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Sunspot cycle characteristics based on the newly revised sunspot number.

ABSTRACT

Effective July 1, 2015, the International sunspot number database was revised to correct a number of inhomogeneities in the sunspot number series. In particular, the revision has affected the overall scale of sunspot number amplitudes and, in some cases, the timing. This study examines the characteristics of sunspot cycles using the newly revised sunspot number and the inferred associations between specific parameters. Also examined is the current ongoing sunspot cycle 24 in comparison with other groupings of sunspot cycles of known ascent duration and cycle length, as well as a look ahead to sunspot cycle 25, the next sunspot cycle. Based on smoothed monthly mean sunspot number, sunspot cycle 24 had its minimum sunspot number amplitude (Rm=2.2) in December 2008 and its maximum sunspot number amplitude (RM=116A) in April 2014, thus yielding an ascent duration (ASC(m-M)) of 64 months. Hence, sunspot cycle 24 can be characterized as a slow-rising sunspot cycle. Because most slow-rising sunspot cycles (9 of 12) have also been sunspot cycles of long-period (PER(m-m) [greater than or equal to] 135 months), sunspot cycle 24 is not expected to end until 2020 or later. Based on the Even-Odd Effect, sunspot cycle 25 is anticipated to have a maximum sunspot number amplitude larger than sunspot cycle 24, measuring about 166 [+ or -] 40 (the [+ or -] 1 standard deviation prediction interval).

INTRODUCTION

Sunspots are the dark, cool regions of enhanced magnetic field that appear on the Sun. They have sizes spanning from the just barely visible to those many times larger than the size of the Earth (Kiepenheuer 1953). Although pretelescopic references to naked-eye observations of 'dark spots' on the Sun are found in ancient Chinese chronicles and Greek, Roman, and Arabic writings, sunspots were not observed telescopically until the early 1600s by Galileo, Scheiner, Harriot, Fabricus, and others (Hoyt and Schatten 1997).

While early observers noted the number of sunspots to vary over time, being more plentiful at certain times and less plentiful or nearly nonexistent at other times, recognition of the 'sunspot cycle' went largely unnoticed until the mid-1800s. In particular, it was the founding work of Samuel Heinrich Schwabe, a German apothecary and amateur astronomer, who having diligently observed the Sun daily from Dessau, Germany, for more than four decades (1826-1868) showed the spottiness of the Sun to wax and wane in number in a somewhat regular manner over a period of about 10 years (Schwabe 1844, Wilson 1998).

Each day throughout the year (averaging about 290 observing days per year) Schwabe counted the number of newly appearing 'clusters of spots' and the number of days when he saw no spots. He found that when the number of spotless days was greatest, the number of clusters of spots was least, and vice versa. He reasoned that the variation in spottiness on the Sun was a regularly recurring phenomenon, having a minimum and a maximum of activity, which we now know today as the sunspot cycle.

Following Schwabe's discovery, Johann Rudolf Wolf, a professional Swiss astronomer from the Bern Observatory and later at Zurich, in 1849 posited his now famous formula for computing sunspot number: R = k(10g + f), where R represents the 'relative' sunspot number; k, a correction factor used to compensate for observing conditions, the size of the telescope, the method of observation, and the individual observer; g, the number of groups (akin to Schwabe's clusters of spots); and f, the number of individual spots discernible on the Sun (Wolf 1861, Waldmeier 1961, Hathaway 2015). During the interval 1849-1893, Wolf was the primary observer for determining R, putting his k factor equal to 1. Wolf counted each spot singly, but initially he did not consider small spots, which are difficult to distinguish when seeing is bad. However, in 1882, the method for counting sunspots was changed to include small spots and the k factor was changed to 0.60 to account for their inclusion. Wolf's successors at Zurich (i.e., Alfred Wolfer 1894-1926, William Otto Brunner 1926-1944, and Max Waldmeier 1945-1979) continued observing the Sun and determining R using the k factor 0.60. However, beginning in 1981 and continuing through the present, the Royal Observatory of Belgium took over the task of determining sunspot number (now called the 'International sunspot number') from the Swiss Federal Observatory.

