Timo Niroma:
Sunspots: From basic to supercycles.

• 4.1. How long is the 11-year cycle?
• TABLE 35. A pyramid of mean lengths from 1755 to 1987.
• 4.2. The rules of Schove interpreted.
• TABLE 36. The 14-year supercycle.
• TABLE 37. Supercycles consisting of 15 to 20 basic cycles.
• 4.3. The Precambrian Elatina formation.
• 4.4. The Gleissberg cycle.
• TABLE 38. The Gleissberg cycle.

- A 2000-year historical perspective.
-- The Roman Empire and its demise.
-- The Mayan Classic Period.
-- When the Nile froze in 829 AD.
-- Why is it Iceland and Greenland and not vice versa?
-- Tambora did not cause it.
-- The spotless century 200 AD.
-- The recent warming caused by Sun.
-- The 200-year weather pattern.
- An autocorrelation analysis.
-- Three variants of 200 years.
-- The basic cycle length.
-- The Gleissberg cycle put into place.
- Some studies showing a 200-year cyclicity.
- The periods of Cole.

- Smoothing sunspot averages in 1768-1992 by one sunspot cycle.
- Smoothing by the Hale cycle.
- Smoothing by the Gleissberg cycle.
- Double smoothing.
- Omitting minima or taking into account only the active parts of the cycle.

- Short supercycles.
- Supercycles from 250 years to a hypercycle of 2289 years.
- The long-range change in magnitudes.
- Stuiver-Braziunas analysis: 9000 years?

PART 4. SUNSPOTS: FROM BASIC TO SUPERCYCLES

4. From basic cycles to supercycles

In the introduction I came to the conclusion that there not really exists a 11-year cycle as a length cycle, it is only a mathematical average of the cycles when the period of inspection is long enough, say 200 years. It became obvious that 10 and 12 years were more preferred than 11 years as a cycle length. But while inspecting the magnitudes of the cycles, the 11-year cycle revealed itself as the most powerful. Still the 12-year-cycle was there, but we got for the 10-year cycle a slightly different value in the two analysis. Length analysis gave 10.3 years, magnitude analysis a cycle of about 9.9 years. But in the long run something like an 11-year cycle exists. It is not much longer than 11 years, but somewhat anyhow. The first question is, is its length rather 11.0 than 11.1 years? The second question is, can we estimate the fourth significant number 11.0x?

One could ask, why not divide the length of n cycles with n. But in the introduction we already saw, that it is not that simple. How large an n do we need? Has the n to be divisible by some m? However, let's try. Let's start with the minimum which occurred in 1986.7. I prefer at the moment not to use the latest cycle that ended in 1996 because of the prevailing disagreement about its length. In 1913-1986 the mean cycle length was 10.4 years. It seems that this century has not even nearly enough cycles for our purpose or we have really badly estimated the mean cycle length.

In the introduction I referred to a value of 11.08 years, which is what Justin Schove has got from his 2000 year data. If the value is 11.07 years, then 15 sunspot cycles would equal 14 Jovian years. 14:13 and 16:15 would lead to mean sunspot cycles of 11.015 and 11.12 years, respectively. I considered them as improbable, but not impossible. In fact, the 15 cycle period is not quite evident in the last 250 years. There is a variation between 12 and 18 cycles if mirrored on the same point in the Jovian orbit. During this period there is some tendency towards 16 cycles per Jovian year, but the variation is great. However none of my hypothesis demands that the relation should be in whole numbers.

Let's continue backwards from 1913 and take six cycles more so that we have the 13 cycles in 1843-1986, which together lasted 143.2 years. The mean value now becomes 11.02 years. But if we add only the two short cycles 7 and 8, the mean length goes down again. The 15 cycles in 1823-1986 lasted 163.4 years, which causes the mean cycle length to be 10.89 years. Even if 15 cycles is a supercycle, at least this 15-cycle group is not in synchrony. A higher order supercycle is needed or the beginning should be a synchpoint.

Next I add the three cycles numbered from 4 to 6, which all end by minima, that contain months of zero Wolfs. The 18 cycles in 1784-1986 lasted together 202.0 years, which causes the mean cycle length to jump up to 11.22 years. The jump of 0.33 years in mean length when going from 15 to 18 cycles suggests, that the value has really not yet stabilized. The whole of our data contains 21 full cycles in 1755-1986. Together their length is 231.5 years, which means a mean cycle length of 11.02 years.

If we add the cycle 22, we get a mean length of 10.96 to 10.98 years depending of whether we use the mathematical minimum or the agreed-upon minimum.

In the next pyramid I have listed all possible mean cycle values with the cycles 1-22, when the amount of the cycles is at least 14.

1. number of cycles
2. lengths beginning with youngest (first figure in each row always ends in 1986, last figure in
each row always begins in 1755, for example: 14 cycles: first one 1833-1986, next 1823-1976,
last one 1755-1913, 21 cycles: 1755-1986). Because the uncertainty of the exact minimum
datum in 1755, I have put values, that include it, into parenthesis.

