The Circular Orbits

The bounded motion of a particle in a central force is in general an ellipse given by

 

where eccentricity e is related to semi-major and minor axes as,

b² = a² (1 - e)²

Hence, if e = 0, b = a, and the ellipse degenerates into a circular orbit. The radius of the circular orbit is,

 

If the particle m is moving in a gravitational force of a fixed mass M, C = GmM, and we get

 

That is, the radial gravitational force is equal to necessary centripetal force required for the particle in the circular orbit.

The planetary data shows that eccentricity e for most of the planets, including earth, is nearly equal to zero. Hence, their orbits are really indistinguishable from circles. However, these circular looking orbits are not concentric with Sun at the center; this is because, these are in fact ellipses, with Sun at the focus.