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Message started by Tony on 11/06/06 at 20:47:38

Title: Alpha Centauri Animation
Post by Tony on 11/06/06 at 20:47:38

Here's an animation I created showing the Kozai Mechanism at work in the Alpha Centauri system.

Two hypothetical planets orbit Alpha Centauri A at distances of 1 AU and 2 AU.  Their orbits are inclined 60 degrees from the orbital plane of Alpha Centauri A and Alpha Centauri B.  This shows about 1350 years, with a screen shot being generated every 2.7 years ~ the period of the outer planet.  The time step used in the simulation was 2048.

Alpha Centauri B is not visible in this animation, but its presence is certainly felt.  The orbit of the outer planet trembles about every 80 years in response to periastron, when the A/B distance decreases to only 11 AU.
http://orbitsimulator.com/gravity/images/acAnimation.GIF

Title: Re: Alpha Centauri Animation
Post by Mal on 11/07/06 at 07:27:18

Interesting that the inner planet doesn't do much... I was wondering what happened if you have more than one planet...

Is the outer planet stabilising the inner planet orbit perhaps? What happens if you put say, Mercury, Venus, Earth, and Mars at the same distances around A (on inclined orbits) as they are around Sol?

Title: Re: Alpha Centauri Animation
Post by Tony on 11/07/06 at 11:05:56

The inner planet is doing the same thing as the outer planet, except much slower.  I gave each planet 1 Earth Mass, so once they get out-of-plane, you've got a double Kozai going on.  I'll try it with Mercury-Mars to see what happens.

Here's the graph of the full sim that produced the animation.  The animation only takes you up to the outer planet's first max ecc.  The graph covers 10x as much time.  I stopped the graph there because The outer planet's eccentricity exceeded 1 and it escaped the system.

http://www.orbitsimulator.com/gravity/images/acGraph01.GIF

Title: Re: Alpha Centauri Animation
Post by Mal on 11/07/06 at 17:59:24

Well that's interesting, I wonder if the extent of the cycling of that outer planet's inc would decrease again given time.

So GravSim does actually realise that the eccentricity can get higher than 1 and thus turn the orbit into a parabola/hyperbola?

This mechanism really puts a crimper on planets around multiple systems... I wonder if you've tried this for that planet they found orbiting the triple system last year:
http://www.space.com/scienceastronomy/050713_triple_sun.html

Though I think it won't be affected much even if the compainions were inclined - it's very close to the star.


BTW, what happens if you have a low mass object in an external orbit (eg. swap the stellar companion and planet)? Does the planet get affected by the companion in the same way?

Title: Re: Alpha Centauri Animation
Post by Tony on 11/07/06 at 21:31:25


Mal wrote:
BTW, what happens if you have a low mass object in an external orbit (eg. swap the stellar companion and planet)? Does the planet get affected by the companion in the same way?

That's an interesting question.  I'd have to modify the code to figure it out.  Although Gravity Simulator gives you the ability to create an object with orbital elements with respect to a system barycenter, it only outputs orbital elements with respect to a single object.  So the data wouldn't be any good.  But at least the plot might yield a clue.  I've got it running now.

Title: Re: Alpha Centauri Animation
Post by Mal on 11/07/06 at 21:55:47

BTW, how do the periods of the oscillations that you get compare to the Pkoz calculation? Do they match, or are they different?

Title: Re: Alpha Centauri Animation
Post by frankuitaalst on 11/07/06 at 22:48:39


Mal wrote:
BTW, how do the periods of the oscillations that you get compare to the Pkoz calculation? Do they match, or are they different?


Interesting question : out of the sims of the 5 earth-moon I always get a "match" , but not totally . Where the period should be rising with distance^3 according to the formula I get a period rising with distance^3.4 , regardless of the time-step I use ( steps between 64 and 512 ) . The factor I become ist always between 3.3 and 3.45 .  This means a difference of about 10% .

I also noticed that the period is not exact or equal between two peaks . This doesn't surprise me because peaks (ecc) occur at slightly different incl and LAN .


Unfortunately I have some difficulties uploading pictures .
:(

Title: Re: Alpha Centauri Animation
Post by Tony on 11/07/06 at 23:19:46


Mal wrote:
BTW, how do the periods of the oscillations that you get compare to the Pkoz calculation? Do they match, or are they different?


