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Timo Niroma:
One possible explanation for the cyclicity in the Sun.

Sunspot cycles and supercycles and their tentative causes.

PART 1. INTRODUCTION 

Table 12. The magnitude and the rise period to maximum

Forewords. 

A speculative hypothesis to explain the Jupiter effect.

1.1. Variation in the length of the sunspot cycles. 

1.2. How do the sunspot minima relate to the Jovian perihelion?

PART 2: Jupiter and the Sun

- Average sunspot magnitude during 19 Jovian years 1762-1987.
- Is the Jovian effect real or an artifact?
- How many Jovian years are needed for the effect to show up?

PART 3: Minima, maxima and medians of the sunspots

- Magnitude minima.
- Magnitude maxima.
- Medians and quartiles.
- The perihelian stability.

PART 4: From basic cycles to supercycles

- How long is the 11-year cycle?
- The rules of Schove interpreted.
- The Precambrian Elatina formation.
- The Gleissberg cycle.

PART 5. The 200-year sunspot cycle is also a weather 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.

PART 6: Searching supercycles by smoothing.

- 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.

PART 7: Summary of supercycles and a hypercycle of 2289 years.

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

PART 8: Organizing the cycles into a web. 

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

Click here to go to "SUNSPOT CYCLES, ARE THEY CAUSED BY VENUS, EARTH
AND JUPITER SYZYGIES?" by Jean-Pierre Desmoulins.
****************************************************************
***************************************************************
PART 1. INTRODUCTION
***************************************************************

1.1. Variation in the length of the sunspot cycles.
* 1.1.1. Two modes of sunspot cycle length distributions.

* Table 1. The cycle lengths since 1745. 
* Table 2. The sunspot cycle lengths classified. 
* Conclusion 1. About the cycle lengths before 1889. 
* Conclusion 2. About the cycle lengths after 1889. 
* Table 3. A probabilistic distribution of the sunspot cycles. 
* Conclusion 3. The sunspot mean length is a mean of means.

1.1.2. How accurate is the estimated sunspot cycle length?
* Table 4. The lengths of sunspot cycles 19-21 based on different
smoothing periods. 
* Table 5. The minima of sunspot cycles 19-22 based on the 5 lowest
months.

1.1.3. Sunspot cycle length estimates extended to 500 years by
auroral numbers.
* Speculation 1. Supercycles of 13-15 basic cycles.

1.1.4. Length of several cycles combined.
* Table 6. 4, 8, and 16 cycles combined. 
* Table 7. Every 15 cycles combined between 1689-1996.

1.2. How do the sunspot minima relate to the Jovian perihelion?
* Table 8. Sunspot minima compared to Jovian perihelion 
* Table 9. The distance between the sunspot minimum and the Jovian
perihelion 
* Table 10. The most attractive distances between minima and
perihelia 
* Conclusion 4. The sunspot minima prefer an area around the Jovian
perihelion 
* Conclusion 5. The distance between minimum and perihelion is
quantisized 
* Conclusion 6. The sunspot minimum and the Jovian perihelion never
exactly meet 
* Table 11. The minima v. Jovian perihelion graphically

1.3. The relation of the length of the cycle to its magnitude
* Table 12. The magnitude and the rise period to maximum 
* Table 13. R(M) compared with the time of rise to maximum 
* Conclusion 7. The maximum possible sunspot number 
* Table 14. The residuals of the actual maximum magnitude compared
with the theoretical value

Forewords

Ever since Heinrich Schwabe and Rudolf Wolf in the years 1843-51
calculated that the amount of the sunspots varied in periods of about
11 years, there has been speculations that some planets, perhaps
Jupiter in the first place, have something to do with the periodicity.
Speculations are based not only on the fact that the orbital period of
the planet Jupiter - 11.86 years - is near the generally accepted
value for the sunspot period - 11.1 years - but also on the facts that
there are no known mechanism that regulates the sunspot activity and
that Jupiter is, besides the Sun, the only body in our planetary
system whose output of energy is greater than the input.

When the planetary effects have been searched as a cause for sunspots,
a gravitational effect is mostly assumed. My theory is purely
statistical so it does not necessitate a theory about the physical
background. But still one can make some speculations. Of course the
gravitational effect of Jupiter makes Sun wobble in its path, which
could in itself in principle cause some turbulence in the Sun's
plasma. However evidence strongly suggests that the sunspots have a
clear electromagnetic nature.

I make a more dramatic suggestion: The electromagnetic fields of Sun
and Jupiter are partly intertwined, sometimes more, sometimes less
during the nearly 12-year orbital revolution of Jupiter. Changes in
eccentricity may then cause long-period changes in Sun's activity. And
one thing we don't know: if the theory of everything combines gravity
and electromagnetic forces the warping of space around Sun would
really cause something extraordinary, like changes in the Sun's
activity. One interesting thing is, that although the main effect of
Jupiter seems to come via the perihelia of Jupiter, also the points
where Jupiter crosses the plane of equator of the Sun, seem to have
some effect.

A speculative hypothesis to explain the Jupiter effect

If we ignore the elliptical orbit of Jupiter around the Sun and
replace it with a more easily grasped model, we can imagine Jupiter as
approaching the Sun 5.93 years and then suddenly reverse the approach
to escape for the next 5.93 years at the moment Jupiter is at its
heliocentric perihelion. Now we see that we can imagine Jupiter's
magnetosphere as approaching the Sun, intruding into it, warping it
and finally intertwin with it, when Jupiter approaches the Sun. During
the perihelion the direction and effect are suddenly reversed. As you
see later, the statistical measurements show that sunspots in average
get more scarce when Jupiter nears the Sun. At the perihelion the
smoothed value has never exceeded 100 Wolfs since we have the monthly
values from 1749. 

Besides during the perihelion, at the moment of the reverse, the solar
wind ceases for a day or two, causing the magnetosphere of Jupiter to
expand enormously. Also the Earth's magnetosphere does that causing a
temporary drop in the temperature of Earth for a week or so. If the
Jupiter's reverse happens during the rise period of Sun's activity, it
dampens the rise, causes the maximum of the ongoing cycle to be low or
moderate and lengthens the cycle period.

The question if the other planets have noticeable effect on the Sun,
remains open. There are hints that they may have. I have however not
studied them irrespective of Jupiter, but the Jupiter effect noticed
does not deny that other planets could have some effect, albeit
smaller than Jupiter.

Still, the prevailing theory, the agreed-upon consensus amongst the
astronomers seems to be against the idea that any planet could
regulate the sunspot behaviour. This is a general phenomenon in the
physical sciences: if you don't understand something, forget it. But
all in all, you must remember, that this is a statistical theory.

1. Introduction
1.1. Variation in the length of the sunspot cycles
1.1.1. Two modes of cycle length distributions

I shall begin by studying the distribution of the lengths of the
sunspot cycles. Unfortunately there are only 23 cycles, beginning in
1745, a little more than 250 years, whose data are reliable enough to
submit themselves to a statistical analysis. That is why they have in
this stage only an introductory character. The analysis in the later
chapters is mostly made using the monthly Wolf values. Then we get rid
of the difficulty with the criteria of defining where a minimum really
was this or that time and instead of 22 or 23 values we have nearly
3000 values.

