## Why Phi? – the resonance of Jupiter’s Galilean moons

Posted: November 26, 2015 by oldbrew in Fibonacci, Phi, solar system dynamics
Tags: ,

Io, Europa and Ganymede – three of Jupiter’s four Galilean moons

The resonance of three of the four Galilean moons of Jupiter is well-known. Or is it?

We’re usually told there’s a 1:2:4 orbital ratio between Ganymede, Europa and Io, but while this is not far from the truth, a closer look shows something else.

The three moons in question – in order from Jupiter – are Io, Europa and Ganymede.
There’s a clear synodic pattern, as this animation shows.

The numbers below refer to orbits of Jupiter, by these three moons:
68 Ganymede = 137 Europa = 275 Io (= 486.51~ days)

Analysing for patterns we find each number increment is ‘double plus one’:
68 Ga = 68 x 2, +1 Eu (= 137)
137 x 2, +1 Eu = 275 Io

With the Ga-Io numbers we find:
68 Ga = 275 Io
68 = 34 x 2
275 = 55 x 5

So this Ganymede-Io conjunction cycle is based on Fibonacci numbers only (2,5,34 and 55).
With these numbers we can resolve the three-body synodic relationship.

Since the ratio of the two consecutive Fibonacci numbers 34 and 55 is almost 1:Phi we can say:
1 Ganymede orbit = 55/34 (~Phi) x 5/2 (= 4.04412~) Io orbits.

Diagram of orbits and synodic periods

This is slightly more than the often quoted 1:4 orbital ratio of these two bodies, which must be inaccurate because division of the two orbit periods (Ganymede/Io) returns 4.0441~, as does 275/68 (4.0441176).

The synodic conjunction numbers derive from the difference in numbers of orbits:
137 Eu – 68 Ga = 69 Eu-Ga conjunctions
275 Io – 137 Eu = 138 Io-Eu (= 69 x 2)
275 Io – 68 Ga = 207 Io-Ga (= 69 x 3)
That confirms the 3:2:1 synodic ratio of the conjunctions.

NB Wikipedia calls it a 4:2:1 ratio but they are referring to the number of orbits, which we find is nearer to 275:137:68.
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.
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This brief youtube video repeats the slight 4:2:1 error.

1. tallbloke says:

Nice work OB. I’ve edited to add the animation.

This confirms what we’ve been saying for some time. It’s the resonant conjunction periods which determine the orbits, not the other way round. That insight gives us a clue as to how we should approach the multi-body problem to discover how the relationship is maintained.

Energy MUST be transferred between the orbital angular momenta of the three bodies for them to stay in synch. That means they MUST be affecting each other’s semi major axes wrt Jupiter.

This is the most clear cut case in the whole system, and the one we must model successfully from first principles in order to unlock the secret of the phi synchronisation of the entire solar system that I laid out in a general overview recently.

2. oldbrew says:

Bear in mind the animation numbers (4:2:1) are NOT accurate , as the post explains.

We say: ’68 Ganymede = 137 Europa = 275 Io (= 486.51~ days)’

8 times this period, less one Europa orbit (i.e. 1095 Europa) = 233 Callisto (the ‘other’ Galilean moon).
1095 Eu x 3.551181d = 3888.5431d
233 Ca x 16.689018d = 3888.5411d

233 is a Fibonacci number.

Update: 233 Callisto is one less Europa orbit (than 8 times the period), 2 less Io orbits (2198 Io = 55 x 40, -2), and half a Ganymede orbit less.

So for Ganymede we need to double the period to get a whole number of orbits.
466 Callisto (233 x 2) = 1087 Ganymede (34 x 32, -1).
1087 – 466 = 621 Ca-Ga conjunctions = 69 x 9 (in just under 16 times the original period, i.e. one Ganymede orbit less: 68 x 16 = 1088).

Note that 69 is also the number used in the original calculation of the 3:2:1 conjunction ratio of the ‘inner three’ Galilean moons.

3. Paul Vaughan says:

I think you guys might like contemplating the fit of your explorations into this aspiring unification:

Evidence of synchronous, decadal to billion year cycles in geological, genetic, and astronomical events (2014)

h/t to Bill Howell for pointing it out.

4. oldbrew says:

PV: in Table 1 of your link the first period is lunar (115 years), and the rest look like multiples of Uranus-Neptune conjunctions, i.e. from 344y onwards (or 172 for the half cycles) just keep multiplying by 3. [1 U-N = 171.4~y]

5. […] Kepler-223’s planets also are in resonance. Planets are in resonance when, for example, every time one of them orbits its sun once, the next one goes around twice. Jupiter’s moons, where the phenomenon was discovered, display resonance [as the Talkshop discussed here]. […]