Cohorts of co-orbital planets

This post is an adventure in world-building. I use N-body simulations to find orbital configurations of planetary systems that (I think) are completely new, pretty awesome and unexpected. It was inspired by discussions with Charles Choi as he was writing this article.

It’s all about co-orbitals, one of my favorite orbital setups

In a co-orbital system, two or more planets share the same orbit. It’s a beautiful and quirky setup. Kind of like eating bacon with ice cream (my favorite celebratory breakfast)!

The classic co-orbital setup includes one massive planet and a second puny one. The puny planet is stable if it stays about 60 degrees in front of or behind the massive one’s orbit, at stable Lagrange points L4 or L5.

Lagrange points of a massive planet (blue) orbiting a star. L4 and L5 are the place where co-orbital planets are most likely to be. L1, L2 and L3 are unstable. From Wikipedia.

In the Solar System, the best example of co-orbitals are Jupiter’s Trojan asteroids. They orbit around L4 and L5 and — fun fact — are thought to have been captured by Jupiter when the giant planets went unstable (more here). NASA’s upcoming Lucy mission plans to fly up close and study several Trojans.

It would be perfectly stable if — instead of Trojan asteroids — Jupiter had an Earth-sized planet sitting at L4. And it could even have a second one at L5. This is one of the ninja moves I used in Building the Ultimate Solar System.

Two planets with similar masses can also share the same orbit if they orbit 60 degrees apart. This means that each is in the other’s L4/L5 Lagrange point. This kind of configuration comes out of our computer simulations, and we expect to find one of these setups among exoplanet systems.

The most extreme case of co-orbital planets are rings of orbiting planets. A ring of planets is stable as long as all the planets have the same mass and they are evenly spaced along the same circular orbit.

This crazy-looking orbital setup — with 42 Earth spaced evenly along Earth’s orbit — is stable for billions of years!  (Explanation here; technical details here).

Rings of planets were at the heart of the Ultimate Engineered Solar System and the Million Earth Solar System. Rings are the most efficient use of orbital real estate.

Let’s explore a side idea: can an arc of co-orbital planets exist? This would be a slice of a ring of planets.

We are not going to just imagine what might exist. We’ll start off that way, then we’ll use computer simulations to test whether our imagination holds up. (I run this kind of simulation in my day job as an astrophysicist).

There are two ways that arcs of planets might be stable, depending on how the planets are spaced out. Either they are spread out at Lagrange points or by “Hill radii” (which I’ll explain).

Let’s go through them one at a time.

Case 1: Co-orbital planets separated by 60 degrees

We already know that it’s stable when Jupiter has an Earth 60 degrees ahead and/or behind its orbit, and when two Earths are separated by 60 degrees along their orbit.

I want to know whether two Neptunes, Saturns or (mega-)Jupiters could share an orbit. And whether 3, 4, 5, or even 6 planets could share the same orbit, spaced by 60 degrees.

Cue the N-body simulations.

These simulations are pretty simple: I input the starting setup of a system of planets, and the code runs the system forward in time, taking into account the gravity between objects. It’s like playing God with planetary systems (or super planet crash).

Technical details: I’ll call a system “stable” if it keeps the same configuration for 100 million years. I know that’s less than 1% of the age of the Universe, but when things go unstable it’s usually fast. Plus, this keeps the simulations from taking too long to run. (Each simulation took between 20 minutes and a couple of hours). We’ll keep the Sun in the middle, and in almost all cases the planets will follow a circular orbit the same size as Earth’s orbit. I’ll just use groups of equal-mass planets to keep things simple (Jupiters with Jupiters and Saturns with Saturns, without mixing). There are some interesting setups with mixed-mass planets — like Klemperer Rosettes — but we’re not going there in this post.

First result: systems with co-orbital Earths or Neptunes are very stable.

Anywhere from 2 to 6 of these planets can follow the same orbit and stay nice and stable. It can be a full ring of 6 planets, or just a slice.

Second result: systems with co-orbital Saturns and Jupiters are pretty stable.

Systems with 2, 3, or 4 Saturns — again separated by 60 degrees along the same orbit — were stable for 100 million years. But systems with 5 or 6 went unstable fast (in about 100 years).

