Equilateral triangles in space

As we learnt in school, some planets, asteroids, comets etc orbit faster than the Earth, for example Venus takes 225 days to orbit the Sun. And some orbit slower, for example Mars takes 687 days to orbit the Sun. Now astronomers have discovered an asteroid that orbits at exactly the same speed as Earth, i.e., it too takes 365 days to go round the Sun.

Now, this is harder that it sounds. It is not possible for another body to orbit just anywhere along the Earth’s orbit, for example if an asteroid orbited the Sun at the same distance as the Earth does but just behind the Earth, then the Earth’s gravitational pull would disrupt its orbit.

However, the French mathematician Joseph Louis Lagrange realised that if the 3 lines between an asteroid, the Sun and the Earth form an equilateral triangle (which implies that all 3 distances, Sun-Earth, Sun-asteroid, and Earth-asteroid, are the same) then the orbit is stable. This then defines two points in the Earth’s orbit: one in front and one behind the Earth where an asteroid can orbit the Earth with a period of 365 days and do so stably, i.e., orbit year after year.

This is a beautiful example of the geometry of the universe. There is a precise correspondence between the asteroid orbiting stably and the distances forming a precise equilaterial triangle. Maybe when you were taught equilateral triangles in maths at school you thought that they were some weird maths thing of no relevance to the real world. If so you were wrong.

Such an asteroid has just been discovered, and announced in the journal Nature. I am not an expert in this but as I understand it a reason it has taken so long to find it – Lagrange predicted these orbits in 1772 – is simple.

Because an asteroid is always 60° ahead (or 60° behind), because all 3 angles of an equilateral triangle are 60°, then it is always in daytime sky – just look at the Sun then rotate your head by 60°. This is different from looking at a star of course, then we just wait till night and look at it then.

This makes detecting an asteroid tricky of course as it only reflects some tiny fraction of the Sun’s light whereas at the same time we are getting the full force of the Sun itself. The asteroid is small, only a few hundred metres across.

But we have found it, and incidentally found that it actually deviates rather a lot from the simple point, i.e., it kind of orbits around the point formed by one corner of the equilaterial triangle, you can see this in the animation of the orbit the authors Connors et al. have made, which is pretty cool.