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This is unabashedly stolen from this recent question:

What would be the structural impact of an oscillating orbit on a planet?

with its sweet sweet gif

from that question

Around this white-hole stands a telluric planet, quite similar to Mars in terms of composition and orbital characteristics, excepted for one thing. Due to some yet-to-understand space history, its orbit is crossing 8 times the neutral-line, as it is oscillating from and to the pull-zone with contrary forces. It forms a pretty star-like shape, as you can see in the toy model I made1 below

:

The planet's orbit on my model, forming a star-like shape

The red planet's orbit forming a star-like shape around the white star-like body.

My question: what force could produce an oscillating orbit like the one shown?
The references question wants to know effects on the planet. The force causing the oscillation is handwaving force. But I want to know if such a orbit is possible. I was thinking about both bodies being charged and an induced magnetic force. I was thinking about diamagnetism.

I am not committed to this being a planet and a sun. Any sinewave orbit is ok but the answer has to explain why.

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    $\begingroup$ I officially approve (and allow) this robbery :p. I'm not as much interested on that issue as you are, but it might open up opportunities. Who knows? $\endgroup$
    – Tortliena
    Jun 7 at 11:47
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    $\begingroup$ I suspect if you had a pair of planets orbiting each other while at the same time orbiting a nearby star, maybe they could end up in a pattern like this? $\endgroup$ Jun 7 at 19:05
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    $\begingroup$ The moon does this around the sun, but with about 12 lobes, not 4 $\endgroup$ Jun 7 at 19:27
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    $\begingroup$ @MadPhysicist: So does the Earth, though with much less amplitude than the Moon! The orbit of the Earth-Moon barycenter makes an ellipse as it orbits the Sun; we just don't think about it much because the barycenter is still inside the Earth. $\endgroup$ Jun 7 at 21:20
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    $\begingroup$ @EricLippert. Of course. The bigger problem is of course that in both cases there's another "thing" tracing out the orbit near the body of interest. The picture strongly implies that OP would like to acheive that orbit with only one "thing" doing the orbiting. $\endgroup$ Jun 7 at 21:22
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The simplest solution would be to have a twin body orbiting. The terrestrial planet has a companion - it could be a micro black hole. The common center of gravity follows an ellipse, but the two bodies follow two epicycloidals.

enter image description here

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    $\begingroup$ For something a little less deadly than a black hole... how about a dark matter planetoid as the partner body? :) $\endgroup$
    – Corey
    Jun 8 at 0:50
  • $\begingroup$ From the model's data I gathered, the distance between the main object and the smaller one can be as small as 60% the maximum distance it can attain. Is this feat still reachable with just another body acting as the oscillator? $\endgroup$
    – Tortliena
    Jun 8 at 9:26
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    $\begingroup$ @Corey: why is a black hole deadly? WIth something like planetary mass, it's not going to evaporate any time soon, and you would have to go very close before tidal effects became dangerous. Seems like a perfect solution to me. $\endgroup$
    – TonyK
    Jun 8 at 11:40
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    $\begingroup$ @Martin-ReinstateMonica: the Hawking radiation from a black hole of planetary mass is so small as to be undetectable. $\endgroup$
    – TonyK
    Jun 8 at 23:47
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    $\begingroup$ @TonyK You are right. I suppose I should have done the math beforehand. Using $P = \frac{\hbar c^6}{15360\pi M^2G^2}$, a moon-massed black hole would be outputting only $6\times10^{-14} W$. Not very deadly indeed. $\endgroup$
    – Poseidaan
    Jun 9 at 7:11
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Any force that is attractive at some distances and repulsive at other closer distances would be able to produce such orbits. As for how to get such a force on Greg Egan's webpage on Orthogonal

https://www.gregegan.net/ORTHOGONAL/ORTHOGONAL.html

He discusses how in a universe with 4 fundamentally similar dimensions and with massive photons the electric force would be attractive at some distances and repulsive at others. So if there was a universe where planets are held in orbit by a force similar to the electric force with massive force carriers and with 4 fundamentally similar dimensions you might be able to get such orbits assuming that the force really is attractive at some distances and repulsive at others.

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    $\begingroup$ +1 just for mentioning Greg Egan's hard s-f universes. $\endgroup$
    – JDługosz
    Jun 7 at 21:11
  • $\begingroup$ I was going to bring up Greg Egan's corrugated electric potentials, but I see you already did. +1. $\endgroup$ Jun 8 at 2:44
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  1. Gravity. If there is a third body forming a binary system with the planet, the combined orbital motions can produce such a shape with appropriate ratios of orbital radii and periods.

