Let

(D.33) 
It follows that
Equation (D.28) transforms into

(D.36) 
Making use of Equation (D.27), this reduces to

(D.37) 
However, it is easily demonstrated that

(D.38) 
Hence, Equation (D.37) yields Radau's equation (Radau 1885),

(D.39) 
where

(D.40) 
Figure: D.1
The function
.

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
socalled reduced Radau equation,

(D.41) 
which can immediately be integrated to give

(D.42) 