- 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?
- Magnitude minima.
- Magnitude maxima.
- Medians and quartiles.
- The perihelian stability.
- How long is the 11-year cycle?
- The rules of Schove interpreted.
- 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?
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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.
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.
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.
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 9.8 xxxxxxxxoooooo 23 1996.6 (12.5 prediction) 24 (2009.1 prediction)
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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.
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
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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.
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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.
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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.
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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.
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
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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.
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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.
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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.
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
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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.
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
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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 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
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 precided by the 9.5 years long cycle 8. Now there has been the cycle 22, whose mathematical length was 9.6 years, with some anomality. Using 15 months smoothing, we get a length of 9.8 years. And it is now following a short cycle, the ongoing 23, which may be shorter than 10 years.
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.
Thus we have alternating 13 and 14 cycles between alternating one or two short cycles following long cycles. Short cycles: -19 and -18, (14), -5, (13) 8, (14) 22 and 23 (pred.).
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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 (less than 10 years) are followed one cycle that is longer than one Jovian year (12.5-13.5 years). Combined this gives 15*13.5 = 202.5, which is a supercycle (see later).
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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.
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
4 cycles combined
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1. 2. 3. 4. 2- 5 1766-1810 44.1 11.0 6- 9 1810-1856 45.4 11.4 10-13 1856-1901 45.7 11.4 14-17 1901-1944 42.5 10.6 18-21 1944-1986 42.5 10.6
The difference of 0.8 years shows here clearly up between the 19th and 20th century.
1. 2. 3. 4. -2- 5 1723-1810 86-88 10.8-11.0 6-13 1810-1901 91.1 11.4 14-21 1901-1986 85.0 10.6
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.
1. 2. 3. 4. -10- 5 1633-1810 176.9 11.06 sum inaccurate by +- one year 6-21 1810-1986 176.1 11.01
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.
So let's see what 15 cycles would give us.
1. 2. 3. 4.
-5- 9 1689-1856 166.5 11.10
-4-10 1699-1867 168.2 11.21
-3-11 1712-1878 166.4 11.09
-2-12 1723-1889 166.1 11.07 phase change I
-1-13 1734-1901 167.7 11.18 last cycle lasted one Jovian year
0-14 1745-1913 168.6 11.24
1-15 1755-1923 168.4 11.23
2-16 1766-1933 167.3 11.15
3-17 1775-1944 168.7 11.25
4-18 1784-1954 169.5 11.30
5-19 1798-1964 166.4 11.09
6-20 1810-1976 166.0 11.07 phase change II
7-21 1823-1986 163.4 10.89 last cycle lasted one Jovian year
8-22 1833-1996 162.7 10.85
(9-23 1843-2009 165.6 11.04 phase change III)
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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.
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.
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 1996.8 1999.4 -2.6 -22% (24 23 2009.1 2011.2 -2.1 -18% prediction)
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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.
Altogether we have here 23 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).
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 23 XX X X 3.2 / 13 23 X 4.0 / 02 18 X X 4.8 / 01 16 X X 5.6 / 00 17 X X APHELION
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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.
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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.
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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).
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
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210987654321098765432109876543210 tenths of years
3 2 1 0 years
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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.
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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.
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.
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.
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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 maximum rounded to nearest 50 Wolfs (each line denoting 50 Wolfs).
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--) | | 24 2011.3 | 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).
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 9.5-11 or 12.5-13.5 years.
Space Science News by NASA (science.nasa.gov/newhome/headlines/ ast16dec99_1.htm) is titled "Solar cycle ups and downs continues to mystify scientists." I see no reason why they should mystify anybody. The Sun is much more variable than previously thought. That should not come as any surprise to scientists.
According to the first prediction including 1999, made by National Geographical Data Center (NGDC) in Boulder, Colorado, in August 1997, was amazingly accurate. The prediction and actual numbers were (in smoothed Wolf numbers):
year month pred. act. 1998 Jun 60 62 1998 Jul 63 65 1998 Aug 66 68 1998 Sep 69 70 1998 Oct 72 71 1998 Nov 76 73 1998 Dec 79 78 1999 Jan 82 83 1999 Feb 84 85 1999 Mar 87 84 1999 Apr 89 85 1999 May 92 90
Only the stagnation from February to April 1999 was not seen in advance. Obviously, because the NGDC does not use one parameter, that according to my theory is essential. But later to that.
However, in March 1998 NGDC made a big mistake. From August 1997 to February 1998 NGDC had said that "May 1996 marks Cycle 22's minimum and the onset of Cycle 23". Based on this premise, the predictions were excellent. But in March NGDC suddenly changed the time of the minimum. With NGDC's own words: "May 1996 marks Cycle 22's mathematical minimum. October 1996 marks the consensus Cycle 22 minimum which NGDC is now using." And has used since. It had a catastrophic effect on the prediction. June 1998 jumped from 60 to 73 (actual 62), November-December 1998 from 76-79 to 106-111 (actual 76-78) and May 1999 from 92 to 136 (actual 90). At the same time December 1999 jumped from 105 to 159 (year 2000 was not yet included).
When the year 2000 prediction first appeared in June 1998, the maximum was predicted due to Mars 2000 with a value of nearly 160.