  2. Fictional physics. Any force that has a natural trough in its radial potential will produce these kinds of orbits. Getting something like that to operate on a solar-system sized scale would be pretty contrived, though.

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    $\begingroup$ Electrostatic force won't do that. Like gravity, it decreases with the square of the distance, so their sum will too. There will be no "potential trough". $\endgroup$
    – Aetol
    Jun 7 at 9:09
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    $\begingroup$ As @Aetol says, a constant electric charge would produce an elliptic orbit. To wooble this way, you would need to change charge, which would amount to building a gigantic electric engine. Good luck powering it. $\endgroup$
    – Pere
    Jun 7 at 16:15
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    $\begingroup$ @Aetol Bother, you're right. $\endgroup$ Jun 7 at 20:39
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The planet is orbiting another massive object, which is itself orbiting the central mass. The two periods are in resonance, which gives you the shape shown.

The terrestrial planet could be orbiting a hypermassive remnant of some kind, so it's actually much larger than the gravitational primary and the inhabitants won't see a huge globe in the sky; might not even know it's there until they develop technology.

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    $\begingroup$ This is honestly the most plausible answer out of all those listed at the time of this writing. It's a little dull, sure, but that tends to be the case for the one answer that doesn't rely on exotic physics. +1. $\endgroup$ Jun 8 at 2:48
  • $\begingroup$ Same question as to LSerni : From the model's data I gathered, the distance between the main object and the smaller one can be as small as 60% the maximum distance it can attain. Is this feat still reachable with just another body acting as the oscillator? $\endgroup$
    – Tortliena
    Jun 8 at 9:27
  • $\begingroup$ In the real world, a supermassive remnant (a neutron star or a black hole) has no faintest chance to be invisible in such a setup. The wind from the main star will create quite a bright accretion disk. $\endgroup$
    – fraxinus
    Jun 8 at 16:07
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This is a special case of a Rosetta orbit. In general relativity, Newton's inverse square law gravity is only an approximation, and in very strong gravitational fields, such as the region just outside the event horizon of a black hole, the orbits are no longer closed ellipses. They instead form a curve called a hypotrochoid in which the orbit "oscillates" radially in a similar way to how you describe. The wiki page on hypotrochoids shows an example that forms a closed five-pointed star.

I'm not totally sure whether a hypotrochoid like this could actually arise as a black hole orbit; it may be there are limits on the parameters allowed that exclude it, or render it unsuitable for life. For a distortion this large, you'd have to be close, and the tidal forces would be fierce. I suspect any reasonable real planet would be torn apart without copious supplies of handwavium. (On the other hand, milder tidal forces would provide a handy source of heat to stop your dark planet turning into a frozen ball of ice without a sun.) And it would in any case require some extreme fine-tuning to get that closed orbit that speaks of a deliberately engineered planetary system, rather than a natural occurence.

Explaining why it has this form would require general relativity, but you can probably see that something like it is intuitively reasonable if you start with an elliptical orbit, and then think about how the black hole's gravity distorts times and distances more strongly when it is close in than when it is far away, giving it a boost of speed at one end of the ellipse and slowing it at the other. That causes the ellipse to steadily rotate.

Your quote mentions white holes. A white hole is a time-reversed black hole, and so the orbits for white holes and black holes are the same. (Strictly, the orbits are time-reversed equivalents, but for the hypotrochoids being discussed, the time-reversal of the orbit looks exactly the same.) Or in the case of an eternal black hole, the white hole and black hole are the past and future of the same structure. So this would seem to apply.

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    $\begingroup$ From what I could find, the shape of orbits in predicted by General Relativity in this situation is called a Schwarchild geodesic. All illustrations I could find of these geodesics have self intersections. Same thing with all the obits I could make with this interactive app (demonstrations.wolfram.com/OrbitsAroundSchwarzschildBlackHoles). Are you sure that every hypotrochoid is a possible Rosetta orbit? $\endgroup$
    – E Tam
    Jun 7 at 18:33
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Actually, it's assumed that most planets already do this. It's called an epicyclic orbit, and comes about when either the planet is radially displaced in some way or the mass of the central object changes.

This is what the orbits of most stars and planets actually look like. Granted, the effect is much smaller for most planets than in this diagram, but there's no reason that you can't hand-wave your planet having a much larger epicycle. I'm much more familiar with it in my field of Galactic astronomy than in solar systems, but I can't think of a physical reason why it's forbidden, just unlikely.

Epicyclic orbits are made when a planet (or star) gains radial motion (towards or away from its central gravity source) on top of its elliptical orbit. This motion causes the planet to oscillate on a small ellipse on top of its orbit, which causes those star-like orbital patterns that you show in your example.

