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i

NASA/TP—2009–216061

Sunspot Activity Near Cycle Minimum 

and What It Might Suggest for Cycle 24, 

the Next Sunspot Cycle

Robert M. Wilson and David H. Hathaway

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

September 2009

National Aeronautics and

Space Administration

Marshall Space Flight Center • MSFC, Alabama 35812
ii

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076 –1320

443–757–5802

This report is also available in electronic form at

<https://www2.sti.nasa.gov>
iii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................ 1

2. RESULTS AND DISCUSSION ........................................................................................ 3

2.1 Parametric Variations Near Cycle Minimum ............................................................... 3

2.2 Parametric variations near cycle maximum .............................................................. 24

2.3 Parametric Comparisons and Correlations .................................................................. 31

3. SUMMARY ...................................................................................................................... 41

REFERENCES ...................................................................................................................... 43
iv
v

LIST OF FIGURES

1. Cyclic variation of (a) SM, (b) Rm, (c) Gm, and (d) Am for cycles 10–24 ................. 4

2. Cyclic variation of (a) P(SM–SM) and (b) P(Rm–Rm) for cycles 10–23 .................... 5

3. Cyclic variation of (a) P(Gm–Gm) and (b) P(Am–Am) for cycles 12–23 .................... 6

4. Monthly variation of (a) latitudinal distribution of spots, (b) L, (c) H, 

and (d) g(h)/g for January 1994–November 2008 ...................................................... 9

5. Monthly variation of (a) S, (b) R, (c) G, and (d) A for January 1994–

 November 2008 ........................................................................................................ 10

6. Monthly variation of A/G for January 1994–November 2008 ................................... 12

7. Cyclic variation of (a) (A/G)m, (b) (A/G)M, and (c) Ratio: 

(A/G)M/(A/G)m for cycles 12–24 ............................................................................... 13

8. Cyclic variation of (a)Lm, (b) LM, (c) Hm, and (d) HM for cycles 12–24 ................. 14

9. Cyclic variation of (a) t(Lm–Rm) and (b) t(Hm–Rm) for cycles 12–24 ....................... 15

10. Cyclic variation of (a) t(Rm–LM) and (b) t(Rm–HM) for cycles 12–23 ..................... 16

11. Cyclic variation of (a) P(Lm–Lm), (b) P(LM–LM), (c) P(Hm–Hm), 

(d) P(HM–HM), and (e) P(0.5–0.5) for cycles 12–23 ................................................. 17

12. Cycle-by-cycle latitudinal distribution of spots for cycles 12–24 for elapsed time 

in months from 36 mo prior to E(Rm) to 24 mo after E(Rm) .................................... 19

13. Cyclic variation of t(FHLS–LLLS) for cycles 12–23 ................................................. 20

14. Cyclic variation of t(FHLS–Rm) for cycles 12–24 ..................................................... 21

15. Cyclic variation of (a) P(FSD–FSD) and (b) P(LSD–LSD) for cycles 10–23 ............. 22

16. Cyclic variation of (a) t(FSD–Rm), (b) t(Rm–LSD), and (c) t(FSD–LSD) 

for cycles 10–24 ......................................................................................................... 23

17. Cyclic variation of (a) RM, (b) GM, and (c) AM for cycles 10–23 ............................. 29
vi

LIST OF FIGURES (Continued)

18. Cyclic variation of (a) P(RM–RM), (b) P(GM–GM), and (c) P(AM–AM) 

for cycles 10–22 ......................................................................................................... 30

19. Cyclic variation of (a) t(GM–RM) and (b) t(AM–RM) for cycles 12–23 .................... 31

20. Cyclic variation of (a) t(LM–RM) and (b) t(HM–RM) for cycles 12–23 .................... 32

21. Cyclic variation of t(Rm-RM)=ASC for cycles 10–23 ............................................... 33

22. Scatter plot of RM versus ASC (the Waldmeier effect) ............................................. 34

23. Scatter plots of (a) RM (cycle n) versus P(Rm–Rm) (cycle n–1), (b) RM

(cycle n) versus P(Gm–Gm) (cycle n–1), (c) RM (cycle n) versus P(Am–Am) 

(cycle n–1), and (d) RM (cycle n) versus P(SM–SM) (cycle n–1) 

(the amplitude-period effect) .................................................................................... 35

24. Scatter plots of (a) RM versus Rm, (b) RM versus Gm, and (c) RM versus Am

(the maximum-minimum effect) ................................................................................ 37

25. Scatter plots of (a) RM versus HM, (b) RM versus GM, and (c) RM versus AM ..... 38

26. Values of H: (a) Scatter plot of HM versus Hm, and (b) superposed 

epoch analysis of H for elapsed time 0–30 mo from E(Rm), giving mean 

of cycles 12–23 and H values for cycles 14 and 19, the smallest and largest 

modern era cycles, respectively .................................................................................. 40
vii

LIST OF TABLES

1. Epochs of occurrence for selected solar cyclic parameters .......................................... 25

2. Values of selected solar cyclic parameters .................................................................. 26

3. Periods for selected solar cyclic parameters ................................................................ 27

4. Elapsed time in months relative to E(Rm) for selected solar cyclic parameters ........... 28
viii
ix

LIST OF ACRONYMS

12-mma 12-month moving average

ASC ascent duration in months, equal to t(Rm–RM)

MSFC Marshall Space Flight Center 

N North latitude

NASA National Aeronautics and Space Administration

NOAA National Oceanic and Atmospheric Administration

RGO Royal Greenwich Observatory

S South latitude

SOON Solar Optical Observing Network

USAF United States Air Force
x

NOMENCLATURE

A 12-mma of monthly mean sunspot area

(A/G)m minimum value of the 12-mma of A/G

(A/G)M maximum value of the 12-mma of A/G

Am minimum value of the 12-mma of A

AM maximum value of the 12-mma of A

cl confidence level

E((A/G)m) epoch of occurrence of (A/G)m

E((A/G)M) epoch of occurrence of (A/G)M

E(Am) epoch of occurrence of Am

E(AM) epoch of occurrence of AM

E(FHLS) epoch of occurrence of FHLS

E(Gm) epoch of occurrence of Gm

E(GM) epoch of occurrence of GM

E(Hm) epoch of occurrence of Hm

E(HM) epoch of occurrence of HM

E(Lm) epoch of occurrence of Lm

E(LM) epoch of occurrence of LM

E(Rm) epoch of occurrence of Rm

E(RM) epoch of occurrence of RM

E(SM) epoch of occurrence of SM
xi

NOMENCLATURE (Continued)

f number of individual sunspots

FHLS month of first high-latitude spot occurrence

FSD month of first spotless day occurrence

g number of groups

G 12-mma of monthly mean number of groups

g(h)/g ratio of monthly mean number of high-latitude groups to monthly 

mean number of groups

Gm minimum value of the 12-mma of G

GM maximum value of the 12-mma of G

H 12-mma of highest latitude spot

Hm minimum value of the 12-mma of H

HM maximum value of the 12-mma of H

k correction factor

L 12-mma of weighted mean latitude of sunspots

LLLS month of last low-latitude spot occurrence

Lm minimum value of the 12-mma of L

LM maximum value of the 12-mma of L

LSD month of last spotless day occurrence

n sunspot cycle number

P probability

P(0.5–0.5) period in months between occurrences of g(h)/g >0.5 of successive cycles

P(Am–Am) period in months between Am occurrences of successive cycles
xii

NOMENCLATURE (Continued)

P(AM–AM) period in months between AM occurrences of successive cycles

P(FSD–FSD) period in months between FSD occurrences of successive cycles

P(Gm–Gm) period in months between Gm occurrences of successive cycles

P(GM–GM) period in months between GM occurrences of successive cycles

P(Hm–Hm) period in months between Hm occurrences of successive cycles

P(HM–HM) period in months between HM occurrences of successive cycles

P(Lm–Lm) period in months between Lm occurrences of successive cycles

P(LM–LM) period in months between LM occurrences of successive cycles

P(LSD–LSD) period in months between LSD occurrences of successive cycles

P(Rm–Rm) period in months between Rm occurrences of successive cycles

r coefficient of correlation; relative sunspot number

r

2 coefficient of determination (a measure of the amount of variance 

explained by the inferred regression)

R 12-mma of monthly mean sunspot number

Rm minimum value of the 12-mma of R

RM maximum value of the 12-mma of R

S 12-mma of monthly mean number of spotless days

se standard error of estimate

sd standard deviation (also given as σ)

SM maximum value of the 12-mma of S

t(AM–RM) elapsed time in months between AM and RM

t(FHLS–LLLS) elapsed time in months between FHLS and LLLS
xiii

NOMENCLATURE (Continued)

t(FHLS-Rm) elapsed time in months between FHLS and Rm

t(FSD–LSD) elapsed time in months between FSD and LSD

t(FSD–Rm) elapsed time in months between FSD and Rm

t(GM–RM) elapsed time in months between GM and RM

t(Hm–Rm) elapsed time in months between Hm and Rm

t(HM–RM) elapsed time in months between HM and RM

t(Lm–Rm) elapsed time in months between Lm and Rm

t(LM–RM) elapsed time in months between LM and RM

t(Rm–HM) elapsed time in months between Rm and HM

t(Rm–LM) elapsed time in months between Rm and LM

t(Rm–LSD) elapsed time in months between Rm and LSD

t(Rm–RM) elapsed time in months between Rm and RM

x independent variable

x* modified independent variable

y dependent variable

y* modified dependent variable

σ standard deviation
xiv
1

TECHNICAL PUBLICATION

Sunspot Activity Near Cycle Minimum and What It Might Suggest

for Cycle 24, the Next Sunspot Cycle

1. INTRODUCTION

About 165 yr ago, Samuel Heinrich Schwabe1

 announced that sunspot activity fluctuated 

with a period of about 10 yr. And so the concept of the sunspot cycle was born. Schwabe determined 

his finding by counting annually the number of spotless days and the number of ‘clusters of spots’ 

(groups) that he observed daily from Dessau, Germany. According to Kiepenheur,2 Schwabe’s result 

did not receive general attention until several years later when Friedrich Wilhelm Alexander Freiherr 

von Humboldt mentioned the finding in volume 3 of his 5-volume series Kosmos, published between 

1845 and 1862 (Humboldt died in 1859). Schwabe, an apothecary by trade, made daily determinations of sunspots for 43 yr, between 1826 and 1868, averaging about 290 observing days per year.3

Schwabe was awarded the Gold Medal of the Royal Astronomical Society for his achievement in 

1857, and in 1868 he was elected a member of the Royal Society.4,5 He died in 1875 at the age of 

85 yr.

