Sun and Jupiter.

Sunspot cycles and supercycles and their tentative causes.

- Table 15. The Wolf number averages during one Jovian year.
- Interpretation 1. Wolf minimum and maximum during one Jovian year.
- Figure 1. 19 Jupiter years indicated by the corresponding monthly Wolf values.
- Figure 2. A fake cycle of 3 months shorter than the Jovian year.

- 2.1.1. Regression analysis.
- Speculation 2. The sunspot number variation during one Jovian year.
- 2.1.2. Exceptional months.

- 2.2.0. The artifact test.
- Table 16. The cycle mean magnitudes from 11 years to Jovian year.
- 2.2.1. The plus-minus analysis.
- Table 17. Plus-minus.
- Conclusion 7. Four types of cycles.
- 2.2.2. The discrimination analysis.
- Table 18. Discrimination.
- Conclusion 8. Three of the cycles are sharp.
- 2.2.3. Difference between the largest and smallest class means.
- Table 19. The classes with the lowest and highest value.
- 2.2.4. The three criteria considered together, a summary.
- Table 20. A summary of cycles.
- Conclusion 9. Intensity cycle times length cycle gives the basic period.
- FIGURE 3. High and low symmetry of cycles between 100 and 170 months.

- 2.3.1. 14 Jovian years.
- Table 21. The magnitude in 14 Jovian year intervals.
- 2.3.2. 19 Jovian years compared to 14 Jovian years.
- Table 22. Magnitudes during 19 and 14 Jovian years ending 1987.
- 2.3.3. The individual Jovian years.
- Table 23. The Jovian-friendliness of the sunspots from 1749 to 1992.
- 2.3.4. The 14 Jovian year set smoothed.
- Table 24. Running means of means in the 14 Jovian year set.
- 2.3.5. Eliminating the supercyclic rise.
- Table 25. The supercyclic rise from 1774 to 1987.
- Table 26. The Jovian year indexed to get rid of the supercyclic rise.

- Magnitude minima.

- Magnitude maxima.

- Medians and quartiles.

- The perihelian stability.

- How long is the 11-year cycle?

- The rules of Schove interpreted.

-- The supercycle of 7 consecutive cycles.

-- The supercycle of 14 consecutive cycles.

- The Precambrian Elatina formation.

- 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 2. SUN AND JUPITER
**

Now it's time to forget the cycles for a while. I will return to them in the chapter 4, when I connect the basic cycle to supercycles. We have hitherto had hints of a possible Jovian effect but no conclusive evidence. Now I will search hints purely by looking at the Wolf values during different periods of the Jovian year.

The analysis is based on the intensity variation of the sunspot behaviour, i.e. the sunspot numbers, mainly the Wolfian ones. Of course, the cycle lengths were also based on them, but now we will use them directly, mainly on a monthly basis, not being tied to any minima or maxima based on some 13 months or any other running means. There is data from a period of 250 years or since 1749, but because of the uncertainty (there are e.g. many missing, and thus interpolated months) I will neglect the decade of 1750 and begin the analysis from the Jovian perihelion in 1762, near the maximum of the cycle 1. The main analysis ends with the first part of the year 1999 with the Jovian perihelion in that year. This analysis thus includes 237 years and 3 months in Earthly terms or exactly 20 Jovian years.

The orbital period of Jupiter round the Sun is 11.862 years or 142.34 months. In this analysis I have used the data of the 20 Jovian years beginning with the perihelion in February 1762 and ending with the perihelion in May 1999 (the last used month is April 1999). I made a computer program that classifies the Wolfian sunspot values according to their distance of the latest perihelion on a monthly basis. I thus have 142 classes each containing 20 observations. In this first analysis I have used the arithmetic mean as the group value for each class. Because the Jovian year has 0.34 months over 142 months, I made the program, to maintain the synchronization, to ignore one month every third Jovian year, a kind of negative Jovian leap year.

If you want to look at the program click here.

The next table contains the means of the 20 Jovian years by months from the perihelion grouped by 10 months, that is N=200. The exceptions are the perihelion (0) and the aphelion (71) months that are listed as such (N=20).

distance the average the change from peri- Wolfian helion in sunspot value months (graphically 0=30) PERIH. (42) ---------------------------------------------- 1- 10 40 xxxxxxxxxx +5 11- 20 51 xxxxxxxxxxxxxxxxxxxxx +11 21- 30 57 xxxxxxxxxxxxxxxxxxxxxxxxxxx +6 31- 40 64 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx +7 41- 50 72 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx +8 51- 60 71 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx -1 61- 70 63 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx -8 APHEL. (65) ---------------------------------------------- 72- 81 61 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx -2 82- 91 58 xxxxxxxxxxxxxxxxxxxxxxxxxxxx -3 92-101 50 xxxxxxxxxxxxxxxxxxxx -8 102-111 44 xxxxxxxxxxxxxx -6 112-121 36 xxxxxx -8 122-131 33 xxx -3 132-141 35 xxxxx +2 PERIH. (42) ----------------------------------------------

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The sunspot value, which has an average value of around 40 Wolfs during and nearly 10 months after the Jovian perihelion, rises in average continuously after that about 0.7 Wolfs per month until Jupiter is about 40-50 months or 3.5-4 years away from its perihelion. The reached sunspot value, about 70 in average, is maintained still in the 50-60 month interval or 4-5 years after the perihelion.

The monthly values show that the Wolf value reaches its maximum value (77) 51 months after the perihelion. Expressed in years this is 4.25, which is very near the mean rise time from minimum to maximum (4.35 years). By chance or has this some deeper meaning?

