Ian Wilson: Could This Be The Climate Smoking Gun?

Posted: April 2, 2013 by tallbloke in climate, Cycles, Dataset, Natural Variation, Solar physics, solar system dynamics, Tides

Here at the talkshop we have recognised from the get-go that with the demise of the pathetically myopic and wholly incorrect co2-driven-climate hypothesis, a better theory will be needed to take it’s place. Now we have some more exciting news from Astrophysicist Ian Wilson. He has discovered a very solid looking link between planetary motion, solar variation, and Earth’s climate. The periods involved also tie in with the work going on in the background here on the Phi/Fibonacci thread. I believe we are getting closer to solving the solar system puzzle, and finding plausible explanations for cyclic climate changes on Earth which are evidenced by various proxy time series. Here, Ian uses the (in)famous bristlecone pine core data. Maybe this was the right proxy with the wrong approach all along. Just for the avoidance of doubt, that’s Prof. Michael Mann I’m alluding to. Excuse me while I spit the taste of his name out of my mouth.

Could This Be The Climate Smoking Gun?
Ian Wilson 2-4-2013

In the paper:

Acta Geodyn. Geomater., Vol. 9, No. 3 (167), 259–268, 2012


Figure 1 shows an amplitude spectrum of the North American temperature over a 2200 year period.

The spectrum is based upon bristle-cone pine series obtained from:

Climatic Change (2005) 70: 465–48

Given the controversial nature of the use of Bristle-cone as a proxy for temperature, I have quoted the section of this paper dealing with methods used in constructing the tree-ring chronology that reflects past temperature variability – see the Appendix at the end of this post. The important thing to note about figure 1 is that, for periods for less than or equal to 500 years, there are prominent spectral peaks at: 89 years, 104 years, ~ 150 years, 208 years, ~ 230 years, ~ 355 years and ~ 500 years. These periods (obtained from proxy temperature data) can be compared with a Fourier spectrum of solar activity qualified by the solar modulation potential and the Fourier spectrum of the planetary torque acting upon the Sun, published by: J. A. Abreu1, J. Beer, A. Ferriz-Mas, K. G. McCracken, and F. Steinhilber Is there a planetary influence on solar activity? A&A 548, A88 (2012) These spectra are based upon a 9400 year time series of solar activity derived from observations of Be10 and C14.

Figure 2


Amazingly, the solar activity spectrum shows peaks at:

89 years, 104 years, 150 years, 208 years, ~ 230 years, ~ 355 years and 506 years.

The planetary torque shows all of these peaks, except those at ~230 & ~ 355 years.
[See Ian Wilson’s previous posts here for more detail on his tidal-torque theory]

I believe that it is extremely unlikely that the spectral peaks for the North American temperatures are almost exactly the same as the spectral peaks for solar activity and planetary tidal torques acting upon the Sun by chance.

I claim that it indicates that the level of solar activity upon the Sun is modulated by planetary tidal-torquing acting upon the convective layer of the Sun and that it is these variations in the level of solar activity that have determined the mean temperatures of the North American continent over the last 2200 years.

Figure 1A


To build a tree-ring chronology that reflects past temperature variability, upper-tree-line Bristlecone Pine (P. aristata), whose ring-width variability is a function of temperature, were sampled at timberline in the San Francisco Peaks (∼3,536 m), where temperature is most limiting to growth (Figure 2). Increment core and sawed samples were collected from living and dead Bristlecone Pine on both Agassiz Peak and Humphreys Peak. Long chronologies were constructed by cross-dating the deadwood samples with the living tree specimens. Prior to AD 659 the chronology is composed entirely from deadwood material. The individual growth rings of each sample were measured to the nearest 0.01 mm. The measured series were converted to standardized tree-ring indices by fitting a modified negative exponential curve, a straight line, or a negatively sloped line to the series.

This process removes the age/size related growth trend and transforms the ring-width measurement values into ring-width index values for each individual ring in each series (Fritts, 1976). Several statistics were calculated to gauge the reliability of the tree-ring series (Cook and Kairiukstis, 1990; Wigley et al., 1984) (Table I). Through conservative standardization techniques and the use of relatively long series, care was taken to preserve low-frequency information in the chronology (Cook et al., 1995). A regional curve standardization (RCS) approach (Briffa et al. 1992a), which has been used in some dendroclimatic studies to resolve multi-decadal to centennial trends, was considered but rejected. In general, RCS was devised for chronologies built from short series that use heavy detrending. The SFP chronology was built from relatively long segment lengths (Table I) and a conservative standardization process was employed. Additionally, we could not meet the necessary assumptions critical to successfully applying this technique: it was impossible to ascertain pith or near-pith dates from the deadwood material due to the irregular growth form of Bristlecone Pine, and we are not able to demonstrate that the age structure of the samples are evenly distributed.

