Seasons and Day/Night Cycles on Horseshoe Habitable Worlds?

Question

In the latest Artifexian video, he talked about terrestrial moons, bringing up great ways they could and would exist. But the video ended with an interesting concept, Horseshoe Habitable Moons.

From Wikipedia:

A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body (such as Earth). The orbital period of the smaller body is very nearly the same as for the larger body, and its path appears to have a horseshoe shape in a rotating reference frame as viewed from the larger object.

The loop is not closed but will drift forward or backward slightly each time, so that the point it circles will appear to move smoothly along the larger body's orbit over a long period of time. When the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, with the larger body between the 'horns'.

Artifexian pointed out the interesting features twin habitable worlds in a coinciding horseshoe orbit and it made me wonder, What would Seasons and day/night cycles on habitable worlds be like if they existed in a horseshoe orbit?

Answer 1

TL;DR: The seasons will have different lentghs depending on the "side" of the horseshoe it is in (there is a slow and a fast "side"), and the day/night cycle will be unchanged.

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For more info on horseshoe orbits, there's this Q&A on physics.SE.

From the perspective of an inertial frame in heliocentric coordinates this asteroid [that has a horseshoe orbit] is in a circular orbit (topologically a circle) around the sun.

This means that the seasons will be the same. It has a moment when it will be nearer the sun (summer) and another when it will be farther away (winter). Depending on its rotational tilt, it will probably also have autumn and spring seasons.

But the length of the seasons will be different, and the order may change too. More on that below.

It probably will have rotation, so day / night cycles are a given. Since it is habitable, it is crucial for the cooling of the planet surface to have a day/night cycle. Again, the horseshoe orbit won't affect it for the most of the time.

When it approaches the main planet, things change a bit:

It will eventually catch up with the Earth, but it is not necessarily gravitationally drawn into the Earth. It interacts with the Earth’s gravity field in its frame with an effective and repulsive potential (...). The gravitational potential plus this effective potential pushes the asteroid into a higher orbital radius. The Lagrange points L4 and L5 act then as attractor points in the rotational frame of the asteroid. Its orbital velocity is now smaller and recedes away from the Earth. Eventually the Earth approaches the asteroid and the process is repeated. ref.

The tides will probably become stronger due to their gravity interaction.

Because it will change its orbital velocity as it "rotates" around L4 and L5, the length of the seasons will change also. It will have a faster and a slower season cycle.

Relative to the sun, the planet never changes direction, only speed. On the slow cycle, the main planet catches up from behind and accelerates the horseshoe-orbit planet. It enters the fast cycle, when it goes around the orbit and approaches the main planet from behind, when it has its speed reduced and goes back to the slow phase. All of this while both bodies orbit the sun.

This is a “hunter-chaser” type of orbit. The thinner the horseshoe is the smaller the angular momentum L is with respect to the Earth at close approach. This means the gravitational attraction can become larger. ref

Answer 2

Interesting question. HEre is what the setup looks like for Saturn's moons Janus and Epimetheus (from https://en.wikipedia.org/wiki/Epimetheus_%28moon%29#Orbit)

The key thing to know is that planets on horseshoe orbits don't strongly affect each other's spin. Both Janus and Epimetheus are tidally locked to Saturn, meaning that they always show the same face to Saturn. So, relative to the Sun, each moon spins once for each orbit it makes around Saturn.

Imagine a more general setup, where the central body is a star and two planets share an orbit in a horseshoe configuration. If the planets are close to the star it is likely they will be tidally locked such that the same side always faces the star. But the planets could well spin at a different rate and that would not be affected by the horseshoe setup. The only really interesting thing is what Artiflexian pointed out, that the other planet would get big in the sky and then retreat.

This is likely a very rare setup -- in running thousands of simulations of planet formation (my day job) I have only encountered this once or twice. Much more common are Trojan planets: configurations where two planets share the same orbit and remain roughly 60 degrees apart -- FYI see the second part of this post: https://planetplanet.net/2014/05/22/building-the-ultimate-solar-system-part-4-two-ninja-moves-moons-and-co-orbital-planets/