Some Interesting Characteristics of Solar Cycle Lengths
G. E. Pease
The historical variation in solar cycle length has recently become a topic of interest to many. This 2009 paper, in particular, has caught my interest:
LONG-TERM VARIABILITY IN THE LENGTH OF THE SOLAR CYCLE
A very complete tabulation of solar cycle lengths from 1610 to 1996 is in Table 2 on page 3 of the paper. An un-trended scatter plot of the SSN lengths is included in Figure 5, page 5 and some plots of “normalized (O-C) Cycle Lengths” on page 7 produced some very pronounced sinusoidal plots that do not bear any resemblance to the raw data. The procedure for computing these plots involved computation of an “average cycle length of 10.95 years derived independently by the FFT and PDM analyses from the sunspot number data.”
Interesting, but I wondered if a more straightforward analysis could possibly yield similar results. Using Excel on the raw data, updated to 2008.9 for the solar minimum times, I obtained this plot:
A fourth order polynomial trend line was used. As I suspected, the raw data trend looks similar at the beginning and end, but shows only one shallow peak at 1800, whereas the elaborate procedure used in the paper produced two very distinct peaks, near 1725 and 1900, for both Pmin and Pmax data!
Here’s the Excel statistical analysis for the Pmin data:
| Pmin | |
| Mean | 11.05833 |
| Standard Error | 0.252774 |
| Median | 10.85 |
| Mode | 11 |
| Standard Deviation | 1.516646 |
| Sample Variance | 2.300214 |
| Kurtosis | 0.226661 |
| Skewness | 0.601997 |
| Range | 6.8 |
| Minimum | 8.2 |
| Maximum | 15 |
| Sum | 398.1 |
| Count | 36 |
Note that Excel agrees with the paper’s mean and standard deviation of 11.0+/-1.5 years.
Here is the Excel plot using solar max times:
As in the first plot, a fourth order polynomial trend line was used. Since cycle 24 max has evidently not yet occurred, for comparison purposes with the first plot I used a “guesstimate” of equal solar cycle length to that of the last Pmin, 12.4 years, which artificially guesses a cycle 24 max time of 2012.7. This was done solely for the purpose of producing an end-time trend comparable to the Pmin end-time trend. The actual Pmax times are historically likely to be significantly shorter or longer than Pmin times, but using the same number for extrapolation purposes does produce a very comparable trendline.
Here are the Pmax data Excel statistics:
| Pmax | |
| Mean | 11.03333333 |
| Standard Error | 0.327448045 |
| Median | 10.85 |
| Mode | 11 |
| Standard Deviation | 1.96468827 |
| Sample Variance | 3.86 |
| Kurtosis | 1.608301154 |
| Skewness | 0.816794177 |
| Range | 9.8 |
| Minimum | 7.3 |
| Maximum | 17.1 |
| Sum | 397.2 |
| Count | 36 |
The mean and median are the same as for the Pmin data, but the standard deviation is greater. The mean and standard deviation again agree with the numbers in the paper. Note the much greater sample variance and kurtosis of Pmax data, compared with Pmin data. Skewness is also slightly greater for the Pmax data.
Tallbloke's Talkshop 
A short comment about solar cycle length.
The first question that should be asked if there is any reason at all to believe that a definite solar cycle length exist expressed by 4-5 digits.
If so there are two factors that have to be resolved. The phase of the solar cycle and the amplitude seem to vary “independantly” and have to be investigated separately. See Appendix 1 in my thesis “Wind Controlled Climate” with title “Analysis of sunspot Cycle Phase Variations – Based on D. Justin Shove´s Proxy Data”.
Let us now assume that there actually exists a well defined solar cycle length. Then it is quite certain that this cycle has to be measured over at least 2000 years to produce a meaningsful average value of the solar cycle length and it will be close to 11.12 years IMO. I have spent much time to investigate if Shove missed any cycle or not in his +2600 year record. IMO he did not and I favour the opinion that a specific solar cycle lenght does exist (at least over a million year or so). One reason being that the cycle length does depend on planetary dynamics and it will be quite stable over a million years but hardly over a billion years.
About the phase variation I wrote:
“…this power spectrum is dominated by a slowly varying component with a “period” around 1700 years.” which is something to ponder about.
SCL(min-to-min) is my favorite solar activity ‘proxy’. It’s simple, you don’t have to worry about the ss counting problems/changes, ±0.5 years accuracy is enough to see some basic correlations. Just plot and compare inverse SCL (frequency) and AMO for example. If AMO goes negative soon, it will further confirm the correlation. Solar activity seems to be FM.
There is always a beautiful wave behind, a regular beat.
Yes adolfo, a wave behind a wave… And a bit of chaos.
Note that the Pmax periods for the three main frequencies are larger than the Pmin periods. Also, the distribution is more skewed in the Pmax distribution. This indicates to me that the definition of solar maximum times should be adjusted to make the Pmax distribution more consistent with the Pmin distribution.
It is exciting to me that the three principal periods in the Pmin frequency distribution correspond very nicely with the Jupiter tide period, the central dynamo period, and the Jupiter-Saturn spring tide period. These were all identified by Dr. Nicola Scafetta in his 2012 paper. Refer to his May 17 post:
Nicola Scafetta: Does the Sun work as a nuclear fusion amplifier of planetary tidal forcing?
The three main central frequencies can be seen in his Figure 8, and are clearly the ones associated with the Pmin frequency distribution of the three predominant periods.
Perhaps the beat decreases when the Sun goes to sleep, making them more spaced. 🙂
In my previous comment I referred to Figure 8 of the post
Nicola Scafetta: Does the Sun work as a nuclear fusion amplifier of planetary tidal forcing?
