Archive for the ‘Maths’ Category


This started as a search for a period when the Sun and the Moon would both complete a whole number of rotations.
The result was:
Solar: 25.38 days * 197 = 4999.860 d
Lunar: 27.321662 * 183 = 4999.864 d
(data sources: see reference notes at end)

Taking these as equivalent, we have 197-183 = 14 ‘beats’.
197 = 14*14, +1
183 = 13*14, +1
4999.864 / 14 = 357.13314 days
357.13314 days * 45/44 = 365.2498 days
45 * 14 (630) beats = 44 * 14 (616) calendar years, difference = 0.022 day

So the beat period of the two rotations is 44/45ths of a year, i.e. the difference in number of rotations is exactly 1 in that length of time.
630 beats = 616 years (630 – 616 = 14)
616/45 = 13.68888 calendar years = 4999.8663 days
184 lunar sidereal months (rotations) = 4999.864 days

Then something else popped up…

The Phi factor:
‘We recover a 22.14-year cycle of the solar dynamo.’ (2016 paper)
See: Why Phi? – modelling the solar cycle

Solar Hale cycle = ~22.14 years (est. mean)
13.68888 * Phi = 22.149~ years
22.14 / 13.68888 = 1.61737 (99.96% of Phi)
(55/34 = 1.617647)

From the same post:
Jupiter-Saturn axial period (J+S) is 8.456146 years.
That’s when the sum of J and S orbital movement in the conjunction period = 1

13.68888 / 8.456146 = 1.618808
Phi = 1.618034

Conclusion:
This cycle of solar and lunar sidereal rotation (SRC) sits at the mid-point of the Phi²:1 ratio between the J+S axial period and the mean solar Hale cycle, i.e. with a Phi ratio to one and inverse Phi to the other.
SRC = (J+S) * Phi
SRC = Hale / Phi
SRC = Hale – (J+S)
(Mean Hale value is assumed)

In a period of 616 years there are 45 SRC.
The period is 44 * 14 years = 45 SRC = 45 * 14 beats.
SRC * (45/44) = 14 years.

Cross-checks:
Carrington rotations per 616 y = 8249
8249 CR / 45 = 4999.865 days

Synodic months per 616 y = 7619
7619 SM / 45 = 4999.856 days
8249 – 7619 = 630 = 45 * 14

45*183 sidereal months = 8235
8235 – 7619 = 616
8249 CR – 8235 Sid.M = 14
Beat period of CR and Sid.M = 616/14 = 44 years = 45 * (13.6888 / 14)
Every 44 years there will be exactly one less lunar rotation (sidereal month) than the number of Carrington rotations.

8249 CR – 7619 synodic months = 630 = 45 * 14
630 – 616 = 14
– – –
The anomalistic year

The beat period of the tropical month and solar sidereal rotation * 45/44 = the anomalistic year.
(27.321582 * 25.38) / (27.321582 – 25.38) = 357.14265 days
45 * 357.14265 = 16071.419 days
44 * 365.259636 = 16071.423 days

The anomalistic year is the time taken for the Earth to complete one revolution with respect to its apsides. The orbit of the Earth is elliptical; the extreme points, called apsides, are the perihelion, where the Earth is closest to the Sun (January 3 in 2011), and the aphelion, where the Earth is farthest from the Sun (July 4 in 2011). The anomalistic year is usually defined as the time between perihelion passages. Its average duration is 365.259636 days (365 d 6 h 13 min 52.6 s) (at the epoch J2011.0).
http://en.wikipedia.org/wiki/Year#Sidereal.2C_tropical.2C_and_anomalistic_years
– – –
Data sources

— Carrington Solar Coordinates:
Richard C. Carrington determined the solar rotation rate by watching low-latitude sunspots in the 1850s. He defined a fixed solar coordinate system that rotates in a sidereal frame exactly once every 25.38 days (Carrington, Observations of the Spots on the Sun, 1863, p 221, 244). The synodic rotation rate varies a little during the year because of the eccentricity of the Earth’s orbit; the mean synodic value is about 27.2753 days.
http://wso.stanford.edu/words/Coordinates.html

— The standard meridian on the sun is defined to be the meridian that passed through the ascending node of the sun’s equator on 1 January 1854 at 1200 UTC and is calculated for the present day by assuming a uniform sidereal period of rotation of 25.38 days (synodic rotation period of 27.2753 days, Carrington rotation).
http://jgiesen.de/sunrot/index.html

