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Binary Eccentricity Test - volunteers needed! (Read 30232 times)
Tony
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Re: Binary Eccentricity Test - volunteers needed!
Reply #30 - 10/26/06 at 23:29:04
 
I got tired of punching the formulas into my calculator so...
 

http://orbitsimulator.com/gravity/KozaiCalc.exe
 
Code:
Private Sub Command1_Click()
Dim M0 As Double, M1 As Double, M2 As Double, A1 As Double, A2 As Double, E2 As Double, Pkoz As Double, P1 As Double
Dim pi As Double

Dim i As Double, eMax As Double, pi2 As Double
On Error Resume Next
pi2 = Atn(1) * 4 * 2
i = Val(txi.Text) / 360 * pi2
eMax = Sqr(1 - (5 / 3) * (Cos(i)) ^ 2)

txEmax.Text = eMax


pi = Atn(1) * 4
M0 = Val(txM0.Text)
If optMsun(0).Value Then M0 = M0 * 1.989E+30
If optMjupiter(0).Value Then M0 = M0 * 1.899E+27
If optMearth(0).Value Then M0 = M0 * 5.97E+24

M1 = Val(txM1.Text)
If optMsun(1).Value Then M1 = M1 * 1.989E+30
If optMjupiter(1).Value Then M1 = M1 * 1.899E+27
If optMearth(1).Value Then M1 = M1 * 5.97E+24


M2 = Val(txM2.Text)
If optMsun(2).Value Then M2 = M2 * 1.989E+30
If optMjupiter(2).Value Then M2 = M2 * 1.899E+27
If optMearth(2).Value Then M2 = M2 * 5.97E+24



A1 = Val(txA1.Text)
If Option1(1).Value Then A1 = A1 * 149597870691#
If Option2(1).Value Then A1 = A1 * 1000
A2 = Val(txA2.Text)
If Option1(2).Value Then A2 = A2 * 149597870691#
If Option2(2).Value Then A2 = A2 * 1000

E2 = Val(txe2.Text)
P1 = 2 * pi * Sqr(A1 ^ 3 / (0.0000000000667 * M0))
Pkoz = P1 * ((M0 + M1) / M2) * (A2 / A1) ^ 3 * (1 - E2 ^ 2) ^ (3 / 2)
txPkoz.Text = Pkoz / 86400 / 365.25

End Sub

 

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Tony
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Re: Binary Eccentricity Test - volunteers needed!
Reply #31 - 10/26/06 at 23:35:47
 
How do you put scales on both axes in Excel  ???  (we need a smily face with question marks too!)
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Re: Binary Eccentricity Test - volunteers needed!
Reply #32 - 10/26/06 at 23:40:40
 
Quote from frankuitaalst   on 10/26/06 at 22:04:58:
Hi Tony , Mal , out of my simulation of the 5 earths I got the following result : the time to get a noticible influence decreases rapidly with distance ... so the sim with 750 might run longer .
Also , as distance increases the frequency of the mechanism decreases with time .
And further a noticed that the maximum eccentricity decreases with distance . F.i. in the 3AU earth simulation the moon didn't get to an ecc of 1.
This makes sense to me , intuitionally ...
In one of the articles a also read that time step should be reduced if eccentrity gets high . I guess especcially in the points were the planet is close to the mother .

I never ran it long enough to see the 3AU complete a cycle, but I'm surprised it didn't approach 1.  Can you give us details, such as time step used and a graph, or the .txt file so Mal or I can make a graph from it?
 
Check your e-mail.  I sent you a link to the beta copy.  You should have a copy if you're doing these types of sims.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #33 - 10/26/06 at 23:48:42
 
Quote from Tony   on 10/26/06 at 23:35:47:
How do you put scales on both axes in Excel ??? (we need a smily face with question marks too!)

 
Select the data line in the graph that you want on the secondary axis, rightclick on it, select "Format Data Series", go to the Axis tab, tick the "secondary axis" checkbox. Then you can adjust the values on the new secondary axis on the right and they'll change independently of any changes on the primary axis on the left.  
 
If you're clever about it you can show one graph above another graph if you manipulate the max and min values of their axes accordingly.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #34 - 10/26/06 at 23:57:19
 
Quote from Tony   on 10/26/06 at 18:47:39:
This is identical to your sim #1, except the secondary is at 200 AU instead of 750 AU. All other values are the same.

The formulas give:
Pkoz=486265 years
emax = .94

 
I notice the graph for inclination seems to be flattening out at 90 degrees...  
 
 
Quote:
It almost seems as if the Kozai Mechanism turns off at a certain distance. Maybe that makes sense. It turns off completely at inclinations <~35 degrees.

 
Well this is the thing - if it does turn off around the 40 degree mark, then that means that if the starting inclination is 75, when the planet's inclination increases to above 35, the effect should cease. This is because the companion inclination isn't changing - so as soon as the planet inclination hits (companion inclination - 40), it should behave as if it's coplanar. So how the heck does the effect continue after that?!
 
Could this be why the effect stops working at a certain distance? Because if you run the simulation long enough, the relative inclinations will become lower than the critical one at which the Kozai effect stops?
 
It's bugging me enough that I may email Takeda and ask him what the inclination is doing in his graph. I wonder if he's just assuming it doesn't change (i.e. so that the relative inclinations are always above the critical angle, so Kozai continues).  
 
