can two planets share the same orbit
Not by the current definition of a planet: https://en.wikipedia.org/wiki/Clearing_the_neighbourhood
Different objects can orbit a celestial body in very similar orbits.
But by that definition, exoplanets are not planets either.
Doesn’t the definition apply only to our solar system? It doesn’t say anything about exoplanets, either way.
Here is the definition: A celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit.
Exoplanets are not planets.
According to the IAU site, that's the "definition of a planet in the Solar System." It makes no claims about objects outside our solar system.
Two plaets could have the same orbit, but stability might be a problem.
(c) has cleared the neighborhood around its orbit.
There is an exception to this rule, however. The L4 and L5 points of any planet, which lay 60 degrees behind and 60 ahead of the planet, tend to collect objects ( case in point, Jupiter's Trojan asteroids). So in these two particular instances, the planet has not cleared the neighborhood of its orbit.
So the question is: would a Planetary sized object be stable at either of these points.
This paper: http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=8093&context=etd_theses
Seems to conclude that it would be possible to have a body at either the L4 or L5 points for the Earth, that has a mass comparable to that of the Moon. For a gas giant like Jupiter, some 333 times more massive than the Earth, I don't foresee a problem with maintaining a body of planetary mass ( satisfying parts (a) and (b) of the definition. at its L4 or L5 point.
Whether such body could be technically called a "planet" by IAU nomenclature is another question. I assume it would depend on whether or not having a single mass of that size there would disrupt the collection of other bodies or gather them into orbit around itself.
It does seem reasonable to me that the definition could be applied for any solar system.
A planet is in orbit around a star and has cleared it's orbit of other substantial objects.
Anything less than that is an asteroid, or a moon
I'm going to go out on a limb and say the OP is probably not after a semantic answer and this [great] technical answer will satisfy him/er.
There are some weird orbits
The bean shape we have companion orbiting the sun 365 day orbit. Cruithne is not a planet but one could have the same orbit.
The Pluto Charon system is a great example. The IAU does not consider Charon to be a planet. The IAU does not consider Pluto to be a planet either. Also Eris-Dysnomia.
There is a lot more if you are asking what "can" happen with organized help. You can use any regular plane geometry figure. equalateral triangle, square, pentagon, hexagon etc. They are highly unstable and unnatural. You can elliptical orbits with the shapes so they pulse in and out. Or you can add a retrograde set.
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