Today I’ll discuss a pretty awesome type of system with just two planets on horseshoe orbits.
OK, giddy-up! (I still will never apologize for my terrible jokes!)
This setup is inspired by two of Saturn’s small moons, Janus and Epimetheus. They each orbit about 150,000 km from Saturn. That’s just beyond Saturn’s main rings and interior to some of its smaller rings. It must make for a great view!
Janus and Epimetheus share the same orbit. But they’re not in a Trojan configuration, with one planet always 60 degrees in front of the other. Their relative position is constantly shifting and it traces out an interesting shape. If you were orbiting Saturn alongside the two moons, Epimetheus traces out a horseshoe shape and Janus traces out a sort of cartoon smile. It looks like this:
Janus spends four years closer to Saturn then Epimetheus, then the two moons pass close to each other and do a U-turn (from a viewpoint that is rotating along with them around Saturn). Then it’s Epimetheus’ turn to be closer to Saturn for 4 years. And it repeats.
Beware! Both Janus and Epimetheus are orbiting Saturn. The horseshoe shape only shows up when viewed from a camera orbiting along with the moons. Here is a comparison of two different viewpoints of the same orbits:
How can Janus and Epimetheus orbit like this? At first glance it seems awfully unstable.
Here is an explanation of exactly what is happening (from this really nice article about their orbits):
This next plot shows Janus’ and Epimetheus’ orbital distances from Saturn over a span of 26 years. As you can see, they stay at a given distance for four years, then have a close passage and swap sides. Every eight years they complete a horseshoe loop cycle. The wiggles during the cycles come from the fact that their orbits are not perfect circles.
Epimetheus’ orbit has larger oscillations than Janus’. This is because Janus is four times as massive as Epimetheus, so Epimetheus gets kicked farther around more.
Let’s look at horseshoe orbits in a more general context. Imagine a single moon orbiting Saturn (or a single planet orbiting a star). The Lagrange points L1-L5 are places where another object can in principle orbit along with the planet. Only L4 and L5 are stable — those are the positions 60 degrees in front of and behind the moon/planet’s orbit. To stay put in one of the other Lagrange points requires continuous fine-tuning. It’s like balancing on one foot on a wobbly stool — in theory you can stay there but in practice you don’t.
In addition to the Lagrange points, this diagram shows the stable co-orbitals paths of small bodies (think, asteroids) that orbit around the star (or Saturn) along with the planet (or moon).
The first interesting thing is that L4 and L5 are not “points” but the center of stable islands. There is a whole class of orbits that wobble about these points. They are sometimes called tadpole orbits. On a tadpole orbit, an object does not remain fixed 60 degrees away from the planet but can instead oscillate (the technical term is “librate”) between being much closer (say, 30ish degrees) and quite a bit farther (up to more than 90 degrees) away. The object can’t reach 180 degrees from the planet — tadpoles are stuck on one side of the planet’s orbit.
The second interesting thing is a class of orbits that do pass from one side the planet’s orbit to the other. These are the horseshoe orbits. They follow circular paths, pass close to the planet (during which they exchange orbital energy with the planet) and then switch to the other side. A horseshoe shape — like my son keeps saying: “well, duh!”
Now let’s build planetary systems with horseshoe orbits.
All we need are two planets and a star. For the star, let’s keep it simple and use the Sun. But what about the planets? One factor is the relative masses of the planets. Things behave differently when there is one big planet and one small one instead of two same-sized planets.
In the first system let’s use two same-sized planets orbiting a Sun in the habitable zone. In this case each planet does a half-horseshoe — or, as I like to think of it, one makes a frowny face while the other one makes a smily face. Here is what it looks like:
The same-sized planet setup can work for two Earths up to two Saturns (with moons). However, for planets much more massive than Saturn, the same-sized horseshoe setup is unstable (details here if you’re interested).
The other extreme is to have one planet that is much more massive than the other. Let’s use Jupiter and Earth. And I’ll throw in a system of large moons around Jupiter for good measure. Here is what it looks like:
Since Jupiter is so much more massive than Earth, its orbit remains more or less fixed whereas the Earth’s follows a nice horseshoe path (shown in yellow).
What would life be like on a planet in a horseshoe system?
The most exciting part of living in a horseshoe system are the encounters. When the two planets pass close to each other they exchange orbital energy and switch sides relative to the star. During that time the other planet can loom huge in the sky. The closest approach between the two planets can be almost as close as the Hill sphere.
For two Earths, that means a closest approach of about 0.01 AU. For scale, the Moon is at about a quarter of that distance and is about a quarter of the size of Earth. That means that at close encounter the other Earth-sized planet would be as big as the full Moon! In the Jupiter-Earth horseshoe system it would be similar: from the point of view of the Earth-sized planet, the gas giant would reach roughly the size of the full Moon in the sky (it would be farther away because its Hill sphere is much bigger, but Jupiter is much larger in size so it compensates). The Jupiter would be so close that its large moons would be quite a sight. However, from the point of view of Jupiter (or those moons), the Earth would be quite a bit smaller, about 5-10 times smaller than the full Moon.
