## Why Phi? – resonant moons of Uranus

Posted: April 7, 2018 by oldbrew in Astronomy, Fibonacci, Phi, solar system dynamics
Tags: , , , 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.
– – –
See also: Why Phi? – Moons of Pluto

Exoplanet system: Trappist-1: seven-planet resonance chain confirmed

1. oldbrew says:

Turning to the outer two of the ‘big five’ Uranus moons:
113 Titania-Oberon conjunctions = 58 days * 48
~58 days is the period identified in the post above.

However, that would not be an exact number of orbits of Uranus for either of these moons.
For that it would be 283 Titania = 183 Oberon = 100 T-O (283 – 183) conjunctions.

But 22 Titania-Umbriel = 58 days * 3 (14 * 3 = 42 Umbriel = 20 Titania).

2. tallbloke says:

Nice work OB. Another 3:2:1 to add to the growing collection.