Figure 13 is actually the one showing the distribution of solar sub-periods. My apologies for any confusion this may have caused.
-Gerry Pease
Thanks Gerry,
It’s always instructive to compare this type of study with that of Timo Niroma, who put a lot of thought into his methodology. http://personal.inet.fi/tiede/tilmari/sunspots.html See table 3.
He too found a clustering of cycle lengths around the Jupter orbital period, and around 10.4 years – the period of an important Jupiter Earth Venus cycle. I think Tim C found a similar hump in his spectrum of the solar daily data which he posted on Scafetta’s thread.
The half period of the Jupiter – Saturn synodic cycle is absent here, which would be expected to be present if J-S tidal effects were dominant. This is one of the factors which leads me to lean towards a hypothesis involving a strong electrodynamic component in cycle length and timing.
Having said that, the ~9.9 year componemt in the periodogram and MEM spectral analysis shouldn’t be dismissed as an artifact of badly collated monthly sunspot data as Tim C seems inclined to IMO.
adolfogiurfa says:
May 23, 2012 at 6:15 pm
Perhaps the beat decreases when the Sun goes to sleep, making them more spaced. 🙂
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
It is definitely true that it is more difficult to identify a specific time for solar minimum when the Sun goes spotless for many months (or years!). Likewise, many solar cycles have relatively flat tops for long periods of time. I think it would be more useful to define solar max as the time of most symmetric peak SSN frequency distribution rather than time of actual SSN smoothed peak.
Tallbloke,
Yes, Timo Niroma did a lot of detailed analysis of sunspot cycle lengths, examining many sample intervals. I certainly agree with his conclusions in the Table 3 section; particularly that “There is no theoretical basis for using the 13-month smoothed values as the marker of the minimum.” A similar statement could be made about finding the markers of the maximum.
I find no discussion of the half period of the Jupiter – Saturn synodic cycle in Timo’s study. I don’t think he was looking for that effect, but I wouldn’t exclude the possibility that this is present in his data, but perhaps escapes notice more from a lack of focus on this as a possible component of some solar cycles than from its actual absence.
Obviously, though, the general Excel statistical analysis I performed on the solar cycles between 1600 and 2010 is not adequate to positively identify exactly what is behind the contents of the various subcycle bins, but there does seem to be a rough correspondence here to Dr. Scafetta’s findings.
Unfortunately, there is no standard long-term solar cycle period data set. The data are quite diverse from study to study, and this blurs any specific numerical identification of common features.
I conclude that there is a real need to find a sound mathematical basis for identifying consistent times of “solar maximum” and “solar minimum,” though such an algorithm almost certainly would be less accurate for past centuries than for the late 20th century and early 21st century.
-Gerry Pease
I’d really like to see Tim’s dataset run through Scafetta’s algorithms so we can compare apples with coarser grained apples and see what drops out. In planetary terms we have the J-S cycle with the key periods 9.93, 11.86 and 19.86 years, and the J-E-V cycles with key periods at 3.244,10.38 and 24 years.
Solar minimum can be more accurately pinpointed than solar max. I agree with you that a method for fixing that based on the centre of a distribution (or maybe on the cusp of the golden section of the distribution?) would be a worthwhile exercise. I mention the golden section because Landscheidt noted that on verge, the rise and fall times of solar cycles are in this ratio.
Vuk’s plot of J-S angle at minimum is worth thinking about too.
“Sir William Thomson (a.k.a. Lord Kelvin) who, despite having done much work on the subject, could not comprehend Maxwell’s work saying:
I never satisfy myself unless I can make myself a mechanical model of a thing. If I can make a mechanical model I can understand it
http://www-groups.dcs.st-and.ac.uk/~history/Projects/Johnson/Chapters/Ch4_4.html
tallbloke says:
May 23, 2012 at 9:58 pm
http://personal.inet.fi/tiede/tilmari/sunspots.html See table 3.
Table 3 is well worth a look. Strange that solar science seems to have missed this
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
Here’s a very interesting 1941 article with some fascinating plots on page 463 extracted from the book “Earth and Sun” by Ellsworth Huntington, which was published in 1923!
http://articles.adsabs.harvard.edu//full/1941PA…..49..462C/0000462.000.html
Figure 1 (Sunspots during Siderial periods of Planets) shows double SSN maxima in the same two parts of the orbits of Jupiter, Saturn, Earth and Venus. The first (a) SSN maximum, 25% of the siderial period past perihelion, is huge for Jupiter and quite pronounced for Saturn as well. The second (b) SSN maximum, about 30% of the sidereal period before the next perihelion is most pronounced for Saturn.
SSN data from 1749-1913 were used for Jupiter and Saturn. Data for Mercury, Earth, and Venus were from 1856-1913. The Mercury plot is quite flat near zero, showing no significant correlation between Mercury’s orbit and sunspot number.
Huntington’s book contains tabulations used for the plots, but I have been unable to find his book or the tabulations online.
-Gerry Pease
If you can’t find the 1941 article using the link in my previous comment, search for
“Planets and Sunspots – ADS”
and a link to it should come up.
Thanks Gerry. I’ll get hold of the book and email you some digitisations of the tables.
Rog, all,
I have put up a draft post on cycle length where this is computed in a very different way. Don’t want to take attention away from here but the same material in a comment is unreasonable.
Not sure what to do so I will let Rog decide. Could be moved to comment, or slide in on an earlier date, ignored, etc.
Tim, I’ve emailed you. We should spend a few days working on that, it is of major interest.
Not a bad idea, talk about it offline.