Orbital Resonances

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Orbital Resonances

Post by Sirius_Alpha on 30th June 2010, 12:53 am

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
φIo, Eu (Io) = λIo - 2λEu + ωIo
(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,
φIo, Eu (Eu) = λIo - 2λEu + ωEu
applies.

Then there's this angle that subtracts the first two to get
φIo, Eu = φIo, Eu (Io) - φIo, Eu (Eu) = ωEu - ωIo

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.

φce0 = λc - 4λe + 3ωc
φce1 = λc - 4λe + 2ωc + ωe
φce2 = λc - 4λe + ωc + 2ωe
φce3 = λc - 4λe + 3ωe
φce = ωc - ωc

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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Re: Orbital Resonances

Post by Lazarus on 8th August 2010, 9:48 am

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:

13 - 3θ12) / 2 = λ1 - 3λ2 + 2λ3

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

Post by Sirius_Alpha on 14th August 2010, 12:05 am

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,
φcb,c = λc - 2λb + ϖc = 5.74 0.85
φcb,b = λc - 2λb + ϖb = 21.9 4.2

What, physically, do both of these angles represent?

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Re: Orbital Resonances

Post by Lazarus on 14th August 2010, 4:16 am

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.
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Re: Orbital Resonances

Post by Roland Borrey on 5th October 2010, 1:31 pm

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
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Re: Orbital Resonances

Post by Sirius_Alpha on 5th October 2010, 1:42 pm

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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