Two planets in a stable horseshoe orbit?

Question

We've seen questions about multiple planets in a single orbit before, using Lagrangian points, through orbiting each other as they go around, etc. I'm looking at the creation of a hypothetical system with two habitable planets orbiting their star in a stable horseshoe orbit.

To sum up a horseshoe orbit: two bodies orbit a star in nearly the same path. One of them is slightly closer to the star than the other, so it will go around a little faster. When it "catches" the second body, it will be accelerated by its gravity. This effectively pushes the first body outward, away from the star, until its orbit is now slightly longer than the second body. The first body will then lag slowly behind until the second body "catches" it, at which point gravity slows the first body and drops it back into the faster orbit.

There's a 2-minute video that animates a horseshoe orbit here; anybody reading this should watch the video, since horseshoe orbits can seem counter-intuitive and are hard to wrap one's head around. Also, this picture from Wikipedia shows two of Saturn's moons in such an orbit, as viewed from a rotating frame (note that Epimetheus is significantly smaller than Janus):

I'm looking for a description of the likely circumstances of such an orbit, given a test case: how often would the planets change places (note that the more frequently this occurs, the greater the change in year length will be), and how long would the transitions take each time? Will there be any significant effects on either planet, such as tidal changes, while they are changing places? Is a horseshoe orbit impossible or unstable in a notably eccentric orbit, or does it take on any unusual characteristics; would the two planets follow the same ellipse or something closer to mirrored ellipses with perihelion/aphelion on opposite sides of the star, for instance?

One particular point I'd like to see addressed in an answer: how much flexibility does a horseshoe orbit have in terms of setting orbital distances? Are certain properties of the orbit fixed the moment one determines mean orbital distance from the star and size of the planets? Or is there room to play around with how far the two planets are in their orbits (tinkering with year length changes, for instance) while still retaining the place-changing characteristic of horseshoe orbits, and if so to what degree? The latter scenario offers more options for worldbuilding, since (if that latitude is great enough) one could then make place-changes happen anywhere from every century to once in five thousand years without changing the planets or mean orbital distance in any way.

I'll provide three hypothetical test cases here for people to work with (of varying difficulty). I'm hoping to keep this question generalized, so other people looking at this question can easily relate it to any comparable system they might try to create themselves; as such, answers that include formulas (and that can thus easily support different figures being plugged in) are appreciated. I'll offer rough estimates on mass and radius that would produce the necessary gravity (assuming Earth density for the planets) in case they're needed. If one answer can address all three cases, great!

Case One (easy):

• Planet A: Earth (in its normal orbit). Mass ~ 5.98 x 10^24, radius ~6400km.
• Planet B: 0.9g Earth-like planet. Mass ~ 4.26 x 10^24, radius ~5700km.

Case Two (moderate):

• Planet A: 1.1g Earth-like planet (in an orbit with eccentricity of 0 and year of 200 Earth days). Mass ~ 7.9 x 10^24, radius ~ 7000km. Assume the star is smaller and dimmer than the Sun to keep this habitable.
• Planet B: 0.5g Earth-like planet. Mass ~ 7.5 x 10^23, radius ~ 3200km. I realize this planet may have atmospheric escape problems.

Case Three (hard):

• Planet A: 1.1g Earth-like planet (in an orbit with eccentricity of 0.1 and year of 500 Earth days). Mass ~ 7.9 x 10^24, radius ~ 7000km. Assume the star is somewhat larger than the Sun to keep this in the habitable zone.
• Planet B: 0.8g Earth-like planet. Mass ~ 3 x 10^24kg, radius ~5100km.

For the purposes of this question, ignore any other planets that might be in the system, although it would be useful to know if either planet in this configuration is capable of supporting one or more moons, or if the horseshoe orbit imposes any significant limits on the moons that can be supported.

This is my first question here, and I recognize that it's probably a very tough one, so please let me know if there's something I need to clarify or edit! I've already made a few edits just waiting for answers, but I won't hesitate to make further changes.

