4.1 Minima and maxima

The dates and values for the cycle minima and maxima are the primary data for many studies of the solar cycle. These data are sensitive to the methods and input data used to find them. Solar activity is inherently noisy and it is evident that there are significant variations in solar activity on time scales shorter than 11 years (see Section 7). Waldmeier (1961) published tables of sunspot numbers along with dates and values of minima and maxima for cycles 1 to 19. McKinnon (1987) extended the data to include cycles 20 and 21. The values they give for sunspot number maxima and minima are those found using the 13-month running mean. However, the dates given for maxima and minima may vary after considering additional indicators. According to McKinnon:

“… maximum is based in part on an average of the times extremes are reached in the monthly mean sunspot number, the smoothed monthly mean sunspot number, and in the monthly mean number of spot groups alone.”

These dates and the values for sunspot cycle maxima are given in Table 1 (the number of groups is multiplied by 12.08 to produce group sunspot numbers that are comparable to the relative sunspot numbers). It is clear from this table that considerably more weight is given to the date provided by the 13-month running mean. The dates provided by Waldmeier and McKinnon are far closer to those given by the 13-month running mean than they are to the average date of the three indicators. (One exception is the date they give for the maximum of cycle 14 which should be half a year earlier by almost any averaging scheme.) The monthly numbers of sunspots and spot groups vary widely and, in fact, should be less reliable indicators and given lesser weight in determining maximum.


Table 1: Dates and values for sunspot cycle maxima.
Cycle
Waldmeier/ McKinnon
13-month Mean Maximum
Monthly Mean Maximum
Monthly Group Maximum
  Date Value Date Value Date Value Date Value
1 1761.5 86.5 1761/06 86.5 1761/05 107.2 1761/05 109.4
2 1769.7 115.8 1769/09 115.8 1769/10 158.2 1771/05 162.5
3 1778.4 158.5 1778/05 158.5 1778/05 238.9 1778/01 144.0
4 1788.1 141.2 1788/02 141.2 1787/12 174.0 1787/12 169.0
5 1805.2 49.2 1805/02 49.2 1804/10 62.3 1805/11 67.0
6 1816.4 48.7 1816/05 48.7 1817/03 96.2 1817/03 57.0
7 1829.9 71.7 1829/11 71.5 1830/04 106.3 1830/04 101.5
8 1837.2 146.9 1837/03 146.9 1836/12 206.2 1837/01 160.7
9 1848.1 131.6 1848/02 131.9 1847/10 180.4 1849/01 130.9
10 1860.1 97.9 1860/02 98.0 1860/07 116.7 1860/07 103.4
11 1870.6 140.5 1870/08 140.3 1870/05 176.0 1870/05 122.3
12 1883.9 74.6 1883/12 74.6 1882/04 95.8 1884/01 86.0
13 1894.1 87.9 1894/01 87.9 1893/08 129.2 1893/08 126.7
14 1907.0? 64.2 1906/02 64.2 1907/02 108.2 1906/07 111.6
15 1917.6 105.4 1917/08 105.4 1917/08 154.5 1917/08 157.0
16 1928.4 78.1 1928/04 78.1 1929/12 108.0 1929/12 121.8
17 1937.4 119.2 1937/04 119.2 1938/07 165.3 1937/02 154.5
18 1947.5 151.8 1947/05 151.8 1947/05 201.3 1947/07 149.3
19 1957.9 201.3 1958/03 201.3 1957/10 253.8 1957/10 222.2
20 1968.9 110.6 1968/11 110.6 1969/03 135.8 1968/05 132.3
21 1979.9 164.5 1979/12 164.5 1979/09 188.4 1979/01 179.4
22     1989/07 158.5 1990/08 200.3 1990/08 195.9
23     2000/04 120.7 2000/07 169.1 2000/07 153.9

The minima in these three indicators have been used along with additional sunspot indicators to determine the dates of minima. The number of spotless days in a month tends to maximize at the time of minimum and the number of new cycle sunspot groups begins to exceed the number of old cycle sunspot groups at the time of minimum. Both Waldmeier and McKinnon suggest using these indicators as well when setting the dates for minima. These dates are given in Table 2 where both the spotless days per month and the number of old cycle and new cycle groups per month are smoothed with the same 13-month mean filter. The average date given in the last column is the average of the 13-month mean minimum date, the 13-month mean spotless days per month maximum date, and the date when the 13-month mean of the number of new cycle groups exceeds the 13-month mean of the number of old cycle groups. For the early cycles, where spotless days and old and new cycle groups are not available, the 13-month mean minimum date is used for those dates in forming the average.


