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Dynamics of 1:2 resonances (Read 7247 times)
frankuitaalst
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Dynamics of 1:2 resonances
Reply #15 - 10/11/08 at 02:52:05
 
Herunder is a comparison between the 1:2 and 2:1 resonance patterns of the two above cases of 360 bodies orbiting sun and Jupiter at 10AU , orbiting insiude and outside respectively ...
One can see the behaviour of the two resonances are totally different .  
Innerresonaces are more stable , show less "drift" and move inwards , while outer resonances move outwards and drift away .  
Depicted is the mean value of the SMA and Ecc of the 360 bodies.  
Another difference between the two types is the initial reaction and the maximum deviation from the initial setting .
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2to1andviceversaResonance.jpg
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frankuitaalst
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Dynamics of resonances
Reply #16 - 10/12/08 at 13:20:16
 
The same way as above I've been calculating the behaviour of some other resonances .  
Surprisingly there are a lot of differences.  
All resonances were calculated with a central sun and a Jupiter at 10Au . It's orbital period is 31 years .  
One can see the 3:1 resonance is very weak . It's hardly changing the eccentricity .  
The 3:2 and 2:1 resonance however are very strong as the eccentricity rises up to 0.1 .
Timescale is expressed in years .  
Resonance periods seem to be 20 ...40 times the orbital period of the disturber .
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ResonancesComparisonEvolution.jpg
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frankuitaalst
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Dynamics of 1:2 resonances
Reply #17 - 10/19/08 at 11:06:30
 
Another representation of the 1:2 Resonance is the time evolution of the orbital elements .  
Pictured herunder is the time-evolution of the  SMA and Ecc of the 360 bodies which are in a 1:2 (in fact 2:1 as they orbit at the inside ) resonance to Jupiter . All bodies have initially zero eccentricity .  
As can be seen both SMA and Ecc show a sine-like behaviour .  
Maximum in SMA corresponds with minimum Ecc and vice versa.  
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EvolutionSMAandEccof1to2Resonance.gif
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Re: Dynamics of 1:2 resonances
Reply #18 - 10/20/08 at 03:23:50
 
This is going to sound simplistic, and I'm probably way off, but here goes anyway...
 
Imagine the orbit of the asteroid as an elastic band and the gravitational pull of Jupiter as a stretching of that band. When is the band going to be stretched the hardest? When the asteroid and Jupiter are closest together. In a 2:1 asteroid:Jupiter resonance the asteroid makes one close approach, and then another when Jupiter has completed half an orbit, then a full orbit and so on. These close encounters are separated by 180 degrees (ie. a straight line) which is like pulling on opposite ends of the rubber band and the originally circular rubber band becomes long and thin. If you had a 3:1 or 4:1 resonance the rubber band would tend to remain "open".
 
I hope you've understood that incoherent ramble. You can put the rubber band back in the drawer now.
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #19 - 10/20/08 at 11:51:52
 
This is indeed exactly what happens to the body .  
It is however remarkable that this stretchng comes to an end as one can see in the pictures above and the stretching tends to go the opposite way again after having reached a maximum eccentricity .
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