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Message started by frankuitaalst on 07/11/08 at 12:14:31

Title: Resonance stability ...
Post by frankuitaalst on 07/11/08 at 12:14:31

I've been playing around with some resonances of bodies in order to find out if resonances are stable .
First attempt is shown hereunder :
System consists of Sun , Earth and a second Earth at 2:1 resonance with Earth , with eccentricity 0.
Watch how the system precesses , especially the inner Earth (ours) . Animation is run for 7000 years . Each frame covers 100 years.

Title: Re: Resonance stability ...
Post by frankuitaalst on 07/11/08 at 12:20:52

The above system is stable , but it's an illusion to think that resonances are "absolute" , at least if the system consists of bodies with comparable mass .
One can see the evolution of the Eccentricity and Semi-Major Axis of both planets.
The system pulsates with a period of about 2250 years .
While the inner planet merely reacts with a change in eccentricyity the outer planet prefers to change its SMA.  Both planets return to their initial state every 2250 years.

Title: Re: Resonance stability ...
Post by Tony on 07/11/08 at 19:58:52

The changes are very subtle.  If you give one or both some eccentricity, they should lock into the 2:1 resonance.  You can view this with the rotating frame.

Title: Re: Resonance stability ...
Post by frankuitaalst on 07/12/08 at 13:05:59

Yes , the changes are very subtile indeed . This may be due to the small masses of the Earth bodies . Do you mean  small eccentricity may lock the bodies even closer ?
I also ran a 3:2 resonance with the inner body at 1AU ( as above ) and also get a smooth oscillation as above . The period of oscillation is shorter . I think this must be due to the fact the bodies are closer in a 3:2 resonance than in a 2:1 resonance .

Title: Re: Resonance stability ...
Post by frankuitaalst on 07/13/08 at 01:37:55

Moving the outer body even closer to the inner body in a 4/3 resonance gives the following result :
The frequency of the oscillation increases , which makes sense ; also the deviation in SMA increases , although still subtile . Both are logical .
So decreasing the distance between the bodies from 2/1 over 3/2 to 4/3 increases the frequency of the oscillations , but also the "overshoot".

Title: Re: Resonance stability ...
Post by frankuitaalst on 07/15/08 at 12:31:14

I've made some more simulations with other resonance ratios and noticed that the amplitude of the 5/3 resonance was much lower than the 3/2 and 4/3 resonance .
Further ratios such as 9/7 , 7/5 have also this low value .
Hereunder an overwiew of the results . Depicted is the SMAmax/SMAoriginal of the outer planet .
This is amazing .
The resonances with odd numbers seem remarkable stable .
I also noticed that for this ratios there was no change in eccentricity .  :-/

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