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Suppose that .... (Read 20564 times)
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Re: Suppose that ....
Reply #15 - 05/17/07 at 16:29:32
 
Quote from frankuitaalst on 05/17/07 at 13:22:19:

Modification : tried to upload the sim , but got a message this isn't possible , instead a screenshot of the system after ca. 1500 years  

Try it again.  I just added .gsim to the allowable extensions.
 
I'm surprised this system is stable.  I'm anxious to try it.
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frankuitaalst
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Re: Suppose that ....
Reply #16 - 05/17/07 at 23:51:34
 
It works , thanks . Sim is running now for 25000 years ...very odd dynamics...but you have to wait for ca.600 years before the fun starts . It looks as if the planets are tied to each other with springs ..pulling and extending ...I haven't seen them com closer than 90° yet .
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Re: Suppose that ....
Reply #17 - 05/18/07 at 00:51:43
 
By mistake I uploaded the 4 Earths at 90° instead of 5 Earths at 72° in the Klemperer configuration .  
Here is the 5 Earths sim . But both have the same type of dynamics ...
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Re: Suppose that ....
Reply #18 - 05/18/07 at 01:06:37
 
Here' s a picture seen from above in the xy frame , centered on one Earth of the 4 Earths sim after 6000 years. All other planets come close to each other and go away aso. The more black the more a planet spends in this region . The innermost circle represents the orbit of the sun (1AU) .
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Klemperer4Earths90.gif
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Re: Suppose that ....
Reply #19 - 05/18/07 at 02:44:35
 
Running the Klemperer System with 4 Earths for a longer time (80000 y ) gives the following evolution of the SMA's :  
The system seems to behave odd , a lot of variation in SMA , but they al stay close within 4/Thousand.  
Plotting the mean SMA gives almost a straight line .  
Is there some correlation between mean SMA and the energy of the system .  
Zooming in into the 2 opposite planets in the 1000-3000 years gives the plot underneath .  
The planets evolve in a "regular" but unpredictable way . I honestly don't understand the behaviour , but I agree with Tony that Earth must be protected somehow against collisions by the horseshoe mechanism .
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Klemperer_4_Earths_90.gif
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Re: Suppose that ....
Reply #20 - 05/18/07 at 12:34:12
 
Is the 4 Klemperer system stable under foreign influence ? To find out Jupiter was added to the simulation . After 6000 years of simulation the system seems to behave as before . The mean SMA however varies more than before ( not visible in the graph , but visible if the SMA Mean is put on a second axis ) .
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Klemperer_4_EarthsJupiter.gif
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Re: Suppose that ....
Reply #21 - 05/19/07 at 00:29:43
 
Is the Klemperer system stable when the planets have different mass ?  
under certain conditions it is .  
Hereunder is a link to the 4 Planets sim .  
1 Planet was given a mass of 3 Earths , the other 0.33 Earths , so quite a difference . As a seed the planet 1 was positionned 5 km "north" .  
The plot heraunder gives the evolution of the SMA of each planet . Interesting is the second plot , zooming into the first years . The system seems to  react immediatelly . The planet in front slows down , the planet behind acclerates , the come close to the big planet ,they are both repelled , ga away , come closer again ... The symmetry is lost after 200 years .  
 
The GravSim (second post herunder ) is even funnier when run in a rotating frame on planet 1. I ran it at 8000-32000 seconds . Watch the planets 2 and 4 coming close and moving away ...
But , wait , take a look then at planet 3 ( opposite of the big ) . It stays a long time at home position , then moves at big speed towards planet 2.....
Negative gravity ?  Shocked  
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4Klemperer_one_big.gif
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Re: Suppose that ....
Reply #22 - 05/19/07 at 00:31:08
 
Heres the GSim file .
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Re: Suppose that ....
Reply #23 - 05/19/07 at 10:52:47
 