Johann Rudolph Wolf, a professional Swiss astronomer, having become aware of Schwabe’s 

discovery, introduced in 1848 his now familiar ‘relative sunspot number,’ r=k (10g+f), where g is the 

number of sunspot groups, f is the number of individually observed sunspots, and k is a correction 

factor (originally having a value of 1 as used by Wolf), dependent upon the observer, observing conditions, size of telescope, etc. Wolf also established an international cadre of observers, which continues through today, to monitor the daily changes in r. By piecing together scattered and incomplete 

older sunspot data, Wolf was able to reconstruct a record of solar activity prior to 1849 (back to 

1700 in terms of yearly average estimates, and back to 1610 in terms of estimated epochs of sunspot 

minimum and maximum), determining the average length of the sunspot cycle to be about 11 yr, 

rather than 10 yr. The record prior to 1849, however, must be viewed circumspectly, owing to incomplete daily records. 2–8 About 1882, Wolf’s successors at Zürich permanently changed the counting 

method to include the smallest observable spots and weighted spots with penumbrae according to 

size and umbral structure, employing k=0.60 for the reduction of the new records to the values 

obtained by Wolf.

In 1874, the Royal Greenwich Observatory (RGO) began routinely (daily) observing the Sun 

photographically (first introduced by Richard Christopher Carrington), measuring the position of 

sunspots on the visible solar disk and the sunspot area, both total and umbral areas. (umbral area 

refers to the darkest portion of sunspots, whereas total area refers to the overall area of sunspots, 

including umbral and penumbral areas, as seen in white light.9

) The RGO, supplemented by observations from Capetown, South Africa, and Kodaikanal, India, continued to record sunspots photographically through 1976, when it concluded its observations. From 1977 onwards, the RGO dataset 
2

has been extended using visual observations as reported in the National Oceanic and Atmospheric 

Administration (NOAA) Space Weather Prediction Center’s Preliminary Report and Forecast of 

Solar Geophysical Data (now called “The Weekly”). The visual observations originate from the 

United States Air Force (USAF) Solar Optical Observing Network (SOON). The augmented RGO 

dataset is easily accessible online at the National Aeronautics and Space Administration (NASA) 

Marshall Space Flight Center (MSFC) solar science website <http://solarscience.msfc.nasa.gov/

greenwch.shtml>. Sunspot areas since 1976 have been increased by 40% to compensate for the difference in photographic and visual determinations of sunspot area.10,11

Since 1981, the task of computing and maintaining the official record of relative sunspot 

numbers has been accomplished by the Royal Observatory of Belgium. The record of reliable continuous sunspot observations now spans 160 yr, from January 1849 to the present, although comparison with Douglas V. Hoyt and Kenneth H. Schatten’s group sunspot number12 suggests reliability 

only since about 1882; i.e., from 1882 the two datasets are virtually identical, but differ, sometimes 

substantially, prior to 1882. Also, the record of reliable continuous sunspot positions and areas now 

spans 125 yr, from May 1874 to the present.

The purpose of this NASA Technical Publication (TP) is to examine more closely a number 

of solar parameters gleaned from the sunspot record, especially how they might relate to cycles 23 

and 24, the just ending and beginning sunspot cycles, since some of the parameters are now very near 

record value, especially as compared to other cycles of the modern era (since about cycle 10).
3

2. RESULTS AND DISCUSSION

2.1 Parametric Variations Near Cycle Minimum

Figure 1 displays the cyclic variation of the 12-mo moving averages (12-mma; also called the 

13-mo running mean or smoothed monthly mean values) of the maximum monthly mean number 

of spotless days (SM, fig. 1(a)), the minimum value of the monthly mean sunspot number (Rm, fig. 

1(b)), the minimum value of the monthly mean number of sunspot groups (Gm, fig. 1(c)), and the 

minimum value of the monthly mean total sunspot area, corrected for foreshortening and expressed 

in millionths of the visible solar hemisphere (Am, fig. 1(d)). For SM and Rm the values span cycles 

10–24, with tentative values given for cycle 24 (i.e., presuming November 2008 to be the epoch of 

cycle minimum for cycle 24), while for Gm and Am the values span cycles 12–24, with tentative values 

given for cycle 24, as well. Also shown are the medians for each parameter. Hence, for SM, Rm, Gm, 

and Am, respectively, the medians are 19.2, 5.1, 0.56 and 50.9. For SM, its tentative value for cycle 

24 is now above the median, while for Rm, Gm, and Am, the tentative values for cycle 24 are below 

their respective medians; i.e., SM varies inversely with Rm, Gm, and Am. The purpose of the small 

arrows for the cycle 24 values is to remind the reader that the values are tentative, presuming cycle 24 

minimum in November 2008, and therefore could slightly change as time progresses.

For cycle 24, the tentative value of SM (24.2) is the fourth highest, below that of cycles 12 

(25.0), 14 (24.3), and 15 (26.2). Similarly, the tentative value of Rm (1.8) for cycle 24 is the second 

smallest, only slightly higher than the value for cycle 15 (1.5); the tentative value of Gm (0.20) is tied 

with cycle 14 as the third smallest, above that of cycles 12 and 15 (both 0.15); and the tentative value 

of Am (7.4) is the second smallest, only slightly higher than the value for cycle 15 (6.9). Clearly, values of SM, Rm, Gm, and Am now being experienced near cycle minimum for cycle 24 are near record 

values based on the reliable record of sunspot activity, not having been seen in about 100 yr.

Now, SM, Gm, and Am occurrences are closely associated with Rm occurrence. For example, 

based on cycles 10–23, 6 of 14 cycles had SM occurring simultaneously with Rm, and 12 of 14 cycles 

had SM occurring within 2 mo (either side) of Rm. The only exceptions were cycles 21 and 22, with 

SM leading Rm by 3 and 6 mo, respectively. Similarly, based on cycles 12–23, 7 of 12 cycles had Gm

occurring simultaneously with Rm, and 9 of 12 cycles had Gm within 3 mo (either side) of Rm. The 

exceptions include cycles 13, 20, and 22, each leading Rm by 5, 4, and 6 mo, respectively. Finally, 

9 of 12 cycles had Am occurring simultaneously with Rm, and 11 of 12 had Am occurring within 

2 mo (either side) of Rm. The lone exception is cycle 21, which had Am lagging Rm by 5 mo. Presuming November 2008 to be cycle 24 minimum, all parameters appear to be reaching their extremes in 

value essentially simultaneously.

Figures 2 and 3 display the variation of the cyclic periods based on the parameters SM, 

Rm, Gm, and Am. As used here, period is defined as the elapsed time in months between successive occurrences of the parameters. Figure 2(a) shows the cyclic variation for the period using SM

(i.e., P(SM–SM)), figure 2(b) shows the cyclic variation for the period using Rm (i.e., P(Rm–Rm)), 
4

10 12 14 16 18 20 22 24

100

150

50

(d)Am

Sunspot Cycle Number

Median

(50.9)

1.0

1.5

0.5

(c)Gm

Median

(0.56)

10

15

5

(b)Rm

Median

(5.1)

10

15

20

25

30

5

(a)SM

Median

(19.2)

Figure 1. Cyclic variation of (a) SM, (b) Rm, (c) Gm, and (d) Am for cycles 10–24.

figure 3(a) shows the cyclic variation for the period using Gm (i.e., P(Gm–Gm)), and figure 3(b) shows 

the cyclic variation for the period using Am (i.e., P(Am–Am)). The horizontal line in each panel is 

the parametric mean. The dashed lines represent class extremes about the means, presuming the existence of two subgroupings— those cycles of shorter period than the mean and those of longer period 

than the mean. Wilson13 previously has suggested that sunspot cycles can be grouped according to
5

10 12 14 16 18 20 22 24

120

110

140

130

150

(b)

P(Rm – Rm)

Sunspot Cycle Number

Mean

(129.6)

120

110

140

130

150

(a)

P(SM – SM)

Mean

(130.7)

Figure 2. Cyclic variation of (a) P(SM–SM) and (b) P(Rm–Rm) for cycles 10–23.

period class when such cycles are described on the basis of 12-mma values, since the parametric 

mean clearly falls in the gap where no parametric values are found; i.e., there is no strong central clustering about the mean. Hence, while solar prognosticators often speak of an 11-yr average period for 

sunspot cycles, the record is better described as consisting of two classes of sunspot cycles: those of 

shorter period and those of longer period. (It should be noted, however, that using a longer smoothing function or a larger binning size negates the bimodal appearances as depicted here.)
6

10 12 14 16 18 20 22 24

120

110

140

130

150

(b)

P(Am – Am)

Sunspot Cycle Number

Mean

(128.1)

120

110

140

130

150

(a)

P(Gm – Gm)

Mean

(128.4)

Figure 3. Cyclic variation of (a) P(Gm–Gm) and (b) P(Am–Am) for cycles 12–23.

For P(SM–SM) and P(Rm–Rm), cycles 10–14, 20, and now 23 are all cycles of longer duration, while cycles 15–19, 21, and 22 are all cycles of shorter duration. Compared to the mean P(SM–

SM) using cycles 10–22 (130.7 mo), the tentative value (148 mo) for cycle 23’s P(SM–SM) is about 

1.8 standard deviations (sd) higher. Statistically speaking, there is only about a 5% chance that cycle 

23’s P(SM–SM) would extend 148 mo or more, inferring an epoch of SM date (E(SM)) for cycle 24 

on or before November 2008 (i.e., 148 mo from July 1996, E(SM) for cycle 23), and there is only a 

1% chance that P(SM–SM) would extend 158 mo or more, inferring E(SM) for cycle 24 on or before 

September 2009.
7

 Compared to the mean P(Rm–Rm) using cycles 10–22 (129.6 mo), the tentative value 

(150 mo) for cycle 23’s P(Rm–Rm) is about 2.2 sd higher. Only a 5% chance existed that cycle 23’s 

P(Rm–Rm) would extend 147 mo or more, which, clearly, it has already done, from cycle 23’s epoch 

of Rm date (E(Rm): May 1996). Only a 1% chance exists that cycle 23 would extend 155 mo or more, 

inferring E(Rm) for cycle 24 on or before April 2009.

Compared to the mean P(Gm–Gm) using cycles 12–22 (128.4 mo), the tentative value 

(147 mo) for cycle 23’s P(Gm–Gm) is about 1.9 sd higher. There is only about a 5% chance that cycle 

23’s P(Gm–Gm) would extend 147 mo or more from cycle 23’s epoch of Gm date (E(Gm): August 

1996), inferring E(Gm) for cycle 24 to be on or before November 2008, and there is only a 1% chance 

that it would extend 156 mo or more, inferring E(Gm) for cycle 24 on or before August 2009.

Last, compared to the mean P(Am–Am) using cycles 12–22 (128.4 mo), the tentative value 

(147 mo) for cycle 23’s P(Am–Am) is about 2.1 sd higher. Again, there was only about a 5% chance 

that cycle 23’s P(Am–Am) would extend 148 mo or more from cycle 23’s epoch of Am date (E(Am): 

May 1996), inferring E(Am) for cycle 24 to have been on or before September 2008, and there is only 

a 1% chance that it would extend 158 mo or more, inferring E(Am) for cycle 24 on or before July 

2009.

