The Trojans – the equilateral triangle in the sky
The Sun, Jupiter and thousands of asteroids form equilateral triangles. Lagrange found this solution in 1772, the first Trojan was discovered in 1906.
Two bodies attracting each other orbit one another on ellipses. Newton solved that completely. With three bodies there is no general solution any more, their paths can become arbitrarily tangled. All the more remarkable that there are two simple arrangements in which three bodies keep their shape for ever: the line and the triangle.
Line and triangle
In 1767 Leonhard Euler found solutions in which the three bodies lie on a straight line and turn together. In 1772 Joseph-Louis Lagrange found the second kind: the three bodies stand at the corners of an equilateral triangle, and the triangle turns as a whole without changing its shape. This holds whatever the three masses are.
If one of the bodies is very small, say an asteroid next to the Sun and Jupiter, there are five places where it can travel along with the two large ones, the Lagrange points. L1, L2 and L3 lie on the line through the Sun and Jupiter, these are Euler’s solutions. L4 and L5 lie 60 degrees ahead of and behind Jupiter on its orbit, and there the Sun, Jupiter and the asteroid form an equilateral triangle.
The Sun (gold), Jupiter (red) and the Trojans at L4 and L5. Rotating with Jupiter, the two triangles stand still and you can see the asteroids swinging slowly around L4 and L5. Seen from outside, the whole figure turns around the Sun. Simplified, with a circular orbit.
The points on the line are unstable. A body nudged slightly there drifts away. The triangle points L4 and L5, however, are stable as long as the larger body is at least about 25 times as heavy as the middle one. The Sun is roughly a thousand times as heavy as Jupiter, so asteroids stay there for billions of years. They do not sit exactly on the points but swing around them in wide arcs, and one cycle of this swinging takes about 150 years.
The Trojans
For more than a hundred years Lagrange’s solution was pure mathematics. In 1906 Max Wolf in Heidelberg discovered an asteroid 60 degrees ahead of Jupiter and named it Achilles. More soon followed, and they were named after the heroes of the Trojan War: ahead of Jupiter, at L4, lies the Greek camp, behind it, at L5, the Trojan camp. Two names were given before the rule was fixed, and so Hector stands in the Greek camp and Patroclus in the Trojan one.
Today well over ten thousand Jupiter Trojans are known. Since 2021 the spacecraft Lucy has been on its way to visit several of them from 2027. They are thought to be unchanged remnants from the time the planets formed.
Triangles everywhere
Trojans are not unique to Jupiter. Neptune and Mars have their own, and even the Earth carries at least two small companions at its point L4, 2010 TK7 and 2020 XL5. Saturn’s moons Tethys and Dione each share their orbits with two tiny moons running 60 degrees ahead and behind.
Spaceflight uses the points on the line. The James Webb Space Telescope sits near the point L2 of the Sun and Earth, 1.5 million kilometres from Earth, and the solar observatory SOHO near L1.
What it means
In the tetractys the point is followed by the line, and the line by the triangle (see From point to solid). The mechanics of the sky shows the same sequence: three bodies can keep their shape only as a line or as a triangle, and only the triangle is stable. The equilateral triangle, the first surface of the Pythagoreans, is the simplest lasting order that three heavenly bodies can form.