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Chasing suns ... (Read 33587 times)
abyssoft
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Re: Chasing suns ...
Reply #15 - 06/10/07 at 20:24:37
 
Alright got a stable one
 
Using the parameters I listed above.
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frankuitaalst
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Re: Chasing suns ...
Reply #16 - 06/12/07 at 11:34:53
 
I recently discovered a paper written bij Simo , who calculated the Figure8 initial conditions .  
Among other periodic choregraphies the "eight-orbit" seems the only stable one for 3 bodies of equal mass.  
In the paper Simo used units for G and masses , but with some deduction I was able to create the following formula :  
 
Distance r between two adjecant bodies ( bodies on one-line ) =r
r = 5*C * Ggrav * Mtot / 4 / 18 / v≤ , with C= 0.4851432 .  
The following formula applies if the initial velocity makes an angle of theta=0.99330597 radians to the x-axis .  
 
I simulated this for different values of Mtot ( total mass) , v (speed ) and this seems to work .  
May sound complex , but the idea is that given the total mass of the system and choosing an initial velocity , the above formula calculates the distance at which the planets have to be placed so that the form the "8".  
 
I think the formula may have other coŽfficients if the angle theta has another value , but this seems to be a good kept secret ( or must be generated by ...I don't know how ) .  
 
If one wants to simulate the following conditions can be applied :  
 
Coordinates:  
Planet 1 :-r ,0,0
Planet 2 : +r,0,0
Planet 3 : 0,0,0
 
Velocities :  
Planet 1 : -v*cos theta , - v*sin theta , 0
Planet 2 : -v*cos theta , - v*sin theta , 0
Planet 3 : 2*v*cos theta , 2* sin theta , 0
 
So the initial conditions look like :  
Planet 3 in the origin. The others are left and right at a distance r  
Planet 3 starts under an angle theta with velocity 2v  
The other two start at an angle theta with velocity -v .  
Herunder is an extrait of the article
 
 
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frankuitaalst
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Re: Chasing suns ...
Reply #17 - 06/12/07 at 22:52:50
 
An amazing feature of the Figure 8 configuration is its stability and its skill to survive small perturbations .  
Changing the initial coordinate of one of its components by a small amount makes it going to rotate , as Tony mentionned .  
Hereunder a sim of this feature . In this simulation the initial x-coordinate of one of them was increased by only 1/1000.  
Remark : This sim was NOT made in rotating frame .
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frankuitaalst
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Re: Chasing suns ...
Reply #18 - 06/13/07 at 08:30:32
 
Changing the initial condition ( x-value) on two of the outer planets results in a wider "8" where the planets change positions ( watch the colours ) .  
If the changes are bigger this results in wider loops quickly evolving to collissions of two of them.
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shellandtube
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Re: Chasing suns ...
Reply #19 - 06/13/07 at 21:50:35
 
This is great!
I wonder if you could scale it up with a large enough distance between the suns to have a small planetary system for each of them?
 
Might have a go at it tonight, seeing as though I am off work.
 
Paul.
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frankuitaalst
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Re: Chasing suns ...
Reply #20 - 06/13/07 at 22:42:47
 
Glad to have you abord . Seems GravSim is expanding to Europe  Smiley
The configuration you suggest might work I think .  
The system is scalable .  
In fact the last sim (rotation )  was done with 3 free floating Earth Planets at a distance of about 0.01 AU . I'll give it a try to add a moon to each of them . See what happens ....
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Re: Chasing suns ...
Reply #21 - 06/13/07 at 22:47:52
 
I added a 0.01 Earth mass planet to one of the suns at 0.05 AU.
It seems stable (after 100 years still there!)
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frankuitaalst
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Re: Chasing suns ...
Reply #22 - 06/14/07 at 12:42:14
 
If a body is added to the system there is a change in angular momentum . The simulation herunder shows what may happen in this case .  
Two small bodies were added (0.01 Mass ) at a distance of 2. They rotate clockwise .  
The system reacts  beginning to rotate clockwise as a whole. So there is an interchange of angular momentum .
Interesting is the fact that the planets of the eight change orbits in this time (visible if one looks at the different color pattern ) . One planet takes the outer orbit for some time , then another takes over ...
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Tony
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Re: Chasing suns ...
Reply #23 - 06/14/07 at 14:21:22
 
http://www.burtleburtle.net/bob/physics/eight.html
Here's a link where someone is describing his simulation, trying a lot of the things people are trying in this thread.
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Re: Chasing suns ...
Reply #24 - 06/16/07 at 03:12:15
 
Been trying to make the 4 body simulation work but no luck so far. Has anyone else had a look?
Let me know if u get anywhere?
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frankuitaalst
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Re: Chasing suns ...
Reply #25 - 06/16/07 at 07:38:29
 
Four-body problem in general are unstable , except in special cases in which 1 mass dominates and a few others . You refer to the 4 body system described in the previous link ? If so , I haven't looked at it , still struggling with the 3 body system which is valid under very narrow conditions . A very important parameter is the angle at which the bodies start . Changing it changes also the relation between masses , velocity and distance .
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Re: Chasing suns ...
Reply #26 - 06/17/07 at 18:03:01
 
Yeah, it had buried in the code what i hoped where some initial conditions, units unspecified.
Couldn't make it work. I believe its stable in the short term with the right conditions.
Was trying to be cleverer than I am.
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frankuitaalst
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Re: Chasing suns ...
Reply #27 - 06/19/07 at 12:58:01
 
In the following article from the alberta University I found the initial conditions for the 4-body orbits with 2 nodes.
http://www.math.ualberta.ca/~bowman/publications/nbody.pdf
I didn't try it myself . It can be hard to simulate because here the masses are set to 1 , as well as the G-constant .  
The initial conditions seem to be symmetric.  
r1 = (1.382857, 0), r2 = (0, 0.157030),
r3 = (−1.382857, 0), r4 = (0,−0.157030),
r˙1 = (0, 0.584873), r˙2 = (1.871935, 0),
r˙3 = (0,−0.584873), r˙4 = (−1.871935, 0).
One can try the conversion scale factor for v=sqrt(G*m/r) , but I'm not sure .  
According to some theoretical work I think this orbit is not stable as according to Simo only the "8" is stable .
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frankuitaalst
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Re: Chasing suns ...
Reply #28 - 06/20/07 at 12:53:41
 
I made a right guess with the scaling formula : vscale= sqrt(Gscale*Mscale/Rscale) .  
Hereunder a system of 4 Earth planets. The outer two were placed at a distance of 0.2 AU .  
Initial data are :  
P1:7500000000,0,0,0,158.509,0
P2:0,851660728,0,507.3214,0,0
P3:-7500000000,0,1,0,-158.509,0
P4:0,-851660728,0,-507.3214,0,0
 
This "double" eight fits quite well for a couple of orbits, but soon a small deviation results in a rather chaotical ( but symmetrical) orbit of the four planets. This symmetry doesn't seem to be broken long after the planets escape from their orbit .
Generating the orbits I noticed that the system is very sensible to small variations . For Earth I used 5.97E+24 kg , For G :0,0000000000667259
Simo may be right also stating that the only stable system is the "8" .
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frankuitaalst
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Re: Chasing suns ...
Reply #29 - 06/20/07 at 14:38:41
 
For the above system the coordinates and velocities are plotted in the picture below . Positions seem to be harmonic , velocities on the other hand are far from commom . No wonder this system is unstable ...
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« Last Edit: 06/20/07 at 22:07:31 by frankuitaalst »  

4bodiesvelocity.gif
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