(G.1) |

Suppose that the aforementioned Keplerian orbit is slightly perturbed; for example, by the presence of a second planet orbiting the Sun. In this case, the planet's modified equation of motion takes the general form

where is a so-calledIf the right-hand sides of Equations (G.3)–(G.5) are set to zero (i.e., if there is no perturbation) then we obtain a Keplerian orbit of the general form

(G.6) | ||

(G.7) | ||

(G.8) | ||

(G.9) | ||

(G.10) | ||

(G.11) |

Let us now take the right-hand sides of Equations (G.3)–(G.5)
into account. In this case, the orbital elements,
, are no longer
constants of the motion. However, provided the perturbation is sufficiently small, we would expect the elements to be relatively
slowly varying functions of time.
The purpose of this appendix is to derive evolution equations for these so-called *osculating orbital elements*.
Our approach is largely based on that of Brouwer and Clemence (1961).