1.   2.
                                             
14 10.91 10.94 11.01 11.14 11.39 11.31 11.22 (11.31)   
15    10.89 11.06 11.09 11.30 11.25 11.15 (11.23)      
16       11.01 11.14 11.25 11.17 11.11 (11.16)         
17          11.08 11.28 11.13 11.04 (11.12)            
18             11.22 11.17 11.01 (11.06)               
19                11.12 11.05 (11.03)                  
20                   11.01 (11.07)                     
21                      (11.02)                        

The best estimate we can get with this method is 11.02 years. Or is it the best estimate? Solely on the number of cycles, yes, on the other grounds, not. We have no special reason to hold 21 cycles to be neither enough nor a representative amount of cycles. In fact, there is no clear tendency if we take the 14, 15 ... 21 last cycles: 10.91, 10.89, 11.01, 11.08, 11.22, 11.12, 11.01, and possibly 11.02 years.

The instability of the 21 cycle value is demonstrated with the two 20 cycle values. In 1755-1976 the mean length was 11.07 years, while in 1766-1986 it was 11.01 years. By replacing 1 cycle in 20 we have a 0.06 year reduction in mean length. And if we take 19 cycles the range is from 11.03 to 11.12 years, almost 0.1 years. The accuracy we are seeking is 0.01 years.

In the following table I have gathered all the mean lengths that are based on at least 18 cycles to see if there emerges any pattern:

11.01 xx
11.02 x
11.03 x
11.05 x
11.06 x
11.07 x
11.12 x
11.17 x
11.22 x

The median of these values is 11.055 years and the mean 11.076. If we omit the two extreme values, we get a range of 11.02-11.12 years or just the range in which we are seeking the value. So we are not nearer it than in the beginning.

Not even the supposed supercycle of 15 cycles (nor 14 or 16 either) does provide any clear answer. The slowness near the perihelion and the more rapid cycle switching near the aphelion is despite of that seen: the mean length is 11.15 years in 1766-1933, 11.30 years in 1784-1954, and 10.9 years in 1823-1986. The 16-cycle period shows a similar rise and fall: from 11.11 to 11.25 to 11.01 years.

There is one thing, however, which could be helpful. The year 1810 could act as a natural synchpoint because it is the only totally zero Wolf year in the whole 250 year data. The pyramid shows that 15 cycles from it lasted in average 11.06 years (the 14- and 16-cycle values are both 11.01 years). This calculation is based on the official minimum 1810.6 (plus the minimum in 1976.5). In the introduction we came into the conclusion that the minimum was rather 1810.5 or even 1810.4. The former causes the mean cycle to have a length of 11.067 years and the latter 11.073. A Jovian synchrony based on a relation 15:14 would require a value of 11.071 years.

The relation can't be 16:15, if we regard the year 1810 as a synchpoint, because the mean value in 1810-1986 is 11.01-11.02 years and this relation would demand an amount of 11.12 years. But of course from other synchpoints, such as the Maunder minimum, it could be adding more room to the possible vacillation of a possibly not-so-constant mean.

Evidently, we need more help. Rudolf Wolf calculated already in 1852 the minimum and maximum times over the Maunder minimum or back to year 1607. Justin Schove has in the 1970s revised these figures and extended them back to 1501 based partly on auroral numbers. Unfortunately, the accuracy of these calculations is plus minus one year or in some few cases plus minus half a year.

One of these more accurate minima is 15 cycles before 1810 or 1645.5. This is an interesting minimum, because it starts the Maunder minimum. The 30 cycles in 1645-1976 lasted 331 years plus minus half a year. This gives a mean value in the range of 11.02-11.05 years. The most probable length also for the last cycle before the Maunder minimum (-10) was one Jovian year, according to Schove calculations.

The next question is, could we continue with the auroral numbers further into the past. The answer is partly yes, partly no. There are auroral observations, that can be used for our purpose, at least from a period of 1500 years. The trouble is, that there are many gaps. The latest of them occurred about 1450-1500. With tree ring analysis it has been verified (Eddy), that the scarcity of the aurorae then was real: this period has been named as the Sporer minimum, second during the second millennium only to the Maunder minimum 200 years later. By combining auroral observations with tree ring analysis we can go back into the past at least 1000, maybe 2000 years.

Justin Schove has made a mainly auroral analysis back to 649 BC. However the first 400 years are mostly a guesswork, so that a more reliable analysis begins in 215 BC (D. Justin Schove: The sunspot cycle, 649 BC to AD 2000. Jour. Geophys. Research 60, 1955). He has calculated that between 215BC and AD 1947 there were 195 sunspot cycles thus giving 11.08 years as the average length. He has also calculated the interval between intense maxima as being 554 years and 50 cycles. This gives also a mean period of 11.08 years. But he has also showed, that there is so much vacillation even in periods of 500 to 1000 years (at least from 11.04 to 11.09 years), that the question is not definitively settled. With much confidence, however, we can say, that the mean sunspot cycle is not less than 11.05 or more than 11.09 years.