The sidereal month is the time between maximum elevations of a fixed star as seen from the Moon. In 1994-1998, it was 27.321662 days.
http://scienceworld.wolfram.com/astronomy/SiderealMonth.html

Lunar precession update

Posted: October 15, 2017 by oldbrew in Fibonacci, Maths, moon, Phi, solar system dynamics
Tags: ,

Credit: NASA


I found out there’s an easy way to simplify one of the lunar charts published on the Talkshop in 2015 on this post:
Why Phi? – some Moon-Earth interactions


In the chart, synodic months (SM) and apsidal cycles (LAC) are multiples of 104:
79664 / 104 = 766
728/104 = 7

The other numbers are not multiples of 104, but if 7 is added to each we get this:
86105 + 7 = 86112 = 828 * 104 (TM)
85377 + 7 = 85384 = 821 * 104 (AM)
5713 + 7 = 5720 = 55 * 104 (FMC)
6441 + 7 = 6448 = 62 * 104 (TY)

TM = tropical months
AM = anomalistic months
SM = synodic months
LAC = lunar apsidal cycles
FMC = full moon cycles
TY = tropical years


Here’s an imaginary alternative chart based on these multiples of
104. [Cross-check: 828 – 766 = 62]

In reality, 55 FMC = just over 62 TY and 7 LAC = just short of 62 TY.
For every 7 apsidal cycles (LAC), there are 766 synodic months (both chart versions).

In the real chart:
For every 104 apsidal cycles, all numbers except SM slip by -1 from being multiples of 104. So after 7*104 LAC all the other totals except SM are ‘reduced’ by 7 each.

In the case of tropical years, 6448 – 7 = 6441 = 19 * 339
19 tropical years = 1 Metonic cycle

If the period had been 6448 TY it would not have been a whole number of Metonic cycles.
Also 6441 * 4 TY (25764) is exactly one year more than 25763 synodic years i.e. the precession cycle, by definition.

Fibonacci: 104 is 13*8, and the modified FMC number is 55 (all Fibonacci numbers).

Phi: we’ve explained elsewhere that the number of full moon cycles in one lunar apsidal cycle is very close to 3*Phi².
We can see from the modified chart that the FMC:LAC ratio of 55:7 is 3 times greater than 55:21 (55/21 = ~Phi²)
– – –
Note – for more discussion of the ~62 year period, try this search:
site:tallbloke.wordpress.com 62 year
[see Google site search box in grey zone on left of this web page]

Credit: NASA


This is from a Q&A on a website linked with Sydney Observatory. We add brief notes at the end.

Lionel asks: Congratulations on your Venus book.

Excellent. I notice that there is a 243 year cycle for Transits of Venus
243 x 365.242 = 224.7 x 395
So far so good. The axial rotation period for Venus is 243.1 days.
Is this a coincidence or is there some underlying geometrical fact that I cannot see?
well-done,

Answer: An interesting and complex question that I address below.

Patterns in the transits of Venus
Let us first look at the patterns in the transits of Venus. We need to note that Venus and the Earth line up with the Sun every 583.92 days or 1.59872 years. This is called the synodic period.

If there was a transit, say the one in June 2004, for another transit to occur, the two planets must not only line up with each other and the Sun, but do so after an integer number of years so that they are back in the right places on each of their orbits.

Venus and Earth fulfil these requirements after five synodic periods = 7.9936 years as this is almost, though not quite, equal to the integer eight. Thus transits of Venus generally occur in pairs eight years apart. However, because of the slight inequality there is no third transit after another eight years.

A more accurate relationship occurs after 152 synodic periods = 243.00544 years or ~395 Venus years. The pattern of Venus transits thus repeats at 243 year intervals (This is the cycle quoted by Lionel in his question above). For example, the first pair of June transits after 8 June 2004 begins on 11 June 2247. Of course, in the meantime there is also a pair of December transits beginning in 2117.

The rotation of Venus
Scientists using radar observations from the 1960s onwards discovered that Venus spins backwards, that is in the opposite direction to its motion around the Sun, at the slow rate of 243.02 days.

