GEO 287 (Feb. 16, 1996):

EVOLUTION OF ORBITAL RESONANCES WITH TIME:
THE URANIAN SATELLITES EXAMPLE

* Equations describing evolution of satellite mean motion
The mean motion (n) describing a satellite orbit changes with time t as a result of tidal dissipation:

where n = satellite's mean motion = 2/P (P = orbital period)
(also called "orbital frequency" or "orbital angular velocity")

k2 = planet's Love number =

Q = tidal dissipation parameter of planet
= rigidity of satellite
M,m = mass of planet, satellite
R,r = radius of planet, satellite
,g = density, gravity of satellite

As Q increases, tidal dissipation lessens, and dn/dt decreases.

* Orbital Resonances
For Io (J1), Europa (J2), and Ganymede (J3):

n1 - 3n2 + 2n3 = 0

describes the Laplace resonance, which forces Io's eccentricity and gives rise to that satellite's volcanoes.

Miranda and Ariel are relatively small bodies, but they show evidence for past endogenic activity. The heat for this activity likely was tidal. Also, Miranda has a high eccentricity, and Miranda, Ariel, and Umbriel all have high inclinations, suggesting present or past orbital interactions.

For the Uranian satellites Miranda (U5), Ariel (U1), Umbriel (U2), Titania (U4), and Oberon (U5), there are currently no resonances of the form:

j5n5 + j1n1 + j2n2 + j3n3 + j4n4 + j5n5 = 0

where the sum of the coefficients must be zero for a stable resonance (see: Squyres, S.W., R.T. Reynolds, and J.J. Lissauer 1985, The enigma of the Uranian satellites' orbital eccentricities, Icarus 61, 218-223).

* How can Uranian satellites be explained?
The satellites may have moved through temporary resonances as their orbits evolved outward due to tidal dissipation.

Peale, S.J. 1988, Speculative histories of the Uranian satellite systems, Icarus 74, 153-171:
Peale (1988) finds that Ariel and Umbriel once may have been involved in a 2:1 resonance. If Miranda's mass is near the upper limit of its error, then Miranda, Ariel, and Umbriel may have been involved in a Laplace-type resonance. The actual capture into and escape from resonance, however, is likely complex and chaotic.

Dermott, S.F., R. Malhotra, and C.D. Murray 1988, Dynamics of the uranian and saturnian satellite systems: A chaotic route to melting Miranda?, Icarus 76, 295-334:
Increase in the satellite orbital radii over time could have caused past resonances of the uranian (and saturnian) satellites' orbits. These resonances would not have been well separated, so chaotic motion would have resulted, and has been simulated numerically. A 3:1 resonance between Miranda and Umbriel may explain Miranda's unusual orbital parameters and geology.

* How might Miranda have been heated?
Greenberg, R., S.K. Croft, D.M. Janes, J.S. Kargel, L.A. Lebofsky, J.I. Lunine, R.L. Marcialis, H.J. Melosh, G.W. Ojakangas, and R.G. Strom 1991, Miranda, in Uranus (J.T. Bergstralh, et al., eds.), pp. 693-735:

Miranda likely formed in a nebula of temperature ~80 K. How can its temperature have risen high enough to have caused its endogenic activity?

1. Accretional heating:
can raise temperature only ~30 K.