Table 2: Dates and values for sunspot cycle minima. The value is always the value of the 13-month mean of the International sunspot number. The dates differ according to the indicator used.
Cycle
13-month Mean Minimum

Waldmeier/ McKinnon

Spotless Days Maximum

New > Old

Average

  Date Value

Date

Date

Date

Date

1 1755/02 8.4

1755.2

 

 

1755/02

2 1766/06 11.2

1766.5

 

 

1766/06

3 1775/06 7.2

1775.5

 

 

1775/06

4 1784/09 9.5

1784.7

 

 

1784/09

5 1798/04 3.2

1798.3

 

 

1798/04

6 1810/08 0.0

1810.6

 

 

1810/08

7 1823/05 0.1

1823.3

1823/02

 

1823/04

8 1833/11 7.3

1833.9

1833/11

 

1833/11

9 1843/07 10.6

1843.5

1843/07

 

1843/07

10 1855/12 3.2

1856.0

1855/12

 

1855/12

11 1867/03 5.2

1867.2

1867/05

 

1867/04

12 1878/12 2.2

1878.9

1878/10

1879/01

1878/12

13 1890/03 5.0

1889.6

1890/02

1889/09

1890/01

14 1902/01 2.7

1901.7

1902/01

1901/11

1901/12

15 1913/07 1.5

1913.6

1913/08

1913/04

1913/06

16 1923/08 5.6

1923.6

1923/10

1923/09

1923/09

17 1933/09 3.5

1933.8

1933/09

1933/11

1933/10

18 1944/02 7.7

1944.2

1944/02

1944/03

1944/02

19 1954/04 3.4

1954.3

1954/04

1954/04

1954/04

20 1964/10 9.6

1964.9

1964/11

1964/08

1964/10

21 1976/03 12.2

1976.5

1975/09

1976/08

1976/03

22 1986/09 12.3

 

1986/03

1986/10

1986/07

23 1996/05 8.0

 

1996/07

1996/12

1996/08

24 2008/12 1.7

 

2008/12

2008/09

2008/11


When available, all three indicators tend to give dates that are fairly close to each other and the average of the three is usually close to the dates provided by Waldmeier and McKinnon. There are, however, two notable exceptions. The dates given by Waldmeier for the minima preceding cycles 13 and 14 are both significantly earlier than the dates given by all three indicators. The cycle 13 minimum date of 1889.6 was adopted from Wolf (1892) while the cycle 14 minimum date of 1901.7 was adopted from Wolfer (1903).

Since many researchers simply adopt the date given by the minimum in the 13-month running mean, the date for the minimum preceding cycle 23 is also problematic. The minimum in smoothed sunspot number came in May of 1996. The maximum in the smoothed number of spotless days per month came in July of 1996. However, the cross-over in the smoothed number of groups from old-cycle to new cycle occurred in December of 1996. Harvey and White (1999) provide a good discussion of the problems in determining cycle minimum and have argued that the minimum for cycle 23 should be taken as September 1996 (based on their determination that new cycle groups exceed old cycle groups in January of 1997). The average of the three indicators gives August of 1996.

Additional problems in assigning dates and values to maxima and minima can be seen when using data other than sunspot numbers. Table 3 lists the dates and values for cycle maxima using the 13-month running mean on sunspot numbers, sunspot areas, and 10.7 cm radio flux. The sunspot areas have been converted to sunspot number equivalents using the relationship shown in Figure 6View Image and the 10.7 cm radio flux has been converted into sunspot number equivalents using Equation (3View Equation). Very significant differences can be seen in the dates. Over the last five cycles the ranges in dates given by the different indices have been: 4, 27, 25, 1, and 22 months.


Table 3: Dates and values of maxima using the 13-month running mean with sunspot number data, sunspot area data, and 10.7 cm radio flux data.
Cycle
13-month Mean Maximum
13-month Mean Sunspot Area
13-month Mean 10.7 cm Flux
  Date Value Date R-Value Date R-Value
1 1761/06 86.5        
2 1769/09 115.8        
3 1778/05 158.5        
4 1788/02 141.2        
5 1805/02 49.2        
6 1816/05 48.7        
7 1829/11 71.5        
8 1837/03 146.9        
9 1848/02 131.9        
10 1860/02 98.0        
11 1870/08 140.3        
12 1883/12 74.6 1883/11 88.3    
13 1894/01 87.9 1894/01 100.4    
14 1906/02 64.2 1905/06 75.4    
15 1917/08 105.4 1917/08 93.0    
16 1928/04 78.1 1926/04 92.3    
17 1937/04 119.2 1937/05 133.3    
18 1947/05 151.8 1947/05 166.5    
19 1958/03 201.3 1957/11 216.5 1958/03 201.2
20 1968/11 110.6 1968/04 100.9 1970/07 109.6
21 1979/12 164.5 1982/01 156.0 1981/05 159.4
22 1989/07 158.5 1989/06 158.5 1989/06 168.0
23 2000/04 120.7 2002/02 126.7 2002/02 152.3

These tables illustrate the problems in determining dates and values for cycle minima and maxima. The crux of the problem is in the short-term variability of solar activity. One solution is to use different smoothing.


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