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Case File Copy NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Report 32-7360 Periodic Orbits in fh e Ell@tic Res f~icted Three-Body Problem CASE FILE COPY JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA July 15, 1 969 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Report 32-1360 Periodic Orbifs in the Elliptic Res tiicted Three-Body Problem R. A. Broucke JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA July 15, 1969 Prepared Under Contract No. NAS 7-100 National Aeronautics and Space Administration Preface The work described in this report was performed by the Mission AnaIysis Division of the Jet Propulsion Laboratory during the period from July 1, 1967 to June 30, 1963. JPL TECHNICAL REPORT 32- 1360 iii Acknowledgment The author wishes to thank several persons with whom he has had constructive conversations concerning the work presented here. In particular, some of the ideas expressed by Dr. A. Deprit of Boeing, Seattle, are at the origin of the work. ilt JPL, special thanks are due to Harry Lass and Carleton Solloway, with whom the problem of the stability of orbits has been discussed in depth. Thanks are also due to Dr. C. Lawson, who has provided the necessary programs for the differential corrections with a least-squares approach, thus avoiding numerical cliKiculties associated with nearly singular matrices; and to Georgia Dvornychenko, who has done much of the programming and computer runs. JPL TECHNICAL REPORT 32-7360 Contents I. Introduction ................ 1 II . Equations of Motion .............. 3 A . The Underlying Two-Body Problem ......... 3 B. Inertial Barycentric Equations of Motion of the Satellite . 4 C. Equations of Motion With Rotating Coordinates . 5 D . Equations of Motion for the Rectilinear Elliptic Restricted Three-Body Problem .......... 7 E . An Alternate Form of Equations of Motion for the Elliptic Restricted Three-Body Problem ......... 8 F . Some Formulas for Coordinate Changes ....... 9 T . Inertial coordinates ((.?l) to rotating coordinates (Y.v) ... 9 2 . Rotating coordinates (2. j7) to inertial coordinates (,t. rl) ...10 3 . Rotating coordinates (2. 7) to pulsating coordinates (x.y) . 10 4 . Pulsating coordinates (x.y) to rotating coordinates (2.p) ...10 5 . lnertial barycentric coordinates ([.r)) to inertial geocentric coordinates (,to.17G) ....... 6 . lnertial barycentric coordinates (<.rl) to inertial selenocentric coordinates (ts.17s) .........10 G . The Five Equilibrium Points of the Elliptic Problem ..... H . The Energy Equation of the Elliptic Restricted Three-Body Problem ............. 13 I . Regularization With Birkhoff Coordinates ........ 13 J . EquationsofMotionWithBirkhoffCoordinates ...... 15 K . Initial Conditions for Eiection Orbits .........F7 Ill . Computation of Periodic Orbits ........... 18 A . lntegration of the Equations of Motion With Recurrent Power Series ............. '18 B. lntegration of the Variational Equations With Recurrent Power Series ............ 20 C. Periodic Orbits in the Elliptic Restricted Three-Body Problem ...23 D. Families of Periodic Orbits in the Elliptic Restricted Three-Body Problem ..........24 E . DifferentialCorrectionsforPeriodicOrbits . 25 F . Properties of the Eigenvalues of the Fundamental Matrix ....26 G. Solution of the Characteristic Equation of the Elliptic Problem ...:27 H . The Seven Types of Stability Characteristics .......28 JPL TECHNlCAL REPORT 32-1360 Contents (contd) I!! I!! . Families of Periodic Orbits .............30 A . Families 12A and 6A of Periodic Orbits ........30 B. Periodic Orbits in the Rectilinear Restricted Three-Body Problem .............32 C. Family 7P ...............39 D. Family 7A ................ 51 E . Family 8P ...............51 F . Family 8A ................ 51 G. Family 1 1 P ................ 66 H . Family 11A ............... 66 I. Family 10P ................ 66 J . A Family of Periodic Collision Orbits .........81 V . Computer Programs .............89 A . Introduction ............... 89 B. Derivative Subroutines ............89 1 . SubroutineDERlVwithinertialcoordinates .......89 2 . Subroutine DERlV with rotating-pulsating coordinates . 90 3 . Subroutine DERlV with regularized Birkhoff coordinates ...90 4 . Subroutine DERlV with rotating-pulsating variables and with variational equations .......... 90 5 . SubroutineERPSVforrecurrentpowerseries . 90 C. Conversion Subroutines ............ 91 D . Differential Correction Subroutines .........92 E . Main Program With Variational Equations and Recurrent Power Series .............93 1 . JN = 1 ................ 94 2 . JN = 2 ................ 94 3.Computingthevariabl e-integrationstep .......94 4 . Logicofthedifferentialcorrection process .......94 5 . Additional operations for periodic orbits .......94 6 . Computer time .............. 