Sunspots: minima, maxima and medians.

Sunspot cycles and supercycles and their tentative causes.

- 3.1.1. How valid is a zero value on monthly basis?
- Table 27. The low Wolf numbers, occurred v. expected.
- 3.1.2. Three lowest minima.
- Table 28. Three lowest 10-month averages.

- Table 29. Three highest 10-month averages.
- Conclusion 10. A 30-month peaceful area around the perihelion.

- 3.3.1. Stability.
- Table 30. The stability of the Wolf values during the Jovian year.
- Conclusion 11. The perihelion month is a gate.
- 3.3.2. Medians compared to means.
- Table 31. Medians and means compared.
- 3.3.3. Is the supercyclic growth in intensity even during the Jovian year?.
- Table 32. Supercyclic growth in medians.

- 3.4.1. The months with the narrowest ranges of values.
- 3.4.2. The perihelian month.
- Table 33. The perihelion values in 10.7 cm in 1963, 1975 and 1987.
- Table 34. Perihelion month values and their possible deviation from the trend.

- 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 3. SUNSPOTS: MINIMA, MAXIMA, AND MEDIANS
**

There are in our data 1762-1987 63 months whose Wolf value is exactly zero (this means 2.3%). If we count only integers and round the range 0.1-0.4 as zero we get 84 months (3.1%).

**The month 39 is the first month from the perihelion when there never has been a low value of
0 or 1.** During the 52 months numbered from 39 to 90 after the perihelion only 5 months have
reached the zero (rounded) value or more properly said 47 or 90% of these months has never
reached the zero value. Amongst the rest 51 months before the perihelion 21 months have
reached zero (rounded) or 30 or 59% have not.

The 61 zero months or the 81 non-zero months distribute thus very unevenly. During the rise after the perihelion every month has at least once reached zero, around aphelion 90% has not reached, and then when we near the perihelion about 60% have not reached zero and 40% have.

There are six months during which the value has never fallen even below 10. First of these months is 40 months after the perihelion and the last 3 months after the aphelion.

But before we draw any further conclusions, let's consider a little about what we really are measuring here. We have seen that the average Wolfian number has been higher in 1951-1987 than in 1786-1821. Therefore we must ask, if we are here analyzing only a few old cycles. Zero nowadays is more zero than it was, say a 150 years ago. If it is now rare, its relative value weighs more than the absolute. And that seems to be case.

To avoid any wrong conclusions, however, it's good to know that the last exact zero month was in 1913, and 44 or 70% of the 63 zeros (exact) were in the years 1807-1824. If we use the rounded zeros there was two in 1933, one in 1944, and the two last ones so far in 1954, but 49 or 58% of them appeared in 1807-1824.

So this means that if there is some trouble with these zero values, we had to cancel the Jovian years 5-7 from our analysis. To test the validity of these values, I made two regression analysis to see, if some zeros should be replaced by negative numbers. The first data contained Wolf values of 0-36, the second Wolf values of 1-36.

If we denote the Wolf number by X, and the expected number of occurrences by Y, the former data produces a regression equation of Y=48.2-0.81X, and the latter an equation of Y=44.0- 0.64X. Because the relation is not strictly linear, the occurrence goes to zero at 59 Wolfs with first, and at 68 Wolfs with the second equation although we know that the value can go over 200.

The reason for using two equations, is that the first caused the value 0 to have a residual of 36 or 75% over the expected value. The validity of the great residual was confirmed by the second: it rose to 40 or nearly double the expected value. There are too many zeros.

