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i

NASA/TP—2010–216433

Predicting the Size and Timing of Sunspot 

Maximum for Cycle 24

Robert M. Wilson

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

June 2010

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 ........................................................................................................................... 3

3. DISCUSSION AND SUMMARY ..................................................................................... 11

REFERENCES ....................................................................................................................... 12
iv
v

LIST OF FIGURES

1. Comparison of cycle 24’s AA values and mean AA values for cycles 12–23 

for elapsed time in months relative to E(Rm), –18 ≤ t ≤ 18 ......................................... 4

2. Cyclic variation of (a) RM and (b) AAm for cycles 12–24 ........................................ 5

3. Scatter plots of (a) RM versus AAm based on single-cycle values (the method 

of Ohl) and (b) RM versus AAm based on 2-cma values .......................................... 6

4. Scatter plot of ASC versus RM (the Waldmeier effect) ............................................. 8

LIST OF TABLES

1. Selected solar cycle parametric values for cycles 10–24 ............................................... 9
vi

LIST OF ACRONYMS AND ABBREVIATIONS

12-mma 12-month moving average

2-cma 2-cycle moving average

AA 12-mma value of the monthly aa-geomagnetic index

AAm minimum value of 12-mma value of the monthly aa-geomagnetic index

ASC ascent duration (from minimum sunspot amplitude to maximum 

sunspot amplitude)

E(AAm) epoch of minimum value of 12-mma value of the monthly aa-geomagnetic 

index occurrence

E(Rm) epoch of minimum sunspot amplitude occurrence

PER period (from minimum sunspot amplitude of cycle n to minimum sunspot 

amplitude of cycle n+1, where n is the sunspot cycle number)

RM maximum sunspot amplitude

Rm minimum sunspot amplitude

TP Technical Publication
vii

NOMENCLATURE

cl confidence level

f fast rising cycle

n sunspot cycle number

P probability

r coefficient of correlation

r2 coefficient of determination

s slow rising cycle

se standard error of estimate

t elapsed time in months from epoch of minimum sunspot amplitude occurrence

x independent variable

y dependent variable

s standard deviation
viii
1

TECHNICAL PUBLICATION

PREDICTING THE SIZE AND TIMING OF SUNSPOT MAXIMUM FOR CYCLE 24

1. INTRODUCTION

Accurately predicting in advance the size and timing of sunspot maximum for an ongoing 

sunspot cycle continues to be one of the long-standing problems in solar physics, one that becomes 

quite relevant every 11 years or so, especially around the time of sunspot minimum (Rm).1–9 Accurate prediction of the size and timing of sunspot cycles is important for a variety of reasons, in 

particular as related to climatic change and the effects of solar cycle-related phenomena, like earthward-directed coronal mass ejections and solar flares, on human space flight and on our modern 

technological systems (satellite orbital dynamics, electrical distribution, airline operations, etc.).10–16

Some 40 years ago, A.I. Ohl noted that the level of geomagnetic activity near Rm provides 

a reasonable estimate for the expected size of the ongoing sunspot cycle,17 usually about 3 years in 

advance of maximum amplitude (RM) occurrence. Since Rm has now been ascertained for cycle 

24, having occurred in December 2008 on the basis of the 12-month moving average (12-mma) of 

monthly mean sunspot number and since a minimum in geomagnetic activity in the vicinity of Rm 

(AAm) occurrence appears to have recently been seen in September 2009, a firm estimate of cycle 

24’s RM using the method of Ohl can now be made.

The purpose of this NASA Technical Publication (TP) is to give an estimate for the expected 

size and timing of the RM for cycle 24, one based on the updated inferred statistically significant 

relationship found between RM and AAm using 12-mma values (method of Ohl). Also, an alternate 

method for predicting cycle 24’s RM using a variation of Ohl’s method, one based on using 2-cycle 

moving averages (2-cmas) of RM and AAm, is given. Finally, the ascent duration (ASC), defined as 

the elapsed time between Rm and RM, is deduced for cycle 24 from the Waldmeier effect using the 

