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Kozai- 5 earths sim (Read 3026 times)
frankuitaalst
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Kozai- 5 earths sim
10/30/06 at 10:18:57
 
here some results for the 5-earth sim .  
I ran them for about 400 years .  
Timestep = 128 sec .  
Tilmestep was perhaps too big for the 4 inner planets , as they lost their moons . The 5th earth kept his moon .  
 
Interesting is the following :  
1. Time before the Kozai becomes "visible" rises about exponentially with AU ( see last Excel-graph ) . This means that the moons of the outer planets must be protected by "time" .  
2. Inclination changes slowly in the beginning , but increases rapidly as the ecc. goes to 1.  
3. After reaching the max ecc the system gets in a cycle with decreasing ecc and decreasing incl , rising again , a.s.o.  
 
It is further interesting to run the sim with a smaller time step in order to obtain the frequency of the fluctations of incl . ( avoiding the moons are lost ) .  
I tried to correlate the frequency of oscillation tothe time needed to obtain full ecc , but data are to few . Such a link must exist surely .  
 
Note : running the sim I observed that the moon of the first earth changed the orientation at ecc=1 , meaning she returned to the earth in the some direction as she come from , meaning rotating clockwise instead of anti-clockwise .  
Really surprising !!!  
 
 
 
 
 
 
 
 
 
 
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Tony
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Re: Kozai- 5 earths sim
Reply #1 - 10/30/06 at 11:47:48
 
Here's the graphs from the other post that go along with Frank's sims.  Notice the interesting eccentricity double peak on the thrid graph.  That's worth some further investigation.
 

 






 

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frankuitaalst
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Re: Kozai- 5 earths sim
Reply #2 - 10/30/06 at 13:14:33
 
As I lost the moons in the timestep 128 sec I ran the sim with Timstep 32 . In this sim also the moons were lost . The results are +/- the same , implying the integration should be correct at Ts= 128.  
I'll post the results tomorrow .  
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Re: Kozai- 5 earths sim
Reply #3 - 10/30/06 at 23:02:01
 
I have a feeling you're losing the moons simply because as ecc approaches 1, your peripsis distance approaches 0, and there's not a time step small enough to handle such a close encounter.  I'll bet the lost moons are whizzing away at unrealistic speeds.  Use View>Distance and Velocity box to see how fast they're moving.
 
So starting with a 90 degree inclination for the moons is useful to observe the first peak at 0.5 Kozai Period, but everything after that point is untrustable, including the double peak (too bad, that's a very cool looking graph).
 
A better alternative might be to start with moons that are 70 or 80 degrees instead of 90.  That way the eccentricity will not reach 1.  Still, a very small time step must be used.
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frankuitaalst
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Re: Kozai- 5 earths sim
Reply #4 - 10/31/06 at 08:44:08
 
you're right Tony about the loosing of the moons . They come to close and fly away , even if the time step is diminuished .  
Herunder are the data for the 32 sec sim (90° angle at start )  
 
 
 
 
Interesting is the fitting of the power curev relating AU and time to Max Ecc. This power curve was hardly influenced by the reduction of the time step .  
 
BTW : in another simulation I experienced also the moons of Mars , Phobos and Deimos have a Kozai trend in their motion .  
 
 

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frankuitaalst
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Re: Kozai- 5 earths sim
Reply #5 - 11/01/06 at 01:30:50
 
Concerning the question whether Uranus moons undergo the Kozai effect I was more or less wrong some days ago . I obtained  a very large period ..., more than the lifetime of the Universe ...
 
According to the formula of Tadeka ( Pkoz ~ P1* ...etc) the moon Umbriel however should have an oscillation period of about 5 * 10+6 years , reaching max eccentricity of ca. 0.98 . ( I guess the formula is valid if the secondary = the sun , the host = uranus and the planet = Umbriel) .This is a very long time for observation and therefor hardly noticeble .
 
 
 
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« Last Edit: 11/01/06 at 02:44:33 by frankuitaalst »  
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Re: Kozai- 5 earths sim
Reply #6 - 11/01/06 at 01:48:17
 
Tadeka formula vs. GravSim :
Applying the Tadeka Formula for the Pkoz period to the 5-earths sim I obtain the following formula ...
 
Pkoz (years )  = 15,479* (AU,earth)^2,9978
 
This means our Earths moon "has" a period of 15.479 years .  
The moon around Earth at 2AU has a period of about 15.49*8 years ...
 
Comparing the obtained formula (GravSim )  for Tmaxecc = 6.89 * (AU)^3.3985 to the formula above gives us that the max ecc. occurs after ~~ the half of the period . This is reasonable .  
 
I can not explain the difference between the exponent ( 2.99 <-> 3.39 ) . Both formulas  match relatively well , but both have their accurancy .
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