Some 20-30 months around the aphelion the average sunspot value seems to hover in average near 60 Wolfs. If this indicates an independence from Jupiter's grip, it means that the Sun is on its own only 15 to 20 % of its lifetime.

50-60 months or 4-5 years before the perihelion the Wolf value begins to drop. But the Wolf minimum, about 30, is reached before the perihelion and has a sudden rise to 40 around the perihelion. Does this have something to do with the fact that Jovian perihelion and sunspot minimum never have coincided?

In fact the rise slowly begins already 9 months before the perihelion, while the minimum is 20-10 months before the perihelion. In average, of course.

A fake cycle of only 3 months shorter than one Jovian years is enough to disrupt the effect in 19 Jovian years:

When **the rise** finally begins 9 months before the perihelion, it is rather steady and linear during
the 60-month or 5-year climbing it has ahead beginning with the Wolf value 28 and ending with
the value 77 51 months after the perihelion. Because of the linearity of the rise, about 7 Wolfs
in every 10 months, I made a regression analysis of these 60 months.

**
If we denote the Wolf number with Y and the month with X,
we get the equation
Y = 35.9 + 0.70X,
which means that
the expected perihelian value is 36 and
the rise is 7.0 Wolfs in ten months.
The correlation is very high, 96.6% and
standard error rather low, 3.3 Wolfs.
**

**
The fall** doesn't look so linear. Despite of this I made a regression analysis of it too. At least we
can analyze the residuals. The end of the fall is an easy choice: 21 months before the perihelion,
when the Wolf value gets its lowest value of all the 142 values, 27, and where begins a smooth
period without any downward or upward trend, but the beginning had to be chosen somewhat
arbitrarily, because there exists no obvious turning point: the fall begins gradually. I chose the
month 76 or 5 months after the aphelion or 66 months before the perihelion with its value of 69.
The fall thus lasts 46 months.

**
The regression equation is now
Y = 120.2 - 0.71X,
which would give the aphelion (not included) a value of 70.
The fall is 7.1 Wolfs in ten months
correlation being 94.4% and
the standard error is 3.4 Wolfs.
**

There are some individual months whose values deviate from the surrounding months more than chance would predict.

The first of these is the perihelion month 0:

-3 30 x -2 36 xxxxxxx -1 37 xxxxxxxx 0 42 xxxxxxxxxxxxx 1 38 xxxxxxxxx 2 37 xxxxxxxx 3 33 xxxx

The rise has a temporary peak exactly at the perihelion. The value 42 is lastingly achieved only in the month 6.

The second type is an abnormal rapid temporary change in the ongoing trend. There is one such during the rise, and one during the fall.

Suddenly during the rise there is a sharp increase from month 28 to month 30.

28 45 x 29 53 xxxxxxxxx 30 61 xxxxxxxxxxxxxxxxx

While the normal increase between the months 28 and 30 would be 1.4 Wolfs, in reality it is over tenfold, or 16 Wolfs normally needing 23 months. The Wolf number had already achieved the value 52-53 and the previous value of 45 was in the month 16 or 12 months earlier. And the value 61 is exceeded only in the month 37 or 7 months later.

The one similar case, now during the fall, occurs during the months 105-107 or 35-37 months before the perihelion, and looks like this:

105 (-37) 53 xxxxxxxxxxxxx 106 (-36) 46 xxxxxx 107 (-35) 41 x

The 12 Wolf decrease during two months is again nearly tenfold the expected 1.4 Wolfs.

One possible cause for these two sudden changes could be the fact, that Jupiter's orbit crosses the solar equator during months 34 and 105.

Finally the residual analysis of the regression equations unveils some lonesome months that are not neatly in the row. The two above equations are used for the rise and the fall. These lonesome values could be ignored as caused by some unreliability in the data, if there were not some regularity between their occurrence. The greatest residuals are 10 Wolfs. In the following list are all residuals whose absolute value exceeds 5:

months: -rise: 0 18 26 28 35 46 51 -fall: 88 105 121

The months 26 to 28 had a speculative explanation above and the months 46 to 56 are the top of the average values. If we dismiss the already discussed months 26-28 and the month 46 as a preliminary for the top, we are left with the following series:

rising months: 0 18 35 51 intervals: 18 17 16 falling months: 88 105 121 intervals: 17 16

The 16-17 month intervals could be a reflection of the mean sunspot period. 1/8 of the supposed mean sunspot period of 11.07-11.08 years is 16.6 months.

Now we have the first hint that the Jovian cycle is real and not a broadened fake reflection of the 11 year period. The Jovian cycle seems to consist 8 times 1/8 of the mean cycle (8*16.6 months) plus 2 times 4.75 month (9.5 months) surplus, one around the aphelion and the other around the perihelion. This surplus is needed because one Jovian year is about 7% longer than the mean sunspot period.

How can then the sunspot period of about 11.07 years synchronize itself with the Jupiter year of 11.86 years, when the two periods differ 0.8 years in length? Actually there seems to be two ways thru which the synchronization and the following relatively smooth rise and fall that has a length of one Jovian year is achieved. Of course this requires at least one full round of n cycles which correspond n+1 Jovian years. The difference of 0.8 years of 7% of the Jovian year was inspected already in the introduction and let to the speculation that there may be needed 15 cycles and 14 Jovian years or 166.1 calendar years. Actually half of that amount will usually begin to show the shape. On the other hand, 30 sunspot cycles and 28 Jovian years may be needed for a more accurate picture, because it seems likely that the synchronization may oscillate between 14-16 cycles and correspondingly between 13 and 15 Jovian years. The 225 years studied is thus enough to show the effect but some 330 years would be needed for a more accurate picture. Now it's easy to understand why the three whole Jovian years 1951-1987, that would give us the more more accurate 10.7 cm flux values, are not enough for this analysis.