The abovementioned article of Justin Schove is very important to the existence of this very study, because the idea of a relation between the Jovian year and the sunspot cycle first occurred to me, when I read the following rules, despite of the fact that Schove does not even mention Jupiter. The final impulse that led me to this study came from the Elatina study (see chapter 4.3). The later critics that has questioned the validity of Elatina strata as a sunspot indicator has led me to study it more deeper and I am ever more convinced that Elatina tells us about the Sun, the pre- Cambrian Sun, whose cyclicity has not much changed during the last 600 or 700 million years.

Schove: "The Sunspot Cycle, 649 BC to AD 2000", page 140: "Seven cycles, measured from minimum to minimum, usually occupy between 77 and 79, but always more than 72 and less than 83 years. Fourteen cycles, measured from minimum to minimum, usually occupy between 154 and 158 years, but always more than 150 and less than 162 years. The longer values mentioned in the two preceding rules, that is, 83 and 162 years, are approached in periods measured forwards from the conclusion of an aurorally rich period... The shorter values, 72 and 150 years, are conversely approached in periods which end near the conclusion of an aurorally rich period... The sunspot cycle is longer in aurorally weak periods and shorter in very active periods."

The last sentence can easily be seen come true when comparing the 19th and 20th century. The 19th century contained mostly long but weak cycles, the 20th century has had short but strong cycles. A correlation does not automatically mean a direct causation, but despite it is interesting to note that the 19th century was colder than the relatively warm 20th century. The same pattern seems to have occurred during the 17th and 18th century. The most renown incident from this period is the Maunder minimum, a spotless and cold period.

I began wondering, what the numbers would be in Jovian years and so I transformed: Seven cycles are usually between 6.5 (6.49) and 6.7 (6.66), but always more than 6.1 (6.07) and less than 7.0 (7.00) Jovian years. And fourteen cycles are usually between 13.0 (12.98) and 13.3 (13.32), but always more than 12.6 (12.65) and less than 13.7 (13.66) Jovian years. The accuracy - plus minus half a year - with which Schove expresses the numbers, is in Jovian years 0.04. Within this limit are seven of the eight numbers expressible in whole Jovian years, in its halves or in its thirds. More than that: in six cases the deviation is at most 0.01 Jovian years, in one case 0.02 Jovian years.

A thought experiment, that first was just one indication of my passions to calculate whenever I see a series of numbers, led to a silencing founding. I had a long time wondered what was behind the sunspot cyclicity, but never before found anything even remotely giving anything than randomness. Now I had two series of numbers that somehow seemed to have something in common. So I went on to analyze what I had found.

One of the eight values was not however close enough to any such value as to be expressible in 1/1, 1/2 or 1/3 Jovian years. Because seven of eight were, the eighth must also have some relation, I reasoned, albeit probably a little more complicated. The 72 years is too much to be 6 Jovian years and too small to be 6 1/3 Jovian years. To assume that Schove had an error here and the real value would be 71 years, would be an easy solution. In that case both seven and fourteen adjacent cycles will vary by one Jovian year. Seven cycles would vary between 6 and 7 Jovian years and fourteen cycles between 12 2/3 and 13 2/3 Jovian years. In fact the upper limit of seven cycles has really been achieved, according to Schove there were between AD 196 and 302 only 9 cycles, which causes the mean cycle length to be 11.8 years during that period (page 145). Schove adds a sic to the figure to remark the reader that it's no error, it's in accord with his rules, albeit the extreme case. But the low limit: error would be too easy an explanation, and besides, Schove seemed to be accurate. So I left that value to rest and wait for an idea to pop up.

At least what Schove calls the usual values are the same for both supercycles (exactly twice as the amount of the cycles is doubled: from 6.5 to 6 2/3 and from 13 to 13 1/3). If we calculate the mean length based on these values we get 11.015 years as the lower limit and 11.30 years as the upper limit. If one looks at the mean lengths in the pyramid in the chapter 4.1, all mean values beginning with 16 cycles are within these limits or from 11.01 to 11.28 years.

So let's elaborate the 72 years and assume that it is as accurate as the other seven values, i.e. the value in Jovian years is plus minus 0.02 Jovyrs of the mathematical product. The division gives 6.070 years, or something between 6.068 and 6.072 years. Now I make a thought experiment and watch where it leads us. Let's assume that it is some whole number fraction of the Jovian year. The only number in these limits is 14. 1/14 equals 0.071. Now if we assume the lower limit to be 6.071 Jovian years, we get as the seven cycle mean lower limit 10.29 years or just the value we got in the introduction to be the mean value of the shorter category cycles. The upper category mean was the same as the Schove upper limit, one Jovian year (even if he does not anywhere notice these Jovian connections).