As for what could generate this orbit, I would suggest a large asteroid striking the planet head-on at early times, causing this perturbation on its orbit. Once the orbit is perturbed, it will stay perturbed for very long periods of time, since it's difficult to shed energy/angular momentum in star/planet systems. If you want the orbit to be even more stable, make this planet the only one in the solar system, since that's the main thing that will "even out" your orbit over time as far as I know.

Another possibility is that the star that the planet is orbiting gained or lost mass very rapidly in some way. This would cause the planet to be moving too fast or too slow for its orbit, and would cause an epicycle to occur. This is the process that is thought to actually generate epicycles in the orbits of disk stars, like the Sun.

EDIT: The processes I suggest would result in an elliptical orbit (which is technically a circular orbit with an epicyclic period half it's orbital period), but wouldn't generate the desired orbit above. For that I believe you will need extra bodies or diffuse mass distributions.

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    $\begingroup$ I don't get it. If you have an impulse such as the asteroid strike, shouldn't that result in a new ellipse? I wonder if the situation with a star in a galaxy (moving within a diffuse mass) is simply not applicable to the simple case of a planet orbiting a sun? $\endgroup$
    – JDługosz
    Jun 7 at 21:14
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    $\begingroup$ Uhh... hanh? The only epicycles I know of in solar-system-scale astrophysics were in Ptolemy's geocentric model of the solar system, which was challenged by Copernicus and debunked by Kepler. Planets orbiting spherical suns travel in ellipses (or, occasionally, other conic sections). Always. Unless there's something else tugging on them (like a moon orbiting a planet). This doesn't apply to stars traveling through galaxies, though, since a galaxy's mass is distributed over its volume, not concentrated in its core. So maybe the OP's planet is orbiting in a protostellar disk? $\endgroup$ Jun 8 at 2:41
  • $\begingroup$ @SomeoneElse37 - An epicycle with exactly half the period of the main cycle produces an ellipse, and all ellipses can be expressed this way. Copernicus did not challenge epicycles. They were integral to his theory - he just placed the sun in the center instead of Earth. It was Tycho Brahe who showed Copernicus' epicycle model was false by taking observations that could not be matched this way. Kepler solved the problem by allowing only ellipses for closed orbits, and moving the sun from the center to a focus. $\endgroup$ Jun 8 at 15:32
  • $\begingroup$ Yeah you're all right, the processes I suggest would cause a change in orbital energy and angular momentum, which would place the planet on a new elliptical orbit. $\endgroup$
    – Tom Donlon
    Jun 8 at 16:06
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This star is a hydrogen fusion star only in its outer shell. Its core is composed of an exotic form of matter which rotates through a higher dimension, such that periodically (some of) its mass gravitationally disappears from our universe, allowing the orbit to expand. Soon afterward and before the planet can settle into its new orbit, the mass returns.

There would be side effects. The diameter of the star fluctuates with the same period as the orbit of the planet and thus so does its brightness/temperature. Unfortunately, the star gets hotter as the planet gets closer, so rather than balancing out any climate concerns this exacerbates them.

This is fictional physics, and there's no evidence that this is possible in the real universe.

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If the planet has no atmosphere, periodic strong volcanism can act as a booster, increasing or reducing momentum and producing the oscillating orbit depending on where it happens with respect to the direction of motion.

On a small scale this happens with comets, where the effect of differential gas emission due to exposure to sun light slightly alter their orbit.

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    $\begingroup$ How many times can this happen? $\endgroup$ Jun 7 at 19:29
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Two of Saturn's moons are particularly strange. Janus and Epimetheus have the unique (in our solar system, at least) distinction of having nearly the same orbit as each other, interacting with each other in a complex dance but never colliding. One orbits slightly closer to Saturn than the other. A closer orbit is slightly shorter, so the inner moon eventually catches up with the outer moon. Once they get close enough to gravitationally interact, their mutual pull on each other will accelerate the inner moon and slow down the outer moon, causing them to essentially swap orbits.

The mathematics behind Janus and Epimetheus result in an orbit swap every four Earth years. If you can play with the numbers (relative masses, distances, orbital periods, etc) such that you get four orbit swaps per local year, you'd have a star-shaped orbit resembling the image in your question.

This mechanic requires the two object's orbits to be very close to each other during both swaps. That means the overall amplitude of the "wave" in your orbit will be a very small percentage of the mean orbital distance (otherwise they'd drift too far away from each other to interact). That's probably good if this is an inhabited planet, since large swings in orbital radius can create hyper-extreme seasonal changes or move the planet out of the habitable zone entirely.

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