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A Mars co-orbital asteroid -- 1999 ND43 (Read 16645 times)
Tony
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A Mars co-orbital asteroid -- 1999 ND43
05/02/07 at 19:32:50
 
Here's an animated GIFof asteroid 1999 ND43 in an rotating frame set to the period of Mars.  Frankuitaalst brought this asteroid to my attention in the "asteroids" thread.  It is co-orbital with Mars.  From the beginning of the animation until about the year 5200 it is in a horseshoe orbit with Mars.  Then, unable to crest the L3 point, it briefly becomes an L5 trojan, before returning to a horseshoe orbit.
 
Earth has a big effect on this asteroid's orbit, speeding up and slowing down its period in the rotating frame.  This is most noticable from the year 5200 until about 8300.  In the next few days I'll post an animation in a rotating frame set to the period of Earth's orbit.
 
The simulation was run at a time step of 32 seconds.  A screenshot was made every 47 years.
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« Last Edit: 05/05/07 at 17:57:03 by Tony »  
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frankuitaalst
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #1 - 05/04/07 at 11:22:46
 
The orbit of this asteroid is remarkable: plotting the distances to Mars and Earth gives the following plot over the first couple of thousand years .  
Although the ast is really bound to Mars it seems to spend more time closer to Earth than to Mars .  
In this time span the mean distance to Mars was much more than the distance to earth ...
I wonder how this ast may evolve ...
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1999ND43Dist.gif
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #2 - 05/05/07 at 01:15:15
 
I restarted the sim of 1999ND43 in order to see how the ast evolves in the long time . The sim runs now at almost 7000 years . But what a mess!!.  
The sim consits of all the planets of the solar system without pluto,neptune and uranus ,but with the smaller minor plantes ceres , pallas , hygiea ...In total there are 14 bodies .  
From the beginning it was clear that some minors are under influence of big changes in their orbits, but look at the picture after 7000 years ! .  
1999ND43 is the fourth body from the sun , almost osculating the orbit of the third (Earth) . The points where it crosses the orbit of Mars seem to evolve . I intend to extend the sim to about 20.000 years .
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1999ND43Above.gif
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #3 - 05/05/07 at 18:09:00
 
This is one facinating asteroid.
 
I added the animated GIF with a rotating period set to Earth's period to the first post in this thread.  It shows that 1999 ND43 is in a 25:47 mean motion resonance with Earth.  This resonance is responsible for the variability it the asteroid's speed in the frame with a rotating period set to Mars' period.
 
After the asteroid encounters Mars in about the year 5000, Mars pulls the asteroid into a slightly higher and slower orbit, breaking it from its 25:47 mean motion resonance with Earth.  But when it encounters Mars again from the other direction in about the year 7800, Mars pulls the asteriod into a slightly lower and faster orbit, and restores the asteroid's 25:47 mean motion resonance with Earth.
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #4 - 05/06/07 at 00:19:50
 
Cool gif ...25:47 means 25 revolutions of te ast vs 47 of earth ? Is really an "odd" number in both senses .
I ended the sim after about 12000 years forward . Looks as if the asteroid has a pure sine curve in his inclination of about 16000 years , also the ecc seems to have this feature . Why 16000 years ??.  
I restarted the sim backwards . When ready I'll paste both sims together and post them .
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« Last Edit: 05/06/07 at 02:18:57 by frankuitaalst »  
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #5 - 05/06/07 at 02:18:05
 
Pasting reverse and forward sim gives the following picture of the asteroid .  
There's a lot of dynamics in the orbit . Remarkable in the past is the time -5000 years where the asteroid slows down and becomes a co-orbital of Mars . In some way we are "lucky" today .  
Furthermore  the asteroids inclination has a mean period of about 16000 years .( Is this somehow related with some product of orbital periods of other planets  ? ) .  
Also in the past the asteroid spended more time near earth then to Mars ( related to the mean distance ) . It seems however that between "now" and 2500 years the influence of Earth is at biggest .
By the way : I have a series of screenshots in gif of the orbit of the Asteroid in which the precession of the perihelion or aphelion becomes clear . I put them together in .ppt format , but the file is rather big . Can this forum accept a .ppt or do I have to convert to animated gif ?
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1999ND43Total.gif
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #6 - 05/06/07 at 18:58:01
 
I get a different result than you do.  Here's my graph through 3700 years.  This graph was made at a time step of 16 seconds.  The animations were made at a time step of 32 seconds.  They seem to agree with each other.  After making an approach to Mars at 3000 years, the asteroid is pulled into higher slower orbit and reverses direction in the rotating frame.  But shortly after, it slows and comes back to Mars.  Perhaps you are taking too large of a time step?
 