The low Wolf numbers or values 0-36 according to the second equation:

1. the Wolf number 2. the occurrence in the 19 Jovian years 1762-1987 3. the expected occurrence (Y=44.0-0.64X) 4. the residual (the actual occurrence frequency minus the expected) 1. 2. 3. 4. 0 84 44 40 **************************************** 1 44 43 1 * 2 38 43 -5 ***** 3 39 42 -3 *** 4 46 41 5 ***** 5 49 41 8 ******** 6 36 40 -4 **** 7 38 40 -2 ** 8 52 39 13 ************* 9 42 38 4 **** 10 33 38 -5 ***** 11 43 37 6 ****** 12 44 36 8 ******** 13 32 36 -4 **** 14 34 35 -1 * 15 28 34 -6 ****** 16 28 34 -6 ****** 17 26 33 -7 ******* 18 26 33 -7 ******* 19 31 32 -1 * 20 24 31 -7 ******* 21 29 31 -2 ** 22 41 30 11 *********** 23 23 29 -6 ****** 24 25 29 -4 **** 25 21 28 -7 ******* 26 32 27 5 ***** 27 30 27 3 *** 28 28 26 2 ** 29 32 26 6 ****** 30 27 25 2 ** 31 28 24 4 **** 32 26 24 2 ** 33 22 23 -1 * 34 23 22 1 * 35 21 22 -1 * 36 20 21 -1 *

I consider the value 0 last and search first for great residuals or deviating trends in the range of 1-36 Wolfs. There are too many values of 8, and possibly of 5 and 12 also. Taking into account the value of k, the Wolf number of 8 actually represents a value of 13 or 1 sunspot group with three spots. 1 sunspot group seems to contain 3-5 spots rather than 1-2 or 6-7 (9 Wolfs is actually 15). The other Wolf number overrepresented is 22, which corresponds to an actual value of 36-37. This can mean 3 groups with 2 spots in each or 2 groups with 8 spots in both or most probably 2-3 groups containing 3-5 spots in each one (remember these are monthly averages, not daily). The values 15-18 seem to be underrepresented. This corresponds to a true value of 25-30. This means that two groups rarely exist by themselves but seem rather to co-exist with three groups. After Wolf number 30 there are no great residuals (with the single exception of 49).

Our two regression equations are valid only until month 40. We must use a third regression equation with values higher than 40.

Because there are 84 zero-months instead of the expected 44, we can say that some zeros are really more zero than others. But because an illusory value of -1 would satisfy all that is needed, we can be assured that our analysis is valid even with low values when they are the averages of, say, 140 or 190 values, as in our case.

I made a comparison between monthly values of Wolf and 10.7 cm during the low period from April 1995 to June 1997. The correlation was .86.

So we can consider the following magnitude minimum and near-minimum table valid:

1. distance from perihelion 2. magnitude minimum, the average of 10 months (except PER and APH) 3. the second lowest values, 10-mo avg 4. the third lowest values, 10-mo avg 5. graf: . = first, * = second added, + = third added 1. 2. 3. 4. 5. PER 0mo 0 2 3----------------------- 1- 10mo 0 1 4 *+++ 11- 20mo 0 0 4 ++++ 21- 30mo 0 0 4 ++++ 31- 40mo 2 5 10 ..***+++++ 41- 50mo 5 9 13 .....****++++ 51- 60mo 6 9 13 ......***++++ 61- 70mo 3 13 19 ...**********++++++ APH 71mo 8 9 11 ---------------------- 72- 81mo 5 8 10 .....***++ 82- 91mo 3 5 9 ...**++++ 92-101mo 2 5 8 ..***+++ 102-111mo 0 3 6 ***+++ 112-121mo 0 2 4 **++ 122-131mo 2 3 6 ..*+++ 132-141mo 2 5 7 ..***++

The distribution looks familiar, even if the maximum is nearer the aphelion than with mean
values. With the second lowest values the five greatest values are 12 months and 4-7 months
before the aphelion, with the third lowest values 12, 7, 5, 4, and 2 months before the aphelion.
With the mean values it was the perihelion that was in different ways in a peculiar position, but
the aphelion didn't show up in no discernible way. **With magnitude minima the prohibition not
to go low gets firmer the nearer we are the aphelion, but rapidly disintegrates after the aphelion.**
This may the reason for the two-stage fall from about month 50 to the period of 20 months
before the perihelion.

The second difference compared to the mean values occurs during the first 35 months. **During
this period there is a very strong low-value, even zero-tendency.** Again there is a change in the
sunspot behavior around this magic amount of months.

