Thoughts: As a rule of thumb, not strict, trojan orbits are more stable
with a planet that does not have massive neighbors both inside and outside
of it. Thus, Jupiter, with nothing all the way in to Mars and being the
most massive planet in the solar system, has a very stable trojan
zone. Neptune, with no significant mass on the outside, likewise has a
pretty stable trojan zone (and even a stable 3:2 resonance zone where Pluto
lives). Uranus is more problematic, with Saturn on the inside and Neptune
outside, and Saturn likewise, with Jupiter inside and Uranus outside. Mars
and Earth are marginal. Mars has nothing outside, but is not very massive;
the Earth has Mars to the outside, but again, it outweighs Mars by a lot so
it sort of doesn't have much on the outside, all the way to Jupiter.

Thinking specifically about the claimed Earth trojan, if you have a look at
the Nature paper, the libration amplitude is almost 180 degrees, that is,
it librates about the 60-deg L4 point all the way from zero to 180 degrees,
with a period of about 400 years. In fact, about 1600 years ago, if you
believe the integrations back that far (you probably shouldn't) it
"horseshoed", and before that was librating about the L5 point. Now here's
the problem: the current argument of perihelion is 45 degrees, and the
eccentricity is substantial. So currently (hundreds to a thousand or so
years), when the object librates to be almost at the Earth's longitude, it
doesn't get very close to the Earth because it is far inside or outside of
the Earth when it is at node, and far above or below the Earth when it is
at 1 AU. But that doesn't last forever. The period of the trojan
libration (~400 years) is not commensurate with the period of circulation
of the argument of perihelion. sooner or later, the libration will carry
it to the Earth's longitude at a time when the argument of perihelion is
near 90 (or 270) degrees, and it will make a series of very close passes by
the Earth, very likely resulting in ejecting it from "trojan"
configuration. Typically the period of perihelion circulation is some tens
of thousands of years, so my guess is that this object probably got parked
in its present configuration some thousands of years ago, and will get
ejected back out some thousands of years in the future. But as properly
noted in the Nature paper, the motion is chaotic over only hundreds to a
thousand years, so one really can't track it 10,000 years and say what will
actually happen or when. But certainly, this object is not in a stable
configuration like the current Jupiter Trojan population.

At 04:32 PM 7/27/2011, Tomas wrote:
>Could we have a uranus trojan on our hands? The first four sets of
>observations available on NEOCP currently give the following elements:
>
>P100Oms
> Perihelion 2053 May 26.294451 TT = 7:04:00 (JD 2471048.794451)
>Epoch 2011 Jul 27.0 TT = JDT 2455769.5 Ur: 0.7678 Find_Orb
>M 180.00780 (2000.0) P Q
>n 0.01178013 Peri. 333.53195 -0.73404352 -0.67826697
>a 19.1294492 Node 163.61868 0.64231709 -0.70952696
>e 0.1244084 Incl. 6.85767 0.22047418 -0.19111616
>P 83.67 H 8.5 G 0.15 q 16.7495849 Q 21.5093135
> From 12 observations 2011 July 25-27; RMS error 0.391 arcseconds
>
>Which would place the object into a zone of stability as described by
>Dvorak, R.; Bazsó, Á.; Zhou, L.-Y. in Where are the Uranus Trojans?
>
>http://adsabs.harvard.edu/abs/2010CeMDA.107...51D
>
>any thoughts
>
>
>
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>
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Alan W. Harris
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Space Science Institute
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