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Message started by frankuitaalst on 10/29/08 at 15:59:37

Title: Origin of Trojans ?
Post by frankuitaalst on 10/29/08 at 15:59:37

Recently I saw an article in which a simulation was described which deals with the possible origin of the Trojans of Jupiter ( I think an article of Piet Hut ) .
The article says Trojans might have been build from coorbitals from Jupiter which stabilize around the L5 and L4 points .
I've set up a system of 1 Jupiter and 360 coorbitals between 30-330° to Jupiter .
Here's the animation of GravSim , covering the first years of the simulation .
The bodies indeed move towards L5 and L4 , but overshoot .
I'll post the simulation itself too .


Title: Re: Origin of Trojans ?
Post by frankuitaalst on 10/30/08 at 09:45:39

Here's the simulation which generated the picture above .
Running the sim for longer tile will result in a more chaotic position of the bodies , although they tend to join up around the L5,L4 points .

Question for Tony : I tried to output the a,e and also big omega .
Big omega result in valid values initially , but from the second value they equal zero .
May I have done something wrong ?

Title: Re: Origin of Trojans ?
Post by Tony on 10/30/08 at 11:31:56

What you've created in this simulation is horseshoe orbits.

Big Omega is longitude of the ascending node.  But your particles don't have any inclination, therefore they can't have an ascending node, so 0 is chosen for them through a piecewise function.

You get a similar problem when you ask for argument of perigee for a round orbit since a round orbit has no perigee.

Title: Re: Origin of Trojans ?
Post by frankuitaalst on 10/30/08 at 14:24:29

Thanks , I should have used small omega ( argument of periapsis) , which works better !

Title: Re: Origin of Trojans ?
Post by frankuitaalst on 10/31/08 at 10:46:06

Representing the above simulation in a time representation shows a horseshoe orbit exhitits a lot of dynamics , even in this simple case of one Jupiter mass orbiting a central sun .
Depicted is the SMA vs Eccentricity relationship of the above 360 bodies as a function of time .
The system starts at zero eccntricity .
The several bodies spread out in SMA while some of them gain eccentricity .
Remarkable is the pulsation in the eccentricity .  

One can say that if one detects a body with a SMA, Eccentrity corresponding to the plot hereunder the body in case can be considered being capable to be a coorbital to the parent planet .  

Title: Re: Origin of Trojans ?
Post by Mal on 10/31/08 at 18:16:04

:o

...

You got me there, I have no idea why it's doing that but it's pretty amazing to watch. It's like it's bouncing off the x-axis! :)

Title: Re: Origin of Trojans ?
Post by frankuitaalst on 11/01/08 at 02:04:51


Mal wrote:
:I have no idea why it's doing that but it's pretty amazing to watch. It's like it's bouncing off the x-axis! :)


So do I  :'( .
But I remember a paragraph in Murray and Dermot where the analogy is described between the resonance pattern and the pendulum . In some paragraphs  they even use pendulum equations to interprete the results of resonances .
After taking a closer look to the graph I can imaging that the particles initially have some energy , just as you pull out a string and then release it . The initial setting must be some deviation from the equilibrium point . When you release the system it starts to oscillate around the equilibrium point . I think this is what the particles are doing in some way .
The equilibrium point in this case may be the L4 and L5 points .

Title: Re: Origin of Trojans ?
Post by frankuitaalst on 11/01/08 at 04:19:01

With the aid of the viewer the behaviour of the above system can be understood a little bit more .
The picture herunder represents the evolution of the eccentricity and the SMA of each individual body .
The latest picture shows the envelope of Ecc vs SMA .
One can see the system spreads out in SMA , and is harmonic in eccentricity.
The particular bouncing and bending in the above animation is caused by the fact the bodies devellop at slightly different frequencies as can be sen in the first frame herunder .  

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