May it be possible for Earth to host Klemperer Moons ?  
Although planets can theoretically share other planets in their orbit , it is difficult for Klemperer Moons in an orbit around Earth to survive .  
Starting with 4 Moons , each with a mass of 1/10 of our moon and gradually decreasing the mass in different simulations the result is  instability resulting in eccentric orbits and collisions .  
As the mass of the moons decreases the lifetime seems to go up .  
The system with 4 moons , each 1/1000 Moon mass survived for 37 years . Other heavier systems crashed after 3 years . The picture shows the orbits of the moons ,each 1/1000 Moon massas , centered on earth .  
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KlempererMoons41o1000.gif
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Re: Suppose that ....
Reply #24 - 05/30/07 at 13:33:09
 
Simulating the 5 Earth Klemperer configuration in a rotating frame gives the following result .
All the planets share the same orbit around the sun , but are 72° separated from each other .  
The first earth is initially situated at 3 o clock . This is the reference planet in the rotating frame.  
 
After a couple of hondred years the planets come into motion relative to each other and move towards the first earth .  
They increase their speed , come close and ....reverse.
This goes on and on .  
The individual planets are not easy to distinguish as they share the same orbit , except for their color .  
The sim spans more than 3000 years ...
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5Earths72degreesRot.gif
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Re: Suppose that ....
Reply #25 - 05/30/07 at 18:00:50
 
I'm not sure what's going on here.  When I try similar simulations, they stay stable for a few hundred years, or more, depending on the time step.  Theoretically, they should be stable indefinately.  So errors introduced by the integration are what cause it break down with time.  But when it breaks down, it should become very chaotic.  They should not stay in the same orbit.  They should become eccentric and they should make close approaches to each other.
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Re: Suppose that ....
Reply #26 - 05/31/07 at 09:38:36
 
Perhaps the most simple simulation is this one : 1 Sun and 2 Earths .  
Both Earths share the same orbit but the second is exactly 180° ahead of the first . The system seems stable and is performing as the 5 Earth system .  
In order to investigate the stability I added Jupiter to the sim .
Will it still be stable ?  
Here's the result in a rotating frame around Earth1 . The system seems remarkable stable for the first 3000 years .  
There is no indication for instability .
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2Earths180JRot.gif
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Re: Suppose that ....
Reply #27 - 05/31/07 at 11:24:14
 
The system above seems to have a remarkable dynamic stability . In the run for ca. 6000 years there seems to be a periodicity of ca.700 years , where the planets approach each other .  
Closest approach is at ca. 24°.  
The second planet spends a lot of time at 180° and then slowly moves away , gaining speed until it gets as close as 24° and then moves the other way back .  
Once arrived at 180° it can ...continue after a while or... return .  
So it sometimes "attacks" from the other site .  
Have no idea were the cycle period nor the 24° come from and how they are related to the system parameters.
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2EarthsJupGraph.gif
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Re: Suppose that ....
Reply #28 - 05/31/07 at 14:30:50
 
The symmetry in the above system is remarkable as can be seen from the plot hereunder .  
Starting at an angle of 180° one tends to slow down , the other is gaining speed . As a result he angle betweem them becomes smaller .  
 At a point at about 50° they are slowing down and speeding up and come as close as 24 degrees , then are moving away again ...
Both planets seem symmetric to each other
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2EarthsJupSMA.gif
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Re: Suppose that ....
Reply #29 - 08/11/07 at 03:15:10
 
What is happening in this simulation ?  Huh
The sim herunder consists out of two solar masses ( Sun and the outer orbit ) .  
They orbit their common barycenter . Distance between both is 1.5 AU.  
So two suns which are on the Earth orbit .  
A really small excentricity is added in their orbits , just as Earths orbit around the sun . There is no central sun .  
A third body is added then with mass 0.03 solar masses . The planet shows itself as the inner circle and is placed just in between the 2 solar masses , also in their barycenter .  
The screenshot is centered on the first sun ( indicated by Sun) .  
One would think this creates a stable orbit for both suns , but see what happens .... Shocked
The orbits of the suns change quite a lot ...
They seem to be stable for long , and then suddenly they "jump"...This is repeated several times.  
At the end of the simulation the planet is ejected .  
Fysically impossible ? Resonance ? Well yes , a kind of .  
I'll post the .gsim file . This may explain a lot .
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2Sunsandaplanet.gif
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