Using his relative sunspot number R, Wolf also sought to reconstruct the variation in solar activity for the earlier period, prior to 1849. In terms of daily sunspot number counts, he was able to determine values back to 1818, although the series has many missing days. In terms of monthly means, he derived values back to 1749, and in terms of yearly means, he determined values back to 1700. While monthly means of sunspot number have been estimated back to 1749, clearly, because of missing data, values prior to 1849 are considered less reliable than those since 1849.

A comparison of Wolfs relative sunspot number with an alternate series of sunspot number, called the 'Group sunspot number' (Hoyt and Schatten 1998a,b) found that the two descriptors of solar activity sometimes were in considerable disagreement (e.g., Hathaway, Wilson, and Reichmann 2002). In order to address these and other inconsistencies, a series of sunspot number workshops were initiated, culminating in a revision of the sunspot number database (Clette et al. 2015, Hathaway 2015). Consequently, the International sunspot number, effective July 1, 2015, was revised, correcting a number of past inhomogeneities in the sunspot number time series, with the most prominent change being the elimination of the k = 0.60 correction factor, which raises the scale of the entire sunspot number series.

This paper examines a number of the characteristics of sunspot cycles and the associations inferred to exist between them using the newly revised International 'smoothed monthly mean' sunspot number series. Smoothed monthly mean sunspot numbers, also called 12-month moving averages or 13-month running means, typically serve as the preferred parameter for describing the monthly variations in solar activity. Also, this study compares the current ongoing sunspot cycle (SC) 24's smoothed monthly mean sunspot numbers against those of earlier SCs and looks ahead towards the next sunspot cycle, SC25, expected to have its onset near the beginning of the next decade.

METHODS AND MATERIALS

As aforementioned, the basis for this study is the use of smoothed monthly mean sunspot numbers, where the smoothed monthly mean sunspot number is simply the average of two consecutive 12-month intervals, with the start of the second 12-month sequence shifted by 1 month from the start of the first 12-month sequence. Another way to look at it is that, for any sequence of 13 months, the smoothed monthly mean sunspot number for the centermost month is equal to twice the sum of the innermost 11 months plus the values at the beginning and end of the 13-month string with the quantity divided by 24. The smoothed monthly mean sunspot numbers for the interval July 1749 through June 2015 form the basis for this study and have been taken directly from the Royal Observatory of Belgium's Solar Influences Data Analysis Center's Web page, found online at <http://sidc.oma.be/silso/DATA/SN_ms_tot_V2.0.txt>.

On the basis of these revised smoothed monthly mean sunspot numbers, a number of descriptive cyclic parameters for SCO through SC24 are determined. Comparisons of the parametric values are then examined using scatterplots and standard linear and nonparametric statistical techniques.

Table 1 provides a summary of the general sunspot cycle characteristics for each SC, including the (1) epochs of minimum and maximum amplitude occurrences (Em and EM, respectively), (2) minimum and maximum amplitude (Rm and RM, respectively), (3) ascent duration, or elapsed time in months from Em to EM (i.e., ASC(m-M)), (4) descent duration, or elapsed time in months from EM SC(n) to Em SC(n+1) (i.e., DES(M-m)), (5) period, or duration in months of the SC (i.e., PER(m-m)), (6) sum of R values over the SC (i.e., Sum R(m-m)), (7) mean R for the SC (i.e., <R> = Sum R(m-m)/PER(m-m)), (8) average slope during the ascent (i.e., Slope (ASC)), and (9) average slope during the descent (i.e., Slope (DES)). Slope(ASC) is computed as (RM-Rm)/ASC(m-M), where all values are for SC(n), and Slope(DES) is computed as (Rm-RM)/(DES(M-m), where Rm is for SC(n+1) and the other values are for SC(n). Also given in the table are the parametric means and standard deviations (sd).

Table 2 provides two-cycle moving averages (2-cma) for the parameters given in Table 1. The purpose for computing the 2-cma values is to determine the local and overall trend in SC parametric values where this trend might possibly relate to the variation associated with the Hale cycle. The Hale cycle refers to the hemispheric magnetic polarity variation over two consecutive sunspot cycles. For any two consecutive sunspot cycles, the even-numbered sunspot cycle (like the current ongoing SC24) has northern hemispheric leading magnetic polarity that is predominantly negative (i.e., southward pointing, or having a magnetic vector pointing into the Sun) and following magnetic polarity that is predominantly positive (i.e., northward pointing, or having a magnetic vector pointing outward, away from the Sun), while it has southern hemispheric leading magnetic polarity that is predominantly positive and following magnetic polarity that is predominantly negative. For SC25 (the odd-numbered following SC), the hemispheric leading and following magnetic polarities will be reversed. Together, this behavioral variation is referred to as the Hale cycle (Hale 1924).

