Are Trojans possible for other planets besides Jupiter?

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Discussion Overview

The discussion revolves around the possibility of Trojan asteroids existing for various celestial bodies beyond Jupiter, including Saturn, Uranus, and Earth's Moon. Participants explore the stability of these potential Trojans, the conditions required for their existence, and the implications of gravitational interactions with other bodies. The conversation includes theoretical considerations, simulations, and inquiries about known Trojans in the solar system.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants note that while there are known Trojans associated with Jupiter, the stability of Trojans around other planets, such as Saturn and Uranus, remains uncertain due to gravitational influences from larger bodies like Jupiter.
  • One participant discusses the Lagrangian condition for stability, suggesting that certain configurations can allow for stable Trojan orbits, but acknowledges that perturbations from other bodies can complicate this stability over time.
  • Another participant argues that Newtonian dynamics likely prevent stable orbits for Trojans inside Mars' orbit, citing the instability of saddle zones close to the Sun.
  • Simulations conducted by participants indicate that Jovian moons can maintain stable Trojans, while Earth's Moon may not be able to support stable Trojans due to its high eccentricity and the strong solar gravitational tide.
  • There is mention of Earth Trojans, such as 3753 Cruithne, which is noted to have a horseshoe orbit, raising questions about the classification of such objects.
  • Participants express interest in the existence of Trojans at Earth's L4 and L5 points, with some noting that while Trojans have been found at Mars, similar discoveries for Earth remain elusive.

Areas of Agreement / Disagreement

Participants generally agree on the existence of Trojans around Jupiter and Mars, but there is no consensus on the stability of Trojans for other planets, particularly regarding Saturn, Uranus, and Earth's Moon. The discussion remains unresolved regarding the conditions necessary for stable Trojan orbits in these contexts.

Contextual Notes

Limitations include the dependence on specific gravitational models and the complexity of interactions between multiple celestial bodies, which may affect the stability of proposed Trojan orbits. The discussion also highlights the need for numerical simulations to explore these dynamics further.

Nereid
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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?
 
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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
 
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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.
 
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.
 
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
 
the Physic freak said:
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
 
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.
 
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?
 
selfAdjoint said:
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
 
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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
 

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