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35812
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i

NASA/TP—2011–216461

An Estimate of the Size and Shape of Sunspot

Cycle 24 Based on Its Early Cycle Behavior

Using the Hathaway-Wilson-Reichmann

Shape-Fitting Function

Robert M. Wilson

Marshall Space Flight Center, Marshall Space Flight Center, Alabama

March 2011

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/login/wt/>
iii

TABLE OF CONTENTS

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

2. RESULTS ........................................................................................................................... 3

3. DISCUSSION AND SUMMARY ..................................................................................... 16

REFERENCES ....................................................................................................................... 18
iv
v

LIST OF FIGURES

1. Variation of (a) 12-mma values of R(t) for cycle 24 (t=0–18) and the mean 

modern era sunspot cycle (t=0–60); (b) the ratio of R(t) cycle 24 values 

to mean modern era sunspot cycle values; and (c) variation of ΔR(t) values 

for cycle 24 and the mean modern era sunspot cycle ................................................... 4

2. RM versus ΔR(t) max ................................................................................................. 5

3. Variation of (a) RM and (b) aam for cycles 12–24 ...................................................... 6

4. RM versus aam (the method of Ohl) .......................................................................... 7

5. ASC versus RM (the Waldmeier effect) ...................................................................... 8

6. Comparison of 12-mma values of R(t) for cycle 24 and (a) the mean of fastrising modern era sunspot cycles; (b) the mean of slow-rising modern era 

sunspot cycles; (c) the mean of short-period modern era sunspot cycles; 

and (d) the mean of long-period modern era sunspot cycles ....................................... 10

7. Values for cycle 24 and the HWR shape-fitting function using RM=70 and 

ASC=56: (a) Comparison of 12-mma values and (b) comparison of the fd (R(t)) .... 13

8. RM versus slope at t=ΔR(t) max ............................................................................... 14
vi
vii

LIST OF TABLES

1. Hindcasts of RM and ASC based on inferred estimates ............................................. 9

2. Ratios of cycle 24 R(t) values to mean values for selected groupings of cycles 

for t=0–18 mo ............................................................................................................ 11

3. Comparison of predicted and observed R(t) values for cycle 24 using selected 

values of RM and ASC in the HWR shape-fitting function ........................................ 12

4. Values of R(t) for t=18–30 mo and slopes .................................................................. 14

5. Statistical parameters for the RM versus R(t) fits for t=18–32 mo ............................. 15
viii

LIST OF ACRONYMS AND ABBREVIATIONS

12-mma 12-month moving average

aam minimum value of the 12-mma of the aa-geomagnetic index

ASC ascent duration

<ASC> mean cycle ascent duration

E(Rm) epoch of minimum amplitude occurrence

E(ΔR(t) max) epoch of maximum month-to-month rate of change in R(t)

fd first difference (same as ΔR(t)) 

FRC fast-rising cycle

HWR Hathaway-Wilson-Reichmann

J January

LPC long-period cycle

PER period

<PER> mean cycle period

R monthly mean sunspot number

RM maximum amplitude

<RM> mean cycle maximum amplitude

Rm minimum amplitude

R(t) 12-mma of monthly mean sunspot number at elapsed time t

ΔR(t) month-to-month change in R(t)

ΔR(t) max maximum month-to-month rate of change in R(t)

SCN sunspot cycle number

SPC short-period cycle

SRC slow-rising cycle

TP Technical Publication
ix

NOMENCLATURE

a parameter in HWR shape-fitting function; also the y-intercept in the regression equation

ad average deviation

b parameter in HWR shape-fitting function; also the slope in the regression equation

c parameter in HWR shape-fitting function

cl confidence level

P probability

r coefficient of correlation

r2 coefficient of determination

se standard error of estimate

t elapse time in months from E(Rm) to E(R(t))

t0 initial starting point

x independent variable in regression equation

y dependent variable in regression equation

y′ alternate dependent variable in the regression equation
x
1

TECHNICAL PUBLICATION

AN ESTIMATE OF THE SIZE AND SHAPE OF SUNSPOT CYCLE 24 BASED 

ON ITS EARLY CYCLE BEHAVIOR USING THE HATHAWAYWILSON-REICHMANN SHAPE-FITTING FUNCTION

1. INTRODUCTION

Having a reliable estimate for the strength and duration of solar activity over the course of 

the solar cycle is important for a variety of reasons. Namely, it provides a decadal overview of the 

expected space weather conditions that likely will prevail years in advance, which is important for 

planning space missions to the moon, asteroids, and planets, and which also is important for climatology, electrical power distribution, and global communication. Additionally, it possibly can 

provide an early indication for establishing the start, end, or continuance of a suspected long-term 

anomalous trend in solar activity (e.g., a continuing secular rise or fall, or the occurrence of another 

Maunder- or Dalton-like minimum or a Grand- or Modern-like maximum).1–15

In this Technical Publication (TP), the very early solar/geomagnetic activity of cycle 24 is 

examined to ascertain the likely level of solar activity over the remainder of the solar cycle. In particular, (1) 12-month moving averages (12-mma) of monthly mean sunspot number (R) for cycle 24 

are compared against 12-mma values for the mean of cycles 10–23 (the modern sunspot era); (2) its 

maximum month-to-month rate of rise (ΔR(t) max), as observed thus far, is compared to the maximum month-to-month rate of rise for the mean cycle;16 (3) the minimum amplitude of the 12-mma 

aa-geomagnetic index (aam) for cycle 24 is compared against those for other sunspot cycles; (4) the 

method of Ohl17–22 is used to predict the expected maximum amplitude (RM) for cycle 24; (5) the 

