Orbital Resonances

With the announement of GJ 876 e and whatnot, I've been studying mean motion resonances.

If I understand this right, for Io and Europa, the resonance is given as
φ_{Io, Eu (Io)} = λ_Io - 2λ_Eu + ω_Io
(I use ω, perhaps inappropriately, for the precession of the longitude of periapsis since I can't do ω + a dot.)

and if I've got this right so far, that is in the reference frame of Europa's orbit, while for the reference frame of Io's orbit,
φ_{Io, Eu (Eu)} = λ_Io - 2λ_Eu + ω_Eu
applies.

Then there's this angle that subtracts the first two to get
φ_{Io, Eu} = φ_{Io, Eu (Io)} - φ_{Io, Eu (Eu)} = ω_Eu - ω_Io

· What is this exactly? It this a measure of the rate at which the two periapsis move apart?

For a 4:1 resonance, there's apparently a much larger set of equations. I took these from the paper describing the discovery of Gliese 876 e.

φ_ce0 = λ_c - 4λ_e + 3ω_c
φ_ce1 = λ_c - 4λ_e + 2ω_c + ω_e
φ_ce2 = λ_c - 4λ_e + ω_c + 2ω_e
φ_ce3 = λ_c - 4λ_e + 3ω_e
φ_ce = ω_c - ω_c

I haven't been yet able to understand why there are as many angles for a 4:1 resonance, what exactly they all describe, and where the coefficients on the periapsis precessions come from.

Help D:?

This article has the details on how to obtain the resonant angle for a given 2-body mean motion resonance.

First you need to find the values of p and q so that the resonance is given by (p+q):p. Then the resonant angle is given by

θ = (p + q)λ₂ - pλ₁ - qϖ₁

and the value of this angle oscillates about some fixed value.

To get the Laplace resonance angle, take the resonant angles for the 2:1 resonance between Io and Europa, and the 4:1 resonance between Io and Ganymede:

θ₁₂ = 2λ₂ - λ₁ - ϖ₁
θ₁₃ = 4λ₃ - λ₁ - 3ϖ₁

Combine these to eliminate ϖ₁:

(θ₁₃ - 3θ₁₂) / 2 = λ₁ - 3λ₂ + 2λ₃

which is the resonant angle for the 1:2:4 Laplace resonance.

I'm afraid I still am not grasping what the critical angles actually, physically represent.

Again calling upon the example of Gliese 876 (just because the numbers are handy), for the 2:1 resonance between b and c,
φ_cb,c = λ_c - 2λ_b + ϖ_c = 5.74° ± 0.85°
φ_cb,b = λ_c - 2λ_b + ϖ_b = 21.9° ± 4.2°

What, physically, do both of these angles represent?

It's not really a geometric angle here that you can draw on the diagram. The mean longitude in terms of the mean anomaly and the longitude of periapsis is:

λ = M + ϖ

The first of the two resonant angles, given by
θ₁ = (p + q)λ₂ - pλ₁ - qϖ₁
= (p + q)M₂ - pM₁ - (p + q)(ϖ₂ - ϖ₁)

The other resonant angle which is given by:
θ₂ = (p + q)λ₂ - pλ₁ - qϖ₂
= (p + q)M₂ - pM₁ - p(ϖ₂ - ϖ₁)

The angle ϖ₂ - ϖ₁ represents the angle between the periapses of the orbits.

The angle (p + q)M₂ - pM₁ describes how the planets are going around their orbits, for orbital periods exactly in the ratio (p+q):p this angle is constant.

Orbital resonnance seems to be a low point in the energy level of the 2 planets ( a little like the L points). And if so should be important in the stability of planetary system.
Does anyone knows of a paper based on that subject

You might try this one.

Lazarus wrote:

This article has the details on how to obtain the resonant angle for a given 2-body mean motion resonance.