Gaussian maximum is that it will happen around 100 days after the peak maximum.
7 Summary and Discussion
Taking into account spotless days, we have represented the ISN time series as the sequence of precisely defined consecutive time intervals, called passive and active intervals, according to whether they contain minimal or maximal daily sunspot numbers. Active intervals correspond to a phase of maximal sunspot numbers (both the peak and the Gaussian maximum are inside an AI), but passive intervals are extended beyond a classically understood minimum phase of the solar cycle and include the declining phase of the previous maximum. The properties of each passive interval, such as characteristic lengths, numbers of spotless days, declining, and rising indices are predictors of the length and amplitudes of the approaching active interval.
We have found clear differences between properties of two types of solar cycles (see Table 3) referred to as the passive (Cycles 9 – 15) and active (Cycles 17 – 23) cycles. Each group contains seven cycles in a row. According to the Hathaway division of cycles into small, medium and large (Hathaway 2010, 2011), the passive cycles are small and medium ones, while the active cycles are large and medium. The distribution of days with spots having a relatively high sunspot number is different in the both types of cycles (see Figure 5). All the three discussed cycle amplitudes (the peak, the Gaussian, and the integrated one) show a variation with time that can be fitted using sinusoidal functions. The periods of these functions (see Table 4) change from 12 to 14 solar 11year periods. These facts indicate that the occurrence of passive and active cycles can be connected with a higher mode of the Gleisberg cycle (Gleissberg 1939; Ogurtsov et al. 2002).
 i)
What is the cause of the apparent difference between passive and active cycles? We can answer that it can be either the specific cycle phase division based on the spotless days or that there is a key physical reason that underlies the different observed distribution of days with a relatively high sunspot number for this group of cycles. The fact that the passive and active cycles occur in a row, and in a number which corresponds to the Gleisberg cycle, indicates a possible underlying physical cause.
 ii)
The cycle overlapping effect revealed by Cameron and Schüssler (2007, 2008) can explain the high fluctuation of the sunspot numbers of the passive cycles during their passive intervals (a relatively large overlapping) as well as the marked reduction of this fluctuation in the case of the active cycles (a small overlapping). This is also confirmed by the large number of spotless days during the passive cycles because their long passive intervals lead to more days without activity even if some days have relatively large sunspot numbers.
However, we cannot say whether changes of the declining index (see Figure 19) are related only to the overlapping of cycles and the Waldmeier effect (Cameron and Schüssler 2007), or whether they are connected also with the meridional plasma flow variations considered by Nandy, MuñozJaramillo, and Martens (2011). Similarly, it is difficult to assume without further study that one of the various possible mechanisms, i.e. the evolution of the polar field around solar minimum, could explain the observed fluctuations of our rising index (Cameron et al. 2010, 2013; DasiEspuig et al. 2010; Jiang et al. 2010; Cameron and Schüssler 2012).
Svalgaard and Kamide (2013) have shown that the polar fields play a crucial role in the solar cycle. Because of solarcycle N–S asymmetries, polarfield reversals do not occur at the same times in both hemispheres (Norton and Gallagher 2010; MuñozJaramillo et al. 2013b, 2013a; Zhao, Landi, and Gibson 2013), and some differences among various passive intervals may be expected. The longterm persistence of a phase leading in one of the hemispheres (Zolotova et al. 2009) can explain a repetition of some noticed properties of passive and active intervals.
The focus of this article is to analyze the daily sunspot time series using the days for which timing and sunspot number are precisely defined, and we have done so. We have, however, found some other facts and characteristics that will be the object of further analysis.
Notes
Acknowledgements
We thank the anonymous referee for constructive comments and suggestions, which much improved the original version of the manuscript.