1. months from perihelion 2. the average of the highest maxima, 10-mo avg. 3. the average of the second highest maxima, 10-mo avg. 4. the average of the third highest maxima, 10-mo avg. 5. 2.-4. graphically (x=hit (rounded to nearest 10 Wolfs), *=two hits o=gaps between hits) 1. 2. 3. 4. 5. PER 0mo 101 85 75 --------------------------- 1- 10mo 92 83 77 *x !!!!! 11- 20mo 139 116 101 xoxox 21- 30mo 153 134 104 xooxox 31- 40mo 146 124 112 xxoox 41- 50mo 170 146 131 xoxox 51- 60mo 182 153 139 xxoox 61- 70mo 186 147 124 xooxooox APH 71mo 211 144 114 --------------------------- 72- 81mo 193 146 111 xoooxooox 82- 91mo 180 141 116 xoxooox 92-101mo 154 120 106 xxoox 102-111mo 146 121 99 xoxoox 112-121mo 128 106 90 xoxox 122-131mo 110 90 69 xoxox 132-141mo 98 81 66 xxox

**
It is immediately obvious that one of the mechanisms that synchronizes the Jovian year with the
11.1-year cycles, is the fact that the maximum during the 10 months after the perihelion is low;
calculated in 10-month intervals the average Wolf number has never exceeded 90. This effect is
enhanced by the fact that during the 10 months before the perihelion the average Wolf number
has never exceeded 100 and during the preceding 10 months only once.
**

If we look at the individual months, the Wolf number has never exceeded 100 during the months 16-15, 13-10, and 3-1 before the perihelion and 1-5 and 8-10 months after the perihelion. Elsewhere it has happened. The Jovian months 11-124 are continuous in this respect with not a single exception.

There seems to be a 30-month peaceful area around the perihelion, which begins about 20 months before the perihelion and ends 10 months after the perihelion. This may be one of the primary mechanisms through which Jupiter affects sunspots. If we take 10-months intervals, the Wolf number during this period has never exceeded 110, but during the other 110 months from the month 10 to the month 120 the Wolf number has in every interval at least once exceeded 130.

****************************************************************I first inspect the stability of the monthly values during the Jovian year. The median is supposed to indicate what is a typical value. The median can fulfill this task the better the more stabile it is. To measure this stability I have used the sixth smallest and the sixth biggest value of the 19 values describing each month (median is in this instance at the same time the 10th smallest and 10th biggest value).

The most usual variance indicators, when using the median as middle value indicator, are the quartiles. Between the upper and lower quartile belongs by definition 50% of the data. In our case that would mean the mean of the 5th and 6th value from both ends. To avoid the dividing, I have used the approximation of the 6th values. These near-quartiles include 47% of the data. The following table contains these values in the years 1762-1987 as 10-month averages (except PER and APH which are of that particular month alone):

1. months from perihelion 2. the 6th smallest value 3. the 6th biggest value 4. difference between 2. and 3. (47 % of the observations) 1. 2. 3. 4. PER 0mo 35- 51=16 -----xxx------------!!!!!--- 1- 10mo 14- 60=46 xxxxxxxxx 11- 20mo 12- 68=56 xxxxxxxxxxxx 21- 30mo 18- 70=52 xxxxxxxxxx 31- 40mo 25- 86=61 xxxxxxxxxxxx 41- 50mo 29-101=72 xxxxxxxxxxxxxx 51- 60mo 29- 94=64 xxxxxxxxxxxxx 61- 70mo 33- 74=41 xxxxxxxx APH 71mo 27- 78=51 ---xxxxxxxxxxx-------------- 72- 81mo 27- 76=49 xxxxxxxxxx 82- 91mo 28- 74=46 xxxxxxxxx 92-101mo 28- 63=35 xxxxxxx 102-111mo 21- 60=40 xxxxxxxx 112-121mo 12- 47=36 xxxxxxx 122-131mo 10- 44=34 xxxxxxx 132-141mo 13- 43=30 xxxxxx

**
The stability of the perihelion month is astounding. Even if viewed on monthly basis the range
of only 16 Wolfs is the smallest of the 142 values.** Similarly the three highest values in the first
10 months after the perihelion was in the narrow range of 77-92.