**The first way** in which the Jovian year is mediated to the sunspot cycle became evident already
in the introduction. **More cycles begin near the perihelion at the favored distances of 0.8 or 1.6-
1.7 years than near the aphelion. This is achieved by the cycle lengths: they are longer when the
minimum begins near the perihelion and shorter when it begins nearer the aphelion.** If we omit
the cycles that begin on the borderline where Jupiter's orbit intersects the Sun's equator (13-15,
1889-1913 and 23, 1996), of the remaining 20 cycles, 14 (3-12, 1775-1878 and 19-22, 1954-
1986) begin on the perihelion side and only 6 (0-2, 1745-1766 and 16-18, 1923-1944) on the
aphelion side. The perihelion cycles last together 156 years (11.2 years per cycle) and the
aphelion cycles 61 years (10.2 years per cycle).

**The second** fact that mediates this phenomenon is that **during the period that begins about 20
months before the perihelion and ends about 10 months after the perihelion the maximum has
never exceeded 100-110 Wolfs,** when during other Jovian months it can exceed 200 (the more
probably the nearer the maximum occurs the aphelion). **Thus perihelion-beginners can have
practically any possible height, but aphelion-beginners are either low or medium, never high.**

But before we can be sure that this still not is some kind of an artifact, we must be sure that the Jovian cycle is no reflection of some nearby cycle, for example a dampened reflection of the mean cycle. This can be proved by showing that the cycles whose lengths are a few months less or more than the orbital period of Jupiter are continuously more disordered than the Jovian cycle so that it has a life of its own. This does not exclude other cycles, as long as between the cycles there is some differentiating gap.

To test this I made a program that generates all the cycles between 103 and 172 months (or from 8 yrs 7 mons to 14 yrs 4 mons) with 1 month interval to cover the whole cycle length range that has ever been observed and a little more. The run was made in seven parts each consisting of ten consecutive months and using the maximum amount of cycles available in the years 1762-1987 allowed by the longest cycle of this ten cycle's packet.The reason for this 7 parts and 10 cycles at time was purely practical: to save the programming effort.

The analysis uses 10 month averages, which means that the surplus months which appear in 9/10 of the cycles must be treated in a special way. The classes that contain 1-5 observations are omitted, and the classes that contain 6-9 observations are included (the sum of course divided with a lesser number than 10). Every cycle of course has one month more than the previous cycle.

The following table contains the cycles from 132 months (11 years) to 143 months (12 years 11 months) so that both the mean sunspot period of 133 months and the Jovian year of 142.3 months are covered.

The distance from the beginning of the cycle (that month plus next 9 months):

0 10 20 30 40 50 60 70 80 90 100 110 120 130 cycle: 132mo 68 61 49 41 34 28 26 26 36 53 73 85 83 - the mean period of ------------------------------------------- 133mo 58 48 39 31 24 24 30 38 52 70 83 80 70 - the sunspot cycle -------------------------------------------- 134mo 46 35 28 24 24 31 41 55 68 84 81 71 63 - 135mo 35 27 26 27 32 42 56 66 78 81 69 63 51 - 136mo 29 30 32 34 41 51 64 76 74 73 63 51 40 (30) 137mo 34 34 36 41 46 60 70 73 70 65 50 42 36 (36) 138mo 38 39 38 45 53 62 72 69 61 52 46 44 45 (43) 139mo 40 36 42 48 55 68 66 60 52 55 52 52 47 (45) 140mo 38 37 45 50 60 64 62 55 56 62 55 53 48 38 141mo 35 42 47 53 59 65 60 61 62 60 55 48 42 34 - 142mo 39 43 52 55 64 67 64 63 61 58 48 43 33 31 - Jovian cycle: the small discrepancies with the earlier distribution 142.3 40 46 51 60 67 69 63 62 59 53 48 39 32 31 - are due to the fact that these means contain perihelion and aphelion 143mo 43 49 54 58 66 62 59 57 54 53 44 36 34 33 -

The rise and fall changes of the mean period cycle and of the Jovian year cycle. The months are synchronized to help the comparison.

mean: 6 8 14 18 13 -3 -10 -12 -10 -9 -8 -7 0 (mean has one class less because of the smaller amount of months) Jovian: 9 6 7 9 7 2 -6 -1 -3 -6 -5 -9 -7 -1

The Jovian cycle doesn't look like a stretched mean cycle, but let's be sure about that.

**
When we take continuously more distance from the 133 month (11.1 year) cycle whether
lengthening or shortening the period, the distribution should deteriorate so that in theory in the
end there is left only an even distribution with same values in each class and in practice a
distribution whose minor fluctuations are purely random. This is unless we encounter a new
cycle. This is the basis for our test. The test is very critical: IF THE JOVIAN CYCLE SHOWS
NO DISENTROPY OF ITS OWN, WE CAN FORGET THE JOVIAN EFFECT.
**

I have used three tests to verify whether a cycle is a real independent cycle: 1. Plus-minus analysis to test that there are only two turning points, 2. analysis of the absolute changes from one class to the other to test that the mean change is greater with this cycle than with the surrounding cycles, and 3. analysis of the difference of the greatest and the smallest class mean to test that the difference within this cycle is greater than within the surrounding classes.

