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Suppose that .... (Read 20586 times)
frankuitaalst
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Suppose that ....
04/06/07 at 13:08:04
 
GravSim is well suited to perform simulations ....in the sense of ...suppose that ..
Suppose that at the "beginning" the sun is circled by a bunch of equidistant planets of equal mass with eccentricity 0 ...Pure symmetry at the beginning .What will happen ?  . Will they stay in their perfect circles , will they interact , collide or will they be ejected ?  
The answer is  : depends upon .  
 
The next picture is the result of 20 planets each having a mass of 10*Mass Earth , perfectly distributed between the sun and 2 AU . This means 200 Earth masses in a distance of 4 AU ! Thats's a lot .
The result is ... chaos in  the long time for the outer planets . The simulation stopped at about 500 years when one of the outer planets collided . The inner planets stay remarkable stable , due to their strong binding to the sun . The outer planets however prefer to go elliptic and even exchange orbits ...searching for stabilty, which they don't find .  
The picture shows the evolution of the SMA of the outer planets from the start of the simulation .  
It seems that the planets tend to form "couples" at the beginning , meaning the gaps between the couples are bigger than the gaps between the initial planets . This causes of course stong interactions between the couples, leading to instability . Planets also may interchange orbits as can be seen in Planet14 and 12 ! .  
Stability definitevly depends upon the mass of the planets . Starting with planets of Earth mass gives a lot more stability in the system ...
 
  
 
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frankuitaalst
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Re: Suppose that ....
Reply #1 - 05/14/07 at 14:09:55
 
Suppose that ...the solar system consists of the sun and 4 earths .  
Each of the 4 planets has equal mass ( Earth mass) , they all are at 1 AU from the sun , orbit perfect in a circle , but are all 90° shifted from each other .  
Perfect geometry .  
What will happen after a "while" ?  
Will they keep into orbit ? Will they collide ? Which one will collide ?  
Not easy to answer . I think the perfect configuration can hold on for a while , as the system is in perfect equilibrium . After some time some small perturbations have to build op to ...??
In order to create some "chaos" , ie equivalent to a real system , in the following sim the first planet was given an offset of only 5000 meters in the y direction . The others were at 90°, 180° and 270°.  
The system is steady for about 500.000 days , then the initial offset results in mutual attractions out of the steady state as can be seen from the picture . Looking from "above" all the planets remain for thousands of years within a small gap , as they were before . However the planets must come somehow in resonance as can be seen .  
I omitted the 3th  and 4th planet in order to keep the plot clear , but it seems to me a system of horseshoe orbits is created.  
Must be fun to see this in a rotating frame .  
Out of the plot one might think there is a certain cycle of about 10000 days ( some 30 years ) .
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Re: Suppose that ....
Reply #2 - 05/14/07 at 17:30:27
 
If the planets were at the lagrange points (say, L4, L5, and L3 relative to one of the planets) then one would expect some stability but I would have thought that the configuration you set up with them separated by 90° wouldn't have been anywhere near that stable...
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Tony
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Re: Suppose that ....
Reply #3 - 05/14/07 at 17:46:45
 
This is similar to a discussion I'm taking part in on the BAUT forum:  http://www.bautforum.com/showthread.php?t=58590
 
Appearently, you can have trojan planets like Mal pointed out as long as their combined mass < about 4% the mass of the star they orbit.  I ran some sims with 2 planets spaced 60 degrees apart that remain intact indefinately.
 
I'm guessing that 3 planets would also be stable, spaced 60 degrees apart.  But how about 4 or more?  I'm going to guess that 4 and 5 are unstable, and 6 is stable, provided the 60 degree seperation.
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Re: Suppose that ....
Reply #4 - 05/14/07 at 22:25:21
 
Tony Since you are a master of sim building I have one that relates to this thread I would like to see.
 
Star =1.45 Msol
6 planets=0.5Mjup
Placement of planets such that  
 
#1 @ 2.6 AU Base planet
 
#2 @ LG1 of Star and #1
#3 @ LG2 of Star and #1
#4 @ LG3 of Star and #1  
#5 @ LG4 of Star and #1
#6 @ LG5 of Star and #1
 
All Should be in 1:1 resonance with #1
Just to see what happens.
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frankuitaalst
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Re: Suppose that ....
Reply #5 - 05/14/07 at 22:47:55
 
your link to the baut forum provides a nice link indeed http://www.physics.montana.edu/faculty/cornish/lagrange.pdf for the situation of the lagrange points , where the mass limit is calculated for a 3-planet system at the L-points .  
I think however that there are solutions also for the 4 planet system as described above . One may think that the 4 planet systems evolves to a 3 planet system (without collision) so that 2 planets oscillate around one of the points .  
Similar as if they played "accordeon". This is just a guess .
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Re: Suppose that ....
Reply #6 - 05/15/07 at 02:55:41
 
Quote from abyssoft on 05/14/07 at 22:25:21:
Tony Since you are a master of sim building I have one that relates to this thread I would like to see.