Outer Planet:
If you look at the graph, it takes 1730 years to reach max eccentricity from its initial 0 eccentricity.  Multiplying this by 2 gives 3460 years as a period.  The computed value is 3400.  Not a bad match since I'm just eyeing the graph for dates.

But then following that, the period gets shorter.  You'll notice that from the starting point where eccentricity was set to 0, it rose slowly at first.  But after bottoming out at the first valley, it doesn't reach eccentricity = 0.  And it rises faster.  Consider the peaks centered on years 1736 to 6920.  It does 3 complete cycles.  (6920-1736)/20 = 1728 years per cycle, almost half of what is expected.

Inner Planet:
From the graph, it takes about 5400 years to reach max ecc from its initial starting ecc of 0.  Doubling this gives 12800 years.  But again, eccentricity takes a long time to start picking up from ecc=0.  Peak-to-peak is only about 6500 years, again about half the original value.  The computed value is 9800 years.

I notice the same thing going on in Figure 2 (page 17) in Takeda's paper.  

0.9 solar mass companion:
Time from begining to first peak is 16 million years.  Doubling gives 32 million years.  But time from 1st peak to 2nd peak shortens considerably to 22 million years.  Considering all 15 cycles on the graph, it has a time averaged period of 20.5 million years.  The computed value is 25.6 million years.

0.08 solar mass companion:
Time to first peak = 186 million years.  Doubling this gives 372 million years.  But the time from the first peak to the 2nd valley = 120 Myrs.  Doubling this gives 240 Myrs.  The computed value is 288 Myrs.

So it seems that the period is not the same from cycle to cycle and depends heavily on how close to ecc=0 the valley comes;  the closer, the longer the period.

This sort of makes sense then the author would use the asymptoticlly equal to symbol.  It's as if to say don't trust this formula for any one specific period, but as cycles approach infinity, the time-averaged period should approach the computed period.  

Title: Re: Alpha Centauri Animation
Post by Tony on 11/07/06 at 23:30:32


frankuitaalst wrote:
Unfortunately I have some difficulties uploading pictures .
:(

Check your private box.  I'll send you a link where you can upload them to my server.

Title: Re: Alpha Centauri Animation
Post by Tony on 11/07/06 at 23:33:04


Mal wrote:
BTW, what happens if you have a low mass object in an external orbit (eg. swap the stellar companion and planet)? Does the planet get affected by the companion in the same way?

This is the 2nd time I'm quoting this sentence.  I started with no eccentricity in either the planet or the 2nd star.  But other than that, the sims are the same with the planet and star reversed.  The results are interesting enough for me to make an animation.  Stay-tuned.  This'll have to run overnight.

Title: Re: Alpha Centauri Animation
Post by Tony on 11/08/06 at 16:57:56

Well, I never did make the animation.  But I can see what is going on.  The planet, inclined 60 degrees from the barycenter, with initial ecc=0 remained 60 degrees inclined and ecc=0 for the dureation of the simulation.  However, its longitude of ascending node advanced non-stop, completing 360 degrees in about 16000 years.

Title: Re: Alpha Centauri Animation
Post by Tony on 11/09/06 at 11:44:03

I placed Mercury, Venus, Earth & Moon, & Mars around Alpha Centauri A, and also around Alpha Centauri B.  Since ACA it is 1.1 times as massive as the Sun, I also added sqrt(1.1) to their velocities so their orbits would retain their solar orbit properties.  Likewise, since ACB is 0.907 times as massive as the Sun, I subtracted sqrt(0.907) from their velocities.

After only 1800 years at a nice slow time step of 64:
Alpha Centauri A
http://www.orbitsimulator.com/gravity/images/ac4p.GIF

Alpha Centauri B
http://www.orbitsimulator.com/gravity/images/ac4p2.GIF
Later, I'll turn this into an animation

Title: Re: Alpha Centauri Animation
Post by Mal on 11/09/06 at 17:01:51

Is the red orbit Mars?

Title: Re: Alpha Centauri Animation
Post by Tony on 11/09/06 at 22:33:43

Yellow - Alpha Centauri (A or B depending on image)
Gray - Mercury
White - Venus
Blue / Gray - Earth / Moon
Red - Mars

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