But even if we had enough values of the cycle lengths for a reliable
statistical analysis, there is another difficulty that is inherent
with the lengths: they are measured from a cycle minimum to the next
minimum, but the definition of a cycle minimum is not based on any
theory. It has been internationally agreed that the minimum is the
month, whose smoothed Wolf number is the smallest. The smoothed values
are calculated effectively as an average value of 13-month running
means (actually as the average of two consecutive 12-month running
means). In case this produces several minima, the number of spotless
days per month is used for help.

Now the situation has changed still more. The last minimum was so
difficult to establish or the smoothed month seemed not to be the
right one, that the scientists decided to use still more criteria to
define when the minimum actually occurred. This leads to the curious
situation that we have two minima for the beginning of cycle 23 and
therefore two different lengths for the cycle 22.

Let's look closer upon the difficulty: the twelve monthly Wolf values
for the minimum year 1996 were: 10, 10, 10, 9, 8, 9, 8, 8, 8, 9, 10,
10. The old mathematical smoothing gives as the minimum month May
(1996.4). But if we take the new criteria into account, the right
month is October (1996.8). Because the beginning of the cycle 22 is
calculated to be 1986.8, we get for the cycle 22 either a length of
9.6 or 10.0 years. But I have doubts even for the accuracy or
rightness of this minimum. (I will discuss that later). However, when
we make comparisons, we must use the same criteria for all. So in the
next table I use the length calculated in the same way as the others
or in this case 9.6 years. One exception I however make, because in
one case the smoothed values do not solve the place of the minimum.
This happened in 1809-1811 when there are more 0 Wolf months than the
13 ones used in the calculations.

Even if we trusted the agreed-upon values as an approximation, we must
give some room for inaccuracy. As a first approximation I estimate the
accuracy of the lengths to be plus minus 0.2 years based on the
inaccuracy of the Wolf estimates and the varying lengths of the months
combined with the rotation period of the sun not exactly matching it
and the use of one tenth of the year instead of the month plus the
arbitrariness of the 13-month smoothing.

The case of a questionable minimum mentioned above is the minimum
which separates the cycles 5 and 6. There is in fact a clear
separation, the year 1810 is the only totally spotless year since the
regular observations began in 1749. The question is about the month of
the minimum, because the number of months whose sunspot number is zero
far exceeds the 13 months used when calculating the running means. The
middle point used instead depends on whether we use as limits 0.0, 0.4
or some higher value (e.g. 10) or instead of using the middle point
use smoothing by a greater value than 13 months. The generally used
minimum is 1810.6, which uses the 0.0 limits. Other estimates give
1810.4 or 1810.5. So in all probability the length of the cycle 5 is
estimated as too high and that of the cycle 6 as too low. I use here
as the limiting value 0.4, because the descent and ascent to and from
that value seem to be clear turning points in the activity. This gives
a minimum date of 1810.4.

TABLE 1. The sunspot cycle lengths since 1745.

The second vertical lines mark the average sunspot period (11.07
years). The third double line marks the Jovian year (11.86 years). The
first lines are a mirror of the Jovian line (10.2-10.3 years). The
lengths are based upon data calculated in Boulder, Colorado except the
1810 minimum. Otherwise the lengths are based partly on the 13 month
smoothed average and partly on the number of spotless days. The length
of the cycle 0 is tentative only.

1. the number of the cycle
2. the minimum as a tenth of the year
3. the length of the cycle numerically
4. the length of the cycle graphically (3 tenths added before and
after)
1.     2.   3. 4.
0 1745.0 10.2 xxxxxxxxxxx??????????
1 1755.2 11.3 xxxxxxxxxxxxxxx|x|xxxxxxx|o|ooooo
2 1766.5  9.0 oooooo
3 1775.5  9.2 xxoooooo
4 1784.7 13.6 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxxxxx|x|xxxxxxxxxxxxxx
5 1798.3 12.1 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxxxxx|o|ooooo
6 1810.4 12.9 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxxxxx|x|xxxxxxxoooooo
7 1823.3 10.6 xxxxxxxxxxxxxxx|x|oooooo
8 1833.9  9.6 xxxxxxoooooo
9 1843.5 12.5 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxxxxx|x|xxxoooooo
10 1856.0 11.2 xxxxxxxxxxxxxxx|x|xxxxxxo|o|oooo
11 1867.2 11.7 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxoooo|o|o
12 1878.9 10.7 xxxxxxxxxxxxxxx|x|xoooooo
13 1889.6 12.1 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxxxxx|o|ooooo
14 1901.7 11.9 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxxxxoo|o|ooo
15 1913.6 10.0 xxxxxxxxxxooooo|o|
16 1923.6 10.2 xxxxxxxxxxxxooo|o|oo
17 1933.8 10.4 xxxxxxxxxxxxxxo|o|oooo
18 1944.2 10.1 xxxxxxxxxxxoooo|o|o
19 1954.3 10.6 xxxxxxxxxxxxxxx|x|oooooo
20 1964.9 11.6 xxxxxxxxxxxxxxx|x|xxxxxxx|x|xxooooo|o|
21 1976.5 10.3 xxxxxxxxxxxxxoo|o|ooo
22 1986.8  9.6 xxxxxxoooooo
23 1996.4

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

Of the cycles 1-22
- 12 are shorter than average
- 2 (9%) may be of average length (cycles 1 and 10)
- 8 are longer than average

Of the longer cycles
- 5 (23%) may have a length of 1 Jovian year (cycles 5, 11, 13, 14 and
20)
- 3 are longer than 1 Jovian year (cycles 4, 6 and 9)

Two things catch the eye. First thing is that only 2 or 9 % of the
cycles seem to have the average length as their length. The lengths
are either clearly longer or clearly shorter than the average length
and there are two favoured lengths, 11.8-11.9 and 10.2-10.3 years. The
other thing is that the long cycles are all old and the recent cycles
are short. There has not been a longer than Jovian year cycle since
the cycle 9 1843-1856. With one exception the 8 Jovian or longer
cycles occupy the 19th century or its immediate vicinity. Of the 11
cycles from 4 to 14 or from 1784 to 1913 7 or 64% belong to this
category. Of the 8 cycles from 15 to 22 or from 1913 to 1996 only 1 or
12.5% belongs to this category.

On the other hand the long or Jovian period 1784-1913 contains no
cycle in the category 10.2-10.3 years, which is so typical to the 20th
century. All the 4 cycles 15-18 or from 1913-1954 plus the cycle 21
1976-1986 belong to this category and the cycle 19 1954-1964 is so
near that it is possible it belongs also here. So 5 or 6 of the 8
cycles or some 70% are in this category.

The range of the cycles in the years 1745-1986 is from 9.0 to 13.6
years. If we count the frequency of aurorae we can go back to the year
1501. The aurorae follow so closely the frequency of sunspots that we
can use them as an estimate of the sunspot cycles. This gives us as
the lowest estimated value in the years 1501-1745 9.3 years and as the
highest one 13.5 years. We can thus be confident that at least during
the last half millennium the cycle length has not been below 9 years
or higher than about 13 and half years (counted from minimum to
minimum).

How do these limits compare to Jovian years? 1.15 Jovyr is 13.6 in our
calendar years. 0.75 Jovyrs is 8.9 calyrs. The shortest cycle is two
thirds of the longest, or if we prefer, the longest cycle is one and a
half times the shortest. Is this a coincidence or does it have some
deeper meaning?