Answer 1

So first of all, we need to define our terms here. To be clear, the orbit relates to the barycenter of the system. The changes are such that Janus is always farther away from Saturn's center of gravity than Epimetheus. When it comes to the barycenter.

I generated a constrained 3-body model, and plotted the distance of the moons from the barycenter, running form 2006 January 1 00:00 TDB, over 20 days. This is a transition in which Janus starts out farther than Epimetheus.

So far so good. In fact, we could just extend this to your problem, scaling all parameters so that they can apply to a very large star, and planets, they will of course not follow your instructions. Now to your example, the gravitation parameters calculated as the fractions ps the Sun and Earth mass are as follow:

1. Star ...... - $0.5403728495401237^{-3}\left[AU\right]^3/\left[Day\right]^2$
2. Planet A - $0.9776461689638198^{-9}\left[AU\right]^3/\left[Day\right]^2$
3. Planet B - $0.7110153956100508^{-9}\left[AU\right]^3/\left[Day\right]^2$

Next the positions and velocities. We will derive the star's Ephemeris, so let's start with Planet A.

Assuming the star has 1.1 times the sun's luminosity dependent on surface which is r squared rather than mass' r cubed, but I'm not changing all this stuff now that I realized this :), this means that for a similar habitableness.

Solve[{L == 1.1 L2, L == 1/r^2, L2 == 1/r2^2}]

r -> 0.953463 r2

So our new orbit is at 1.04881 times Earth's orbital radius, or simply 1.04881AU. Which we will use as the initial x-component for position, with y- and z-components 0.

For an eccentricity of (near) 0 with respect to the barycenter, it further needs a velocity of: $$\sqrt{a r}\approx\sqrt{\mu_{star}/r_{star}}=0.02216411615734939\left[AU\right]/\left[Day\right]$$ Unfortunately this is inconsistent with the idea of a 500 day orbit, so I will ignore this here.

Now to Planet B, let's give it the position and velocity that Planet A will have one day later. We can multiply the resultant position and velocity by some factor, and be assured that they remain consistent and that crossing events take place because of the planets' gravitational pull towards each other.

The result can give.... interesting orbits. For instance, when Planet B has initial orbital radius of 0.995 times and initial velocity of 1.005 times that of Planet A: Distance of planets from the barycenter:

One orbit:

Using $t=500$ yields: {{r -> 1.68759, v -> 0.0178942}}. We can then continue as before.

Unfortunately, the effect which you are looking for, I'm not finding experimentally, I suspect the ratio beween the Sun and the planet's masses is too slim, if I draw the planets close enough together, they just end up sling-shotting each other out of orbit.

So I sadly don't think it's going to work at this distance from the sun.

You can have interesting orbits, orbits that cross each other, but not this.

Answer 2

There is no such thing as a "stable horseshoe orbit".

Horseshoe orbits can be observed in reality, of course - but how stable are they?

"We find that horseshoe co-orbitals are generally long lived (and potentially stable) for systems with primary-to-secondary mass ratios larger than about 1200". That's the case for Earth and Trojan asteroids, but not a suitable solution to the OP's question.

What about Epimetheu and Janus? This horseshoe orbit can possibly have a lifetime of more than 10Gyr (10^10 years). For comparison: Earth is about 4.5Gyr old and the universe about 13.7Gyr. So plenty of time to develop life...right?

Unfortunately, with a central body of a larger mass (like the sun), the stability/lifetimes of the horseshoe orbits suffer. For a central mass equal to that of Jupiter, the horseshoe orbits will have a lifetime measured in Myr (million years).

Source and further reading: https://pdfs.semanticscholar.org/6e46/9b3abad04b18beed9b0044b45f11f327f647.pdf

Answer 3

Up front, I have zero background in this so I offer no hope of explaining any of the following to you, however, to add to Feyre's answer, these initial conditions may be useful to help determine if the orbit you want is possible. They are derived from this discussion thread around this paper.

An earlier paper on the same subject is here.

Maybe Feyre or others can help to utilize these resources and come to a meaningful conclusion.