More recently, because new-cycle spots differentiate themselves from old-cycle spots by their 

magnetic configuration, where even-numbered sunspot cycles have positive leading polarity in the 

northern hemisphere (opposite in the southern hemisphere), and odd-numbered sunspot cycles have 

negative leading polarity in the northern hemisphere (opposite in the southern hemisphere), this 

characteristic is sometimes employed in the determination of sunspot minimum; i.e., the epoch date 

when the monthly number of new-cycle spots first dominates over old-cycle spots.14 Unfortunately, 

because only recent cycles have sufficient magnetic configuration determinations to be used in the 

determination of new cycle spots, some other measure must serve as a proxy for new-cycle spots. An 

alternative to using the magnetic configuration for the determination of new-cycle spots might be 

one using the latitudinal distribution of sunspots near cycle minimum.

The latitudinal distribution of sunspots changes over the sunspot cycle. This was first noticed 

by Carrington in 1858 and confirmed first by Wolf (also in 1858) and later by Gustav Spörer (in 

1867) and Father Pietro Angelo Secchi (in 1870).2,5 The first spots of a new sunspot cycle usually 

begin appearing at latitudes above 25–30 deg N or S, and the last spots of the old sunspot cycle usually appear near the Sun’s equator. The depiction of the resulting pattern of plotting the latitudinal 

position of sunspots against time takes on the appearance of butterfly wings; hence, the name ‘butterfly diagram’ is often used to describe it. Edward Walter Maunder was the first (in 1904) to show 

the full content and regularity of the pattern.15

Another way of depicting the latitudinal variation over a sunspot cycle is to fold the two 

hemispheres together.16,17 As an example, figure 4(a) displays such a plot for the interval encompassing cycle 23 (January 1994–June 2009). The plot clearly shows the first occurrence of new-cycle, highlatitude spots in May 1996 for cycle 23 and the last old-cycle, low-latitude (near the equator) spots 

around April 1998 for cycle 22, suggesting an overlap of the two cycles of about 2 yr. It also clearly 

defines the latitudinal envelope of spots over the entire cycle. Noticeable is the expansion of the 
8

latitudinal distribution of spots as the cycle progresses from sunspot minimum to maximum and the 

flattening and declining of the upper envelope of the distribution especially after sunspot maximum, 

with the lowest latitude new-cycle spot (0 deg) first occurring in October 2000. The highest latitude 

new-cycle spot (50 deg) for cycle 23 occurred in early July 2001, slightly more than 1 yr past sunspot 

maximum (the spot was at 49 deg latitude in late June 2001), although one at 48 deg occurred in 

March 1998, about 2 yr prior to sunspot maximum.

Also shown are the variations of the 12-mma of the weighted mean latitude of sunspots: 

Figure 4(b) shows L, weighted by sunspot area, and figure 4(c) shows the highest latitude spot, H. 

The minimum and maximum values of L and H and their occurrence dates are identified. Figure 

4(d) shows the variation of the ratio g(h)/g, where g(h) is the monthly mean number of high-latitude 

groups (i.e., those having latitudes ≥25 deg) and g is the monthly mean number of all sunspot groups. 

Hence, the ratio identifies when new-cycle, high-latitude groups first become dominant contributors 

to solar activity (i.e., when g(h)/g >0.5), which for cycle 23 occurred in June 1997, about 1 yr past 

sunspot minimum.

For cycle 23, which had Rm (8.0) in May 1996 and RM (120.8) in April 2000, Lm (7.8 deg) 

occurred in December 1995 and LM (23.1 deg) occurred in February 1998. Hm (16.1 deg) occurred in 

October 1995 and HM (38.4 deg) occurred in December 1999. As already mentioned, the ratio g(h)/g

was first greater than 0.5 in June 1997, and its peak (0.67) occurred in September 1997. Although a 

high-latitude (26-deg) spot was observed in March 1995, its magnetic configuration was that of old 

cycle 22 rather than new cycle 23. The first true new-cycle, high-latitude (38-deg) spot was seen in 

May 1996, occurring simultaneously with Rm. (The epoch of occurrence of g(h)/g >0.5 varies markedly with respect to Rm. Five of 12 cycles have values preceding Rm by 2 to 8 mo, and 6 of 12 cycles 

have values lagging Rm by 3 to 13 mo. Only cycle 16 had a value occurring simultaneously with Rm. 

Presuming Rm in November 2008 for cycle 24, g(h)/g >0.5 first preceded Rm by 2 mo.)

For cycle 24, Lm (6.6 deg) occurred in May 2007 and Hm (11.7 deg) occurred in June 2007. 

Rm has not, as yet, been officially discerned, although a value of R=1.8 was seen in November 2008, 

which is near record value (cycle 15 had an Rm=1.5 in August 1913). Values in November 2008 

measured 16.0 deg for L and 20.8 deg for H. The ratio g(h)/g was first greater than 0.5 in September 2008 and its peak (1.00) occurred in November and December 2008. Although a high-latitude 

(26-deg) spot was observed in October 2006, its magnetic configuration was that of old cycle 23 

rather than new cycle 24. The first true new-cycle, high-latitude (30-deg) spot was observed in January 2008. (Previously, Wilson18 used the first occurrence of a high-latitude spot to estimate the timing of cycle minimum.)

For comparison, figure 5 displays the 12-mma values for S (fig. 5(a)), R (fig. 5(b)), 

G (fig. 5(c)) and A (fig. 5(d)) for the interval January 1994–November 2008. For cycle 23, SM measured 13.8 in July 1996, occurring just 2 mo past Rm (8.0 in May 1996), and RM measured 120.8, 

occurring in April 2000. A slightly smaller secondary burst (R=115.5) of solar activity is observed 

to have peaked in November 2001, giving cycle 23 a double-humped appearance, such an appearance also being seen in cycles 20 and 22. The double-humped appearance is also found in G and A, 

although the later-occurring burst is the larger for these solar parameters. While smaller peaks in G

and A occurred essentially simultaneously with RM, the true peaks in G and A followed RM, being 
9

Calendar Date (yr – mo)

40

35

30

25

20

15

H (deg) g (h)/g

J J

1994 1996 1998

Cycle 23

Cycle 23 Cycle 24

Cycle 23

Cycle 24

E(Lm)

E(Lm)

7.8

6.6

2000 2002 2004 2006 2008

J J J J J J J J J J J J J J J

5

10

20

25

20

15

(b)

J J

1994 1996 1998 2000 2002 2004 2006 2008

J J J J J J J J J J J J J J J (c)

J J

1994 1996 1998 2000 2002 2004 2006 2008

J J J J J J J J J J J J J J J (d)

10

0.0

0.5

1.0

L (deg)

E(Hm)

E(HM)

E(Hm)

E(LM) 

J J

1994 1996 1998

E(Rm) = 8.0

2000 2002 2004 2006 2008

J J J J J J J J J J J J J J J

60

40

(a)Latitude (deg)

E(Rm) = 120.8

11.7

16.1

38.4

23.1

Figure 4. Monthly variation of (a) latitudinal distribution of spots, (b) L, (c) H, 

and (d) g(h)/g for January 1994–November 2008.
10

A

Calendar Date (yr–mo)

J J

1994 1996 1998 2000

Cycle 23

70.3

0.81

E(Am)

2002 2004 2006 2008

J J J J J J J J J J J J J J J (d)

500

1,000

1,500

2,000

2,500

8.0

E(AM)

G

J J

1994 1996 1998 2000

E Cycle 23 (Gm)

2002 2004 2006 2008

(c) J J J J J J J J J J J J J J

5

10

15

E(GM)

120.8

R

J J

1994 1996 1998 2000

E(Rm) Cycle 23

2002 2004 2006 2008

(b) J J J J J J J J J J J J J J J

20

40

60

80

100

120

E(RM)

13.8 S

J J

1994 1996 1998 2000

Cycle 23 Cycle 24

E(SM)

2002 2004 2006 2008

(a) J J J J J J J J J J J J J J J

5

10

15

20

25

30

10.23

2,048.8

Figure 5. Monthly variation of (a) S, (b) R, (c) G, and (d) A for January 1994–November 2008,
11

associated with the secondary burst of activity in R. GM measured 10.23 in December 2001 and AM

measured 2048.8 in February 2002. During the first peak, G measured 9.75 in March and April 2000, 

while A measured 1712.2 in April 2000. Thus, the secondary peak, which was about 5% smaller than 

RM, was about 5% larger as seen in G and 20% larger as seen in A.

Figure 6 plots the ratio of A/G for the same interval (January 1994–November 2008). Clearly, 

the area per group varied over cycle 23 by a factor greater than 3, from its minimum of 70.4 in February 1997 to 228.8 in March 2004. The values now being experienced (about 37.0) are below those 

seen in the mid-1990s, and, in fact, are lower than that seen in any previously observed cycle since 

cycle 12, dropping below the previous record held by cycle 15, which measured 46.0 in August 1913 

(see fig. 7). Six of 12 cycles have (A/G)m occurring simultaneously with Rm, and 10 of 12 have (A/G)

m occurring (either side) within 5 mo of Rm. The two exceptions are cycles 14 and 23, with lead 

and lag times of 23 and 9 mo, respectively. For cycle 24, A/G appears to be near minimum value in 

November 2008.

Figure 7 plots the minimum, maximum, and ratio values of A/G based on their 12-mma 

values for cycles 12–24, with cycle 24’s values being considered tentative. Median values of 79.2 and 

225.35 millionths of a visible solar hemisphere per group are found for (A/G)m and (A/G)M, respectively, for cycles 12–23. The ratio of (A/G)M/(A/G)m spans about 2.0 (cycle 13) to 4.1 (cycles 15 and 

19) and has a median of 2.85. Correlation analysis (not shown) reveals only a marginally statistically 

significant (confidence level cl >90%) relationship between (A/G)M and (A/G)m.

Figure 8 plots the cyclic variation of Lm (fig. 8(a)), LM (fig. 8(b)), Hm (fig. 8(c)), and HM

(fig. 8(d)). Also shown are the medians. While Lm and LM appear to show only slight variation with 

time (i.e., their values appear rather flat over time), in contrast, Hm and HM appear to have increased 

significantly in value between cycles 12 and 22, but now seem to be in decline. If this is true, then 

one expects cycle 24’s HM to fall below the median (38.35 deg). Previously, Wilson and Hathaway19

found a strong statistical relationship to exist between a cycle’s maximum amplitude RM and HM, 

one that suggests cycle 24’s RM to be below average size if its HM is below median value and larger 

than average size if its HM is above median value. Presently, H measures only about 21 deg, with HM

not expected until late 2010 or later. (More will be said about the relationship between RM and HM

in section 2.3.) 

Figure 9 displays the elapsed time in months between the occurrences of Lm and Rm (t(Lm–

Rm)) (fig. 9(a)) and between the occurrences of Hm and Rm (t(Hm–Rm)) (fig. 9(b)) for cycles 12–24. 

Lm and Hm have always preceded Rm, by 2–29 mo and 3–25 mo, respectively. The medians based on 

cycles 12–23 are 13 and 11.5 mo, respectively. Presuming November 2008 to be E(Rm) for cycle 24, its 

Lm and Hm both precede E(Rm) by 18 and 17 mo, respectively. For future sunspot cycles, including 

cycle 24’s values, one anticipates that once Lm and Hm have been observed, Rm should be expected 

to follow within 6 to18 mo, having proved true for 7 of 13 cycles using Lm and 9 of 13 cycles using 

Hm.

