Gravity Simulator
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl
General >> Discussion >> Dynamics of 1:2 resonances
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1223493229

Message started by frankuitaalst on 10/08/08 at 12:13:45

Title: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 .

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 .  

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 ...
 

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 .


Title: Re: Dynamics of 1:2 resonances
Post by Mal on 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?

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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

Title: Re: Dynamics of 1:2 resonances
Post by Mal on 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?

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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  ;) )

Title: Re: Dynamics of 1:2 resonances
Post by Tony on 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.

Title: Re: Dynamics of 1:2 resonances
Post by Mal on 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.

Title: Dynamics of 1:2 resonances
Post by frankuitaalst on 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

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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

Title: Dynamics of 1:2 resonances- Birds in space....
Post by frankuitaalst on 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 .

Title: Dynamics of 1:2 resonances - inverse system
Post by frankuitaalst on 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 .

Title: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 .

Title: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 .

Title: Dynamics of resonances
Post by frankuitaalst on 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 .

Title: Dynamics of 1:2 resonances
Post by frankuitaalst on 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.

Title: Re: Dynamics of 1:2 resonances
Post by Nexus on 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.

Title: Re: Dynamics of 1:2 resonances
Post by frankuitaalst on 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 .

Gravity Simulator » Powered by YaBB 2.1!
YaBB © 2000-2005. All Rights Reserved.