Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1990-09 Analysis of a perturbation solution of the main problem in artificial satellite theory Krambeck, Scott D. Monterey, California: Naval Postgraduate School http://hdl.handle.net/10945/34905 NAVAL POSTGRADUATE SCHOOL Monterey, California NrM DTic N~ ELECTE '1 OCT 3 1991 jVSTArZS 4 CJADXU'J THESIS ANALYSIS OF A PERTURBATION SOLUTION OF THE MAIN PROBLEM IN ARTIFICIAL SATELLITE TH'IEORY by Scott David Krambeck September 1990 Thesis Advisor Donald A. Danielson Approved for public release; distribution is unlimited. 91-12244 "". Unclassified security classification of this page REPORT DOCUMENTATION PAGE Ia Report Security Classification Unclassified 2a Security Classification Authority 2b Declassification Downgrading Schedule 4 Performing Organization Report Number(s) 6a Name of Performing Organization 6b Office Symbol Naval Postgraduate School (if applicable) 31 lb Restrictive Markings 3 Distribution!Availability of Report Approved for public release; distribution is unlimited. 5 Monitoring-Qrganization Report Number(s) 17a Name of Monitoring O-ganization // Naval Postgraduate SchooPl.. ( 6c Address (city, state, and ZIP code) Monterey, CA 93943-5000 7b Address (city, state, and ZIP cod) Monterey, CA 93943-5000 Sa Name of FundingSponsoring Organization Sb Office Symbol (ifapplicable) 8c Address (city, state, and ZIP code) '9ProcuremenUnstrumentldentification Number 10 Source of Funding Numbers ,_Program Element No Project No ITask No I Work Unit Accession No ANALYSIS OF A PERTURBATION SOLUTION OF THE MAIN PROBLEM IN ARTIFICIAL SATELLITE THEORY 12 Personal Author(s) Scott David Krambeck I1 Title (tnclude securihy classificatlon) 3aType of Report 14 Date of Report (year, month, day) 13b Time Covered 15 Page Count Engneer's Thesis From To September 1990 142 16 Supplementary Notation The views expressed in this thesis are those of the author and do not reflet the official policy or po- sition of the Department of Defense or the U.S. Government. 18 Subject Terms (continue on reverse if necessary and Identify by block number) 17 Cosati Codes rield Group Subgroup Oblateness, Perturbation, First Order Solution, Numerical Solution Comparisou, Measured Satellite Data Comparison. 19 Abstract (continue on reverse 1( necessary and identify by block number) The development of a universal solution of the main problem in artificial satellite theory has only recently been accomplished with the aid of high powered computers. The solution to this long standing problem is an analytical expression that is sinilar in form to the two-body solution. An analysis is presented in %hich the solution is compared with the tv'.o-body solution, a proven numerical solution, and actual measured satellite data. The solution is shoNn to be sianificantly more accurate than the two-body solution. The theoretical accurac.y of the solution is confined. The solution compares extremely well with a proven numerical solution for at least 41 orbits Nith a relatihe error on the order of O. The solution compares extremely well with measured satellite data for satellites in near Earth orbits. For a satellite hi orbit at an altitude of approximately 1000 kilometers, the solution reduces the error of the twso-bod, solution by about 9 5 "o. For satellites m orbit at semisynchronous and geosynchronous altitudes, the solution reduces the error of the two-bodN solution b) at least 50%. The solution is free of singularities and is valid for all eccentricities and inclinations. 20 Distribution;Availablty of Abstract N unclassified unlimited 0 same as report 22a Name of Responsible Individual Donald A. Danielson DD FORM 1473,84 MAR 21 Abstract Security Classification 0 DTIC users Unclassilicd 22b Telephone (Include Area code) 22c Office Symbol (408) 646-2622 MA ld 83 APR edition may be used until exhausted All other editions are obsolete security classification of this page Unclassified Approved for public release; distribution is unlimited. Analysis of a Perturbation Solution of the Main Problem in Artificial Satellite Theory by Scott David Krambeck Lieutenant, United States Navy B.S., Iowa State University, 1982 Submitted in partial fulfillment of the requirements for the degrees of MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING and AERONAUTICAL AND ASTRONAUTICAL ENGINEER from the NAVAL POSTGRADUATE SCHOOL September 1990 Author: Scott David Krambeck Approved by:b6 Donald A. Danielson, Thesis Advisor Reader Deptmnt of Aero atcsa DEAN OF FACULTY AND GRADUATE STUDIES Hi Astronautics ABSTRACT The development of a universal solution of the main problem in artificial satellite theory has only recently been accomplished with the aid of high powered computers. The solution-to this long standing problem is an analytical expression that is similar in -form to the two-body solution. An analysis is presented in which the solution is compared with the two-body solution, a proven numerical solution, and actual measured satellite