Consider a planet of mass $m$ and relative position vector ${\bf r}$ that is orbiting around the Sun, whose mass is $M$. The planet's equation of motion is written (see Section 4.16)

$\displaystyle \ddot{{\bf r}} + \mu\,\frac{\bf r}{r^{\,3}} = {\bf0},$ (G.1)

where $\mu= G\,(M+m)$. As described in Chapter 4, the solution to this equation is a Keplerian ellipse whose properties are fully determined after six integrals of the motion, known as orbital elements, are specified.

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

$\displaystyle \ddot{{\bf r}} + \mu\,\frac{\bf r}{r^{\,3}} = \nabla {\cal R},$ (G.2)

where ${\cal R}({\bf r})$ is a so-called disturbing function that fully describes the perturbation. Adopting the standard Cartesian coordinate system $X$, $Y$, $Z$, described in Section 4.12, we see that the preceding equation yields

$\displaystyle \ddot{X}+ \mu\,\frac{X}{r^{\,3}}$ $\displaystyle = \frac{\partial {\cal R}}{\partial X},$ (G.3)
$\displaystyle \ddot{Y}+ \mu\,\frac{Y}{r^{\,3}}$ $\displaystyle = \frac{\partial {\cal R}}{\partial Y},$ (G.4)
$\displaystyle \ddot{Z}+ \mu\,\frac{Z}{r^{\,3}}$ $\displaystyle = \frac{\partial {\cal R}}{\partial Z},$ (G.5)

where $r=(X^{\,2}+Y^{\,2}+Z^{\,2})^{1/2}$.

If 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

$\displaystyle X$ $\displaystyle =f_1(c_1,c_2,c_3,c_4,c_5,c_6,t),$ (G.6)
$\displaystyle Y$ $\displaystyle =f_2(c_1,c_2,c_3,c_4,c_5,c_6,t),$ (G.7)
$\displaystyle Z$ $\displaystyle =f_3(c_1,c_2,c_3,c_4,c_5,c_6,t),$ (G.8)
$\displaystyle \dot{X}$ $\displaystyle =g_1(c_1,c_2,c_3,c_4,c_5,c_6,t),$ (G.9)
$\displaystyle \dot{Y}$ $\displaystyle =g_2(c_1,c_2,c_3,c_4,c_5,c_6,t),$ (G.10)
$\displaystyle \dot{Z}$ $\displaystyle =g_3(c_1,c_2,c_3,c_4,c_5,c_6,t).$ (G.11)

Here, $c_1,\cdots,c_6$ are the six constant elements that determine the orbit. (See Section 4.12.) It follows that

$\displaystyle g_k = \frac{\partial f_k}{\partial t},$ (G.12)

for $k=1,2,3$.

Let us now take the right-hand sides of Equations (G.3)–(G.5) into account. In this case, the orbital elements, $c_1,\dots,c_6$, 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).