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%)

Kepler-223 has another resonance: 2 d = 3 c (> 99.88%)

That means 8 b = 6 c = 4 d = 3 e (orbits)

The equivalent Saturnian moon numbers are: 32 Mi = 22 En = 16 Te = 11 Di

(For the moons to match K-223 exactly, the orbit numbers would have to be 32,24,16 and 12)

Turning to the syzygies, we can create charts to show the patterns:

Saturn chart

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The first three moons have the synodic ratio (yellow underlines):

6 Te-En = 10 En-Mi = 16 Mi-Te which is a 3:5:8 ratio (3,5 and 8 are Fibonacci numbers).

Accuracy 99.05 – 99.75%

Kepler-223 chart

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The planets c,d, and e have the synodic ratio (yellow underlines):

1 d-e = 2 c-d = 3 c-e (1,2 and 3 are Fibonacci numbers).

Accuracy 99.71 – 99.81%

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*Related*

Phi is nature’s way of pulling order from chaos. It is the counter to entropy that has created everything from atoms to Life.

I’ve elaborated my speculation on phi in the form of a conversation – with myself, if you like.

http://brindabella.id.au/ftp/GoldenRatioAndPhi-bit.pdf

Dai – yes, some examples here…

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#section3

See also: KAM theory.

https://tallbloke.wordpress.com/2017/05/10/kam-theorem-and-its-application-to-the-stability-of-the-solar-system-why-phi/

https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem

– – –

From the Kepler-223 chart we also get:

2 b-c = 3 c-e = 5 b-e

b,c and c,e are ‘neighbour’ planets and b,e are not. The non-neighbour pair is always the sum of the two neighbours.

In the Fibonacci series any two consecutive F numbers always sum to the next one in the series, e.g. 2+3=5.

Turning to b,c and d the chart shows:

2 b-c = 2 c-d = 4 b-d

therefore

1 b-c = 1 c-d = 2 b-d

The Fibonacci series starts like this: 0,1,1,2,3,5,8,13 etc.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html

African violet (from the first link) – the positions of the numbers on the right show maximum leaf exposure to sunlight and rain.

more at:

http://blog.world-mysteries.com/science/nature-fibonacci-numbers-and-the-golden-ratio/

You can see phi as fibonacci relative to year of 365 days with Venus where the synod is 8/5 and the orbit is 8/13. If you follow this trough you see that the solar year effectively reciprocates the orbit and the GM has that unique property of reciprocated without losing its fractional part.

Other causes of resonance are whole numbers, the most perfect being musical interval ratios, seen commonly between moons of giants (famously Jupiter’s naked eye moons)

Since my Matrix of Creation (2002), the musical resonance of the two visible outer planets, Jupiter and Saturn, as 9/8 and 16/15 relative to the lunar year i.e. whole and semitone intervals, took until 2014 to place properly in ancient near eastern tuning theory (see my Harmonic Origins of the World).

Obviously Phi relations derive from its fractional uniqueness whilst musical ratios derive from integer relations within resonance providing a stable state for the group of moons.

Artist’s impression of Saturn, its rings and major icy moons—from Mimas to Rhea

A number of features in Saturn’s rings are related to resonances with Mimas. Mimas is responsible for clearing the material from the Cassini Division, the gap between Saturn’s two widest rings, the A Ring and B Ring. Particles in the Huygens Gap at the inner edge of the Cassini division are in a 2:1 orbital resonance with Mimas. They orbit twice for each orbit of Mimas. The repeated pulls by Mimas on the Cassini division particles, always in the same direction in space, force them into new orbits outside the gap. The boundary between the C and B ring is in a 3:1 resonance with Mimas.https://en.wikipedia.org/wiki/Mimas_(moon)#Orbital_resonances

Mimas clearing the Cassini Division has echoes of Jupiter clearing the Kirkwood Gaps.

See: https://en.wikipedia.org/wiki/Kirkwood_gap