Elliptical orbits

[methane]

I'd like to know why planetary orbits are elliptical and how Newton's work demonstrated this? Does the eccentricity of the ellipse depend on the orbiting bodies initial velocity before being captured in the orbit?

[Clanad]

Although Newton's work had its profound application to the eccentricities noted in the orbits of all planetary bodies, it was actually Johannes Kepler that formulated his three laws of planetary orbit. He showed that the eccentricity of each elliptical was due to gravitational perturbations... i.e., the interaction of two or more bodies gravitationally. Each can be measured and changes predicted adhering to his laws:
1. The orbit of every planet is an ellipse with the sun at one of the foci.
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time.
3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.

The interaction with Newton rests, primarily on Kepler's Third Law... This law, when applied to circular orbits where the acceleration is proportional to a�P−2, shows that the acceleration is proportional to a�a−3 = a-2, in accordance with Newton's law of gravitation. (With assistance from Astro-tom)...

[mibn2cweus]

Consider two objects in space that lack sufficient velocity to escape mutual gravitational influence. If they fail to collide (if there approach trajectories do not intersect) they establish some form of orbit. A nearly circular orbit is but one of an almost limitless orbital shape configurations and as the odds would have it are relatively rare.

shoot a cannonball into orbit.

[heathfield]

It will depend not only upon the initial velocity, but also the masses and thus the mutual gravitational attraction of the two bodies.