The lower quartile has a low value of 10-14 Wolfs 30 months before and 20 months after the
perihelion. **The lower quartile of the perihelion month, 35, is really anomalous. It's even higher
than any 10-month interval lower quartile.** This is in keeping with the rise of the perihelion
month above its surroundings by 10 Wolfs in average. A high value of 25-33 Wolfs of the lower
quartile is maintained 40 months before and 30 months after the aphelion.

Here we have a gate effect, that may be one of the primary mechanisms in the Jovian effect. The perihelion never has a high value, but seldom does it either have a low value.

***************************************************************There is a two-phase rise of the upper quartile beginning 10 months before the perihelion and ending 40-50 after the perihelion. The interruption is just before the month 30. Before that there is a rise rate of about 8 Wolfs per 10 months, after that it is doubled. After the maximum value of about 100 there is a drop to 74-76 Wolfs than is maintained over the aphelion and about 20 months after it. The fall rate 20-50 months before the perihelion has then an average rate of 9 Wolfs in 10 months

**
Now we see that the perihelion month is in line of the rise from -10 to 20 months, which was not
the case with the lower quartile. At the low end there is only a tendency to avoid low values, but
at the high end there seems to be an absolute limit at about 100 Wolfs.
**

The two very stabile months, the perihelion month and the month 35, are treated separately in the chapter 3.4.

1. months from perihelion 2. average 3. median 4. average minus median 5. graphically: + = median, * = average, x = difference 1. 2. 3. 4. 5. PER 0mo (42 43 -1) ------------------------------------------- 1- 10mo 40 41 -1 *+ x 11- 20mo 47 42 5 +--* xxxxx 21- 30mo 52 47 5 +-* xxxxx 31- 40mo 60 57 3 +* xxx 41- 50mo 68 59 9 +---* xxxxxxxxx 51- 60mo 68 57 11 +----* xxxxxxxxxxx 61- 70mo 63 49 13 +------* xxxxxxxxxxxxx APH 71mo (61 41 20) ------------------------------------------ 72- 81mo 62 54 8 +---* xxxxxxxx 82- 91mo 59 55 4 +-* xxxx 92-101mo 51 41 10 +----* xxxxxxxxxx 102-111mo 46 32 14 +------* xxxxxxxxxxxxxx 112-121mo 36 20 16 +-------* xxxxxxxxxxxxxxxx 122-131mo 32 21 11 +----* xxxxxxxxxxx 132-141mo 33 25 8 +---* xxxxxxxx

A difference between the median and the average would mean asymmetry of the distribution, symmetrical distributions would make the values equal. Because with one exception the average is greater than the median, this means the high wing is more expanded than the low wing. This may of course be the result of the measuring stick, the Wolfian formula. We already measured its validity with low values, but whether it exaggerates the high values, we don't know. There are so scanty high values measured by 10.7 cm, that to demonstrate for example an exponential value for the measurement value while the actual intensity rises only linearly, is at the moment statistically difficult.

But medians can give as a hint. With layman's language they are the typical values as long as the quartiles are not too different. Because of this I introduced the quartiles first. Now it's time to make an analysis based on the medians. At the same time I compare them with the means.

We can see that there is a big asymmetry just before the aphelion and 20-40 months before the perihelion. At the same time these are just the two periods during the Jovian year when there is a deep fall in sunspot intensity. In fact medians show two peaks, when means have only one peak at about month 45. The secondary low is just before the aphelion. The means have a continous fall, albeit it is very slow at this point. The explanation is offered by the maxima or high values that have their peak at the aphelion, when both means and medians peak in month 45 or 25 months before the aphelion.

