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Chasing suns ... (Read 33602 times)
frankuitaalst
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Re: Chasing suns ...
Reply #30 - 06/23/07 at 10:38:31
 
On the edge of stability :
Adding a velocity component on the Figure 8 in the z-direction gives separate orbits for the 3 bodies . They seem to follow each of them their own path within the "8" . This run hereunder is about the maximum the system can hold . Increasing the z-velocity a little bit more makes the bodies to leave their path .
BTW : running the Figure 8 centered on 1 planet gives an amazing figure . The other planets do not move in a closed path but follow a curved line from left to right and reverse , following the same line...
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Figure8z.gif
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frankuitaalst
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Re: Chasing suns ...
Reply #31 - 06/24/07 at 11:39:47
 
An interesting feature of the figure 8 arises when small masses are added in a polar orbit . A Kozai mechanism seems to be created this way .  
In the sim hereunder 2 small masses were added to the system , each in a polar orbit ( mass =0.01 original) , distance = 10* radius of the system .  
The masses can be seen as the green lines as they cross in front or behind .  
The 8 system starts to oscillate for a while and then gets inclined also , coming in a sort of polar orbit itself .  
The change of inclination can best be seen in the z-frame .
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Figure8polar.gif
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frankuitaalst
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Re: Chasing suns ...
Reply #32 - 06/30/07 at 10:53:11
 
In an article "New Periodic Orbits for the n-Body Problem" by Prof Cristopher Moore I found the initial coordinates for a 3 body choreography called the Criss-Cross .  
After scaling the conditions to real world dimensions of a solar system this may be an initial set :  
Mass : 5.9736e+24
"Rx Ry Rz Vx Vy Vz of planets"
"E1", 1.61385E+10,0,0,0,3.1802e+01,0
"E2", -1.06425E+09,0,0,0,-2.00810e+02,0
"E3", -1.50744E+10,0,0,0,1.69008e+02,0
This means : 3 Earth planets about 0.2 AU distance.. Note that all planets start at the X-axes and have initial vertical  velocities .  
Here's the animated gif . The period is ca. 18 years .  
The system seems to be stable , according to the article.
It's amazing that one planet almost has a circular orbit , while the others have a very weird orbit .  
 
If one intends to scale the system with other distances or mass this can be done as following :  
Multiply the distances fi. with factor a . Multiply the masses with factor b. The velocities then should be multiplied by factor sqrt (b/a).  
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CrissCross.gif
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frankuitaalst
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Re: Chasing suns ...
Reply #33 - 06/30/07 at 11:26:48
 
Adding a velocity component in the z-direction of 3m/s (10% of the original ) for the both outer planets , in order to check the stability , gives also a stable system and starts to rotate .
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CrissCrossz.gif
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Tony
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Re: Chasing suns ...
Reply #34 - 06/30/07 at 11:29:53
 
very neat stuff, Frank  Shocked
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Tony
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Re: Chasing suns ...
Reply #35 - 06/30/07 at 12:18:59
 
I set this up using your numbers:
http://www.orbitsimulator.com/gravity/simulations/crisscross.gsim
 
It's amazing that this is stable.
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frankuitaalst
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Re: Chasing suns ...
Reply #36 - 06/30/07 at 12:23:00
 
This was quickly done !  Smiley. May be interesting to play with it changing the params slightly off or adding a small body somewhere.
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frankuitaalst
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Re: Chasing suns ...
Reply #37 - 06/30/07 at 14:41:56
 
Plotting the velocities and the SMA of this remarkable system gives the following pictures .
The total absolute velocity ( and hence the kinetic energy ) is almost constant , as well as the velocity of the third body which is close to a circular orbit .
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CrissCrossvelocity.gif
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EDG
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Re: Chasing suns ...
Reply #38 - 06/30/07 at 15:54:35
 
Wow  Shocked  
 
These wacky orbits are just amazing... they can't occur naturally though can they? I can't imagine how unlikely it must be to just happen to get planets orbiting like this.
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frankuitaalst
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Re: Chasing suns ...
Reply #39 - 06/30/07 at 16:59:59
 
These are merely inventions of mathematicians... I don't know if they exist somewhere in the real world .  
Another example ( and this is really a beauty  ... Tongue  Tongue) comes from the work of Chen "Action minimizing periodic ...solutions in the N-Body problem " who gives the unit initial conditions in his work .  
Herunder is a sim of 3 equal Earth masses. I put them at ca. 10 E+9 meters .  
The system seems to be stable ! , although I notice a small rotation ( perhaps due to round off as the initial values have only 4 significant numbers ) . I'll post the initial conditions seperately ...
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Chen3.gif
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frankuitaalst
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Re: Chasing suns ...
Reply #40 - 06/30/07 at 17:18:46
 
The initial values of the flower pattern of Chen are :  
Mass : 5.9736e+24
"Rx Ry Rz Vx Vy Vz of planets"
"E1", 8.682e+9,0,0,0,-8.49456e+1,0
"E2", 2.106e+9,0,0,0,2.70144e+2,0
"E3", -1.0788e+10,0,0,0,-1.85198e+2,0
By curiosity I made a screenshot of this system , centered on the last planet ...
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Chen3centered.gif
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Re: Chasing suns ...
Reply #41 - 06/30/07 at 17:43:06
 
These are amazing.  But as Frank and Mal have figured out, they're unlikely to occure in real life as the planets must be perfectly equal in mass.
 
I set up the one you call "a beauty".  Here it is:
 
http://www.orbitsimulator.com/gravity/simulations/beauty.gsim
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frankuitaalst
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Re: Chasing suns ...
Reply #42 - 06/30/07 at 18:09:06
 
Thanks for the set-up . I tried a rotating frame but somehow I got a "zero" as rotating period . Is it possible that Gravsim needs a central mass to calculate the orbit period ?
Concerning the equallity of masses : in the article of Chen there are some patterns for different masses in the range of 1...10.  
I don't know if the orbits are stable . Will soon try to create some .  
But these mathematicians really try hard ( I think sometimes by trial and error ) to create such a system , so I think the chances are very very small that such a system can exist , I mean has naturally evolved . The deviations from the positions , velocities and mass ratios must be really small in order to get a stable system .  
What kind of process in the universe can fi. create three almost equal masses at an appropriate distance and the right velocities ?  
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Tony
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Re: Chasing suns ...
Reply #43 - 06/30/07 at 18:17:53
 
Yes, Gravity Simulator is expecting some sort of a central mass so it can compute a semi-major axis and hence a period.  You can set the period if you want by timing how long the system takes to do 1 loop and entering this in seconds into the rotating period box.
 
Keep 'em coming.  These are fun.
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frankuitaalst
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Re: Chasing suns ...
Reply #44 - 07/01/07 at 02:42:26
 
The flower pattern of Chen for 3 planets as above , and in a rotating frame with the last planet ( the circular orbiting one ) gives the following unexpected picture :
Both other planets follow in this frame the same circular orbit .  
The last planet stays fixed near the left side of the picture (not visible due to resizing ).  
 
All of the systems described above have a centre of gravity at 0,0,0 . This means that the orbital period can be calculated , by lack of a central planet , around a virtual fixed point at 0,0,0.
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Chen3RotFrame3.gif
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