# Why Phi? – resonant moons of Uranus

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.

Orbit periods of the relevant Uranus moons:
Miranda = 1.413479 days [* 41 = 57.952639 d]
Ariel = 2.520379 d [* 23 = 57.968717 d]
Umbriel = 4.144177 d [* 14 = 58.018478 d]
Source: Wikipedia

As a very close approximation: 41 Miranda = 23 Ariel = 14 Umbriel
The number of conjunctions (syzygies) in a given period of any two moons is the difference between the number of their respective orbits of the parent body (in this case: Uranus) in that period.

Synodic periods (average time between syzygy with Uranus):
Miranda-Ariel = 3.21845 d [* 18 = 57.9321 d]
Ariel-Umbriel = 6.432386 d [* 9 = 57.891474 d]
Miranda-Umbriel = 2.145132 d [* 27 = 57.918564 d]

27 M-U (41-14) = 18 M-A (41-23) = 9 A-U (23-14)
therefore, dividing by 9:
3 M-U = 2 M-A = 1 A-U

Note that 81 (9²) Uranus rotations take 58.185 days, only slightly more than the time for the three moons to complete their 41:23:14 synchronous orbits/rotations.
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