expected value of its RM.
2
3

2. RESULTS

Figure 1 displays an epoch analysis of the variation of the 12-mma values of the AA-geomagnetic index in the vicinity of sunspot minimum (–18≤ t ≤ 18, where t is the elapsed time in months 

relative to E(Rm), the epoch of Rm), with the solid line representing the mean values of the AA 

index for cycles 12–23 and the filled circles representing values for cycle 24 (determined by the British Geological Survey).18 Across the top and to the right are denoted the occurrences (E(AAm)) and 

values (AAm) of the minimum AA-geomagnetic index for cycles 12–23. The minimum value of the 

mean occurs at about t=4–6 months, having a value of 15.5 nT. Cycles 12, 13, 15, 17, 20, 21, and 22 

had their AAm occurrences within the interval t=0–9 months, while cycles 16, 18, 19, and 23 had 

their AAm occurrences during the interval t=12–17 months. Only cycle 14 had its AAm occurrence 

prior to E(Rm). Also, cycles 17–19 and 21–23 had AAm values above the mean, while cycles 12–16 

and 20 had AAm values below the mean.

Cycle 24’s AAm appears to have minimized at t=9 months (i.e., September 2009), having a 

value of about 8.4 nT. This value is well below the mean and, in fact, is now the smallest value on 

record, falling below that previously found for cycle 14 (8.9 nT), the smallest sunspot cycle of the 

modern record (64.2). (The reader should note that all AA values prior to 1957 have been increased 

by 3 nT to account for changes in the repositioning of the magnetometers used in the determination 

of the AA index.19-20)

Figure 2(a) shows the variation of RM values for cycles 10–23, where the thin jagged line 

represents the actual cyclic values and the heavy smoother line represents the 2-cma values of RM 

(i.e., the trend line). In figure 2a, the filled circles represent longer duration cycles, those of minimumto-minimum period (PER) 11 years or more in length, while the filled triangles represent shorter 

duration cycles, those of PER less than 11 years in length. The little fs and ss appearing beside each 

cyclic value denote the ascent duration class for each cycle, where fast-rising (f) cycles have ASC<48 

months and slow-rising (s) cycles have ASC≥ 48 months. Interestingly, all f cycles tend to lie above 

the 2-cma trend line except cycle 18, and all s cycles tend to fall below the 2-cma trend line except 

cycle 15. Also, all even-numbered cycles tend to lie below the 2-cma trend line except cycle 22, and 

all odd-numbered cycles tend to lie above the 2-cma trend line without exception. The implication is 

that cycle 24 probably will be a slow-rising cycle with RM below its 2-cma trend line value, once that 

value becomes known (unknown until cycle 25’s RM is observed).

Figure 2(b) shows the corresponding cyclic AAm and 2-cma values of AAm, where the filled 

square represents the provisional minimum value of AAm for cycle 24. Comparison of the two 

curves reveals an unmistakable coupled behavior, especially as described using the 2-cma values. 

Increasing/decreasing values of AAm are strongly associated with increasing/decreasing values of 

RM. The suggestion then is that, because AAm for cycle 24 has decreased in value relative to cycle 

23’s AAm, RM for cycle 24, likewise, will decrease in value relative to cycle 23’s RM (120.8).
4

–18 –12 –6 0 6 12 18

5

10

Cycle 24

AAm(cycle 24)

30252015

Elapsed Time in Months from E(Rm), t

AA(12-mma)

19182217232016151214

2113

14 14 14 15 12 22 21 20 20 17 16 16 18 23 23 13

15

E(AAm)

22 21 19

AAm

mean (12-23)

Figure 1. Comparison of cycle 24’s AA values and mean AA values for cycles 12–23 for elapsed time

in months relative to E(Rm), –18≤ t ≤18.
5

10 15 20 25

5

10

15

20

50

100

f

f

f

f

f

f

f

f

s

s

s

s

s

s

(a)

(b)

150

200

Sunspot Cycle Number

AAm RM

Figure 2. Cyclic variation of (a) RM and (b) AAm for cycles 12–24.

Figure 3(a) displays the scatter plot of RM versus AAm for cycles 12–23 (the method of Ohl). 