In the following table I have put the cycle lengths into categories so
that we can see the distribution more clearly and at the same time I
have got rid of the false impression that the 0.1 years accuracy gives
us. As the interval for the classes I selected 0.8 years, because it
is the time between the average and the 10+ year cycle as well as the
time difference between one Jovian year and the average cycle. As the
middle category I selected the class that got the least number of hits
or that gap around the average length.

TABLE 2. The sunspot cycle lengths classified.

x = the 7 cycles 1-7 or 1755-1833
o = the 5 cycles 8-12 or 1833-1889
v = the 10 cycles 13-22 or 1889-1996
length      no and kind of cycles
8.4- 9.1 x
9.2- 9.9 xov
10.0-10.7 xovvvvvv
10.8-11.5 xo
11.6-12.3 xovvv
12.4-13.1 xo
13.2-13.9 x

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

The cycles 1-7 in the years 1755-1833 are both evenly and broadly
distributed so that every class gets one and only one hit. The cycles
8-12 in the years 1833-1889 are almost similarly divided except that
the two extreme classes on both ends do not get a hit. The cycles
13-22 divide in two classes, the longer one has an average length of
11.87 years and the shorter one either 10.17 or 10.23 depending on
which value we use for the cycle 22. If we leave it out, we get an
average length of 10.27 years. Thus we can hypothesize that the 20th
century cycles have had only two possible lengths allowing an accuracy
of 0.4 years.

CONCLUSION 1.

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

The sunspot lengths had during the years 1755-1889 an even and flat
distribution containing the whole known range from 9 to 13.5 years.

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

CONCLUSION 2.

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

The sunspot lengths had during the years 1889-1996 only two possible
values, either 10.2-10.3 years or 11.8-11.9 years. In Jovian years
these are 0.86-0.87 and 1 years.

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

In the next table I have drawn lengths of the cycles so that the
"official" value gets four points, the nearest value three points, the
tenths of years whose distance is 0.2 years get two points and finally
one point is given at the distance of 0.3 years. This should
compensate for the inaccuracy of the values. For the years 5 and 6 I
have used the calibrated values, for the cycle 22 the traditional
value. The tentative cycle 0 (10.2 years) is added with a "o"
notation.

TABLE 3. A probability distribution of the sunspot lengths.

yrs   points
8.7 x
8.8 xx
8.9 xxxx
9.0 xxxxxx
9.1 xxxxxx
9.2 xxxxxx
9.3 xxxxxx
9.4 xxxxxx
9.5 xxxxxxx
9.6 xxxxxxxx
9.7 xxxxxxx
9.8 xxxxxxx
9.9 xxxxxxxxo
10.0 xxxxxxxxxxoo
10.1 xxxxxxxxxxxxxooo
10.2 xxxxxxxxxxxxxxoooo
10.3 xxxxxxxxxxxxxxxooo
10.4 xxxxxxxxxxxxxxxoo
10.5 xxxxxxxxxxxxxxo
10.6 xxxxxxxxxxxxxx
10.7 xxxxxxxxxxx
10.8 xxxxxxx
10.9 xxxxx
11.0 xxxx
11.1 xxxxx
11.2 xxxxxxx
11.3 xxxxxxxx
11.4 xxxxxxxx
11.5 xxxxxxxx
11.6 xxxxxxxxx
11.7 xxxxxxxxx
11.8 xxxxxxxxxx
11.9 xxxxxxxxxxx
12.0 xxxxxxxxxx
12.1 xxxxxxxxxx
12.2 xxxxxxxx
12.3 xxxxxx
12.4 xxxxx
12.5 xxxx
12.6 xxxx
12.7 xxxx
12.8 xxxx
12.9 xxxx
13.0 xxx
13.1 xx
13.2 x
13.3 x
13.4 xx
13.5 xxx
13.6 xxxx
13.7 xxx
13.8 xx
13.9 x

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

The highest frequency goes to years 10.3-10.4. If we include the cycle
0, the highest frequency goes to years 10.2-10.3. If we use the
alternative value for the cycle 22, we also get the high at 10.2-10.3
years. Actually the frequency distribution has two peaks. The other
one, a little lower than the 10.2-10.4 year peak, is at the year 11.9
years, or 1 Jovian year. What is interesting is that the lowest
frequency between these two peaks goes to the year 11.0 which is very
close to the mean length of the sunspots.

CONCLUSION 3.

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

The mean sunspot length, 11.1 years, is not a preferred length for a
single sunspot cycle. It is rather the mean of either a flat
distribution (1755-1889 = 11.2 years) or a bimodal distribution
(1889-1986 = 10.8 years) or a combination of these two modes of
distribution.

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

1.1.2. How accurate is the estimated sunspot cycle length?

There is no theoretical basis for using the 13-month smoothed values
as the marker of the minimum. This has actualized with the difficulty
of defining when the minimum between cycles 22 and 23 occurred. The
Solar Cycle Prediction Panel decided in New Mexico in September 1997,
that "when observations permit, a date selected as ... a cycle minimum
... is based in part on an average of the times when extremes are
reached (1) in the monthly mean sunspot number, (2) in the smoothed
monthly mean sunspot number, ... (3) in the monthly mean number of
spot groups alone ... (4) the number of spotless days and (5) the
frequency of occurrence of old and new cycle spot groups.

First I inspect minima by varying the number of months used in
smoothing (the above definition takes no explicit stand on the
number). In the following table I have used the 10.7 cm flux instead
of Wolf numbers, because they are more objective than the Wolf number,
which after all is based on subjective observations. The flux is
adjusted to 1 AU. Unfortunately the first whole year from which we
have these radio fluxes is 1948, so that they cover only four whole
sunspot periods, the periods 19-22. First I count the lengths of the
cycles 19-21 by varying the smoothing periods and leave the cycle 22
for a later study.

TABLE 4. The lengths of sunspot cycles 19-21 based on different smoothing
periods

running  method of      cycle19 cycle20 cycle21
period    measurement

13 mo    Wolf              10.6    11.6    10.3

9 mo    10.7 flux         10.3    11.8    10.5
11 mo    10.7 flux         10.2    12.0    10.3
13 mo    10.7 flux         10.4    11.7    10.3
15 mo    10.7 flux         10.3    11.8    10.3

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

We can see that the lengths vary as much as 0.4 years as we assumed in
the previous tables. If we only replace Wolf by flux and keep the 13
month as the smoothing period, the cycle 19 is shortened from 10.6 to
10.4 and the cycle 20 lenghtened from 11.6 to 11.7 years. These are
more in line with the long term trends from 1889 and the cycles 19 and
21 are practically of the same length. By increasing the smoothing
period from 13 to 15 months, they are equal, 10.3 years and the cycle
20 has a length of one Jovian year.

In the following table I have calculated lowest months as tenths of
year of the four minima from which there exist 10.7 cm flux values.
The purpose is to investigate the sharpness of the minima plus the
possibility of several minima or a prolonged minimum.