Systems with 2 or 3 Jupiters were stable, but systems with 4 or more Jupiters were unstable really fast (in less than 100 years).

Third result: co-orbital stability breaks for really massive planets.

Systems with 3 planets that are each 3 times Jupiter’s mass, separated by 60 degrees, are stable. But increase the mass to just 3.2 times Jupiter’s mass and they’re unstable.

Next I ran simulations to test how massive two co-orbital planets (60 degrees apart) could be. This kind of system is stable with planets up to 14 Jupiter masses, but unstable at 15 Jupiter masses.

Why do systems stop being stable when they become too massive? In simple terms, there is just too much gravity. It’s like a chain of people walking down the street holding hands. The harder they pull on each other (the stronger the gravitational pull, or the more massive the planets), the more easily the chain will break.

Are moons of these planets stable? I didn’t test this because it requires a different type of code (and it takes much longer). But moons are just bit players in the stability game, so the should be stable.

Let’s use what we’ve learned to improve our old friend the Ultimate Solar System.

We’ll focus on Ultimate system number 2. In that system, the habitable zone was dominated by four Jupiters on concentric orbits. Most of the habitable real estate was on the moons of those Jupiters:

Ultimate Solar System 2. Each orbit around the star harbors a gas giant orbited by five large moons. There is also a binary Earth in both the leading and trailing Trojan points with the gas giant (60 degrees in front of and behind the giant planet in its orbit around the star).

Each orbit was shared by one Jupiter (with five moons) as well as two binary Earths, 60 degrees in front of and behind the Jupiter’s orbit. That’s 9 habitable worlds per orbit.

Now we know that a single orbit can hold three Jupiters or four Saturns. And there’s no reason each of those gas giant planets can’t have a system of moons. That makes 15-20 habitable worlds (moon) sharing a single orbit.

But how many of these orbits can fit inside the habitable zone?

Systems of gas giants are naturally spaced in orbital resonance. In 2:1 resonance, the outer planet completes one orbit for every two orbits of the inner one, and the planets re-align in the same spot each time.

With more than one planet per orbit, resonances between orbits become less stable. Four Jupiters orbiting the Sun, each in 3:2 resonance with its neighbors, is stable. But with 3 Jupiter per orbit, it is unstable.

I did not exhaustively search for the configurations that would optimize the number of gas giants in the habitable zone. But I did find that three orbits hosting 3 Saturns each can be stable if each orbit is in 2:1 resonance with the next one out. Three such orbits can fit in the habitable zone.

Let’s call this Ultimate Cohort System (number 1 — more will follow below).

The first Ultimate Cohort System, with a total of 45 habitable worlds — all moons of Saturn-mass gas giants.

A cohort is a group of people that share something in common. A cohort of planets falls in between a single planet (or a simple pair of co-orbitals) and a ring of many planets.

But, even though I’m tempted, we can’t abbreviate a cohort of co-orbitals as “co-horbitals” because that would just be ridiculous!

Case 2: Co-orbital planets spaced by Hill radii

The Hill sphere is the ball-shaped region around a planet in which the planet’s gravity dominates over the star’s. Moons must orbit within a planet’s Hill radius (and moons-of-moons within a moon’s Hill radius):

A moon orbiting a planet orbiting a star.  In each panel the “camera” is orbiting along with the planet or moon. The thin lines show stable orbits and the five Lagrange points are labeled (although only L4 and L5 are stable).  Adapted from Domingos & Winter (2005).

To remain stable and avoid crashing into each other, planets must stay far enough apart. Stability is guaranteed when two planets are more than about 4 Hill radii apart. Systems with more planets have to be more widely spaced.

When building rings of planets (a la Ultimate Engineered Solar System), stability requires that planets are far enough apart along their shared orbit rather than in-between orbits. To ensure that the ring is stable, the planets must all be the same mass and be perfectly evenly-spaced.

This made me wonder: can two or more Earths share the same orbit simply by being far enough apart, without worrying about Lagrange points or perfectly-spaced rings?

Let’s find out, using the same kind of N-body simulations as before.

Experiment 1. How close together can 2 or 3 Earths be along a shared orbit?

I ran a set of simulations with 2 Earths on the same orbit and another set with 3 Earths on the same orbit. In each simulation I only changed how far apart the planets were at the start.