The thin horizontal and vertical lines are the medians for RM and AAm, respectively. The results 

of statistical testing using Fisher’s exact test for 2×2 contingency tables is shown in the lower-right 

portion of figure 3(a).21 The probability (P) of obtaining the observed distribution or one more 

suggestive of a departure from independence (chance) is computed to be P=0.1%, based on the 

sample size of 12 sunspot/geomagnetic cycles, suggesting a very strong association between the two 

parameters. The diagonal line running from the lower left to the upper right is the inferred regression
6

5 10

5

5 0

0

15

15

14 13

16

17

y

19

18

21

22

20

20 25

50

(b)

(a)

100

150

200

AAm (2-cma)

RM (2-cma)

 y = –35.447 + 10.479x

 r = 0.972, r2 = 0.944

se = 8.7, cl >99.9%

⇒ p = 0.4% 

5 10

6

6 0

0

15

15

14

12

20

13

16

17

y

18

22 21

19

23

20 25

50

100

150

200

AAm

RM

 y = –30.492 + 10.134x

 r = 0.926, r2 = 0.857

 se = 16.6, cl >99.9%

⇒ p = 0.1% 

Figure 3. Scatter plots of (a) RM versus AAm based on single-cycle values (the method 

of Ohl) and (b) RM versus AAm based on 2-cma values.

based on linear regression analysis with the line given as y=–30.492+10.134x, where y represents 

the predicted value of RM and x represents the observed value of AAm. The inferred correlation has a coefficient of correlation (r) equal to 0.926 and a coefficient of determination (r2) equal 
7

to 0.857, suggesting that about 86% of the variance in RM can be explained simply by the variation 

of AAm alone. The standard error of estimate (se) is estimated to be about 16.6 units of smoothed 

monthly mean sunspot number and the inferred linear correlation is found to have a confidence level 

(cl)>99.9%, meaning that the inferred correlation is considered statistically very important.

Figure 3(b) is similar to figure 3(a), except now it is based on using the 2-cma values of RM 

and AAm (a variation of the method of Ohl). Although it is based on two fewer cycles, the inferred 

regression (based on linear regression analysis) is inferred to be slightly stronger, having r=0.972, 

r2=0.944, and se=8.7 units of smoothed monthly mean sunspot number. The inferred regression is 

y=–35.447+10.479x.

Using AAm=8.4 nT for cycle 24, RM for cycle 24 is expected to lie within the lower-left 

quadrant of figure 3(a). The inferred regression equation yields RM=54.6±16.6 (i.e., the ±1 –σ prediction interval). Hence, there is a 75% chance that cycle 24’s RM will measure ≤66.2, a 95% chance 

that it will measure ≤84.7, and only a 1% chance that it will measure ≥100.5.

On the basis of the observed 2-cma of AAm for cycle 23, equal to (17.5+2(15.8)+8.4)/4=

14.4 nT, one expects the 2-cma of RM for cycle 23 to lie in the lower-left quadrant of figure 3(b). The 

inferred regression equation yields the 2-cma of RM for cycle 23 to be about 115.5±8.7 (the ±1 –s

prediction interval). Hence, RM for cycle 24 can be crudely estimated to be RM=4(115.5±8.7)–

2(120.8) –158.5=61.9±34.8. Because the average deviation between predicted and observed 2-cma 

values of RM is about 6.5, cycle 24’s RM, based on the 2-cma correlation, is estimated to be about 

62±26.

Together, the two predictions strongly indicate that the RM for cycle 24 will be considerably smaller than was observed for cycle 23 (120.8), measuring probably about 55–62 in terms of 

smoothed monthly mean sunspot number. Using these values, one can easily estimate the timing of 

RM occurrence relative to E(Rm) for cycle 24 using the Waldmeier effect, a loose but statistically 

significant relationship between RM and ASC.22

Figure 4 depicts the Waldmeier effect, plotting ASC versus RM, where the number appearing beside each plotted point is the sunspot cycle number. Statistical testing, using Fisher’s exact 

test for 2×2 contingency tables, reveals a strong association between the two parameters, having 

P=0.2%, meaning that the P of obtaining the observed table or one more suggestive of independence is only 0.2%. Thus, smaller RM cycles tend to have ASC≥48 months (relative to 

E(Rm)), while larger RM cycles tend to have ASC<48 months. Of the smaller RM cycles, all 

have had ASC≥ 48 months, except cycle 13, with all smaller RM cycles having ASC≥47 months. 