94 F . A Regularized Main Program ...........95 References ..................124 JPL TECHNICAL REPORT 32-7360 Contents (contdl 1 . Orbit variables and defining equations ........ 2 . Recurrence relations for coefficients of power series ..... 3 . Properties of the seven stability regions ........ 4 . Initial conditions for family 6A .......... 5 . Initial conditions for family 12A .......... 6 . Initial conditions for 13 periodic orbits ......... 7 . ~nitialandfinalconditionsfor13periodicorbits ...... 8 . Initial and final conditions for family 5P ........ 9 . Initial and final conditions for family 9P ........ 10 . Initial and final conditions for family 4P ........ 11. Initial and final conditions for family 3P ........ 12 . Initial and final conditions for family 7P ........ 13 . Initial and final conditions for family 7A ........ 14. Initial and final conditions for family 8P ........ 15 . Initial and final conditions for family 8A ........ 16 . Initial and final conditions for family 11P ........ 17. Initial and final conditions for family 1 1A ........ 18 . Initial and final conditions for family 10P ........ 19 . Finalconditionsforperiodiccollisionorbits ....... 20 . Families of periodic orbits .......... 21 . List of subroutines .............. 22 . Computer program ............ Figures 1 . Configuration of masses MI. M. and libration point L. ....12 2 . Seven stability regions .............29 3 . Root configuration for stability regions ........29 4 . Circular restricted three-body problem: Strb'mgren's Class C of periodic orbits around Ll (equal masses; e = 0; rotating axes) .............30 5 . Elliptic restricted three-body problem: periodic orbit for mass ratio p = 0.5 (equal masses; e = 0.74. rotating axes) ............31 JPL TECHNICAL REPORT 32- 1360 Contents (contd) Figures (contd) 6 . Elliptic restricted three-body problem: periodic orbits for constant mass ratio p = 0.5 (equal masses) and variable eccentricity (e = 0.0 to 1 .0. inertial axes) . 31 7 . Family 6A of periodic orbits-rectilinear elliptic restricted three-body problem : periodic orbits for constant eccentricity (e = 1 .0) and variable mass ratio (p = 0.50 to 0.1 66; inertial axes) .......32 8 . Family 12A of periodic orbits ........... 35 9 . Diagram of initial conditions for periodic orbits . 39 10 . The 13 periodic orbits in the barycentric (inertial) coordinate system .............. 40 1 1 . The first six periodic orbits in the geocentric and the selenocentric coordinate systems .........41 12 . Families 5P. 9P. 4P. and 3P of periodic orbits .......43 13 . Family 7P of periodic orbits ........48 14. Family 7A of periodic orbits ....... 15 . Family 8 of periodic orbits ........ 16. Family 8P of periodic orbits ....... 17. Family 8A of periodic orbits ....... 18 . Family 1 1 of periodic orbits ....... 19. Family 11P of periodic orbits ....... 20 . Family 1 1 A of periodic orbits ....... 21 . Family 10P of periodic orbits ....... 22 . Ejection orbits (from m2 = p) with 0 = 0.0, p = 0.5 . 23 . Family of ejection orbits (e vs p) ...... 24. Family of ejection orbits (e vs E) ...... 25 . An extreme orbit of the family of periodic collision orbits (e = 0) ........ 26 . An extreme orbit of the family of periodic collision orbits (e = 0.98) .......... 27. Periodic collision orbits in different coordinate systems JPL TECHNICAL REPORT 32- 1360 Abstract Results of a detailed study of the two-dimensional, elliptic restricted three- body problem are presented. The equations of motion and the variational equations have been solved with recurrent power series. Several computer programs have been prepared to use the recurrent power series for the generation of families of periodic orbits. Some of these programs have been regularized with the well-known Birkhoff regularization. A total of 15 families of periodic orbits (1127 orbits) are described. The linear stability of several of these orbits has been computed; that of periodic orbits has been studied theoretically for the elliptic problem and for any nonintegrable, nonconservative, dynamical system with two degrees of freedom, and it has been found that seven types of stability or instability exist. Periodic orbits have been obtained for both the earth-moon mass ratio and the Striimgren mass ratio (equal masses). The whole range of eccentricities e from 0 to 1 (including 1) has also been studied, and it is shown that some periodic orbits exist for all of these eccentricities. A family of periodic collision orbits is also described. JPL TECHNICAL REPORT 32- 1360 Periodic Orbits in the Elliptic Restricted Three-Body Problem I. Introduction The elliptic problem may be considered as the proto- type of all nonconservative, nonintegrable dyncrrnical
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