The mystery of the missing planets

This article was originally posted on Nautilus’ blog (see here). I’ve spruced it up with some poems along the way and a little more detail about my own pet theory at the end.

There is an unsolved problem I want to tell you about: The case of the missing Trojans. You might be thinking of the mythical horse with soldiers hidden inside. Or maybe you’re thinking of a sports team. Or a type of computer virus, or, let’s be honest, of the condoms. (Note that I said, “Case of the missing Trojans,” not, “the Missing case of Trojans.”)

But there is another type of Trojan: an orbit. A swarm of thousands of asteroids share an orbit with Jupiter. They are not right next to the planet, like moons, but rather clustered in two clumps, one ahead of Jupiter on its orbit and one behind. The leading clump is called the “Greeks” and the trailing clump the “Trojans,” but they are usually just lumped together and simply called Jupiter’s Trojans. Many of them are named after mythological figures from the Trojan wars.

solar system Trojans breaker
The inner Solar System. The planets are labeled and the blue lines show their orbits. The small dots are asteroids. The main asteroid belt is shown in white. The green dots—called “Greeks” and “Trojans”—what we call Jupiter’s “Trojans.”Wikipedia user Mdf

Jupiter is more than 300 times more massive than Earth. Its gravity has almost completely emptied out the asteroid belt, which we think once contained 1,000 times more material than it does now. How can these Trojans survive so close to a gas giant and yet remain on stable orbits?

Jupiter's massive. It's got a bright aura.
And swarming companions like sharks have remora
The Trojans all follow in Jupiter's path
But keep at a distance, avoiding its wrath...

It turns out that this was figured out way back in 1772 by French mathematician Joseph-Louis Lagrange. If we visualize a planet’s orbit around a star, there are a few special points where an extra body can orbit at the same rate as the planet. These are called the Lagrange points, and there are five of them, named L1 through L5.

Lagrangian point schematic breaker
Lagrange points of a planet (blue) orbiting a star. L4 and L5 are the place where Trojan planets can survive. The other points (L1, L2 and L3) and not stable.  Credit: WikipediaWikipedia user Cmglee

Only two of the Lagrange points are stable: L4 and L5. A body that is located at or near these points will happily orbit the Sun, remaining roughly 60 degrees in front or behind Jupiter. (More precisely, the extra body will actually oscillate with a typical amplitude of about 20 degrees around its Lagrange point, and there is even a class of orbit called a “horseshoe” that goes all the way around between L4 and L5.) Jupiter has Trojans simply because its L4 and L5 are stable despite the gravitational kicks of the other planets. They are islands of stability in a huge unstable ocean. If asteroids were sprinkled uniformly across the inner Solar System, the only ones that would survive near Jupiter are the Trojans.

Jupiter is not the only planet in the Solar System to have Trojans, although it has by far the most. Uranus (1 Trojan) and Neptune (13) each have them. So does Mars (7). And Earth even has one! Surprisingly, Saturn does not have any; the reason appears to be that Jupiter’s gravity de-stabilizes Saturn’s Trojans on million year timescales. Saturn’s Trojan island has sunk under the ocean of instability (though Saturn’s moons actually have four Trojan), as has Venus’.

Let’s take this Trojan idea a step farther. The objects trapped near the stable islands at L4 and L5 don’t need to be puny little asteroids. They can be full-grown planets! If two planets lie in the same orbit, separated by 60 degrees, they form a pair of Trojan planets.

