AN ANALYSIS OF MOON-TO -EARTH TRAJECTORIES P. A. Penzo 30 October 1961 \ (NASA-CR-132100) AN ANALYSIS OF N73-72541 MCCN- IO-E AETH TFA JECTO R IES (Space Technoloqy Inc.) 93 p Labs., Unclas 1 00/99 03694 I / SPACE TECHFJOLOaY LABORATORIES, INC. P.O. Box 95001,Los Angeles 45,California 897 6 - 00 08 - RU - 00 0 30 October 1961 AN ANALYSIS OF MOON- TO-EARTH TRAJECTORIES P. A. Penzo Prepared for JET PROPULSION LABORATORY California Institute of Technology a Contract No. 950045 Approved E. %,T& E. H. Tompkins Associate Manager Systems Analysis Department SPACE TECHNOLOGY LABORATORIES, INC. P. 0. Box 95001 Los Angeles 45, California 8976- 0008-RU-000 Page ii CONTENTS Page I. IN TROD U C TION 1 A. The Trajectory Model 2 B. Applications of the Analytic Program 5 11. THE ANALYTIC PROGR-AM 6 A. Independent Parameters 6 B. Program Logic 10 C. Sensitivity Coefficient Routine 20 111. PROGRAM ACCURACY 22 A. Preliminary Study 22 B. Correction Scheme 26 C. Evaluation of Tau 28 D. Final Accuracy 32 IV. TRAJECTORY ANALYSIS 38 A. Earth Phase Analysis 38 B. Moon Phase Analysis 55 C. Sensitivity Coefficient Analysis 71 REFERENCES 88 8976-0008-RU-000 Page iii ACKNOWLEDGEMENTS The author wishes to thank the programmers I. Kliger, C. C. Tonies and G. Hanson for their extensive effort in developing the logic and program- ming and checking out the Analytic Lunar Return Program. He is grateful to Mrs. L. J. Martin who generated and plotted the majority of the data presented here and who carried out the investigations discussed in Section 111. Finally, he is indebted to E. H. Tompkins for checking and editing the report to its final form. 897 6 - 00 08 - RU - 000 Page iv GENERAL NOTATION (j = 1,2, 3) = coordinates of position and velocity with respect to the xj' uj earth (equatorial). yj, v (j = 1, 2, 3) = coordinates of position and velocity with respect to the j moon (equatorial). "primes" attached to position and velocity denote selenographic coordinates "bars" above position and velocity symbols denote vectors = right ascension and eclination 1, P = selenographic longitude and latitude r) = angles measured in the trajectory plane; single subscript - from perifocus e = angles measured in the equatorial plane a, e, is L! = normal conic elements (equatorial) H, J = angular momentum in earth phase; moon phase P = flight path angle measured from the vertical A = azimuth angle L = geographic longitude "bars" above quantities other than position and velocity coordinates denote those with respect to the moon = Julian Date of launch; impact Dr# Di h t = time measured from day of launch (0 GMT) h t' = time measured from day of impact (0 GMT) T = time measured from perifocus (single subscript), time measured between two points (double subscript) 89 7 6 - 00 08 - RU - 0 0 0 Page v GENERAL NOTATION (Continued) Subs c ript s : 0 = launch point b = burnout point S = point of exit from MSA i = point of impact (touchdown,) r = point of re-entry m = quantities referring to the moon 89 7 6 - 0 0 08 - RU - 0 00 Page 1 I. INTRODUCTION The present United States space program for manned lunar exploration has made it necessary to conduct thorough investigations of all trajectory and guidance aspects of lunar operations. Generally, such operations may be divided into three classes: (1) earth- to-moon trajectories in which a spacecraft is transferred from earth to the lunar s3rface or an orbit about the moon, (2) lunar rsigrn, or moo_nItp- earth trajectories where the spacecraft is launched from the surface of the moon or from a lunar orbit and returns to a designated landing site on earth, with prescribed re-entry conditions, and (3) circumlunar trajectories in which the spacecraft is launched from earth, passes within a specified distance of the moon, and returns to earth with or without an added impulse in the vicinity of the moon. This report is concerned with the second class of lunar trajectories. Its specific purpose is to provide an insight into the parametric relationships and geometric constraints existing among all of the principal trajectory variables. The procedure which was used to explore these relationships was to first develop an analytic model and an associated computer program which accu- rately describe three-dimensional moon- to-earth trajectories, and then to employ this computer program to make an extensive study of the trajectory properties. The report has been divided into four sections which are essentially independent and these may be read in an order other than as presented here, if desired. The remainder of Section I discusses the nature and application of the analytic model. Section I1 gives a complete description of the ”Analytic Lunar Return Program’’ which was used to