guidance, flight mechanics and trajectory optimization

where the subscript 70 refers to the condition at the time of periapsis
passage.
If the nonlinear terms are omitted and the target orbit is circular,
Equation (1.66) is identical to Equation (1.49) and a solution for x, y, a
can be found in terms of the initial conditions x,, yO, zO, 4, fO, &, by
.the use of the transition matrix Equation (1.58). If this solution is de-
noted by the subscript 1'~" then
The solution to the nonlinear equation will be defined in terms of this
solution and small corrections; that is, the solutions to the set (1.66)
have the form
If this solution set is substituted in (1.66), a set of differential
equations for the variables 6~ , 6% , and 62 is produced. This set is
then simplified by neglecting the s;naller terms such as x,6x , =6x ,
ec , eZv, etc. The resulting differential equations are
8.2 +8-i? = 3~52, -3ezc cfm(r-7,)
This set of equations is linear in terms of the known. forcing functions;
therefore, the solution is straightforward. For convenience the solution is
given in two parts indicated by
where the superscript 0 denotes the solution when the target orbit is
circular, and the superscript e denotes the effect of small eccentricity on
the solution. These solutions are given by
Sxp=Aof + /f&n z +/~+KI r tAjP& 2 r +/4~cooZ r + A5%.
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