Waldmeier effect23,24 is used to estimate the ascent duration (ASC) (i.e., the elapsed time between 

minimum and maximum amplitude occurrences) for cycle 24; (6) 12-mma averages of monthly 

mean sunspot number are compared against 12-mma mean values for selected groupings of sunspot 

cycles (e.g., slow- and fast-rising cycles and long- and short-period cycles); and (7) 12-mma values of monthly mean sunspot number are compared against selected Hathaway-Wilson-Reichmann 

(HWR) shape-fitting functions25 to determine the best fit for the very early behavior of cycle 24.
2
3

2. RESULTS

Figure 1 shows (a) R(t) for cycle 24 (the filled circles, elapsed time t=0–18 mo) and those for 

the mean modern era sunspot cycle (the smoothed line, t=0–60 mo, based on an epoch analysis of 

cycles 10–23 using minimum amplitude occurrence as the common epoch for comparison), (b) the 

ratio of cycle 24 R(t) values to those of the mean modern era sunspot cycle, and (c) the month-tomonth rate of rise (ΔR(t)) using the 12-mma values of cycle 24 and the mean modern era sunspot 

cycle. Also shown are RM, ASC, ΔR(t) max, and E(ΔR(t) max) for the modern era sunspot cycles. 

Clearly, cycle 14 is the smallest modern era sunspot cycle (RM = 64.2) yet observed, while cycle 19 is 

the largest (201.3). Also, cycle 12 is the slowest rising modern era sunspot cycle (ASC = 60 mo) yet 

observed, while cycle 22 is the fastest rising (34 mo). Of the 14 modern era sunspot cycles, 7 have had 

RM ≤110.6 (cycle 20) and 8 have had ASC ≤47 mo (cycles 19 and 23).

Regarding ΔR(t) max, cycle 14 has the smallest maximum month-to-month rate of rise (3.9) 

yet observed, while cycle 19 has the largest (10.8). Cycle 12 has the earliest occurring ΔR(t) max 

(14 mo) yet observed, while cycle 15 has the latest occurring ΔR(t) max (42 mo). Of the 14 modern 

era sunspot cycles, 8 have had ΔR(t) max ≤6.4 (cycles 17 and 20), 7 have had ΔR(t) max occurring 

between 14 and 24 mo, and 7 have had ΔR(t) max occurring between 30 and 42 mo.

Thus far, R(t) values for cycle 24 have remained well below that of the mean modern era 

sunspot cycle, having a ratio against the mean cycle (R(t) cycle 24 to R(t) mean cycle) <0.5, inferring 

that, if this ratio continues through the occurrence of maximum amplitude, RM for cycle 24 will be 

<60. Likewise, the maximum ΔR(t) yet observed for cycle 24 has remained ≤1.7, a rate less than half 

the maximum month-to-month rate of rise observed for cycle 14 (3.9), the smallest cycle of the modern era. Together, these observations suggest that cycle 24 potentially might become the new smallest 

cycle of the modern era,26 and, possibly herald the occurrence of another Maunder- or Dalton-like 

minimum.
4

 

2218

2113101612

19 (201.3)

21 (164.5)

22 (158.5)

18 (151.8)

11 (140.5)

23 (120.8)

RM

ASC

17 (119.2)

20 (110.6)

15 (105.4)

10 (97.9)

13 (87.9)

16 (78.1)

12 (74.6)

14 (64.2)

19 (10.8)

22 (10.6)

11 (9.5)

18 (8.6) 15 (8.5) 21 (8.1)

17, 20 (6.4)

16 (6.3)

23 (5.5) 13 (4.6) 10, 12 (4.1)

14 (3.9)

∆R(t) max

E(∆R(t) max)

19, 23 15 14, 20

1117

1213222111181014, 17 2315

16, 19, 20

200

175

∆R(t) R(t) R(24)/R(mean)

150

125

100

75

50

25

(a)

1.5

1.0

0.5

0

(b)

12

6

0

0 10 20 30

t

(c) 40 50 60

0

0 10 20 30 40 50 60

0 10 20 30 40 50 60

Figure 1. Variation of (a) 12-mma values of R(t) for cycle 24 (t=0–18) and the mean 

modern era sunspot cycle (t=0–60); (b) the ratio of R(t) cycle 24 values 

to mean modern era sunspot cycle values; and (c) variation of ΔR(t) values 

for cycle 24 and the mean modern era sunspot cycle.
5

Figure 2 depicts RM versus ΔR(t) max for the modern era sunspot cycles, where ΔR(t) max 

is the maximum rate of growth in R during the ascending phase of the sunspot cycle (ΔR(t) max for 

cycle 14 actually is 4.5; however, this value occurred after the occurrence of its maximum amplitude). 

A strong preferential linear correlation is indicated, one having a coefficient of correlation r=0.86, 

a standard error of estimate se=21.3, and a confidence level cl >99.9% for the inferred regression. 

The inferred regression equation is y=22.041+14.032x, where y represents RM and x represents 

ΔR(t) max, and the inferred regression can explain about 73% of the variance in RM (i.e., the coefficient of determination r 2=0.73). For cycle 24, because the maximum value for ΔR(t) is presently 

only 1.7 (through t=18 mo), one infers cycle 24’s RM=45.9±21.3 (the ±1 se prediction interval). 

Clearly, however, since t has only recently entered the window when one expects to find ΔR(t) max 

(t ≥14 mo), one can not know with complete certainty that ΔR(t) max for cycle 24 has indeed been 

observed. The value of 1.7 could easily be exceeded in the coming months, which, if true, would 

indicate a higher RM for cycle 24. Fisher’s exact test27 for 2×2 contingency tables shows that the 

probability of obtaining the observed result (6:1:5:2), or one more suggestive of a departure from 

independence, is P=5.1%. Thus, RM for cycle 24 is expected to lie within the lower-left quadrant 

(i.e., RM<114.9) if ΔR(t) max <6.4. (In fig. 2 and succeeding figs. 3–5 and 8, filled circles denote 

slow-rising, long-period cycles—those having ASC ≥48 mo and PER ≥132 mo; unfilled circles denote 

fast-rising, long-period cycles; filled triangles denote slow-rising, short-period cycles; and unfilled 

triangles denote fast-rising, short-period cycles.)