Figure 10 shows the elapsed time in months between the occurrences of Rm and LM (t(Rm–

LM)) (fig. 10(a)) and between the occurrences of Rm and HM (t(Rm–HM)) (fig. 10(b)) for cycles 

12–23. LM and HM have always followed Rm by 4–24 and 14–52 mo, respectively. The medians are 
12

A/G

Calendar Date (yr–mo)

Cycle 23

J J

1994 1996 1998 2000 2002 2004 2006 2008

J J J J J J J J J J J J J J J

50

100

150

200

250

E((A/G)m)

E((A/G)M)

70.4

228.8

Figure 6. Monthly variation of A/G for January 1994–November 2008.
13

12 14 16 18 20 22 24

4

5

1

2

3

Ratio = (

A/G)M/(A/G)m

Sunspot Cycle Number

Median (2.85)

200

300

100

(b)

(c)

(A/G)M

Median (225.35)

200

100

(a)

(A/G)m

Median (79.2)

Figure 7. Cyclic variation of (a) (A/G)m, (b) (A/G)M, and (c) Ratio: (A/G)M/(A/G)m

for cycles 12–24.
14

12 14 16 18 20 22 24

50

35

30

40

45

(d)HM (deg) Hm (deg)

Sunspot Cycle Number

Median (38.35)

25

25

20

20

15

10

5 (c)

Median (15.2)

30

(b)LM (deg)

Median (23.15)

5

10

(a)Lm (deg)

Median (7.5)

Figure 8. Cyclic variation of (a)Lm, (b) LM, (c) Hm, and (d) HM for cycles 12–24.

13 and 32 mo, respectively. For cycle 24, its LM is expected to follow November 2008 (presuming 

this to be E(Rm) for cycle 24) by about 8–15 mo, having proved true for 8 of 12 cycles, and HM is 

expected to follow by about 23–37 mo, having proved true for 8 of 12 cycles. 

Figure 11 depicts the periods for each of the L and H parameters and when higher latitudes 

dominate (g(h)/g >0.5): P(Lm–Lm) (fig. 11(a)), P(LM–LM) (fig. 11(b)), P(Hm–Hm) (fig. 11(c)), 

P(HM–HM) (fig. 11(d)), and P(0.5–0.5) (fig. 11(e)). Also shown are the means. Interestingly, the 

gap—delineated by the dashed lines—noticed in periods based on S, R, G, and A is also apparent in periods determined using Lm and Hm. Four of 5 cycles of longer than mean P(Lm–Lm) 
15

10 12 14 16 18 20 22 24

–20

–30

–10

(b)

t(Hm – Rm)

Sunspot Cycle Number

Median (11.5)

–20

–30

–10

(a)

t(Lm – Rm)

Median (13)

Figure 9. Cyclic variation of (a) t(Lm–Rm) and (b) t(Hm–Rm) for cycles 12–24.

or P(Hm–Hm) (also P(0.5–0.5)) have longer periods as determined using S, R, G, and A. The only 

exception is cycle 16 (the exception is cycle 21 based on P(0.5–0.5)). Similarly, 6 of 7 cycles of shorter 

than mean P(Lm–Lm) or P(Hm–Hm) (also P(0.5–0.5)) have shorter periods as determined using S, 

R, G, and A; the only exception is cycle 12. Thus, because the epochs of Lm and Hm (not true for 

E(g(h)/g >0.5)) precede minimum values of R, G, and A and maximum values of S, both P(Lm–Lm) 

and P(Hm–Hm) might be used to guess the period class for an ongoing sunspot cycle (i.e., a cycle of 

shorter or longer period), having proved correct for 10 of 12 sunspot cycles (7 of 7 cycles for cycles 

17–23). For cycle 23, its P(Lm–Lm) and P(Hm–Hm) measured, respectively, 143 and 140 mo, suggesting cycle 23 to have P(Rm–Rm) longer than the mean and longer than or equal to 135 mo, the 

lower extreme of longer period sunspot cycles based on R. This has proven correct since P(Rm–Rm) 

for cycle 23 now measures at least 150 mo through October 2008, presuming E(Rm) for cycle 24 in 

November 2008.
16

10 12 14 16 18 20 22 24

20

30

40

50

60

10

(b)

t(Rm – HM)

Sunspot Cycle Number

Median (32)

20

30

10

(a)

t(Rm – LM)

Median (13)

Figure 10. Cyclic variation of (a) t(Rm–LM) and (b) t(Rm–HM) for cycles 12–23.
17

Sunspot Cycle Number

P(0.5 – 0.5)

12 14 16 18 20 22 24

Mean (130.1)

110

130

150

(e)

P(HM – HM)

Mean (129.9)

120

100

140

160

(d)

P(Hm – Hm)

Mean (128.8)

120

100

140

160

(c)

P(LM – LM)

130 Mean (129.8)

110

150

(b)

P(Lm – Lm)

130 Mean (128.5)

110

150

170

(a)

Figure 11. Cyclic variation of (a) P(Lm–Lm), (b) P(LM–LM), (c) P(Hm–Hm), 

(d) P(HM–HM), and (e) P(0.5–0.5) for cycles 12–23.
18

For cycle 23, LM occurred in February 1998. Based on the median period for LM (about 

130 mo), LM for cycle 24 should be expected about December 2008 (February 1998+130 mo). The 

value of L in November 2008, however, measures only 16.0 deg, well below all previous LM values. 

The lowest, 20.9 deg, occurred in November 1902 during cycle 14; see figure 8(b). The November 

2008 value of L suggests, perhaps, that LM for cycle 24 still lies several months ahead. Because 

cycle 14 has the longest P(LM–LM), equal to 149 mo (fig. 11(b)), one infers that LM for cycle 24 

probably should be expected within the next year, before July 2010. However, from figure 10(a), 

one expects LM to follow Rm by about 8–15 mo, suggesting LM will occur anytime between July 

2010 and February 2011, which, if true, implies that P(LM–LM) for cycle 23 will be record setting 

(>149 mo).

Likewise, HM for cycle 23 occurred in December 1999. Based on the median period for HM

(about 130 mo), HM for cycle 24 should be expected about October 2009. The value of H in November 2008 measured about 21 deg—again, far lower than all previous HM values (the lowest observed 

HM measured 31.5 deg during cycle 12; see fig. 8(d)). The shortest P(HM–HM) measures 107 mo 

(cycle 15; see fig. 11(d), inferring HM for cycle 24 on or after November 2008 (December 1999+

107 mo). So, HM for cycle 24, likewise, appears to lie several months ahead, probably sometime 

before the end of 2010, although from figure 10(b), one expects HM to follow Rm by about 23–

37 mo, suggesting HM would occur anytime between October 2010 and December 2011. If true, this 

suggests P(HM–HM) for cycle 23 would be about 130–144 mo.

Other periods of interest include those based on the elapsed time between the first spotless 

days (FSD) of successive cycles, the last spotless days (LSD) of successive cycles, and the first newcycle, high-latitude spots (FHLS) of successive cycles, as well as the elapsed time between FSD and 

Rm and between FHLS and Rm. FSD is defined here as the first spotless day (i.e., when no spots are 

reported on the Sun as recorded by the Royal Observatory of Belgium, the caretakers of the International Sunspot Number) that occurs after the preceding cycle’s RM. LSD is defined here as the last 

spotless day prior to the ongoing cycle’s RM. Without magnetic configuration determinations, the 

first occurrence of the new cycle’s FHLS is inherently speculative. Here, however, FHLS is taken to 

be the first occurrence of a high-latitude spot (≥25 deg latitude) during the decline of the cycle, within 

the last 3 yr of the cycle, and always either before or simultaneous with Rm occurrence, especially 

after a lengthy continuous run of lower-latitude spot occurrences. Using the folded hemispheric representation of the latitudinal distribution of sunspots facilitates the selection of FHLS, as shown in 

figure 12 for cycles 12–24, plotted relative to Rm occurrence. For cycle 24, its tentative representation 

presumes Rm occurrence in November 2008.

Figure 12 shows the latitudinal distribution of sunspots within 3 yr prior to sunspot minimum through 2 yr post sunspot minimum for cycles 12–24. Notice that all cycles have FHLS either 

prior to or simultaneously with Rm occurrence. Notice also that the overlap of old and new cycles 

tends to be about 1–4 yr, i.e., the elapsed time from FHLS of the new cycle to the last lowest-latitude 

spot (LLLS) of the old cycle near the equator. While the actual FHLSs for cycles 21 and 22 seem 

somewhat ambiguous (i.e., the occurrences might be slightly closer to their E(Rm) dates), for the 

remainder of the cycles FHLS appears well determined.

Figure 13 depicts the cyclic variation of the elapsed time in months from FHLS to LLLS (i.e., 

t(FHLS–LLLS)) individually for cycles 12–23, with tentative values shown for cycle 23 (>157 mo, 
19

20

(a) (b) (c)

Cycle 12 E(Rm) Cycle 13 E(Rm) Cycle 14 E(Rm)

E(FHLS)

60

40

Latitude (deg)

E(FHLS) E(FHLS)

20

(d) (e) (f)

Cycle 15 Cycle 16 Cycle 17

E(FHLS)

60

40

Latitude (deg) Latitude (deg) Latitude (deg) Latitude (deg)

E(FHLS)

E(FHLS)

20

(g) (h) (i)

Cycle 18 Cycle 19 Cycle 20

E(FHLS)

60

40

E(FHLS) E(FHLS)

20

(j) (k) (l)

Cycle 21 Cycle 22 Cycle 23

E(FHLS)?

60

40

E(FHLS)? E(FHLS)

20

(m)

Cycle 24*

E(FHLS)

60

40

–36 –24 –12

Elapsed time in months from E(Rm)

*assumes E(Rm) in November 2008

0 12 24

–36 –24 –12

Elapsed time in months from E(Rm)

0 12 24 –36 –24 –12

Elapsed time in months from E(Rm)

0 12 24

Figure 12. Cycle-by-cycle latitudinal distribution of spots for cycles 12–24 for elapsed time 

in months from 36 mo prior to E(Rm) to 24 mo after E(Rm).
20

12 14 16 18 20 22 24

125

150

175

200

t(FHLS – LLLS)

Sunspot Cycle Number

Median (162)

?

?

?

?

?