The arithmetic mean of these two values (10.286 and 11.862 years) is 11.074 years or in the very range of values, we have assumed for the mean cycle length. 72 years would be exactly 6.5 mean cycles. We have now good grounds to accept also this value even if it does not fulfill the symmetry of the other seven values. But why should it? There surely are other forces than Jupiter that impact the sunspot cycle and this asymmetry is the way to cope with it.

During the last 400 years every 15th cycle (among some others) is one Jovian year long, and is a synchpoint at the same time: cycle -10 (1633-1645) was the last one before the Maunder minimum, cycle 5 (1798-1810) ended in a spotless year, and cycle 20 (1964-1976) followed a record-high cycle. Furthermore, all these cycles have their minima about 10 months after the Jovian perihelion. By the way, the "extra" 0.07 Jovian years over 6, is 0.83 Earthly years, which we in introduction noticed to be the most favored difference between Jovian perihelion and sunspot minimum.

Lastly I will test the fourteen cycle rule with our 240-year data. The limits were 150 and 162 years or 12.667 and 13.667 Jovian years. Typically the values were between 154 and 158 years or 13.00 and 13.333 Jovian years. For the minimum of the cycle change 22/23 has been used the traditionally calculated 1996.4 (9.6 years), not the agreed-upon value.

TABLE 36. The 14-year supercycle

1. cycles
2. years
3. total length in Earthly years
4. mean length in Earthly years
5. total length in Jovian years
6. change of 5. from the previous supercycle
7. the maximum R(M), average of the 14 cycles
8. the total length graphically (the limits: *=149 and 163, usual=:)

  1.        2.    3.    4.    5.    6.  7.  8.

1-14 1755-1913 158.4 11.31 13.35 +0.14 101 *    :   x    *
2-15 1766-1923 157.1 11.22 13.24 -0.11 102 *    :  x:    *
3-16 1775-1933 158.3 11.31 13.35 +0.11 100 *    :   x    *
4-17 1784-1944 159.5 11.39 13.45 +0.10  97 *    :   : x  *
5-18 1798-1954 155.9 11.14 13.14 -0.31  98 *    : x :    *
6-19 1810-1964 154.1 11.01 12.99 -0.15 109 *    x   :    *
7-20 1823-1976 153.2 10.94 12.92 -0.07 113 *   x:   :    *
8-21 1833-1986 152.8 10.91 12.88 -0.04 120 *   x:   :    *
9-22 1843-1996 152.9 10.92 12.89 +0.01 120 *   x:   :    *

After 1755 the supercycle(14) has been at its maximum length in years 1784-1944 beginning with a cycle whose minimum is near the perihelion and ending with a cycle whose minimum is near the aphelion. With 15 cycles (see introduction) the maximum was in years 1784-1954. The mean lengths are 11.39 and 11.30 years, respectively. The maximum supercycle(14) has the lowest mean R(M) or 97 Wolfs. The supercyclic rise of R(M) begins to show immediately after the pre-1810 time has been left behind. The mean R(M) has had a continuous rise as long as the period has had a continuous shortening in length.

But now both the length and magnitude has become to a standstill. Since the pre-1810 years were out the total length reached and went below its usual value of 13 Jovyr, but has not reached its lower limit of 12.67. respondingly when the pre-1810 years were included, the mean maximum R(M) was in the narrow range of 97-102, but climbed rapidly to 120, when the cycle reached the 1980's. This could be the real reason for the warming of the climate in the 1990's.

Every super-cycle(14) except the last one since 1784-1944 has been shorter than the previous one, but the rate of shortening has decreased to half of its previous value at every cycle, so that we have a series of -0.31, -0.15, -0.07, and -0.04. At the same time the maximum Wolf number increased by 22 or about 7 Wolfs per cycle. But now both values have become to a halt.

If it holds (see introduction) that every 15th cycle (among some others) is of the length of one Jovian year and if we calculate a relation of 14:15 between the Sun and Jupiter, it means that the mean sunspot cycle equals 11.071 years. From this follows that we have a supercycle of supercycles, whose length is 14*15 or 210 cycles or 2325 years. If we omit every 15th cycle or the Jovian-year-length synchpoints, we are left with 196 cycles. Their mean length can now be derived from the equation 210x+14*(11.86-x)=2325, which gives x equals 11.015 years.

Above I calculated that the mean length of such 14 cycles, that are surrounded by two Jovian years, is 11.01 years, and the mean length of 15 cycles I have estimated to be near 11.07 years. If the year 1810 is a synchpoint, we get as the supercycle(15) length 11.07 and the whole supercycle gets a length of 166.05 years (1810.45 as the most probable minimum). 14 Jovian years is 166.07 E-years. The supercycle(14) length is 11.01 years. After all the upside down top of our length pyramid (chapter 4.1) seems to more valid than we at the beginning thought.

If we take from our pyramid the four nearest values to the most probable mean length of 11.07, we get:

1. number of cycles
2. cycles
3. years
4. combined length
5. mean length 

1.    2.        3.    4.    5.

