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.
Does the classification affect the Physics involved?
Sure they can.
Saturn has two moons in coorbital arrangement. Same is possible for planets.
It's not as simple as the two satellites following the same path for ever though, is it? They interact, That sort of mutual gravitational effect also means that geostationary satellites need to be spaced a significant distance apart to minimise the amount of fuel needed to cancel their mutual effect and keep them stationary in the same orbital spot.
Before devling into the question of co-orbing planets, let's consider a better definition of a planet than the offical IAU one given. It seems a better definintion would be that a planet is a natural object in space that is spheroidal in shape and is not undergoing fusion in its interor and therefore is not self luminous. Size, compostion, orbital parameters, and location have no bearing. The added stuff about having "cleared its neighborhood", what ever that means, is limiting and confusing and should be discarded.
As for whether two planets can share the same orbit, considering that a planet is a natural spheroidal, none self luminous object in space, I can think of no good reason why not. We have examples in the Solar System, although they involve moons and small irregular natural satellites. Tethys and Dione, which orbit Saturn, share their orbits respectively with Calypso and Telesto and with Helene and Polydeuces, located at either the leading or trailing Lagrange point. It may be speculation, but perhaps somewhere in the Galaxy, orbiting some star, there's a planet the size of Mars, lead or trailed by one the size of Mimas or Enceladus, or one the size of Earth lead or trailed by one the size of Europa. Seems possible if conditions were right during formation. We'll have to wait and see.
Basically they do. The periodic change in their orbits' radius is less than the diameters of the satellites.
And they have to be at Lagrange points for this? That has now made things clearer for me.
In this case, no. They are in mutual "horseshoe" orbits. One is in a lower orbit and thus is faster, it very slowly approaches another, they they exchange momentum (since they attract), and now the leading object is in a lower orbit and thus moves faster, and thus runs away. Counterintuitive.
I have read about horseshoe orbits but to they really pass that close to each other? The diagrams don't seem to suggest that.
They come close enough to start feeling a weak mutual attraction. Which is not really very close for planetoid-sized objects.
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