Figure 13. Cyclic variation of t(FHLS–LLLS) for cycles 12–23.

through June 2009) and those of questionable length identified using question marks. The median 

is 162 mo (about 13.5 yr), and the range is 142 mo (cycle 15) to 190 mo (cycle 14). Interestingly, for 

every cycle, t(FHLS–LLLS) is longer than its corresponding P(Rm–Rm), averaging about 37 mo 

longer (range: 22–55 mo). This suggests that LLLS for cycle 23 probably remains well in the future, 

since the difference between the presumed E(Rm) for cycle 24 (November 2008) and June 2009 is only 

7 mo. The epoch of LLLS for cycle 23 seems likely not to occur until after about September 2010, 

unless cycle 23 proves to be a statistical outlier. Also interesting is that 4 of 5 cycles with t(FHLS–

LLLS) ≥173 mo are longer-period sunspot cycles (the exception is cycle 21), and 6 of 7 cycles with 

t(FHLS–LLLS) ≤162 mo are shorter-period cycles (the exception is cycle 23, having t(FHLS–LLLS) 

>157 mo).
21

Figure 14 plots the variation of the elapsed time in months from the occurrence of FHLS

to Rm for cycles 12–24 (t(FHLS–Rm)), where cycle 24’s elapsed time presumes Rm occurrence in 

November 2008. Because the occurrences for FHLS for cycles 21 and 22 are somewhat uncertain, for 

these two cycles the elapsed time could be slightly shorter (hence, the use of question marks for their 

t(FHLS–Rm) values). The median is –14.5 mo and, including cycle 24, 5 of the past 6 cycles have had 

t(FHLS–Rm) values shorter than the median. All of the cycles have had FHLS occurrence within 

2 yr (leading) of Rm occurrence, except, possibly, cycle 21. (Presuming Rm in November 2008 for 

cycle 24, t(FHLS–Rm) equals –10 mo.)

12 14 16 18 20 22 24

–30

–10

–20

t(FHLS – Rm)

Sunspot Cycle Number

Median (–14.5) ?

?

Figure 14. Cyclic variation of t(FHLS–Rm) for cycles 12–24.

Figure 15 plots the variation of the periods based on FSD and LSD: P(FSD–FSD) in figure 15(a) and P(LSD–LSD) in figure 15(b). For P(FSD–FSD), its mean is about 132 mo with the hint 

of a downward trend between cycles 12 and 23 and with 5 of the past 6 cycles having P(FSD–FSD) 

below the mean value. If the trend continues, then FSD for cycle 25 should occur before January 

2015 (January 2004+132 mo). For P(LSD–LSD), its mean is about 129 mo, so one infers LSD for 

cycle 23 about October 2008, which has already passed. Spotless days continue to be seen in the 

overlap of cycles 23 and 24 (through July 2009), so one infers t(LSD–LSD) for cycle 23 >138 mo. 

Cycle 23’s t(LSD–LSD) is now the third longest ever seen, although still far shorter than that seen 

for cycles 13 (170 mo) and 15 (163 mo). Should cycle 23 be closely akin to these cycles, spotless days 

might continue to be seen during cycle 24 for another 2–3 yr.

Figure 16 displays the variation of the leading elapsed time between FSD and Rm occurrences 

(t(FSD–Rm), fig. 16(a)), the lagging elapsed time between Rm and LSD occurrences (t(Rm–LSD), 

fig. 16(b)), and the elapsed time between FSD and LSD occurrences (t(FSD–LSD), fig. 16(c)) 

for cycles 10–24. Concerning t(FSD–Rm), prior to cycle 24, its value was –62 mo or more for
22

10 12 14 16 18 20 22 24

140

180

160

120

100

(b)

P(LSD – LSD)

Sunspot Cycle Number

Mean (129.0)

140

160

120

100 (a)

P(FSD – FSD)

Mean (132.0)

Figure 15. Cyclic variation of (a) P(FSD–FSD) and (b) P(LSD–LSD) for cycles 10–23.

cycles 10–15 and –40 mo or less for cycles 16–23; i.e., a large gap is perceived between the earlier 

cycles and the more recent cycles. Presuming Rm occurrence for cycle 24 to be November 2008, its 

t(FSD–Rm) measures –58 mo, a value more akin to the earlier cycles than the more recent cycles, 

but 4 mo shorter than the upper extreme of the perceived gap. Could cycle 24 be the start of another 

extended interval of t(FSD–Rm) measuring about 5 yr or longer?

 Concerning t(Rm–LSD), 7 of the past 7 cycles have had values below the mean (26.5 mo). 

So, if cycle 24’s Rm occurrence is indeed November 2008, then LSD for cycle 24 should occur before 

February 2011 and probably before May 2010, based on the average length for the past 7 cycles. 

Through July 2009, t(Rm–LSD) measured 8 mo, a value below the 10 mo observed for cycle 22, 

so spotless days should continue at least another 2 mo and probably longer; i.e., LSD for cycle 24 

should not be expected before September 2009, based on the recent behavior of t(Rm–LSD).
23

Sunspot Cycle Number

t(FSD – LSD)

10 12 14 16 18 20 22 24

Mean (76.7)

40

60

80

100

120

140

t(FSD – Rm)

Mean (–50.1)

–20

–40

–60

–80

–100

t(Rm – LSD)

Mean (26.5)

40

20

(b)

(a)

(c)

60

Figure 16. Cyclic variation of (a) t(FSD–Rm), (b) t(Rm–LSD), and (c) t(FSD–LSD) 

for cycles 10–24.
24

 Concerning t(FSD–LSD), it too has behavioral traits suggestive of characteristic differences 

between the earlier cycles 10–15 and the more recent cycles 16–23. Cycles 10–15 have t(FSD–LSD) 

longer than the mean (about 77 mo), while cycles 16–23 have t(FSD–LSD) shorter than the mean. 

In fact, 7 of the past 7 cycles have values averaging only about 51 mo, ranging between 44 and 

58 mo. Hence, if cycle 24 is akin to the more recent cycles, then the length of time in months from 

FSD to LSD should be less than 77 mo, implying LSD for cycle 24 before June 2010 (January 2004+

77 mo). Presently, spotless days have been observed for 66 mo, so, presuming cycle 24 to be more 

akin to recent cycles, a rapid transition to fewer spotless days should now be underway, with LSD for 

cycle 24 expected within 12 mo. However, if cycle 24 is more akin to the earlier cycles, as suggested 

by t(FSD–Rm), then LSD for cycle 24 will be delayed until after June 2010, more likely after about 

December 2010, and possibly as late as December 2012, based on the average length of t(FSD–LSD) 

for cycles 10–15, which measures about 107 mo. (Wilson20 and Wilson and Hathaway21–23 have provided additional material concerning spotless days and their relationship to the sunspot cycle and the 

prediction of sunspot minimum.)

For convenience, included here are tables 1–4, which identify, respectively, the various epochs 

of occurrence for the selected solar cyclic parameters, the values for the selected solar cyclic parameters, the periods for the selected solar cyclic parameters, and the elapsed times relative to E(Rm) for 

the selected solar cyclic parameters.

2.2 Parametric variations near cycle maximum

Figure 17 plots the cyclic variation of RM (fig. 17(a)), GM (fig. 17(b)), and AM (fig. 17(c)) for 

cycles 10–23. The medians are 114.9, 9.9, and 1,834.1, respectively. Cycle 23 had values just above 

the medians. For each parameter, correlation analysis suggests a statistically significant rise (secular 

variation) during the observed intervals, given by the diagonal lines. Presuming the inferred trend 

continues, cycle 24 is expected to have RM=158.1 ± 34.5 (± 1σ accuracy), GM=14.2 ± 2.3 (± 1σ

accuracy), and AM=2,769.7 ± 598.1 (± 1σ accuracy). However, if the trends are now downward, 

possibly having peaked about cycle 19, then one expects values for the parameters to fall below the 

medians. (Wilson24 and Hathaway, Wilson, and Reichmann12 have previously drawn attention to the 

secular rise in sunspot number.)

Figure 18 shows the periods associated with each of the parameters: P(RM–RM) (fig. 18(a)), 

P(GM–GM) (fig. 18(b)), and P(AM–AM) (fig. 18(c)). The medians are 128, 127, and 126 mo, respectively. Eight of 13 cycles have P(RM–RM)=128 ± 6 mo, with 2 of the past 3 cycles falling within 

the bounds, suggesting cycle 24’s RM would occur about December 2010 (±6 mo), given E(RM) of 

April 2000 for cycle 23. However, this seems far too early, given that E(Rm) for cycle 24 has only 

just occurred (about November 2008). To accept P(RM–RM)=128 ± 6 mo implies a very fast rise 

for cycle 24, one measuring just 25 ± 6 mo, which is shorter than ever seen. The fastest rise to date 

(34 mo) was observed for cycle 22, the third largest cycle on record (RM=158.5). A more modest rise 

of 48 mo gives E(RM) for cycle 24 about November 2012 and yields P(RM–RM)=151 mo, which 

would make it the second longest P(RM–RM) on record, slightly shorter than observed for cycle 11’s 

P(RM–RM).
25

Table 1. Epochs of occurrence for selected solar cyclic parameters.

Cycle E(Rm) E(SM) E(Gm) E(Am) E(g(h)/g >0.5) E(Lm) E(Hm) E(RM) E(GM) E(AM) E(LM) E(HM) E(FSD) E(LSD) E(FHLS) E(LLLS) E((A/G)m) E((A/G)M)

10 12–1855 12–1855 – – – 02–1860 – – – – 05–1849 04–1858 – – – –

11 03–1867 05–1867 – – – 08–1870 – – – – 10–1861 07–1869 – 03–1880 – –

12 12–1878 10–1878 12–1878 12–1878 06–1879 09–1878 09–1878 12–1883 03–1884 11–1883 02–1880 11–1880 05–1873 09–1883 04–1877 11–1891 07–1878 03–1883

13 03–1890 02–1890 10–1889 02–1890 12–1889 02–1888 05–1888 01–1894 09–1893 01–1894 07–1890 04–1892 01–1885 12–1891 12–1888 01–1904 03–1890 02–1897

14 01–1902 01–1902 01–1902 01–1902 05–1902 01–1901 12–1900 02–1906 02–1906 06–1905 11–1902 03–1905 11–1895 07–1905 01–1900 11–1915? 02–1900 06–1905

15 08–1913 08–1913 08–1913 08–1913 02–1913 06–1912 06–1912 08–1917 07–1917 08–1917 04–1914 11–1915 10–1906 10–1916 12–1912 10–1924? 08–1913 10–1921

16 08–1923 10–1923 08–1923 08–1923 08–1923 10–1921 07–1921 04–1928 06–1928 06–1928 05–1924 10–1924 04–1920 07–1926 10–1921 03–1935? 08–1923 04–1926

17 09–1933 09–1933 09–1933 09–1933 03–1934 07–1933 02–1933 04–1937 04–1937 05–1937 09–1935 01–1938 09–1930 07–1935 01–1932 07–1945 09–1933 07–1943

18 02–1944 02–1944 02–1944 04–1944 07–1944 11–1942 10–1942 05–1947 11–1947 05–1947 12–1945 02–1947 11–1941 09–1945 04–1942 05–1955 05–1944 12–194619 04–1954 04–1954 04–1954 04–1954 02–1954 10–1952 06–1953 03–1958 03–1958 11–1957 02–1955 01–1957 12–1950 10–1955 01–1954 06–1966 05–1954 10–1957

20 10–1964 11–1964 06–1964 10–1964 07–1964 05–1962 03–1963 11–1968 06–1970 04–1968 01–1966 05–1967 11–1961 08–1966 10–1963 03–1978 10–1964 06–196821 06–1976 03–1976 03–1976 11–1976 10–1975 11–1975 08–1975 12–1979 06–1979 10–1981 07–1977 08–1978 07–1973 07–1977 09–1973? 07–1988 11–1976 01–1982

22 09–1986 03–1986 03–1986 09–1986 12–1986 03–1986 11–1985 07–1989 02–1991 05–1989 10–1987 05–1990 11–1983 07–1987 07–1985? 04–1998 09–1986 05–198923 05–1996 07–1996 08–1996 05–1996 06–1997 12–1995 10–1995 04–2000 12–2001 02–2002 02–1998 12–1999 04–1994 01–1998 05–1996* – 02–1997 03–2004

24 (11–2008) (11–2008) (11–2008) (11–2008) 09–2008 05–2007 06–2007 – – – – – 01–2004 – 01–2008* – 11–2008?

