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Chasing suns ... (Read 33625 times)
frankuitaalst
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Chasing suns ...
06/07/07 at 00:21:02
 
Can celestial bodies count ?  
Yes , indeed .  
 3 bodies , regardless of their mass ,but equal, can form a figure "8" , when they have proper initial conditions .  
See the article http://www.arxiv.org/PS_cache/math/pdf/0011/0011268v1.pdf.  
Herunder is an animation of such an orbit .  
The sim consists of 3 solar masses , given proper initial conditions . The screen field is about 1 Au .  
As can be seen the suns chase each other in an endless loop , forming the figure "8" .  
The configuration seems to be stable ...
 
Its not easy to find the initial conditions . I used this  
 
Sun mass : 0.9e+30 kg  
Coordinates and velocities : x,y,z,vx,vy,vz
"Sun1",36375163500,-9115782375,0,18648.1474,17294.6292,0
"Sun2",-36375163500,9115782375,0,18648.1474,17294.6292,0
"Sun3",0,0,0,-37296.2948,-34589.2584,0
 
The article refers to other amazing orbits , such as 3D orbits , but I found this one already very amazing ...
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Figure8Anim.gif
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frankuitaalst
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Re: Chasing suns ...
Reply #1 - 06/07/07 at 02:42:42
 
Is this configuration above stable under external influences ?  
To find out I added an Earth to the system at 1AU .  
The system itself seems stable (Sim was run for 10 years ) . The orbit of Erath  of course is influenced by the motion of its suns .  
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Figure8EarthAnim.gif
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frankuitaalst
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Re: Chasing suns ...
Reply #2 - 06/07/07 at 10:24:14
 
The 3 bodies above share the same orbit and are in a time-lag of 120° in their period .  
How does their position and velocity evolve in their orbit .?  
The picture herunder represents the change of the position and velocity as a function of time ( t in sec) .  
The velocity seems to be maximal in the origin where the orbit crosses. None of the functions seem to be an ordinary sine or cosine.
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3ChasingSuns.gif
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Sublyme
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Re: Chasing suns ...
Reply #3 - 06/07/07 at 15:48:11
 
I've been having a hell of a time setting this system up myself...  I tend to get only 2 bodies orbiting each other, everytime, and they're not really orbiting...  More like flying off towards doom.  Are you sure all your numbers are right, because I've put in everything you have, and I get the same results everytime.  Although this could just be a case of me using KM/s instead of M/s, etc.
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frankuitaalst
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Re: Chasing suns ...
Reply #4 - 06/07/07 at 15:53:44
 
The units I use are m , kg , sec .  
The data mentionned are from the input file . So they are correct .  
Please check your units .
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Re: Chasing suns ...
Reply #5 - 06/07/07 at 19:14:56
 
Quote from Sublyme on 06/07/07 at 15:48:11:
I've been having a hell of a time setting this system up myself...  I tend to get only 2 bodies orbiting each other, everytime, and they're not really orbiting...  More like flying off towards doom.  Are you sure all your numbers are right, because I've put in everything you have, and I get the same results everytime.  Although this could just be a case of me using KM/s instead of M/s, etc.

It was tricky for me too, but I finally got it.  Thanks for the numbers, Frank.  Where did you get them?
 
 
Here is a link to my simulation
http://orbitsimulator.com/gravity/simulations/figure8.gsim
 
I set it up like this:
Pause the simulation
File > New
Object > Edit Objects, rename "Center", mass=0, size=0
Object > Create Objects, 3 objects, Reference object = "Center", mass = 0.9e30 kg, Semi-major axis = 10, leave all other value at their defaults.
Object > Edit Objects, rename "Sun 1", "Sun 2", "Sun 3", and enter Frank's numbers for each Sun.  He gives them as meters and meters/second.  Make sure you hit "Apply" after editing each object's vectors.  Then change Sun 1's and Sun 2's reference object to Sun 3.  Then delete the object "Center".
Press the "F" button on the Graphics Options window to put the simulation in floating mode.  Choose Sun 3 as the Focus object.
 
File > Save As... give it a name.
Unpause and enjoy...
 
This one is fun to play with.  I can run it up to time step 8192 before it flies apart.
Try putting in "absolute mode" by pressing the "A" button on the "Graphics Options" window to lock Sun 3 to the middle of the screen.
 
Also, try turning off trails, and in the "Preferences Menu" increase the minimum object size to 4.
 