To create the mean site chronology, the annual standardized indices of tree growth were averaged. A single chronology was developed from samples collected at two sites on Humphreys Peak and one site on Agassiz Peak. The SFP chronology extends from 663 BC–AD 1997. In total, 234 series (130 trees) are used. The period before 266 BC is considered less reliable than the rest of the chronology as six or fewer series cover this interval. The climate data used in the temperature reconstruction calibration are from the Fort Valley Experimental Research Station, which is part of the United States Historical Climatology Network. The station data, from an elevation of 2,239 m and approximately 4.5 km from the high elevation SFP tree-ring sites, span the period 1909–1994.
end quote:

  1. tallbloke says:

    Nota bene

    89*Phi^2 =233.005 Both 89 and 233 are Fibonacci numbers
    2*89*SQRT(2-1/Phi)=209.251 This is around the De Vries cycle period


  2. cosmic says:


    I thought tree rings were basically a flawed temperature proxy. Too many other factors affect them, light, rainfall, nutrient availability, and they are formed in three months of the year.

  3. tallbloke says:

    cosmic: Yes. I tend to think of them as a climate-for-trees proxy rather than a temperature proxy. However, all is not lost, because in general the level of solar activity is quite a good indicator of better (warmer, sunny, moist) and worse (colder, overcast, dry) climatic conditions. Given that those climatic conditions affect the rate of tree growth and thus ring width, the same works in reverse. Ring width is a reasonable indicator of solar activity levels. There was a post on this a while back, I’ll see if I can find it.

    Here we go, there are a few;

  4. I’m not a massive fan of alternative theories for why the world appears to be warming at the moment, although I do consider them. Solar influence is something I’ve looked at only in passing up to now (mostly via posts on the talkshop) but it’s not something which has particularly struck me as having any more or less credibility than the CO2-is-the-control-knob narrative.

    Those sceptical of AGW scepticism (IYSWIM) would seem most likely counter this with something like: “There seems to be a tendency to criticise the use of certain proxies when “the team” do it or when it gives an answer sceptics don’t like, but to embrace them when the results confirm our own theories.”

    But as Roger says (comment at 4pm, 2/4/2013), whether or not temperature is the driving force, if there is a genuine correlation as close as this one appears to be then it almost doesn’t matter what the trees are measuring. That “the team” think it’s temperature, and are using it to produce a “history” of temperature, is enough to make the discovery of such a correlation worthwhile.

    Interesting stuff.

  5. tchannon says:


    Q: does a tree growing on the equator have rings?

    Take as rhetorical, food for thought.

  6. tchannon says:

    Its been suggested the bristlecones are probably showing water availability and also perhaps related the effect of damage such as from fire.

    Temperature has a somewhat vague relationship.

    I wonder too how fogs relate, which spins off into yet more into possible extraterrestrial effects, a word chosen deliberately to widen the possibility to more than only solar effect. A commonality.

  7. tallbloke says:

    Tim, yes. You’ll notice the second link in my comment above was to a tree ring study which (unexpectedly) found a link to cosmic rays

    It’s worth a read from the climategate angle too.

    In the email sent by Grace to his colleagues, we read
    “Dear Colleagues
    We have found a correlation between tree rings and galactic cosmic radiation:

    “This is an unexpected result, for which we don’t yet have a good explanation. I hope doesn’t result in scientific excommunication!

    The email is priceless because Grace was so afraid of being excommunicated by his AGW colleagues that he didn’t mention Svensmark’s name in the article, and also did not mention the known link between the galactic cosmic ray flux cycle and the solar cycle. The cosmic ray maximum flux times shown in Figure 3 of the article all occur at times of minimum solar sunspot activity (1965, 1977, 1987, and 1997).