References
 Cameron, R.H., Schüssler, M.: 2007, Astrophys. J. 659, 801. ADSCrossRefGoogle Scholar
 Cameron, R., Schüssler, M.: 2008, Astrophys. J. 685, 1291. ADSCrossRefGoogle Scholar
 Cameron, R.H., Schüssler, M.: 2012, Astron. Astrophys. 548, A57. CrossRefGoogle Scholar
 Cameron, R.H., DasEspuig, M., Jiang, J., Işik, E., Schmitt, D., Schüssler, M.: 2013, Astron. Astrophys. 557, A141. ADSCrossRefGoogle Scholar
 Cameron, R.H., Jiang, J., Schmitt, D., Schüssler, M.: 2010, Astrophys. J. 719, 264. ADSCrossRefGoogle Scholar
 Clette, F., Berghmans, D., Vanlommel, P., Van der Linden, R.A.M., Koeckelenbergh, A., Wauters, L.: 2007, Adv. Space Res. 40, 919. ADSCrossRefGoogle Scholar
 DasiEspuig, M., Solanki, S.K., Krivova, N.A., Cameron, R.H., Peñuela, T.: 2010, Astron. Astrophys. 518, A97. ADSCrossRefGoogle Scholar
 Dikpati, M., Gilman, P.A., de Toma, G.: 2008, Astrophys. J. 673, L99. ADSCrossRefGoogle Scholar
 Gleissberg, W.: 1939, Observatory 62, 158. ADSGoogle Scholar
 Gnevyshev, M.N., Ohl, A.I.: 1948, Astron. Zh. 25, 8. Google Scholar
 Harvey, K.L., White, O.R.: 1999, J. Geophys. Res. 104, 19759. ADSCrossRefGoogle Scholar
 Hathaway, D.H.: 2010, Living Rev. Solar Phys. 7, 1. ADSCrossRefGoogle Scholar
 Hathaway, D.H.: 2011, Solar Phys. 273, 221. ADSCrossRefGoogle Scholar
 Hathaway, D.H., Wilson, R.M., Reichmann, E.J.: 1999, J. Geophys. Res. 104, 22375. ADSCrossRefGoogle Scholar
 Hathaway, D.H., Wilson, R.M., Reichmann, E.J.: 2002, Solar Phys. 211, 357. ADSCrossRefGoogle Scholar
 Jiang, J., Işik, E., Cameron, R.H., Schmitt, D., Schüssler, M.: 2010, Astrophys. J. 717, 597. ADSCrossRefGoogle Scholar
 Kane, R.P.: 2008, Solar Phys. 248, 203. ADSCrossRefGoogle Scholar
 Karak, B.B., Choudhuri, A.R.: 2011, Mon. Not. Roy. Astron. Soc. 410, 1503. ADSGoogle Scholar
 Kopecky, M.: 1950, Bull. Astron. Inst. Czechoslov. 2, 14. ADSGoogle Scholar
 Lantos, P.: 2000, Solar Phys. 196, 221. ADSCrossRefGoogle Scholar
 McKinnon, J.A.: 1987, UAG Repots UAG95, National Geophysical Data Center, NOAA, Boulder, CO. Google Scholar
 MuñozJaramillo, A., Balmaceda, L.A., DeLuca, E.E.: 2013a, Phys. Rev. Lett. 11, 041106. ADSCrossRefGoogle Scholar
 MuñozJaramillo, A., DasiEspuig, M., Balmaceda, L.A., DeLuca, E.E.: 2013b, Astrophys. J. Lett. 767, L25. ADSCrossRefGoogle Scholar
 Nandy, D., MuñozJaramillo, A., Martens, P.C.H.: 2011, Nature 471, 80. ADSCrossRefGoogle Scholar
 Norton, A.A., Gallagher, J.C.: 2010, Solar Phys. 261, 193. ADSCrossRefGoogle Scholar
 Ogurtsov, M.G., Nagovitsyn, Y.A., Kocharov, G.E., Jungner, H.: 2002, Solar Phys. 211, 371. ADSCrossRefGoogle Scholar
 Podladchikova, T., Lefebvre, B., van der Linden, R.: 2008, J. Atmos. SolarTerr. Phys. 70, 277. ADSCrossRefGoogle Scholar
 Svalgaard, L., Kamide, Y.: 2013, Astrophys. J. 763, 23. ADSCrossRefGoogle Scholar
 Waldmeier, M.: 1935, Astron. Mitt. Eidgenöss. Sternwarte Zür. 14(133), 105. ADSGoogle Scholar
 Waldmeier, M.: 1939, Astron. Mitt. Eidgenöss. Sternwarte Zür. 14(138), 470. ADSGoogle Scholar
 Waldmeier, M.: 1961, The SunspotActivity in the Years 1610 – 1960, Schulthess, Zürich. Google Scholar
 Wilson, R.M.: 1995, Solar Phys. 152, 197. ADSCrossRefGoogle Scholar
 Wilson, R.M.: 1998, Solar Phys. 182, 217. ADSCrossRefGoogle Scholar
 Wilson, R.M., Hathaway, D.H.: 2005, NASA Technical Report, NASA/TP2005213608. Google Scholar
 Wilson, R.M., Hathaway, D.H.: 2006, NASA Technical Report, NASA/TP2006214601. Google Scholar
 Wilson, R.M., Hathaway, D.H.: 2007, NASA Technical Report, NASA/TP2007215134. Google Scholar
 Wolf, R.: 1861, Mon. Not. Roy. Astron. Soc. 21, 77. ADSGoogle Scholar
 Zhao, L., Landi, E., Gibson, S.E.: 2013, Astrophys. J. 773, 157. ADSCrossRefGoogle Scholar
 Zięba, S., Masłowski, J., Michalec, A., Michałek, G., Kułak, A.: 2006, Astrophys. J. 653, 1517. ADSCrossRefGoogle Scholar
 Zięba, S., Nieckarz, Z.: 2012, Solar Phys. 278, 457. ADSCrossRefGoogle Scholar
 Zolotova, N.V., Ponyavin, D.I., Marwan, N., Kurths, J.: 2009, Astron. Astrophys. 503, 197. ADSCrossRefGoogle Scholar
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