guidance, flight mechanics and trajectory optimization

and assume an elliptical reference orbit. Note that the reference system is
centered at the target. Now, using the derivative of UJ and xl as in
Section 2.1.3.1 the equations are found to be:
Note that moving the origin to the target vehicle has the effect of causing a
K term to occur in all three equations. The second-order form of these equations
as used by Anthony and Sasaki (Reference 1.2) is obtained by introducing
changes of scale for both distance and time. In.this reference, the semi-major
axis, aT' of the elliptical
thus, x = aTX19 y = apl$
to mean anomaly, M, and
reference is used as a normalizing variable;
z = aTzlt and = aTq. The time is then changed
d/dM is represente 'a with the open dot, O, as
before. Including the second-order terms on the right, the equations become:
For circular orbits the equations simplify to equations of exactly the
same form as Equations 1.20 thus 'indicating the equivalence of the two origins
(either active or target) for rendezvous. Thus, one finds
2-3n”x -2nj =a,
Y
+Zfl;r: =a Y
it + n2t =a *
For the final two forms of the equations, consider that 2 is expressed in a
set centered at the target in an elliptical orbit and oriented in a fixed set
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1.25