Figure 1 shows the cyclic variation of (a) Rm, (b) RM, (c) ASC(m-M), and (d) DES(M-m). In this and the following two figures, the thin jagged line is the cyclic parametric value, and the thick less-jagged line is the 2-cma parametric value. Rm is found to vary between 0 (SC6) and 18.6 (SC2), having a mean of 9.3 and sd= 5.7; i?Mis found to vary between 81.2 (SC6) and 285.0 (SC19), having a mean of 178.7 and sd=57.8; ASC{m-M) is found to vary between 35 (SC3) and 82 (SC5) months, having a mean of 52.3 months and sd= 13.5 months; and DES(M-m) is found to vary between 48 (SC7) and 122 (SC4) months, having a mean of 79.8 months and sd-15.3 months. Instead, if one uses only the most reliably known cycles (i.e., SC10-SC24, often called the 'modern era' of sunspot cycles), the means (and sd) measure, respectively, 8.5 (4.7), 180.3 (50.9), 49.3 (8.2), and 82.9 (9.0). The behavior of the 2-cma line suggests that the parametric variations, especially for Rm and RM, appear to be episodic, with the next few-to-several sunspot cycles perhaps having smaller amplitudes (and possibly longer ascent and shorter descent durations), as compared to the respective means. For SC24, its Rm (=2.2) is the third smallest overall and the smallest during the modern era, while its RM(= 116.4) is the fourth smallest overall and the second smallest during the modern era; its ascent duration (=64 months) is the fifth longest overall and the longest during the modern era (1 month longer than occurred in SC23).

Figure 2 displays the cyclic variation of (a) PER(m-m), (b) Sum and (c) <R>. PER(m-m) has values spanning 108 (SC2) to 163 (SC4) months, having a mean of 132.4 months and sd= 14.2 months; Sum R(m-m) has values spanning 4,700.9 (SC6) to 16,813.8 (SC4), having a mean of 11,055.2 and sd= 3,203.0; and <R> has values spanning 30.5 (SC6) to 129.2 (SC19), having a mean of 84.6 and sd=25.6. Using only the modern era SC, the means (and sd) are, respectively, 131.1 (9.4) months, 11,340.9 (2,633.8), and 87.1 (22.2). Interestingly, while the sunspot cycle often is described as being 11 years (132 months) in length, close inspection reveals that cycles have always had PER{m-m) either shorter than 127 months or longer than 134 months, without exception (Wilson 1987a). Shorter period cycles (11) average about 120 months in length (sd=6.1 months) and longer period cycles (12) average about 144 months in length (sd= 8.8 months). As with Figure 1, these particular parameters appear to show episodic behavior, with perhaps longer PER(m-m) and smaller Sum R{m-m) and <R> expected for the next few-to-several sunspot cycles.

Figure 3 depicts the cyclic variation of (a) Slope(ASC) and (b) Slope (DES). Slope (ASC) is found to span 0.935 (SC4) to 7.209 (SC3), having a mean of 3.619 and sd=1.759, while Slope(DES) is found to span -0.965 (SC6) to -3.268 (SC3), having a mean of-2.163 and sd=0.625. Using only the modern era SC, the means (and sd) are, respectively, 3.665 (1.426) and -2.138 (0.585). SC24 has the fifth smallest Slope(ASC) overall and the smallest Slope(ASC) during the modern era. As with the preceding charts, these particular parameters appear to show episodic behavior, with Slope (ASC) values possibly being below average and Slope (DES) values above average for the next few-to-several sunspot cycles.

