Archive for the ‘Fibonacci’ Category

Jupiter – the dominant planet in the solar system

The aim here is to show a Lucas number based pattern in five rows of synodic data, then add in a note on Mercury as well.

There’s also a strong Fibonacci number element to this, as shown below.

The results can be linked back to earlier posts on planetary harmonics involving the Lucas and Fibonacci series (use ‘search this site’ box on our home page).


Continuing our recent series of posts, with Uranus-Neptune conjunction data an obvious starting point for the table is where the difference between the number of Neptune orbits and U-N synods is 1.

647 U-N takes a long time (~110,900 years) but the accuracy of the whole number matches is very high.

Lucas no. (7 here) is fixed, and Fibonacci nos. follow the correct sequence (given their start no.).
Full Fib. series starts: 0,1,1,2,3,5,8,13,21…etc.
Multiplier: 0,1,1,2,3
Addition: 1,1,2,3,5

The Neptune orbits are multiples of 26 with the same Fibonacci adjustment:
Add 0,1,1,2,3 to the Neptune column numbers to get an exact multiple of 26 (which will be the pattern number in the last column).


Distances not to scale.

This is an easy data table to interpret.

The Uranus orbits are all Fibonacci numbers, and the synodic conjunctions are all a 3* multiple of Fibonacci numbers.
[Fibonacci series starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …etc.]

In addition, the difference between the two is always a Lucas number. And that’s it for Saturn-Uranus, which would make for a very short blog post.

But it’s possible to go further.


We’re now looking for a pattern arising from the Jupiter-Saturn synodic conjunctions and the orbit periods.

Focussing on the numbers of Jupiter orbits that are equal, or nearly equal, to an exact number of Saturn orbits (years), a pattern can be found by first subtracting the number of conjunctions from the number of Saturn orbits.


A simple pattern emerges when looking at the Earth-Mars synodic conjunctions.

Focussing on the numbers of Mars orbits that are equal, or almost equal, to an exact number of Earth orbits (years), the pattern can be found by subtracting the number of conjunctions from the number of Mars orbits.

The difference between the two sets of numbers follows the Fibonacci series, which is strongly related to the golden ratio.


Jupiter – the dominant planet in the solar system

The aim here is to show a Lucas number based pattern in seven rows of synodic data.
There’s also a Fibonacci number element to this, as shown below.
The results can be linked back to an earlier post on planetary harmonics (see below).

The nearest Lucas number equation leading to the Jupiter orbit period in years is:
76/7 + 1 = 11.857142 (1, 7 and 76 are Lucas numbers).
The actual orbit period is 11.862615 years (> 99.95% match).
[Planetary data source]

It turns out that 7 Jupiter orbits take slightly over 83 years, while 76 Jupiter-Earth (J-E) synodic conjunctions take almost exactly 83 years. One J-E synod occurs every 1.09206 years. (83/76 = 1.0921052).


Earth from the Moon [image credit: NASA]

Part 3

To recap, the Lucas series starts: 2, 1, 3, 4, 7, 11, 18, 29 … (adding the last two numbers each time to find the next number in the series).

Note: for clarity, the three parts of this mini-series should be read in order (links below).

Since Part 1 showed that 7 Jupiter-Saturn conjunctions (J-S) = 11 * 13 lunar tropical years (LTY), and from Part 2 we know that 363 LTY = 353 Earth tropical years (TY), these numbers of occurrences can be integrated by applying another multiple of 13:
363 = 3*11*11 LTY
353 * 13 TY = 3*11*11*13 LTY = 3*7*11 J-S

7 and 11 are Lucas numbers.
13 is a Fibonacci number.
3 belongs to both series.


Image credit: Wikipedia

In Part 1 the period of time in question was 13 lunar tropical years (LTY). Here we show how this relates to the Jupiter-Saturn conjunction and other significant periods.

13 LTY = 169 * 27.321582 days (lunar orbit period) = 4617.3473 days
1 Jupiter-Saturn conjunction = 19.865036 years * 365.25636 days = 7255.8307 days
[planetary data source]

Jupiter-Saturn-Earth orbits chart

This gives a ratio of exactly 11:7 as follows:
11 * 4617.3473 = 50790.82 days
7 * 7255.8307 = 50790.814 days
7 and 11 are Lucas numbers.

Multiplying by 3 (which is both a Fibonacci and a Lucas number), the results from Part 1 can now be ‘plugged in’ to the chart on the right from a previous blog post, which is based on multiples of 21 (3 * 7) Jupiter-Saturn conjunctions, as the chart on the right shows.

Quoting from that post:
The synodic periods all occur in multiples of six, and one sixth of 2503 years is 417.1666 years which is 21 J-S, 382 J-E, 403 S-E and two de Vries cycles.

Updating that, the matching periods now are:
2 de Vries cycles
21 Jupiter-Saturn conjunctions (3 * 7)
382 Jupiter-Earth conjunctions
403 Saturn-Earth conjunctions (13 * 31)
352 Chandler wobbles (11 * 32)
429 Lunar tropical years (11 * 39 or 13 * 33)
627 Venus rotations (11 * 57 or 19 * 33)

Therefore: 31 Saturn-Earth = 33 Lunar tropical years.