Star =1.45 Msol
6 planets=0.5Mjup
Placement of planets such that

#1 @ 2.6 AU Base planet

#2 @ LG1 of Star and #1
#3 @ LG2 of Star and #1
#4 @ LG3 of Star and #1
#5 @ LG4 of Star and #1
#6 @ LG5 of Star and #1

All Should be in 1:1 resonance with #1
Just to see what happens.

 
Gravity Simulator does not provide an easy way to do this.  So this is a good simulation to practice your algegra and trig.  Start with a drawing of your system:

Now all you need to do is compute the x,y positions (call them r), and the x,y velocities (call them v).
 
Given is that m1 is 2.6 AU from the star.  m3, m4, and m5 are the same distance too:

 
The distance from the star to m1 and m2 can be computed by adding and subtracting the m0's Hill Sphere radius:

 
Now we need to break these distances up into their x and y components.  This is easy for 0, 1, 2, & 3 as they lie on the x-axis.  Just pay attention to the diagram to make sure you get your signs correct:

 
This requires a little bit of 4th grade trig for objects 4 & 5:

 
Now we need the velocities.  Objects 0, 3, 4, & 5 can be computed using the circular velocity formula:

 
Now we need to break these velocities up into their x and y components.  This is easy for 0 & 3 as they lie on the x-axis:

 
And requires a little bit of trig for 4 & 5:

 
Now we need the velocities of objects 1 & 2.  The circular velocity formula won't work here as object 1 will travel a little slower than normal since it is in the L1 point and must have the same period as object 0. And object 2 will travel a little faster than normal since it is in the L2 point and must have the same period as object 0. So we can get its velocity by dividing distance travelled by its period.  Its distance travelled is simply its distance from the star times 2 pi:

 
Now we need to break the velocities into their x and y components.  Since they lie on the x-axis, this is easy:

 
This gives us everything we need to start plugging in the numbers:
r1=388954463796.6-388954463796.6*(9.49304340314795E+26/(3*2.88392952200079E+30))
^(1/3)=370333651756.755
r2=388954463796.6+388954463796.6*(9.49304340314795E+26/(3*2.88392952200079E+30))
^(1/3)=407575275836.445
 
r4x=388954463796.6*cos(60)=194477231898.3
r4y=388954463796.6*sin(60)=336844446563.21
 
v0=v3=v4=v5=sqr((G*(9.49304340314795E+26+2.88392952200079E+30))/ 388954463796.6)=22245.5130780415
 
v4x=-v5x=22245.5130780415*sin(60)=19265.1794458029
 
v4y=v5y=22245.5130780415*cos(60)=11122.7565390208
 
v1=370333651756.755/sqr(388954463796.6^3/(G*(9.49304340314795E+26+2.883929522000
79E+30)))=21180.5310394943
v2=407575275836.445/sqr(388954463796.6^3/(G*(9.49304340314795E+26+2.883929522000
79E+30)))=23310.4951165886
 
 
Now make a table of each objects position and velocity vectors:
rx(0)=388954463796.6
ry(0)=0
vx(0)=0
vy(0)=22245.5130780415
 
rx(1)=370333651756.755
ry(1)=0
vx(1)=0
vy(1)=21180.5310394943
 
rx(2)=407575275836.445
ry(2)=0
vx(2)=0
vy(2)=23310.4951165886
 
rx(3)=-388954463796.6
ry(3)=0
vx(3)=0
vy(3)=-22245.5130780415
 
rx(4)=194477231898.3
ry(4)=336844446563.21
vx(4)=-19265.1794458029
vy(4)=11122.7565390208
 
rx(5)=194477231898.3
ry(5)=-336844446563.21
vx(5)=19265.1794458029
vy(5)=11122.7565390208
 
 
Open Gravity Simulator, create a system with a star 1.45 solar masses, place 6 objects, each 0.5 Jupiter masses in orbit around it, and use Objects > Edit Objects to enter the position and velocity vectors.
 
You will end up with
http://orbitsimulator.com/gravity/simulations/lagrange5.gsim
Before you run it, take your best guess as to what will happen.  I took my best guess, and I was correct  Wink
 
Gravity Simulator uses the programming convention that down is positive in the y-axis, while my diagram above used the standard math notation that up is positive in the y-axis, hence this diagram is inverted.

