It follows that

(D.34) | ||||||

and | (D.35) |

Equation (D.28) transforms into

(D.36) |

Making use of Equation (D.27), this reduces to

However, it is easily demonstrated that

(D.38) |

Hence, Equation (D.37) yields

(D.39) |

where

Figure D.1 shows the function
. It should be noted that
is necessarily positive, because
is a monotonically increasing function. Moreover,
on the surface of the Earth, and the largest value of
for any planet in the solar system is about
(Cook 1980). It is thus clear, from the figure, that it is an excellent approximation to take
for such bodies. This leads to the
so-called *reduced Radau equation*,

(D.41) |

which can immediately be integrated to give