Constellations of co-orbital planets

It’s time to revisit one of my favorite topics: co-orbital planetary systems, in which more than one planet share the same orbit around a star.

(Side note: co-orbital systems can exist on different size scales — many stars can share an orbit around a black hole, or many moons share an orbit around a planet, and those are also really cool).

Back in 2020 I wrote a blog post entitled Cohorts of co-orbital planets. The basic idea is summed up in this image:

The punchline is that three planets can share the same orbit without being stuck in Lagrange points. The stable L4 and L5 Lagrange points are located 60 degrees on either side of a planet’s orbit — that is where, for example, Jupiter’s Trojan asteroids have lived since the Solar System formed. Planets can remain stable orbiting 60 degrees apart from each other on the same orbit, in each other’s L4/L5 points (for more, see here).

To write “cohorts of co-orbital planets” I ran some N-body simulations to figure out which systems were stable and which were not. The idea of cohorts of co-orbital seemed to be new: there were no published papers in the literature that quite explained this phenomenon.

On a philosophical level, I loved the idea of “cohorts” — planets sharing the same orbit around their star. If they always stayed close to each other, these co-orbital planets would be cosmic buddies, keeping each other company in this lonely Universe.

But I soon discovered that almost everything I knew about “cohorts” was completely wrong.

I wanted to go deeper in my exploration of co-orbital cohorts. First, I recruited some of my favorite collaborators: Dimitri Veras, Matt Clement, Andre Izidoro, David Kipping, and Vikki Meadows. Each one a scientific titan (see below).

Next, I ran more simulations and studied them closely. What I found was that the cohorts I had found were not cohorts at all! A cohort is a group that stays together. But the planets in my simulations did not just remain evenly separated as they went around their stars. Instead, they were moving relative to each other, sometimes seeming to come closer together before making “U-turns” and changing direction.

In fact, they were following horseshoe orbits (which I’ve used to build weird planetary systems before). But not a type of horseshoe orbit that anyone had seen before… (cue dramatic music)

Our paper on “horseshoe constellation” systems was just published, so it’s time to share what this is all about.

Let me illustrate this idea with a series of animations.

Let’s start with the simplest case. Below is an animation of a planet like Earth orbiting a star like the Sun. The “inertial frame” (left panel) shows what this system would look like through a telescope staring down at the star, with the planet going around the star. The “co-moving frame” shows what it would look like if your telescope was rotating along with the average speed of the planet. Now the planet appears to be stationary, because our telescope is following it along its orbit. This is the boring case — things will get more interesting with more planets. [Note: the planet’s size is greatly exaggerated compared to the size of its orbit in this animation and all the others coming up.]

Let’s add in another planet to spice things up. The next animation shows two Earths sharing an orbit around a Sun-like star. In the inertial frame both planets are just going around their Sun (although time is moving much faster than in the 1-planet case). But notice how the two planets are not actually sharing an orbit. First the blue planet is a tiny bit closer to the Sun, then the two planets come close to each other (although they are still far from colliding), and then they switch orbits, with the red planet closer to the Sun. (Note that the actual distance to the star is exaggerated to make it easier to see).

Now the co-moving frame becomes really helpful. Here the ‘telescope’ is moving at the average speed of the two planets combined, but neither planet is moving at this average speed! Instead, one planet is a little closer to the star (and therefore orbits a little faster that this average speed) and the other is a little farther away (and orbits a little slower). When the two planets get close together they give each other a gravitational kick — the outer planet gets slowed down and ‘falls’ closer to the star, whereas the inner planet is accelerated and pushed out away from the star. The net effect is that they swap orbital distances. (For more on horseshoe orbits, see here).

In the co-moving frame, each planet traces out the shape of a horseshoe — hence the name horseshoe orbit. But it takes about 50 years for the planets to trace out these horseshoes, which is why the animation moves a lot faster than the single-planet animation.

What about systems with more than two planets?