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Re: Suppose that ....
Reply #7 - 05/15/07 at 10:16:33
 
WOW  Shocked
 
I definately did not expect that configuration to be that unstable.  It collapsed in less then a year.
 
The 4 planet configuration theat you mentioned earlier that might be stable collapses in less then 100 years.  undecided
 
The 3 planet configuration appears completely stable unless even the small perturber is present.  Ie eventually the trojan asteroids will eventual be booted.
 
The 6 planet configuration is also stable unless even the small perturber is present.  This is actually more sensitive to pertubations.
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Re: Suppose that ....
Reply #8 - 05/15/07 at 11:14:00
 
Wow , that was quick !  
I started the sim Lagrange5 and saw as abyssoft mentions that the planets escape . Lowering the iteration time (up to 8 sec after first running at 64  ) gave somehow other pictures , so I decided to run it with Picard Integrator . The diameters were set to zero , so no collision takes place .  
Heres a picture of the start of the simulation after some 700 days .  
The system was further run for 80 years and shows a lot of chaos . It seems however that some planets are bound to each other , as the iteration step is decreasing periodically . It looks as if the m0 and m2 build some kind of an earth-moon system with high eccentricity ....
The interference of both can be seen where the dots became black .  
Has anyone yet come up with ...euh...a negative SMA ? I did here and I don't know what it means !!.  
Can it be a curvature away from the sun ?
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Re: Suppose that ....
Reply #9 - 05/15/07 at 11:26:29
 
[quote author=abyssoft link=1175890089/0#7 date=1179249393]WOW  Shocked
The 4 planet configuration theat you mentioned earlier that might be stable collapses in less then 100 years.  undecided
quote]

What timestep did you use in your sim and what accurancy of the initial velocities ? If the initial velocities are not accurate up to 1m/s or less a planet can quickly gain a lot of distance over 1 year .  
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Re: Suppose that ....
Reply #10 - 05/15/07 at 12:42:44
 
I made some additional calculations about the 4 Earths at 90° . Data were generated with the Picard integrator ( same data as above ) . I wondered how the planets stay related to each other .  
Planet 1 is at 0° , Planet 2  is at 90° aso ... initially . From the picture hereunder it is clear that after some time (1000 years ) planet 2 moves away , planet 4 gets closer ,  
then they change : planet 2 gets closer , planet 4 goes away . But then ... (after 1250 years ) planet 2 comes closer , also planet 4. Then they get both further away ... The cycle seems to repeat somehow . I looks like planet 1 is the middle of an "accordeon" . Both planets approach and get further away ...aso  
Distances are expressed in AU .
The minimum distance is about 0.3 AU . This corresponds with ca. 17°
Remarkable is the cycle , some 330 years ...
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Re: Suppose that ....
Reply #11 - 05/15/07 at 22:55:30
 
Concerning the Lagrange5 Sim : this system is chaotic . As mentionned I ran it with the P-Integrator , setting the diameters to zero , so no collision occurs .  
Herunder are some screenshots :  
Especially the motion of m0 and m2 are hard to describe : they obtain fabulous accelerations and stay close to each other , by while getting as close as 288 km !!; later  even closer . In the real world this isn't possible of course , but as they are point massas  they must accelerate then with :  
6.6E-11*9.5*E+26 /288000^2 is ca. 780km/s2 .  
this is fabulous . I think thats the reason why the system is hard to stabilize.  
In reallity the worlds are breaking down as they sweep around each other .  
The other half jupiters remain more or less in their orbits , although some are exchanging orbits ( see yellow and brown SMA line ).
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Re: Suppose that ....
Reply #12 - 05/16/07 at 02:14:49
 
Quote from abyssoft on 05/15/07 at 10:16:33:
...The 3 planet configuration appears completely stable unless even the small perturber is present.  Ie eventually the trojan asteroids will eventual be booted...

The 60-60-60 configuration is quite stable, even in the presence of perturbers.  Saturn's moons, Tethys, Telesto, and Calypso are an example.  The 60-60 configuration is also stable.  
 
The 120-120-120 configuration (delete objects 0, 1, & 2) is stable, and will last indefinately unless a perturber is present, as is the 60-60-60-60-60-60 system.  
 
The [object 0, 1, & 2] system is stable unless a perturber is present, however, even the smallest pertabution, which an n-body simulation will induce, will de-stabalize it.  So objects in the L1 and L2 points usually won't last even a single orbit.
 