TABLE 5. The minima of sunspot cycles 19-22 based on the 5 lowest months

method of    cycle19 cycle20 cycle21 cycle22
measurement

13 month running minimum
Wolf  1954.3  1964.9  1976.5  1986.8
flux  1954.3  1964.7  1976.4  1986.7
15 month running minimum
flux  1954.3  1964.6  1976.4  1986.6
5 lowest smoothed months (as tenths of years)
flux  1954.0  1964.5  1976.2  1986.0 begins
flux  1954.4  1964.9  1976.5  1986.1 ends
flux                          1986.5 begins
flux                          1986.7 ends

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

The minima of the minima 19, 20, and 21 are clear and relatively
sharp, so there is not much room for speculation. Only the Wolf value
of the cycle 20 minimum looks a little late. If we use the mid-month
between the begin and end times instead of the average, we get for the
cycle 19 a length of 10.5 years when the "official" Wolf value is 10.6
and the smoothed flux value is 10.3 or 10.4 years. For the cycle 20 we
get 11.65 years, the official Wolf value being 11.6 and the smoothed
flux value 11.7 or 11.8 years, all values being very near one Jovian
year. The most probable lengths can be counted by using as the minima
1954.3, 1964.7 and 1976.4 giving lengths of 10.3 and 11.7 years for
cycles 19 and 20, respectively.

The case of the minimum of the cycle 22 is instead not so clear-cut.
The "official" time of the minimum is calculated as 1986.8, but the
first months of the year 1986 are almost equally low The monthly
smoothed Wolf numbers for the year 1986 are 14, 13, 13, 14, 14, 14,
14, 13, 12, 13, 15, 16. (I would count the minimum as 1986.7 (8.5/12)
but NGDC Boulder uses 1986.8 (9/12).) Now if we measure the length of
the cycle 21 from the first minimum of the cycle 22, we get 9.6 years
as the length for the cycle 21 instead of the "official" 10.3 years
counted from the second minimum. Now we have a problem. If we be
purely mathematical, the length of the cycle 21 is 10.3 years. But if
we search for causes behind the cycles, we must take into account that
this minimum had a secondary, albeit smaller, minimum 6-7 months or
0.5-0.6 years earlier.

The problem recurs during the minimum that ends the cycle 22 and
begins the cycle 23, but maybe in a more dramatical fashion. I cite
the "Summary Report of the Second Meeting of the Solar Cycle
Prediction Panel" NSO/Sacramento, New Mexico: "The most problematic
point discussed was the time of the cycle minimum. While the
traditional numerical prescription as well as other measures of solar
irradiance and activity agree that May, 1996, was the minimum smoothed
month of the cycle, Karen Harvey noted that there are several factors
that argue that this date is misleading as a fiducial for cycle 23
onset. In particular, no new cycle spots were observed before this
month - a situation never before recorded.

The date of minimum is expected to represent that time when new cycle
activity becomes dominant; that is the new cycle should have been in
progress as the old cycle declined, the minimum then marking the
crossover. But for cycle 23, new cycle regions did not outnumber old
cycle regions until December, 1996. The resurgence of activity in the
months following May is due to old cycle regions. Another factor that
indicates May is a misleading date is that the maximum number of
spotless days occurred in September and October, 1996. ..." Thus "the
Panel agreed that October, 1996, was the effective onset of cycle 23."

NGDC, Boulder gives now two minima for the cycle 23 onset: May 1996 as
the mathematical minimum and October 1996 as the consensus minimum.
For predictions NGDC uses the latter. The first minimum gives for the
cycle 22 a length of 9.6 years, the latter gives 10.0 years.

I prefer the use of straight 15 month smoothing instead of the
indirect 13 month smoothing, because it seems to give the most stabile
and sensible point of minimum. When it is used, we get as the minimum
month August 1996 (8.7), and the second lowest July and June. If we
are using the 10.7 cm flux, we get minimum in July 1996 (the second
lowest month is June). This gives some credence for the proposition
that May really is too early. But in a similar way October is too
late. It seems a little odd choice to nominate the last of the low
months as the minimum of minimum. The 12 month list of the year 1996
is 12, 4, 9, 5, 6(*), 12, 9, 14(***), 2, 2(**), 19, 13 (Wolf) or 72,
70, 70, 70, 71(*), 72, 74(***), 74, 70, 69(**), 79, 75 (10.7 cm flux).
* denotes the traditional method, ** the "consensus" method and *** my
15 month smoothing.

On this basis I suggest 1996.6 as the beginning for the cycle 23. I
earlier suggested 1986.6 as the beginning for the cycle 22, so this
leads to 10 years, the same length as now seems to be "official". The
fact that the neaOn this basis I suggest 1996.6 as the beginning for
the cycle 23. I earlier suggested 1986.6 as the beginning for the
cycle 22, so this leads to 10 years, the same length as now seems to
be "official". The fact that the nearest favorite lengths seem to be
9.6 and 10.3 years, depends on the smoothing window. 15 months begins
to be longest possible, because from 17 month onwards the bias caused
by the asymmetric form of the cycle, slow descent but rapid rise,
begins to take its toll.

So we see here a deviation in 1996 from the bimodal pattern that
otherwise reigned through the 20th century:
method of    cycle20        cycle21        cycle22        cycle23
measurement

13 month running minimum
Wolf  1964.9 (11.6y) 1976.5 (10.3y) 1986.8 ( 9.6y) 1996.4
flux  1964.7 (11.7y) 1976.4 (10.3y) 1986.7 ( 9.8y) 1996.5
15 month running minimum
flux  1964.6 (11.8y) 1976.4 (10.2y) 1986.6 (10.0y) 1996.6
5 lowest smoothed months
(as tenths of years)
flux  1964.5 begins  1976.2         1986.0 begins  1996.2
flux  1964.9 ends    1976.5         1986.1 ends    1996.3
flux                                1986.5 begins  1996.5
flux                                1986.7 ends    1996.6

1.1.3. Sunspot cycle length estimates extended to
500 years by auroral numbers

Now that we are convinced about the inaccuracy of the length of the
sunspot cycles, even about the newest one, we possibly dare to use the
aurorae for substitute to get more cycles. One particular reason for
this is the Maunder minimum which occurred during the latter part of
the 17th century. There were very few sunspots, during the last decade
of that century virtually none.

If we begin with the Jovian year cycles, according to Justin Schove,
there may have been one in the 17th century, namely -10 or 1633-1645,
the last one before the Maunder minimum. Now cycles 5 or 1798-1810 and
20 or 1964-1976 seem also to have been near one Jovian year long. Here
we have something to begin with. The difference between these cycles
is 15 and the length 165-166 years. 14 Jovian years is 166.07 calendar
years. Are 1645, 1810 and 1976 synchpoint years having between them 15
sunspot cycles equalling 14 Jovian years and beginning a new phase in
the Sun? 1645 started the Maunder minimum that lasted about 60 years,
1810 started a very low and cold period, that albeit lasted only one
decade. 1976 started a Hale period of two prolonged and high maxima
and minima which mean increasing warmth.

One other thing seems also interesting. According to Schove, the
period that ended the Maunder minimum, -4 from 1699 to 1712, seems to
have had the maximum length observed or 13.5 years. The other cycles
longer than one Jovian year were 4 (1784-1798), 6 (1810-1823) and 9
(1843- 1856). What is interesting is that these cycles or cycle groups
(4-6) preceded anomalously short cycles. The cycle -5, during which
was the coldest phase at least since 1600 in Europe, lasted only about
9 years. The cycles 2 and 3 both had a length of 9 years, but the
following three cycles had a mean length of almost 13 years. The 12.5
years long cycle 9 was presided by the 9.5 half years long cycle 8.

