guidance, flight mechanics and trajectory optimization

For all the equations given to this point, the reference orbit can be
elliptical. If the path is circular, a further simplification occurs for each
form presented. The set with time as the independent variable (Eq. 1.12)
becomes (rl = r. = constant):
The remaining sets (Eqs. 1.14, 1.16, and 1.18) reduce to a single set because
of the equality of the three anomalistic variables (M = 8 = E) for circular
orbits. This set is
It is to be noticed that for linear systems and a circular reference orbit
the set of equations has constant coefficients and is, therefore, easily
integrated for the case of coast arcs (g = .- a = 2).
As already mentioned for the. sets of equations in terms of true anomaly or
eccentric anomaly, the second of the three equations possesses an immediate
first integral for the no-thrust situation. This integral is a representation
of the constant difference in the angular momentum per unit mass for the two
vehicles. Thus, for the no-thrust case, there must exist three more independent
integrals consisting of simple combinations of 51, t2, 6'1, rt2 represent-
ing constant differences in other elliptical orbit elements (e.g., semi-major
axes, arguments of perigee, times of perigee passage). This concept, in fact,
yields a method for obtaining the integrals to the sets.
2.1.3.2 Distance Forms of the Equations
In the first place, let
10
1.20b
1.2oc
1.21a
1.2lb
1.21c
1.22