Figure 4 shows the scatterplots of RM versus (a) Rm and (b) ASC(m-M). In this chart and succeeding charts the numbered filled circles identify the specific SC. For RM versus Rm, one infers that a preferential association exists between the two parameters, such that large (small) Rm usually is indicative of large (small) RM and vice versa (Wilson 1984, Wilson, Hathaway, and Reichmann 1996a, Wilson, Hathaway, and Reichmann 1998, Hathaway, Wilson, and Reichmann 1999). Of course, exceptions occur as noted for SCI, SC 10, SC 17, SC19, SC20, and SC23. Based on linear regression analysis, the inferred regression is y=126.379+5.637x, where y is RM and x is Rm, having a coefficient of correlation r=0.557, a coefficient of determination [r.sup.2]=0.310 (meaning that 31% of the variance can be explained by the inferred regression), a standard error of estimate se=49.063, and a t statistic equal to 3.143, inferring that the regression is statistically important ([sigma]= 1%). On the basis of nonparametric statistical tests (Gibbons 1993, Everitt 1977) (i.e., Spearman [rho]s, Kendall [tau] or [[tau].sub.b], and the results of Fisher's exact test for the observed 2x2 contingency table determined by the parametric medians, the thin vertical and horizontal lines), the inferred association between RM and Rm is statistically important at [sigma]=2%. Based on the inferred regression, SC24's Rm=2.2 would have suggested RM=138.8 [+ or -] 49.1 (the [+ or -] 1 se prediction interval). The actual value for SC24 measured 116.4, some 22.4 units smaller than the predicted value, but within the [+ or -] 1 se prediction interval.

A stronger inverse association is found between RM and ASC(m-M). This association is inferred to be highly statistically important at [sigma] < 0.1% (Wilson 1988, Wilson, Hathaway, and Reichmann 1996b). Only SC9, SC 10, and SC 13 fail to adhere strictly to the paradigm of faster (slower) rising SC being indicative of larger (smaller) maximum amplitude and vice versa. Had one assumed that SC24 would be a slow-rising cycle (i.e., ASC(m-M) [greater than or equal to] 49 months), one would have predicted its RM <183.3. Had one used the estimate of RM-139 for SC24 (from the RM versus Rm association, given above), one would have predicted ASC(m-M)=64 months for SC24, which turned out to be true.

Figure 5 displays the scatterplots of (a) PER(m-m) versus RM for the same cycle n and (b) RM (cycle n+1) versus PER(m-m) for cycle n. Regarding the former scatterplot, using all SC, one finds no statistically important association to exist between the parameters. However, if one omits SC2 and SC4 (the extremes of PER{m-m)), then one infers a possibly statistically important association. The RM for SC24 (=116.4) is marked along the abscissa. Clearly, a strong hint exists that SC24 will have PER(m-m) equal to or longer than 135 months (the median, shown as the horizontal line), since 8 of 11 SC having RM < 186.2 (the median, shown as the vertical line) have been cycles of longer period. Using the inferred linear regression, one predicts SC24 will have PER(m-m) = 139 [+ or -] 11 months, the [+ or -]1 se prediction interval.

Regarding the latter scatterplot, clearly, once the PER(m-m) becomes known for SC24, this appears to provide an early estimate for the expected size of the next sunspot cycle, SC25 (Hathaway, Wilson, and Reichmann 1994). Presuming SC24 is indeed a long-period sunspot cycle, one expects RM 186.2 for SC25. Presuming SC24 has PER(m-m)= 139 months (from above), one predicts RM 161 [+ or -] 2, which suggests that SC25 likely will be larger than SC24. (The inferred linear regression, as well as the [r.sub.s] and [[tau].sub.b], all suggest that the inferred association between RM (cycle n+1) and PER(m-m) (cycle n) is statistically important.)

Figure 6 depicts the scatterplots against RM of (a) Sum R(m-m) and (b) <R>. Both associations are inferred to be highly statistically important ([sigma] < 0.1%). Based on SC24's RM (=116.4), one expects both Sum R(m-m) and <R> to lie within the lower-left quadrant of each scatterplot, near the inferred regression line. Hence, one expects Sum R(m-m) and <R> to be about 7,750.7 [+ or -] 1,344.4 and 56.9 [+ or -] 7.3, respectively (the [+ or -] 1 se prediction interval) for SC24. Through June 2015, Sum R(m-m) = 5,266.4 and <R> = 67.5 (<R> will continue to decrease as the cycle progresses).