    His fear of being excommunicated proved to be well founded, as he was apparently shunned by the AGW gang after publication of the article, in which he had revealed himself to be a traitor to the AGW Cause by including this reference:

    Svensmark H, Friis-Christensen E. 1997. Variation of cosmic ray flux
    and global cloud coverage – a missing link in solar-climate relationships.
    Journal of Atmospheric and Solar-Terrestrial Physics 59: 1225-

  8. Doug Proctor says:

    cosmic says: April 2, 2013 at 3:52 pm

    The problem you/we’re facing is that “health” is the combination of many things or at least some things. If the solar cycles above cause even different things under the same forcing mechanisms, and if each is “unhealthy” or “healthy” by itself, then the tree-ring pattern will show up despite a non-unique causitive agent in the environment that affects tree-ring growth.

    I guess it is like growing up in a ghetto: poverty, violence, despair, broken families, bad schools and defeated attitudes individually are different, but several together will create the future personal troubles you might have.

    Seems to me, anyway. A demonstration that correlation may not mean causation but can be used to predict future/other observations, i.e. has predictive value.

  9. tallbloke says:

    Derek: As I recall, Steve McIntyre had less of a problem with the raw data than he did with the statistical games Mann played with them. Also, IIRC it was the strip-bark pines which were really problematic. Remember too, that whereas the hockey-jockeys were creating long contiguous time series out of dubiously massaged series, Ian is using a spectral analysis of bristlecone data collated by non-hockey-team members here. There can be no pre-selection bias, because they weren’t to know Ian was coming along years later to see if the peaks in their spectral analysis matched planetary periodicities and proxies of solar activity levels. Whereas ‘the team’ had a pretty clear idea of what they wanted the data to say before they chose their series. In fact one of them is on record as saying paleodendroclimatology is a unique discipline because it chooses those series to use from those which show good correlation to temperature. You couldn’t make it up…

  10. cosmic says:

    I’m always wary of fooling ourselves with things we want to hear.

  11. Chaeremon says:

    Hello, I appreciate and respect this tremendous work and the comparison with (tree) proxy data. But let me add (hopefully uncontroversial): instead of funding the AGW disaster I wish we already had a comparable hypothesis for tree rings (as proxy), like from the following (exemplary) site. There the trees live under and depend on the cloud layer (amounts to substantially reduced sunlight, excellent for comparing cyclic phenomenae).

    “[the Agana site’s] laurel forest inhabited areas are on the windward sides of the upper Canary Islands at an altitude of 600-1400 m above sea level in the trade winds zone. The cloud layer is absolutely vital for this type of forest on the otherwise very dry Canaries. The laurel tree species are unable to limit transpiration and thus rely on a good supply of moisture [ref.], both in winter and in summer, when the humidity drops from 80% to about 40% [ref.] [T]he direct sunlight is, by shading of clouds, reduced by nearly half …” Excerpt translated by me from the Biology/Botanic thesis of Maren Evers, Johannes Gutenberg University Mainz, 2003, “Vegetationsstruktur auf unterschiedlich alten Lorbeerwaldregenerationsflaechen im Norden Teneriffas.”

  12. Richard111 says:

    Interesting. I am currently reading a book by John L. Casey titled COLD SUN in which he espouses The Theory of Relational Cycles of Solar Activity (The RC Theory). This seems to involve primarily a Bi-Centennial Cycle of 206 years. He is very convinced the sun is moving into a period of hibernation which started with SC24 and will result in the Earth experiencing several decades of cold. He points out that there are fewer farmers in the world today than there were during the Dalton Minimum. Given current population levels this could be a disaster.

  13. tallbloke says:

    Chaeremon,nice translation thanks. Makarieva points out that forests create their own microclimate with evapo-transpiration. In Teneriffe, you can have warmth and cloud on the lee side of Mt Teide.

    Richard111. The de vries cycle features in the abreu et al paper. Tim Channon was on the case a couple of years earlier:

    This article I wrote sheds some light too:

  14. Roger Andrews says:

    “Could this be the climate smoking gun?”

    It’s actually just one of a very large number of “smoking guns” that show correlations between solar/planetary/lunar variables and climatic and related variables here on Earth, many of which have already been presented on this blog. Because the sun is the basic driver of the Earth’s climate these correlations aren’t in the least surprising. It would in fact be a lot more surprising if they didn’t exist.

    “Here at the talkshop we have recognised from the get-go that with the demise of the pathetically myopic and wholly incorrect co2-driven-climate hypothesis, a better theory will be needed to take it’s place.”

