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Eccentricity in binaries - simulation 2 (Read 11523 times)
EDG
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Eccentricity in binaries - simulation 2
10/29/06 at 10:07:04
 
New simulation here:
 
 
-----------------------
Simulation (2)
 
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=0.000°, 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 = 75°, long of asc. node = 0°, arg of perifocus = 0°, mean anomaly = 140°. (red)
 
Timestep = 512 secs.
Start time = year 1 1 0 , time 0 0 0  
-----------------------
 
Should be quicker to run, hopefully the timestep is low enough to rule out any problems... going to run it for 500,000 years, the kozai cycle should be 335,795 years.
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Re: Eccentricity in binaries - simulation 2
Reply #1 - 10/29/06 at 11:45:44
 
Quote from Mal   on 10/29/06 at 10:07:04:
New simulation here:


-----------------------
Simulation (2)

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=0.000°, 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 = 75°, long of asc. node = 0°, arg of perifocus = 0°, mean anomaly = 140°. (red)

Timestep = 512 secs.
Start time = year 1 1 0 , time 0 0 0
-----------------------

Should be quicker to run, hopefully the timestep is low enough to rule out any problems... going to run it for 500,000 years, the kozai cycle should be 335,795 years.

 
I observed in my sim that the time to reach the first max. ecc  is appr. 2.5 times the cycle of the Kozai . This could mean that you'll need about 800.000 years to reach max ecc and then the cycle begins .  
 
Frank  
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Re: Eccentricity in binaries - simulation 2
Reply #2 - 10/29/06 at 15:41:12
 
Well this is a bit odd. I'm taking samples every 100 years, and this is what I get for planet inclination and eccentricity:
 

 
Those spikes are not always equidistant - they're 2900 years +/- 100 years apart, with no apparent pattern to the variation (sometimes it's 2800, sometimes it's 2900, sometimes it's 3000). I thought they might be the orbital period of the companion (ie due to close approaches when it's at perihelion?) but if I've done the calculation right the companion's orbital period is 3952 years. That said, I'm not sure what happens if the masses of the primary and compaion are similar, does that shorten the orbital period?
 
EDIT: I got GravSim to calculate the period and it agrees that it's about the same as I calculated above - so I dunno what that 3000 year period is!
 
Note that this is after only 30,000 years or so, so it's still early days.
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« Last Edit: 10/29/06 at 19:13:11 by EDG »  

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Re: Eccentricity in binaries - simulation 2
Reply #3 - 10/29/06 at 18:30:56
 
More oddness.  
 

 
Top show the planet's orbit, bottom shows the companion's. Note that the spikes in eccentricity in both bodies align. There's also sharp dips in the Arg of Peri of the planet too at the same time (and smaller upward kinks in the LAN of the planet).
 
Wonder what's going on??
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Re: Eccentricity in binaries - simulation 2
Reply #4 - 10/29/06 at 19:47:35
 
Got it!
 
I was watching the simulation around the time of the next encounter (33300 years), and it IS the closest approach of the companion.  
 
At 33300 the companion was at closest approach to the star, and the planet was directly between it and the primary. So it's obviously getting a sharp gravitational tug from this, which bumps up its eccentricity, but once the companion has passed the eccentricity isn't as low as it was before the encounter. So the planet eccentricity gets ratcheted higher with each pass. I've only seen this one encounter - I'm guessing the different height of the spikes is because the planet and companion aren't exactly lined up in each encounter?
 
Though I'm still not sure why it's every 2800-3000 years if the orbital period of the companion is longer than that...
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Re: Eccentricity in binaries - simulation 2
Reply #5 - 10/29/06 at 20:25:23
 
Quote from Mal   on 10/29/06 at 19:47:35:

Though I'm still not sure why it's every 2800-3000 years if the orbital period of the companion is longer than that...

 
It's because you've helped uncover a bug in the beta's period formula  embarrassed
 
The forumla it uses is the generalized formula for when M1 >> M2:
 

 
In this case M1 is not >> M2, and it needs to use M1+M2 for M in the formula.
 
The real answer for period in of the companion is .... drumroll please.....
 
 
2868 years
 
You can use this formula in Excel to get the correct period.  1.9 represents 1.9 solar masses which is M1 + M2:
 
=2*3.14159 *SQRT(( K2*1000)^3/(0.000000000066725985*(1.98891691172467E+30*1.9)))
 
And here's a utility you might find helpful.  It's a javascript period calculator:
http://orbitsimulator.com/gravity/articles/PeriodCalculator.html
 
Also, you can have Gravity Simulator compute an object's period using the "Rotating Frame Adjustment" window.  Choose the object and it will display its period in seconds.  It gets this one right as it uses M1+M2.  Then just hit cancel so you don't go into Rotating Frame mode.
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Re: Eccentricity in binaries - simulation 2
Reply #6 - 10/29/06 at 22:16:30
 
Aha, I thought the formula was a bit different when the masses of the bodies were similar. Huzzah for experimentation! Smiley
 
This is cool, I feel like I'm doing real Science! here Wink.
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Re: Eccentricity in binaries - simulation 2
Reply #7 - 10/29/06 at 22:23:36
 
Quote from Mal   on 10/29/06 at 22:16:30:

This is cool, I feel like I'm doing real Science! here Wink.

 
You are!  How many people in the world, besides the authors of the quoted papers, do you think know more about the Kozai Mechanism than you?
 
The real science will be when we can figure out why the formulas break down at large distances.  Does it have to do with the Kozai Mechanism, or with our methods?
 