15  6-20 1810-1976 166.0 11.07
17  5-21 1798-1986 188.4 11.08
18  1-18 1755-1954 199.0 11.06
20  1-20 1755-1976 221.3 11.07

In the light of some other analysis at least the supercycles of 15, 18 and 20 cycles or 166, 199 and 221 years are essential and real supercycles.

In 1982 a 680 million years old Precambrian formation in Elatina, South Australia, was drilled to get cores of a rhytmic lamination. G. E. Williams interprets these laminations as being caused by the activity variation of the Sun. The continuous laminations, 9.4 meter thick, span about 19000 years, almost 100 times more than our data. The cited Elatina values are from the article "The Solar Cycle in Precambrian Time" by George Williams, published in August 1986 in Scientific American 255/2 (main values are from page 87).

Before analyzing those figures, I must comment an article published in Nature 335/6193. The article, named "The lunar orbit in the late Precambrian and the Elatina sandstone laminae" (C. P. Sonett, S. A. Finney and C. R. Williams, 1988), says in the end of the abstract (page 806): "Prompted by a letter from G. E. Williams, summarizing new field data from the Adelaide region, South Australia, which imply that the Elatina varves may be tidal, we re-examined the dark-band spectrum and propose a luni-solar tidal interaction model as the source of the laminae."

Let's begin with the critics.

1. "The existence of a tidal cutoff is inferred directly from the about 11.6 light laminae per dark band. [Abstract: The formation 10-m layer of graded sandstone, consists of a pattern of light bands of periodically varying thickness. A sequence of dark bands separates, on average, 11.6 light bands; the spacing of these dark bands exhibites a rich and complex spectrum.] If these represent semi-diurnal layers, then a full complement of 31 layers should be deposited in a PLO [partial lunar orbit], whereas less than half of this is seen. Even a full tidal inequality would lead to deposition of 15 laminae."

This being the hard proof and the model itself being largely a speculation, there really is, to my mind, no need to change the original overall theory of G. E. Williams, which says that the periodicity is based on the activity of the Sun. The first question that arises to me is, has the sunspot period (or Sun activity period) decreased from 11.6 years to 11.1 years in 700 million years? A drop of 4 % would not be odd, because it is only 0.07 years per 100 million years. The Sun has burned a huge amount of hydrogen during the last 700 million years, its weight and gravity has changed and possible a 4 % change in planetary orbits is neither too much.

2. "From the power spectrum of the dark bands it is deduced that the amplitude modulation repeats with a period of 26.25 PLO. Thus there are 14.12 sidereal lunar months."

Assuming of course that the dark bands represent tidal periods. One other possibility is that they represent the Jovian-solar activity. 14 is anyway a very central amount in my study, albeit I speak of Jovian years, the critics of lunar months.

3. "The apsidal rotation ... has a sidereal period of 9.43+-0.14 years." "Estimates of the apsidal period of the Precambrian Moon vary from 9.92 to 9.26 yr depending upon the choice of spectral line..."

I got also these values of 9.3 to 9.9 years. But my values are deduced from the solar cyclicity. So there is a rival explanation.

So, I begin with the assumption, that the original viewpoint of George Williams is right, and the Elatina lamination is of solar origin. If the figures don't fit with figures from other independent studies, so that's that, and they are not solar (or the parameters have changed significantly in 700 million years). But there is nothing in the Nature article, that would make a lunar effect more probable than the solar. In the following I will call the study published in Nature as SFW-study (based on the names of the researchers). Williams refers to George E. Williams, as well as the "original" study refers to his study in 1986.

Williams has identified seven supercycles in his data. If we define that supercycles, whose lengths are multiples of each other, make a supercycle set, there are four such sets in the Williams data. In this chapter I will study only one of these sets, consisting of three super- cycles, and postpone the treatment of the remaining four supercycles to later chapters. The two longest supercycles in this set are also the two longest supercycles in the whole Elatina data. Their interest lies in the fact that they seem to be equal to or multiple of one of the two most studied supercycles in the present day data, namely the Gleissberg cycle (the other is the 200-year cycle). The Gleissberg cycle itself most probably obeys the 7 cycle rule of Schove.

Because I assume the laminations to represent years, I use in the following the word "year" instead of "lamination". If the equation proved wrong, it would show up regardless of the word used.

The three supercycles I will study in this connection have a duration of 314, 157 and 79 years. It seems evident that the set is one of n*78.5 years, where n = 1,2,4. The Gleissberg cycle is most often referred to as lasting 78 years, which is very well in accord with the Schove rule, that seven cycles are usually between 77 and 79 years. But first I will study the multiples of it, the 157 and the 314 year supercycles.

The 314-year supercycle, the longest one identified in the Elatina study, seems to be very basic to it. It corresponds to 26.1 cycles in the original study, which obviously is the same as 26.25 PLO in the SFW-study. Equivalently the 157 years would correspond to 13.05 cycles. This means that the basic cycle length should be 12 years, as Williams suggests. However, the SFW study got a basic cycle length of 11.6 years. Because the Jovian year is between these two values, lets look what it would give.