They soon realised that means that Venus, almost but not quite, shows the same face towards the Earth each time the planets are lined up with each other and the Sun. Somehow there is a resonance between the motion of the Earth around the Sun and Venus’ spin around its axis. Scientists are unsure why this is the case, but one suggestion is that Venus is more massive on the face turned towards the Earth at those times and consequently it was gravitationally captured by the Earth.

How is it worked out that Venus shows the same face towards the Earth each time they line up? The quoted value of 243.02 days is with respect to distant stars. With a little arithmetic (taking inverses) we can easily convert that value to the rotation period with respect to the Sun or, in other words, to the day on Venus. It is 116.75 (Earth) days. Five of those periods equal 583.75 days, which is almost the same as the 583.92 day synodic period. So each time the planets line up Venus shows almost the same face to the Sun and hence the same face to the Earth, which is always on those occasions on the opposite side of Venus.

Coincidence or not
As Lionel points out it is interesting that transits of Venus repeat in a cycle of 243 years while the rotation period of Venus with respect to the stars is 243 days, The above detailed discussion indicates that there is no obvious connection that gives rise to the same number in each case. However, the calculations all depend on many of the same factors such as the orbital periods of Venus and the Earth so maybe there was a chance that the same number should recur.

Note the values quoted above are from the NASA Venus Fact Sheet.

Source: Are transits and the rotation of Venus linked? – Observations
– – –
Talkshop notes

Re: ‘Five of those periods equal 583.75 days, which is almost the same as the 583.92 day synodic period.’ [‘Venus and the Earth line up with the Sun every 583.92 days or 1.59872 years’]

Note 1: 23 solar rotations @ 25.38 days = 583.74 days
This also looks like a resonance, this time between the Sun and the Venus day.
. . .
Re: Venus and Earth fulfil these requirements after five synodic periods = 7.9936 years
A more accurate relationship occurs after 152 synodic periods = 243.00544 years or ~395 Venus years.

Note 2: using their own data, 157 synodic periods is more accurate, i.e. closer to a whole number of Earth orbits.
1.59872 * 152 = 243.00534 years (as stated in their notes)
1.59872 * 157 = 250.99904 years (~408 Venus years)
Of course that would be an ‘extra’ five synodic periods = 7.9936 years.

That may contradict the official ‘wisdom’ but there it is. It was discussed in some detail in this 2015 Talkshop post (some readers may find the comments to be of interest):
Why Phi? – a Venus transit cycle model

Why Phi? – the rainbow angle

Posted: September 3, 2017 by oldbrew in Maths, Measurement, Phi, weather
Tags:

The rainbow angle [credit: Hong Kong Observatory]


The minimum deviation angle for the primary bow [of a rainbow] is 137.5° according to Wikipedia. This is known as the rainbow angle. A circle is 360 degrees, so the ratio of the rainbow angle to the circle is therefore the square of the golden ratio i.e. 137.5:360 = 1:2.61818~.
– – –
Hong Kong Observatory has some useful explanatory text and graphics (rounding 137.5 to 138 degrees) titled:
Why is the region outside the primary rainbow much darker than that inside the primary rainbow?
Written by : SIU Kai-chee (summer intern) and HUNG Fan-yiu

Let’s first look at Figure 1, which shows sun rays entering a water drop and going through refraction and reflection.

The ray (ray no. 1) passing through the centre goes directly backward on reflection, i.e. a change in direction of 180 degrees.

For ray no. 2, this angle becomes smaller, following the rules of refraction and reflection.

For the next (ray no. 3) the angle continues to decrease, so on and so forth. This trend does not continue for long, however.

(more…)

Some Pythagorean triples [credit: Cmglee / Wikipedia]


Could Babylonian base-60 maths be about to make a comeback? The tablet has been dated to between 1822 and 1762 BC and is based on Pythagorean triples, as Phys.org reports. It uses ‘a novel kind of trigonometry based on ratios, not angles and circles’.

UNSW Sydney scientists have discovered the purpose of a famous 3700-year old Babylonian clay tablet, revealing it is the world’s oldest and most accurate trigonometric table, possibly used by ancient mathematical scribes to calculate how to construct palaces and temples and build canals.

The new research shows the Babylonians beat the Greeks to the invention of trigonometry – the study of triangles – by more than 1000 years, and reveals an ancient mathematical sophistication that had been hidden until now.

