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

16 TWO-BODY ORBITAL MECHANICS Ch.1
the potential energy term? The value of this constant will depend on
the zero reference of potential energy. In other words, at what distance,
r, do you want to say the potential energy is zero? This is obviously
arbitrary. In your elementary physics courses it was convenient to '
choose ground level or the surface of the earth as the zero datum for
potential energy, in which case an object lying at the bottom of a deep
well was found to have a negative potential energy. If we wish to retain
the surface of the large mass, e.g. the earth, as our zero reference we
would choose c = , where rEB is the radius of the earth. This would be
perfectly legitimate but since c is arbitrary, why not se t it equal to
zero? Setting c equal to zero is equivalent to choosing our zero
reference for potential energy at infinity. The price we pay for this
simplification is that the potential energy of a satellite (now simply - ¥>
will always be negative.
We conclude, therefore, that the specific mechanical energy, , of a
satellite which is the sum of its kinetic energy per unit mass and its
potential energy per unit mass remains constant along its orbit , neither
increasing nor decreasing as a result of its motion. The expression for
is
(1.4-2)
1.4.2 Conservation of angular momentum. The angular momentum
constant of the motion is obtained as follows:
1. Cross multiply equation (1.34) by r
r x f + r x r = O.
r
2. Since in general a x a= 0, the second term vanishes and
r X f = O.
3. Noticing that .fL (rxf) = f Xf + rxr the equation above becomes
dt '
-.iL (r X r) = 0
dt
or -.iL (rxv) = O.
dt