**
Medians and means agree best during the first 40 months after the perihelion, although also even
here is a difference: the means begin to rise only a few months after the perihelion, when the
median rise begins about 20 months after the perihelion. This confirms our conclusion that the
peaceful time period actually lasts this time, but some very aggressive cycles may "spoil" this
peace.
**

The median fall from about 60 months to about 20 months before the perihelion (from 55 to 20 Wolfs) is really dramatic. It parallels similar fall in means, but occurs faster and goes deeper. The reason may be analogous to the delay in rise: some cycles still in their best strength do not easily go down despite the Jovian perihelion nearing.

What then does it mean that the medians and means agree during the ten months after the aphelion. It means that the peak values are missing. The distribution is very near the so called normal (Gaussian) distribution, not elongated to the high side.

Lastly I list two 13 Jovian year distributions (instead of 14 years to avoid dividing while calculating the median) to see if the growing of the Wolf number seen in means, minima and maxima is even, that is, does it happen throughout the Jovian year. The first distribution includes the years 1762-1916, and the second the years 1833-1987.

1. months from perihelion 2. median in 1762-1916 3. median in 1833-1987 4. the difference 5. graphically: + = older median, * = younger median, x = the difference 1. 2. 3. 4. 5. PER 0mo (42 52 10) --------------------------------------------- 1- 10mo 39 51 12 + * xxxxxxxxxxxx 11- 20mo 46 59 13 + * xxxxxxxxxxxxx 21- 30mo 53 60 7 +* xxxxxxx 31- 40mo 67 78 11 + * xxxxxxxxxxx 41- 50mo 64 85 21 + * xxxxxxxxxxxxxxxxxxxxx 51- 60mo 57 78 21 + * xxxxxxxxxxxxxxxxxxxxx 61- 70mo 48 62 14 + * xxxxxxxxxxxxxx APH 71mo (41 55 14) --------------------------------------------- 72- 81mo 49 57 8 +* xxxxxxxx 82- 91mo 49 58 9 + * xxxxxxxxx 92-101mo 36 48 8 + * xxxxxxxx 102-111mo 25 34 9 + * xxxxxxxxx 112-121mo 13 27 14 + * xxxxxxxxxxxxxx 122-131mo 14 26 12 + * xxxxxxxxxxxx 132-141mo 22 39 17 + * xxxxxxxxxxxxxxxxx

The average value had a rise of 8.5 Wolfs in 5 Jovian years. The median value has a rise of 13 Wolfs in 5 Jovian years. So the supercyclic rise is a very deep-rooted phenomenon during the years 1762-1987. The variations are not statistically significant, so we can consider the supercyclic growth even throughout the Jovian year.

It is still somewhat mysterious how Jupiter affects the sunspots, but one hint is offered by those individual months whose stability overrides that of its surroundings thus showing a tendency towards some particular value at that point of the Jovian year. With the 14 Jovian year averages we found two such months: the perihelion month and the month 35 or 1/4 of the Jovian year. With 19 Jovian years we saw a sharp rise from month 28 to month 30 plus the rise during the perihelion.

**
If we look at the individual months we see that the perihelion month 0 is the most stabile month
if we count the mid-47 % of the values. The range is only 16 Wolfs or from 35 to 51. The range
of the next month or the month 1 is already 45 Wolfs.** Almost as narrow as the perihelion,
namely 17 Wolfs (from 20 to 37) is the range of the month 107 or -35 months or 1/4 Jovian years
before the perihelion. All other months have at least a range of 20 Wolfs the top one being held
by month 51, namely 91 Wolfs. Month 51 is just at the breakpoint, where the mean Wolf value
has reached its highest value, and a standstill (means) or a fall (medians) begins.

**
In the case of the perihelion month the stability is caused by the lower limit being "too" high, 35,
when the next month has a lower limit of only 14 Wolfs.** In the case of the month -35 the
stability is caused by the upper limit being "too" low, 37, when the next month has an upper limit
of 61 Wolfs. The interval between these two gates is 3/4 of the Jovian year or the lower limit for
the cycle lengths at least since 1750 and probably since 1500. The maximum rate of rise and fall
is followed these months after a delay of about 15 months, not immediately.