Known as Plimpton 322, the small tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, academic, diplomat and antiquities dealer Edgar Banks, the person on whom the fictional character Indiana Jones was based.

(more…)

The Catch-22 of Energy Storage

Posted: July 27, 2017 by tallbloke in Analysis, Energy, Maths, wind
Tags:

.

H/T @hockeyschtick1 for this great article on the non-viability of wind/solar as large-scale replacement for fossil/nuclear. Now can we scrap the CCA please?

 

Brave New Climate

Pick up a research paper on battery technology, fuel cells, energy storage technologies or any of the advanced materials science used in these fields, and you will likely find somewhere in the introductory paragraphs a throwaway line about its application to the storage of renewable energy.  Energy storage makes sense for enabling a transition away from fossil fuels to more intermittent sources like wind and solar, and the storage problem presents a meaningful challenge for chemists and materials scientists… Or does it?


Guest Post by John Morgan. John is Chief Scientist at a Sydney startup developing smart grid and grid scale energy storage technologies.  He is Adjunct Professor in the School of Electrical and Computer Engineering at RMIT, holds a PhD in Physical Chemistry, and is an experienced industrial R&D leader.  You can follow John on twitter at @JohnDPMorgan First published in Chemistry in Australia .

Several recent analyses of the…

View original post 1,723 more words

Ned and Karl have finally got some big exposure to the general public for their paradigm shifting breakthrough in geo and astro-physics. World Net Daily front page stories are read by over a million people. This is a great step forward for recognition of their work.
ned-karl-wnd3

Study blows Greenhouse Theory out of the water

7-9-2017 By Alex Newman for World Net Daily

BOZEMAN, Mont. – A new scientific paper contends the entire foundation of the man-made global-warming theory – the assumption that greenhouse gases warm the atmosphere by trapping heat – is wrong.

If confirmed, the study’s findings would crush the entire “climate change” movement to restrict CO2 emissions, the authors assert

Some experts contacted by WND criticized the paper, while others advised caution.

Still others suggested that the claimed discovery represents a massive leap forward in human understanding – a “new paradigm.”

The paper argues that concentrations of CO2 and other supposed “greenhouse gases” in the atmosphere have virtually no effect on the earth’s temperature.

(more…)

N-KFig_4

Back in late 2011, the Talkshop splashed the story on a ‘Unified Theory of Climate’  developed by PhD physicists Ned Nikolov and Karl Zeller. They set out to show that the ‘greenhouse effect’ is not a phenomenon arising out of the absorption and reemission of outgoing long-wave radiation by the atmosphere (as thought for 190 years), but is a form of compression heating controlled by solar radiation and the total atmospheric pressure at the Earth’s surface. Pressure is in turn a product of the gas mass contained in a column of air above a unit surface area, and the planet’s gravitational effect on that mass.

It’s been a long and treacherous road involving many revisions and refinements of the original study. On several occasions the manuscript was rejected unread, but Ned and Karl have finally got their greatly improved and expanded paper published. This latest version is a tour de force strengthened by the rigors of criticism from an army of peer reviewers at several journals along the way.

Using dimensional analysis (a classical technique for inferring physically meaningful relationships from measured data), they show that the long-term global equilibrium surface temperature of bodies in the solar system as diverse as Venus, the Moon, Earth, Mars, Titan and Triton can accurately be described using only two predictors: the mean distance from the Sun and the total atmospheric surface pressure. This type of cross-planetary analysis using vetted NASA observations has not been conducted by any other authors. It represents the first and only attempt in the history of climate science to assess Earth’s surface temperature in the context of a cosmic physical continuum defined by actual planetary-scale observations. The result is a new insight that planetary climates are independent of the infrared optical depth of their atmospheres arising from their composition, and that the long-wave ‘back radiation’ is actually a product of the atmospheric thermal effect rather than a cause for it.

dimensional

(more…)

shining_sun

With sadness, I’m sharing the news that my Talkshop co-blogger Tim Channon passed away on Friday. Tim had been bravely battling with cancer for some time, and was still upbeat and lively-minded when I spoke with him last week. Since then unfortunately, medical complications set in.

