Is it probable that planets will stop orbiting in ellipses?

I saw this video, but I had the impression that orbiting in an ellipse or a circle is quite a coincidence - it depends on how the planet got into the Solar System. Because I think that either all active forces balance in a circle, or simply all the planets with too much eccentricity burn in the Sun, or to break away from the Sun's gravitational field. Is it true?

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Related: physics.stackexchange.com/q/12140/2451 and links therein. – Qmechanic Apr 22 '14 at 11:01
"orbiting in an ellipse or a circle is quite a coincidence - it depends on how the planet got into the solar system" Is incorrect. It follows from the form of the force law and the fact that the system is bound and nothing else. – dmckee Apr 22 '14 at 13:12
Orbits are conic sections. An orbital path that isn't circular or elliptical is either parabolic (i.e. crashing into the Sun) or hyperbolic (i.e. leaving the solar system, never to be seen again). In either case, they're not really orbits any more. – Matt Brennan Apr 22 '14 at 14:26
I don't think parabolic entails crashing in to the sun. The sun will be at the focus of the parabola, which is not on the parabola. What happens with a parabolic "orbit" is that the object will depart along a path that is parallel (in the limit) to the one it approached at. – Theodore Norvell Apr 22 '14 at 17:14
@dmckee I would argue it is a double coincidence. Its either only because the force field is Keplerian (in a galaxy a typical orbit is not elliptical). Its quasi circular because it was put there by viscous forces during the formation process. – chris May 3 '14 at 9:23

In a planetary system you would expect the orbits of the large bodies to be almost but not quite circular. This is because there are two opposing effects: tidal forces tend to make orbits circular but perturbations between planets tends to make orbits elliptical.

In a two body system, just the star and one planet, tidal deformation of the planet and the star will dissipate energy and the effect of this is to make the planet move outwards. Exactly this effect is currently causing the Moon to recede from the Earth at around 4 cm/year. The tidal forces vary strongly with distance, so in an elliptical orbit the tidal dissipation is much greater when the planet is nearest the star than when it's farthest away. The end result is the the orbit gradually becomes circular.

With only two bodies the orbit will remain circular, however when other planets are present the planets all exert gravitational forces on each other and they perturb each others orbits. The end result is that the orbits become slightly elliptical, and also also that the eccentricity of the orbits is continually changing. For example the eccentricity of Earth's orbit varies between 0.000055 and 0.0679 over a period of about half a million years.

Small bodies like asteroids and comets can have wildly elliptical orbits, but over the long term such orbits aren't stable because close approaches to large bodies like planets are likely to eject the asteroids from the Solar System. In general stable elliptical orbits are only possible when they are stabilised by a resonance, for example as Pluto's orbit is stabilised by a 3:2 resonance with Neptune.

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The recession of the Moon from the Earth illustrates tidal forces of the three-body system (Moon-Earth-Sun) and is not a two-body system phenomenon. – hardmath Apr 23 '14 at 0:50

A planet orbiting in a true circle would be an extreme coincidence, but there are infinitely more ellipses than circles.

However, an ellipse is not a coincidence. Any object that doesn't have enough kinetic energy to esacpe the gravity of a star will have an elliptical orbit (in the approximation of Newtonian gravity, no additional objects, spherical shapes).

If there is more kinetic energy than needed to escape, the path would be hyperbolic, not elliptical.

The possiblity of burning up in the Sun does happen to comets sometimes.

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A minor query: isn't it the case that a parabolic orbit occurs when kinetic energy exactly equals escape energy? – Carl Witthoft Apr 22 '14 at 11:31
yes, that's right – DavePhD Apr 22 '14 at 11:33
Then again, the same argument that explains why there are no circular orbits also explains why there are no parabolic orbits: The chance is mathematically zero. Two numbers would have to be exactly identical but they are statistically independent and continuously distributed. – MSalters Apr 22 '14 at 20:34