Viewed from one horseshoe Earth, the other planet would grow in the sky until it reached the size of the full Moon. The encounter is short compared to the time for a full horseshoe to complete, but could still last weeks (we’ll get to that). Our home planet settles onto its new orbit, and the other planet slowly recedes into the distance, growing fainter and smaller over a period of years. Eventually the other planet passes behind the Sun out of sight, and remains undetectable for a good chunk of the horseshoe before approaching again from the opposite side, slowly coming in for the next close encounter.
The length of a horseshoe cycle depends on how wide the horseshoe is. The planet closer to the star moves faster than the planet farther out, so after a certain amount of time the planets catch up to each other. The width of the horseshoe orbit can vary — there are a lot of different horseshoe paths in the Lagrange point image above. The maximum width is a simple function of the planet’s mass. Like the Hill sphere, the width of the horseshoe region scales with the cube root of the planet’s mass. Jupiter is 318 times Earth’s mass, so its maximum horseshoe width is about seven times larger than Earth’s.
For two horseshoe Earths, the widest-possible horseshoe extends from 99.5% of Earth’s current orbital distance out to 1.005%. That’s not a big shift, so things would move slowly. It would take about 33 years between encounters on the widest-possible horseshoe configuration — that’s the one with the closest approach between the planets too. It would take longer between encounters on closer horseshoes. The small shift in orbital distance in the two-Earth horseshoe system means that the planets’ climates wouldn’t be particularly affected by the switches between sides of the horseshoes.
For an Earth-Jupiter system things are more interesting. This is because Jupiter’s horseshoe region is much wider than Earth’s. There is a significant (7%) difference between the distance to the Sun along the inner and outer part of the Earth-sized planet’s orbital path. That may not sound like much, but it’s a 15% difference in energy received from the star. If the present-day Earth’s solar energy budget were increased by just 11% our planet would enter a runaway greenhouse and we would be doomed. This means that the climate on the Earth-sized planet in the Jupiter-Earth horseshoe system could oscillate between two very different states. This is not like the seasons: the whole planet would bounce between getting a lot more and a lot less energy from the Sun. It would take about 18 years between encounters, between jumps from a cold to a hot state. Given its much larger mass, the Jupiter’s orbit would be basically unaffected by the encounters.
Things would be more extreme in a mega-Jupiter+Earth system with an even wider horseshoe region. For a ten Jupiter-mass gas giant, the Earth-sized planet would be twice as wide, creating a 30% difference in solar energy between the different sides of the Earth’s horseshoe, with just 9 years between encounters. Going to more massive planets enters the brown dwarf regime, and we already know how strange it would be to be on a planet orbiting a brown dwarf.
No horseshoe planetary systems have been found (yet). But horseshoe orbits have been found in a few different situations beyond Janus and Epimetheus. Earth itself has a few known transient horseshoe asteroids.
Computer simulations have found that in systems with a large planet, moons in horseshoe configurations can form naturally (details here). And people are looking for horseshoe exoplanets (see here), although none is in the bag yet.
Horseshoe systems as a setting for science fiction
What do horseshoe systems offer for storytelling? Let me count them:
- The co-existence of multiple planets in the habitable zone of the same star, on very close orbits. Two planets, two rival alien species (even more if there are large moons involved)!
- The occasional close encounter between two potentially habitable planets (or many habitable moons). The alien species must prepare for their once-per-generation encounter with the enemy alien planet. An opportunity to lob some bombs on the competition from close range? Or how about a tragic alien love story: two lovers on opposite sides of the horseshoe tracks that can only be together during the short close encounter phase?
- A climate that may oscillate between two very different states on a timescale of roughly 10-100 years (longer around bluer stars, shorter around redder stars). The accelerating greenhouse emissions on one horseshoe Earth keep pushing it closer and closer to climate disaster every time it is on the close side of the horseshoe. Now it’s a race against the clock to fix things up, and the only people with knowledge of how to engineer the climate live on — you guessed it — the other horseshoe planet!
There you have it: Horseshoe planetary systems. Boom!
Questions? Comments? Words of wisdom?
- A really nice article on the orbits of Janus and Epimetheus by Emily Lakdawalla
- Wikipedia article on Horseshoe orbits
- Other types of co-orbital setups: Trojans and rings of planets
- An article I wrote for Nautil.us on the absence of Trojan exoplanets (to date)
- Blog post from oklo.org about looking for exoplanets in horseshoe configurations
- MOJO video of the day: How common are Solar Systems (see also here)?