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Tony
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Re: Chasing suns ...
Reply #6 - 06/07/07 at 22:31:59
 
This thing is way fun.  Even pertubations don't shake it.  Open the trust box and give any object a 1 km nudge in any direction.  That should be enough to bring down any house of cards.  But all it does is introduce a rotation to the pattern.
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frankuitaalst
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Re: Chasing suns ...
Reply #7 - 06/07/07 at 22:48:00
 
Yes , they start to rotate if the initial velocities change by small amounts , but don't mess with the masses ... Grin.Is it possible to give an impression of the rotation you get there ?  
I changed masses of two upwards , lowering the third one ; this is stable in the beginning but after a while the system degenerates . So the masses must be equal .  I think it is almost impossible to find such a system in the real universe due to this fact alone ...
It was not easy to find the initial settings .... I used the formula in the article , scaled it up to about 0.5 AU ( I guess ) with the initial sun mass . The system starts good but velocities were not right ; so I reduced after some trials the masses to 0.9 e+30 instead of 1.99e+30. This is whats described above ...
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Re: Chasing suns ...
Reply #8 - 06/08/07 at 00:47:31
 
Wow, that's just crazy!  Huh
 
Could this configuration conceivably arise naturally?
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frankuitaalst
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Re: Chasing suns ...
Reply #9 - 06/09/07 at 00:59:53
 
I don't think this configuration can exist in the universe in large amounts as the three masses have to be equal in a narrow band .  
I found an article dealing with this problem .  
Usefull herein is that it gives a formula to calculate initial conditions .
 
http://arxiv.org/PS_cache/math-ph/pdf/0304/0304014v3.pdf
Here's a screenshot of the formula :  
An interesting parameter to play with is the theta , must influence the direction of intersection , also influencing the stretching of the orbit .  
It may take some scaling as they use as gravitational contstant 1 instead of G= ...
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Figure8Expression.gif
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Re: Chasing suns ...
Reply #10 - 06/09/07 at 12:53:13
 
Care to figure out the initial conditions in the systems described here?:
http://www.santafe.edu/~moore/gallery.html
 
They start with the 3 objects in a figure 8, then move up to 21 masses in a figure 8.  It then gets crazier from there.
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Re: Chasing suns ...
Reply #11 - 06/10/07 at 12:31:45
 
This one is definately fun to play with and remarkable stable.
 
I went with a 1 Mjup @ 2.14080352 AU this would place it smack dab in the middle of the Habitable zone for the system. The Kicker on this one is I have the planet orbiting @ 90 deg to the stellar plane.
 
I then added 4 moons in orbit of the 1Mjup body @ Rhill/30, Rhill/10, Rhill/3.33333_, Rhil/1.11111_  
@ Inclinations 5,15,25,35 to the stellar plane respectively.
 
The result is rather stunning, the planet's orbit is stable but rather unusual in that it occilates from 90 to -90 deg.  The figure 8 stellar orbital path is now rotating on all 3 axises (sp?)
 
The outer moon was lost @ ~ 185 yrs in to the sim ejected from the planet family.
the 2nd moon was accreted by the planet @ about the same time this destroyed the stability of the moon system. 8(
 
the only good part is the outer moon was not completely lost, as it moved into a stable orbit outside the jovian orbit.
I will try with fewer moons more inline with the planets orbital plane
 
3 Moons @ Rhill/30, Rhill/10, Rhill/3.33333; 85, 80, 75
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Re: Chasing suns ...
Reply #12 - 06/10/07 at 12:58:13
 
Image is from 90 deg above initial stellar plane
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frankuitaalst
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Re: Chasing suns ...
Reply #13 - 06/10/07 at 14:21:53
 
The orbits described in the link above are really exciting , beginning with the 21 planet 8.  
I'm not capable yet to define the initial conditions .  
There are a lot of papers dealing with te figure8 but most of them are mathematical and use unit paramets as fi . G=1 , r=1 . So it's a hell of a job to find out how the formulas look for real G and for distances different from 1m !
Maybe there are papers available , but in this field of interest most of them require a payment or are written in .ps ( which I can't run ) (Is .ps associated with Visio ? ).  
Playing with parameters I have to conclude that the initial conditions must be exact (ie. there seems to be a unique angle Theta which generates the figure 8 , given a certain speed v and a distance r ).  
If the angle is not exact the planet passes under or above the eight .  
For the figure8 with 3 planets I found the formula : v = sqrt(5*G*m/12/r). I haven't found yet the formula for theta .  
 
Someone knows a good program to convert .ps files to .pdf ?
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« Last Edit: 06/11/07 at 13:02:51 by frankuitaalst »  
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Re: Chasing suns ...
Reply #14 - 06/10/07 at 14:55:30
 
It's silly how some of the articles want to charge you $35 just to view the article?  A scientist is likely to work for an instutution with a subscription.  And I doubt hobbiests like us would pay that much for a single article.  Who is their market? Do they get any sales?
 
If I need to read anything for which a subscription is charged, I just go to the library at San Francisco State University, a 10 minute walk from my house.  They have a subscription, so their internet terminals deliver the article.  And upstairs in the archives I can look at the hardcopy if I want.
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