Orbital Resonances
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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:?
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 2body 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_{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.
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 dF star
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Registration date : 20080612
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_{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?
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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Sirius_Alpha Admin
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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_{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.
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 dF star
 Number of posts : 3079
Registration date : 20080612
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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Registration date : 20100921
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 2body mean motion resonance.
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Sirius_Alpha Admin
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