Quoting from Part 1 of the post:
353 Earth tropical years (ETY) = 363 Lunar tropical years = 10 beats

363 LTY = 33 * 11
Therefore: 363 LTY = 31 * 11 (341) Saturn-Earth conjunctions (= synodic periods).

Quoting from another earlier post – Sidorenkov and the lunar or tidal year:
4719 LM = 128930.54 days  [note: 4719 = 363 * 13 i.e. 363 LTY]
4366 SM = 128930.55 days
4727 CR = 128930.34 days
5080 SSR = 128930.40 days
353 TY = 128930.49 days

(see post re. abbreviations).

NASA’s Saturn Fact Sheet says re. Saturn-Earth:
Synodic period (days) 378.09

TY = tropical years
128930.49 days / 341 S-E = 378.09527 days
This ties Saturn to Sidorenkov’s 353 year period, which is therefore 11/13ths of 21 J-S.
Also: 11/13ths of 429 LTY = 363 LTY.

In the graphic the full period is 126 J-S, described as 6 * 21.
It could also be described as 7 * 18, which are Lucas numbers.


The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large [credit: Wikipedia]

Here we show numerical connections between the Moon, the Earth and Venus. These will be carried forward into part 2 of the post. The focus is on the smaller Lucas numbers (3-18).

Wikipedia says: The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values.

A look at the numbers:
19 Venus rotations = 169 (13²) lunar rotations
Lunar tropical year = 13 lunar rotations / orbits (1 rotation = 1 orbit)
So: 19 Venus rotations = 13 Lunar tropical years
(13 is a Fibonacci number. The Lunar tropical year is derived from the nearest whole number of lunar orbits to one Earth orbit.)

169 * 27.321582 = 4617.3473 days (Data source)
19 * 243.018 = 4617.342 days (Data source)

Now we bring in the Chandler wobble:
13*3 = 39
39 Lunar tropical years = 32 Chandler wobbles
19*3 = 57

Referring to the chart on the right:
7 and 18 are Lucas numbers.
This theme will continue in part 2 of the post.

(32 + 57 = 89 axial, and 89 is a Fibonacci number. In 1/89th of the period the sum of CW and Ve(r) occurrences is 1).

Re. the period of the Chandler wobble:
39 LTY / 32 CW = (169 * 3 * 27.321582) / 32 = 432.8763 days

Or, if we say 27 Chandler wobbles = 32 Earth tropical years:
(365.24219 * 32) / 27 = 432.8796 days

The two results are almost identical (Wikipedia rounds it to 433 days).

353 Earth tropical years (ETY) = 363 Lunar tropical years = 10 beats
1 beat = 35.3 ETY which is linked to the Chandler Wobble

These numbers also feed into part 2 of the post, with more planetary links.

Orbital (top line) and synodic relationships of Kepler-107, plus cross-checks

The system has four planets: b,c,d, and e.

The chart to the right is a model of the close orbital relationships of these four recently announced short-period (from 3.18 to 14.75 days) exoplanets.

It can be broken down like this:
b:c = 20:13
c:d = 13:8
d:e = 24:13 (= 8:13 ratio, *3)
b:d = 5:2
c:e = 3:1
(1,2,3,5,8, and 13 are Fibonacci numbers)

Image credit:

It turns out that the previous post was only one half of the lunar evection story, so this post is the other half.

There are two variations to lunar evection, namely evection in longitude (the subject of the previous post) and evection in latitude, which ‘generates a perturbation in the lunar ecliptic latitude’ (source).

It’s found that the first is tied to the full moon cycle and the second to the draconic year.


Why Phi? – a lunar evection model

Posted: November 16, 2018 by oldbrew in Fibonacci, moon, Phi, solar system dynamics
Tags: ,

Apogee = position furthest away from Earth. Earth. Perihelion = position closest to the sun. Moon. Perigee = position closest to Earth. Sun. Aphelion = position furthest away from the sun. (Eccentricities greatly exaggerated!)

Lunar evection has been described as the solar perturbation of the lunar orbit.

One lunar evection is the beat period of the synodic month and the full moon cycle. The result is that it should average about 31.811938 days (45809.19 minutes).

Comparing synodic months (SM), anomalistic months (AM), and lunar evections (LE) with the full moon cycle (FMC) we find:
1 FMC = 13.944335 SM
1 FMC = 13.944335 + 1 = 14.944335 AM
1 FMC = 13.944335 – 1 = 12.944335 LE

Since 0.944335 * 18 = 16.9983 = 99.99% of 17, and 18 – 17 = 1, we can say for our model:
18 FMC = 233 LE (18*13, -1) = 251 SM (18*14, -1) = 269 AM (18*15, -1)
See: 3 – Matching synodic and anomalistic months.

Three of Saturn’s moons — Tethys, Enceladus and Mimas — as seen from NASA’s Cassini spacecraft [image credit: NASA/JPL]

This is a comparison of the orbital patterns of Saturn’s four inner moons with the four exoplanets of the Kepler-223 system. Similarities pose interesting questions for planetary theorists.