    I fully agree that CO2 didn’t cause the recent warming, but demonstrating a long-term sun/climate connection doesn’t disprove a theory which in essence claims that CO2 forcing has swamped solar and other natural forcing only over the last 40 years or so. If we want to disprove AGW we have to show either that the solar forcing over this period is underestimated or that the CO2 forcing is overestimated, or both. And I think we’ve already done that.

  15. acckkii says:

    Reblogged this on acckkii.

  16. Gary says:

    How does it look with ice core data? Confirmation with a totally different proxy would be very helpful.

  17. tallbloke says:

    Gary: Don’t know about ice cores but the hurricane free shores of Hudson Bay and Siberia tell a supporting tale going back thousands of years. Evidence of beach ridges left by perodic tidal highs are left high and dry on the uplifting surface rebounding from the vanished weight of melted glaciers. There is a ridge every ~45 years, with progressively bigger ones at ~90, ~180, ~360 years. More here:


  18. Roger Andrews says:

    Surface air temperature for the last 1420 years reconstructed using ice cores sampled from a Greenland glacier (“Ice core temperature”):

    Summer surface air temperature for the last 1400 (500-1900) years reconstructed from the tree rings of Scots pine:

    From: http://www.fao.org/docrep/005/y2787e/y2787e06.htm

  19. oldbrew says:

    ’89 years, 104 years, 150 years, 208 years, ~ 230 years, ~ 355 years and 506 years’

    89 x 4 = 356. 89 is about 8 average solar cycles of 11 years.

  20. tallbloke says:

    Oldbrew: Yes. And the midpoint between 150 and 208 is … 179

  21. Ninderthana says:


    Thank you for re-blogging my post. A few points:

    # I think you (and some other posters) are spot-on-the-money saying that the Bristle-cones proxies are probably representative of a mix of climate parameters that promote/impede the growth of these trees (e.g. moisture, temperature, cumulative sunshine etc.).

    # If you use half-the Hallstatt Cycle ~ 1157 years to amplitude modulate the Jose Cycle ~ 178.7 years you get:

    (1157 x 178.7) / (1157 – 178.7) = 211 years – positive side lobe

    (1157 x 178.7) / (1157 + 178.7) = 155 years – negative side lobe

  22. tallbloke says:

    Ian, I just realised I forgot to link your blog, apologies, I will correct that.
    Along with behind the scenes developments with the Phi/Fibonacci work, we have a complete integration of solar sytem periods, solar variation and climate variation coming together. It will form the basis of our presentation at the September conference I am organising in Leeds for the 5th &6th September. More news on that soon. Speakers include Nicola Scafetta, Nils Axel Morner, Christopher Monckton, Hans Jelbring, Jerki Kipponen, Richard Holle doing a digital presentation running throughout, and me presenting this work on your behalf.

  23. Ninderthana says:

    Rog, Excellent, it looks as though your bringing out some of the big-guns in September. Hopefully, this will let the World know that there is an alternative to a CO2 driven climate.

  24. Ninderthana says:

    I think it is important to note that the paper:

    Cyril RON, Yavor CHAPANOV and Jan VONDRÁK
    Acta Geodyn. Geomater., Vol. 9, No. 3 (167), 259–268, 2012

    also points to a possible linking mechanism between the level of solar activity and the Earth’s climate. This link was first pointed out by Nikolay Sidorenkov and his Russian collaborators.

    The Earth’s LOD (= the time derivative of (UT1 – TT)) is heavily influenced by the redistribution
    of water between the oceans and the polar ice caps. This is why the establishment of the link between variations in the mean-sea-level (MSL), UT1-TT and level of solar activity is so important
    in this paper.

  25. tallbloke says:

    Ian, Two media representatives including one from a major European organisation have signed up so far.

    Is it cool for me to make the Cyril RON, Yavor CHAPANOV and Jan VONDRÁK paper available here as a free download?

  26. Ninderthana says:


    As far as I can tell, the paper (in pdf format) is freely available to download from

  27. Ninderthana says:

    You could also try this URL:

    Click to access 2_Ron.pdf

  28. Tenuc says:

    Roger Andrews says:
    April 2, 2013 at 7:03 pm

    [Could this be the climate smoking gun]? – “It’s actually just one of a very large number of “smoking guns” that show correlations between solar/planetary/lunar variables and climatic and related variables here on Earth, many of which have already been presented on this blog. Because the sun is the basic driver of the Earth’s climate these correlations aren’t in the least surprising. It would in fact be a lot more surprising if they didn’t exist…”

    One explanation for this is that a significant proportion of total system energy comes from outside the system itself (~15% according to Miles Mathis, who had the original idea). Regular variation in the amount of galactic energy hitting our solar system then provides an explanation of the root cause of all the ‘smoking guns’ we observe.

