Here are some asteroids viewed in a rotating frame of reference where Jupiter almost stands still. The Trojans, in green, are asteroids that stay near the Lagrange points 60° ahead or behind Jupiter. They go around the Sun once each time Jupiter orbits the Sun. But the Hildas, in purple, go around the Sun 3 times while Jupiter goes around twice. We say they’re in a 3:2 resonance with Jupiter.
The Hildas seem to be moving in a triangular pattern. But actually each one takes an elliptical orbit around the Sun. There are three kinds of ellipses. Two go farthest from the Sun near the Lagrange points, while one goes farthest from the Sun opposite Jupiter. Although the whole triangle of Hildas is nearly equilateral, it’s not quite. The side between the two Lagrange points is a bit different from the two other sides. You can also see the whole triangle pulsing as Jupiter moves in and out!
These animated gifs were made by Petr Scheirich, and you can have hours of fun looking at his website: http://sajri.astronomy.cz/asteroidgroups/groups.htm
There’s a lot to say about Trojans and Lagrange points, but let me talk about Hildas. Over 1,100 Hildas have been found, the being Hilda, named after the discoverer’s daughter. It’s big - 175 kilometers in diameter - but not very bright, because it’s made of ancient stuff containing lots of carbon, similar to the nucleus of a comet.
The Hildas don’t form a ‘true’ asteroid family, because they aren’t fragments of a single parent object. Instead, they’re a ‘dynamical’ family: they’re defined by having similar orbits. Any Hilda’s orbit has an eccentricity less than 0.3, an inclination less than 20°, and a semi-major axis between 3.7 AU and 4.2 AU. Remember, the semi-major axis of an ellipse is half the distance between the farthest points.
So, the Hildas are outside the main asteroid belt, which lies between the 4:1 resonance with Jupiter at 2.1 AU and the 2:1 resonance at 3.0 AU.
The density of Hildas near the triangle’s corners is more than twice the density on the sides. The reason is that the Hildas move more slowly when they’re farther from the Sun! So, they stay near the corners for an average of 5.0-5.5 years, but move along the sides of the triangle more quickly, for 2.5 to 3.0 years. The overall period of the Hildas is about 7.9 years, which is 2/3 the period of Jupiter.
Credit: John Baez, Petr Scheirich