The plus-minus analysis is a dichotomous analysis: the cycle is either approved or disproved. The rules for the approval are simple: 1. The cycle has only two turning points. 2. Zero is allowed only at turning points. 3. Only one zero per turning point is allowed.

good 116mo + + + - - - 0 + + - 0 117mo + - - - - - + + + + + X 118mo - - - - - + + + + + + X 119mo - - - - 0 + + + + 0 - X 120mo - - - - + + + + - - - - X 121mo - - + + + + + + - - - - X 122mo + + + + + + 0 - - - - - X 123mo + 0 + - 0 - + + - - - + 124mo - - - - - - + - - + + + 125mo - - - - + + + + + + - - 126mo - - + - + + + + + - - - - 127mo + + 0 + + + + - - - - - - 128mo + + + + + + - - - - - - - X 129mo + + + + - - - - - - - + + X 130mo + + - - - - - - - - + + + X 131mo - - - - - - - - - + + + + X 132mo - - - - - - - 0 + + + + - X the average ------------------------------------------------------- 133mo - - - - - 0 + + + + + - - X sunspot cycle ----------------------------------------------------- 134mo - - - - 0 + + + + + - - - X 135mo - - - + + + + + + + - - - X 136mo - + + + + + + + - - - - - - X 137mo - 0 + + + + + + - - - - - 0 138mo - + - + + + + - - - - - + - 139mo - - + + + + - - - + - 0 - - 140mo 0 - + + + + - - + + - - - - 141mo + + + + + + - + + - - - - - 142mo + + + + + + - - - - - - - - X the Jovian cycle -------------------------------------------------- 142.3 + + + + + + - - - - - - - - X the Jovian cycle -------------------------------------------------- 143mo + + + + + - - - - - - - - - X 144mo + + + + - - - - - - - - - + X 145mo + + + - - - + - - - - 0 + + 146mo + + - - - - - - - - - + + + + X 147mo + - - - + - - - - + + + + + + 148mo + - - - + - - + + + - + - - + 149mo - - - + - - + + + + + - - + 0 150mo + - - + + + + 0 + 0 - - - - 0 151mo - - + + + + + + - - - - - 0 - 152mo - + + + + + - - + - - 0 - - 0 153mo + + + + + + - + - - - - - - + 154mo - + + + + - - - - - - - - + + 155mo + + + - + - - - 0 - - 0 0 + +

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Months whose cyclicity are acceptable, are 111-112 (outside the table), 117-122, 128-136, and
142-144. In years they are 9.25-9.3, 9.75-10.2, 10.7-11.3, and 11.8-12.0. Thus we have here both
the average length 11.0+-0.3 and the Jovian year 11.9+-0.1. The most conspicuos cycle governing
the lengths, the 10.3 year cycle, however does not appear at all when using the magnitude.
Instead there is a strong intensity cycle around 9.9-10.0 years that doesn't appear in the cycle
lengths. **Now we have four types of cycles: one that shows up both in intensity and actual lengths
(one Jovian year, 11.9 calyrs), one that shows up in intensity and only in theoretical lengths (the
mean length, 11.1 yrs), one that shows up only in actual lengths but not in intensity (10.3 yrs),
and one that shows up only in intensity but not in the lengths (9.9 yrs).** There may be one more
cycle of the last type at 9.3 yrs.

****************************************************************

The mean of the absolute changes between 10 month intervals. The lower the value, the more similar are the values. Zero means no cyclicity with this interval, and a value below one means only statistical noise. Cycles appear as peaks.

109mo 2.6 ************* 110mo 3.5 ****************** 111mo 5.1 ************************** 112mo 5.8 ***************************** 113mo 6.0 ****************************** 114mo 3.8 ******************* 115mo 3.6 ****************** 116mo 4.1 ********************* 117mo 6.0 ****************************** 118mo 7.8 *************************************** 119mo 8.7 ******************************************** 120mo 8.0 **************************************** 121mo 6.7 ********************************** 122mo 4.3 ********************** 123mo 3.4 ***************** 124mo 3.9 ******************** 125mo 4.4 ********************** 126mo 5.4 *************************** 127mo 6.6 ********************************* 128mo 6.5 ********************************* 129mo 6.6 ********************************* 130mo 6.9 *********************************** 131mo 8.8 ******************************************** 132mo 9.1 ********************************************** the average ------------------------------------------------- 133mo 9.1 ********************************************** sunspot cycle ----------------------------------------------- 134mo 9.2 ********************************************** 135mo 8.5 ******************************************* 136mo 6.7 ********************************** 137mo 5.8 ***************************** 138mo 5.0 ************************* 139mo 5.1 ************************** 140mo 4.8 ************************ 141mo 4.8 ************************ 142mo 5.2 ************************** the Jovian cycle -------------------------------------------- 142.3 5.4 *************************** the Jovian cycle -------------------------------------------- 143mo 4.7 ************************ 144mo 4.4 ********************** 145mo 4.3 ********************** 146mo 3.6 ****************** 147mo 3.3 ***************** 148mo 3.4 ***************** 149mo 2.4 ************ 150mo 2.5 ************* 151mo 2.6 ************* 152mo 3.1 **************** 153mo 2.9 *************** 154mo 3.1 **************** 155mo 2.4 ************ 156mo 2.9 *************** 157mo 2.5 ************* 158mo 1.9 **********

The first cycle appears with a period of 112-113 months or 9.3-9.4 years. The next cycle at month 119 or 9.9 years is relatively sharp. 123 months would correspond the primary cycle length, 10.2-10.3 years, but in intensity it is completely non-existent. The most powerful intensity cycle is at months 132-134, which corresponds to the mean cycle length: 133 months would be 11.08 years. From month 135 (11.25 years) to month 138 (11.5 years) there is a sharp decrease in the discrimination power, but then it comes to a halt, and has a short but sharp rise exactly at one Jovian year, 11.86 years or 142.3 months. Then it begins to fall again until a rise in the month 152, two times the distance of the Jovian year from the mean cycle. The first four of these five cycles are acceptable by the plus-minus criterion. The last is not, and can be considered as a reflection of the Jovian year.