This leads to an interesting speculation. If we rely on the Schove
estimates based on aurorae, there has since 1500 been a tendency
between intervals of 13 or 14 cycles to have the pairing either of one
or two 9 to 9.5 years cycle plus immediately after them one 12.5-13.5
year cycle. Thus there are -19 and -18 about 1534-1554, the Maunder
minimum minimum beginning in about 1689 (the cycle -5), and the cycle
8 in 1833-1843. And these cycles are always followed by one cycle
whose length is longer than one Jovian year. So the cycles -17
(1554-1567), -4 (at the end of Maunder minimum 1699-1712) and 9
(1843-1856) belong to this category. The anomalous low cycles 5 and 6
are also preceded by the short cycles 2 and 3 and the long cycle 4
following also this schema thus proving that there are these pairs as
well as Jovian length cycles (11, 13, 14) also outside these
supercycles, but not vice versa: the supercycle still persists.

If the above supercyclicity is a real phenomenon and not a temporary
one (15 cycles between Jovian cycles and 13.5 between the short-long
pairs), the cycle 22 should be about 9.5 years long, as the
traditional calculation also gives. From this follows that the cycle
23 will be longer than one Jovian year. (i. e. the short cycle 22
belongs to the series -5, 8, 22 with alternating 13 and 14 cycle
intervals requiring this from the next cycle). This makes a supercycle
of 2260 years (15*13.6*11.071).

SPECULATION 1.

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

Every 15 cycle has a length of 1 Jovian year. With intervals of 13 or
14 cycles there are 2 or 3 cycles such that 1 or 2 short cycles (9 to
9.5 years) are followed one cycle that is longer than one Jovian year
(12.5-13.5 years).

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

1.1.4. Length of several cycles combined

Finally I will join consecutive cycles to see what kind of large scale
variations there exist. I have next calculated the combined and
average length of 4, 8, and 16 consecutive cycles using as the
breakpoint the spotless year 1810. In the longer cycles I have used
the Schove estimates based on aurorae.

TABLE 6. Several cycles combined

1. the cycles
2. the years of minimum (from start of the first to the end of the
last)
3. the combined length
4. the average length as Earth years
5. how many Jovian years are required to include n+1 cycles, if the
amount of the Jovian years is denoted by n?
4 cycles combined
*****************
1.        2.   3.   4. 5.

2- 5 1766-1810 44.1 11.0 14
6- 9 1810-1856 45.4 11.4 23
10-13 1856-1901 45.7 11.4 27
14-17 1901-1944 42.5 10.6 10
18-21 1944-1986 42.5 10.6 10

The difference of 0.8 years shows here clearly up between the 19th and
20th century.
8 cycles combined
*****************
1.      2.    3.        4.    5.

-2- 5 1723-1810 86-88 10.8-11.0 11-13
6-13 1810-1901  91.1      11.4    25
14-21 1901-1986  85.0      10.6    10

1810-1901 three of the eight cycles have a length of one Jovian year,
none is near the 10.2-10.3 years. 1901-1986 only one of the eight
cycles has the length of one Jovian year, all the others belong to the
10.2-10.3 year type. 200 years are needed to give the generally
accepted value of the cycle length, a little over 11 years. The first
8 cycles is in this table show that there are at least three types of
supercycles. The interesting question is did 1986 began a new
supercycle, and if so, is it similar to the first one in this table,
or a new, fourth type.
16 cycles combined
******************
1.        2.    3.    4.   5.

-10- 5 1633-1810 176.9 11.06 14.7 sum inaccurate by +- one year
6-21 1810-1986 176.1 11.01 13.9

This is a most interesting table because it captures the Maunder
minimum. Unfortunately the first supercycle is very inaccurate.
Although 1633 seems to be the best choice for the minimum year, 1632
and 1634 are almost as possible. So this means a total length anywhere
between 176 and 178 years. Could be the same as the latter, but still
older records favor the mean of 11.07 years, so probably it had
slightly longer cycles in average. On the other hand no-one can say if
16 cycles is any good indicator for a supercycles. The Jovian cycles
seem to appear in 15 cycle intervals, and the pairs of short/long
cycles in 13/14 cycle intervals.

TABLE 7. 15 cycles combined

So let's see what 15 cycles would give us.
1.        2.    3.    4. 5.

-5- 9 1689-1856 166.5 11.10 16
-4-10 1699-1867 168.2 11.21 18
-3-11 1712-1878 166.4 11.09 15
-2-12 1723-1889 166.1 11.07 15 phase change I
-1-13 1734-1901 167.7 11.18 17 last cycle lasted one Jovian year
0-14 1745-1913 168.6 11.24 19
1-15 1755-1923 168.4 11.23 19
2-16 1766-1933 167.3 11.15 17
3-17 1775-1944 168.7 11.25 19
4-18 1784-1954 169.5 11.30 21
5-19 1798-1964 166.4 11.09 15
6-20 1810-1976 166.0 11.07 15 phase change II
7-21 1823-1986 163.4 10.89 12 last cycle lasted one Jovian year
8-22 1833-1996 162.5 10.83 12

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

If we calculate the length beginning from the spotless year 1810, we
get for the average sunspot cycle length a value, that adds
probability to the hypothesis that 15 sunspot cycles are equal to 14
Jovian years. This relation, if true, is exact only in average and in
practice it is exact only when calculated from such synchronization
points as the year 1810 and the beginning of the Maunder minimum in
1645 15 cycles earlier. Is 1976 the third syncpoint in our data? 1645
and 1810 began cold spells, 1976 seems to begin a warm spell.

1.2. How do the sunspot minima relate to the Jovian perihelion?

Now I make some questions. Because the mean sunspot length (11.07
years) is shorter than the Jovian year (11.86 years) the distance
between these two events should change in average 0.79 years or 6.7%
per Jovian year. But now we know, that the most preferred length is a
little less than 10.3 years. If we assume symmetricity around the mean
length (at least it is not far), we can use the value 10.28 years, and
thus we can expect that the average length should increase by 1.6
years or 13.3% per Jovian year. This is interrupted sometimes by a
cycle, whose length is 1 Jovian year. The first question is, do the
perihelia of these Jovian years situate at some standard distance
compared to the sunspot minima or are they randomly distributed? The
other question is, how do the short/long cycles relate to Jovian
perihelia.