    One of Mile’s short papers on this topic available free of charge here…


  29. oldbrew says:

    Ninderthana says; ‘ If you use half-the Hallstatt Cycle ~ 1157 years’

    1157 = 89 x 13 i.e. 13 ‘average’ solar cycles
    (34² = 1156)
    89 / 8 years = 11.125 years = 1 average solar cycle
    2 x 89 = 178 years = Jose cycle

    2, 8, 13, 34 and 89 are all Fibonacci numbers.

  30. tallbloke says:

    OB: 144 and 233 and 377 are also Fibonacci numbers. Notice they are diverge from the numbers in the spectra of solar activity and tree growth both sides of the de Vries cycle length though. But it is interesting that 377 + 144 = 511 is close to the 505 figure. Lanscheidt thought there was a basic solar period around 37 years IIRC. Roy Martin found a pattern of solar activity related to planetary motion at 55.15 years.

  31. oldbrew says:

    The reason 89/8 works is it relates directly to solar cycle periods. Looking at it like that, the 55.15 could be 89/8 x 5 i.e. 5 solar cycles. Solar cycles in the last century or two have tended to be a bit shorter than the long-term average.

  32. Chaeremon says:

    @Roger Andrews: thanks for the link to fao.org diagrams, very impressive (what else can be expected from proficient naval people 😉

    @oldbrew, tallbloke: perhaps a bit OT but can I ask: astrophysicists have (re-)modeled our planetary system (and much more in cosmology) on base of powers of 2 in 1 equation template (with Kepler’s as lead), their thesis predicts not only the asteroid belt but also an unoccupied orbit between Jupiter and Saturn. From your Fibonacci cycle maths, where would you place such an additional orbit in AU, if there were any?

  33. oldbrew says:

    @ Chaeremon

    If the radii of the main bodies in the solar system are in balance, as I suggest here…

    …something would have to split in two to make the extra orbiting body. Jupiter and Saturn seem too big (i.e. the majority of all the orbiting planetary mass) to allow anything in between them, so it would probably have to be 3 planets replacing Saturn and Uranus, with roughly the same mass as those two.

    However as Jupiter and Saturn have a magnetic interaction and very similar rotation rates, the whole basis of the solar system might unravel in that case.

  34. ferdberple says:

    Very interesting result. If you can get a similar result with some other widely spaced proxies it would be hard to argue this is co-incidence.

  35. ferdberple says:

    enuc says:
    April 3, 2013 at 8:12 am
    One of Mile’s short papers on this topic available free of charge here…
    Wow! I had to double check the numbers. Miles brings up a very interesting point. The core of the sun is nowhere near dense enough to support fusion under conventional theories. Creating fusion in the lab should be child’s play. Why do we need a A bomb to create the necessary pressure and temps? Something is missing.

  36. ferdberple says:

    Off topic. I was reading up on absolute rotation as a consequence of one of the posts on ether. fascinating topic. It appears Einstein recognized that gravitational frame dragging implies that Mach was right. The gravity of the universe establishes an absolute frame of reference for rotation – in effect gravity is the ether.

  37. Chaeremon says:

    ferdberple said “… implies that Mach was right.” Well, I think that the rate of propagation (here at home and around us in the universe) is rather not off topic in this thread. It’s very subtle but at the base of arguments presented in virtually all of the above. For some distraction towards Mach I suggest “Inertia, Mach’s Principle and Expansion of Space” (PIRT X, London, 2006) by Heikki Sipilae, available from

    Click to access Sipila.pdf

    Spilae (and consorts) seem not interested in just the next “academically rightful” interpretation of Einstein’s worst-than college-level maths outside of reality. Serious work on rate of propagation is always a good read and has well above average fresh and forthright ideas (like Miles Mathis’).

    But fasten your seat belts, it’s about observations e.g. “… force[s] affect[s] the accelerated object without any delay.” and heresy for believers in Holy Gravitational Waves.