 ∆R(t) max RM

50

0 5 10

1 6

15

17 23

18

21

19

y

22

11

20

10

13

12

14

16

5 2

100

150

200

 y = 22.041 + 14.032x

r = 0.856, r 2 0.732

se = 21.3, cl >99.9%

⇒P = 5.1%

Figure 2. RM versus ΔR(t) max.
6

Figure 3 shows the cyclic variation of (a) RM and (b) the minimum value of the 12-mma aageomagnetic index (aam) for the modern era sunspot cycle, where SCN refers to the sunspot cycle 

number. Noticeable is a seemingly strong preferential behavioral correlation between RM and aam, 

one depicting a rise in the level of solar activity between cycles 14 and 19, a dip in cycle 20, resurgence in cycle 21, and a decline thereafter. The horizontal lines denote the parametric medians and 

the unfilled box (cycle 24) simply denotes that presently the nature of cycle 24 remains unknown (i.e., 

whether its ASC is one typical of fast- or slow-rising cycles and whether its PER is one typical of 

long- or short-period cycles). While true, because of the inferred closely correlated behavior between 

RM and aam, one strongly suspects that cycle 24 will have RM below the median (114.9), being of 

similar size to early cycles 12 and 14.

13.8

114.9

aam RM

5

10

15

20

50

(a)

(b)

100

200

150

SCN

SCN

10 15 20 25

10 15 20 25

Figure 3. Variation of (a) RM and (b) aam for cycles 12–24.
7

Figure 4 compares RM and aam for cycles 12–23, the so-called method of Ohl. Some 

40 years ago, Ohl17,18 noted that the minimum level of geomagnetic activity in the vicinity of sunspot cycle minimum amplitude (Rm) occurrence provides a simple means for estimating the later 

occurring expected size (maximum amplitude RM) of the ongoing sunspot cycle, usually about 

3 years in advance. The observed scatter plot suggests a very strong preferential linear relationship 

between RM and aam, one having r=0.93 and se=16.6. Hence, given the value of aam for a sunspot cycle, one can infer its expected size usually to within about 11%, on average. For cycle 24, its 

observed aam value of 8.4 (the lowest on record, denoted by the downward-pointing arrow along 

the x-axis and occurring at t=8 and 9 mo past E(Rm)) suggests RM=54.6±16.6 (based on the 

inferred regression), 54.6±12.8 (based on the average deviation, ad), or 54.6±6 (based on the 11% 

average percentage of deviation; the maximum observed percentage of deviation for the modern 

era sunspot cycles is about 22%). From the inferred regression, one finds that there is only a 1% 

chance that RM for cycle 24 will exceed 100.5. Hence, RM for cycle 24 very probably will lie within 

the lower-left quadrant of figure 4, unless, of course, cycle 24 should turn out to be a statistical outlier. (It should be noted that aam usually occurs between t=0–8 mo following E(Rm), true for 7 of 

12 sunspot cycles, including cycles 12, 13, 15, 17, 20, and 21; only cycle 14 had its aam prior to E(Rm), 

having occurred at t= –8, –5, and –4 mo, and cycles 16, 18, 19, and 23 had a later occurring aam, at 

t=12 –17 mo.)

5 10

Cycle 24

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, r 2 = 0.857

 se = 16.6, cl >99.9%

⇒ P = 0.1% 

Figure 4. RM versus aam (the method of Ohl).

Figure 5 depicts the scatter plot of ASC versus RM, demonstrating the so-called Waldmeier 

effect.23,24 Namely, stronger cycles tend to rise to maximum amplitude more quickly than weaker 

cycles. Based on the observed 2×2 contingency table, the probability of obtaining the observed
8

RM

ASC

30

35

40

45

50

55

60

50 100 150 200

6 0

21

19

48

22

18

17

15

16

23

y

y ́

11

10

12 114.9

14 20

13

1 7

 y = 59.770 – 0.111x

 r = –0.658, r 2 0.433

se = 5.3, cl >98%

RM(24) = 55 ⇒ ASC = 54 ± 5 mo

⇒P = 0.2%

 y ́ = 70.460 – 0.206x

 r = –0.884, r 2 = 0.781

se = 3.5, cl >99.9%

RM(24) = 55 ⇒ ASC(24) = 59 ± 4 mo

Excludes cycles

14 and 19.

Figure 5. ASC versus RM (the Waldmeier effect).

result, or one more suggestive of a departure from independence, is P=0.2% (from Fisher’s exact test 

for 2×2 contingency tables). Hence, a sunspot cycle having RM <114.9 is strongly expected to have 

ASC ≥48 mo. Because the predicted value for cycle 24’s RM is <114.9 (based on the method of Ohl), 

it follows that one expects cycle 24 to be a slow-rising cycle (i.e., ASC ≥48 mo), inferring E(RM) for 

cycle 24 after December 2012.

Also shown in figure 5 are two inferred regression lines, one using all modern era sunspot 

cycles (y) and the other using all modern era sunspot cycles except the extremes (y′, cycles 14 and 19). 

Based on the predicted value of RM=55, one infers ASC=54±5 (using y) or ASC=59±4 (using y′) 

for cycle 24. Hence, cycle 24’s RM is expected to occur sometime in 2013 to early 2014 (the prediction 

intervals are ±1 se.)

