Trojan Planets

[eburacum45]

Now then; a little puzzle;
everything I have heard in the past seems to indicate that planets in the trojan points of other planets do not remain there indefinitely unless they are very much smaller than the primary planet; this is why trojan asteroids are very small compared to the planet they are associated with, be it Jupiter, Neptune or Mars.
In a similar vein, the object that may have hit the Earth long ago and caused the creation of our Moon was large compared to the Earth, perhaps as large as Mars; but it could not remain there because of the inherent instability of the system.

However these people (Dvorak et al, Swartz et al) have carried out simulations of the possibility of trojan planets in various known extrasolar systems;
http://www.ias.u-psud.fr/medoc/cw6/contribCW6/dvorak_poster.pdf
http://www.liebertonline.com/doi/pdf/10.1089/ast.2005.5.579?cookieSet=1
and they seem to say that a hypothetical trojan planet might be of arbritary size;



We concluded that the stable zone of hypothetical Trojan planets does not depend on their mass

Before I include such planets in my worldbuilding activities does anyone have any dissenting views? Can trojan planets really be stable at any mass?

[tony873004]

I believe there's a ratio between the combined masses of the planets and the Star. If the ratio (?something like 25:1?) is not exceeded, it should be stable for long periods of time. So it is possible to have 2-3 planets of equal masses spaced 60 degrees apart.

I'm not sure if it is known why they don't exist in our solar system.

[grant hutchison]

Yeah, Laughlin and Chambers' Extrasolar Trojans: The Viability and Detectability of Planets in the 1:1 Resonance (http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2002AJ....124..592L&db_key=AST&) investigated this:

Call the mass of our giant planet m1 and the mass of the trojan planet m2; total system mass (primary+giant+trojan) is M.
If m2 = 0, then the trojan position is stable if:

m1/M < (9 - root[69])/18 = 0.038520896...

According to Laughlin and Chambers, if m1 = m2, the pair of mutual trojans is stable if:

(m1+m2)/M < (6 - 4*root[2])/9 = 0.038127305...

And for all intermediate values of m2 ( 0 < m2 < m1), the stability threshold for (m1+m2)/M lies between the two limits above.

So the star's mass is relevant. There's also the possibility that massive objects will get into a horseshoe rather than a tadpole orbit: your guys write that they haven't specified the exact nature of the 1:1 resonance.

Grant Hutchison

[eburacum45]

A horseshoe orbit like Cruithne, eh? Tricky one...

I think I quite like the third option-


The third configuration that we examine is more exotic and involves a pair of planets that exchange angular momentum in a manner that allows them to indefinitely avoid close encounters. An illustrative example of this resonance occurs when one planet has a highly eccentric orbit while the other planet moves on a nearly circular orbit; the periapses are in alignment, and conjunctions occur near periapse.

I wonder how close they could get and remain stable? A planet which periodically shows a disk would be very entertaining.

[grant hutchison]

A horseshoe orbit like Cruithne, eh? Tricky one...I quite like the idea. A big superjovian and a terrestrial world: the exchange of energy would mean that the terrestrial moved quite dramatically inwards or outwards after each close approach to the superjovian. So you'd have "super-seasons", lasting many years, superimposed on the normal seasonal cycle. Interesting to imagine what evolution might come up with to cope with that.

Grant Hutchison

[Disinfo Agent]

Have you ever read Brian Aldiss' Heliconia trilogy?

[eburacum45]

Yes. I have;
The planet Helliconia orbits a small sun-like star Batalix, which is in an elliptical orbit around a giant star Freyr.
Because of the short life spans of giant stars such an arrangement wouldn't occur in nature; the ecology on Helliconia would not have enough time to evolve.
But if a Helliconia-type planet was terraformed and an artificial ecology introduced, it would work fine.

[publiusr]

I wonder what it would take to get a large object in a trojan orbit.

An unrelated question. If an extra-solar impactor were to strike Venus and spin it up--while opening (widening) its orbit up a bit--how wide an orbit could it have without disturbing Earths orbit much?