	 * based on actual magnetic configuration
26

Table 2. Values of selected solar cyclic parameters.

Cycle Rm SM Gm Am Lm Hm RM GM AM LM HM FHLS (A/G)m (A/G)M

10 3.2 24.1 – – – – 97.9 – – – – – – –

11 5.2 20.4 – – – – 140.5 – – – – – – –

12 2.2 25.0 0.15 15.1 7.3 7.8 74.6 6.29 1,406.6 21.8 31.5 28.6 98.6 240.2

13 5.0 17.9 0.48 56.5 6.6 12.6 87.9 8.66 1,610.6 24.0 32.8 35.5 103.3 209.3

14 2.6 24.3 0.20 31.7 6.1 9.4 64.2 5.74 1,192.3 20.9 32.2 38.4 83.7 225.4

15 1.5 26.2 0.15 6.9 6.1 9.9 105.4 9.58 1,570.9 23.2 40.5 26.3 46.0 186.7

16 5.6 17.8 0.63 49.8 7.6 18.5 78.1 7.07 1,369.9 24.4 34.8 34.0? 79.0 234.5

17 3.4 20.8 0.41 27.7 8.0 12.5 119.2 10.69 2,160.1 24.0 36.5 48.0 67.6 225.3

18 7.7 15.0 0.74 101.8 8.0 16.8 151.8 11.85 2,713.5 22.7 40.5 30.3 116.1 251.1

19 3.4 21.4 0.36 23.5 7.5 15.2 201.3 14.82 3,547.7 26.9 43.6 37.8 60.5 246.3

20 9.6 9.9 1.05 51.9 8.9 17.9 110.6 9.33 1,619.3 22.3 38.3 35.0 48.1 192.4

21 12.2 9.4 1.09 138.2 8.4 19.6 164.5 14.88 2,524.6 21.7 40.5 35.4? 106.3 223.4

22 12.3 11.7 1.00 88.9 6.5 21.7 158.5 13.57 2,592.7 25.2 42.4 30? 79.4 202.2

23 8.0 13.8 0.81 70.3 7.8 16.1 120.8 10.23 2,048.8 23.1 38.4 38 70.4 228.8

24 (1.8) (24.2) (0.20) (7.4) 6.6 11.7 – – – – – 30 (37.0) –

Regarding P(GM–GM), 6 of 11 cycles have P(GM–GM)=127 ± 6 mo, with 2 of the past 

3 cycles falling within the bounds, implying cycle 24’s GM would occur about July 2012 (±6 mo), 

given E(GM) of December 2001 for cycle 23. Regarding P(AM–AM), only 5 of 11 cycles have P(AM–

AM)=126 ± 6 mo, with none of the past 3 cycles falling within the bounds. Such a range implies AM

for cycle 24 would occur about August 2012 (±6 mo), given E(AM) of February 2002 for cycle 23.

Figure 19 displays the elapsed time in months between the occurrences of GM and RM

(t(GM–RM)) (fig. 19(a)) and between AM and RM (t(AM–RM)) (fig. 19(b)), where negative values 

indicate parametric leading of E(RM). The medians are 1 and zero, respectively, indicating that GM

and AM usually occur closely in time with E(RM). For GM, 9 of 12 cycles have E(GM) within 6 mo 

of E(RM), with cycles 20, 22, and 23 lagging substantially (19–20 mo). For AM, 10 of 12 cycles have 

E(AM) within 8 mo of E(RM), with cycles 21 and 23 lagging substantially (22 mo). For both parameters, however, slightly smaller values peaked when RM occurred.

Figure 20 depicts the elapsed time in months between the occurrences of LM and RM (t(LM–

RM)) (fig. 20(a)) and between HM and RM (t(HMRM)) (fig. 20(b)). Again, negative values indicate 

parametric leading of E(RM). For t(LM–RM), its median is –35.5 mo, and all values are negative, indicating that LM always leads RM. Six of the past 7 cycles have had values shorter than the 

median (averaging about –26 mo), with 7 of 7 being shorter than –37 mo. Thus, when LM occurs for 

cycle 24, probably sometime in late 2010, one expects RM to follow within about 2–3 yr.

For t(HM–RM), its median is –15 mo, and all except two cycles (cycles 17 and 22) have had 

HM leading RM. Six of 12 cycles have HM leading RM by about 15 ± 6 mo. So, again, once HM has 

been discerned for cycle 24 (unless, of course, cycle 24 happens to behave similarly to cycles 17 and 

22), one expects RM to follow within about 1 yr.
27

Table 3. Periods for selected solar cyclic parameters.

Cycle P(Rm–Rm) P(SM–SM) P(Gm–Gm) P(Am–Am) P(0.5–0.5) P(RM–RM) P(GM–GM) P(AM–AM) P(FSD–FSD) P(LSD–LSD) P(Lm–Lm) P(LM–LM) P(Hm–Hm) P(HM–HM)

10 135 137 – – – 126 – – 145 135 – – – –

11 141 137 – – – 160 – – 139 170 – – – –

12 135 136 130 134 126 121 114 122 140 99 113 125 116 137

13 142 143 147 143 149 145 149 137 130 163 147 148 151 155

14 139 139 139 139 139 138 137 146 131 135 137 149 138 128

15 120 119 120 120 126 128 131 130 158 117 112 121 109 107

16 121 125 121 121 127 108 106 107 125 108 141 136 139 159

17 125 122 125 127 124 121 127 120 134 122 112 123 116 109

18 122 122 122 120 115 130 124 126 109 121 119 110 128 119

19 126 127 122 126 125 128 147 125 131 130 115 131 117 124

20 140 148 141 145 135 133 108 162 140 131 162 138 149 135

21 123 120 120 118 134 115 140 91 124 120 124 123 123 141

22 116 124 125 116 126 129 130 153 125 126 117 124 119 115

23 (150) (148) (147) (150) 135 – – – 117 (137) 143 – 140 –
28

Table 4. Elapsed time in months relative to E(Rm) for selected solar cyclic parameters.

Cycle E(Rm) t(SM–Rm) t(FSD–Rm) t(Rm–LSD) t(Gm–Rm) t(Am–Rm) t((A/G)m–Rm) t(g(h)/g>0.5) t(FHLS–Rm) t(Rm–RM) t(Lm–Rm) t(Hm–Rm) t(Rm–LM) t(Rm–HM) t(LM–RM) t(HM–RM)

10 12–1855 0 –79 28 – – – – – 50 – – – – – –

11 03–1867 2 –65 28 – – – – – 41 – – – – – –

12 12–1878 –2 –67 55 0 0 –5 6 –20 60 –3 –3 14 23 –46 –37

13 03–1890 –1 –62 21 –5 –1 0 –3 –15 46 –25 –22 4 25 –42 –21

14 01–1902 0 –74 42 0 0 –23 4 –24 49 –12 –13 10 37 –39 –11

15 08–1913 0 –82 38 0 0 0 –6 –8 48 –14 –14 8 27 –40 –21

16 08–1923 2 –40 35 0 0 0 0 –22 56 –22 –25 9 14 –47 –42

17 09–1933 0 –36 22 0 0 0 6 –20 43 –2 –7 24 52 –19 9

18 02–1944 0 –27 19 0 2 3 5 –22 39 –15 –16 22 36 –17 –3

19 04–1954 0 –40 18 0 0 1 –2 –3 47 –18 –10 10 33 –37 –14

20 10–1964 1 –35 22 –4 0 0 –3 –12 49 –29 –19 15 31 –34 –18

21 06–1976 –3 –35 13 –3 5 5 –8 –33? 42 –7 –10 13 26 –29 –16

22 09–1986 –6 –34 10 –6 0 0 3 –14? 34 –6 –10 13 44 –21 10

23 05–1996 2 –25 20 3 0 9 13 0 47 –5 –7 21 43 –26 –4

24 11–2008 0 –58 – 0 0 0 –2 –10 – –18 –17 – – – –

Note: Minus sign means leading E(Rm) and E(RM), and the question mark means that the actual time might be slightly shorter.
29

Sunspot Cycle Number

AM

10 12 14 16 18 20 22 24

Median (1,834.1)

1,000

2,000

3,000

4,000

(c)

y

y = 37.6 + 113.84x

r = 0.584, r2 = 0.292

se = 598.1, cl>95%

GM

Median (9.9)

5

10

15

(b)

y

y = –0.462 + 0.611x

r = 0.709, r2 = 0.503

se = 2.3, cl >99%

RM

Median (114.9)

50

100

150

200

(a)

y

y = 35.2 + 5.12x

r = 0.543, r2 = 0.295

se = 34.5, cl >95%

Figure 17. Cyclic variation of (a) RM, (b) GM, and (c) AM for cycles 10–23.
30

Sunspot Cycle Number

P(AM – AM)

10 12 14 16 18 20 22 24

Median (126)

100

120

140

160

(c)

P(GM– GM)

Median (127)

110

130

150

(b)

P(RM– RM)

Median (128)

120

100

140

160

(a)

Figure 18. Cyclic variation of (a) P(RM–RM), (b) P(GM–GM), and (c) P(AM–AM) 

for cycles 10–22.
31

Sunspot Cycle Number

t(AM – RM)

10 12 14 16 18 20 22 24

Median (0)

30

20

10

0

–10 (b)

t(GM – RM)

Median (1)

20

10

0

–10 (a)

Figure 19. Cyclic variation of (a) t(GM–RM) and (b) t(AM–RM) for cycles 12–23.

2.3 Parametric Comparisons and Correlations

Figure 21 shows the cyclic variation of the elapsed time, in months, between Rm and RM

(t(Rm–RM)), which often is simply called the ascent duration (ASC). The median is 47 mo, with 7 of 

the past 7 cycles having a value of 34–49 mo (averaging about 43 mo). In fact, 11 of 14 cycles have 

ASC=39–50 mo, with only cycles 22 (ASC=34 mo, the fastest), 12 (ASC=60 mo, the slowest), and 16 

(56 mo) having ASCs that fall outside the limits. Because cycle 24’s Rm is believed to have occurred 

about November 2008, one strongly expects its RM to follow within the next 39–50 mo, implying 

E(RM) for cycle 24 about February 2012 to January 2013.
32

Sunspot Cycle Number

t(HM – RM)

10 12 14 16 18 20 22 24

Median (–15)

–30

–40

–20

–10

0

10 (b)

t(LM – RM)

Median (–35.5)

–50

–40

–30

–20

–10 (a)

Figure 20. Cyclic variation of (a) t(LM–RM) and (b) t(HM–RM) for cycles 12–23.
33

Sunspot Cycle Number

t(Rm – RM) = ASC

10 12 14 16 18 20 22 24

Median (47)

60

50

40

30

Figure 21. Cyclic variation of t(Rm–RM)=ASC for cycles 10–23.