TABLE 8. Sunspot minima compared to Jovian perihelion

1. the number of the sunspot period
2. the year of Jupiter (the numbering is my own)
3. the minimum time of the spots (the beginning of the cycle)
4. the nearest Jovian perihelion
5. the difference between sunspot minimum and Jovian perihelion
6. the previous as a percentage of the Jovian year
7. the change in the previous value
8. remarks; the saying "over aphelion/perihelion" means that the
minimum is situated on the different side of the aphelion/perihelion
compared with its previous position
1. 2.    3.     4.   5.   6.   7.        8.
0  0 1745.0 1750.2 -5.2 -44% -14% slightly unreliable
----------------------------------over aphelion
1  1 1755.2 1750.2  5.0  42%  -5%
2  2 1766.5 1762.1  4.4  37% -24% very short
3  3 1775.5 1774.0  1.5  13% -22% very short
----------------------------------over perihelion
4  4 1784.7 1785.8 -1.1  -9%  14% very long
----------------------------------back
5  5 1798.3 1797.7  0.6   5%   2% very low, Jovian cycle
6  6 1810.4 1809.6  0.8   7%   9% very low
7  7 1823.3 1821.4  1.9  16% -11%
8  8 1833.9 1833.3  0.6   5% -18%
----------------------------------over perihelion
9  9 1843.5 1845.1 -1.6 -13%   5%
10 10 1856.0 1857.0 -1.0  -8%  -6%
11 11 1867.2 1868.9 -1.7 -14%  -1% Jovian cycle
12 12 1878.9 1880.7 -1.8 -15% -10%
13 13 1889.6 1892.6 -3.0 -25%   2% Jovian cycle
14 14 1901.7 1904.4 -2.7 -23%   -  Jovian cycle
15 15 1913.6 1916.3 -2.7 -23% -16%
16 16 1923.6 1928.2 -4.6 -39% -14%
----------------------------------over aphelion
17 16 1933.8 1928.2  5.6  47% -12%
18 17 1944.2 1940.0  4.2  35% -15%
19 18 1954.3 1951.9  2.4  20% -12% very high
20 19 1964.7 1963.8  0.9   8%   -  Jovian cycle
21 20 1976.5 1975.6  0.9   8% -14%
----------------------------------over perihelion
22 21 1986.8 1987.5 -0.7  -6% -18%
23 22 1996.4 1999.4 -3.0 -25%

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

Now I divide this perihelion dance, as I like to call it, into 5
periods and 3 types. Types are: A. Minima that occur clearly when
Jupiter is nearer its aphelion than its perihelion. B. Minima that
occur very near halfway between perihelion and aphelion, i.e. 2.97 or
approximately at the distance of 3 years. C. Minima that occur when
Jupiter is clearly nearer its perihelion than its aphelion.

Periods are as follows:

Period 1. The 3 cycles from 0 to 2 (possibly including also the cycle
-1). Type is A. This phase ends and following phase begins with the
shortest known cycles (both are only 9 years long) as if the minimum
had a great hurry to synchronize with the Jovian perihelion.

Period 2. Consists of the next 10 cycles, from 3 to 12 (the first one
began in 1775 and the last one in 1878). They all are type C. The
minimum oscillates on both sides of the perihelion as if the
perihelion had some attractive force. The first minimum is after, the
second before, the next 4 again after and the last 4 before the
perihelion. But one thing seems odd taking into account this
attraction: none of the 10 minima occurs exactly at the perihelion.
The cycles 3 and 4 hold the all time (since 1750) record of length
difference (9 and 13.5 years) and the cycle 5 and 6 the all time
(since the Maunder minimum) record of low spot number (both had a
maximum of about 50 Wolfs) and between them is the only spotless year,
1810, since the Maunder minimum. The 10 cycles beginning during this
phase has a mean length of 11.4 years.

Period 3. Consists of the next 3 cycles, from 13 to 15 (the first one
began in 1889 and the last one in 1913). They are of type B. The
standstill halfway between perihelion and aphelion of cause means that
the first two cycles of this period last 1 Jovian year. What caused
the standstill at this point? One guess is that Jupiter intersects the
plane of Sun's equator at 85.4 degrees from perihelion, which is 24%
of the orbit. What caused the attraction to Jovian perihelion to
loosen? One guess would be Saturn. Saturn has its perihelion during
the last cycle of period 2, or the cycle 12, 1885.6 or near a Jovian
aphelion. Maybe a coincidence, but the next perihelion of Saturn
occurs during the cycle 15, 1915.1

Period 4. Consists of the next 3 cycles, from 16 to 18 (the first one
began in 1923 and the last one in 1944). They are of type A. What is
interesting is that the previous phase that was type A (period 1)
contained most probably 4 cycles and this one contains 3 cycles,
whereas the type C period 2 contained 10 cycles. If we include the
intermediary cycles 15 and 19, these five cycles all belong to the
10.3 year length category.

Period 5. Consists of the 4 cycles from 19 to 22. The beginning cycle
19 has an all time record (at least for 400 years) in height, 200
Wolfs. They are of type C. In one respect they behave in a similar
attraction/avoidance way towards the Jovian perihelion as the C type
cycles during period 2. Continuing the straight 10.3 year course began
in 1913 by the cycle 15 would have caused the cycle 21 have its
beginning minimum coincide with the Jovian perihelion. Instead the
cycle 20 is prolonged to 1 Jovian year, so that the minimum of cycle
21 is postponed to occur almost a year after the perihelion. But the
prolonged minimum beginning the cycle 22 occurs on its straight place
as if nothing happened between, only that it occurs before the
perihelion.

Cycle 23. An intermediary of type B. All my rules indicate that the
cycle is longer than one Jovian year, most probably 12.5 years.

Altogether we have here 24 minima. 6 or 25% of them (0-2 or 1745-1766
and 16-18 or 1923- 1944) are of type A (on the aphelion side), 4 or
17% (13-15 and 23) are of type B (on the borderline), and 14 (3-12 or
1775-1878 and 19-22 or 1954-1986) or 58% are of type C (on the
perihelion side). The distance between the two type A groups is 178
years or 15 Jovian years or 16 cycles.Type B is always before the
perihelion, A and C may be on both sides.

In the next table I have calculated the distance of the minimum from
the Jovian perihelion and used the 0.8 year classification. I have
omitted the sign and used the value as an absolute value to get the
distance to the nearest perihelion regardless of whether the minimum
or the perihelion was first. The maximum distance is of course half a
Jovian year (5.93 years).

TABLE 9. The distance between the sunspot minimum and the Jovian perihelion

1. distance +- 0.4 years
2. the cycles
3. the cycles in time sequence
1. / 2.                                        3.

0.0                                                      PERIHELION
0.8 / 04 05 06 08 10 20 21 22      XXX X X          XXX
1.6 / 03 07 09 11 12              X   X X XX
2.4 / 14 15 19                               XX    X
3.2 / 13 23                                 X          X
4.0 / 02 18                      X                X
4.8 / 01 16                     X               X
5.6 / 00 17                    X                 X      APHELION

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

The affection that the minimum shows to the Jovian perihelion seems
very obvious, but it shows itself in a rather peculiar way: it avoids
the exact time of the perihelion and prefers instead a distance of
about 0.6-0.9 years.

If we set the limit of the distance from the perihelion to +-2 years
or +-17% or 1/3 of the Jovian year, 13 cycles are inside this region,
whereas the remaining 2/3 Jovyrs or almost 8 calyrs contain only 11
cycles.

To see the probability of chance producing this distribution, I made a
binomial test. I tested the hypothesis that by chance 13 or more
minima out of 24 are at a distance of 0.5-2.0 years of the perihelion.

**************************************************************
The probability of success = 0.25 ((2*(2.0-0.5))/11.86=0.253)
Number of trials = 24 (minima)
Number of successes = 13 (or more minima inside this region)
Probability of success by chance = 0.0016 or 0.16% or 1 to 625.
***************************************************************

So we can accept our hypothesis, that the phenomenon is real, with a
probability of 99.84%. 