Furthermore, 5 of 7 smaller RM cycles have also been cycles of longer duration (PER>11 years), 

the only exceptions being cycles 15 and 16. Because cycle 24 is predicted to be a smaller RM cycle, 

one infers that it too likely will be a slow riser (ASC≥ 48 months) and quite possibly be a cycle of 

longer duration. This suggests that RM occurrence for cycle 24 very likely will occur after December 

2012, since E(Rm) for cycle 24 occurred in December 2008, and that cycle 24 very likely will end 

sometime in 2020 or later.
8

Also shown in figure 4 is the inferred regression, ignoring cycles 14 and 19, the extremes 

in RM and considered here as statistical outliers with respect to the Waldmeier effect. Presuming 

RM=55–62 for cycle 24, the ASC for cycle 24 is estimated to be about 58–59±4 months (the ±1–s

prediction interval). Thus, RM for cycle 24 is, indeed, expected to occur sometime in 2013–14 unless, 

of course, cycle 24 turns out to be a statistical outlier. Interestingly, cycles 14 and 19, the cycles of 

extremes in terms of RM, are 5 cycles apart, and cycle 24 follows cycle 19 by exactly 5 cycles. Will 

cycle 24 also be a statistical outlier with respect to the Waldmeier effect? (The reader should note that 

cycle 9, which precedes cycle 14 by 5 cycles, also appears to be a statistical outlier with respect to the 

Waldmeier effect, having RM=131.6 and ASC=55 months.)

50

0

1 7

6

100

13 23

14

10

15

16

17

11

y*

19

18

21

22

20

150 200

30

35

40

45

50

55

60

RM

ASC

 y*= 70.46 – 0.206x

 r = –0.884, r2 = 0.781

 se = 3.5, cl >99.9%

*ignores cycles 14 and 19

12

⇒ p = 0.2% 

Figure 4. Scatter plot of ASC versus RM (the Waldmeier effect).

For convenience, table 1 is included to provide the reader with the exact values and dates 

for the selected cyclic parameters associated with the modern era sunspot cycles used in plotting 

figures 1–4.
9

Table 1. Selected solar cycle parametric values for cycles 10–24.

Cycle RM Rm E(Rm) ASC PER AA(Rm) AAm* RM(2-cma) AAm*(2-cma)

10 97.9 3.2 12-1855 50 135 – – 117.0 –

11 140.5 5.2 03-1867 41 141 – – 113.4 –

12 74.6 2.2 12-1878 60 135 9.8 9.7 94.4 –

13 87.9 5.0 03-1890 46 142 14.2 13.6 78.7 11.5

14 64.2 2.6 01-1902 49 139 9.1 8.9 80.4 10.7

15 105.4 1.5 08-1913 48 120 11.2 11.2 88.3 10.9

16 78.1 5.6 08-1923 56 121 13.4 12.4 95.2 13.1

17 119.2 3.4 09-1933 43 125 18.6 16.2 117.1 16.0

18 151.8 7.7 02-1944 39 122 26.2 19.3 156.0 18.7

19 201.3 3.4 04-1954 47 126 20.9 19.9 166.3 18.2

20 110.6 9.6 10-1964 49 140 15.2 13.8 146.8 16.8

21 164.5 12.2 06-1976 42 123 22.3 19.6 149.5 17.6

22 158.5 12.3 09-1986 34 116 18.3 17.5 150.6 17.6

23 120.8 8.0 05-1996 47 151 18.8 15.8 – 14.4 

24 – 1.7 12-2008 – – 10.4 8.4 – –

*means in the vicinity of E(Rm). For cycle 21, its true AAm occurred about 46 months after E(Rm), measuring 17.2. 
10
11

3. DISCUSSION AND SUMMARY

Several years ago during the declining portion of cycle 23, efforts were made to reach a consensus prediction for the size and timing of sunspot cycle 24. The outcome was a split prediction, 

some favoring a large RM and quick rise and others favoring a more subdued RM and slower rise. 