A comparison of RM values against ASC produces the well-known Waldmeier effect, where 

faster-rising cycles tend to be larger-amplitude cycles and slower-rising cycles tend to be smalleramplitude cycles. Figure 22 displays the Waldmeier effect, both in terms of using ASC as the determining factor and using RM as the determining factor. The numbers beside the plotted points are 

the sunspot cycle numbers. Expressed as a 2 × 2 contingency table, the probability P of obtaining the 

observed result, or one more suggestive of a departure from independence (chance), is found using 

Fisher’s exact test for 2 × 2 contingency tables25 to be 5.1%. Hence, there exists strong statistical evidence that slower-rising cycles tend to be associated with smaller-amplitude cycles (i.e., they tend to 

populate the lower-right quadrant of the scatter plot), while faster-rising cycles tend to be associated 

with larger-amplitude cycles (i.e., they tend to populate the upper-left quadrant of the scatter plot). 

Expressed as regression equations (i.e., performing linear correlation analysis and ignoring cycles 14 

and 19, the lower and upper extremes of RM, presumed to be statistical outliers), the inferred correlation coefficient is r=–0.884, the coefficient of determination is r2=0.781 (meaning that more than 

three-fourths of the variance can be explained by the inferred regressions), and the confidence level 

of the inferred regressions is cl >99.9%. Hence, correlation analysis likewise supports the existence 

of a strong relationship between RM and ASC. Should cycle 24 be a slow riser, one expects its RM

to be average to lower-than-average size, while, in contrast, should cycle 24 be a fast riser, one expects 

its RM to be average to above-average size. Unfortunately, sunspot cycles usually do not reveal their 

true character until about 1–2 yr past sunspot minimum.26–34

Because sunspot minimum for cycle 24 seems very close at hand and, in fact, may have already 

occurred in November 2008, the period of cycle 23 (i.e., P(Rm–Rm)) can be more accurately gauged, 

as well as the minimum values of Rm, Gm, and Am for cycle 24. These values can then be used in 

the amplitude-period35 and maximum-minimum12 effects to estimate RM for cycle 24. Figure 23 

shows the amplitude-period effect for estimating RM for an ongoing sunspot cycle using P(Rm–Rm), 

P(Gm–Gm), P(Am–Am), and P(SM–SM). It compares RM for cycle n (the ongoing cycle) against 

periods for cycle n–1 (the preceding cycle). Based on P(Rm–Rm) and P(SM–SM), Fisher’s exact test 
34

30

50

100

150

200

40 50 60

5 2

1 6

x*

y*

ASC (x)

RM (y)

y* = 292.9 – 3.79x

r = –0.884, r2 = 0.781

se = 15.1, cl > 99.9%

x* = 70.5 – 0.21y

r = –0.884, r2 = 0.781

se = 3.5, cl > 99.9%

21

18

11

23

20

15

10

16

14

12

13

19

17

22 * Ignores cycles 14 and 19

P = 5.1%

Figure 22. Scatter plot of RM versus ASC (the Waldmeier effect).

for 2 × 2 contingency tables has a probability P of obtaining the observed result, or one more suggestive of a departure from independence, equal to 7.8%. This suggests a marginally statistically significant association between maximum amplitude of the ongoing sunspot cycle and the preceding cycle’s 

period length. Five of 6 preceding cycles of shorter than median period length have had ongoing 

cycles of larger than median maximum amplitude, and 5 of 7 preceding cycles of longer than median 

period length have had smaller than median maximum amplitude. Only cycle pairs 10/11, 15/16, and 

20/21 fail to adhere to the inferred association pattern. Cycle 23’s P(Rm–Rm) and P(SM–SM) both 

measure longer than median period length, suggesting that cycle 24’s maximum amplitude probably 
35

100 120 140 160 180 100 120 140 160 180

100 120 140 160 180

5 2

1 5

50

y*

150

100

200

P(Rm – Rm) (cycle n – 1)

RM (cycle n)

y* = 497.9 – 2.91x

r = –0.660, r2 = 0.435

se = 31.6, cl > 95%

RM(24) = 61.9 ± 57.9

20/21

10/11

15/16 * Ignores cycles 15/16 and 20/21

P = 7.8%

(a) (b)

3 3

2 3

y*

P(Gm – Gm) (cycle n – 1)

y* = 516.5 – 3.07x

r = –0.696, r2 = 0.484

se = 31.4, cl > 95%

RM(24) = 65.8 ± 59.5

20/21

15/16 * Ignores cycles 15/16 and 20/21

P = 60.8 %

100 120 140 160 180

4 2

1 4

50

y*

150

100

200

P(Am – Am) (cycle n – 1)

RM (cycle n)

y* = 505.2 – 3.00x

r = –0.703, r 2 = 0.494

se = 30.9, cl > 95%

RM(24) = 55.8 ± 58.6

20/21

15/16 * Ignores cycles 15/16 and 20/21

P = 17.5%

(c) (d)

5 2

1 5

y*

P(SM – SM) (cycle n – 1)

y* = 596.7 – 3.65x

r = –0.753, r 2 = 0.568

se = 27.7, cl > 99%

RM(24) = 59.9 ± 50.7

20/21

10/11

15/16 * Ignores cycles 15/16 and 20/21

P = 7.8 %

Figure 23. Scatter plots of (a) RM (cycle n) versus P(Rm–Rm) (cycle n–1), (b) RM (cycle n)

versus P(Gm–Gm) (cycle n–1), (c) RM (cycle n) versus P(Am–Am) (cycle n–1), 

and (d) RM (cycle n) versus P(SM–SM) (cycle n–1) (the amplitude-period effect).
36

will be smaller than median size. Less statistically important results are obtained using P(Gm–Gm) 

and P(Am–Am), which also are based on two fewer cycle pairs.

On the basis of linear correlation analysis, ignoring cycle pairs 15/16 and 20/21 (filled squares), 

which have the largest deviations from the inferred regression lines, one finds statistically important 

associations between RM and period length for all parametric periods (i.e., P(Rm–Rm), P(Gm–Gm), 

P(Am–Am), and P(SM–SM)). The most statistically important inferred regression is the one linking 

RM for cycle n with P(SM–SM) for cycle n–1, having r=–0.753. Applying cycle 23’s P(SM–SM), 

equal to 147 mo, one computes the 90% prediction interval for cycle 24’s RM to be 59.9 ± 50.7, inferring only a 5% chance that cycle 24 will have RM larger than 110.6. Averaging the four estimates 

yields an RM of about 61 ± 57 for cycle 24, suggesting an upper limit of about 118 for cycle 24.

Figure 24 displays the maximum-minimum effect for estimating RM of an ongoing sunspot 

cycle (cycle 24) using minimum amplitudes: i.e., (a) Rm, (b) Gm, and (c) Am). Again, both Fisher’s 

exact test for 2 × 2 contingency tables and linear correlation analysis (ignoring cycle 19) have been 

employed in the estimate of RM for cycle 24. Based on Rm, cycle 24’s RM is estimated to be about 

85 ± 42 (the 90% prediction interval), while, based on Gm and Am, RM for cycle 24 is estimated 

to be about 83 ± 46 and 98 ± 42, respectively. Averaging the three estimates yields an estimate for 

cycle 24’s RM of about 89 ± 43, implying an upper limit of about 132 and a lower limit of about 46 

for cycle 24’s RM. Thus, cycle 24 seems more likely to be a cycle of average to slightly below average 

size, in contrast to being potentially one of the largest cycles on record.36–40 Combining the results 

of the amplitude-period and maximum-minimum effects, the best guess for cycle 24’s RM is about 

82 ± 36.

Figure 25 shows the scatter plots of RM versus HM (fig. 25(a)), RM versus GM (fig. 25(b)), 

and RM versus AM (fig. 25(c)). All scatter plots are statistically significant, whether expressed as 

2 × 2 contingency tables or as cls of inferred regression fits. Hence, by monitoring the monthly values of H, G, and/or A, one can effect monthly projections for RM. As an example, HM for cycle 

24 is expected to lie somewhere in the range of 31–44 deg. Cycles having HM less than the median 

(38.35 deg) usually (5 of 6) have RM less than the median (114.9), with all having RM ≤119.2. Cycles 

having HM greater than the median usually (5 of 6) have RM greater than the median, with all 

having RM ≥105.4. The value of H in November 2008 measures about 21 deg, but is continuing to 

rise. Similarly, cycles having GM less than the median (9.91) always (6 of 6) have RM less than the 

median, actually ≤110.6, while cycles having GM greater than the median always (6 of 6) have RM

greater than the median, actually ≥119.2. Cycles having AM less than the median (1,834.1) always 

(6 of 6) have RM less than the median, actually ≤110.6, while cycles having AM greater than the 

median always (6 of 6) have RM greater than the median, actually ≥119.2.

Interestingly, 10 of 12 cycles fit the paradigm of being larger (or smaller) than median size 

(114.9) when H exceeds (or never exceeds) median value (38.35 deg), failing for cycles 15 (HM=

40.5 deg at t=27 mo and RM=105.4 at t=48 mo) and 17 (HM=36.5 deg at t=52 mo and RM=119.2 

at 43 mo). The larger than median cycles that fit the paradigm include cycles 18, 19, and 21–23. 

These cycles had H values that first exceeded median value at t=29, 14, 16, 6, and 43 mo, respectively. Hence, if H exceeds its median value, especially within the first 1–2 yr following E(Rm), one 

very probably expects the cycle to be larger than median size. Applying the Spearman rank correlation test41 to cycles 18, 19, and 21–23 (ranking the t values when H first exceeded median value and 
37

Figure 24. Scatter plots of (a) RM versus Rm, (b) RM versus Gm, and (c) RM versus Am (the maximum-minimum effect).

5 10 15

2 5

5 2

50

y*

200150100

Rm

RM

RM(24) = 85.2 ± 41.8

19

y* = 73.2 + 6.65xr = 0.737, r2 = 0.543

se = 23.3, cl > 99.9%

* Ignores cycle 19

P = 14.3%

(a) 0.5 1.0 1.5

2 4

4 2

y*

Gm

RM(24) = 83.4 ± 46.1

19

y* = 69.2 + 70.62xr = 0.727, r2 = 0.529

se = 25.2, cl > 98%

* Ignores cycle 19

P = 28.4%

(b) 50 100 150

2 4

4 2

y*

Am

RM(24) = 97.8 ± 41.7

19

y* = 72.2 + 0.69xr = 0.786, r2 = 0.618

se = 22.7, cl > 99.5%

* Ignores cycle 19

P = 28.4%

(c)
38

Figure 25. Scatter plots of (a) RM versus HM, (b) RM versus GM, and (c) RM versus AM.