26.1/26.25 and 13.05/13.125 Jovian cycles would equal supercycle lengths of 309.6/311.4 and 154.8/155.7 years, respectively. According to the Schove rule of 14 sunspot cycles, the usual limits were between 154 and 158 years, with a mean value of 155.2 years, so the fit would be excellent.

But back to the Elatina data. Williams continues: "The cycles containing the thickest varves occur on the average every 26 cycles... In contrast, the cycles containing the greatest number of varves occur on the average about every 13 cycles. Intriguingly, the two rhythms are negatively correlated: the thickest varves tend to occur in the cycles having the fewest varves, that is, the cycles of shortest duration. In other words, strong cycles tend to be brief and weak cycles tend to be long."

This is just the situation today, 700 million years later. Strong and brief cycles in the 18th and 20th century, weak and long cycles in the 17th and 19th century. If we draw the lower limit of R(M) as 150 for a strong cycle, we get the cycles 3, 18, 19, 21 and 22 or 1775-1784 and 1944- 1996 (with an interruption in the middle). If we draw the upper limit of R(M) as 100 for a weak cycle, we get the cycles 5-16 or 1798-1933 (with 4 interruptions). We don't have enough cycles to make a meaningful comparison with Williams data, but at least the tendency is the same and besides Williams has a long fork of 60 years in which his 300-year cycle oscillates.

As concerns the lengths we can say, that the short cycles continued until 1784, the long cycles reigned until 1913 (with a few exceptions), and since then there has again been short cycles with only one exception. So the negative correlation between the length and magnitude is as clear as it was 700 million years ago.

Now again Williams: "The identified periodicities include the 'Elatina cycle', a prominent rhythm averaging about 26.1 12-'year' cycles in duration, or about 275 to 335 'years'." If a corresponding transformation as above is made to these values, we get 272 and 331 years. From these values we get for the mean cycle a lower limit of 10.4-10.5 years and an upper limit of 11.7-11.8 years. These are very near the today's two favorite values, 10.3 and 11.8-11.9 years, that we got already in the introduction.

That Williams is really measuring sunspot cycles gets still more credence from his measurements that the mean period varies between 9 and 14 years, as it varies in the nowaday data. Is Williams only simplifying things, when he says the mean is 12 years, when his critics says it is 11.6 years. Maybe Williams prefer to use the nearest whole numbers, while his critics uses the more accurate 11.6 years. That may also explain the discrepancy of 26.1 v. 26.25 cycles between the two studies.

But let Williams continue: "Superposed on the Elatina cycle is a sawtooth, or zigzag, pattern resulting from the characteristic alternation of relatively thick and relatively thin 12-'year' cycles. The pattern exhibits 180-degree phase shifts (that is, the thick-thin alternation reverses) at intervals ranging from nine to 23 cycles; the average interval between shifts is 14.6 cycles. The mean period for a 360-degree change of phase is therefore 29.2 cycles, or 3.1 cycles longer than the mean period of the Elatina cycle. Hence the positions of features of the sawtooth rhythm gradually change along the sequence with respect to the features of the Elatina cycle."

Now this is a little difficult to explain. That's however no reason to omit it. A tentative guess would be that this has something to do with the rising tendency of the R(M) that has been evident in supercycles. With Schove rules we had a 14 Jovian year rise since 1798 (chapter 4.2).

One final thing about the Williams remarks. On page 89 we read: "In particular the Elatina data show deep minima in cycle heights about every 275 to 335 'years', but nowhere in the 19000 'years' of contiguous signals is there a cessation of cyclicity comparable to the alleged Maunder minimum. This finding may imply that the Maunder minimum is an artifact of insufficient or inaccurate data." This echoes a common mistake about the Maunder minimum, i.e. that it was wholly or nearly wholly spotless. In fact only the 1690's where nearly and some years wholly spotless. At all other times during the Maunder minimum from 1645 to its end (generally regarded as 1715, but I prefer 1705) there was some activity at least during the cycle maxima. Schove has estimated, that the R(M) values in the five maxima in about 1650-1694 varied from 20 to 35. The figure on page 85 in the Williams article shows just this kind of cyclicity.

Now let's take our synchpoint, the spotless year 1810, and see what happens if we add/subtract these Schove/Elatina supercycles of 155/310 years. 1810-310 = 1500, which is the minimum of the Sporer minimum. John Eddy: "The Maunder Minimum" in Science 192, 1976, page 1196: "The information available at present allows one to describe the history of the sun in the last millennium as follows: a possible Grand Maximum in the 12th century, a protracted fall to a century-long minimum around 1500, a short rise to 'normal', and then fall to the shorter, deeper Maunder minimum, after which there has been a steady rise in the envelope of solar activity." "The earlier minimum, which we might call the Sporer minimum, persisted by our 10-parts-per- mil criterion from about 1460 through 1550." 1500-310 = 1190 or the Medieval Grand Maximum." A phase shift has occurred. But between the Napoleonic minimum of 1810's and the Sporer minimum of around 1500 there was the millennium record of cold and spotlessness of the Maunder minimum. How about it?

