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Dynamics of 1:2 resonances (Read 7245 times)
frankuitaalst
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Dynamics of 1:2 resonances
10/08/08 at 12:13:45
 
In the discussion "Kirckwood gaps " there's enough evidence that asteroids avoid the 1:2 resonance with Jupiter.  
Whats the mechanism and how does a system evolve which is initially in a 1:2 resonance ?  
To find out I created the following simulation : 360 bodies were put in a 1:2 resonance to Jupiter at 0 eccentricity and 0 inclination .  
The sim starts in rotating frame to Jupiter .
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #1 - 10/08/08 at 12:21:44
 
The following animation of the 300 generated screenshots  covers 150 years of the above system .  
One can see that Jupiter slowly disturbs the path of the asteroids .  
The orginal circular envelope becomes more and more elliptical .  
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1to2ResonancetoJupiter.gif
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #2 - 10/08/08 at 12:35:01
 
After the initial 150 years the system quickly gains more eccentricity in the next 75 years .  
Even knots appear and the orginal circular envelope will be disturbed further ...  
  
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1to2ResonancetoJupiterPart2.gif
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #3 - 10/08/08 at 12:45:25
 
The following animation shows the SMA vs Eccentricity evolution of the 360 bodies  .  
All bodies start at 1 point at 0 eccentricity .
 
The resonance mechanism first induces a spreading of the SMA of the bodies and  increases the eccentricity of each body .  
Each body is differently affected , resulting in this curious ever changing patterns .  
The animation covers 70 years .  
The effect of the resonance is that the bodies are repelled from their 1:2 resonance , althus creating a "gap" .  
 
If the simulation is run for much longer time the bodies tend to approach again the 1:2 resonance and will be repelled again.  
Due to perturbations of other planets however the bodies don't reach their initial configuration anymore but drift more and more away .
 
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1to2ResonAe.gif
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Re: Dynamics of 1:2 resonances
Reply #4 - 10/08/08 at 13:53:41
 
Fascinating stuff... I wonder if the frequency of the "flicks" on the last graph is equal to Jupiter's orbital period?
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #5 - 10/08/08 at 14:02:33
 
I think it's related to Jupiters period in a 1:2 mode , as far as I can count .  
Also the knots in the other animation show a near 1:2 pattern
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Re: Dynamics of 1:2 resonances
Reply #6 - 10/08/08 at 15:51:42
 
Just to be clear, are you actually talking about a 2:1 resonance here (i.e. the objects are at 3.28 AU and have half its orbital period)? Or do you really mean a 1:2 resonance, where they're outside Jupiter's orbit (at 8.26 AU) and have twice its orbital period?
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #7 - 10/08/08 at 22:38:40
 
The simulation deals with a resonance where the bodies are inside Jupiters orbit ( 2:1 resonance as you call it ) .  
In the animation Jupiter is visible at the bottom at 5 'o clock .  
( Might be interesting to simulate the other resonance too  Wink )
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Re: Dynamics of 1:2 resonances
Reply #8 - 10/08/08 at 23:10:03
 
I think the resonance is defined by the order in which you phrase the objects.  
 
example1:
Neptune is in a 3:2 resonance with Pluto.  i.e. Neptune completes 3 orbits for every 2 of Pluto.
 
example2:
Pluto is in a 2:3 resonance with Neptune.  i.e. Pluto completes 2 orbits for every 3 of Neptune.
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Re: Dynamics of 1:2 resonances
Reply #9 - 10/08/08 at 23:38:20
 
Yes, but in this case if he's talking about asteroids within Jupiter's orbit then it's more obvious to say he's talking about the 2:1 resonance with Jupiter.
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frankuitaalst
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Dynamics of 1:2 resonances
Reply #10 - 10/09/08 at 10:40:06
 
How does a 2:1 resonance to jupiter evolves without disturbances of other planets ?  
The animation shows 360 bodies around a Jupiter sized planet at 10AU in a 2:1 resonance .  
As in the previous simulation the initial circular envelope evolves to an elliptical shaped envelope. Animation covers about 200 years after the system was about 500 years old.  
Jupiter itself is at 10 'o clock in this rotating frame
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1to2ResonRotating.gif
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frankuitaalst
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Re: Dynamics of 1:2 resonances
Reply #11 - 10/09/08 at 10:45:05
 
The animation herunder shows how the 360 bodies evolve out of their circular orbit in a 2:1 resonance to Jupiter .  
After some time ( 30 years) the bodies all evolve towards a lower SMA at a higher eccentricity .  
This remarkable pattern goes on and on for some cycles ...
 
Hint : if the animation is slow on the screen it helps to click the icon to open a separate window
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1to2ResonanceDetail.gif
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frankuitaalst
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Dynamics of 1:2 resonances- Birds in space....
Reply #12 - 10/09/08 at 10:51:18
 
The above "ideal" resonance system shows another interesting feature :  
 
The increase of eccentricity and lowering of SMA comes to an end after some time.  
Then the resonance mechanism seems to send the system back  to the original state , and the cycle repeats al over ....  
 
The animation herunder covers 1000 years .
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« Last Edit: 10/09/08 at 12:53:42 by frankuitaalst »  

1to2Resonanclong.gif
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Dynamics of 1:2 resonances - inverse system
Reply #13 - 10/10/08 at 11:07:46
 
Above animations deal with an asteroid belt in an inner 1:2 resonance , where the Jupiter is at the outside .  
What happens if the 1:2 resonating asteroids are at the OUTside of Jupiters orbit ?  
The animation herunder shows in detail the Ecc vs SMA evolution of an initial ring of bodies at 1:2 resonance .
 
The picture is totally different from the above system .  
Although the system also evolves to an elliptical envelope the major difference is that the SMA in this case increases with time , instead of decreasing .  
Also the orbits of the asteroids vary a lot more ( the system is loosely bound ) .  
Also after having reached the maximum eccentricity the system returns back to the original "zero" eccentricity , but the variation between the individual asteroids is a lot more than in the previous case .
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2to1Resonancedetal.gif
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frankuitaalst
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Dynamics of 1:2 resonances
Reply #14 - 10/10/08 at 14:08:45
 
The 1:2 resonance in which the bodies are inside Jupiters orbit show a remarkable feature as can be seen in the graph hereunder .  
 
Represented is the Mean Value of SMA and Eccentricity of the 360 bodies  
The graph clearly shows that SMA and Eccentricity are changing periodically with time , but whats more  the SMA and Ecc move along a sort of parabolic .
The top of the parabole corresponds with the 1:2 resonance .
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EvolutionSMAvsEccentricity_1to2Resonance.jpg
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