45 40 35 30 (a) (b) (c)

1 5

5 1

50

y* 200

150100

HM

RM

y* = –250.9 + 9.94x

r = 0.951, r2 = 0.905

se = 14.2, cl > 99.9%

15

* Ignores cycle 15

P = 4.0%

15 10 5

0 6

6 0

y

GM

y* = –14.3 + 13.11x

r = 0.970, r2 = 0.942

se = 10.6, cl > 99.9%

P = 0.1%

1,000 1,500 2,000 2,500 3,000 3,500 4,000

0 6

6 0

y

AM

y = 1.8 + 0.06x

r = 0.973, r2 = 0.947

se = 11.4, cl > 99.9%

P = 0.1%
39

the later occurring RM values) hints of an almost marginally significant inverse association between 

the two parameters, such that the quicker that H exceeds its median maximum value, the larger the 

expected maximum amplitude for the cycle.

Figure 26(a) displays the scatter plot of HM versus Hm for cycles 12–23, with the arrow representing the Hm value for cycle 24. Given are the results of Fisher’s exact test for 2 × 2 contingency 

tables and correlation analysis, ignoring cycles 15 and 16 (the filled squares). The measured value 

of Hm for cycle 24 (11.7 deg in June 2007) suggests that cycle 24 probably will fall short of median 

value (38.35 deg), having an expected value of about 35.0 ± 4.6 deg (the 90% prediction interval). 

Consequently, cycle 24 is expected to have an RM of only average to below-average size (i.e., about 

97 ± 46, applying the value of 35.0 ± 4.6 deg to the inferred regression given in fig. 25(a)) rather than 

average to above-average size.

Figure 26(b) gives the mean variation of cycles 12–23 for elapsed time in months from E(Rm) 

to 30 mo past E(Rm) based on superposed epoch analysis. Also shown for comparison are the values 

of H from E(Rm) for cycles 14 (the smallest cycle of the modern era) and 19 (the largest cycle of the 

modern era, as well as the H values for cycles 12–24 at E(Rm), where cycle 24’s H value at E(Rm) is 

marked by a small arrow and presumes E(Rm) in November 2008. The value of H at E(Rm) for cycle 

24 plainly is lower than the mean at E(Rm) and is in the middle range of H values at E(Rm). HM

values for cycles 12–23 are also displayed. It is apparent that if cycle 24 is destined to be a largeramplitude cycle, its H must rapidly increase in value within the first year or so, rising above the mean 

of cycles 12–23. If it does not rise rapidly exceeding the mean, it seems highly unlikely that cycle 24 

will be a larger-amplitude cycle.
40

10 20 30

10

20

30

40

50

(b)

H

10 20

2

2

4

4

P = 28.4%

30

20

30

15

16

40

50

(a)HM

Elapsed time in months from E(Rm)

Hm

H(E(Rm))

20

16

13 21

23

22

24

19

18

14, 15

17

12

y = 25.4 + 0.82x

r = 0.839, r 2 = 0.704

se = 2.5, cl >99.5%

HM = 35.0 ± 4.6

y

HM

19

19

Mean (12–23)

22

15, 18, 21

23

20

17

16

13

12 14

14

Figure 26. Values of H: (a) Scatter plot of HM versus Hm, and (b) superposed epoch

analysis of H for elapsed time 0–30 mo from E(Rm), giving mean of

cycles 12–23 and H values for cycles 14 and 19, the smallest and largest 

modern era cycles, respectively.
41

3. SUMMARY

Twelve-mma values of cyclic parameters (e.g., S, R, G, A, and A/G) are reflective of cyclic 

minimum conditions in late 2008. Presuming sunspot minimum in November 2008 (Rm ≈1.8), 

cycle 23 has a minimum-to-minimum length of 150 mo, the longest of all modern era sunspot cycles 

(from cycle 10). The first spotless day after cycle maximum for cycle 23 and before the start of 

cycle 24 occurred in January 2004; the first true new-cycle, high-latitude spot occurred in January 

2008; and the monthly mean number of new-cycle groups first dominated the monthly mean number 

of old-cycle groups in September 2008. The 12-mma values of L and H were at minimum value in 

May and June 2007, respectively, with current values of L and H continuing to rise. The overlap of 

cycles 23–24 now spans at least 16 mo (from January 2008 through April 2009), but will possibly be 

longer if old-cycle spots continue to be seen later in 2009 (overlapping of 1–4 yr has been seen in 

previous cycles based on FHLS and LLLS determinations). 

Spotless days have continued to be seen through at least August 2009, now spanning at least 

68 mo. If the current period of spotless days (i.e., t(FSD–LSD)) is more akin to those for early cycles 

10–15 than more recent cycles 16–23, spotless days will continue to be seen for at least another year 

or more, perhaps another 2–3 yr. In November 2008, the 12-mma of S measured 24.2, which is the 

fourth highest number, close to but below that for cycles 12 (25.0), 14 (24.3), and 15 (26.2). Because 

SM closely associates with the occurrence of Rm, having occurred simultaneously with Rm for 6 of 

14 cycles and within 2 mo of Rm for 12 of 14 cycles (failing for cycles 21 and 22, with SM leading 

Rm by 3 and 6 mo, respectively, for these cycles) and because of the recent increase in solar activity in early July 2009, the minimum for cycle 24 is expected to be established in late 2008 (probably 

November or December 2008).

Cycle 23 is now known to be a cycle of longer period, more akin to early cycles 10–14 and 20 

rather than to cycles 15–19, 21, and 22, which are of shorter period. On the basis of P(Lm–Lm) and 

P(Hm–Hm), cycle 23 could have been projected to be a cycle of longer period some time ago, since 

Lm and Hm usually occur well before Rm. Previously, 4 of 5 cycles of longer than median P(Lm–

Lm) or P(Hm–Hm) have been cycles of longer period, as determined using R (i.e., P(Rm–Rm), and 

6 of 7 cycles of shorter than median P(Lm–Lm) or P(Hm–Hm) have been cycles of shorter period as 

determined using R. Only cycles 12 and 16 failed to agree with this paradigm.

While cycles are generally believed to average about 11 yr in length between minimum occurrences using 12-mma of R, the distribution of cycle lengths clearly is better described as bimodal, 

consisting of shorter and longer period cycles with an 8-mo gap separating the lower and upper 

extremes of the inferred cyclic groupings. While true, interestingly, the length of individual cycles as 

determined using t(FHLS–LLLS) has a median of 162 mo, ranging between 142 and 190 mo, suggesting, perhaps, that the real average length of an individual sunspot cycle—i.e., from its FHLS to 

its LLLS—is much longer than that found using sunspot number. Perhaps it is the strength of the 

following new cycle (i.e., the rapidity of new cycle spot formation and manifestation) that determines 
42

the apparent period of the just-ending old cycle (i.e., being either of shorter or longer period as 

determined from R). Based on R, the overlap of old and new cycles is about 1–4 yr (often 3 yr), with 

the shortest overlap (about 16 mo) being associated with the ending of cycle 18 and the beginning of 

cycle 19, the largest cycle of the modern era. The current overlap for cycles 23/24 measures at least 

16 mo through April 2009 (no old-cycle, near-equatorial spots were seen in May or June 2009).

Current values for the period of cycle 23 (≈150 mo) and the minimum amplitude (≈1.8) of 

cycle 24 can be used in the amplitude-period and maximum-minimum effects to estimate the size of 

cycle 24. Both strongly suggest that cycle 24 will be a cycle of average to below-average size, unless 

it proves to be a statistical outlier. Furthermore, an average to below-average size cycle suggests an 

average to slower-than-average ASC (from the Waldmeier effect) for cycle 24. Consequently, given 

minimum amplitude for cycle 24 in late 2008, one anticipates its maximum amplitude in late 2012 to 

early 2013, unless, again, cycle 24 proves to be a statistical outlier. The longer cycle 23 persists, the 

greater the chance that cycle 24 will be only of average to below-average size. There is only about a 

1% chance that cycle 23 will persist 155 mo or more, implying E(Rm) for cycle 24 on or before April 

2009.

Continuous monitoring of H, G, and A will provide strong evidence as to whether cycle 24 will 

be only of average to below-average size or larger-than-average size. For example, should H exceed 

38.35 deg during its rise from minimum to maximum, one expects cycle 24 to be of larger amplitude. 

The same is true should G and A exceed their median maximum values (9.9 and 1,834.1, respectively). The value of H in November 2008 measures only ≈21 deg (up from its minimum value of 

11.7 deg in June 2007), but is continuing to rise, with maximum value expected within the next 1–2 yr. 

If H does not exceed median maximum value, then one expects RM for cycle 24 to be only of average 

to below-average size. Based on the inferred relationship between HM and Hm (ignoring cycles 15 

and 16), one expects HM for cycle 24 to measure about 35.0 ± 4.6 deg (the 90% prediction interval). 

Surprisingly, there is a hint that the quicker that HM exceeds the median, the larger the expected size 

of the cycle; e.g., cycles 19, 21, and 22, the three largest cycles of the modern era, exceeded median 

value in only 6–16 mo from E(Rm). So, it is with great anticipation that solar prognosticators await 

the unveiling of cycle 24. Will it be a statistical outlier and thus be larger than expected from the 

amplitude-period and maximum-minimum effects, suggesting a continuing rise of the secular trend 

(RM=158.1 ± 34.5, ± 1σ accuracy)? Or, has the trend now reversed and cycle 24 can be expected to, 

instead, be part of a long-term downward trend over the next several sunspot cycles? Only time will 

tell.
43

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45

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Sunspot Activity Near Cycle Minimum and What It Might 

Suggest for Cycle 24, the Next Sunspot Cycle

Robert M. Wilson and David H. Hathaway

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

Subject Category 92

Availability: NASA CASI (443–757–5802)

Prepared by the Science and Mission Systems Office, Science and Exploration Research Office

M–1267

Technical Publication

NASA/TP—2009–216061

Sun, sunspot cycle, solar activity, solar cycle prediction, solar cycle 24, latitudinal distribution of spots, maximumminimum effect, amplitude-period effect

01–09–2009

U U U UU 60

In late 2008, 12-month moving averages of sunspot number, number of spotless days, number of groups, area of sunspots, and 

area per group were reflective of sunspot cycle minimum conditions for cycle 24, these values being of or near record value. The 

first spotless day occurred in January 2004 and the first new-cycle, high-latitude spot was reported in January 2008, although oldcycle, low-latitude spots have continued to be seen through April 2009, yielding an overlap of old and new cycle spots of at least 

16 mo. New-cycle spots first became dominant over old-cycle spots in September 2008. The minimum value of the weighted mean 

latitude of sunspots occurred in May 2007, measuring 6.6 deg, and the minimum value of the highest-latitude spot followed in 

June 2007, measuring 11.7 deg. A cycle length of at least 150 mo is inferred for cycle 23, making it the longest cycle of the modern 

era. Based on both the maximum-minimum and amplitude-period relationships, cycle 24 is expected to be only of average to 

below-average size, peaking probably in late 2012 to early 2013, unless it proves to be a statistical outlier.

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