Figure 7 shows the scatterplot of Slope (DES) versus Slope (ASC). The association is inferred to be highly statistically important. Based on SC24's Slope (ASC) (=1.784), one expects its Slope (DES) to lie within the upper-left quadrant of the scatterplot, near the inferred regression line. Hence, one expects Slope (DES) = -1.587 [+ or -] 0.383 for SC24. Presuming SC24 is indeed a long-period cycle (i.e., PER(m-m) [greater than or equal to] 135 months), one expects SC24's DES(M-m) [greater than or equal to] 71 months, inferring that Em for SC25 should not be expected until March 2020 or later. Presuming Rm = 9.3 (the mean of SC1 through SC24) for SC25 and Slope (DES) = -1.204 (the lower extreme of the [+ or -] 1 se prediction interval), one computes DES(M-m) = 89 months for SC24, implying PER(m-m)= 153 months for SC24, slightly longer than the 148 months experienced for SC23. Consequently, Em for SC25 would occur later near August 2021, if true.

Figure 8 contrasts cycle maxima based on RM(odd-following) versus RM(even-leading) sunspot cycles, the so-called Even-Odd Effect (or Gnevyshev Effect; Gnevyshev and Ohl 1948, Hathaway, Wilson, and Reichmann 2002). Excluding cycle pair SC4/SC5 (clearly, a statistical outlier with respect to the Even-Odd Effect), one infers a statistically important linear association to exist between even- and odd-numbered sunspot cycles, such that the odd-following cycle tends to be larger than the preceding even-leading cycle, with 8 of 11 cycle pairs adhering strictly to this pattern. Shown in the figure are the 1:1 line and the inferred regression line. Based on the observed RM for SC24 (=116.4), one estimates RM= 166.1 [+ or -] 40.3 for SC25, presuming of course that cycle pair SC24/SC25 obeys the Even-Odd Effect and is not a statistical outlier like SC4/SC5.

Figure 9 shows the scatterplot of PER(m-m) versus ASC(m-M). Noticeable is that there seems to be an 8-month gap in the distribution of PER(m-m), with cycles having periods either longer than 134 months (12 cycles) or shorter than 127 months (11 cycles). The gap centers on the mean cycle length of 132 months. Based on the Fisher's exact test for the observed 2x2 contingency table (determined using the parametric medians), one finds that the probability of obtaining the observed result, or one more suggestive of a departure from independence (chance), is computed to be P=3%. Hence, fast-rising cycles tend to be cycles of shorter period (denoted here as FS, 8 of 11 cycles) and slow-rising cycles tend to be cycles of longer period (denoted here as SL, 9 of 12 cycles). The other two cycle groupings are designated here as FL (i.e., fast-rising, long-period cycles, 3 of 11 cycles) and SS (i.e., slow-rising, short-period cycles, 3 of 12 cycles). Because SC24 is known to be a slow-rising SC, one expects that it likely will also be a cycle of longer period (i.e., it likely should be designated as an SL cycle).

Figure 10 depicts the variation in R for the interval January 2006 through June 2015, both as (a) the combined hemispheric sunspot number and (b) separately as north (N) and south (S) hemispheric sunspot numbers. Obviously, SC24 is double-peaked with the initial peak of 98.3 occurring in March 2012 and the main peak of 116.4 occurring in April 2014. R values now are in steady decline, although sunspot cycles often have several bumps in activity during the decline of a cycle. The initial peak is driven primarily by solar activity that occurred in the northern hemisphere, which actually peaked earlier in August-September 2011 (RM(N) = 60.1), whereas the main peak is driven primarily by solar activity that occurred in the southern hemisphere, which peaked in April 2014 (RM(S) = 80.4). (The northern minimum Rm(N) = 0.6 occurred in December 2007-March 2008, while the southern minimum Rm(S) = 0.8 occurred in December 2008. Hemispheric R is available only since January 1992.)

Figure 11 displays the latitudinal locations of sunspots using the modified 'butterfly diagram' for the interval January 2006 through December 2010, an interval that encompasses sunspot minimum for SC24. The purpose of this chart is to show that old and new cycle sunspots typically overlap each other in the vicinity of sunspot minimum by 1-3 years (Howard 1977, Wilson 1987b). The first spots of a new cycle typically form at higher latitudes ([greater than or equal to] 25 deg), while old cycle spots typically are found at lower latitudes closer to the equator near the end of a sunspot cycle. The first spots associated with SC24 certainly are apparent beginning in January 2008, while the last of the old spots for SC23 appears to have ended in April 2009, suggesting an overlap of about 16 months. (The usual butterfly diagram is plotted showing both northern and southern hemispheres separately. The modified butterfly diagram merely combines the two hemispheres into one, making it easier to see the overlap and the occurrence of higher latitude spots. Strictly speaking, for a spot to be considered a new or old cycle spot, it should have the appropriate magnetic configuration of leading and following spots, which can be determined from magnetograph observations.)

