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Quasi-orbitals of Equal Mass (Read 3075 times)
phoenixshade
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Quasi-orbitals of Equal Mass
01/21/11 at 13:31:13
 
Near 1:1 resonances are fascinating to me. This little simulation is based on something that spontaneously arose in a planetary-accretion simulation I was running. Two Mercury-like planets evolved and shared nearly identical orbits, except in one of them the perihelion was rotated by about 90. They briefly settled into a quasi-orbital pattern, but unlike many quasi orbitals, these had roughly equal mass. Unfortunately this situation was swiftly disrupted by the nearby formation of a hot-Neptune, so I recreated it in isolation.
 
This is approximately what happened:
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phoenixshade
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Re: Quasi-orbitals of Equal Mass
Reply #1 - 01/21/11 at 14:10:42
 
The trailing planet tends to circularize the orbit of the leading, then they switch roles. The entire cycle has a period of about 40 sidereal years:
 

 
 
Changes in semi-major axis over time due to orbital libration. There are two obvious components; the low-frequency component has a period of 9.26 sidereal years; the high-frequency component has a period of 0.975 years.
 

 
The high-frequency component seems to have a relation to the precession rate. At each local minimum for a, ΔM = 2Δω. I don't know why this would be, and it may just be a coincidence over the period of observation. A suggested investigation would be to try changing the initial e holding all other parameters constant, which would alter the precession rates, then see whether this relationship holds. I will probably do this in the near future and post results here.
 
 
On an unrelated note, here's a look at where they "cross" each other in the quasi-orbit:
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« Last Edit: 01/22/11 at 12:10:41 by phoenixshade »  

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phoenixshade
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Re: Quasi-orbitals of Equal Mass
Reply #2 - 01/21/11 at 14:28:46
 
And finally, the orbital elements (and the gsim file):
 
Objectdiam.massaeωM
Sun13920001 sun
English150002 earths0.5 AU0.0500
French150002 earths0.5 AU0.0590271

 
Other elements (inclination, long of node) are 0.
 
As an aside, in the original planetary accretion sim, these planets eventually settled into a remarkably stable 13:12 resonance, with the inner planet in a near-circular orbit and the outer in a moderately eccentric one, crossing inside of the orbit of the inner planet. I suspected the hot-Neptune was somehow maintaining this resonance, but even in isolation they remained stable.
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Tony
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Re: Quasi-orbitals of Equal Mass
Reply #3 - 01/21/11 at 23:03:48
 
This sim is amazing.  I've run your gsim file and 5300 years (and counting) later they're still quasi!
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frankuitaalst
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Re: Quasi-orbitals of Equal Mass
Reply #4 - 01/23/11 at 01:10:55
 
this sim is really amazing . Never saw such thing ! Did these orbits really evolve out of a bunch ? Do you have any reference what the original setup was in tis case ?  
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phoenixshade
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Re: Quasi-orbitals of Equal Mass
Reply #5 - 01/23/11 at 20:29:55
 
Quote from frankuitaalst on 01/23/11 at 01:10:55:
this sim is really amazing . Never saw such thing ! Did these orbits really evolve out of a bunch ? Do you have any reference what the original setup was in tis case ?

 
Unfortunately, I do not have the exact initial conditions. The objects were generated randomly with masses of 0.025 earths and diameters of 3000 km. By the time the quasi-orbitals formed, almost everything inside of 0.7 AU had been swallowed up or driven out by a planet with a mass of almost 3 earths orbiting at that distance. This pair was nearly coorbital at a distance of about 0.21 AU with masses of about 0.25. There was a third object in an eccentric orbit that crossed the coorbitals (Mass 0.15, SMA~0.5, e~0.6). Unfortunately I didn't see it happen, but I was able to contrive a situation that gives rise to such a quasi-orbital system, attached.
 
Keep the stepsize at 4 seconds or lower; otherwise there is too much error introduced. (At 8 seconds, a quasi-orbital forms but is quickly disrupted by the third body; at 16 the quasi-orbital never even arises.) If it's rendering too slowly, try lowering the graphics interval to 1000. Unless you're running gravsim on a dinosaur, you'll get a reasonable simulation speed.
 
At 0000-7-4 2:57, the third body makes a close pass just as the coorbitals make their closest approach, resulting in a stable quasi-orbital system. I have not checked to see how long it lasts, but it survives a series of close passes that almost destroys it around 50 years in. I'm currently running this to instability (and capturing data for graphs and such).
 
EDIT:Well, that didn't take long. At around 60 years or so, the third body breaks the resonance with another near pass. I may upload some graphs later if I find anything particularly interesting. One thing I have noticed is several clear jumps where you can see energy transfers due to close encounters.
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« Last Edit: 01/24/11 at 08:24:13 by phoenixshade »  
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