Solar System Formation & Dynamics

Keplerian orbits are solutions to the two body problem in Newtonian gravitational dynamics. Under Newtonian gravity, two point masses that are gravitationally bound to each other describe closed elliptical orbits in space. Their mutual separation vector also describes an ellipse. The Keplerian orbital elements that describe the shape of the orbit are,

• The semi-major axis a of the ellipse. Geometrically, this is half of the longest distance across the ellipse.
• The eccentricity e of the ellipse.
• The inclination i of the orbit relative to some plane.

(There are also angles which describe the orientation of the orbit in space, but these are less important for our purposes.) For our purposes, a number of basic properties of Keplerian orbits are important.

The period of the orbit is solely a function of the masses and semi-major axis. For a Solar mass star, this is given by Kepler’s third law in the simple form,

,

where the period is measured in years and the semi-major axis in AU. More generally, the mean angular velocity is given by

and the period is .

The pericenter distance (the distance of closest approach between the body and the star) is given by . The apocenter distance (the greatest separation) is .

The total energy of the orbit per unit mass is,

.

Energy is conserved, so at any point around the orbit the sum of the kinetic and potential energies must add up to the total energy,

.

This equation allows us to calculate the speed of a body on a Keplerian orbit at any distance from the star.