1810-155 = 1655, which is the first minimum during the Maunder minimum and begins the second lowest cycle during the MM secondary only to the 1690's. 1810+155 = 1965, which is a minimum after the highest known cycle, and begins a Jovian length cycle, which according to my definitions is a synchpoint cycle. Thereafter there has been one Jovian year and short cycles plus a warming climate.

1655-310 = 1345 or the minimum of the Middle Age minimum according to Eddy (Fig. 5 on p. 1195 in his article) and various auroral data studies (for example Gleissberg and Siscoe).

The third supercycle in this Elatina main supercycle set is the 78.5 year (79 'year') cycle. It is the only one in the set, that is well-known in the present-day data. The 155/157 year cycle was observed by Schove, but otherwise there are very few results, that can be interpreted as being influenced by this supercycle. Any 310/314 year cycle is hardly ever mentioned. This is not surprising. First of all, our more or less accurate data covers 250 years (since 1749), and the definitively less accurate data 380 years (since 1610). Secondly, the cycles of 78.5/155/310 years are primarily length cycles and only secondarily intensity cycles, which make it difficult to observe them in auroral numbers. Thirdly, the length of the cycles in this set are greatly variable, being exact only from a synchpoint to another synchpoint.

But the shortest of the three cycles is easily seen in the nowaday data.

This well-known cycle of about 80 years is called the Gleissberg cycle. Most often it is considered as a 78-year cycle. The limits for it are 72 and 83 years according to the Schove rules, or 6.07 and 7 Jovian years. If we make to the Elatina value (assuming it to be 78.5 (not 79) years) the same transformation (11.862/12 * x) as to the other two values, we get 77.6 years or 6.54 Jovian years. It is within the Schove 'normal' limits of 77-79 years.

Even if the data convincingly show, that the Gleissberg cycle is always between 6 1/14 and 7 Jovian years, usually between 6 1/2 and 6 2/3 Jovian years and in the average about 77.6 years, there are many researchers seeking one exact value and they are getting different results. Many of them are not satisfied with the usually referenced value of 78 years.

The 155 year cycle also shows up in these shortenings and lengthenings oscillating between 145 and 165 years or from 10.3 to 11.8 years per cycle. 155+-10 years is the distance from near-perihelion to near-perihelion or from aphelion to aphelion the exact length depending on whether more time is spent on the side of the perihelion or the aphelion. Actually the theoretical supercycle seems to be 14 Jovian years or 166.1 years, but because of other factors are also contributing to the Sun's cyclicity (Venus, Saturn, internal factors), the actual supercycle is 155.2 years for 14 cycles.

The Gleissberg cycle lower limit of 72 years corresponds to seven cycles, whose average length is 10.3 years. These are cycles, who are at a greater distance than 2 years from the Jovian perihelion. The Gleissberg upper limit of 83 years is exactly 7 Jovian years and represents the perihelion-seeking cycles.

A. Kiral has summarized in his article "Autocorrelation and solar cycles" (Istanbul Univ. Obs. Publ. 80, 1961) various lengths established for the Gleissberg cycle: "Solar activity, and solar frequency in particular, has for a long time been known to be regulated by two fundamental cycles: the 11-year cycle discovered by Schwabe, and the 80-year cycle, postulated as early as 1862 by Wolf, which has been the subject of several investigations by Gleissberg and others... By harmonic analysis and successive approximation Kimura managed to find, among others, a period of 82.2 years, little different from the 81-year period of Wolf... In view of the impossibili ty of representing the series of observations as a Fourier series, Gleissberg has applied to the lists of Schove an arithmetic method defined and christened secular smoothing. Thanks to this method he proved the existence of the long period of 80 years, and he was also able to determine its value as 78.8+-3.3 years... We will try for our part to apply to the Schove series the method of auto-correlation... The power spectrum clearly reveals the secondary periods of two, nine, and eighteen 11-year cycles (i.e., 22, 100, and 200 years) with the fundamental period of seven cycles... The effective existence of periods of ... 22, 78, 100, and 200 years of solar activity can thus be confirmed..."

T. W. Cole has got a power spectrum, that shows a primary cycle of 196 years, and a secondary cycle of 78.5 years. I will, however, be back with his analysis later (in chapter 5).

It's no surprise, that various researchers have achieved various values for this cycle, and that the material needs a special kind of treatment before opening up in an autocorrelation analysis. The cycle has used practically all of its space between 6 and 7 Jovian years since 1749. So the value gotten is very sensitive to the time interval from which the analyzed data is derived.

Lastly I test the Gleissberg cycle with the data of the last 250 years. The limits were 72 and 83 years or 6.07 to 7 Jovian years.

1. cycles
2. years             
3. total length
4. mean length                            
5. total length in Jovian years
6. change in 5.
7. the mean maximum R(M)
8. the total length graphically (the limits: *=71, *=84)
9. obs!