What went wrong? We can easily see that this prediction will not be fulfilled. The Sun is is variable, but not that chaotic. The behavior of the Sun should be calculated mathematically, not by voting a consensus.
According to my theory (personal.inet.fi/tiede/tilmari/sunspots.html) the smoothed sunspot Wolf value cannot exceed 90 during 20 months before Jupiter's perihelion, 100 exactly at the perihelion, and 110 10 months after the perihelion. Jupiter's perihelion was in May 1999. The actual Wolf value was near maximum allowable, 90.
Now, when the influence of the choice of the time of minimum has continuously diminished and been replaced by the actual rise of the cycle 23, NGDC's predictions look again more reliable. The predicted value is 111-113 from February 2000 to Mars 2001, the peak being somewhere between Mars to November 2000. This sounds reasonable.
But back to 1998-1999. The raw Wolf value which hovered between 60-90 in autumn 1998, stagnated to 62-69 from January to April 1999. Why? According to my theory this is the Jovian effect: when Jupiter nears its perihelion and the sunspot cycle is rising, it sets back to its prevailing minimum level and stays there until the perihelion is reached.
Then the May perihelion causes a temporary rise, in this case 106 in May and 137 in June. After that follows the low period, far lower than the predicted 140-160. In this case the quiet period reaches the maximum time, which causes the maximum in 2000 to be far lower (some 110) than the general prediction has been.
If we look back the 250 years that we have more or less reliable figures, what is mystifying in the Sun's behavior in 1998-2000? If we look back the 2000 years that has been estimated from undirect sources, we have still less to wonder about. The Sun really is variable, but still obeys some strict rules.
What is happening in the Sun is a change to a new mode: the Gleissberg cycle has reached its minimum (71 years) and now there will be a reversal to longer and lower cycles. When the 200-year cycle is also nearing its turning point (2010-2030), my theory also predicts a turn of the global warming to a global chill. The warm winters of 1990's correlate very well with the highest smoothed sunspot activity since at least 1700's, and probable since 1200's. But that's another story.
Another interesting paper by NASA was titled "The Day the Solar Wind Disappeared" (science.nasa.gov/newhome/headlines/ast13dec99_1.htm). I quote: "From May 10-12, 1999, the solar wind that blows constantly from the Sun virtually disappeared -- the most drastic and longest-lasting decrease ever observed." ... "Starting late on May 10 and continuing through the early hours of May 12, NASA's ACE and Wind spacecraft each observed that the density of the solar wind dropped by more than 98%." ... "According to observations from the ACE spacecraft, the density of helium in the solar wind dropped to less than 0.1% of its normal value, and heavier ions, held back by the Sun's gravity, apparently could not escape from the Sun at all."
According to NGDC, the Wolf sunspot number got a jump a few days earlier: 08 May the number was 151 and 09 May it was 149, when the first-May value was 75 and the peak was followed by a decrease, so that the value was 85 in May 24. Compare these to the smoothed value of May, 90.
According to my theory the Jupiter's perihelion very much regulates the sunspot cycles. The last Jovian perihelion occurred just during those days in May 1999, when the Sun appeared to behave abnormally. But the effect does not include only those few days, it affects the whole cycle.
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 1755.2 1961 11.3 6.3 87
2 1766.5 1769 9.0 3.2 116
3 1775.5 1778 9.2 2.9 159
4 1784.7 1788 13.6 3.4 141
5 1798.3 1805 12.1 6.7 49 maximum is 17 years after the preceding
6 1810.4 1816 12.9 6.0 49 (record low with the preceding one)
7 1823.3 1829 10.6 6.6 72
8 1833.9 1837 9.6 3.3 147 maximum is 7 years after the preceding
9 1843.5 1848 12.5 4.6 132
10 1856.0 1860 11.2 4.1 98
11 1867.2 1870 11.7 3.4 141
12 1878.9 1883 10.7 5.0 75
13 1889.6 1894 12.1 4.5 88
14 1901.7 1907 11.9 5.3 64
15 1913.6 1917 10.0 4.0 105
16 1923.6 1928 10.2 4.8 78
17 1933.8 1937 10.4 3.6 119
18 1944.2 1947 10.1 3.3 152
19 1954.3 1957 10.4 3.7 201 (record high)
20 1964.9 1968 11.8 4.7 111
21 1976.5 1979 10.3 3.4 165
22 1986.8 1989 9.8 2.8 159
23 1996.6 2000 12.5?3.8 121
(24 2009.1 ?)
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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).
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
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Regression analysis:
If Y = sunspot number and X = length of rise, then
Y = 236-28X
correlation = -0.82
s.d.=40
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3yrs 151
4yrs 124
5yrs 96
6yrs 68
7yrs 41
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
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deviation(Wolfs)-31-30-29-28-27-26-25-24-23-22-21-20-19-18-17-16-15
Gap 1 (-14...-2)
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deviation (Wolfs) -1 0 1 2 3 4 5 6 7 8 9
Gap 2 (10...19)
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deviation (Wolfs) 20 21 22 23 24 25 26 27 28
Gap 3 (29...67)
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deviation (Wolfs) 68 69 70
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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.
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beginning.
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