Moral:  in nature we won't find:
  • any objects in the L1 or L2 or L3 points of any other object.  
  • any triple system in the 120-120-120 configuration.  
  • any 6-body system in the 60-60-60-60-60-60 configuration  

But we may find pleanty of:
  • 60-60 systems
  • 60-60-60 systems

 
Quote from frankuitaalst on 05/15/07 at 22:55:30:
Concerning the Lagrange5 Sim : this system is chaotic...

Yes, this thing collapsed under chaos.  That's why I don't see any reason to follow the system after the collapse.  Give it slightly different starting conditions (run it at a different time step for a brief period of time), and you'll see drastically different results.   I imagine that sometimes you'll get orbiting pairs, sometimes you won't.  I would guess that even the orbiting pairs are short-lived over the course of 10s of thousands of years.
 
If you want to get an intuitive feel for what the L1 point is, consider the following:  The L1 point is stable in 2 axes, and unstable in 1 axis.  Picture 3 axes, one that is tangent to the orbit, one at a right angle to this, towards and away from the Sun, and one that is up/down (z-axis), hence you can't see it in the x-y plane diagrams in this thread.  This point is stable on the z-axis, and on the axis in the direction of motion, but unstable on the solar radial axis.
 
If perturbed on the solar radial axis, its drift will accelerate and it will quickly depart the L1 region.  But if perturbed on the axis tangent to the orbit, or on the z-axis, it will oscillate about this axis.  When it oscillates about both of these axes, it is said to have a Lissajous orbit.  The SOHO space telescope has a Lissajous orbit.
 
An object in the L1 point will feel an acceleration towards the Sun, and 180 degrees away,towards the Planet.  The star's pull is much stronger.  Adding these vectors together, it feels a pull towards the star that is slightly weaker that what it would normally feel for its distance, making the star seem "appearantly" less massive.  Objects in orbit around less massive stars move slower in circular orbits than objects in orbit around more massive stars.  When the "appearant" mass of the star matches what the circular velocity formula says it should be for an orbit with the velocity of Planet 0, you are in the L1 point.
 
This is as close to a video game as Gravity Simulator might come:

  • Pause the simulator
  • Open lagrange5.gsim
  • Delete Planets 2, 3, 4, 5
  • Edit Planet 1 and set it's mass to 0.
  • Open a thrust box: menu View > add Thrust Box
  • Open a distance and velocity box: menu View > add Distance and Velocity box
  • In both boxes, Choose Planet 1, with a reference object as the Sun
  • Unpause the simulation
  • Adjust the zoom so you can see Planet 0 and Planet 1
  • Increase the time step to 32 seconds.  As you get better, you'll be able to go even higher
  • On the thrust box, using ONLY the TOWARDS and AWAY buttons, attempt to keep the radial velocity as close to 0 as possible
  • If you are successful, Planet 1 will orbit the Sun with the same mean period as Planet 0.  If you drift ahead of, or behind Planet 0, you will naturally be pulled back due to the stability on this axis.  
  • See how many orbits you can complete.
  • When you get tired of doing this, simply stop correcting the orbit, and watch how quickly your object departs the L1 point.
  • When you're in the mood to do this again, try instead with an object in the L2 point.  Same instructions, except delete Planets 1,3,4, & 5, and use the trust box and distance box for planet 2.

 
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Re: Suppose that ....
Reply #13 - 05/16/07 at 13:52:42
 
Googling for some Lagrangian motions I found this link  
http://articles.adsabs.harvard.edu/full/seri/AJ.../0067//0000162.000.html
It's about  the "Klemperer Rosette" theory. Haven't read it yet fully but seems interesting as it is all about stability in this particular configuration . Seems that under certain conditions of massas a uniform n body configuration can be dynamically in an orbit around central body . The 90° Earths sim above is a special case .  Seems , as far as I've read it ,that a configuration in a regular polygon is dynamically stable ...
And here's an Java Applet about the same Klemperer Rosette :
http://www.burtleburtle.net/bob/physics/kempler.html
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« Last Edit: 05/16/07 at 17:38:51 by frankuitaalst »  
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Re: Suppose that ....
Reply #14 - 05/17/07 at 13:22:19
 
Heres the GravSim Simulation with 5 Earths ,72° separated from each other in the Klemberer configuration . The sim runs best at ca. 32000 s timestep . I used the rotating frame with Planet 1 as reference . It's nice to see how the system (other planets ) come close to planet 1 and then get away , decreasing the arc of 360° to 120° , then increasing again . Odd dynamic stability ...  
 
Modification : tried to upload the sim , but got a message this isn't possible , instead a screenshot of the system after ca. 1500 years  
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« Last Edit: 05/17/07 at 14:40:47 by frankuitaalst »  

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