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Plus-minus and the discrimination analysis support each other. Only the values are now sharper.
**The first cycle seems to be very near 112 months or 9.3 years. The second cycle sharpens near
119 months or 9.9 years. The third cycle is not so sharp, but still sharper than in previous
analysis, 132-134 months or 11.00 to 11.17 years, the mean of sunspot cycle length, when there
are at least 15-20 cycles.
**

**
The fourth cycle is even sharper than the 9.9 years cycle, but here is used an accuracy of 0.1
months or 3 days. It is the Jovian year. Even a change of 0.3 months or 9 days begins to
deteriorate the cycle. The strongest evidence about a Jovian cycle and a Jovian influence on
sunspots we can get.**

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This analysis is redundant with the above discrimination analysis but used to check the slight irregularity caused by the last class in nine cases out of ten not containing ten months. Besides this, this analysis also has the advantage that it takes only in account the really extreme classes, and is not affected by any asymmetry. It's the amount of difference that counts. The continuity of the rise and fall was checked via the plus-minus analysis.

I first made a similar histogram as in the discrimination analysis, but because the resulting diagram was almost identical, I omitted it. The cycles of 112 and 113 months (9.3-9.4 years) got an almost equal value: 32 and 33. The second cycle peaked equally at 119 and 120 months (implicating about 9.95 years) with value 48. The third cycle was equally broad compared with discrimination analysis: the mean cycle from 132 to 134 months or somewhere around 11.1 years. It was also here the highest one, the difference between hi and low was 59-60 Wolfs.

The Jovian cycle also repeated itself in the same form as in the previous analysis. On monthly basis, the 142 months had the highest difference, 36 Wolfs, but with the accuracy of 0.1 months or 3 days, the exact Jovian year or 142.3 months, was again the highest, 38 Wolfs. 143 months dropped already to 33 Wolfs.

Around 160 months the changes begin to approach what can be expected by noise, by random fluctuations.

The lows between these cycles are at months 109, 115, and 123, which in years are 9.1, 9.6, and 10.25, respectively. Interestingly they correspond to cycle lengths.

So I made a diagram that shows only the classes with the smallest and largest mean so that the values are vertically on the relative position, that is dictated by its value. The maximum separation means a cycle, the nearing of the values means that only reflections exist at that month.

months 106 40 57 107 40 61 108 42 60 109 44 54 110 41 58 111 36 64 112 32 64 113 33 66 114 40 61 115 43 57 116 41 61 117 32 65 118 28 71 119 27 75 120 27 75 121 30 70 122 35 61 123 40 55 124 42 61 125 39 65 126 37 71 127 33 76 128 30 72 129 27 70 130 26 71 131 24 81 132 26 85 the average -------------------------------------------------------- 133 24 83 sunspot cycle ------------------------------------------------------ 134 24 84 135 26 81 136 29 76 137 34 73 138 38 72 139 36 68 140 37 64 141 34 65 142 31 67 the Jovian cycle ------------------------------------------------ 142.3 31 69 the Jovian cycle ------------------------------------------------ 143 33 66 144 34 65 145 35 65 146 37 64 147 41 63 148 40 61

The Jovian year has the sharpest difference, the average one the widest.

The above methods are surprisingly unanimous about where the cycles lie.

Now it's time to look at the three analysis together to get as accurate a picture as one can get with this method. The first value and x denote the discrimination analysis. The second value and c denote the difference between the greatest and the smallest mean. Ok means that the cycle is approved by the plus-minus analysis.

110mo 3.5 17 xxxc 111mo 5.1 28 xxxxxcc ok 112mo 5.8 32 xxxxxccc ok 113mo 6.0 33 xxxxxxccc 114mo 3.8 21 xxxcc 115mo 3.6 14 xxxc 116mo 4.1 20 xxxxcc 117mo 6.0 33 xxxxxxccc ok 118mo 7.8 43 xxxxxxxcccc ok 119mo 8.7 48 xxxxxxxxcccc ok 120mo 8.0 48 xxxxxxxxcccc ok 121mo 6.7 40 xxxxxxcccc ok 122mo 4.3 26 xxxxcc ok 123mo 3.4 15 xxxc 124mo 3.9 19 xxxc 125mo 4.4 26 xxxxcc 126mo 5.4 34 xxxxxccc 127mo 6.6 43 xxxxxxcccc 128mo 6.5 42 xxxxxxcccc ok 129mo 6.6 43 xxxxxxcccc ok 130mo 6.9 45 xxxxxxcccc ok 131mo 8.8 57 xxxxxxxxccccc ok 132mo 9.1 59 xxxxxxxxxccccc ok the average ------------------------------------------------- 133mo 9.1 59 xxxxxxxxxccccc ok sunspot cycle ----- ----------------------------------------- 134mo 9.2 60 xxxxxxxxxcccccc ok 135mo 8.5 55 xxxxxxxxccccc ok 136mo 6.7 47 xxxxxxcccc ok 137mo 5.8 39 xxxxxccc 138mo 5.0 34 xxxxxccc 139mo 5.1 32 xxxxxccc 140mo 4.8 27 xxxxcc 141mo 4.8 31 xxxxccc 142mo 5.2 36 xxxxxccc ok the Jovian cycle -------------------------------------------- 142.3 5.4 38 xxxxxccc ok the Jovian cycle -------------------------------------------- 143mo 4.7 33 xxxxccc ok 144mo 4.4 31 xxxxccc ok 145mo 4.3 30 xxxxccc 146mo 3.6 27 xxxcc ok 147mo 3.3 22 xxxcc 148mo 3.4 19 xxxc 149mo 2.4 15 xxc 150mo 2.5 18 xxc 151mo 2.6 20 xxcc 152mo 3.1 21 xxxcc 153mo 2.9 20 xxcc 154mo 3.1 21 xxxcc 155mo 2.4 17 xxc 156mo 2.9 16 xxc 157mo 2.5 13 xxc 158mo 1.9 10 xc