Table 1 summarizes hindcasts of RM and ASC based on the method of Ohl and the Waldmeier effect. Given in the table are the observed values of aam, RM, and ASC for cycles 12–23 and 

the aam value for cycle 24. The predicted value of RM is deduced using the inferred preferential 

regression from the method of Ohl (fig. 4). The predicted value of ASC is deduced using the inferred 

preferential regression from the Waldmeier effect (y, fig. 5) based on the predicted value of RM from 

the method of Ohl. The differences (observed minus predicted) are identified for both RM and ASC. 

The average deviation (ad) appears at the bottom of the differences.
9

Table 1. Hindcasts of RM and ASC based on inferred estimates.

Cycle

Observed Predicted Differences

aam RM ASC RM ASC RM ASC

12

13

14

15

16

17

18

19

20

21

22

23

24

9.7

13.6

8.9

11.2

12.4

16.2

19.3

19.9

13.8

19.6

17.5

15.8

8.4

74.6

87.9

64.2

105.4

78.1

119.2

151.8

201.3

110.6

164.5

158.5

120.8

–

60

46

49

48

56

43

39

47

49

42

34

47

–

67.8

107.3

59.7

83.0

95.2

133.7

165.1

171.2

109.4

168.1

146.9

129.6

54.6

52

48

53

51

49

45

41

41

48

41

43

45

54

6.8

–19.4

4.5

22.4

–17.1

–14.5

–13.3

30.1

1.2

–3.6

11.6

–8.8

–

8

–2

–4

–3

7

–2

–2

6

1

1

–9

2

–

ad 12.8 4

Figure 6 compares the observed early rise of cycle 24 (the filled circles) with the mean of 

(a) fast-rising cycles (FRC), (b) slow-rising cycles (SRC), (c) short-period cycles (SPC), and (d) 

long-period cycles (LPC), where the means are computed for t=0–36 mo using E(Rm) as the common epoch for comparison. Also given in subpanels (a)–(d) are the mean values of RM, ASC, and 

PER for each grouping of sunspot cycles. Hence, for the FRC group, on average, <RM>=144±34, 

<ASC> =42±4 mo, and <PER> = 131±12 mo (±1 se prediction intervals). For the SRC group, the 

means are, respectively, about 89±19, 52±5, and 132±9; for the SPC group, the means are, respectively, about 140±41, 44±7, and 122±3; and for the LPC group, the means are, respectively, about 

100±27, 49±6, and 140±5. Noticeable is that the R values for cycle 24 have remained well below the 

monthly means of all cycle groupings for t=0–18 mo, inferring that RM for cycle 24 very likely will 

be far smaller than indicated by the group cycle means.

Table 2 gives the ratios of R(t) values for cycle 24 to the mean cycle and to each grouping 

of sunspot cycles for t=0–18 mo. As previously noted, in comparison to the mean cycle, cycle 24 is 

behaving as if its R(t) values will remain below 0.5, inferring that its RM will be <60. Compared to 

the FRC group, the ratio has remained below about 0.4, also inferring RM<60. Similarly, for the 

other groups, SRC, SPC, and LPC, the ratios have remained below about 0.6, 0.4, and 0.5, respectively, inferring RM<55. However, because ΔR(t) max probably has not yet been observed, one 

suspects that the ratios will become slightly larger, indicating a slightly higher predicted RM for 

cycle 24. Recall that first difference R(t) values (fd R(t)) have only recently entered the window when 

one finds ΔR(t) max (t ≥14 mo), so the maximum rate of rise probably has not yet been seen. (The 

window extends from 14 to 42 mo, based on the modern era sunspot cycles, with the inflection point 

occurring, on average, about 28 mo past E(Rm); 7 of 14 inflection points have occurred between 

t=14–24 mo and 10 of 14 inflection points have occurred between t=14–34 mo, with only cycles 14, 

15, 17 and 23 having maximum month-to-month rates of rise longer than 3 years past minimum 

amplitude.)
10

Long-period cycle (PER ≥ 132 mo)

mean (LPC)

mean (SPC)

<RM> = 99.5 ± 26.6

<ASC> = 49 ± 6

<PER> = 140 ± 5

Short-period cycle (PER < 132 mo)

<RM> = 139.8 ± 41.4

<ASC> = 44 ± 7

<PER> = 122 ± 3

mean (SRC)

Slow-rising cycle (ASC ≥ 48 mo)

<RM> = 88.5 ± 18.7

<ASC> = 52 ± 5

<PER> = 132 ± 9

mean (FRC)

Fast-rising cycle (ASC < 48 mo)

<RM> = 143.1 ± 34.4

<ASC> = 42 ± 4

<PER> = 131 ± 12

t (Elapsed Time in Months from E(Rm))

R

(d)

50

0

0 6 12 18 24 30 36

100

150

(c)

(b)

(a)

t (Elapsed Time in Months from E(Rm))

R

50

0

0 6 12 18 24 30 36

100

150

t (Elapsed Time in Months from E(Rm))

R

50

0

0 6 12 18 24 30 36

100

150

t (Elapsed Time in Months from E(Rm))

R

50

0

0 6 12 18 24 30 36

100

150

Figure 6. Comparison of 12-mma values of R(t) for cycle 24 and (a) the mean 

of fast-rising modern era sunspot cycles; (b) the mean of slow-rising 

modern era sunspot cycles; (c) the mean of short-period modern era 

sunspot cycles; and (d) the mean of long-period modern era sunspot cycles.
11

Table 2. Ratios of cycle 24 R(t) values to mean values for selected 

groupings of cycles for t=0–18 mo.