The last minimum which obeyed both the attraction and the avoidance
was the one between cycles 20 and 21 (1976).

In the following histogram I have drawn the absolute distances of
minima from the Jovian perihelion calculated with accuracy limits as
plus minus 0.2 years (i.e. five marks for each minimum). The vertical
line represents the perihelion and the unit is one tenth of a year.
Included is only the more populated area beginning with 3 years (3.2
with limit).

TABLE 10. The most attractive distances between minima and perihelia

The lowest row denotes years, the second lowest row tenths of years
and the higher rows correspond hits.
x
xxx
xx     xxxx
xx         xxxx   xxxxxx
xxxxx     xxxxx   xxxxxxx
xxxxxxx     xxxxxxxxxxxxxxxx
xxxxxxxxxxx xxxxxxxxxxxxxxxxx
---------------------------------
210987654321098765432109876543210 tenths of years
3         2         1         0 years

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

It is easily seen that there are three favored distances from the
perihelion: 0.8 years, 1.6-1.7 years and 2.8-2.9 years. The
simultaneous hate and love of the perihelion is here clearly evident.
0.8 years is interestingly the distance from the mean of the sunspot
cycle length both to the 10.3 year category lengths and the 1 Jovian
years length (11.9 years). Similarly 1.6 years is exactly the
difference between 1 Jovian year and most favored sunspot cycle length
of 10.3 years. One lonely minimum (19) is at the distance of 2.4
years. The peak at 2.8-2.9 may have something to do with the Jovian
orbit intersecting the plane of the Sun's equator 2.9 years before its
perihelion. If that's true this distance occurs only before the
perihelion, never after.

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

CONCLUSION 4. The sunspot minima prefer an area around the Jovian perihelion

At a distance beginning 0.5 years and ending 2 years before or after
the Jovian perihelion is an area that attracts the the sunspot
minimum.

CONCLUSION 5. The distance between minimum and perihelion is quantisized

Taking into account the inaccuracies while measuring the exact
location of the sunspot minima, we have good grounds to assume that
inside 3 years before or after the Jovian perihelion there are only 3
possible distances that the minimum may occupy. The distances are 0.8,
1.6, (possibly also 2.4) and 2.8-2.9 years.

CONCLUSION 6. The sunspot minimum and the Jovian perihelion never exactly meet

If we take the last 250 years as representative, we can conclude that
there never is an exact match of the sunspot minimum and the Jovian
perihelion.

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

Finally I have drawn the perihelion dance of the minima as a graphical
representation. The first column is the number of the beginning cycle,
the column is the datum of the Jovian perihelion and X marks the
minimum. The middle | marks the perihelion of Jupiter (that year in
the second column), the side ones are the aphelia. Between the aphelia
of Jovian years 15 and 16 are two minima. X is followed by 1-4
horizontal lines that mark the height of the minimum rounded to
nearest 50 Wolfs (each line denoting 50 Wolfs).

TABLE 11. THE MINIMA V. JOVIAN PERIHELION GRAPHICALLY

Aph.              Perih.              Aph.
Min Year/perih.
1    1750.2 |                  |              X--)|
2    1762.1 |                  |            X--)  |
3    1774.0 |                  |   X---)          |
4    1785.8 |             X---)|                  |
5    1797.7 |                  | X-)              |
6    1809.6 |                  | X-)              |
7    1821.4 |                  |    X-)           |
8    1833.3 |                  | X---)            |
9    1845.1 |            X---) |                  |
10    1857.0 |              X--)|                  |
11    1868.9 |            X---) |                  |
12    1880.7 |           X-)    |                  |
13    1892.6 |        X--)      |                  |
14    1904.4 |         X-)      |                  |
15    1916.3 |         X--)     |                  |
16/17 1928.2 |   X--)           |                X-|
18    1940.0 |-)                |           X---)  |
19    1951.9 |                  |      X----)      |
20    1963.8 |                  | X--)             |
21    1975.6 |                  | X---)            |
22    1987.5 |               X--|)                 |
23    1999.4 |        X         |                  |

The shortening of the sunspot cycles is here clearly seen. The minima
of the cycles 2 and 18 are nearly at even length after the perihelion,
nearer the aphelion. The distance is 16 cycles or 178 years or 11.1
years per cycle. But the cycles 20 and 21 behave as the cycles 5, 6
and 8, the minimum is at the magical distance of 0.8 years after the
perihelion. The first two cycles have a distance of 15 cycles or 166
years or 14 Jovian years. But the most recent minimum, that of cycle
23 occurs already halfway between the perihelion and the aphelion, at
the point where Jupiter's orbit intersects the plane of Sun's equator.
The previous minimum that did this occurred only 10 cycles or 107
years earlier. So this drops the mean cycle length during this period
to 10.7 years. Remark the avoidance of the perihelion by the cycle 21
(the avoidance previously shown by cycles 5, 6 and 8).

EVERY OTHER CYCLE

Aph.              Perih.              Aph.
Min Year/perih.
2    1762.1 |                  |            X--)  |
4    1785.8 |             X---)|                  |
6    1809.6 |                  | X-)              |
8    1833.3 |                  | X---)            |
10    1857.0 |              X--)|                  |
12    1880.7 |           X-)    |                  |
14    1904.4 |         X-)      |                  |
16    1928.2 |   X--)           |                  |
18    1940.0 |                  |           X---)  |
20    1963.8 |                  | X--)             |
22    1987.5 |               X--|)                 |

The pairing of the cycles is now more clearly evident. Cycles 2 and
18, 6 and 20, and 10 and 22 seem to correspond each other, but cycle 4
has no pair and the recurrence of cycle 6 as cycle 8 has neither any
correspondence. The cycles 2 to 4 were shorter than cycles 16 to 22,
the nine-year cycles of the 18th century are exactly twice as short as
the ten-year cycles in the 20th century if we count the distance from
the Jovian perihelion. But this is true also internally: while the
cycles 2 and 18 began at a distance of 4.3 years after the perihelion,
the cycles 6 and 20 began 0.9 years after it or the distance
diminished by 3.4 years in 4 and 2 cycles, respectively (0.8-0.9 years
per cycle), but the corresponding distance from 6 to 10 and from 20 to
22 took half of the previous value, or 1.7 years when it crossed the
perihelion. Both had a standstill and the previous one also a change
of direction.

But what happens next? We already know that the minimum of the cycle
23 pairs with that of the cycle 14, but how about the cycle 24? If the
cycle 23 behaves according to the rules (every 13th/14th/15th, see
earlier) that have prevailed since at least 1500, it can't have a
Jovian year length which means that its length is either 9.5 or
12.5-13.5 years.

1.3. The relation of the length of the cycle to its magnitude

Besides the length there are other variables that characterize the
sunspot cycles, especially the magnitude or intensity of the cycle.
The oldest and still the most common way to measure it is to calculate
the so called sunspot number. The sunspot number is derived from the
equation R=k(f+10G), where f is the number of spots, G is the number
of spot groups and k is a factor of the observer that depends e.g. on
the type and size of the telescope used plus the opacity of the sky,
the so-called seeing. The timing of the minimum was not based on any
theory, and so is the case also with this number. Nevertheless, it is
the only choice available in this kind of study that penetrates
centuries, because the availability of the results with alternate
methods are at most 50 years. It's accuracy is debatable but for our
purposes it is enough. The situation can be improved by using smoothed
and average values as is commonly done and as I will also mostly do.