I’m glad you asked! Below is a simulation with 4 planets — each one Earth mass — sharing an orbit around a star like the Sun. You’ll see that all of the planets are on horseshoe orbits, but things look a little different compared with the two planet case. Here, each planet can’t drift nearly as far without running into another planet, because there are more planets occupying the same orbit. When two planets approach each other, they still make horseshoe “U-turns” in the co-moving frame, and then head in the opposite direction. You can see in the inertial frame that planets take turns being a little closer or farther from the star, and swap places each time they encounter a neighbor.

Each planet only encounters its next-door neighbors.

If we number the planets 1, 2, 3, and 4 in order around the orbit, planet 1 sometimes encounters planet 2 and other times planet 4, but never comes that close to planet 3. Planet 2 undergoes encounters with planets 1 and 3 but never planet 4, and so on.

Let’s take this setup even further. The next animation shows a simulation in which six planets (again, each the same mass as Earth) share an orbit. The planets follow the same pattern as in the 4-planet case, but the orbit that they share is more crowded. The simulation starts with them all bunched up on one side (in the co-moving frame), but then they spread out and undergo a series of encounters with their neighbors, each of which leads to a horseshoe U-turn and a swap in orbital distance from the star (visible in the inertial frame).

As before, each planet only interacts with its next-door neighbors, but now there are more planets. So, if we order the planets 1 through 6, then planet 1 only encounters planets 2 and 6 but never planets 3, 4, or 5, while planet 2 only encounters planets 1 and 3 but never planets 4, 5, or 6, and so on.

Now let’s take it to 12. The next animation shows a 12-Earth horseshoe system in the co-moving frame (the inertial frame gets too messy with this many planets). As in the 6-planet case, the planets all start off on one side of the star and then spread out (in the co-moving frame). They follow the same pattern, encountering their next-door neighbors and undergoing horseshoe U-turns. With so many planets on one orbit the system starts to get a little cramped, and often the planets don’t go very far before running into their neighbors.

I couldn’t stop there, so I took it all the way up to 24! This next animation shows 24 Earths sharing an orbit the same size as Earth’s. The orbit has become so crowded that the planets barely have room to budge. Viewed in this co-moving frame, each planet only wobbles back and forth by about 15 degrees, so there is no sign of the horseshoe shape that we saw in the 2-planet case. Nonetheless, the dynamics is the same and is still governed by gravitational encounters between the planets that lead to horseshoe U-turns.

How much farther could this go? Could even more planets fit into a horseshoe constellation like this one?

The highest number of planets that I tested was 24. There is another type of co-orbital ring in which the planets don’t undergo horseshoe U-turns and instead just remain fixed in place in a co-moving frame. Those are the ones that I’ve used in the past to build giant planetary systems with hundreds (to millions) of planets orbiting a single star (or black hole). That type of co-orbital ring can have up to 42 Earths sharing a single orbit, and if kicked hard enough (by a rogue protoplanet) can transition to a horseshoe-like state. (I’ll discuss this more below talking about long-term stability; it’s actually the idea behind of an offshoot scientific paper that we’re publishing). So, I think the absolute maximum is 42 (isn’t that always the answer?).

What about systems in which the planets don’t all have the same mass? This is the case for the only known horseshoe observed in nature: two moons of Saturn — Janus and Epimetheus — are currently locked in a horseshoe that was only discovered in 1980. Since Janus is more massive than Epimetheus, it doesn’t move as far after the two moons encounter each other, so Epimetheus traces out a big horseshoe while Janus just drifts a little ways.

This is reminiscent of a blog post I once wrote about what it would be like to live in a horseshoe-type system. I considered a couple of scenarios, including the case where an Earth-mass planet was in a horseshoe configuration along with a Jupiter-mass planet. The Earth ended up doing almost all of the moving around (traced in yellow in the image below, in a co-moving frame of course) while the Jupiter, being 300 times more massive, basically stayed put.

I ran simulations of systems that included planets with masses of both 1 and 10 Earth masses. Just like for an Earth co-orbiting in a horseshoe setup with a Jupiter, the lower-mass planet makes much bigger oscillations in orbital distance, and so it’s these small planets that end up tracing out the horseshoe while the more massive planets just move slowly. This is the case for Janus and Epimetheus too.

