Bate, Mueller, and White - Fundamentals of Astrodynamics ... - UL FGG

12 TWO-BODY ORBITAL MECHANICS Ch. 1
2. There are no external nor internal forces acting on the system
other than the gravitational forces which act along the line joining the
centers of the two bodies.
1.3.2 The Equation of Relative Motion. Before we may apply
Newton's second law to deter mine the equation of relative motion of
these two bodies, we must find an inertial (unaccelerated and
nonrotating) reference fr ame for the purpose of measuring the motion
or the lack of it. Newton described this inertial reference frame by
saying that it was fixed in absolute space, which "in its own nature,
without relation to anything external, remains always similar and
irrnnovable."
s However, he failed to indicate how one found this frame
which was absolutely at rest. For the time being, let us carry on with
our investigation of the relative motion by assuming that we have found
such an inertial reference frame and then later return to a discussion of
the consequences of the fact that in reality all we can ever find is an
"almost" inertial reference frame.
Consider the system of two bodies of mass M and m illustrated in
Figure 1.3-1. Let (X', Y', Z') be an inertial set of rectangular cartesian
coordinates. Let (X, Y, Z) be a set of nonrotating coordinates parallel
to (X', Y', Z') and having an origin coincident with the body of mass M.
The position vectors of the bodies M and m with respect to the set (X',
Y', Z') are rM a nd rm respectively. Note that we have defined
r = r m - rM•
Now we can apply Newton's laws in the inertial frame (X', Y', Z') and
obtain
- mf = GMm 1..
m
r 2
r
and =
Mf
GMm L
M
r 2
r
The above equations may be written:
and
f m =_-.GMr
r 3
tM= r .
r
(1.3-1)
(1.3-2)