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Message started by frankuitaalst on 10/30/06 at 10:18:57

Title: Kozai- 5 earths sim
Post by frankuitaalst on 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 !!!  

Title: Re: Kozai- 5 earths sim
Post by Tony on 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.


Title: Re: Kozai- 5 earths sim
Post by frankuitaalst on 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 .

Title: Re: Kozai- 5 earths sim
Post by Tony on 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.

Title: Re: Kozai- 5 earths sim
Post by frankuitaalst on 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 .

Title: Re: Kozai- 5 earths sim
Post by frankuitaalst on 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 .

Title: Re: Kozai- 5 earths sim
Post by frankuitaalst on 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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