In this next example simulation, there are two 1 Earth-mass planets and two 10 Earth-mass planets. One of the 1 Earth-mass planets ends up getting kicked back and forth by its massive neighbors and completes a nice horseshoe (although it doesn’t make it all the way around the bend every time). The other 1 Earth-mass planet remains sandwiched between its 10 Earth-mass neighbors.

I also tested systems with 6, 8, and 10 planets alternating between 1 and 10 Earth masses. The only ones that were sometimes stable were the 6-planet systems. So, it seems like the only way to build elaborate horseshoe constellation systems is with equal-mass planets (although nothing prevents them from having moons).

The Solar System could easily host one or more horseshoe planets.

In addition to simple systems that all shared a single orbit, I tested some more exotic configurations. The most interesting ones were built on the Solar System.

It turns out that the Solar System could easily host another Earth is placed on the same orbit as our own. Two more Earths (for a total of three) can even fit, doing a horseshoe dance, and remain stable for billions of years. Likewise, up to three Venuses could share Venus’ current orbit, two Marses or two Mercuries. I couldn’t find a configuration in which all four rocky planets had horseshoe companions, but Venus and Earth can both have them at the same time. Imagine a couple extra terrestrial planets in the Solar System!

From a philosophical point of view, this shows that there is almost always “leftover space” in planetary systems. It’s been shown that all eight Solar System planets would remain perfectly stable if an additional planet as massive as Earth (or even more massive) magically appeared within the asteroid belt (see here and here). Jupiter’s L4 and L5 (Trojan) Lagrange points could each host a planet as well (instead of all those asteroids). And now we’ve shown that the rocky planets could each share their own orbits with other planets.

Clearly, the process of planet formation is inefficient in that it does not fill up every available nook of orbital real estate, instead leaving stable niches scattered among the planets’ orbits. This leaves a lot of room for both building giant, majestic systems with hundreds to millions of planets (as I do on this blog), and also for orbital engineering. If we find extremely exotic systems around another star, it’s quite possible that they could be artificial in nature (perhaps serving as cosmic time capsules).

Enough philosophizing for now — back to the horseshoes.

The orbits of “horseshoe constellation” systems remain stable for billions of years.

I ran simulations of a subset of these systems for 10 billion years (most of the age of the Universe) and they just chugged along happily.

But, you might wonder, a star like the Sun will become a red giant when it reaches an age of about 10 billion years. Won’t that destabilize horseshoe planets?

No, it turns out horseshoe constellations remain nice and stable during this phase. We already saw this in a previous post (A cosmic time capsule). This image shows the evolution of the orbital distances of each of the five planets in a 5-planet horseshoe constellation (which I called a “cohort” at the time) as their Sun-like star evolves from the main sequence until becoming a white dwarf (see here for more on stellar evolution). What you can see is that the planets’ orbits expand as the star loses mass, but all of the planets remain at the same orbital distance, in their horseshoe constellation.

You might still be wondering whether some other perturbations would destabilize horseshoe constellation systems. With so many planets sharing the same orbit, it can’t be that hard to knock them around, can it?

I ran a bunch more simulations to figure out exactly what it takes to break a co-orbital system of planets. I tested two kinds of co-orbital planet rings: horseshoe constellations and stationary rings. Stationary rings are like the ones in the Ultimate Engineered Solar System. What’s different about stationary rings of planets is that all the planets stay the same distance away from each other. So, in a co-moving frame, they look, well, stationary.

We found two key results. First, rings of co-orbital, Earth-mass planets can be destabilized by a rogue planet that zooms by if that rogue planet is more massive than the Moon (which is about 1% of Earth’s mass). Second, kicking a stationary ring of planets often turns it into a horseshoe constellation. Here is an animation of a co-orbital ring of 9 planets that is kicked (at about 30,570 years) and starts drifting. (The rogue protoplanet is not shown, but its first victim is the gray planet in the bottom left):

This system was clearly kicked from its stationary perch down into horseshoe-land.

This means that there aren’t really two different types of co-orbital rings of planets. Horseshoe constellations and “stationary rings” are the same beast. Horseshoe constellations are just in a more excited state than stationary ring systems. Ring systems are like scoops of ice cream nicely floating on a big glass of milky. Horseshoe systems are a shaken-up version, like a milkshake.