Tim was one of a kind. A humorous, thoughtful and technically brilliant individual. His contribution to our understanding of cyclic phenomena through the analysis software he wrote propelled me into my own research. His patient recording of weather data and survey of UK weather stations demonstrate the depth of interest and passion he had for bringing facts to bear on the climate debate. His dedication, skill and good natured rebukes against uninformed speculation and bad theory puts him in the Pantheon of great sceptical thinkers and scientists.

Tim will be missed and remembered.

_____________________________________

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Saturn and the lunar year

Posted: January 28, 2017 by oldbrew in Maths, solar system dynamics
Tags:

Comparison of Saturn and Earth [image credit: NASA]

Comparison of Saturn and Earth [image credit: NASA]


In a recent post: Sidorenkov and the lunar or tidal year we were looking at the match between tropical years and periods of 13 lunar months (i.e. the lunar, or tidal, year):
353 tropical years = 363 tidal years (where 1 lunar year = 13 lunar tropical months)

Here we want to see if Saturn links to the lunar year.
From the JPL ephemeris [target body: Saturn] we have:
Saturn orbit period = 10755.698 days

Jupiter-Saturn-Earth orbits  chart

Jupiter-Saturn-Earth orbits chart


From another post we produced a chart [right] based on 85 Saturn orbit periods:
85 x 10755.698 days = 914234.33 days

One tidal year = 13 x 27.321582 days = 355.18056 days
914234.33 / 355.18056 = 2573.9987 tidal years (2574)
So 85 Saturn orbits = 2574 tidal years

Since 2574 is divisible by 6 (= 429) we can use the chart to say:
403 Saturn-Earth conjunctions (S-E) = 429 tidal years
or, dividing by 13:
31 S-E = 33 tidal years
therefore, multiplying by 11:
341 S-E = 353 tropical years = 363 tidal years (the original match, see above)
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[click on image to enlarge]

[click on image to enlarge]


Another one to add to the ‘how and why did they do that?’ list of ancient sites. Years of research lie ahead.

Imagine you are about to plan and construct a building that involves several complicated geometrical shapes, but you aren’t allowed to write down any numbers or notes as you do it. For most of us, this would be impossible.

Yet, new research from Arizona State University has revealed that the ancient Southwestern Pueblo people, who had no written language or written number system, were able to do just that – and used these skills to build sophisticated architectural complexes, reports Phys.org.

Dr. Sherry Towers, a professor with the ASU Simon A. Levin Mathematical, Computational and Modeling Sciences Center, uncovered these findings while spending several years studying the Sun Temple archaeological site in Mesa Verde National Park in Colorado, constructed around A.D. 1200.

“The site is known to have been an important focus of ceremony in the region for the ancestral Pueblo peoples, including solstice observations,” Towers says. “My original interest in the site involved looking at whether it was used for observing stars as well.”

However, as Towers delved deeper into the site’s layout and architecture, interesting patterns began to emerge.
(more…)

Sidorenkov and the lunar or tidal year

Posted: November 27, 2016 by oldbrew in climate, Cycles, Maths, moon
Tags: ,

Credit: reference.com

Credit: reference.com


This is an attempt to understand via the numbers the concept proposed by Russian researcher Sidorenkov of a lunar year interacting with the terrestrial year to produce an effect of a ‘quasi-35 year’ climate cycle.

Sidorenkov in his paper ON THE SEPARATION OF SOLAR AND LUNAR CYCLES says:

The lunisolar tides repeat with a period of 355 days,
which is known as the tidal year. This period is also
manifested as a cycle of repeated eclipses. Meteorological
characteristics (pressure, temperature, cloudiness, etc.)
vary with a period of 355 days. The interference of these
tidal oscillations and the usual annual 365-day oscillations
generates beats in the annual amplitude of meteorological
characteristics with a period of about 35 years (Sidorenkov
and Sumerova, 2012b). The quasi 35-year variations in
cloudiness lead to oscillations of the radiation balance
over terrestrial regions. As a result of these quasi-
35-year beats, the climate, for example, over European
Russia alternates between “continental” with dominant
cold winters and hot summers (such as from 1963 to 1975
and from 1995 to 2014) and “maritime” with frequent
warm winters and cool summers (such as from 1956 to
1962 and from 1976 to 1994)

(more…)

Tim writes,

Published 27th February 2016 Climate Audit published a small article in honour of Kreiss to do with atmospheric modelling, the hydrostatic balance… or not.

