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Kozai - sim 3 (Read 11517 times)
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Kozai - sim 3
11/03/06 at 18:22:05
 
New sim here. Just seeing if it makes any difference to the output if the planet is at 75 degree inclination and the star and companion are at 0 degrees.  
 
I'm also running it at 1024 because I'm impatient Wink.
 
 
--------------------
Simulation (3)
 
Primary Star: 1 solar mass, 1 solar radius (yellow)
 
Planet: SMA = 5 AU, mass = 1 Jupiter mass, radius = 69,911 km , e=0.000, i=75, long of asc. node = 0, arg of perifocus = 0, mean anomaly = 16. (purple)
 
Companion: SMA = 250 AU, mass = 0.9 solar masses, radius = 626400 km (0.9 solar), e = 0.800, i = 0, long of asc. node = 0, arg of perifocus = 0, mean anomaly = 140. (red)
 
Timestep = 1024 secs.
Start time = year 1 1 0 , time 0 0 0
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Re: Kozai - sim 3
Reply #1 - 11/03/06 at 20:36:00
 
This is what I did to test the Excel formula I gave you.  With the companion orbiting in the ecliptic, now planet inclination will give you inclination from the plane of the secondary.
 
I merged 2 spreadsheets together so I could compare:
  • 1. The inclination of the planet in the "secondary = 75 degree" sim
  • 2. The inclination of the planet relative to the secondary in the "secondary = 75 degree" sim using the Excel formula
  • 3. The inclination of the planet relative to the ecliptic in the "secondary = 0 degree" sim
  • 4. The inclination of the planet relative to the secondary in the "secondary = 0 degree" sim using the Excel formula.

 
I found that the 1st one was much different than the other 3.  #2 & #3 were within a degree of agreement.  This is because the secondary does get tugged around a little bit, so it is often a degree or so away from the ecliptic.
 
And I found that #2 & #4 were in perfect agreement out to about 4 decimal places, demonstrating that the Excel formula does the trick for giving you inclination between 2 non-ecliptic planes.
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Re: Kozai - sim 3
Reply #2 - 11/03/06 at 22:38:39
 
Huh. That's interesting...  
 
After 80,000 years, the planet's eccentricity is going up as expected, but the inclination is pretty constant. And the companion's inclination is also hovering around zero. So neither object's inclination is doing much at all now!  
 
By this point in the other simulations the planet's inclination was going up at a constant rate.
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Re: Kozai - sim 3
Reply #3 - 11/04/06 at 00:57:05
 
131,100 years:
 
Planet eccentricity still rising (0.13 now). Inclination constant. LAN and Arg of Peri are decreasing slowly with time.  
 
Companion ecc and inc are level (with slight variation). LAN and AoP are roughly constant (0 +/- about 1.5)
 
This is weird. Shouldn't the inclination of one of these be changing?
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Re: Kozai - sim 3
Reply #4 - 11/04/06 at 07:39:28
 
278,000 years.
 

 
Well look at that. The planet inclination looks like it's doing what it's supposed to be doing after all - but all the action occurs when the eccentricity peaks. But it goes from 75 down to 40 and then back up again.
 
The bottom graph is the companion inc (left axis) and ecc (right axis).
 
I'll let this get to the next peak so we can see how long the cycle is, but it looks like Tony's correct about the inclination cycle being visible in the graph when the planet is the body that's inclined.
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Re: Kozai - sim 3
Reply #5 - 11/04/06 at 10:49:27
 
I'm running the sim you describe here at 1024 time step, except I put the planet in 0 inc and the secondary in 75 deg.
 
Maybe you already ran this, but just to double check I'm running it now.
 
After 100,000 years, the familar increase in inclination in a linear, stair-steping line is showing.
 
I suspect the differences are related to how inclination is read in each case.  If I apply the Excel formula from the other post to my run, I suspect my inclination graph will look like yours, and that's the correct way to read it.  I'll let you know in 100 thousand years  Wink.
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Re: Kozai - sim 3
Reply #6 - 11/04/06 at 11:01:37
 
Quote from Tony   on 11/04/06 at 10:49:27:
I'm running the sim you describe here at 1024 time step, except I put the planet in 0 inc and the secondary in 75 deg.

Maybe you already ran this, but just to double check I'm running it now.

 
Yeah, that's the second sim I ran, though I used the 512 timestep for it.  
 