data. The solution is shown to be significantly more accurate than the two-body solution. The theoretical accuracy of the solution is confirmed. The solution compares extremely well with a proven numerical solution for at least 41 orbits with a relative error on the order of P0. The solution compares extremely well with measured satellite data for satellites in near Earth orbits. For a satellite in orbit at an altitude of approximately 1000-kilometers, the solution reduces the error of the two-body solution by about 95%. For satellites in orbit at semisynchronous and geosynchronous altitudes, the solution reduces the error of the two-body solution by at least 50%. The solution is free of singularities and is valid for all eccentricities and inclinations. COpy I '0 DiU i t. n TIB r cml I -i Avallabllt .P ! iiiRA" IDi t a a . - TABLE OF CONTENTS -1. INTRODUCTION............................................. II. BACKGROUND ............................................. A. ORBITAL KINEMATICS........ ............................ B. EQUATIONS OF MOTION ................................... C. SOLUTION ............................................... D. SIMPLIFIED SOLUTION................................... E. THE CRITICAL INCLINATIONS.............................12 F. SPECIFIC MECHANICAL ENERGY ........................... 111. METHOD OF ANALYSI1S.................................... A. OR'3ITAL PARAMETERS.................... ............... 1. Argument of Latitude (60) ................................ I 2 2 4 7 10 14 16 16 16 2. Radius Magnitude ( r)...................................18 18 3. -Inclination (i)........................................ 4. Longitude of the Ascending Node (f2 )........................18 B. ROMBERG INTEGRATION TECH-NIQUE.......................19 IV. 22 METH-OD OF COMPARISON................................. A. NUMERICAL INTEGRATION COMPARISON..................22 1. Delta Radius Vector................ I.....................22 23 2. Earth Arc Angle ........................................ 24 3. -Delta Omega, Delta Inclination, Delta Theta ................... 24 4.- Relative Errors......................................... 24 5. Track Errors........................................... B. MEASURED DATA COMPARISON ........................... 27 V. RESULTS .................................................. A. NUMERICAL INTEGRATION COMvPARISON..................28 I. Delta Radius Vector Comparison............................29 2. Earth Arc Angle Comparison...............................30 iv 28 3. Delta Omega Comparison ................................. 4. Delta Inclination Comparison ............................... 5. Delta Theta Comparison .................................. 6. Delta Theta Relative Error Comparison ....................... 30 31 31 7. Delta Radius Relative Error Comparison ....................... 8. Radial Track Error Comparison ............................. 9. Along Track Error -Comparison .............................. 10. Cross Track Error Comparison ............................. B. MEASURED DATA COMPARISON ........................... 32 32 32 33 33 31 1. Near Earth Orbit Comparison ............................... 2. Semisynchronous Orbit Comparison .......................... 3. Geosynchronous Orbit Comparison ........................... VI. CONCLUSIONS AND RECOMMENDATIONS 34 36 38 .................... 40 APPENDIX A. NUMERICAL SOLUTION COMPARISON RESULTS ...... 42 APPENDIX B. NEAR EARTH ORBIT COMPARISON RESULTS ......... 70 APPENDIX C. SEMISYNCHRONOUS ORBIT COMPARISON RESULTS ... 83 APPENDIX D. GEOSYNCHRONOUS ORBIT COMPARISON RESULTS APPENDIX E. COMPUTER PROGRAM ' .......................... .. 94 105 LIST OF REFERENCES .......................................... 126 INITIAL DISTRIBUTION LIST 128 ................................... LIST OF TABLES Table 1. A SCHEMATIC OF ROMBERG INTEGRATION................20 21 Table 2. ROMBERG INTEGRATION.............................. vi LIST OF FIGURES Figure 1. Spherical coordinate system ................................ Figure 2. Orbital plane . ............................................ Figure 3. Delta radius vector and Earth arc angle ....................... 2 3 23 Figure 4. Track errors . ............................................ Figure 5. Delta radius vector (1 day) ................................. Figure 6. Delta radius vector (1 day) ................................. 26 43 44 Figure 7. Figure 8. Figure 9. Figure 10. Figure 11. Figure 12. Delta radius-vector (3 days) ................................. Earth arc angle (I day) .................................... Earth arc angle (1 day) .................................... 45 Earth arc angle (3 days) ................................... Delta omega (I day) . ..................................... Delta omega (3 days) ..................................... 