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Kozai - sim 3 (Read 11499 times)
EDG
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Re: Kozai - sim 3
Reply #15 - 11/05/06 at 14:59:08
 
Quote from Tony   on 11/05/06 at 13:08:10:
You can see what you've entered by pressing View > Add Orbital Elements Box.

 
 
Looks OK then...
Though the planet starts at 196° for the Arg of Peri for some reason. And the companion starts at 180° for the Arg of Peri too. But I entered them both as 0°. Any reason for this?
 
It says the mean anomaly for the planet is 0° too, and I'm sure I entered 16° for it. Though I could have entered that for the Arg of Peri, and if the program thinks that 180° is 0° for that then that might explain why the Arg of Peri is 180+16...
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Re: Kozai - sim 3
Reply #16 - 11/05/06 at 20:38:51
 
Quote from Mal   on 11/05/06 at 14:59:08:



Looks OK then...
Though the planet starts at 196° for the Arg of Peri for some reason. And the companion starts at 180° for the Arg of Peri too. But I entered them both as 0°. Any reason for this?

It says the mean anomaly for the planet is 0° too, and I'm sure I entered 16° for it. Though I could have entered that for the Arg of Peri, and if the program thinks that 180° is 0° for that then that might explain why the Arg of Peri is 180+16...

When you enter 0 for the argument of perigee in an orbit whose ecc = 0, you telling it to make a perfectly round orbit, with the closest point 0 degrees from the Longitude of the Ascending node.  But since a perfect circle has no closest point, this is a meaningless entry.  When you run the sim, even for a couple of timesteps, small amounts of eccentricity are introduced into your perfectly round orbit, and Gravity Simulator picks up on this.  With the ecc very low, the argument of periapsis can rapidly and chaotically shift.
 
As for the companion, even though it has a lot of eccentricity, you entered 0 for the inclination.  So its Longitude of the Ascending node is undefined.  And since Argument of Periapsis is degrees away from this undefined point, it too is undefined, or poorly defined, so it also chaotically dances all over the place too.
 
The same applies to mean anomoly.  It's a measurement off the Longitude of Ascending node in an orbit that has an undefined Ascending node since its inclination is 0.  So Mean Anomoly chaotically flops all over the place too.
 
When I re-do the edit box to include orbital elements, I may also provide True Longitude which should be more intuitive since it is a measure of degrees from a fixed line.  And I may add Longitude of Perihelion, as it too would be a measurement against a fixed value rather than a dynamic one.
 
As for why the simulated Kozai period doesn't match the computed one, I'm not sure.  It's within an order of magnitude of correct.  You’re expecting 220,000 and getting 330,000.  That’s a lot different than expecting 220,000 and getting 3000, or getting 40,000,000.  It would be interesting to know what the author meant by “~”.  Perhaps an e-mail to the author may shed some light on this.  Astronomers are famous for “order of magnitude” predictions.  
 
Or perhaps the difference has to do with general relativity.  Gravity Simulator does not model it (not yet at least  Wink ).  It only propogates objects forward in time, using Newton’s laws, based on their initial positions and velocities. The paper talks about GR being a big perturber to the Kozai Mechanism.  But I don’t imagine it would have this large of an effect since GR only perturbs Mercury’s Longitude of perihelion by 43 arcseconds per century.  And Mercury is much closer to the Sun than the simulated planet at periastron.
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Re: Kozai - sim 3
Reply #17 - 11/07/06 at 00:02:58
 
After reading the papers a little more carefully, I don't think the difference has to do with GR.  The formula for Kozai period was credited to a paper authored by Ford et. al. 2000.  Here's a link to that paper:
http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v535n1/40691/40691.web.p df
Equation 36 on page 6 gives the formula for Kozai period, and in the text explaining the formula, it says Quote:
The period of eccentricity oscillations is given approximately by Pe~P1((m0+m1)/m2)*(a2/a1)^3*(1-e2^2)^3/2, where P1 is the orbital period of the inner binary (Mazeh & Shaham 1978 ).  This expression should be multiplied by a coefficient of order unity that can be obtained using Weierstrass's zeta function as shown by Kozai (1962).

Does anyone know what a coefficient of order unity is?
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Re: Kozai - sim 3
Reply #18 - 11/07/06 at 07:24:56
 
Quote from Tony   on 11/07/06 at 00:02:58:
Does anyone know what a coefficient of order unity is?

