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planet eccentricities in binary systems (Read 11308 times)
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planet eccentricities in binary systems
10/22/06 at 23:36:14
 
This paper is bugging the heck out of me:
http://arxiv.org/abs/astro-ph/0510296
 
Basically they're claiming that a very distant (> 500 AU) brown dwarf or companion star can pump up the eccentricity of a planet close to the primary star, if the companion is on a very inclined (> 40°) orbit. This sounds crazy to me  - but what's crazier is that the magnitude of the eccentricity pumping is determined solely by the difference in inclination of the orbits, and not by mass or distance or anything (those just determine the period of the oscillation).  
 
Can Gravity Simulator simulate this?
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Re: planet eccentricities in binary systems
Reply #1 - 10/23/06 at 07:22:54
 
Inclination test
 
I tried running this overnight with a 65k timestep, and recorded the first million year's worth of evolution, and I got no noticeable change in eccentricity of the inner gas giant (beyond the usual small scale cycles), and a very very slight increase in its inclination. I just got some more data points for around the 10 million year mark and the inclination of the gas giant has gone from 0.00 (at the start) to 0.02 (at 1 million years) to 0.24 degrees (at 11 million years) - the eccentricity hasn't changed though.
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Re: planet eccentricities in binary systems
Reply #2 - 10/23/06 at 08:02:07
 
try running the same sim with a 16K time step for 60 myr taking note every 5 myr
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Re: planet eccentricities in binary systems
Reply #3 - 10/23/06 at 10:34:16
 
I've heard of this before.  It's related to the Kozai mechanism.  I have not yet simulated it for binary stars, but on the "Simulations" link there is an article called "Kozai Mechanism".  It shows 2 examples with Gravity Simulator.  The first places the Moon in a circular polar orbit around Earth.  All seems fine for a few orbits, but then eccentricity is pumped into the Moon's orbit.  Within a decade, its perigee is below the surface of the Earth  Wink
 
The next example doesn't actually simulate the effect, but shows what is theorized to be caused by the effect.  The jovian system of moons contains no moons with inclinations above about 60 degrees.  The reasoning...  If they existed, the Kozai mechanism would pump eccentricity into them until their perijoves were in the vicinity of the Galilean Moons.  Then their days are seriously numbered (think 5-planet system  Smiley ).
 
Comparing a binary star system to the Earth / Moon simulation, the two stars are analgous to the Sun / Earth, and the planet is analgous to the Moon.  There's no reason Gravity Simulator shouldn't be able to handle this, although the time scale required to see results may be very long.
 
Notice in this simulation that it takes a while for the action to begin.  The Moon orbits the Earth normally for several orbits before you notice any significant eccentricity develop.  But once it does, the rate accelerates.
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Re: planet eccentricities in binary systems
Reply #4 - 10/23/06 at 10:47:47
 
Just a thought...
 
Sometimes, when a simulation is difficult because of the amount of time needed, or the amount of objects needed, you can play some games to get good results.
 
For example, in the case of the Kozai Mechanism, you already know that Gravity Simulator can quickly model an exaggerated version as demonstrated by placing the Moon in a polar Earth orbit.  It takes about 10 years for the eccentricity to reach 0.9.  Enter this data into excel (1 AU, 10 years).  Next, create a new simulation with an identical Earth / Moon system at 2 AU.  Give it a go and plot your data.  With only a few data points you should be able to determine if your function is growing linearly, or as a polynomial or exponential function.  Graph it in Excel, add a trendline and have it display the formula for the trend line.  Now you can simply extrapolate the curve / line to values too large to simulate.
 
Try it again, but this time instead of plotting semi-major axis against time to exceed 0.9, try initial inclination vs. time, or vs. max eccentricity.  Use the same technique in Excel.  Soon you'll have a series of curves that you may be able to use to extrapolate any condition.
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Re: planet eccentricities in binary systems
Reply #5 - 10/23/06 at 11:58:16
 
Right, but the problem I have with this is the idea that an inlcined brown dwarf 500 AU from the star can screw around with the eccentricities of a planet orbiting really close to the star. That sounds really counter-intuitive to me.
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Re: planet eccentricities in binary systems
Reply #6 - 10/23/06 at 12:09:20
 
That's what makes this so interesting  Grin
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Re: planet eccentricities in binary systems
Reply #7 - 10/23/06 at 14:03:52
 
It makes it annoying too, when you're trying to write a worldbuilding program! Wink
 
That said, it only holds if the BD or companion star is on a very inclined orbit (45° or more) relative to the planet/ecliptic. I'm not entirely convinced that's likely - wouldn't they have formed in the same plane because the companions form from the same disk of material as the primary and planets?
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Re: planet eccentricities in binary systems
Reply #8 - 10/23/06 at 14:19:22
 
Yes, but wide systems are not very well protected against disturbances beyond the system.  Over time lots of crazy stuff can happen to distant objects.  Consider Xena (aka Iris, 2003 UB313).  It's inclination is almost 45 degrees.  
 