I have calculated the lengths of the four cycles that appeared in this analysis by using weighted averages counting the two or three highest values and their immediate neigbours. Results are as follows:

cycle length in yrs by 2.2.2. 2.2.3. I 9.37 9.37 II 9.95 9.96 III 11.08 11.08 IV 11.84 11.84

It appears that it doesn't matter which one of the two methods we use. The statistical error (95%) margins are +-0.03 years, so with this confidence limit we get for the cycle values 9.34-9.40, 9.93-9.98, 11.05-11.11 and 11.81-11.87 years.

Because the main cycles while considering lengths from minima to minima were about 10.3 years and one Jovian year, a new cycle at 9.9-10.0 years in intensity was unexpected (as well as the disappearance of the 10.2-10.3 year cycle).

To see how it looks like, I drew a graphical picture with a cycle length of 119 months (9.92 years). A very clear dichotomy between low and high values is what shows up. The first part is low except in two cases, where there is a permanent uplift in the Wolf value level. The second part is usually high and excepting the two rises of level all the high values lie in the second part.

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2.3.1. 14 Jovian years

The above analysis was based on 19 Jovian years or 225.3 Earthly years. 14 Jovian years, exactly or about, correspond to 15 sunspot cycles. If the relation would be 13:14, the mean cycle would be 11.015 years, if it would be 15:16, the mean would be 11.12 years. For the relation to be exactly 14:15, the mean cycle should be 11.071 years. Most scientists prefer a value between 11.07 and 11.08 years.

So solely on the grounds 14 Jovians/15 cycles we have in our analysis 36% too much data, which can cause bias. On the other side we saw in our perihelion dance, that at least two consecutive supercycles do not resemble each other, so we should rather say, that we have only 68% of the needed 28 Jovians/30 cycles period. But now we have a complication. We are rather interested in the magnitudes than the cycle lengths. But later we do seem, that there is a supercycle of about 212 years. Thus we should have 106% of the needed data. Or do we? The first obstacle is that the 200-year supercycles have only some characteristics in common, for example the supercyclic rise needs a hypercycle. There could be one around 1060 years (5 times 212?). So we don't actually know how representative our data are. But let's elaborate a little with what we do have.

Now we have in each category 140 observations (months) instead of 190, and I have chosen to leave the years 1762-1774 out because there are so many missing days. One interesting thing that we can try to find out is, is there any tendency of the magnitude to change with the time.

1. months from the perihelion (N=140 except for PER and APH, N=14) 2.-6. 1774-1940, 1785-1951, 1797-1963, 1809-1975, 1821-1987 1. months 2. 3. 4. 5. 6. PER 0mo (40 41 41 43 46)------- 1- 10mo 41 43 41 41 42 11- 20mo 54 57 50 50 52 21- 30mo 59 61 51 53 55 31- 40mo 66 65 56 59 66 41- 50mo 69 61 57 61 71 51- 60mo 61 51 55 59 69 61- 70mo 50 43 51 56 64 APH 71mo (45 39 51 52 61)------- 72- 81mo 41 41 50 54 61 82- 91mo 38 43 52 54 58 92-101mo 33 40 46 49 50 102-111mo 31 38 42 45 46 112-121mo 27 34 36 38 38 122-131mo 27 32 33 36 35 132-141mo 32 36 37 38 39 PER 0mo (40 41 41 43 46)-------

**We can see that the magnitude preserves its stability only during the first 10 months after the
perihelion, when its value is 41-43.** However the last 10 months before perihelion also show
stability, if we ignore the first period, the four last ones are in the range of 36-39 Wolfs.

We have one interesting thing left. **From the aphelion until 10 months before the
perihelion the Wolf number increases or at least preserves its value from every interval to the
next (with one minor exception) in every class.** In absolute figures the increases are 20, 20, 17,
15, 11 and 8 Wolfs. In fact this tendency begins already before the aphelion, after the highest mean value has been reached. The last 10-month period before aphelion has a rise amount of 14 Wolfs. What is odd is that there is no corresponding
rise during months from 10 to 50. The superrcyclic rise happens during months from 55 to 130.

During the 50 months beginning 60 months before the perihelion there is the drop in the
magnitude, that is familiar to us from the 19 Jovyr analysis. It occurs during every interval
disregarding the horizontal growth in intensity. **The intensity is lowest during the 10 to 20
months before the perihelion disregarding the intensity growing from 27 to 35-36. The fall begins 50-60 months before the
perihelion despite the intensity growing from 38 to 58 Wolfs.**

**After perihelion the intensity grows in every supercycle until about 40 month's distance.** The
horizontal time-dependent growth is however only seen after the Jupiter-induced growth ceases.