As you might imagine, systems in which the planets started close together were unstable faster than those that started farther apart. And when the planets were far enough apart, the system stayed stable.

Here is a graph showing the results for the 3-Earth case:

The graph shows that in order for a system of 3 co-orbital Earths to be stable for at least 100 million years, the planets have to be at least 20 Hill radii apart.

You might wonder, would a system that was stable for 100 million years go unstable in a billion years? To test this I would have to burn a lot more computing time to find out. From past experience, the answer is: probably not. Usually a system that’s going to go unstable does it quickly. And the farther apart the planets are, the lower the chances of a later instability.

[Side note: there is a roughly 1% chance that the Solar System itself will go unstable in the next 5 billion years. That was determined using this same kind of N-body simulations while taking into account the uncertainties in the exact positions of the planets.]

Here is what a cohort of 3 Earths separated by 20 Hill radii looks like.

The most compact configuration of three Earths sharing an orbit around the Sun. Each planet is 20 Hill radii from its neighbors, about 11.5 degrees apart in angle.

These three planets are a real cohort, buddies sharing the same orbit!

Compared with the 3-Earth case, a system with 2 Earths can even closer. A 2-Earth cohort is stable when the planets are as close as 8 Hill radii.

Experiment 2: what combinations of cohorts could orbit the same star?

I ran a few simulations with 4 or more Earths sharing the same orbit. I found that 4 Earths need to be spaced by at least 25 Hill radii to be stable. And when Earths are spaced by 30 Hill radii (17.2 degrees along their orbit), at least 12 can share the same orbit.

[Side note: in rings — like in the Ultimate Engineered Solar System — the planets can be quite a bit closer, about 15 Hill radii apart. In a ring, the gravity from a neighboring planet on one side is cancelled out by the gravity from the neighbor on the other side. In a cohort, the overall strength of the gravity of planetary neighbors must be smaller because there is not always another planet to balance things out. ]

Here are a couple of stable co-orbital systems containing 6 and 12 Earths. In the 6-Earth cohort, the planets span roughly a quarter of the entire orbit. In the 12-Earth cohort they span more than half.

Cohorts can also be spread out along the same orbit. For example, one orbit can fit at least three different cohorts of 3 Earths each (spaced by 34 Hill radii), or four cohorts of two Earths (spaced by 30 Hill radii).

A system can contain multiple orbits, each of which is populated by cohorts of co-orbital planets.

As an example, let’s triple-up the Solar System’s rocky planets. Let’s make a system with three Venuses, three Earths and three Marses. And, to spice things up, let’s make all of the planets have the same mass and size as Earth.

A simulation shows that this setup is stable.

Although I’ve used images of Venus and Mars, in the simulation all of the planets on each orbit were Earth’s mass. Note that the angle between planets is the same on each orbit because the Hill radius scales with the orbital size.

There is a vast landscape of possible planetary systems with many orbits hosting cohorts of planets. I have only explored a tiny piece of that landscape but found some pretty cool (stable) systems.

I didn’t try to optimize the number of planets that could fit in the habitable zone. But I still felt compelled to beat out our old friend Ultimate Solar System 1, which had 24 planets in the habitable zone.

Here is Ultimate Cohort System 2, with three orbits in the habitable zone. Each orbit has four separate cohorts of two Earths, spread apart by 90 degrees. This makes a total of 24 habitable zone planets.

Ultimate cohort system 2 has a total of 24 planets in the habitable zone, with each orbit hosting four cohorts of 2 closely-spaced Earths.

Ultimate Cohort System 3 also has three orbits in the habitable zone, but each orbit contains a mega-cohort of 12 Earths that spans more than 180 degrees of the orbit.

Here each ring of orbits makes a beautiful crescent shape!

To have Ultimate Cohort Systems 2 and 3 remain stable, each orbit had to be widely-enough spaced relative to its neighbors. These systems are stable when the orbits are located at 1, 1.5 and 2.25 Astronomical Units (50 Hill radii between orbits), but not if they are closer to each other.

The outer orbit is close to the too-cold boundary of the traditionally-defined habitable zone. In practice, whether a planet at 2.25 AU can have liquid water will depend mainly on the thickness and composition of the planet’s atmosphere, and whether it can provide enough greenhouse heating.