 
In the system you describe, at 500 AU, its very likely that a close stellar passage when the star system was young and still in its birth cluster, would pump lots of eccentricity and inclination into the wide system.
 
Interesting to note that Uranus, with its almost polar tilt, doesn't display signs of the Kozai mechanism in its system of moons.  Or does it?
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Re: planet eccentricities in binary systems
Reply #9 - 10/23/06 at 14:33:55
 

Concerning the last question : I think the Kozai comes only in action if the inclination is big  related to the plane of motion of the mother planet . The oriontation of spin around its axe doesn't matter ....
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Re: planet eccentricities in binary systems
Reply #10 - 10/23/06 at 15:40:08
 
Sorry, I wasn't too clear with the way I phrased it.  You're right.  The tilt has nothing to do with it.  But since Uranus' moons orbit in the same plane of its equator, the tilt implies that the moons' are in steep orbits with respect to the plane of Uranus' orbit.
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Re: planet eccentricities in binary systems
Reply #11 - 10/23/06 at 17:19:50
 
25 million years into it, and the inclination of the gg has gone from 0.26 to 0.56 in the 14 million years or so since this morning. Eccentricity is still cycling in exactly the same way as it did before though, at 0.6 +/- 0.05 - that hasn't changed at all each time I've looked at it.
 
So... over time, inclination is increasing from zero, but eccentricity is just staying stable (on average).  
 
The inclined BD seems to be the same as it ever was.
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Re: planet eccentricities in binary systems
Reply #12 - 10/23/06 at 22:45:03
 
Quote from Tony   on 10/23/06 at 15:40:08:
Sorry, I wasn't too clear with the way I phrased it.  You're right.  The tilt has nothing to do with it.  But since Uranus' moons orbit in the same plane of its equator, the tilt implies that the moons' are in steep orbits with respect to the plane of Uranus' orbit.

you're right , I overlooked this fact . Maybe the Kozai is not "visible" in this case while the orbital period of Uranus is much longer than earths . I estimate that the Kozai force must be proportional to 1/omega^2 .
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Re: planet eccentricities in binary systems
Reply #13 - 10/23/06 at 22:46:18
 
Check out how sensitive the Kozai Mechanism is to distance.  I ran a simulation similar to the one on the website where the Moon was placed in polar orbit around Earth.  But in this sim, I had 5 Earth systems:
 
Earth 1: 1 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 2: 1.5 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 3: 2 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 4: 2.5 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 5: 3 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
And the data:
 

 
The Moons of Earth 2-5 are all crammed together on the bottom, barely visible with the scale high enough to show Earth 1's Moon full range.  I did not give the Earth's any size in this simulation to avoid collisions.  So it is interesting to note that the Kozai Mechanism does not pump ecc>1 into a system.  Once it approaches 1, it backs off, and I'm guessing periodically goes back to close to 0 and repeats.
 
Leaving the data for Earth 1's moon off the chart to do a better job of plotting the moons for Earth 2-5 gives this:
 

Again, the Moon from the closest Earth significantly outpaces the others in its eccentricity gain.
 
I think I'll run a similar sim tommorow, but with Earth SMA values more closely surrounding 1 AU.
 
This was run at a slow time step of 16 (not 16K !), for only about 7 years.
 
Try it yourself:
http://www.orbitsimulator.com/gravity/simulations/kozaitest.gsim
 
 
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Re: planet eccentricities in binary systems
Reply #14 - 10/24/06 at 07:44:39
 
Still running at 41 Ma, and the inclination of the planet is up to 0.9 degrees. still no change in eccentricity.  
 
Maybe I should try this with an actual star as the companion instead of a BD - apparently that makes the period shorter.
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Re: planet eccentricities in binary systems
Reply #15 - 10/24/06 at 08:10:53
 
OK, I'm trying again but with the companion replaced by a 1 solar mass star. Timestep is 32k.
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Re: planet eccentricities in binary systems
Reply #16 - 10/24/06 at 11:39:21
 
Tony, on those earth/moon plots, it looks like the eccentricity in the Earth1 case is increasing significantly over the course of one orbit when it's approaching 1.000! (each tick at the bottom is every four months, so that's four orbits, right?).
 
According to the paper the max eccentricity should be the same for all of them, the only difference is the period of the oscillation.
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Re: planet eccentricities in binary systems
Reply #17 - 10/24/06 at 11:54:11
 
Nice sim with the 5 earths .  
I ran it and noticed the following :  
 
1. For the first time a saw the moon at AU 1 turn the other way around the earth ! This happened at the longest elongation point  ( after about 6.5 years ) where the moon choosed the other side to "round" the corner . Then the cycle repeated ...
 
2. The other earths have the same phenomenon , but it takes a lot longer ....
 
3. Max eccentricity occurs at about :  
 
AU1 : 6.5 y
AU1.5 : 13 y
AU2: 28 y
AU 2.5 : 74 y  
AU 3 : 158 y
 
 
plotting this into excel gives : time,max exc ~~ 1.10 exp(1.65 AU )  
 
For Uranus ( AU19.22 ) this gives : 7...e+13 years !! Longer than the lifetime of the universe  
 
please note that the red simulations were done at higher time step .  
 