As the Universe gets older and older, the odds of any given planetary system being kicked around (and shaken up) get higher and higher. So, the odds of finding a horseshoe constellation increase and the odds of finding a stationary ring system decrease. Milkshakes for the win!

Horseshoe constellation systems are detectable with present-day technology.

We can find planets as small as Earth around other stars using methods like the transit and radial velocity techniques (see here for a refresher on planet-finding methods). We’ve found some pretty interesting orbital configurations, like systems of planets in chains of orbital resonances (a la Trappist-1), and planets on very stretched-out, eccentric orbits.

Several groups have been looking but no one has found co-orbital planets — yet. (Side note: my personal guess that ESA’s PLATO mission will be the first to detect them, unless it has to scale-down its duration — this is because tides often destabilize the orbits of co-orbitals on orbits very close to their stars, where most exoplanets have been found).

The easiest way to find a horseshoe constellation around another star would be to see the system in transit. In that case, each planet would pass along our line of sight to the host star and block a little bit of starlight.

Credit: ESA.

A horseshoe constellation would produce a *lot* of dips in the star’s brightness from all of those planets passing in front. It would be a big challenge to use those dips to figure out the orbital structure of the system, for a couple of reasons. First, the planets would all be the same size (as required to remain stable), so it would be hard to know which dips corresponded to which planet. In a system where each planet has its own orbit and size, it’s easy to tell the planets apart, but not in this case. Second, the exact timing of planetary transits would slowly shift. A single planet transits like a clock on the hour — always at a consistent rate. But in horseshoe constellations the planets are constantly shifting their relative positions, which would confuse that rhythm.

In our paper we flipped that idea on its head and showed that it’s possible to use the planets’ shifting positions to confirm that the planets are indeed in a horseshoe constellation. The trick is to watch a system long enough, taking careful measurements of the exact timing of each planetary transit, to see several of the horseshoe U-turns, when the planets’ drift changes direction. The best-case scenario is for horseshoe constellations that are close enough to their stars that horseshoe U-turns happen often but not so close that tidal interactions with the star make the system unstable.

How might horseshoe constellations form?

Another way to frame this question is: can horseshoe constellations form naturally? Unfortunately, the answer falls into the galaxy-sized box labeled “no one knows (yet)”. We do know that the Janus-Epimetheus system formed around Saturn, and there are models that can explain their formation with standard, natural processes. However, no one has shown how more than one planet could end up orbiting a star in a horseshoe configuration, let alone how a constellation of 3, or 4, or 12, or 24 planets could form. One can imagine a giant ring of dust around a star in which pockets locally collapse to form planets sharing an orbit. But is it really plausible to envision the planets ending up in a stable co-orbital configuration? (Remember that a single, Moon-sized object out of place would disrupt the system). Maybe it’s possible, but I am guessing that it’s ridonkulously rare for horseshoe constellations to form naturally.

The alternative is that horseshoe constellations might be produced by orbital engineering done by highly advanced civilizations. If such a civilization wanted to leave a mark of its existence that would be both long-lived and detectable from astronomical distances, a horseshoe constellation would be a good candidate. This is similar to the setup invoked in A Cosmic Time Capsule — in a previous study we called these SETI beacons. It’s hard to know exactly how or why an advanced civilization would build a horseshoe constellation. But, given that this type of system is both stable and unlikely to form naturally, horseshoe constellations are a solid candidate for SETI beacons.

We should definitely keep our eyes out for horseshoe constellation exoplanet systems. If we find one, it will either revolutionize planet formation or be the signpost (or time capsule) built by a highly-advanced alien civilization. It’s a win-win!

Questions? Comments? Words of wisdom?

Additional Resources

4 thoughts on “Constellations of co-orbital planets

  1. How close do the two Earth mass planets get, and how much does their solar distance change?

    1. The absolute closest two Earth-mass planets can get during a horseshoe encounter is about 5 Hill radii, which translates to about 0.05 au for Earths at 1 au from a star like the Sun. The maximum change in orbital distance is about half a Hill radius in either direction, so oscillating between about 0.995 and 1.005 au from the Sun.

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