Unperturbed planetary orbit

(I.1) |

Let , , be a Cartesian coordinate system in a reference frame whose origin corresponds to the location of the Sun, and which is such that the planet's unperturbed orbit lies in the plane , with the angular momentum vector pointing in the positive -direction, and the perihelion situated on the positive -axis. Let , , be a cylindrical coordinate system in the same reference frame.

We know from the analysis of Chapter 4 that

where and . Moreover, the planet's mean orbital angular velocity is its orbital energy per unit mass is its orbital angular momentum per unit mass is where and its eccentricity vector is Here, and are the planet's orbital major radius and eccentricity, respectively. Note that, for the unperturbed orbit, the quantities , , , , , and are all constant in time. We also have where , , and are the planet's true anomaly, eccentric anomaly, and mean anomaly, respectively. (See Chapter 4.)