1. months from perihelion

2. 1821-1987 (14 Jovian years, N=140, except in PER and APH)

3. 1762-1987 (19 Jovian years, N=190, except in PER and APH)

4. difference (last 14 Jovyrs minus last 19 Jovyrs)

1. 2. 3. 4. PER 0mo (46 42)--------- 1- 10mo 42 40 / +2 11- 20mo 52 47 / +5 21- 30mo 55 52 / +3 31- 40mo 66 60 / +6 41- 50mo 71 68 / +3 51- 60mo 69 68 / +1 61- 70mo 64 63 / +1 APH 71mo (61 61)--------- 72- 81mo 61 62 / -1 82- 91mo 58 59 / -1 92-101mo 50 51 / -1 102-111mo 46 46 / - 112-121mo 38 36 / +2 122-131mo 35 32 / +3 132-141mo 39 33 / +6

It does not make much difference whether we use the intensity cycle of 19 Jovian years or the
length cycle of 14 Jovian years. The cycles may not be representative, because they don't begin
at any synchronization years, but despite of that we can see that there is in both cases a clear
Jovian effect. **The magnitude reaches its peak, around 70 Wolfs, 40-50 months after the
perihelion and is at its lowest, a little over 30 Wolfs, 10-20 months before the perihelion. The
higher magnitude shows up where the Jovian effect is at its clearest: 50 months after the
perihelion and 30 months before the perihelion. During the 60 months in-between there is no
difference, no rise of the magnitude.**

If we look at the 19 Jovian years, we can see that there are years that apparently do not show any influence of Jupiter, in fact some seem opposite to the trend that we have sketched above. This must also be readily apparent from the fact that the mean sunspot cycle is about 7% shorter than one Jovian year.

How is it then possible that putting 19 Jovian years together we have a clear Jovian year influence? How is it possible that even if we divide the material into 14 Jovian year sets, the effect still clearly shows up in every one of them? The shape is also not affected of the slightly rising tendency in the average intensity.

Theoretically we have two possibilities. Either only part of the sunspot cycles are affected or all are affected, but with different amount. The first possibility means that there must be more cycles that respond to Jupiter than cycles that are immune to it. The second possibility means that the height of the cycle varies according to the relative positions of the cycle minimum and the Jovian perihelion.

To solve this problem I made a computer program, that drew altogether 1330 Jovian cycles in all the 190 possible combinations beginning with 19 times 1 cycle, continuing with 18 times 2 cycles etc. until drawing 2 times 18 cycles and lastly 1 time 19 cycles.

**
The analysis shows that 10-11 consecutive cycles are always able to show the Jupiter-effect.
**

But the individual cycles vary from apparently immune cycles to cycles that behave as the above average suggests, plus cycles that exaggerate the effect. Or maybe we'd better say that some cycles show the effect very clearly, some a little dampened and some seem not to be affected at all.

In the next table I have measured the affection or hostility of each Jovian year (added with the year 0 and the last whole Jovian year 20 (1987-1999) so that we have 21 Jovian years) by subtracting the Wolf value preceding the aphelion from the Wolf value preceding the perihelion. If the difference is positive, the cycle is affected, if it is negative, the cycle is immune. There are 15 positive (from 1774 to 1892, from 1904 to 1916 and from 1951) and 6 negative years (before 1774 from 1892 to 1904 and from 1916 to 1951). This would be enough to transmit the Jupiter-effect.

The affection was seen already in chapter 1: the most preferred position for the minimum was about 8 or 17 months away from the perihelion. If the minimum situated nearer the aphelion than the perihelion, it hurried speedily to a new position causing the cycle to be short until it was again near the perihelion.

But there is also a second factor that contributes to the Jupiter-effect: **The maximum Wolf
number near the perihelion is 80 in a 10-month interval and 90-100 for an individual month.** The
Wolf number can exceed 200 Wolfs, but only after 25 months have elapsed from the perihelion,
and not long after the aphelion. The aphelion environment may be low or high, but the perihelion
environment is always low, even if the cycle maximum occurs near the perihelion.

1. the year of the perihelion 2. Wolf(PER+5yrs)-Wolf(PER-1yrs) 1. 2. graphically (every * = 10 Wolfs) 0 1750 -71 ******* 1 1762 -48 ***** 2 1774 91 ********* 3 1785 80 ******** 4 1797 29 *** 5 1809 6 * 6 1821 20 ** 7 1833 75 ******** 8 1845 52 ***** 9 1857 55 ****** 10 1868 59 ****** 11 1880 46 ***** 12 1892 -10 * 13 1904 20 ** 14 1916 -21 ** 15 1928 -63 ****** 16 1940 -56 ****** 17 1951 58 ****** 18 1963 68 ******* 19 1975 120 ************ 20 1987 81 ******** 21 1999 -20 **

The average friendliness is +28 Wolfs. The friendlines has been high from 1774 to 1797, from 1833 to 1892 and from 1951 to 1999.