Figure 12 shows the variation of R for the interval 12 months prior to sunspot minimum to the end of the cycle for specific groupings of cycles: (a) FS and FL cycles and (b) SS and SL cycles. Also plotted is the variation of R for SC24 (the filled circles) in Figure 12(b). Noticeable is that SC24 is a cycle that is smaller overall than either of the means for SS and SL cycles. While the peak of SC24 (at t = 64 months) occurs at the same time as the peak for SL cycles, its RM is some 14.8 units smaller than the mean SL cycle (116.4 versus 131.2). For t = 78 months (June 2015), the difference has grown to more than 30 units. (The nine SL cycles have PER(mm) values that range between 135 and 154 months, suggesting that onset for SC25 should not be expected until about March 2020 to October 2021, presuming SC24 is indeed an SL cycle.)

In conclusion, this study has examined sunspot cycle characteristics using the newly revised sunspot numbers and the inferred associations between them. It appears that the current ongoing sunspot cycle SC24 might be designated a slow-rising, long-period cycle (i.e., SL). In terms of its minimum amplitude (Rm = 2.2), it is ranked as the smallest in the modern era of sunspot cycles (SC10-SC24). Its maximum amplitude (RM = 116.4) is ranked as the second smallest, and its ascent duration (ASC(m-M) = 64 months) is the longest (1 month longer than occurred in SC23) in the modern era. SC25, the next sunspot cycle, is not expected to have its onset until March 2020 or later (probably sometime between March 2020 and October 2021). Based on the Even-Odd Effect, one anticipates SC25 to have an RM larger than was observed for SC24 (=116.4).

LITERATURE CITED

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Robert M. Wilson

NASA Marshall Space Flight Center, NSSTC, Huntsville, Alabama

Corresponding Author: Robert Wilson (robert.m.wilson@nasa.gov)

RESULTS AND DISCUSSION

Table 1. Summary of sunspot cycle parametric characteristics

Cycle     Em       Rm       EM        RM     ASC (M-m)   DES (M-m)

0         --        --    1750-04   154.3       --          59
1       1755-03   14.0    1761-06   144.1       75          60
2       1766-06   18.6    1769-09   193.0       39          69
3       1775-06   12.0    1778-05   264.3       35          76
4       1784-09   15.9    1788-02   235.3       41          122
5       1798-04    5.3    1805-02    82.0       82          65
6       1810-07     --    1816-05    81.2       70          84
7       1823-05    0.1    1829-11   119.2       78          48
8       1833-11   12.2    1837-03   244.9       40          76
9       1843-07   17.6    1848-02   219.9       55          94
10      1855-12    6.0    1860-02   186.2       50          85
11      1867-03    9.9    1870-08   234.0       41          100
12      1878-12    3.7    1883-12   124.4       60          75
13      1890-03    8.3    1894-01   146.5       46          96
14      1902-01    4.5    1906-02   107.1       49          89
15      1913-07    2.5    1917-08   175.7       49          72
16      1923-08    9.4    1928-04   130.2       56          65
17      1933-09    5.8    1937-04   198.6       43          82
18      1944-02   12.9    1947-05   218.7       39          83
19      1954-04    5.1    1958-03   285.0       47          79
20      1964-10   14.3    1968-11   156.6       49          88
21      1976-03   17.8    1979-12   232.9       45          81
22      1986-09   13.5    1989-11   212.5       38          81
23      1996-08   11.2    2001-11   180.3       63          85
24 *    2008-12    2.2    2014-04   116.4       64          --
mean               9.3              178.7      52.3        79.8
sd                 5.7               57.8      13.5        15.3

Cycle   PER (m-m)   Sum R (m-m)    <R>    Slope (ASC)   Slope (DES)

