Are Trojans possible for other planets besides Jupiter?

[Nereid]

There are some 2,000 http://cfa-www.harvard.edu/iau/lists/JupiterTrojans.html" (so far).

There are two Tethys Trojans (Telesto and Calypso), and one Dione (Helene).

I'm pretty sure it's easy to show that any Galilean (Io, Europa, Ganymede, Callisto) Trojan could not be in a stable orbit. And neither would any Mercurian, Venusian, Plutonian, or any of any smaller body (Himalia, for example, or Phobos).

But what about the larger bodies - are Saturnian or Uranian Trojans unstable (due to Jupiter and Saturn, respectively)? Trojans of our Moon? Titanian or Tritonian (sp?) Trojans?

Simulations - digital orreries - are now likely good enough to test stability (within Newtonian gravity anyway; maybe GR would need to be included for Mercury or Venus?); but could some sensible constraints be put on these analytically?

 

[Garth]

Some years ago I did some work on the Jovian Trojans: The orbits of Trojan asteroids.

The Lagrangian condition for stability of three bodies m₁, m₂, & m₃ is they should all be equidistant (equilateral triangle)
that m₁ > 25m₂, and m₃ << m₁ & m₂.

m₃ is stable in the sense that it can oscillate around to equilibrium point L₄ or L₅ in co-rotating 'teardrop' or 'horseshoe' orbits, see Lagrange points 1-5 of the Sun-Earth system.


The other three in-line points of equilibrium L₁, L₂, and L₃ are unstable points of equilibrium, although the Earth ones may be good for a hundred years or so and are used by space probes such as WMAP at L₂.

Perturbations from other bodies may cause the oscillating orbits to become unstable over long periods of time although it is not trivial to work out which ones, numerical simulations are required.

GR effects on their stability are not significant in the Sun's gravitational field.

Garth

 

[Chronos]

Agreed, Garth, but Newtonian dynamics, IMO, virtually forbid stable orbits inside the orbit of Mars. The saddle zones are very unstable that close to the sun.

 

[tony873004]

Neat paper Garth! I've only glanced at it, but I'll read it later.

Based on Nerid's question, I'm running a simulation on the Jovian moons. Each moon has a test particle in its L4 and L5 point. It's positioned precisely, so they sit still rather than trace horseshoe or tadpole (teardrop) orbits.

After 500 years (~40 Jupiter years) they're all still centered in their L points, with no sign of instability.

Here's a screen shot of that system 383 years into the simulation.
http://orbitsimulator.com/PF/galTrojans.GIF

I imagine all the other objects you mention can have stable trojans, except Earth's Moon. It's orbit sits near the edge of the Earth's stability radius (~1/3 of the Hill Sphere). The solar gravitational tide is strong through the Earth / Moon system. The Moon's high eccentricity doesn't help either. A trojan would have a harder time keeping 60 degrees ahead of, or 60 degrees behind something that keeps speeding up and slowing down.

I've simulated this too. Here's some screen shots of objects in the Earth / Moon L4 & 5 points. Objects perfectly placed on the Moon's L4 and L5 begin to drift immediately, forming tadpole orbits around the L point. After a few months, they are tracing wide chaotic orbits around the L points, and after a few more months they completely escape the L points. These screen shots are in a rotating frame whose period approximately equals the Moon's orbital period.

http://orbitsimulator.com/PF/lunaTrojans1.GIF
http://orbitsimulator.com/PF/lunaTrojans2.GIF
http://orbitsimulator.com/PF/lunaTrojans3.GIF

The longest I've been able to keep an object orbiting the L4 or 5 points of the Moon is about 20 years. I did this by flooding the L4 region with about 100 objects and timimg how long it took them be ejected from the L4 point. The winner took 20 years. Of course a spacecraft positioned there could stay indefinately with minor correction burns.

 

[the Physic freak]

hi all, i m quite a beginner of this topic, can someone kindly tell me about trojans, i have no clue what it is, and i m willing to know...thx

 

[Garth]

hi all, i m quite a beginner of this topic, can someone kindly tell me about trojans, i have no clue what it is, and i m willing to know...thx

Hi Pf!
You could start with a Wikipedia article here: Trojan asteroid

Garth

 

[the Physic freak]

thx for the website, i now know a bit about the trojan asteroids, before i never knew there is asteroids on the orbit of jupiter, i only know about the main asteroid belt separating the inner space and outer space of the solar system.

 

[selfAdjoint]

Since the wiki site says trojans have been discovered in the orbit of Mars (which I didn't know), has anyone looked for or found trojan asteroids at the L4 and L5 points of Earth's orbit?

 

[Garth]

Since the wiki site says trojans have been discovered in the orbit of Mars (which I didn't know), has anyone looked for or found trojan asteroids at the L4 and L5 points of Earth's orbit?

Asteroid 3753 Cruithne is an Earth Trojan on a horseshoe orbit centred on L₁. There may be Earth Trojans at L₄ and L₅ but they have not been discovered yet as far as I know. They will be ~ 1AU away, as far away as the Main belt, and so fairly faint if small and dark.

Garth

 

[tony873004]

Mars has a few. The most famous is named Eureka. They actually come closer to Earth than Earth's trojans would, and unlike Earth trojans, they will have an opposition where they lie almost directly opposite the Sun.

Here's a screen shot of Mars' trojans in a frame rotating with Mars' orbital period:
http://www.orbitsimulator.com/gravity/eureka.GIF
I think I got the list from Wikipedia, but interestingly, a couple of the asteroids in this image do cross the position Mars which should exclude them from being called trojans.

Someone once told me that Cruithne was not considered a trojan because it crosses the Earth's L3 point into a full horeshoe orbit. I'd be more inclined to think that Garth is right, and it should be called a trojan.

Here's an animation of Cruithne's orbit in a frame rotating with Earth's period. Only one side of its kidney-bean shaped orbit has "apparent repulsive properties". Due to inclination, the other side, while it looks close in the animation, has a lot of z-axis distance that prevents Earth's gravity from getting a good grip.
http://www.orbitsimulator.com/gravity/acruithne.GIF

Earth also has 2002 AA29 in a horseshoe orbit. This one is interesting because sometimes it leaves the horseshoe configuration and enters a quasi-orbit around the Earth. Here's a screenshot of its orbit: http://www.orbitsimulator.com/gravity/2002aa29_1.GIF

Earth also has man-made trojan; the Spitzer Space Telescope. Here's an image of its orbit: http://www.orbitsimulator.com/gravity/spitzer.GIF