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #7 - 05/06/07 at 22:58:29
 
You're right ... After the first "peak" our results don't match .  
However the peakvalue of 1.002 matches .  
Perhaps some parameters were different . I used 14 bodies up to Saturn ( Nept, Uranus , Pluto : not ) . Moon and Earth were two bodies . Maybe the value of G was different . I used g = 6.67259E-11
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #8 - 05/06/07 at 23:44:31
 
I used Mercury, Venus, Earth, Moon, Jupiter barycenter, Saturn barycenter, Uranus barycenter and Pluto barycenter.  I'm using the same value of mu (G * sun mass) as in the Apophis simulation where our results were identical.
 
Can you try the first 4000 years at a slower time step?  That probably won't take too long using Picard. Wink
 
Also, very important in this simulation are the masses of Earth and Mars.  I used (kilograms):
5.97369125232006E+24 Earth
6.41853544762209E+23 Mars
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #9 - 05/07/07 at 09:49:39
 
After T=6500 my 16 second run starts to differ from my 32 second run.
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #10 - 05/07/07 at 10:07:22
 
By how much ?  
I restartet the sim with the new value of Earth and Mars Mass , and the factor as used by your JPL contact , still with 14 bodies . Now I'm running at 5 m accurancy . The time step is 118 hours , whereas at 10 m acc the timestep is 121 hours . The timestep is given by the motion of the moon .  
If I would delete the moon and replace it with 1 body the timestep should increase .
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #11 - 05/07/07 at 10:29:59
 
Don't delete the Moon because the asteroid makes numerous close approaches to the Earth.  If you wanted to delete the Moon, it's not as easy as simply deleting it.  You would need to replace Earth with an accurate value for the mass, position and velocity of the Earth barycenter.  This is what I did with Jupiter, Saturn, Uranus, Neptune, and Pluto.  The asteroid makes no close passages to these objects, so it is justified for these planets.
 
Quote:
By how much?

If you look at my animation, the asteroid travels counter-clockwise around Mars' orbit until it encounters Mars.  Then it travels clocwise away from Mars about 90 degrees before reversing direction and heading back to Mars.  After Mars pushes it away for the 2nd time, it travels completely around Mars' orbit in a clockwise direction.  My 16 second run shows it makes 2 attempts to get past the 90 degree point before travelling all the way around the orbit.
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #12 - 05/07/07 at 22:54:53
 
I ran the sim for 4000 years at accurancy 5m and the params you gave me . Basically I got the same result as before. The ast makes his first jump at 2863 years to a higher level .Herunder the result at acc 5m .  In my sim I also take Ceres , Pallas ...Juno in consideration . I saw that after some time one of them crosses the orbit of the ast . Maybe this is the reason ...I started the same sim now , but without the Minor planets . Curious what comes out . KYI.
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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #13 - 05/08/07 at 09:09:31
 
I ran the sim as mentionned above without the small minor planets . There is no significant change. The only difference a notice is that the program performed 15 days more to perform 300000 iterations . I think his is due to the fact that the program didn't have to slow down while there were less bodies. I think we have to check our initial conditions . Herunder the sim . By curiousity I added the velocity of the asteroid relative to Mars . I really am surprised to see that this relative velocity seems to follow the distance to Mars .
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« Last Edit: 05/08/07 at 10:25:26 by frankuitaalst »  

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Re: A Mars co-orbital asteroid -- 1999 ND43
Reply #14 - 05/09/07 at 11:50:29
 
Searching for our differences I zoomed in to the timespan where the asteroid jumps to higher/lower levels . This jumps are induced by close approaches to earth as can be seen from the pink lines . The first jump occurs in 2859+2007 years . Do the close approaches correspond?
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