 
I think it means "about 1". Unity = 1. (why they don't just say that is beyond me. Scientists have a talent for overcomplicating things Wink )
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Re: Kozai - sim 3
Reply #19 - 11/07/06 at 12:50:14
 
Quote from Mal   on 11/05/06 at 08:40:08:


Yes, but this is way beyond the error caused by approximation.

 
Reading a bit more into this, the symbol the author uses is a tilda (~) on top of a dash (-).  It does not mean approximately equal to.  It means "asymptotically equal".  The symbol for approximately equal to are two tildas (~) on top of each other.
 

 
I'm not sure what asymptotically equal is.  I know what an asymptote is as it relates to a curve.  But I'm not sure how it relates in this situation.
 
We have noticed that the difference between the simulation and the computed values get better at smaller distances, perhaps converging on perfection at a very small value.  That's what an asymptote does on a graph.  So perhaps that's how it applies here.
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Re: Kozai - sim 3
Reply #20 - 11/07/06 at 13:22:36
 
Concerning the ~ sign and the meaning in the article I think the author meant "approximately equal" .  
As I look an the formulas from which the formula is deducted I think there is a lot of nonlinearity in it .  
 
For example if one says the period of a suspended beam is ~ Ct*square (l/g) he is right  , beacuse saying = Ct*square (l/g) neglects the existence of nonlinear terms in the equation of motion .  
 
"assymptotically equal" could mean equal in the long term , but I don't think this is meant here.  
 
Concerning the Weierstrass zeta funktion , this is really not clear at all ,because the Weierstrass Zeta funktion is a function of  more than one variable . God ( Kozai) knows which variables .
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Re: Kozai - sim 3
Reply #21 - 11/07/06 at 17:44:20
 
Either way - obsessing over what the symbol means doesn't change the fact that 220,000 is not 335,000 years. Usually if they just mean "of the same order of magnitude as the result of the equation" they'll just say that.  
 
I still think something fishy is going on here - I emailed Genya about it but he's not replied (I'm finding that scientists are surprisingly bad at responding to inquiries about their work in a timely manner from the general (educated) public. I guess graduates have more time on their hands than the professors though. It's odd though because if someone asked me about my work I'd be very keen to discuss it with someone who actually showed an interest!).  
 
BTW, Tony - do you actually have another algorithm to hand that can replace the existing Euler one with in GravSim? It'd be interesting to see how much of a difference a different numbercrunching algorithm would make to the results.
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Re: Kozai - sim 3
Reply #22 - 11/07/06 at 20:30:04
 
Quote from Mal   on 11/07/06 at 17:44:20:
BTW, Tony - do you actually have another algorithm to hand that can replace the existing Euler one with in GravSim? It'd be interesting to see how much of a difference a different numbercrunching algorithm would make to the results.

Not yet, I want to write an RK-4.  But using it wouldn't be any more accurate than using Euler's method at a slower timestep.
 
For example, Euler's method at a timestep of 64 almost perfectly predicts the precession rate of Mercury due to Newtonian Gravity.  
 
I'm anxious to see what the author says.  Additionally, I'll ask my Astrobiology teacher, Debra Fischer.  Her name appears in the references of both of Genya's papers.  No class until Monday though.
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Re: Kozai - sim 3
Reply #23 - 11/08/06 at 08:51:13
 
Well, after all that it looks like I was actualy mistaken - Genya emailed me back and said that the equation should give a result that is actually 'in the same order of magnitude' as what it should be. So 220k is as good as 335k.  
 
That said, in my defence that wasn't made too clear in the paper Wink. He also was rather impressed that Gravsim can show the kozai mechanism too, so thumbs up from him there Smiley
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Re: Kozai - sim 3
Reply #24 - 11/08/06 at 12:16:49
 
I'm glad you got a response.  Did Genya see all the nice graphs you and frankuitaalst put together?
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Re: Kozai - sim 3
Reply #25 - 11/08/06 at 12:56:26
 
Quote from Tony   on 11/08/06 at 12:16:49:
I'm glad you got a response. Did Genya see all the nice graphs you and frankuitaalst put together?

 
I sent him that comparison graph showing the inclination with the transformation equation applied, he was glad everything worked!
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