Your 750 AU sim was qualitatively correct, but the timing was seriously off.  My 400 AU sim was qualitatively correct, and the timing was roughly correct.  All sims less than 400 AU had good to perfect timings.
 
 
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Re: Eccentricity in binaries - simulation 2
Reply #8 - 10/29/06 at 22:31:02
 
Well I looked at the close approach at 62000 years and the companion and planet weren't lined up then. So it's definitely not repeating the exact configuration each time.  
 
I wonder what the inverted spikes are though - sometimes the eccentricity suddenly drops and then shoots up and back down a bit to get to the next step. I'm guessing the planet is on the opposite side of the sun to the companion's pericentre there?
 
So was this happening in the 750 AU sim too, but we couldn't see it because of the timestep? And is this what the Kozai mechanism really is? Because this "tugging at closest approach" thing seems more like a normal direct orbital mechanics situation, I thought the Kozai mechanism was not such a direct thing.  
 
I'm wondering if the timings were off in the more distant ones because of the timesteps. For the closer ones we can use shorter timesteps and not die of old age waiting for th results. For the further ones we have to use larger timesteps if we want to see the results in a few days, and that might be messing things up?
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Re: Eccentricity in binaries - simulation 2
Reply #9 - 10/29/06 at 22:37:10
 
Quote from Mal   on 10/29/06 at 22:31:02:

So was this happening in the 750 AU sim too, but we couldn't see it because of the timestep?

 
Not the time step of the simulation, but the time step of the data output.  You're now asking for output to be plotted every 100 years instead of every several thousand years.  So the spikes are just completely skipped over, and even the stair-stepping is just reduced to fuzz on the curve plotted ever few thousand years.  By asking for data every 100 years, you're zooming in on the fuzz.
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Re: Eccentricity in binaries - simulation 2
Reply #10 - 10/29/06 at 22:45:10
 
Quote from Mal   on 10/29/06 at 22:31:02:

I'm wondering if the timings were off in the more distant ones because of the timesteps. For the closer ones we can use shorter timesteps and not die of old age waiting for the results. For the further ones we have to use larger timesteps if we want to see the results in a few days, and that might be messing things up?

 
Patience is a virtue.  But that said, I think the correct approach is to start with short doable sims to try to spot a trend.  Maybe make a graph of how accurate the formula is vs. AU of companion.  Then derive a formula based on the graph and use it to predict the Kozai period for a high AU secondary.  Then perform a sim that might just take a few weeks to run.
 
Just some random thoughts...
Gravity Simulator takes up 50% of your computer's CPU cycles.  I've asked on coding forums how to increase this, but have never got a solution.  But because of this, you can easily check e-mail, surf the web, etc with gravity simulator running in the background.
 
Or... you can run 2 sims at the same time with little or no comprimise in the speed vs. running only 1 sim.  And e-mail, etc, isn't impossible, only sluggish, when running 2 instances of Gravity Simulator at the same time.
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Re: Eccentricity in binaries - simulation 2
Reply #11 - 10/30/06 at 07:05:58
 
156,000 years:
 

 
I hope the eccentricity isn't flattening out yet - we're nearly at halfway through the cycle and it's still nowhere near 0.94... and the inclination is at 32 degrees and still rising.
 
I think what SHOULD be happening is that the eccentricity goes up to 0.94 around 170,000 years, and then the inclination reaches about 35 degrees at the same time and starts going down when eccentricity reaches its peak. I'm not sure if that's going to happen here...
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Re: Eccentricity in binaries - simulation 2
Reply #12 - 10/30/06 at 20:51:28
 
306,200 years (planet):
 

 
Well this looks a bit promising. The eccentricity peak's at the about the right height (0.934-ish). I presume the next peak is going to be 335,000 years from that point.  
 
The LAN's obviously dropped through 0 degrees and looped round to 360 degrees and still dropping, which explains the vertical line there. Though I have no clue whatsoever what is going on with the Inclination (yellow) or Arg of Peri (cyan). The Inc peaks at about 136 degrees and then starts dropping down again.
 
The companion inclination and eccentricity isn't doing much, still just varying slightly around its starting values.  
 
I got nothin'. Anyone got any ideas what's going on here?
 
Is there any way we can calculate the relative inclination here?
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Re: Eccentricity in binaries - simulation 2
Reply #13 - 10/30/06 at 23:05:41
 
I doubt the companion is going to be affected much by the planet.  Though it would be interesting to see what would happen if two stars orbited at 2.5 AU, and a Jupiter-mass companion orbited the pair, out of plane from a distance.  I don't think the formulas account for such a scenerio.
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Re: Eccentricity in binaries - simulation 2
Reply #14 - 10/30/06 at 23:10:14
 
Quote from Tony   on 10/30/06 at 23:05:41:
I doubt the companion is going to be affected much by the planet. Though it would be interesting to see what would happen if two stars orbited at 2.5 AU, and a Jupiter-mass companion orbited the pair, out of plane from a distance. I don't think the formulas account for such a scenerio.

 
I wouldn't expect it to be affected by the planet, but I don't think we're seeing the cyclic variation in inclination that we're supposed to be seeing here. But what's weird is that the eccentricity seems to be doing what it's supposed to now.  
 
Could there be something missing in gravity simulator's code here that prevents it from doing the inclination cycle properly? Or is the inclination actually doing the right thing here, but we're just not seeing it because we're not looking at the relative inclination (which presumably is calculated in a more complex way than just "companion inclination minus the planet inclination").
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