t

 Mean

RM = 119.7

SRC

RM = 88.5

FRC

RM = 143.1

 LPC

RM = 99.5

 SPC

RM = 139.8

 0

 1

 2

 3

 4

 5

 6

 7

 8

 9

10

11

12

13

14

15

16

17

18

0.29

0.29

0.27

0.27

0.27

0.26

0.27

0.33

0.39

0.45

0.45

0.43

0.41

0.41

0.42

0.43

0.43

0.43

0.41

0.41

0.42

0.39

0.36

0.35

0.34

0.36

0.42

0.51

0.60

0.61

0.59

0.49

0.50

0.57

0.51

0.58

0.58

0.55

0.24

0.24

0.22

0.22

0.23

0.22

0.23

0.28

0.33

0.38

0.37

0.35

0.34

0.34

0.35

0.36

0.37

0.36

0.35

0.33

0.33

0.32

0.31

0.32

0.30

0.32

0.38

0.45

0.51

0.51

0.48

0.46

0.46

0.47

0.49

0.49

0.49

0.47

0.26

0.26

0.23

0.23

0.24

0.22

0.24

0.29

0.34

0.39

0.40

0.38

0.37

0.37

0.38

0.39

0.39

0.39

0.36

Previously, HWR25 described a simple function, consisting of two parameters (RM and 

ASC), that appears to adequately reproduce the general shape (using sunspot number) of each sunspot cycle. The formulation (adapted for using 12-mma values of R) is

R(t)=a(t– t

0)3/{exp[(t– t0)2/b2] – c} , (1)

where a=(RM)b–3/0.504, b=(ASC)/1.081, c=0.71, and t0 is the initial starting point (E(Rm)). Using 

selected values of RM and ASC, one can generate numerous shapes for the sunspot cycle curve and 

compare them to actual values of R during the early rising phase of the sunspot cycle for a determination of the most appropriate fit.

Table 3 summarizes six cases that were examined for comparison to the early growth phase 

of cycle 24. The computed values using the HWR shape-fitting function extend 2 yr, although at 

the time of writing, R values for cycle 24 are known only t=18 mo. The closest fit to cycle 24 values 

appears to be that of case 6, which uses RM=70 and ASC=56 mo.
12

Table 3. Comparison of predicted and observed R(t) values for cycle 24 using selected 

values of RM and ASC in the HWR shape-fitting function.

t

R(t)(Predicted)

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 R(t)(Observed)

 0

 1

 2

 3

 4

 5

 6

 7

 8

 9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

0.0

0.0

0.0

0.1

0.1

0.3

0.4

0.7

1.0

1.4

1.8

2.4

3.0

3.7

4.5

5.3

6.3

7.3

8.3

9.4

10.6

11.7

13.0

14.2

15.5

0.0

0.0

0.0

0.0

0.1

0.2

0.3

0.5

0.7

1.0

1.3

1.7

2.2

2.7

3.3

3.9

4.6

5.4

6.2

7.1

8.0

9.0

10.0

11.0

12.1

0.0

0.0

0.0

0.1

0.2

0.4

0.6

1.0

1.4

2.0

2.6

3.4

4.3

5.3

6.5

7.7

9.0

10.4

11.9

13.5

15.1

16.8

18.5

20.2

22.0

0.0

0.0

0.0

0.1

0.1

0.3

0.5

0.8

1.1

1.5

2.1

2.7

3.4

4.2

5.1

6.1

7.2

8.4

9.7

11.0

12.4

13.8

15.3

16.9

18.4

0.0

0.0

0.0

0.1

0.3

0.5

0.9

1.4

2.0

2.8

3.7

4.8

6.1

7.5

9.1

10.7

12.6

14.5

16.5

18.7

20.9

23.1

25.4

27.8

30.1

0.0

0.0

0.0

0.1

0.2

0.4

0.7

1.1

1.6

2.3

3.0

4.0

5.0

6.2

7.5

8.9

10.5

12.2

13.9

15.8

17.7

19.7

21.8

23.9

26.1

1.7

1.8

1.9

2.0

2.2

2.3

2.7

3.6

4.8

6.2

7.1

7.6

8.3

9.3

10.6

12.3

14.0

15.5

16.4

–

–

–

–

–

–

Case 1: RM = 40 and ASC = 55 mo, implying a = 0.000603 and b = 50.8788 in the HWR fit

Case 2: RM = 40 and ASC = 62 mo, implying a = 0.000421 and b = 57.3543 in the HWR fit

Case 3: RM = 55 and ASC = 54 mo, implying a = 0.000875 and b = 49.9537 in the HWR fit

Case 4: RM = 55 and ASC = 59 mo, implying a = 0.000671 and b = 54.5791 in the HWR fit

Case 5: RM = 70 and ASC = 52 mo, implying a = 0.001248 and b = 48.1036 in the HWR fit

Case 6: RM = 70 and ASC = 56 mo, implying a = 0.000999 and b = 51.8039 in the HWR fit

Figure 7 depicts an initial projection for cycle 24 based on case 6 for t=0–140 mo. The box 

spans 49–72 in terms of R(t) and 49–63 mo in terms of t, being based on predictions of RM and 

ASC for cycle 24 discussed earlier in the text. Thus, cycle 24 is expected to be at or near RM sometime between January 2013 and March 2014, being of comparable size to that of cycle 14. With the 

inclusion of additional observations of R over the next year or so, one should be able to significantly 

improve the projection of solar activity for the remainder of cycle 24. For example, determining the 

maximum rate of growth during the ascending phase of cycle 24 and determining exactly when it 

occurs relative to E(Rm) (the inflection point) should greatly improve the prediction of RM and the 

overall fit using the HWR shape-fitting function.
13

J 2009 J 2010 J 2011 J 2012 J 2013 J 2014 J 2015 2016 2017 2018 2019 2020

Calendar Year–Month

J J J J J J

Cycle 24 Projection

(b) t

(a)R(t) fd (R(t)) –12

–8

–4

4

8

25

0

50

75

100

12

0

0 12 24 36 48 60 72 84 96 108 120 132

t

0 12 24 36 48 60 72 84 96 108 120 132

Figure 7. Values for cycle 24 and the HWR shape-fitting function using RM=70 and ASC=56: 

(a) Comparison of 12-mma values and (b) comparison of the fd (R(t)).