Let’s leave the Ultimate-ness there. We could pack planets more tightly by having neighboring orbits revolve in opposite directions around the star (like in the Ultimate Retrograde System), or include many stars (like in the Ultimate 16-star system). But cohort systems will never be as tightly-packed as rings of planets (like the Ultimate Engineered system).

I am not sure whether planets in cohorts could have large moons. I suspect that the added gravitational kicks would make moons become unstable in time, but more complicated (and much longer) simulations are needed to find out.

What would a cohort of planets look like in the sky?

In the most tightly-packed cohort with 2 Earths separated by 8 Hill radii, the planets are 0.08 astronomical units apart along their orbit. That’s about 32 times the distance between the Earth and Moon.

Earth is 4 times bigger than the Moon, so sitting on one planet in a 2-planet cohort, your neighbor would be about 1/8th the size of the full Moon. That’s bigger (and much brighter) than Venus ever gets in the sky.

Now imagine that you lived in Ultimate Cohort System 3. Let’s say your home planet lies within the middle cohort of the system.

The other planets in the system would look like strings of pearls in the sky.

Planets in a cohort would follow a line across the sky, in the same way as the motion of the planets and the Sun and Moon in our sky. This is because all of the planets would be in the same plane (or close to it).

Each planet would have a phase between crescent and full. Venus goes through phases that are easily seen with binoculars or a small telescope. The planets in different cohorts would show similar phases, and since they are spread along the same orbit, it would be like seeing snapshots of Venus’ phases but all at the same time!

From the Astronomy Picture of the Day

Phases of planets in a cohort closer to the Sun would look a lot like the phases of Venus. But phases of planets in your own cohort would be a little different because you would never see thin crescents, or any planet less than half-full. And the phases of planets in a more distant cohort would always be closer to full.

Cohorts maintain their relative configuration as they orbit their Sun. This means that the other planets in your own cohort would always be in the same place relative to the Sun (although would move compared to the background stars).

For example, your closest (and brightest and biggest) neighboring planets would be visible high in the sky at sunset (for the planet trailing you in the cohort) and at sunrise (for the planet leading you in the cohort).

The next neighbors would appears just a little closer to the Sun than your closest neighbors.

Planets in your own cohort would always have the same phase. Your closest neighboring planets would always be half-illuminated.

Planets in neighboring cohorts would be a bit smaller and fainter than the closest neighbors in your cohort. They would also move in the sky relative to the Sun. Since all cohorts share a common orbital plane, planets in other cohorts would sometimes be eclipsed by planets in your own cohort!

With all of these bright moving objects in the sky, one can only imagine what kind of legends would arise on a planet in a cohort system. The distant stars would pale in comparison, and the study of planetary science would far outpace the study of more distant objects like stars and galaxies.

If you lived on a moon in Ultimate Cohort System 1, your home gas giant would dominate the sky. There would be a lot of interesting differences compared with Earth (some of which are discussed here). The other members of the cohort would be bright companions that would stay fixed relative to the Sun.

Science fiction possibilities within cohort planetary systems

Like a close-knit group of friends, a cohort of planets shares a bond that is stronger than its connection with other planets in the system.

The strongest bond would be with your closest neighboring planet (or planets), especially if you lived in a closely-packed cohort.

With such bright, constant reminders in the sky, neighboring planets might take on God-like status. A civilization would develop with these planets at the forefront of their mind.

As new technology was developed, it would all be pointed toward those neighboring worlds. Early telescopes would detect that the planets were only half-lit, and might even detect global-scale events like giant volcanic eruptions or giant storms. In time, telescopes would map the planet’s surface as it rotated and search for signs of life.

Nearby worlds would be the first targets for space exploration. Satellites would be placed in orbit, looking down at this world’s surface and beaming it back in full resolution.

Neighboring worlds would make ideal space colonies (surface conditions permitting). And once a civilization was spread over two planets, it would be ready to take over the entire system.

Imagine you lived on a planet in the Ultimate Cohort 2 system, with a single nearby neighboring planet. After colonizing your companion world, the next targets would be the other cohorts sharing your orbit. Next, domination of the entire system…

Now imagine that both planets of a 2-Earth cohort system were inhabited, but there was a world war on one of them. Would one faction send signals to its neighboring world asking for help? How would the other planet see and react to the war?

