# Orbital Resonances

## 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

(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

applies.

Then there's this angle that subtracts the first two to get

φ

· 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

φ

φ

φ

φ

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:?

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:?

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**Sirius_Alpha**- Admin
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## Re: Orbital Resonances

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)λ

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:

θ

θ

Combine these to eliminate ϖ

(θ

which is the resonant angle for the 1:2:4 Laplace 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)λ

_{2}- pλ_{1}- qϖ_{1}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:

θ

_{12}= 2λ_{2}- λ_{1}- ϖ_{1}θ

_{13}= 4λ_{3}- λ_{1}- 3ϖ_{1}Combine these to eliminate ϖ

_{1}:(θ

_{13}- 3θ_{12}) / 2 = λ_{1}- 3λ_{2}+ 2λ_{3}which is the resonant angle for the 1:2:4 Laplace resonance.

**Lazarus**- dG star
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## Re: Orbital Resonances

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

φ

What, physically, do both of these angles 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?

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## Re: Orbital Resonances

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

θ

= (p + q)M

The other resonant angle which is given by:

θ

= (p + q)M

The angle ϖ

The angle (p + q)M

λ = M + ϖ

The first of the two resonant angles, given by

θ

_{1}= (p + q)λ_{2}- pλ_{1}- qϖ_{1}= (p + q)M

_{2}- pM_{1}- (p + q)(ϖ_{2}- ϖ_{1})The other resonant angle which is given by:

θ

_{2}= (p + q)λ_{2}- pλ_{1}- qϖ_{2}= (p + q)M

_{2}- pM_{1}- p(ϖ_{2}- ϖ_{1})The angle ϖ

_{2}- ϖ_{1}represents the angle between the periapses of the orbits.The angle (p + q)M

_{2}- pM_{1}describes how the planets are going around their orbits, for orbital periods exactly in the ratio (p+q):p this angle is constant.**Lazarus**- dG star
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## Re: Orbital Resonances

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

Does anyone knows of a paper based on that subject

**Roland Borrey**- Asteroid
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## Re: Orbital Resonances

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.

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