generate information for the tra- jectory study. This material has been included since the model and computer program have other important uses besides the parametric study, and the discussion of the program logic itself displays many features of moon-to- earth trajectories. Section I11 deals with the program accuracy when compared to an n-body integration program, and describes a method by which this accuracy was greatly improved. The final section examines many of the 897 6 - 0008 - RU - 0 00 Page 2 characteristics of moon-to-earth trajectories including required lunar launch conditions, geometric constraints among variables (such as allowable launch dates and re-entry locations), launch- to-re-entry error coefficients, and midcourse correction coefficients. Much of this information is presented graphically and may be used by the reader to analyse particular lunar return flights. A. THE TRAJECTORY MODEL The analytic model upon which this study is based was first presented >;< by V. A. Egorov in 1956 [ 11. In this model, all motion in cislunar space, i. e., motion in the gravitational field of the earth-moon system, is consid- ered to be the result of two independent inverse-square force fields, that due to the earth and that due to the moon. Thus the perturbations of the sun and the planets are ignored. Further, Egorov divides earth-moon space into two regions such that only the moon's gravitational field is effective in one region and only the earth's gravitational field is effective in the remain- ing region. The dividing surface is defined as the locus of points at which the ratio between the force with which the earth perturbs the motion of a third body and the force of attraction of the moon is equal to the ratio between the perturbing force of the moon and the force of attraction of the earth. For the earth-moon system, this surface is approximately a sphere whose center is coincident with the center of the moon. The radius of this sphere is given by r = 0. C7r 2/5 5 31, 000 nautical miles S where r = distance of the moon from the earth, and m/M = ratio of the m mass of the moon to the mass of the earth. Henceforth, this sphere will be referred to as the moon's sphere of action, or the MSA. *Bracketed numbers refer to the list of references. 8 97 6 - 0008 - RU - 0 00 Page 3 Due to the eccentricity of the lunar orbit, which is about 0. 06, the distance of the moon from the earth will vary by about 10 percent during a lunar month. To be precise, the above value of rs should change by this amount; however, the effects of the original simplifying assumptions will outweigh those due to variations in rs.. Since each of the two regions defined above contains only the force field of its respective body, which is assumed to be an inverse square force field, ail motion in the model will consist of conic sections. For the particular class of trajectories delt with in this report, the motion will initiate in the vicinity of the moon, or within the MSA, and terminate near the earth. This will require that the trajectorypass through the surface bounding the MSA. During the period in which the vehicle is within the MSA the moon has rotated through an angle about the earth. This is equivalent to the earth rotating about the moon through the same angle where the MSA is assumed fixed in inertial space. Also, since the same lunar face remains pointed to the earth, except for librations, the moon will seem to revolve about its axis through the same angle within the sphere of action. For typical moon-to-earth trajectories this angle will be about 6 degrees. To an outside observer, the conic within the MSA will be non-rotating. This effect is shown in Figure 1. Once the vehicle has arrived at the surface of the MSA, it is necessary to transform its position and velocity to an earth-centered inertial coordinate system. This can be easily accomplished if the position and velocity of the moon are known at the time the vehicle passes through the surface. Specifically the earth referenced coordinates are given by, u =;tu m -- where- (y, -v) are the vehicle's position and velocity referenced to the moon and (xm, um) are the moon's position and velocity referenced to the earth. All of these variables are, of course, three dimensional vectors. As shown in Figure 1, to an outside observer there will be no discontinuity in position 8976-0008-RU-000 Page 4 SPHERE OF ACTION AT LAUNCH -1 / \ SPHERE OF ACTION AT EXIT / \ \ I \ I \ I \\\ I POINT Figure 1. Schematic of Moon-to-Earth Flight. 8976-0008-RU-000 Page 5 but there is an apparent discontinuity in the velocity since we have drawn both moon-frame inertial and earth-frame inertial phases of the trajectory in the same picture.