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General >> Discussion >> Kozai (kinda) - sim 4
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Message started by Mal on 11/09/06 at 23:38:03

Title: Kozai (kinda) - sim 4
Post by Mal on 11/09/06 at 23:38:03

Just for the hell of it, I decided to do the Sim 3 run (http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?board=news;action=display;num=1162606926), but with neither the planet nor the companion in an inclined orbit. So they're in the same plane, but the companion is in an orbit with eccentricity of 0.8.

So far it looks a bit curious. The planet's eccentricity peaked around 0.1 around 210,000 years and is going back down - I dunno if it'll come back up again later on, I'll continue to run the program to see what happens. The planet's Arg of Peri is merrily rotating slowly through 360 over time, inclination is still at zero.

Title: Re: Kozai (kinda) - sim 4
Post by Tony on 11/10/06 at 01:03:59

I don't think the rise to 0.1 ecc and its subsequent decline will be periodic, but rather chaotic.  But I'm anxious to hear the results.  It may be similar to what Jupiter does to Earth and especially to Mars.  The planes are too similar for the Kozai mechanism to work its magic.  But Earth's and Mars' eccentricity does vary between a max ecc and nearly circular.

As far as inclination, if both the planet and companion were started with exactly 0 for the inclination, then inclination will never change, guaranteed.  You've simply set up a 2-D simulation.  Inclination can only change when something torques up or down on an object, which is impossible if all objects have a value of 0 for their z-axis position component.  Consider the following formula from the source code:
Quote:
fz = (dz / D) * f;

The acceleration on the z-axis equals the distance on the z axis divided by total distance times total acceleration.  dz, the distance between 2 objects on the z-axis, will always be zero if all your objects have 0 for their z-axis position component.

Title: Re: Kozai (kinda) - sim 4
Post by Mal on 11/10/06 at 07:16:17

Oh, I figured the inclinations would remain at zero. I just wanted to see if the eccentricity cycled, and so far it looks like it does - it goes from 0 to 0.1 to 0 in about 450,000 years and appears to be repeating itself... I'll post a graph tonight.

Title: Re: Kozai (kinda) - sim 4
Post by Mal on 11/10/06 at 20:52:06

872,400 years (planet eccentricity and Arg of Peri):

http://www.evildrganymede.net/temp/sim4.gif

I changed to a 2048 second timestep around 760,000 years, doesn't seem to have made any noticeable difference to anything.

Title: Re: Kozai (kinda) - sim 4
Post by Tony on 11/10/06 at 21:54:51

I was wrong about it being chaotic rather than periodic.  It's interesting to watch Gravsim get confused on Arg of Peri when eccentricity drops to 0, giving its periapsis formula undefined or poorly-defined numbers to work with.

Your simulation identifies a few of the components in the Milankovitch cycle: a cycle thought responsible for periodic ice ages.  As Earth's eccentricity waxes and wanes due to Jupiter having a similar effect as your companion star, it affects how much solar radiation we receive.  The northern hemisphere of Earth and the southern hemisphere do not absorb sunlight equally as the north has more land (though I forget if that makes it better or worse at absorption).  Currently, the northern hemisphere is fartest from the Sun in its summer.  So dampening out eccentricity minimizes this contrast.

Likewise, having our perihelion march around the Sun alternates which hemisphere is closest during its summer.

There's other components to this cycle not included here.  The tilt of Earth's axis has a 26,000 year precession.

All these factors together come into and out of phase in a period of about 100,000 years, and it is theorized that this leads to cyclic climate change.

Title: Re: Kozai (kinda) - sim 4
Post by Mal on 11/10/06 at 22:27:11

Oh yeah, I didn't make that connection... I was actually doing this to see if there was still eccentricity cycling if the inclinations were zero, and there clearly is. So I guess the Kozai mechanism is just this eccentricity cycling plus another layer of cyclicity that involves the inclination?

I wonder what would happen if you had a relative inclination of say 20 degrees - less than the critical one for the Kozai cycle. Would we just end up with a graph like this one still? I'll give it a go and see what happens.

Title: Re: Kozai (kinda) - sim 4
Post by Tony on 11/10/06 at 22:54:14

you might want to try closer to the kozai critical value of 39 degrees ( I think ).

Kozai mechanism is where you trade ecc for inc.  Below this value, you may get ecc cycling, but not at the expense of inc.  Curious to see your results.

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