If we take 7 mid-values instead of 9 (covering 37% of the values), there exists 24 months whose range(7) is smaller than 20. 22 of these appear during the last 57 months before the perihelion, one is the perihelion month and the one remaining is the month 35 or 1/4 Jovian year after the perihelion. It's the only month whose range(7) is below 20 between the perihelion and the aphelion, namely only 13 Wolfs. The seven mid-values range of the perihelion month is still the narrowest also in this range category, namely 7 Wolfs from 40 to 47. The greatest width(7), 72 Wolfs, is due to the month 50.

Only the perihelion month and the month of 1/4 Jovian year deviate clearly from the surroun- dings. The perihelion shows also here its lifting tendency: the month 1 has a low limit of 23 Wolfs. The month -35 has "too" low a value: the upper limit of the next month is 82 Wolfs. So however we measure the values, the same picture of the Jovian year emerges:

**
1. month -35: a downward influence
2. the fall accelerates until the month -21
3. perihelion month 0: a lifting influence
4. the rise rate has its maximum in months 3-15
5. month 35: a downward influence
6. the rising stops near the month 45
**

For the Jupiter-effect to be evident usually at least 11-12 Jovian years are needed. The perihelion month is however so special a case, that one could expect some effect to show up even in individual cases.

We can begin with the four perihelia from which there exists 10.7 cm data, which supposedly is more accurate than the somewhat subjective Wolf number. These four perihelia occurred 1951, 1963, 1975 and 1987. There is no observable change in 1951, 1963 and 1987. This is easily explainable, because the value is typical for the perihelion value without any adjustment.In these cases it is enough that the perihelion value is in-line with the on-going trend. In 1951 and 1963 the trend was upwards, in 1987 downwards.

But in 1975 the sunspot activity was already low although the minimum of the minimum was in 1976. The perihelian value was too low to obey "our" rule. So it had a temporary rise:

The 1963, 1975 and 1987 values graphically: 0=65, +x=5 (rounded). For example x=70, xx=75 etc. The 1999 values graphically: 0=100, +x=10 (rounded). For example x=110, xx=120 etc. The perihelion month plus five preceding and five following months. 1963 1975 1987 1999 xxx x x xxxxx (Dec 1998) xxxxx x xx xxxx xxxx x xxxx xxxx xxx x xxxxx xx xxxx xxx xxx xx xxxx-----xxxxxx----xxxx------xxxxx----perihelion xxxx xxx xxxxx xxxxxxxx xxx xx xxxx xxxxxxx xx xxx xxxxxx xxxxxxxx xx x xxxxxxx xxxx xx x xxxxx xxxxxx

**
In the 19 Jovian years since 1762, the perihelion month Wolf value has varied from 0 to 101.
This is a typical range during the 30 months beginning 20 months before and ending 10 months
after the perihelion.** The maximum is far lower than during the other 110 months. In comparison,
the aphelion month Wolf value has varied between 8 and 211.

**
NGDC, Boulder Colorado predicted that the smoothed value of May 1999, coinciding with the
perihelion of Jupiter, overrides 130 in its monthly predictions from March to September 1998. It lowered its prediction to between 120-130 in October and November, and to lower than 110 in December 1998. The actual value then was 90 in May 1999, when the perihelion occurred. As I remarked on these web pages already in June 1998, the Wolf value cannot exceed 100 in May 1999 because of the Jovian perihelion.
**

From this follows that the cycle 23 can't be high, because the cycle maximum and Jovian perihelion are so near each other. I predicted a maximum value below 110 in the first part of 2000. In fact the maximum was 121 in April 2000. The "official" prediction was 160. My prediction was based on the Jovian perihelion being only one year earlier.

The near-quartile range of the perihelion month was from 35 to 51 Wolfs. Our analysis ended in 1987 without including the newest perihelion month of July 1987. If we add its value of 26 Wolfs and correspondingly the lowest value omitted above this 9 mid-Jovyr range, we get a range of 26-59 Wolfs, which thus covers 11/20 or 55% of the data.

