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Chasing suns ... (Read 33579 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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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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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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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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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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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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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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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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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 / v² , 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 coëfficients 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Re: Chasing suns ...
Reply #34 - 06/30/07 at 11:29:53
 
very neat stuff, Frank  Shocked
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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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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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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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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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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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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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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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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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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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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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Re: Chasing suns ...
Reply #45 - 07/01/07 at 10:33:52
 
Actually, I realized the flower-shaped one is just 3 normal orbits, and would probably remain stable even if the masses were not equal.  It consists of two binary orbits.  The first one consists of two equal-massed (but they don't have to be) planets orbiting their common barycenter in circular orbits.  The second one consists of a planet and the barycenter of the first planet pair orbiting the barycenter of that system.  Try watching the simulation with the trails turned off.  You'll see what I mean.
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Re: Chasing suns ...
Reply #46 - 07/01/07 at 12:20:50
 
You're right Tony ...amazing how the trails can be ...complexing things. I also think that changing masses can work ...Maybe some work to do  Wink .  
Here's another one , also from Chen . In this case 4 equal masses start in the y-direction , also at given distances . The system starts good , but due to the low accurancy ( remember the 4 significant digits ) the orbits change ...
However the system stays stable . The period of this system is about 8 years . It was run for more than 500 years .  
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Reply #47 - 07/04/07 at 12:38:01
 
The system above is not stable in the long term , although it seemed to be so . This may be a result of the "poor" accurancy of the initial conditions , or may be inherent . I don't know.  
Here are the initial conditons ( from the work of dr Chen ) .  
"Earth",5.9736e+24 kg  
"Rx Ry Rz Vx Vy Vz of planets"
"E", 1.10153e+10,0,0,0,-2.82011e+0,0
"E", 3.9966e+9,0,0,0,3.52200e+2,0
"E", -3.9966e+9,0,0,0,-3.52200e+2,0
"E", -1.10153e+10,0,0,0,2.82011e+0,0
 
Another 4 body system is this one (also from the same author :  
"Earth,5.9736e+24
"Rx Ry Rz Vx Vy Vz of planets"
"E", 9.70470e+9,0,0,0,2.83396e+01,0
"E", 2.19975e+9,0,0,0,4.04237e+02,0
"E", -2.19975e+9,0,0,0,-4.04237e+02,0
"E", -9.70470e+9,0,0,0,-2.83396e+01,0
The system has a period af about 6 years and keeps its form for 100 years , but suddenly the planets escape from their orbit , evolving to a typical 4-body motion as can be seen hereunder .
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Re: Chasing suns ...
Reply #48 - 07/04/07 at 13:58:24
 
Here is finally an example of a choreography with distinct masses ( ref : Chen )  Smiley:  
The planets have resp. 1 , 5 and 10 Earth masses . The inner planets have the biggest masses .  
Initial conditions are :  
"Earth1",5.9736e+24
"Earth2",2.9868e+25
"Earth3",5.9736e+25
"Rx Ry Rz Vx Vy Vz of planets"
"E", 1.8e+10,0,0,0,6.30687e+2,0
"E", 9.85935e+9,0,0,0,-4.55923e+2,0
"E", -6.72975e+9,0,0,0,1.64893e+2,0
The system seems stable . A small rotation is noticible , as a kind of precession .  
 
Chen defined several categories of orbits , depending upon the cyclic permutation .  
This orbit belongs to the same category as the criss cross .
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Re: Chasing suns ...
Reply #49 - 07/04/07 at 23:56:28
 
OK, this is getting really silly now Smiley
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Reply #50 - 07/05/07 at 08:27:21
 
The following choreography is a member of the same set as the flower  
It consists of 3 bodies . The first has 1 Earth masses , the 2nd 5 and the 3th 10 Earth masses.  
The outerst planet is at 2.54565e+10 m from the center .  
Orbital period is about 11 years .  
The system seems quite stable .  
Amazing how the 5 Earth mass body rotates in almost a square ...
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Re: Chasing suns ...
Reply #51 - 07/05/07 at 10:21:14
 
Hereunder is a simulation with 4 distinct masses (1 ,5,8,2 Earth Masses ) from right to left . The outermost planet is 2.54 e+10 m from the center . Orbit period is about 12 years
It's a piity this system is unstable and degenerates quickly .  
From all the 4 body choreographies none of them was stable till now.  
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Re: Chasing suns ...
Reply #52 - 07/07/07 at 02:38:44
 
Peanuts have arrived ..!!! Smiley
From the work of Chen who investigated retrograde orbits comes the following simulation .  
Two big masses ( 1 and 2 Earth masses in this case ) circle aroud each other . A third mass (0.01 Earths ) circles around them , or tries to. In fact it is pushed into a patatoo or peanut orbit .  
The orbit seems stable .  
 
Masses und coordinates used here are :  
"Earth1",1.19472e+25
"Earth2",5.9736e+22
"Earth3",5.9736e+24
"Rx Ry Rz Vx Vy Vz of planets"
"E", 3.31350e+9,0,0,0,1.16179e+02,0
"E", 1.19010e+10,0,0,0,-1.75825e+2,0
"E", -6.7470e+9,0,0,0,-2.30597e+2,0
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Re: Chasing suns ...
Reply #53 - 07/07/07 at 03:09:35
 
The peanut system above seems to me to be possible in the real universe for several reasons :  
first double stars are quite common in the universe , second other orbits are possible with other sun masses , and third the peanut orbit seems stable if the initial conditions of the peanutplanet are changed ...
Herunder is a picture of the same system when the initial velocity was changed by more than 2% . The peanut orbit simply rotates but stays stable ...  
 
Moreover the system remains stable ( although the orbit rotates ) if a velocity component in the z-direction of the peanutplanet is added ( vz of 10% vy was added fi ) .
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Re: Chasing suns ...
Reply #54 - 11/12/07 at 11:34:07
 
Thread restored
There's a few threads that were generating error messages when viewed, including this one.  This thread should work again.
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