Page 240 -

223

where B is a second arbitrary constant. The matching of (5.68) and (5.69) with (5.65)
(using the results quoted in (4.54)) follows directly. From (5.68) and (5.65) we find
that matching is possible if

from (5.69) and (5.65) we obtain

which implies the connection formula A = 2B. (We also see that, if C = O(1), then A
and B are as

Our final example under this heading is related to problem E5.1, but now placed
in a celestial context. Our presentation is based on that given by Kevorkian & Cole
(1981, 1996).

E5.10 Slow decay of a satellite orbit
The equations for a satellite in orbit around a primary (in the absence of all other
masses), with a drag proportional to the are first written down in terms of
polar coordinates, These are then transformed to and (where
t is time), and finally—this is Laplace’s important observation—to and we
obtain the non-dimensional equations

where is a measure of the drag coefficient on the satellite. We seek a solution
of this pair of equations, for subject to the initial conditions i.e. conditions
prescribed at what we will call

Here, v at i.e. t = 0) is the initial component of the velocity vector in
the we assume that this is given such that v > 1 and that it is independent
of
The form of equations (5.70), and our experience with problems of this type,
suggests that we should introduce new variables (multiple scales) and
more general choices for T (e.g. or are unnecessary in this
problem. Before we proceed, observe that equations (5.70) contain only through