Quote:
I ran this at time step 16K, which was too fast once the planet's ecc approached 0.94, so I don't trust the graph beyond what's shown.

 
What would be ideal is some kind of automatic, dynamic adjustment by the program, so it notices that the velocity of the object is getting too fast for the stated timestep, and adjusts the timestep accordingly to compensate.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #35 - 10/27/06 at 00:00:06
 
On your graph, if the slope of the ecc line keeps increasing, it'll be close to vertical soon and will quickly climb to the 0.94 value it's predicted to have.
 
The curve at this point looks like it can be approximated as an exponential.  In reality its sinusoidal.  Can you get Excel to draw you an exponential trend line and display the equation? Then use that equation to extrapolate to 25 million years.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #36 - 10/27/06 at 00:09:51
 
Quote from Mal   on 10/26/06 at 23:57:19:

...It's bugging me enough that I may email Takeda and ask him what the inclination is doing in his graph. I wonder if he's just assuming it doesn't change (i.e. so that the relative inclinations are always above the critical angle, so Kozai continues).

...
What would be ideal is some kind of automatic, dynamic adjustment by the program, so it notices that the velocity of the object is getting too fast for the stated timestep, and adjusts the timestep accordingly to compensate.

 
Let me ponder this for a day.  It might have to do with a combination of inc and ecc.  But you will notice that if you plug values under ~35 degrees into the formula for eMax that you end up with a negative sign under the radical which generates an error (or in the case of the above mentioned calculator generates a 0 as the program is told to ignore errors.)
 
You can use Autopilot to adjust the time step, providing you know when you want the time step to change.  In simulations such as this, it will work since you know from the formulas when ecc will peak (1/2 the period), so you can compute this time, add and subtract 20% and adjust the timestep accordingly using Autopilot.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #37 - 10/27/06 at 01:27:02
 
I'm 1 million years into a sim with secondary sma=400, and unlike my sma=200 sim, its behaving a lot like the sma=750 sim.  The ecc is averaging a value without picking up steam.  4 million years is the predicted Kozai period.  So it only has 1 million more years to reach its peak at the half-cycle.  It better pick up the pace and fast...
 
Results tommorow morning  Grin
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Re: Binary Eccentricity Test - volunteers needed!
Reply #38 - 10/27/06 at 07:24:31
 
Quote from Tony   on 10/27/06 at 00:00:06:
On your graph, if the slope of the ecc line keeps increasing, it'll be close to vertical soon and will quickly climb to the 0.94 value it's predicted to have.

The curve at this point looks like it can be approximated as an exponential. In reality its sinusoidal. Can you get Excel to draw you an exponential trend line and display the equation? Then use that equation to extrapolate to 25 million years.

 
I can't get an exponential one that fits, but a 3rd order polynomial claims that it'll be 0.125 at 25 million years.
 
At 11.34 Ma, planet eccentricity is now at 0.019 (de/dt still increasing), inclination is 30 degrees, and arg of pericentre is converging to about 98.5 degrees.  
 
(companion inclination is still at 75 degrees)
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Re: Binary Eccentricity Test - volunteers needed!
Reply #39 - 10/27/06 at 10:05:36
 
My 400 AU sim ultimately gained lots of eccentricity.  But it peaked at ~0.8 instead of ~0.9 as predicted by the formula, and it peaked a little late.  The formula had it peaking (assuming peak to be 1/2 period) at just under 2 million years.
 
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Re: Binary Eccentricity Test - volunteers needed!
Reply #40 - 10/27/06 at 11:18:56
 
It's interesting that the inclination is still increasing after the eccentricity peaks - plus, it's well within the critical relative inclination. I emailed Dr Takeda to ask about what he thinks the inclination should be doing, and pointed him to this thread too.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #41 - 10/27/06 at 17:22:09
 
14.89 million year update:
 
Planet eccentricity still rising, but it's still only at about 0.044 - has a long way to go to reach 0.8ish!
 
Planet inclination still rising, it's nearly 40 now.
 
Planet argument of pericentre is still converging, looks like it's actually going to converge on 100.
 
No real change in companion properties.  
 
Hrm.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #42 - 10/27/06 at 18:09:52
 
14.89 million years seems too long.  With a Kozai Period of 25 million years it should have peaked at around 12 million years.
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Re: Binary Eccentricity Test - volunteers needed!
Reply #43 - 10/27/06 at 20:12:46
 
Quote from Tony   on 10/27/06 at 18:09:52:
14.89 million years seems too long. With a Kozai Period of 25 million years it should have peaked at around 12 million years.

 
The graph looks like it peaks around 19 Ma. But even at 15.9 Ma, the eccentricity is only about 0.054, and I suspect that it's not going to get much higher than 0.1 in the end.  
 
Have you tried the run I'm doing, Tony? I'm sure I entered everything in properly, but verification that someone else is getting the same results as I am would reassure me somewhat...
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Re: Binary Eccentricity Test - volunteers needed!
Reply #44 - 10/27/06 at 20:39:48
 
I started, but when my results were the same as yours I gave up in favor of a smaller semi-major axes for the secondary just to see if a trend develops.
 
But I can try yours again.  Just to verify:
 
primary: 1 Msun
planet: 2.5 AU, 0 ecc, 0 inc
secondary: 750 AU, .8 ecc, 75 inc.
 
What time step are you using?
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