0          --               --      --        --          -2.378
1          135         9,414.8    69.7       1.735        -2.092
2          108        10,726.8    99.3       4.472        -2.623
3          111        12,352.5    111.3      7.209        -3.268
4          163        16,813.8    103.2      5.351        -1.885
5          147         5,701.9    38.8       0.935        -1.262
6          154         4,700.9    30.5       1.160        -0.965
7          126         7,947.3    63.1       1.527        -2.229
8          116        13,014.0    112.2      5.818        -2.991
9          149        14,825.1    99.5       3.678        -2.276
10         135        12,452.9    92.2       3.604        -2.074
11         141        12,532.2    88.9       5.466        -2.303
12         135         7,653.2    56.7       2.012        -1.548
13         142         9,231.2    65.0       3.004        -1.479
14         138         7,429.6    53.8       2.094        -1.175
15         121         8,877.1    73.4       3.535        -2.310
16         121         8,251.4    68.2       2.157        -1.914
17         125        11,992.9    95.9       4.484        -2.265
18         122        13,297.3    109.0      5.277        -2.573
19         126        16,276.6    129.2      5.955        -3.427
20         137        11,909.0    86.9       2.904        -1.577
21         126        14,020.7    111.3      4.780        -2.709
22         119        12,642.9    106.2      5.237        -2.485
23         148        12,205.6    82.5       2.684        -2.095
24 *       --               --      --       1.784          --
mean      132.4       11,055.2    84.6       3.619        -2.163
sd        14.2         3,203.0    25.6       1.759         0.625

* Means SC24 is still ongoing.

Table 2. Two-cycle moving averages of selected sunspot cycle parameters

Cycle    Rm       RM     ASC (m-M)    DES (M-m)   PER (m-m)

1         --    158.9           --        62.0          --
2       15.8    198.6         47.0        68.5       115.5
3       14.6    239.2         37.5        85.8       123.3
4       12.3    204.2         49.8        96.3       146.0
5        6.6    120.1         68.8        84.0       152.8
6        1.4     90.9         75.0        70.3       145.3
7        3.1    141.1         66.5        64.0       130.5
8       10.5    207.2         53.3        73.5       126.8
9       13.4    217.7         50.0        87.3       137.3
10       9.9    206.6         49.0        91.0       140.0
11       7.4    194.7         48.0        90.0       138.0
12       6.4    157.3         51.8        86.5       138.3
13       6.2    131.1         50.3        89.0       139.3
14       5.0    134.1         48.3        86.5       134.8
15       4.7    147.2         50.8        74.5       125.3
16       6.8    158.7         51.0        71.0       122.0
17       8.5    186.5         45.3        78.0       123.3
18       9.2    230.3         42.0        81.8       123.8
19       9.4    236.3         45.5        82.3       127.8
20      12.9    207.8         47.5        84.0       131.5
21      15.9    208.7         44.3        82.8       127.0
22      14.0    209.6         46.0        82.0       128.0
23 *     9.5    172.4         57.0          --          --

Cycle   SumR (m-m)    <R>     Slope (ASC)   Slope (DES)

1              --       --            --         -2.296
2        10,805.2     94.9         4.472         -2.652
3        13,061.4    106.3         6.060         -2.761
4        12,920.5     89.1         4.712         -2.075
5         8,229.6     52.8         2.095         -1.344
6         5,762.8     40.7         1,196         -1.355
7         8,402.4     67.2         2.508         -2.104
8        12,200.1     96.8         4.210         -2.622
9        13,779.3    100.9         4.195         -2.404
10       13,065.8     93.2         4.088         -2.182
11       11,292.6     81.7         4.137         -2.057
12        9,267.5     66.8         3.124         -1.720
13        8,386.3     60.1         2.529         -1.420
14        8,241.9     61.5         2.682         -1.535
15        8,358.8     67.2         2.830         -1.927
16        9,343.2     76.4         3.083         -2.101
17       11,383.6     92.3         4.101         -2.254
18       13,716.0    110.8         5.248         -2.710
19       14,439.9    113.6         5.023         -2.751
20       13,528.8    103.6         4.136         -2.323
21       13,148.3    103.9         4.425         -2.370
22       12,878.0    101.6         4.485         -2.444
23 *           --       --         3.097            --

* Means that SC23 2-cma values are incomplete.
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Author:Wilson, Robert M.
Publication:Journal of the Alabama Academy of Science
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Date:Jul 1, 2015
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