The first four of Saturn’s seven major moons – known as the inner large moons – are Mimas, Enceladus, Tethys and Dione (Mi,En,Te and Di).

The star Kepler-223 has four known planets:
b, c, d, and e.

When comparing their orbital periods, there are obvious resonances (% accuracy shown):
Saturn: 2 Mi = 1 Te (> 99.84%) and 2 En = 1 Di (> 99.87%)
K-223: 2 c = 1 e (>99.87%) and 2 b = 1 d (> 99.86%)


Here we find a match between the orbit numbers of Jupiter, Saturn and Uranus and see what that might tell us about certain patterns in the solar system.

715 U = 60072.044 years
2040 S = 60072.895 years
5064 J = 60072.282 years
Data source: Nasa/JPL – Planets and Pluto: Physical Characteristics

The Jupiter-Saturn part of the chart derives directly from this earlier Talkshop post:
Why Phi? – Jupiter, Saturn and the de Vries cycle


A montage of Uranus’ large moons and one smaller moon: from left to right Puck, Miranda, Ariel, Umbriel, Titania and Oberon. Size proportions are correct. [image credit: Vzb83 @ Wikipedia (from originals taken by NASA’s Voyager 2)]

The five major moons of Uranus in ascending distance from the planet are:
Miranda, Ariel, Umbriel, Titania and Oberon

Of these, the first three exhibit a synodic resonance similar to that of Jupiter’s Galilean moons, as we showed here:
Why Phi? – the resonance of Jupiter’s Galilean moons

Quoting from that post:
The only exact ratio is between the synodic periods which is 3:2:1.
It isn’t necessary to have an exact 4:2:1 orbit ratio in order to get a 3:2:1 synodic ratio.


Cyclones in Jupiter’s atmosphere [image credit: NASA]

Octagon and pentagon (8:5) shapes at the poles, with groups of cyclones in a 9:6 (= 3:2) polar ratio. Fascinating.

Jupiter’s poles are blanketed by geometric clusters of cyclones and its atmosphere is deeper than scientists suspected, says

These are just some of the discoveries reported by four international research teams Wednesday, based on observations by NASA’s Juno spacecraft circling Jupiter.

One group uncovered a constellation of nine cyclones over Jupiter’s north pole and six over the south pole. The wind speeds exceed Category 5 hurricane strength in places, reaching 220 mph (350 kph).

The massive storms haven’t changed position much—or merged—since observations began.


Giant planets of the solar system [image credit:]

This post on the ice giants Uranus and Neptune follows on from this one:
Why Phi? – Jupiter, Saturn and the inner solar system

The main focus will be on Uranus. A planetary conjunction of three bodies (e.g. two planets and the Sun, in line) is also known as a syzygy.

Here’s the notation for the table shown below:
J-S = Jupiter-Saturn conjunctions
S-U = Saturn-Uranus conjunctions
U-N = Uranus-Neptune conjunctions

Each of the columns: U, S-U, J-S shows a Fibonacci progression.

Accuracy of best match is between 99.965% and 99.991%.

Quoting Wikipedia: ‘The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected.’
The Greek letter φ (phi) represents the golden ratio.


Exoplanet – NASA impression

YZ Ceti is a recently discovered star with three known planets (b,c and d) orbiting very close to it. Although some types of mean motion resonance, or near resonance, are quite common e.g. 2:1 or 3:2 conjunction ratios, this one is a bit different.

The orbit periods in days are:
YZ Ceti b = 1.96876 d
YZ Ceti c = 3.06008 d
YZ Ceti d = 4.65627 d

This gives these conjunction periods:
c-d = 8.9266052 d
b-c = 5.5204368 d
b-d = 3.4109931 d
(Note the first two digits on each line.)

Nearest matching period:
34 c-d = 303.50457 d
55 b-c = 303.62403 d
89 b-d = 303.57838 d

34,55 and 89 are Fibonacci numbers.
Therefore the conjunction ratios are linked to the golden ratio (Phi).

Phi = 1.618034
(c-d) / (b-c) = 1.6170106
(b-c) / (b-d) = 1.618425

Data source:

From left, Mercury, Venus, Earth and Mars. [Credit: Lunar and Planetary Institute]

The planetary theory aspect appears a bit later, but first a brief review of some relevant details.

In this Talkshop post: Why Phi? – a triple conjunction comparison we said:
(1) What is the period of a Jupiter(J)-Saturn(S)-Earth(E) (JSE) triple conjunction?
JSE = 21 J-S or 382 J-E or 403 S-E conjunctions (21+382 = 403) in 417.166 years (as an average or mean value).

(2) What is the period of a Jupiter(J)-Saturn(S)-Venus(V) (JSV) triple conjunction?
JSV = 13 J-S or 398 J-V or 411 S-V conjunctions (13+398 = 411) in 258.245 years (as an average or mean value).

Since JSV = 13 J-S and JSE = 21 J-S, the ratio of JSV:JSE is 13:21 exactly (in theory).

As these are consecutive Fibonacci numbers, the ratio is almost 1:Phi or the golden ratio.

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: 62 year
[see Google site search box in grey zone on left of this web page]