I fell into a bit of a self-nerd-sniping loop while reading up a bit more on Juno's mission to Jupiter, and stumbled across this brilliant GIF which demonstrates the way that the gas giant's gravitational pull influences the asteroids near its orbit.

Notice the green clusters that share Jupiter's orbit, but either preceding or following by about 60 degrees? Those are the Jupiter trojans. A trojan, in this sense, is simply an asteroid which shares its orbit with a planet. They manage this by hanging out at a Lagrange point, a spot in an orbital system of two large bodies where a small object can maintain a stable orbit relative to the larger bodies. For each orbital system, there are five Legrange points (named L1-L5) where the combined gravitational pull of the two large bodies provides exactly the centripetal force needed to orbit with them. L1-L3 sit on a line connecting the two large bodies (one on either side of the system and one in between), while L4 and L5 exist on the smaller-large body's orbital path; L4 is 60 degrees ahead of the large body and L5 is 60 degrees behind. It is at L4 and L5 that the Jupiter trojans can be found.

Jupiter trojans are named after heroes of the Trojan War, with those clustered around L4 named for Greek heroes ("the Greek camp") and those near L5 named for the heroes of Troy ("the Trojan camp").

You can also see the magenta asteroids that move between the Jupiter-Sun L3, L4, and L5 points. I'm sure there's a bunch of fancy math to describe this movement, but I ran out of time to track it down this afternoon... feel free to chime in if you know what's up!

Fun fact: Earth has a trojan too! 2010 TK7 is a 300m-diameter asteroid found at the Earth-Sun L4 point.

More reading:
https://en.wikipedia.org/wiki/Jupiter_trojan
https://en.wikipedia.org/wiki/Lagrangian_point
https://en.wikipedia.org/wiki/Earth_trojan

Great visualization of the interplay between gravitational fields which causes the Lagrange points:
https://en.m.wikipedia.org/wiki/Jupiter_trojan#/media/File%3ALagrange_points.jpg
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