Previously, Wilson and Hathaway16 examined various aspects of using the inflection point 

during the ascending phase to determine predictions of the size and timing of maximum amplitude 

for sunspot cycles. In particular, they noted that a highly statistically significant association exists 

between RM and the slope of the line drawn between E(Rm) and E(ΔR(t) max). Figure 8 displays 

the scatter plot of RM versus the slope of the line drawn between E(Rm) and E(ΔR(t) max). The 

inferred preferential regression has r=0.97 and se=11, suggesting a slightly tighter fit than the fit 

for RM versus ΔR(t) max (fig. 2). Interesting is that all cycles having a slope larger than the median 

(2.25) at E(ΔR(t) max) have RM >114.9 and are fast-rising cycles, with 5 of 7 also being short-period 

cycles, while all cycles having a slope smaller than the median have RM<114.9, with 6 of 7 being 

slow-rising cycles and 5 of 7 being long-period cycles. Thus, the slope measured at E(ΔR(t) max) 

provides a strong indication of the size of the ongoing sunspot cycle (14 of 14 cycles), the quickness of its rise to maximum amplitude (13 of 14 cycles), and the likely duration of the cycle (10 of 

14 cycles).
14

1.0 2.0 3.0 4.0

7

7 0

0

15

14

12

10

20

16 13

17

y

21 22

11

18

19

23

2.25

114.9

50

0

100

150

200

Slope at t = ∆R(t) max

RM

 y = 17.945 + 41.301x

 r = 0.966, r 2 = 0.934

 se = 10.6, cl >99.9%

⇒ P < 0.1% 

Figure 8. RM versus slope at t=ΔR(t) max.

Unfortunately, one does not know precisely when ΔR(t) max occurs until it actually has been 

observed. Instead, one can only monitor the month-to-month changes, applying the values in the formulations of figures 2 and 8 to deduce possible values for RM. Table 4 summarizes (a) the observed 

R(t) values for cycles 10–23 for t=18–32 mo and (b) the resultant slopes, and table 5 provides the 

statistical parameters associated with fits of RM versus R(t). For comparison, the R(18) value and 

its associated slope for cycle 24 is included in table 4. Plainly, cycle 24’s R(18) value (16.4) and associated slope (0.82) are the smallest on record. Applying the R(18) value in the inferred regression for 

R(18) values, one computes RM for cycle 24 to be about 74±26.

Table 4. Values of R(t) for t=18–30 mo and slopes.

a. R(t) Values

Cycle Rm RM R(18) R(19) R(20) R(21) R(22) R(23) R(24) R(25) R(26) R(27) R(28) R(29) R(30) R(31) R(32)

10 3.2 97.9 21.5 23.8 26.0 29.4 32.7 34.3 36.0 38.6 41.7 44.8 48.5 51.5 53.6 56.7 60.7

11 5.2 140.5 45.8 47.1 50.5 56.9 61.4 64.6 68.0 69.4 70.1 72.4 74.6 77.6 84.3 93.8 101.7

12 2.2 74.6 31.3 32.8 34.4 36.8 39.5 41.6 43.6 47.0 49.7 49.6 49.9 51.8 53.5 54.6 55.6

13 5.0 87.9 45.3 50.0 53.7 56.5 58.4 62.0 65.2 66.4 68.1 71.0 73.2 73.4 73.9 75.3 76.3

14 2.6 64.2 25.4 26.6 27.9 29.6 31.4 33.5 35.5 37.7 39.7 41.1 41.5 41.6 42.9 46.4 49.8

15 1.5 105.4 34.8 38.9 42.3 45.3 46.9 48.3 49.8 51.5 53.9 56.9 58.6 57.8 55.6 54.0 53.7

16 5.6 78.1 27.1 29.3 32.6 35.9 40.9 47.2 51.8 55.6 57.7 58.9 60.9 62.6 64.1 65.1 65.2

17 3.4 119.2 22.0 25.6 29.9 34.2 37.9 42.0 46.5 51.3 55.0 57.2 59.0 62.2 65.9 68.8 72.5

18 7.7 151.8 38.6 43.9 48.1 52.1 56.0 60.6 67.0 72.9 76.8 81.4 88.6 95.3 100.2 104.3 109.6

19 3.4 201.3 64.4 73.0 81.0 88.8 98.5 109.3 118.7 127.4 136.9 145.5 149.6 151.5 155.8 159.6 164.3

20 9.6 110.6 37.4 40.7 44.7 50.3 56.7 63.1 67.6 70.2 72.7 75.0 78.8 82.2 84.6 87.5 91.3

21 12.2 164.5 56.9 61.3 64.5 69.6 76.9 83.2 89.3 97.4 104.0 108.4 111.1 113.3 117.7 123.7 130.9

22 12.3 158.5 71.3 77.5 83.8 93.7 104.3 113.7 121.2 125.3 130.4 137.6 142.0 145.0 149.7 153.5 156.9

23 8.0 120.8 35.0 39.0 43.7 48.9 53.4 56.5 59.4 62.5 65.4 67.8 69.5 70.5 73.0 77.9 82.6