Until the year 1980 the values were gathered in Zurich. Today the
international values are handled by National Geophysical Data Center
in Boulder, Colorado. The values are originally counted in Belgium
based on several tens of observation stations.

Besides these so-called Wolf numbers, named so after the developer of
this method, a widely used value is the 10.7 centimeter (2800 Mhz)
radio flux. It tells the energy of the sun as fractions of Joule per
time, area and frequency interval. The frequency interval of 2800 MHz
is adjusted to 1 Astronomical Unit. Originally the measurements were
made at Ottawa, today they are measured in British Columbia. The
correlation between this flux and the Wolf number is astoundingly good
on both monthly and yearly basis. The main reason this study uses Wolf
instead of the flux is that we have monthly Wolf numbers since 1749,
but flux numbers only from 1947. We can't afford to lose 200 years.
And besides 50 years are too short a time period for this analysis.

I begin with a table containing 1. the number of the cycle, 2. the
year of the minimum, 3. the year of the naximum, 4. the length of the
cycle, 5. the period from minimum to maximum, and 6. the maximum
sunspot number based on the 13-month smoothed averages. 

TABLE 12. THE MAGNITUDE AND THE RISE PERIOD TO MAXIMUM.

1.   2.   3.   4.  5.  6.

1 1755 1961 11.3 6.3  87
2 1766 1769  9.0 3.2 116
3 1775 1778  9.2 2.9 159
4 1784 1788 13.6 3.4 141
5 1798 1805 12.1 6.7  49 maximum is 17 years after the preceding one
6 1810 1816 12.9 6.0  49 (record low with the preceding one)
7 1823 1829 10.6 6.6  72
8 1833 1837  9.6 3.3 147 maximum is 7 years after the preceding one
9 1843 1848 12.5 4.6 132
10 1856 1860 11.2 4.1  98
11 1867 1870 11.7 3.4 141
12 1878 1883 10.7 5.0  75
13 1889 1894 12.1 4.5  88
14 1901 1907 11.9 5.3  64
15 1913 1917 10.0 4.0 105
16 1923 1928 10.2 4.8  78
17 1933 1937 10.4 3.6 119
18 1944 1947 10.1 3.3 152
19 1954 1957 10.4 3.7 201 (record high)
20 1964 1968 11.8 4.7 111
21 1976 1979 10.3 3.4 165
22 1986 1989  9.6 2.8 159
23 1996 2000 12.5?3.9 121
24 2009.0?

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

As one can see from the table, the rise to maximum is not symmetrical
with the fall from it. Only 3 of the 22 rises exceed 50% or 6 years of
the total cycle length. These begin the cycles 1, 5 and 7. Cycles 1
and 7 precede 9-year cycles, cycle 5 is the other of the two lowest
cycles since 1750. Besides these the cycles 6, 12, 14 and 16 are
nearly symmetrical, time of rise being about 45% of the total. The
cycle 6 is the other record-low cycle and the other three cycles are
the coming, middle and leaving ones as regards the minimum standstill
when the Jovian intersects the Sun's plane. The rest or 15 of the 22
cycles are clearly asymmetrical the time of rise being typically about
one third of the total time or 3.5 years with the ten year type cycle.
The average of all the 22 cycles is 4.4 years or very nearly 40 % of
the total time.

In the following analysis the smoothed Wolfian magnitude maximum is
denoted R(M).

TABLE 13. R(M) compared with the time of rise to maximum

length of rise 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 years
R(M)
40- 59                                        XX
60- 79                        X       XX      X
80- 99                        XX              X
100-119                XX      XX
120-139                      length of rise 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0
-6.9 years
R(M)
40- 59                                        XX
60- 79                        X       XX      X
80- 99                        XX              X
100-119                XX      XX
120-139                        X
140-159        XX      XXXX
160-179                X
180-199
200-219                X

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

Regression analysis:

If Y = sunspot number and X = length of rise, then

Y = 236-28X

correlation = -0.82
s.d.=40

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

CONCLUSION 7. THE MAXIMUM POSSIBLE SUNSPOT NUMBER

If there were no time of rise, the maximum sunspot number would be
236, which can thus be regarded as the theoretical upper limit (the
highest empirically observed sunspot number is the 201 of the cycle
19).

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

Every year added will lower the maximum sunspot number with 28 Wolfs:
Correspondingly the maximum rise of time would be 7.1 years (the
empirical record is the 6.7 years of the low cycle 5, which ended with
the spotless year 1810). So we have:

risetime R(M)

3yrs 151
4yrs 124
5yrs 96
6yrs 68
7yrs 41

First of all, we can make three remarks. The maximum of the first of
the two lowest cycles or 5, is 17 years after the previous maximum,
which is an all-time record (since 1750). The highest known cycle or
the cycle 19 had its maximum at the aphelion. The spotless minimum,
that follows the Jovian cycle 5, begins its nearly two years long
period of spotless months exactly at the perihelion.

Besides these there is one more feature that is interesting. If one
classifies the residuals (the real value of R(M) minus the theoretical
value), we get the following grouping (two points to the exact
risetime and one point to the previous and the following because of
the inaccuracy): 

TABLE 14. The residuals of the actual maximum magnitude compared with the
theoretical value

XXXXXX
XXXXXXXXX      XXX
XXXXXXXXX   XXXXXX
XXX            XXXXXXXXXXXXXXXXXXXXXXXX   XXX
XXXXXXXXX         XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

deviation(Wolfs)-31-30-29-28-27-26-25-24-23-22-21-20-19-18-17-16-15

Gap 1 (-14...-2)

XXXX
XXXX
XXXX
XXXXXX        XXXX
XXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXXXXX
deviation (Wolfs) -1 0 1 2 3 4 5 6 7 8 9

Gap 2 (10...19)

XXXXXX
XXX      XXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXXXXXXXXXX

deviation (Wolfs)  20 21 22 23 24 25 26 27 28

Gap 3 (29...67)

XXX
XXXXXXXXX

deviation (Wolfs)  68 69 70

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

So we have residuals about -24...-23, 0...1, 24-25 and about 69. Could
the residuals be -24, 0, 24, 48 and 72 (possibly also -72) and what
could cause this period of 24 Wolfs? I have no answer to this
question, but because this is a statistical study, I try to catch
every regularity, even if I have no explanation.

Click here to get to PART 2.

CONTENTS

- Average sunspot magnitude during 19 Jovian years 1762-1987.
- Is the Jovian effect real or an artifact?
- How many Jovian years are needed for the effect to show up?

Click here to get to PART 3.

CONTENTS

- Magnitude minima.
- Magnitude maxima.
- Medians and quartiles.
- The perihelian stability.

Click here to get to PART 4.

CONTENTS

- 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.

Click here to get to PART 5.

CONTENTS

- 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.
- 200-year cyclicity and a temperature correlation.
- The periods of Cole.

Click here to get to PART 6.

CONTENTS

- 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.

Click here to get to PART 7.

CONTENTS

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

Click here to get to PART 8.

CONTENTS

- Organizing the cycles into a web.

Go to the
beginning. 

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