If you lived on a habitable moon in Ultimate Cohort System 1, the story would be similar. The first targets for exploration and colonization would be the other moons of your home gas giant. And then, system domination!

Of course, these ideas just scratch the surface. I’m sure there are some fascinating stories to be told in cohort planetary systems…

There you have it — cohort planetary systems!

As far as I am aware, cohort systems are new. I have not seen any studies showing that systems like this can exist and be stable (although it’s entirely possible that such studies exist — please let me know if you are aware of any). I might write a scientific paper following the general outline of this blog, showing that cohort systems exist and outlining their stability. It will be the first blog post that will precede a scientific result for me.

Thanks for joining me for the ride — I hope you enjoyed the post!

Additional resources

More about the book here. Amazon link here.

24 thoughts on “Cohorts of co-orbital planets

  1. Sweet, some more cool systems. Did you coin the term “cohort” as it applies to planets? Wait are they considered planets? Won’t the planets catch up and fall away from each other like Janus and Epimetheus?

  2. I do think that yes, I am coining the term “cohorts” of planets sharing an orbit. And yes, these are still “planets” in my mind, although the definition of a planet is still a little bit up for grabs. And no– the planets won’t catch up to each other because they share the same orbit and so move at the same rate around their Sun. For horseshoe orbits (like Janus and Epimetheus), one planet is slightly closer to the star and so orbits slightly faster, then the two planets exchange places and repeat the cycle. Here the planets all move at the same rate.

  3. Your illustrations of “cohorts” show planets of different size. Does this mean the co-orbiting planets can be of different mass?

    1. Oui, bien sur. Sur le cote droit de la page principale, il y a la possibilité de traduire chaque article avec Google Translate. Il faut selectionner “French” du menu et bim!

    1. Good question! Systems of two planets in each other’s L4/L5 points form systematically in simulations. We expect to find them among exoplanets. I remember one simulation where we even got 3 planets at 0, 60 and 120 degrees along the same orbit (although it’s rare).

      For two — any maybe even three — planets sharing an orbit but not in Lagrange points, I think that it’s possible for them to form naturally. More than that I would not expect.

    1. No I wouldn’t expect the ultimate cohort systems to form naturally. I think systems could form with each of the ingredients in these systems, but I wouldn’t expect them all to happen in a single system.

  4. Replace the star with a supermassive black hole. Replace planet cohorts with star cohorts. Place a planet in an orbit between the star cohorts. If you use only the lopsided cohorts on the outside occasionally all of the stars will be on one side of the plane and you get NIGHTFALL.

    Alternately you can use a couple of the three by three cohorts in the outer orbits. Replace the outer two members of one of the triplets in each ring with white dwarfs. Add a giant moon with an eccentric and inclined orbit to occasionally eclipse the outer stars when they line up.

    If the white dwarfs need to be eclipsed for NIGHTFALL, add Saturns orbiting each, the rings beginning the eclipses can add extra drama.

    If you want a handful of people to miss NIGHTFALL, make the giant moon is too small to eclipse the entire planet, and the planet cold enough that the only people to miss NIGHTFALL are Eskimos and a handful of others visiting the polar regions.

  5. A little confused by all this. If one can throw down so many big planets near arbitrarily on the same orbit then what is special about Lagrange points? What of these calculations that say co-orbitals are unstable if planets are within factor if 25 of each other in mass?

    1. Lagrange points are big islands of stability, especially for the case of a giant planet and much smaller bodies (asteroids or even planets). There is a big chunk of orbital space that is stabilized at the Lagrange L4 and L5 points. In contrast, you can think of cohorts as small narrow stable zones — they are not infinitesimally small, but the width of these stable zones is puny compared with the giant Lagrange-stable islands. Also, co-orbitals seem OK with different masses (preliminary tests).

  6. Ultimate System 1 looks awsome, but what about tides. volcanic activity, radiation, day/night cycle on the moons? Can you please describe it in more detail?

      1. Thank you! What about volcanoes? Io, for example, doesn’t look hospitable. Why doesn’t this happen in your system? What size and mass are the moons? ( iam talking about ultimate cohort system 1)

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