To see the individual patterns I made a table of all the reliable perihelia since 1773 to show, whether the perihelion month has deviated from the on-going trend and in what way. Jovian year 2, with a perihelion in January 1762 has been omitted for two reasons. Firstly the observations at that time were few and unreliable and secondly the perihelion is at the end of the month, and February could as well be used.

1. Jovian year (see introduction) 2. perihelion month 3. the Wolf number of the perihelion month 4. the Wolf number of the preceding or the following month in case the perihelion event occurs earlier than the 10th or later than the 20th day of the perihelion month 5. the mean of the two months in cases the perihelion event not in the middle of one month 6. median of the months 6-10 before the perihelion 7. median of the months 6-10 after the perihelion 8. the trend (6 minus 7) 9. a comment 1. 2. 3. 4. 5. 6. 7. 8. 9. 3 11.1773 41 43 =42 / 41 18 = -23 a delay of the fall 4 10.1785 47 32 =40 / 9 85 = +76 in line 5 08.1797 6 6 = 6 / 8 1 = -7 in line 6 07.1809 0 / 11 0 THE REC-LONG 0-PERIOD BEGINS 07.1809 7 05.1821 2 2 = 2 / 9 1 = -8 a fall too early 8 03.1833 12 3 = 8 / 14 8 = -6 a fall too early 9 02.1845 44 26 =35 / 21 39 = +18 a rise too early 10 12.1856 7 14 =11 / 5 22 = +17 in line 11 11.1868 59 / 27 81 = +54 in line 12 09.1880 66 43 =55 / 19 52 = +33 a rise too early 13 07.1892 77 101 =89 / 52 75 = +23 a temporary rise 14 06.1904 42 40 =41 / 39 55 = +16 a delay of the rise 15 04.1916 72 / 69 66 = -3 in line 16 03.1928 85 74 =80 / 59 61 = +2 a temporary rise 17 01.1940 51 59 =55 /101 67 = -34 a temporary fall 18 11.1951 52 46 =49 / 60 36 = -24 in line 19 09.1963 39 35 =37 / 23 9 = -14 a temporary rise 20 08.1975 40 / 21 12 = -9 a temporary rise 21 07.1987 33 18 =26 / 10 60 = +50 in line

In seven cases (37%) the perihelion value is strictly in line. It's however difficult to say what this proportion would have been if we assume that the whole surroundings (plus minus 10 months) already had adjusted to the nearing perihelion. So this is a case of an added adjustment to an already somewhat adjusted environment.

The other cases (this list gives only some direction, it is not statistically significant because of the small amount of cases):

1. A delay, 2 cases, one of fall, the other of rise. The prevailing value is already 39-41, or the favoured value for a perihelian value, which is maintained over the perihelion.

2. A fall too early, 2 cases, in the years 1821 and 1833 or after the very low decade of 1810's. The prevailing value is 9-14, which collapses.

3. A rise too early, 2 cases. The prevailing value is 19-21 and the trend is upwards.

4. A temporary rise, 4 cases. In two cases the prevailing value is high, 52-59 and in two cases low, 21-23. In the former case the cycle is just achieving its maximum, in the two latter ones, its minimum.

5. A temporary fall, 1 case. This is the perihelion of 1940, when the prevailing value was anomalously high, a record of its period, namely 101. It bends to 51-55 as if to honor the perihelion, but soon after reaches 70.

6. The fall to zero in 1809, which lasted 24 consecutive months, a record long period.

Go to the

beginning of organizing the cycles into a web.

Go to the

beginning of the summary of the supercycles and a 2289-year hypercycle.

- 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 searching supercycles by smoothing.

- By one basic cycle smoothed sunspot averages in 1768-1992.

- 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.

- A 2000-year historical review.

- An autocorrelation analysis.

- Some studies showing a 200-year cyclicity.

- The periods of Cole.

Go to the

beginning of the basic cycles to supercycles.

Go to the

beginning of this part.

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