24 1.7 – 16.4
15

Table 4. Values of R(t) for t=18–30 mo and slopes. (Continued)

b. Slopes

Cycle R(18) R(19) R(20) R(21) R(22) R(23) R(24) R(25) R(26) R(27) R(28) R(29) R(30) R(31) R(32)

10 1.02 1.08 1.14 1.25 1.34 1.35 1.37 1.42 1.48 1.54 1.62 1.67 1.68 1.73 1.80

11 2.26 2.21 2.27 2.46 2.55 2.58 2.62 2.57 2.50 2.49 2.48 2.50 2.64 2.86 3.02

12 1.62 1.61 1.61 1.65 1.70 1.71 1.73 1.79 1.83 1.76 1.70 1.71 1.71 1.69 1.67

13 2.24 2.37 2.44 2.45 2.43 2.48 2.51 2.46 2.43 2.44 2.44 2.36 2.30 2.27 2.23

14 1.27 1.26 1.27 1.29 1.31 1.34 1.37 1.40 1.43 1.43 1.39 1.34 1.34 1.41 1.48

15 1.85 1.97 2.04 2.09 2.06 2.03 2.01 2.00 2.02 2.05 2.04 1.94 1.80 1.69 1.63

16 1.19 1.25 1.35 1.44 1.86 1.81 1.93 2.00 2.00 1.97 1.98 1.97 1.95 1.92 1.86

17 1.03 1.17 1.33 1.47 1.57 1.68 1.80 1.92 1.98 1.99 1.99 2.03 2.08 2.11 2.16

18 1.72 1.91 2.02 2.11 2.20 2.30 2.47 2.61 2.66 2.73 2.89 3.02 3.08 3.12 3.18

19 3.39 3.66 3.88 4.07 4.32 4.60 4.80 4.96 5.13 5.26 5.22 5.11 5.08 5.04 5.03

20 1.54 1.64 1.76 1.94 2.14 2.33 2.42 2.42 2.43 2.42 2.47 2.50 2.50 2.51 2.55

21 2.48 2.58 2.62 2.73 2.94 3.09 3.21 3.41 3.53 3.56 3.53 3.49 3.52 3.60 3.71

22 3.28 3.43 3.58 3.88 4.18 4.41 4.54 4.52 4.54 4.64 4.63 4.58 4.58 4.55 4.52

23 1.50 1.63 1.79 1.95 2.06 2.11 2.14 2.18 2.21 2.21 2.20 2.16 2.17 2.25 2.33

24 0.82

Note: Slope is computed as [R(t) – Rm]/t.

Table 5. Statistical parameters for the RM 

versus R(t) fits for t=18–32 mo.

t r r × r a b se cl (%)

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

0.771

0.796

0.814

0.826

0.834

0.833

0.843

0.860

0.869

0.875

0.883

0.892

0.901

0.911

0.922

0.595

0.633

0.662

0.683

0.695

0.694

0.711

0.740

0.755

0.765

0.779

0.795

0.812

0.830

0.850

41.327

39.255

36.751

35.872

37.156

39.227

39.202

37.122

37.263

38.601

37.405

35.557

35.008

33.203

31.656

1.970

1.847

1.751

1.611

1.453

1.310

1.225

1.187

1.129

1.063

1.041

1.036

1.009

0.991

0.969

26.123

24.887

23.815

23.214

22.711

22.637

22.085

21.039

20.260

19.932

19.457

18.677

17.737

16.997

16.059

99.8

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9

99.9
16

3. DISCUSSION AND SUMMARY

The term ‘sunspot cycle’ refers to the quasi-periodic waxing and waning of sunspots over 

decadal time. While sunspots have been viewed telescopically since Galileo, the existence of the sunspot cycle itself has been recognized as being a real feature of the Sun only since the early-to-mid 

1800s.28–32 Sunspot cycles are numbered sequentially from minimum amplitude to minimum amplitude, with the present sunspot cycle being number 24 (this numbering based on the reconstruction 

performed by Rudolf Wolf33), having had its minimum amplitude in December 2008. At present, 

cycle 24 is in its rising phase towards maximum amplitude, which for the modern era sunspot cycles 

has followed minimum amplitude by 3–5 yr, dependent upon the strength of the cycle (strong cycles 

peaking earlier than weak cycles). Following maximum amplitude, which really is better described as 

a period of several years of enhanced solar activity rather than a specific point in time, the activity 

level will slowly subside until a level once again indicative of minimum amplitude is encountered. 

At minimum, a 1- to 3-yr overlap of old and new cycle spots34 typically is seen, with the old cycle 

spots at low solar latitudes and the new cycle spots at higher solar latitudes (≥30 deg; see Wilson and 

Hathaway24).

Since the Maunder minimum of the late 1600s (a prolonged interval of time of several decades 

when few sunspots were reported), there has been a general upward secular increase with time in 

solar activity.1,35 While other Maunder-like minima have been proposed, like the Dalton minimum 

(cycles 5 and 6, in the early 1800s) and the Modern minimum (cycles 12–14, in the late 1800s to 

early 1900s), the overall trend since the Maunder minimum has been generally upward, with a broad 

peak believed to have occurred about cycles 18–22 (the so-called Modern maximum, the mid-to-late 

1900s). The trend in solar activity now, however, appears to be in decline, since cycle 23 was smaller 

than cycle 22, which also was smaller than cycle 21 (at least, in terms of 12-mma values of R).

Predictions for the size (and timing) of cycle 24 based on a variety of techniques,36 including precursor, climatological, dynamo, neural network, and spectral models, have resulted in a wide 

range of values, from the very small to the very large. Obviously, some of these predictions will be 

deemed crudely successful, while others will fail miserably, thereby, leading, perhaps, to a better 

understanding of the physics involved in the generation of the solar cycle37–40 and more accurate 

prediction of future sunspot cycles.