Pesnell provides a summary of some of these early attempts at predicting the size and timing of cycle 

24,8 as do Obridko and Shelting.7 Additionally, a useful Web site for comparing the many predictions that have been made for cycle 24 is readily available for access.9

While many methods for predicting the latter-occurring RM of a sunspot cycle have been 

proffered, the method of Ohl seems to be one of the most reliable, having a ±1 –s uncertainty of 

about 17 units of smoothed monthly mean sunspot number. The difficulty associated with obtaining 

a singular consensus prediction of RM for cycle 24 was largely caused by its delayed minimum. Early 

on, because cycle 23 had an RM larger than the mean amplitude, statistically speaking, it could be 

argued that it was destined to be a cycle of shorter duration. If true, its end (and consequently the 

conventional onset of cycle 24) was anticipated to occur prior to May 2007.23–26

Because the minimum for cycle 24 appeared most imminent in 2006, because a maximum in 

geomagnetic activity had occurred in August 2003 (the highest ever recorded for the 12-mma of AA), 

and because there exists a statistically significant relationship between the maximum of geomagnetic 

activity during the declining portion of an ongoing sunspot cycle and the size of the following cycle, 

these observations became the basis for the prediction of a large RM for cycle 24, which was also 

supportive of a particular flux-transport dynamo model prediction and of evidence for the deep 

meridional flow setting the sunspot cycle period.20, 27–32 It has since become apparent that E(Rm) 

for cycle 24 would occur later than 2006–2007, inferring that the initial prediction for cycle 24’s RM 

was premature.33–39

In this NASA TP, it has been shown that AAm for cycle 24 appears likely to have occurred 

in September 2009, some 9 months past cycle 24’s E(Rm), measuring about 8.4 nT, a value smaller 

than the smallest previous value on record (8.9 nT for cycle 14, also the smallest sunspot cycle on 

record). The consequence of this is that, unless cycle 24 proves to be a statistical outlier, its RM is 

now anticipated to be much smaller in size than previously forecast, possibly becoming the smallest 

of the modern era.40, 41 The method of Ohl predicts RM=54.6±16.6 (the ±1 –s prediction interval), while using a variation of the Ohl method (2-cma values), RM for cycle 24 is predicted to be 

about 62±26. A value of RM equal to 55–62 suggests from the Waldmeier effect that cycle 24 will 

be a slow-rising cycle, peaking probably about 58–59±4 months (the ±1–s prediction interval) after 

E(Rm) or sometime in 2013–14. Likewise, because cycle 24 is anticipated to be a slow-rising, smallamplitude sunspot cycle, statistically speaking, it should also be a cycle of longer duration, inferring 

that the onset of cycle 25 should not be expected until sometime in 2020 or later. Predicting the overall shape of cycle 24, however, will not be particularly reliable for at least another 2–3 years,42–44 as 

will be predicting its effects on the space weather environment.45–47
12

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34. Wilson, R.M.; and Hathaway, D.H.: “Using the Inflection Points and Rates of Growth and 

Decay to Predict Levels of Solar Activity,” NASA/TP—2008–215473, Marshall Space Flight 

Center, AL, September 2008.

35. Wilson, R.M.; and Hathaway, D.H.: “Anticipating Cycle 24 Minimum and Its Consequences: 

An Update,” NASA/TP—2008–215576, Marshall Space Flight Center, AL, October 2008.

36. Wilson, R.M.; and Hathaway, D.H.: “On the Period-Amplitude and Amplitude-Period Relationships,” NASA/TP—2008–215580, Marshall Space Flight Center, AL, November 2008.

37. Wilson, R.M.; and Hathaway, D.H.: “Predicting the Size of Sunspot Cycle 24 on the Basis of 

Single- and Bi-Variate Geomagnetic Precursor Methods,” NASA/TP—2009–215687, Marshall 

Space Flight Center, AL, February 2009.

