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In our solar system, with the exception of Pluto all planets follow a relatively circular orbit around the Sun, at the same inclination. They also all rotate in the same direction, none are 'retrograde'.

Solar System
An image of the orbits of our solar system.

How and why have they all formed into these orbits? Is this reflected in most orbital systems? What factors affect the orbits of newly formed systems?

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Its considered as a strong indication that they formed from the same proto-planetary nebula, a disc made of gas and dust which was rotating in the same direction. The forming protoplanets within that nebula would have radiated away the component of their motion not consistent with a circular trajectory through collisions, viscosity or tidal torquing. –  chris Mar 16 at 15:28
    
@chris that sounds like answer, go ahead and make it one ;) –  Vedant Chandra Mar 16 at 19:06
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The combination of the conservation of angular momentum and gravity give you the inclination and direction. Gravity will condense material in the axis direction of the angular momentum, and collisions and gravitational interactions will damp oscillations in that direction, forming a disk of material. The angular momentum resists gravity in the plane perpendicular to the axis, maintaining the extent of the disk. Over time, the direction of rotation of the angular momentum is the direction of the vast majority of the material, due to collisions and interactions with that smaller population going the wrong way on a one-way street.

The circularity of the orbits is the result of a more dynamical process. Hopefully someone else here can explain it better than the following. My simplistic understanding is that a bunch of orbits crossing each other is not stable. Planets and planetesimals have their orbits changed continuously until such time as they settle into orbits that have fewer interactions with other bodies. Sort of a natural selection. In the long haul, this results in a relatively stable configuration with circular zones that tend to not interfere with the other circular zones. In fact, this is now part of the definition of the word "planet", in that to be called such, a planet needs to clear the neighborhood of its orbit. Though its not clear how it could occur, a bunch of elliptical zones that happen to be co-aligned would also not be stable since orbits precess due to various influences.

Where the material was less dense than where the planets are currently must certainly have many objects in eccentric, retrograde, and/or highly inclined orbits. Pluto is the first hint at that, with a relatively high inclination and eccentricity compared to the planets. Sedna is far more eccentric.

As for other systems, I'm sure there must be some oddballs out there. However the Kepler mission has observed mostly circular orbits for planets around other stars.

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Inner moons of the gas giants have very near circular orbits. Is that a result of intense interactions, or does the huge gravity dominance of the planet have a direct effect too? Triton has a retrograde orbit and has an eccentricity of zero (within measurement error). How could that be? –  LocalFluff Mar 17 at 11:27
    
Wow, I find it really interesting how planetary formation sounds so much like evolution. Thanks a lot for the answer! –  Vedant Chandra Mar 17 at 12:02
    
I don't think we know what happened with Triton. The last theory I heard was that Triton was part of pair of bodies orbiting the Sun (kind of like Pluto and Charon), and that Triton was captured by Neptune with the other body thrown out. –  Mark Adler Mar 17 at 15:13
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