In this TP, the size and timing of cycle 24 has been estimated based on its early cycle behavior 

and application of the HWR shape-fitting function.25 It is apparent that cycle 24 seems destined 

to be a cycle weaker in strength than was seen in cycle 23, which had RM=120.8. If true, then the 

apparent downward trend in solar activity that has been underway since about cycles 21 or 22 continues and, perhaps, will result in the occurrence of another minimum of sunspot activity (like that 

which occurred during the Maunder, Dalton, or Modern minima).
17

Cycle 24’s early behavior through t=18 mo has continued to fall well below the activity level 

associated with the mean of cycles 10–23. At t=18 mo past minimum (recall that 12-mma values 

trail current calendar time by 6 mo), the ratio of cycle 24’s 12-mma value of R to that of the mean 

cycle is 0.41, inferring that cycle 24 will have an RM less than half the size of the mean modern era 

sunspot cycle, or about 50, provided that the ratio remains the same through the occurrence of its 

maximum amplitude. Mitigating this, however, is the belief that the maximum rate of change in the 

month-to-month R values has not yet been seen, which, if true, suggests that a larger ratio (>0.41) 

and larger RM (>50) will be seen in the months ahead for cycle 24.

To date (through t=18 mo), the maximum month-to-month rate of change (i.e., the fd) of R 

has been 1.7, occurring at t=14 and 15 mo, just within the window of when one expects to observe 

ΔR(t) max. On average, ΔR(t) max occurs about t=28 mo past minimum amplitude, with 7 of 14 

modern era cycles having ΔR(t) max occurring between t=14 and 24 mo and 7 of 14 modern era 

cycles having ΔR(t) max occurring between t=30 and 42 mo. The value of 1.7 for ΔR(t) max suggests RM=45.9±21.3 for cycle 24. However, because ΔR(t) max for cycle 24 probably has not yet 

occurred, a higher RM is expected. Indeed, the observed value of aam for cycle 24 (8.4) suggests 

a slightly higher RM for cycle 24, namely, RM=54.6±16.6, with only a 1% chance that RM will 

exceed 100.5. The value of RM=55 suggests ASC=54±5 mo (from the regression based on all modern era sunspot cycle) or 59±4 mo (using the regression that ignores the extreme cycles 14 and 19) for 

cycle 24. Therefore, it appears highly likely that cycle 24 ultimately will be described as a weak, slowrising sunspot cycle, having RM smaller and ASC longer than was observed in preceding cycle 23, 

potentially, becoming the smallest and the slowest rising cycle of the modern era (comparable in size 

to cycle 14 and comparable in ascent duration to cycles 12 and 16), which would be consistent with 

the ‘low’ panel consensus prediction.36,38,41–47

Comparison of the early cycle 24 R(t) behavior (through t=18 mo) with the mean cycle 

curves for specific groupings of cycles (i.e., FRC, SRC, SPC, and LPC) certainly supports this belief. 

For all comparisons, R(t) values for cycle 24 fall well below the mean cycle curves, being most divergent for FRC and SPC, again, suggestive that cycle 24 probably is a weak, slow-rising cycle that also 

is a long-period cycle. If true, then maximum amplitude for cycle 24 will occur after December 2012 

and new cycle 25 will not begin until after December 2019.

Application of the HWR shape-fitting function to the early cycle behavior of cycle 24 suggests that its behavior is consistent with a sunspot cycle having RM=70 and ASC=56 mo. This 

initial projection can be improved upon once ΔR(t) max is observed (probably sometime in 2011), 

which should result in a more accurate estimation of RM.
18

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22

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An Estimate of the Size and Shape of Sunspot Cycle 24 

Based on Its Early Cycle Behavior Using the HathawayWilson-Reichmann Shape-Fitting Function

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

Subject Category 92

Availability: NASA CASI (443–757–5802)

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

M–1310

Technical Publication

NASA/TP—2011–216461

Sun, sunspot cycle prediction, cycle 24, method of Ohl, Waldmeier effect, Hathaway-Wilson-Reichmann 

shape-fitting function, inflection points

01–03–2011

UU 32

NASA

U U U

On the basis of 12-month moving averages (12-mma) of monthly mean sunspot number (R), sunspot cycle 24 had its minimum amplitude 

(Rm=1.7) in December 2008. At 12 mo past minimum, R measured 8.3, and at 18 mo past minimum, it measured 16.4. Thus far, the maximum 

month-to-month rate of rise in 12-mma values of monthly mean sunspot number (ΔR(t) max) has been 1.7, having occurred at elapsed times past 

minimum amplitude (t) of 14 and 15 mo. Compared to other sunspot cycles of the modern era, cycle 24’s Rm and ΔR(t) max (as observed so far) 

are the smallest on record, suggesting that it likely will be a slow-rising, long-period sunspot cycle of below average maximum amplitude (RM). 

Supporting this view is the now observed relative strength of cycle 24’s geomagnetic minimum amplitude as measured using the 12-mma value 

of the aa-geomagnetic index (aam=8.4), which also is the smallest on record, having occurred at t equals 8 and 9 mo. From the method of Ohl 

(the inferred preferential association between RM and aam), one predicts RM=55±17 (the ±1 se prediction interval) for cycle 24. Furthermore, 

from the Waldmeier effect (the inferred preferential association between the ascent duration (ASC) and RM) one predicts an ASC longer than 

48 mo for cycle 24; hence, maximum amplitude occurrence should be after December 2012. Application of the Hathaway-Wilson-Reichmann 

shape-fitting function, using an RM=70 and ASC=56 mo, is found to adequately fit the early sunspot number growth of cycle 24.

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