   1.        2.   3.   4.   5.    6.   7.    8.             9.
 
 1- 7 1755-1833 78.7 11.2 6.63 +0.03  96 *     : x    *
 2- 8 1766-1843 77.0 11.0 6.49 -0.14 105 *     x :    * near mean 
 3- 9 1775-1856 80.5 11.5 6.79 +0.30 107 *     : : x  * diff < 2yrs
 4-10 1784-1867 82.5 11.8 6.95 +0.16  98 *     : :   x* diff < 2yrs
 5-11 1798-1878 80.6 11.5 6.79 -0.17  98 *     : : x  * diff < 2yrs
 6-12 1810-1889 79.0 11.3 6.66 -0.13 102 *     : x    *
 7-13 1823-1901 78.4 11.2 6.61 -0.05 107 *     :x:    *
 8-14 1833-1913 79.7 11.4 6.72 +0.11 106 *     : :x   *
 9-15 1843-1923 80.1 11.4 6.75 +0.03 100 *     : :x   *
10-16 1856-1933 77.8 11.1 6.56 -0.19  93 *     :x:    * near mean 
11-17 1867-1944 77.0 11.0 6.49 -0.07  96 *     x :    * near mean 
12-18 1878-1954 75.3 10.8 6.35 -0.14  97 *   x : :    *
13-19 1889-1964 75.1 10.7 6.33 -0.02 115 *   x : :    *
14-20 1901-1976 74.8 10.7 6.31 -0.02 119 *   x : :    *
15-21 1913-1986 73.1 10.4 6.16 -0.15 133 * x   : :    *
16-22 1923-1996 72.8 10.4 6.14 -0.02 141 * x   : :    * 

During the last 240 years the supercycle has had its maximum value in the years 1784-1867. If the Schove rule applies, the time period of 82.5 years or 6.95 Jovyrs is at the same time practically at the theoretical maximum of 7 Jovyrs. This is in the middle of the ten consecutive cycles 3-12, whose minima all are within 2 years of the Jovian perihelion. All the seven last Gleissberg cycles after the 1843-1923 cycle have been shorter than the previous one. According to the Schove rule, this can't continue long, because the last cycle length of 72.8 years or 6.14 Jovyrs is already approaching the lower limit of 6.07 Jovyrs. The next Gleissberg omits the cycle 16 that lasted 10.2 years from 1923-1933. This means a lower limit of 9.4 years for the cycle 23, because then the lower limit of Gleissberg is reached.

R(M) has been low or medium from the start of the measurements with cycle 1 in 1755 during 12 Gleissbergs, of which the last ended in 1954. It has oscillated between 93 (1856-1933) and 107 (twice). There has been a rising tendency during the last six cycles since the minimum value. Especially the two last Gleissbergs have been high, 133 and 141. The next cycle omitted had an R(M) of only 78, so there are good chances that mean R(M) will still climb on the next round.

The winter temperature (January plus February) in Helsinki had a 27-year average of -6.6 degrees in 1835-1861. It climbed to a high -5.2 during 1889-1915, then had a slight cold trend and were back to -5.2, from where it has climbed to -4.6 degrees in 1970-1996. Both the late warming trend and the standstill/chilling are in accord with both the Gleissberg length and the Gleissberg R(M), which likewise had a standstill/backward trend before the beginning of the late shortening/rising.

The Gleissberg cycle that begins at our synchpoint, in 1810, lasted 6 2/3 Jovyrs. The Gleissberg cycle that ended 154 years later, lasted 6 1/3 Jovyrs. The Gleissberg cycle that began after the Maunder minimum (1712-1775) had a length of 72.2+-0.5 years or between 6.04 and 6.13 Jovian years or in the range of the shortest possible.

The Gleissberg cycle twice lasts 6.5 Jovian years: these are the cycles 2-8 and 11-17. In both cases the middle one of the seven cycles is the lowest in 100 years, which is also the interval of these supercycles. In this way the Gleissberg cycle is connected to the 200-year cycle.

Go to the
beginning of PART 8: Organizing the cycles into a web.

Go to the
beginning of PART 7: Summary of supercycles and a hypercycle of 2289 years.

Includes
- Short supercycles.
- Supercycles from 250 years to a hypercycle of 2289 years.
- The long-range change in magnitudes.
- Stuiver-Braziunas analysis: 9000 years?

Go to the
beginning of PART6: Searching supercycles by smoothing.

Includes
- Smoothing sunspot averages in 1768-1992 by one sunspot cycle.
- Smoothing by the Hale cycle.
- Smoothing by the Gleissberg cycle.
- Double smoothing.
- Omitting minima or taking into account only the active parts of the cycle.

Go to the
beginning of the 200-year cycle.

Go to the
beginning of this part.

Go to the
beginning of the minima, maxima and medians.

Go to the
beginning of the avg. influence of Jupiter.

Go to the
beginning of the sunspots.

Comments should be addressed to timo.niroma@pp.inet.fi