Quote:
I suspect the differences are related to how inclination is read in each case. If I apply the Excel formula from the other post to my run, I suspect my inclination graph will look like yours, and that's the correct way to read it. I'll let you know in 100 thousand years Wink.

 
I'm just doing it this way to verify that the inclination cycle can be seen if you have the companion at 0 degrees and the planet at 75 like you said earlier. And it looks like it does.
 
Still, I wonder if there can be a way in GravSim to specify an ecliptic plane? Maybe a checkbox when you're entering an object that you can tick to say "this is the ecliptic plane of the system - all inclinations are measured relative to this"? And if no plane is specified then the ecliptic is assumed to be whatever it is in the current version of the program?
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Re: Kozai - sim 3
Reply #7 - 11/04/06 at 16:38:53
 
I goofed up in setting up my output file.  I asked for once a year for 30000 years, so no data after 30000 years.
 
Do you still have the spreadsheet from your 512K run?  Can you zip up both of your spreadsheets and send them to me so I can play with them?
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Re: Kozai - sim 3
Reply #8 - 11/04/06 at 21:02:47
 
549,600 years.
 

 
Well this turned out to be more like it.  
 
Only problem is that the period of the oscillation is about 220,000 years, not 335,000 years like it's supposed to be. No idea why.
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Re: Kozai - sim 3
Reply #9 - 11/04/06 at 21:28:59
 
Now I'm curious to see a similar graph from your 512K run, but with the planet and secondary inclinations swapped.
 
Ecc should be the same, so any difference will expose the time step as the source of the difference.
 
Inclination should be the same if you create a new column and use the Excel formula I posted, rather than Gravity Simulator's default output.
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Re: Kozai - sim 3
Reply #10 - 11/04/06 at 22:51:32
 
Quote from Tony   on 11/04/06 at 21:28:59:
Now I'm curious to see a similar graph from your 512K run, but with the planet and secondary inclinations swapped.
 
 
You already have:
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?board=news;action=display;num =1162145224
 
I'll see if the formula works out the same.
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Re: Kozai - sim 3
Reply #11 - 11/04/06 at 23:08:36
 

 
I'd say we have a match Smiley
 
The blue line is the unmodified inclination from the 512 timestep run, with the comp starting at 75 and the planet at 0.  
 
The pink line is the inclination from the 512 run with your excel formula added to it (oddly enough, I managed to paste the formula into exactly the right column for the cells to be correct!)
 
The yellow line is the unmodified inclination from the 1k run with the comp starting at 0 and the planet at 75.
 
Looks like the formula works! 8) (you'll probably want to keep that in a safe place!)  
 
 
The period is still a problem though. I don't get 220,000 years from the calculation if I swap the planet and companion numbers around, so it can't be that.
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Re: Kozai - sim 3
Reply #12 - 11/05/06 at 01:21:30
 
Quote from Mal   on 11/04/06 at 23:08:36:

The period is still a problem though. I don't get 220,000 years from the calculation if I swap the planet and companion numbers around, so it can't be that.

Remember, the author of the paper that gave the formulas used the ~ sign instead of the = sign.
 
Looks like there's still more to be learned about the Kozai Mechanism  Shocked
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Re: Kozai - sim 3
Reply #13 - 11/05/06 at 08:40:08
 
Quote from Tony   on 11/05/06 at 01:21:30:

Remember, the author of the paper that gave the formulas used the ~ sign instead of the = sign.

 
Yes, but this is way beyond the error caused by approximation. If the formula predicted 335,000 years and we ended up with 330,000 years I'd agree, but this is so far off it that something's got to be wrong somewhere. Either the formula is incorrect, or GravSim is doing something wrong, or the parameters in the simulation were entered incorrectly.
 
Unfortunately there's no way to verify the latter, because GravSim's "Edit Object" window doesn't allow you to see the parameters you entered. I can use it to see if the masses for each object were correct (and they are), but it only gives me current distances (once I "pythagorize" them), not semimajor axis or inclination or eccentricity or anything else I actually entered to create the object. Again, I think the edit window needs to be made a lot more userfriendly and relevant - being able to change an object's xyz location or velocity components really isn't useful.
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Re: Kozai - sim 3
Reply #14 - 11/05/06 at 13:08:10
 
You can see what you've entered by pressing View > Add Orbital Elements Box.
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