# Orbits of Trojan Asteroids

I understand that the Trojan points are located 60 degrees ahead and behind a planet in its orbit. However, since there are quite a number of Trojans in Jupiter's orbit, they cannot all be exactly at that point. Presumably they orbit around it.

So what sort of orbits do they have? Are they circular or anywhere near so, or are they elongated along the orbit of Jupiter? If the latter how far can they get from the 60 degree point?

As an example, look at Earth's only confirmed Trojan

By Phoenix7777 - Own work Data source: HORIZONS System, JPL, NASA, CC BY-SA 4.0, source

Now, to understand what is happening here. The yellow dot is the sun. The blue dot is the Earth. Although the Earth is orbiting the sun, the "camera" is turning so that it appears that the Earth is roughly stationary (it wobbles slightly due to the eccentricty of the earth's orbit)

The pink dot is the asteroid 2010 TK7. It has an orbit that is eccentric, and so sometimes it is much closer to the sun than the Earth, and at other times it is further, but its orbit takes nearly exactly one year. However the exact shape of the orbit changes, sometimes it moves around the sun in slightly less than a year, so it starts to catch up with the Earth, but as it nears the Earth, the Earth's gravity tends to pull it forward, and out, causing it to slow down and move away from the earth. The whole cycle takes hundreds of years.

Note, the actual shape of the orbits are elliptical about the sun, the odd shapes is a result of the camera turning at one revolution per year.

This kind of orbit is said to librate about the L4 point. It is called a "tadpole orbit" Trojans don't have to remain exactly at the L4 point, they can stably orbit in a tadpole orbit around the L4 point.

• I wish I could upvote multiple times! I never saw that cool GIF before, it is really a very intutive representation of how Trojans move. Apr 8, 2021 at 21:03
• @B--rian of how a somewhat unusual one with a whole lot of energy moves.
– uhoh
Apr 9, 2021 at 0:53
• Yes, chosen as an extreme case. Apr 9, 2021 at 3:56
• To clarify this image, the L4 and L5 Lagrange points (60 degrees around in the same orbit aren't absolutely stable. What happens is that as an object moves away from them, gravity leads them to orbit around the L4/L5 points, rather than increasingly drift from them. So they're kind of "metastable", the objects won't stay literally there at those exact points, but they will end up "around" there. Apr 9, 2021 at 7:53
• @Stilez: That's not quite what "metastable" means; see en.wikipedia.org/wiki/Metastability. The behavior that you describe, with orbits around the point, is exactly what you'd expect from a stable Lagrange point. Apr 10, 2021 at 19:18

Trojan asteroids are in roughly circular orbits around the Sun at roughly the same distance as Jupiter, that are in 1:1 resonance with Jupiter and stay very roughly 60 degrees away from it.

Scott Manley's video below shows two classes of asteroids in resonance with Jupiter. The first one shown is confusing because it is in a 3:2 resonance and in the rotating frame it looks like they are cycling between L3, L4 and L5. Skip ahead to 33 seconds and you can see what "normal Trojan asteroids" do. Most of them stay within +/-20 degrees of L4 or L5, only a few exotic stragglers go farther than that away from their Lagrange points. There is some out-of-plane motion as well, as there is for all asteroids.

@JamesK's answer showing a rather exotic asteroid in 1:1 resonance with Earth is an extreme case, but the GIF does help to give some illustration of the back-and-forthness, even though it's pretty exaggerated compared to what normally happens.

After watching, go back to the beginning and see the more confusing 3:2 resonance orbits.

update: There's this!