Elliptic orbit

It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The total energy of the orbit is given by

At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore:

This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E:

For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above:

The general equation of an ellipse under these assumptions using vectors is:

This can be done in cartesian coordinates using the following procedure:

Because at least six variables are absolutely required to completely represent an elliptic orbit with this set of parameters, then six variables are required to represent an orbit with any set of parameters. Another set of six parameters that are commonly used are the orbital elements.

The radial elliptic trajectory is the solution of a two-body problem with at some instant zero speed, as in the case of dropping an object (neglecting air resistance).