Things can be optimized but above indicates that the Kozai is surely  dependent upon distance  
 
Another parameter surely involved is the distance of the moon from the mother planet ...
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Re: planet eccentricities in binary systems
Reply #18 - 10/24/06 at 12:01:56
 
I forgot to insert the table ,
 
     AU            Time(yr)
Earth      1            6
Earth      1,5            13
Earth      2            28
Earth      2,5            74
Earth      3            158
 
Uranus      19,22            7,36925E+13
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Re: planet eccentricities in binary systems
Reply #19 - 10/24/06 at 12:05:42
 
Yes, it is gaining lots of ecc per orbit as the slope of the curve gets steep.  It's outputing the data every 10 days.
 
Since I placed them in 90 degree orbits, I would expect to get the maximum possible eccentricity, which appears to be just under 1 (0.994).  I would guess that if I placed the moons in orbits with 80 degree inclinations that max ecc would be a bit less.
 
I'm doing a new run.  I'm up to 18 years now and the ecc of Earth1's moon does indeed drop back down almost to zero before rising again to max ecc of almost 1.  Earth2's moon is starting to climb on the graph.  I'm guessing that it too will approach 1.  If so, it sounds like that conclusion is correct, that the max ecc will be the same, but the period of the oscillation will differ, and as this illustrates, by quite a bit.  I'll post it later tonight.  Maybe Earth3's moon will be making a little noise by then too.
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Re: planet eccentricities in binary systems
Reply #20 - 10/24/06 at 12:53:28
 
OK... the real test (for me anyway) is what happens when you have the close planet orbiting a star with a companion and > 500 AU. I didn't see any effect with the BD at 500AU other than a 1 degree increase in *inclination* of the planet after 45 million year, and no noticeable effect on its eccentricity at all. So if the effect is real, it's happening on a *really* long timescale in this case.
 
We'll see what it does when I get back home and check the 1 solar mass companion run. It should be a lot more noticeable there.
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Re: planet eccentricities in binary systems
Reply #21 - 10/24/06 at 16:00:46
 
A close planet is probably better protected than a distant one.  Like Frank said, the distance probably matters a great deal.  The Earth's moon is not buried deep within Earth's Hill Sphere.  Rather, its 1/3 of the way to the edge.  This gives us an excuse for another simulation!
 
I'm trying the 5-Earth sim over again at a timestep of 4 since 16 is a bit fast for the close approaches at max eccentricity.  I want to see if I get the same numbers as Frank.
 
I would guess that the quickest way to get the numbers on a 500 AU system would be to start with the companion only a few AU away, record the time to max ecc, then move the companion out a few AU, record... and plot a graph, extrapolating out to 500 AU.  It'll probably be similar to Frank's conclusion about Uranus' moons being protected from Kozai for timescales that exceed the life of the universe.
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Re: planet eccentricities in binary systems
Reply #22 - 10/24/06 at 16:28:26
 
Well I'd just like to see if I can replicate that graph from the paper I linked to in the first post here. I can't recall what parameters I put into the initial gsim though, I might have to regenerate it again with the exact numbers from Figure 1 in the paper. But with a 1 solar mass companion at an inclination of 75° at 750 AU, and a planet at 2.5 AU, IIRC the paper claimed that the eccentricity cycle should be about 20 million years in length. Though interestingly the caption on Figure 1 says the companion eccentricity is 0.8 too. I would have started off with that having a circular orbit myself (that's another thing to test I guess - what effect does the eccentricity of the inclined companion have?).
 
I'm still not sure what to make of the slight increase in inclination of the planet though in my initial simulation.
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Re: planet eccentricities in binary systems
Reply #23 - 10/24/06 at 17:28:32
 
Quote from Mal   on 10/24/06 at 08:10:53:
OK, I'm trying again but with the companion replaced by a 1 solar mass star. Timestep is 32k.

 
Home now. Here's the results so far... the eccentricity of the planet is varying on a regular pattern on a 400000 year cycle, but it's only varying between 0.63 and 0.58 (initial ecc was 0.60). Planet inclination has increased monotonically from 0 degrees to about 12.8 degrees over time (this is over 6.7 million years).
 
So it looks like increasing the companion mass makes the planetary inclination rise faster, unless that's down to the shorter timestep?  
 
I'll start this again with the numbers in the paper and see what happens - I think I can afford a 65k timestep here since the planet's at 2.5 AU there. The inclination increase is strange though - the eccentricity isn't doing anything at all really.
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Re: planet eccentricities in binary systems
Reply #24 - 10/24/06 at 23:49:37
 
http://arxiv.org/abs/astro-ph/0502404
 
Found Takeda and Rosio's 2005 paper, it's similar to the paper I linked to in the first post but also includes how to calculate the period of the eccentricity oscillation!
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Re: planet eccentricities in binary systems
Reply #25 - 10/27/06 at 16:10:42
 
I've just finished the five earths sim up to max ecc for the fifth earth . Seams interesting . I will send the data as a plot of time tomorrow ...
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