  38. peter azlac says:

    Re the value of bristlecone pine data, it is interesting that Libby and Pandolfi, comparing Japanese cedar with bristlecones, found results similar to yours in the 1970’s and linked them to the ‘tidal stresses of the Sun–Moon–Earth system’
    Isotopic tree thermometers: Correlation with radiocarbon
    1. Leona Marshall Libby
    2. Louis J. Pandolfi

    “We have obtained evidence that trees store the record of climate in their rings. In each ring the ratios of the stable isotopes of hydrogen and oxygen vary in proportion to the air temperature when the ring was formed because the isotopic composition of rain and atmospheric CO2 varies with temperature. In this paper the stable isotope variations of hydrogen and oxygen in a Japanese cedar have been correlated with the secular variations of radiocarbon measured in bristlecone pines by Suess (1970). We find significant negative correlations for both isotope ratios over the last 1800 years. The inference is that the small-scale (∼1%) variations in 14C concentrations in tree rings are related to climate variations. In our data we find periodicities of 58, 68, 90, 96, 154, 174, 204, and 272 years. Because our samples are averaged over 5 years each, we are not able to detect the 21-year sunspot cycle in the present data. The Suess samples averaged over about 25 years each reveal a periodicity of 183 years, in agreement with our periodicity of 174 years.

    Climate periods in tree, ice and tides
    Leona Marshall Libby* & Louis J. Pandolfi†
    *Environmental Science and Engineering, University of California, Los Angeles, California 90024
    †Department of Chemistry and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024
    Ten climate periods found in stable isotope ratios of oxygen and hydrogen, measured in 1,800 yr of Japanese cedar rings, agree with climate periods found in 800 yr of Greenland ice and with periods computed from the tidal stresses of the Sun–Moon–Earth system, and with periods found in the 14C record of the bristlecone pine sequence of southern California. The Greenland oxygen ratios have previously been found to have opposite phase to the 14C ratios of the bristlecones, and we have found also an opposite phase between oxygen and hydrogen ratios in the Japanese cedar on the one hand and 14C in bristlecones on the other hand.

  39. tallbloke says:

    Peter A: Many thanks for that paper. I know Ian will have something to say about it, because he made an important discovery regarding the synchronicity of Lunar periods with other solar system periods. The paper you’ve highlighted could form an inportant link in the chain of reasoning which supports Ian’s hypothesis.

  40. steveta_uk says:

    From Miles Mathis’ “Hole at the Center of the Sun”:

    If we take the numbers from Wikipedia as correct, we find that “Its magnetic field is at less than half strength compared to the minimum of 22 years ago.” Well, that doesn’t make any sense. You can’t compare one minimum to another. They must mean it is at half strength compared to some maximum.

    Can Miles really be that ignorant of solar cycles? Any fool can see that the Wiki reference to the “minimum of 22 years ago” refers to the minimum of that particular cyle – not of all time.

    Well, apparently one fool couldn’t see it.

  41. Paul Vaughan says:

    Hiroko Miyahara: Solar Activity and Climate

  42. Paul Vaughan says:

    …looks like that fixed what WP apparently scrambled [turned out http was missing] — also see:

    Click to access 2k_Miyahara_SORCE_brief.pdf

    …and her other work.

    Her common sense interpreting stats is refreshing — and reassuring about the capacity of the intellectual forerunners of the human race.

    What a beautiful breath of fresh air in comparison with the usual ignorantly &/or deceptively scrambled solar-terrestrial distortion relentlessly pushed by climate discussion dark agents.

  43. tchannon says:

    I don’t need persuading about the magnetic effect but the mechanism is the thing which needs clarification, In cloud? One of the possibilities.

  44. Paul Vaughan says:

    Ninderthana (April 2, 2013 at 10:18 pm) wrote:
    “If you use half-the Hallstatt Cycle ~ 1157 years to amplitude modulate the Jose Cycle ~ 178.7 years you get:

    (1157 x 178.7) / (1157 – 178.7) = 211 years – positive side lobe

    (1157 x 178.7) / (1157 + 178.7) = 155 years – negative side lobe”

    / (164.888325 – 84.05119028)
    = 171.4442259

    / (171.4442259 – 164.888325)
    = 4312.016244

    / (4312.016244 – 171.4442259)
    = 178.5430331

    / (178.5430331 – 171.4442259)
    = 4312.016244

    / (178.5430331 + 171.4442259)
    = 87.46081839

    (4312.016244) / 4 = 1078.004061

    / (1078.004061 – 174.9216368)
    = 208.8029063

    / (1078.004061 + 174.9216368)
    = 150.5007321

    Derivation of the other periods (355, 506, 230) is too tedious for concise presentation.