2.-6. series 1-5

7.-11. changes inside each series from the previous values

12. median of all changes

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. PER00 37 39 39 41 43 / +8 +6 +5 +5 +7 / +6 10mo 46 49 45 45 46 / +9 +10 +6 +4 +3 / +6 20mo 58 61 53 53 55 / +12 +12 +8 +8 +9 / +9 30mo 64 64 55 58 62 / +6 +3 +2 +5 +7 / +5 40mo 67 64 56 61 70 / +3 - +1 +3 +8 / +3 50mo 67 57 56 61 72 / - -7 - - +2 / - 60mo 53 45 51 55 65 / -14 -12 -5 -6 -7 / -7 APH71 44 41 52 56 63 / -9 -4 +1 +1 -2 / -2 81mo 39 41 50 55 59 / -5 - -2 -1 -4 / -2 91mo 36 43 51 53 54 / -3 +2 +1 -2 -5 / -2 101mo 31 40 45 48 51 / -5 -3 -6 -5 -3 / -5 111mo 30 39 40 42 41 / -1 -1 -5 -6 -10 / -5 121mo 23 30 32 33 34 / -7 -9 -8 -9 -7 / -8 131mo 29 33 34 36 36 / +6 +3 +2 +3 +2 / +3 PER00 37 39 39 41 43 / +8 +6 +5 +5 +7 / +6

Calculated with running means the low in the five sets occur 21, 21, 17-22, 21, and 19-20 months
before the perihelion. **So 21 months or 1.75 years (15% of one Jovian year) seems to be a very
stable place for the minimum.** The corresponding Wolf numbers are 23, 30, 32, 33, and 33. This
seems to be the point where the quieting Sun again catches turbulence from Jupiter.

The rise rate grows first to 6 Wolfs per 10 months, but gets a kick to about 9 Wolfs some 20
months after the perihelion which is gradually calmed down during the drastic months of 30-35
months after the perihelion. **Then the rise rate slows down until it comes to a halt 40-50 months
after the perihelion:** the high in the five sets occur 44-47, 31-32, 44-45, 39-48, and 48-51 months
after the perihelion. The corresponding Wolf numbers are 70, 67, 58, 62, and 72. The early
second set seems to point to the fact that 14 Jovian years are not always enough to be a
representative sample.

Then there happens an odd phenomenon. There is a rapid and sharp fall just before the aphelion. The aphelion, however, dampens the speed. The speed of fall slowly accelerates again reaching a value of 8 Wolfs per 10 months about 20 months before the perihelion when it comes to a sudden halt.

**Both the rise and the fall are accelerating phenomena, albeit with different characters.** The rise
accelerates first and decelerates then. The fall, if we ignore the "premature" fall before the
aphelion, accelerates first very slowly and after getting speed comes to the sudden halt.

The mean Wolf number (the mean of all the 142 means) has increased throughout the series of 14 Jovian years. In the next analysis has been added the sixth (or first 14 Jovian year 1762-1774), which contains so many missing months that I have avoided it in many analysis. Because of the wealth of material here, I have included it in this analysis. In general it seems very similar to the next set.

From 1774 the mean of means has been steadily growing, by 1.5 Wolfs per Jovian year since 1797.

set years Wolf increment 0. 1762-1928 44.9 1. 1774-1940 44.8 -0.2 2. 1785-1951 46.0 +1.2 3. 1797-1963 47.0 +1.0 4. 1809-1975 49.5 +2.5 5. 1821-1987 53.3 +3.8 6. 1833-1999 56.4 +3.1

Next I made an index for each of the six series, using the five month running means, so that the mean value of each series is 100, to get rid of this growing tendency of the Wolfian number. This supercyclic phenomenon is a further indication of more Jovian years needed. In the table is listed every 10th of the running values and the change from the previous value.

month index change PER 82 ***** / +14 ******* 10mo 98 ********* / +16 ******** 20mo 119 ************* / +21 *********** 30mo 129 *************** / +10 ***** 40mo 136 **************** / +7 **** 50mo 134 **************** / -2 * 60mo 113 ************ / -21 *********** APH 106 ********** / -7 **** 81mo 100 ********* / -6 *** 91mo 97 ******** / -3 ** 101mo 86 ****** / -11 ****** 111mo 77 **** / -9 ***** 121mo 60 * / -17 ********* 131mo 68 *** / -8 **** PER 82 ***** / +14 *******

The overall picture doesn't change from the previous one. The rise begins 10-15 months before the perihelion, reaches its maximum speed about 15 months after the perihelion and the maximum value about 45 months after the perihelion. The fall has its first sharp part just before the aphelion. The second maximum rate of the fall is reached 20-30 months before the perihelion and the Wolf number is at its minimum 15-20 before the perihelion.

The most interesting phases according to the index:

1. fall from 82 to 79 from perihelion to month 3

2. rise from 79 to 116 from month 3 to month 16 (100 in month 11)

3. rise from 117 to 135 from month 27 to month 31

4. a setback to 128-131 from month 33 to month 37 ("deviating mos")

5. MAXIMUM of 140 in months 44-45

--) the next months are months before the perihelion (aphelion = 71)

6. fall from 74 to 60 from month 27 to month 21

7. MINIMUM of 60-61 in months 17-21

8. rise from 75 to 82 from month 3 to perihelion

The rise rate reaches its highest speed from month 3 to month 16 after the perihelion. The rise lasts 61 months, the fall 76 month making the relation as 45:55.

Go to the

Part 8: Organizing the cycles into a web.

Go to the

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

Part 6: 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

Part 5: The 200-year cycle.

Includes

- A 2000-year historical review.

- An autocorrelation analysis.

- 200-year cyclicity and temperature correlations.

- The periods of Cole.

Go to the

Part 4: From basic cycles to supercycles.

Includes

- How long is the 11-year cycle?

- The rules of Schove interpreted.

- The Precambrian Elatina formation.

- The Gleissberg cycle.

Go to the

Part 3: Minima, maxima and medians.

Go to the

beginning of this part.

Go to the

beginning of the sunspots.

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