48 49 Figure 13. Delta inclination (I day) ................................... Figure 14. Delta inclination (3 days) .................................. Figure 15. Delta theta (I day) ....................................... Figure 16. Delta theta (1 day) ....................................... Figure 17. Delta theta (3 days) ..................................... Figure 18. Delta theta relative error (I day) ............................. Figure 19. Delta theta relative error (I day) ............................. Figure 20. Delta theta relative error (3 days) ............................ Figure 21. Delta radius relative error (1 day) ............................ Figure 22. Delta radius relative error (1 day) ............................ Figure 23. Delta radius relative error (3 days) ............................ Figure 24. Radial track error (1 day) .................................. Figure 25. Radial track error (I day) ................................... Figure 26. Radial track error (3 days) ................................. Figure 27. Along track error (I day) .................................. Figure 28. Along track error (I day) .................................. Figure 29. Along track error (3 days) .................................. Figure 30. Cross track error (I day) ................................... Figure 31. Cross track error (3 days) .................................. vii 46 47 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 Figure 32. Delta radius -vector (21 days) ............................... Figure 33. Earth arc angle (21 days) .................................. Figure 34. Delta omega (21 days) .................................... Figure 35. Delta omega-(21 days) ...................................... Figure 36. Delta inclination (21 days) ................................. Figure 37. Delta theta (21 days) ...................................... Figure 38. Delta theta relative error (21 days) ........................... Figure 39. Delta radius relative error (21 days) ........................... Figure 40. Radial track error (21 days) ................................. Figure 41. Along track error-(21 days) ................................. Figure 42. Cross track error (21 days) .................................. Figure 43. Cross track error (21 days) ................................. Figure 44. Delta radius vector (30 days) ................................ Figure 45. Earth arc angle (30 days) .................................. Figure 46. Delta omega (30 days) .................................... Figure 47. Delta inclination (30 days) .................................. Figure 48. Delta theta (30 days) ...................................... Figure 49. Delta theta relative error (30 days) ........................... 71 72 Figure 50. Delta radius relative error (30 days) ........................... Figure 51. Radial track error (30 days) ................................. Figure 52. Along track error (30 days) ................................. Figure 53. Cross track error (30 days) ................. ............... Figure 54. Delta radius vector (28 days) ................................ Figure 55. Earth arc angle (2S days) .................................. 90 91 Figure Figure Figure Figure 56. 57. 58. 59. Figure Figure Figure Figure 60. 61. 62. 63. Delta Delta Delta Delta omega (28 days) .................................... inclination (28 days) ................................. theta (28 days) ...................................... theta relative error (28 days) .......................... Delta radius relative error (28 days) .......................... Radial track error (28 days) ................................ Along track error (28 days) ................................ Cross track error (2S days) ................................ VIIi 73 74 75 76 77 78 79 80 81 82 84 85 86 87 88 89 92 93 95 96 97 93 99 100 101 102 103 104 LIST OF SYMBOLS A TE along track error a semi-major axis ao initial value of a B. rotating orthonormal base vectors b. fixed orthonormal base vectors ( n = 1 ,2,3 CTE cross track error C Cos io e eccentricity eo initial value of e f function G universal gravitational constant ( G = 6.662 x 10- 2 km3lkg-s" ) h angular momentum h0 initial value of h I integration result i inclination io initial value of i J normalized ( n = 1,2,3 ) ) J. 3 12R ) 2 p;coefficient of the n zonal harmonic of a planet's gravitational potential Al planet mass ( Earth, Al = 5.983 19478 x 10' kg ) O center of the planet and the coordinate system O order p serni-latus rectum (p PO R = I/GAV) initial value of p (po = ao(l - c.) equatorial radius of a planet ( Earth, R = 637S.137 kin) R TE radial track error r radial vector from the center of a planet to the satellite r. radial vector from the center of a planet to the satell'te for the reference solution J2 ( J - = a(l - e) ix (p (po = h 2 /GM) r magnitude of the radial vector from the center of a-planet to the satellite r. magnitude of the radial vector from the center of a planet to the satellite for the reference solution S satellite s sin i T t specific kinetic energy time to u initial value of t p.fr V specific potential energy y variable used in the r, i, and Q equations Z Greenwich mean time (ZULU time) Oright ascension angle fi declination or latitude PO Y initial value offil fixed direction of the-vernal equinox A a finite increment 0 argument of latitude 00 initial value of 0 O argument of latitude for the reference solution T 0 Earth arc angle longitude of the ascending node -20 to initial value of Q argument of periapsis too initial value of co x ACKNOWLEDGMENT Sir Issac Newton once remarked, -If I have seen a little farther than others it is because I have stood on the shoulders of giants.