Trojan planets are commonly found in computer simulations that try to explain how planets orbiting other stars (“exoplanets”) are created. Many of the known exoplanets have orbits very close to their stars. Astronomers think that many of these planets formed farther away and were pushed closer in by a process called orbital migration. It is the disks of gas and dust from which the planets were born that are causing the planets to migrate. And it is during this migration that adjacent planets can be trapped in Trojan configurations. Trojan planets don’t form during every simulation but they do pop up every third or fourth time.

trojan planet sim
Snapshot from a computer simulation of a pair of Trojan planets—each 10 times as massive as Earth—migrating in a gaseous protoplanetary disk. The image shows the wakes created by the planets in the disk; the planets are at the center of the light green swirls, at the center of the wakes. Gory details here.Arnaud Pierens

So we think that Trojan planets should exist. We don’t have any in the Solar System, but Trojan planets should definitely exist around other stars, and they should be detectable with current telescopes! The best tools for the job are Kepler and TESS, NASA’s planet-finding space telescopes. So what have they found?

Drumroll…and a big frowny face. Kepler has not found any Trojan planets. There was one announced discovery of a four-planet system called KOI-730 (now named Kepler-223) with two Trojan planets sharing an orbit plus two more planets, one interior and one exterior to the Trojan pair. 

koi orbit

Unfortunately, this system was later determined to be in a different configuration with no Trojan planets. Bummer. TESS also had a promising candidate that didn’t pan out (although that system — TOI-178 — contains 6 planets in a chain of orbital resonances).

The search for Trojan exoplanets has been ongoing for several years, and it continues, but things are looking grim. Trojan planets are supposed to be relatively easy to detect, so their absence stands out. It’s a cosmic mystery: The case of the missing Trojans!

So why aren’t we finding any Trojan planets? There are two possibilities: Either the Trojans are there but we aren’t finding them, or they simply aren’t there.

Could Trojan pairs be harder to find than we thought? Well, two planets on the same orbit each produce a signal with the same frequency and, if you aren’t expecting this, it can make both planets look like noise. But we are expecting it. Plus, the gravitational tug of war between Trojan planets also changes the phase of their signal, but we know this, too, and can actually use it as a tool to try to find Trojans.

Where are those Trojans? Just where are they hiding?
Are they at the store, or maybe hang-gliding?
We think they exist — just what have we missed?
Well, every good story should end with a twist…

Perhaps Trojan configurations are destroyed either as planets migrate inward or, for planets very close to their stars, because of tidal interactions between the planets and star. However, these destruction mechanisms should only affect a small fraction of the planets that Kepler is finding.

Let me share my own pet theory. Planets migrate inward in cohorts, often with Trojan pairs. But when the gaseous protoplanetary disk dissipates, the orbits of the migrated planets usually go haywire. There is a phase of giant collisions between planets that both destroys Trojan pairs and re-organizes the orbits of the survivors. This model matches what we see in the dataset of exoplanets, so that’s encouraging. Plus, once in a while, planets migrate in and do not go haywire, like the multi-resonant Trappist-1 system.

Based on this chain of logic, multi-resonant systems are where we should be searching for Trojan planets. Of course, only a fraction of multi-resonant systems will have Trojans, and we think they need to be pretty far from their stars to avoid those nasty tidal effects. If Trojans need to be at least 1/4 to 1/2 an Astronomical Unit from their stars to survive, then we just need to wait. We’ll probably need to find 10 or so, since about one in ten simulations forms a Trojan planet. Kepler could have found these systems if it had survived, but TESS doesn’t look at any star long enough. So what’s next? ESA’s PLATO mission (planned launch in 2026) will be able to find them, but only after several years because it takes a lot of data to really nail down the resonances.

Luckily the TROY project is dedicated to this, so let’s hope they solve this mystery and find the missing Trojans!


2 thoughts on “The mystery of the missing planets

  1. Interesting! Could your theory explain what is thought to have happened with the Earth and Theia?

    1. There is one theory that suggests that the Moon-forming impactor Theia hung around in a Trojan-like orbit of the proto-Earth for a long time before eventually colliding (….129.1724B/abstract). It was proposed because the impact speed would generally be quite low in that scenario, which was required to match the Moon in early simulations of the giant impact. In newer simulations, a much wider range of impacts is possible, so there is no compelling need for this scenario (although it’s still plausible).

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