Comments follow from Names. I think the comments are worthy of a read and decode by Talkshop readers who have an interest in this kind of topic.

Gerry Browning: In Memory of Professor Heinz Kreiss

Gerry Browning writes:

The Correct System of Equations for Climate and Weather Models

(more…)

My thanks to talkshop contributor and PRP author R.J. Salvador for sending me an updated prediction for changes in LOD during 2016. This plot has been produced using R.J.s model, which has been developed using the planetary periodicities we have been working on here at the talkshop over the last few years.

Updated LOD Forecast

R.J. has kindly agreed to send a monthly update showing the progress of the model output against IERS observations as the year progresses. This is real science in progress. Creating a hypothesis, building a model, and testing it against reality.

(more…)

Phi and the Great Pyramid of Khufu

Posted: November 19, 2015 by oldbrew in Maths, Measurement, Phi
Tags:
Great Pyramid of Giza from a 19th-century stereopticon card photo [credit: Wikipedia]

Great Pyramid of Giza from a 19th-century stereopticon card photo [credit: Wikipedia]

Let’s have a look at some numbers for the Great Pyramid.

Source: Building the Great Pyramid (aka Cheops)
Copyright 2006 Franz Löhner and Teresa Zuberbühler

Dimensions as designed (in Egyptian royal cubits):
Length: 440
Height: 280
Slope: 356

Original dimensions as built (a,h and c in the pyramid diagram below):
Length: 230.36m (half = 115.18m)
Height: 146.59m
Slope: 186.42m

(more…)

Santorini_moon2
What is a Saros? Quoting Wikipedia:
‘One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur’

‘It takes between 1226 and 1550 years for the members of a saros series to traverse the Earth’s surface from north to south (or vice-versa)’

Only a few lines to go … (more…)

One equation for earth temperature

Posted: November 29, 2014 by tchannon in atmosphere, Gravity, Maths

At The Hockey Schitick MS has posted a brave article

The Greenhouse Equation

This seems to be a culmination of a series of articles.

[UPDATE: and another article showing a fit against Standard Atmosphere http://hockeyschtick.blogspot.com/2014/11/the-greenhouse-equation-predicts.html /UPDATE]

Image

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Nicola Scafetta has emailed me to let us know he has a new paper in press which adresses critiques of our solar-planetary theory. I can’t do justice to presenting this work by illustrating this post with figures from the paper using my cellphone, but this a seriously impressive piece of work which Nicola generously shares with Talkshop readers via a link below the break. Nicola writes:

I just would like to share my latest paper
 
Nicola Scafetta, 2014. Discussion on the spectral coherence between planetary, solar and climate  oscillations: a reply to some critiques.

Astrophysics and Space Science in press.

http://link.springer.com/article/10.1007%2Fs10509-014-2111-8

For those who followed this research, the paper strongly rebuts some interesting critiques of the planetary theory of solar and climate variation made by Holm andCauquoin et al. that emerged in the literature during the first months of the 2014. (It also rebuts the very improper and unprofessional criticism made by Anthony Watts)

(more…)

Solar system planets [image credit: BBC]

Solar system planets by size
[image credit: BBC]

Using a simple formula we’re now able to see the link between Jupiter and Saturn orbits. As in the original giant planets update post the numbers are quite large but easily connected to the ‘Why Phi’ concept.

Using the orbit period of each planet we derive the synodic period, i.e. the time taken for the path of faster planet Jupiter to ‘catch up’ with slower planet Saturn. If we call the orbit periods J and S and the synodic period J-S we can make two calculations:
(360 / S) x J-S and (360 / J) x J-S.
The difference between the two results should be 360.

(more…)

Why Phi: giant planets update

Posted: October 26, 2014 by oldbrew in Maths, solar system dynamics

Solar system planets [image credit: BBC]

Solar system planets by size
[image credit: BBC]

The Why Phi series started with a discussion around ‘Relations between the Fibonacci Series and Solar System Orbits’ (link below)

Now NASA has updated its planetary data as of 20th October 2014. With the revised numbers (see below for source) we have to re-calculate the orbital relationships of the giant planets.

Amending the original numbers we find:
89 Jupiter-Saturn conjunctions = almost 1768 years
In Fibonacci numbers: 1768 = 34 x 13 x 2²

And there’s more…

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