Why Phi? – the resonance of Jupiter’s 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.

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.

This brief youtube video repeats the slight 4:2:1 error.