The gravitational potential varies very rapidly near the Earth or Moon but
relatively slowly at a great distance. For computational efficiency therefore
one needs to use a small temporal stepsize h near the Earth and a greater
one further away. If you write a computer program to solve this problem using
runge.c directly (not recommended) you will find that it inevitably
happens that the spacecraft gets too close to the Earth, the temporal stepsize
is too great, the calculation overshoots and gives the spacecraft too great
a kinetic energy and the spacecraft careers out of the Earth-Moon system at
the speed of light.
What is needed is a routine that checks how rapidly the solution is varying
and automatically adjusts the temporal step size appropriately, i.e. it adapts
the step size to the problem. finteg.c (in the same directory as hamil.c)
is such a routine for runge.c . The way finteg.c works is, you suggest
a stepsize h, finteg.c uses runge.c to compute a solution
for stepsizes h and 
. From these solutions finteg works
out how rapidly the solution error is changeing with h and knowing that in
runge the error is fifth order, it computes the required h to keep the
error below some tolerance tol say. Note that finteg.c is not in the standard
form of C code. It has been translated directly from the Fortran version. It
uses goto statements usually greatly deprecated in C programming. This is
in keeping with the spirit of this course to use C as higher level language
only and not e.g. to use pointers. If you would like to read a more detailed
description of how finteg works, ask the lecturer for a copy of a handout.
In fact finteg is not the most sophisticated adaptive step size routine. It
uses the same tolerance test for each of the variables so you don't want the different
variables to differ in magnitude by too many orders of magnitude. In this problem
this objective can be attained by using units of 
in length and time
in hours. You should do this. Check that in these units the numerical value of
the gravitational constant is 
(whereas in SI units
).