38. Wilson, R.M.; and Hathaway, D.H.: “Sunspot Activity Near Cycle Minimum and What It 

Might Suggest for Cycle 24, the Next Sunspot Cycle,” NASA/TP—2009–216061, Marshall 

Space Flight Center, AL, September 2009.

39. Hathaway, D.H.; and Wilson, R.M.: “Recent Geomagnetic Activity Indicates Small Amplitude 

for Sunspot Cycle 24,” (in preparation), 2010.
15

40. Svalgaard, L.; Cliver, E.W.; and Kamide, Y.: “Sunspot Cycle 24: Smallest Cycle in 100 Years?” 

Geophys. Res. Lett., Vol. 32, p. L01104, 2005.

41. Choudhuri, A.R.; Chatterjee, P.; and Jiang, J.: “Predicting Solar Cycle 24 With a Dynamo 

Model,” Phys. Rev. Lett., Vol. 98, p. 131,103, 2007.

42. Hathaway, D.H.; Wilson, R.M.; and Reichmann, E.J.: “The Shape of the Sunspot Cycle,” Solar 

Phys., Vol. 151, p. 177, 1994.

43. Wilson, R.M.; Hathaway, D.H.; Reichmann, E.J.: “On the Correlation Between Maximum 

Amplitude and Smoothed Monthly Mean Sunspot Number During the Rise of the Cycle 

(From t=0–48 Months Past Sunspot Minimum),” NASA/TP—1998–208591, Marshall Space 

Flight Center, AL, August 1998.

44. Wilson, R.M. and Hathaway, D.H.: “Application of the Maximum Amplitude-Early Rise Correlation to Cycle 23,” NASA/TP—2004–213281, Marshall Space Flight Center, AL, June 2004.

45. Wilson, R.M.: “On the Relationship Between Transient Velocity of Interplanetary Shocks and 

Solar Active Processes,” Planet. Space Sci., Vol. 44, p. 441, 1996.

46. Hathaway, D.H.; and Wilson, R.M.: “What the Sunspot Record Tells Us About Space Climate,” Solar Phys., Vol. 224, p. 5, 2004.

47. Wilson, R.M.; and Hathaway, D.H.: “On the Relationship Between Solar Wind Speed, Earthward-Directed Coronal Mass Ejections, Geomagnetic Activity, and the Sunspot Cycle Using 

12-Month Moving Averages,” NASA/TP—2008–215413, Marshall Space Flight Center, AL, 

June 2008.
16

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Predicting the Size and Timing of Sunspot Maximum for Cycle 24

Robert M. Wilson

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

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Availability: NASA CASI (443–757–5802)

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

M–1282

Technical Publication

NASA/TP—2010–216433

solar prediction, the sunspot cycle, cycle 24, method of Ohl

01-06-2010

UU 24

NASA

U U U

For cycle 24, the minimum value of the 12-month moving average (12-mma) of the AA-geomagnetic index in the vicinity of sunspot 

minimum (AAm) appears to have occurred in September 2009, measuring about 8.4 nT and following sunspot minimum by 9 months. 

This is the lowest value of AAm ever recorded, falling below that of 8.9 nT, previously attributed to cycle 14, which also is the smallest 

maximum amplitude (RM) cycle of the modern era (RM=64.2). Based on the method of Ohl (the preferential association between RM 

and AAm for an ongoing cycle), one expects cycle 24 to have RM=55±17 (the ±1 –σ prediction interval). Instead, using a variation 

of Ohl’s method, one based on using 2-cycle moving averages (2-cma), one expects cycle 23’s 2-cma of RM to be about 115.5±8.7 (the 

±1–  σ prediction interval), inferring an RM of about 62±35 for cycle 24. Hence, it seems clear that cycle 24 will be smaller in size than 

was seen in cycle 23 (RM=120.8) and, likely, will be comparable in size to that of cycle 14. From the Waldmeier effect (the preferential 

association between the ascent duration (ASC) and RM for an ongoing cycle), one expects cycle 24 to be a slow-rising cycle (ASC≥ 48 

months), having RM occurrence after December 2012, unless it turns out to be a statistical outlier.

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