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

14 TWO·BODY ORBITAL MECHANICS Ch. 1
some planet or the sun, the mass of the orbiting body, m, will be much
less than that of the central body, M. Hence we see that
G(M+m) :::: :: GM.
It is convenient to define a parameter, Il (mu), called the gravitational
parameter as
Il == GM.
Then equation 1.3·3 becomes
I t"+ * r = 0·1
(1.3·4)
Equation (1.3 4) is the two·body equation of motion that we will use
for the remainder of the text. Remember that the results obtained from
equation (1.34) will be only as accurate as the assumptions (1) and (2)
and the assumption that M m. If m is not much less than M, then
G (M + m ) must be used in place of Il (so defined by some authors). Il
will have a different value for each major attracting body. Values for
the earth and sun are listed in the appendix and values for other planets
are included in Chapter 8.
1.4 CONSTANTS OF THE MOTION
Before attempting to solve the equation of motion to obtain the
trajectory of a satellite we shall derive some useful information about
the nature of orbital motion. If you think about the model we have
created, namely a small mass moving in a gravitational field whose force
is always directed toward the center of a larger mass, you would
probably arrive intuitively at the conclusions we will shortly confirm by
rigorous mathematical proofs. From your previous knowledge of
physics and mechanics you know that a gravitational field is
"conservative." That is, an object moving under the influence of gravity
alone does not lose or gain mechanical energy but only exchanges one
form of energy, "kinetic," for another form called "potential energy."
You also know that it takes a tangential component of force to change
the angular momentum of a system in rotational motion about some
center of rotation. Since the gravitational force is always directed
radially toward the center of the large mass we would expect that the
angular momentum of the satellite about the center of our reference