Orbits of Jupiters moons. Maths question.

[imhimmelkommter]

A math question. How many years would it take to complete a cycle of all four moons of Jupiter back into conjunction, coming back to the same position? There must be a formula out there that can work this out. I am very curious what that sidereal cycle would be. clear skies, Michael

[Tropo-Bob]

I am not sure that they are in the perfect alignment shown in the attachment.
However, if they are, the answer is 750.099 earth days.

This is derived by multiplying the orbital periods of each of the 4 moons.
That is Io's (1.769) x Europa (3.551) x Ganymede (7.155) x Callisto (16.689).

[imhimmelkommter]

Thank you Bob, would calculus apply here, to get an answer. I ask this so I can post it on some maths Facebook groups.

[Tropo-Bob]

Ha, I wish to change my answer.

I was thinking of how common denominators would affect the answer during lunch and I suddenly realised that Jupiter's 3 inner moons were said to be in a fixed orbital resonance.

See: http://www.planetary.org/multimedia/space-images/charts/orbital-resonances-of-galilean-moons.html

If this is strictly true (I find this confusing because when I do the maths, the times are slightly out); then the 3 inner moons repeat their same pattern every orbit of Ganymede. (also note from the website illustration that they never seem to line up in a straight line!)

Anyway, the resonance means that one would only multiply Ganymedes orbital time by Callisto's to obtain the answer: 7.155 *16.689 = 119.409 days.

Maybe... because it also depends on frames of reference. Sidereal cycle was mentioned, but I have ignored any affect of Jupiter's 12 year orbit around the Sun, as that would do my head in!