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Additional info for Space Trajectories. A Symposium Sponsored by the American Astronautical Society, the Advanced Research Projects Agency, and Radiation Incorporated
Example text
Speed focusing. lunar orbit at a later time and, on the other hand, a slightly higher speed results in a shorter transit time and also in an intersection with the lunar orbit at an earlier time. Clearly, there will be a particular set of trajectories where these two effects will cancel (see Fig. 7) and errors in speed will not re sult in large errors in impacting the moon. , and Walters at Aeronutronic. If the lunar-vehicle orbit plane is inclined to the lunar orbit plane, then this focusing effect is not as useful because the deviating trajectories will pass over or under the moon.
Definition of the eccentric anomaly. There exist a number of other formulae that are useful for two-body orbit analysis. , P-Q shown m Fig. 4 . If two new variables D = ττ/\/μ and F = iE are introduced for 37 ASTRODYNABnCS FIG. 4 . Orbit plane coordinates. the parabola and hyperbola respectively, then a tabulation of useful for mulae for obtaining x«, x«, y«, i«, r, r, and Μ can be obtained as is shown in Table I. In connection with this table, it should be noted that the term q [which stands for the perifocal distance = a ( l — e)] is still defined for the TABLE I Two-BoDY FORMULAE Ellipse Χω a(cos Ε — é) sm y» α Vi - Λ / ο Vi r r Μ g - öV2 -D/r sin Ε V2qD VaesiaE = nit - T) ^ Ε - earn Ε a(cosh F — e) > - V sinhF -α - a V 2 - 1 sinh F V ^ V ¡ 2 ^ -e^coaE a(l - e eos E) Hyperbola Parabola q + 2>V2 D/T = kit - Τ) = qD - Z ) V 6 coshF a(l - e cosh F) e sinn F ViÄ/(-a)^(i-T) + e sinh F - F 38 ROBERT Μ.
WHITNEY WHITNEY: I think I didn't quite make the point clear that we have to remember that there are restrictions set by the engineers. We do need a few precise analytic solu tions but we don't need very many of them. What we need are powerful techniques for getting down close. PINES: We have one. We have a contract with Wright Air Development Center in which we take one of the few three-body problems that can be solved; namely, the one with two fixed centers. This is a three-body problem which has a closed solution.