- In completing this endeavor there are -several giants I wish to acknowledge. My special thanks to Professor Don Danielson for his encouragement, inspiration, and patience as he guided me through this project. I will be forever grateful for his confidence and trust. Several words of thanks go to LTC Jim Snider, USA, and LT Chris Sagovac, USN, whose research cleared the path such that this thesis could be attempted. Their numerous and timely suggestions were extremely helpful. I must thank John Rodell from the Colorado Center for Astrodynamic Research (CCAR) for producing the numerical data needed in this analysis. His quick response to my requests we'e greatly appreciated. I must also thank CAPT Greg Petrick, USAF, and IstLT Bruce Hibbert, USAF. from the First Satellite Control Squadron (ISCS) for supplying the measured satellite data needed in this analysis. I extend my sincere appreciation for their help and cooperation. I convey special gratitude to my lovely wife Susan for her support, sacrifice, and understanding which made the completion of this thesis a reality. .'d I. INTRODUCTION With the development of any new method or theory the question always arises on whether the approach is valid or practical for ordinary use. This is particularly true in the prediction of satellite motion. Ever since Sir Issac Newton's discovery of the law of universal gravitation, new methods have been developed to better predict the motion of the heavenly bodies. Usually the method contains one or more restrictions that limits the practical use of the solution. The goal, of course, is to develop a solution that is valid and possesses no restrictions. Recently, such a solution has been formulated. This analysis continues the work that was begun by Snider [Ref. 1] and Sagovac [Ref. 2]. From their research, a higher order universal solution of the motion of an artificial satellite around an oblate planet was developed. The solution is free of singularities aPd is theoretical valid for all orbital parameters. The purpose and scope of this work is to compare the solution with proven numerical solutions and actual measured satellite data in order to determine whether the theoretical work is valid and practical. The first chapter summarizes the development of the theory and presents the solution in its entirety. Also-included is a somewhat less accurate simplified solution. An explanation of the solution near the critical inclinations is presented. The chapter concludes with a discussion on the conservation of specific mechanical energy. The next chapter describes the method of analysis and explains the type of integration routine exercised in the evaluation. The method of comparison is presented next in which the error parameters are described in detail. The results follow which include both a detailed discussion and a graphic representation. The analysis is objective in nature and designed to demonstrate both the advantages and disadvantages of the theory and the solution. Before the solution can be applied extensively, a general understanding of its strengths and weaknesses must be determined. II. BACKGROUND A. ORBITAL KINEMATICS A reference system for a planet in spherical coordinates ( r, a, /3) is shown in Figure 1. The radial distance (r) is measured from the center of the planet (0) to the satellite (S). The line (0 y) is in a direction fixed with respect to an inertial coordinate system. For Earth, the line (0 y) is in the direction of the vernal equinox. The right ascension angle (a) is measured in the planet's equatorial plane eastward from the line (0 Y). The declination or latitude (P) is the angle measured northward from the equator. The position vector (r) of the satellite in the spherical coordinate system is: r = r(coscccosfl) b,+r(sincccosfl) b 2 +r(sin3) b 3 (1) where ( b , b2, b3 ) are orthonormal base vectors fixed in the direction shown. polar axis equatorialplane Figure 1. _3 b0 Spherical coordinate system. A reference system for a satellite in polar coordinates (r, 0) within a rotating orbital plane is shown in Figure 2. The satellite's position and velocity vectors are contained within the orbital plane. The argument of latitude (0) is measured in the orbital plane 2 from the ascending node to the satellite. The inclination (i) of the orbital plane is the angle measured between the equatorial plane and the orbital plane. The longitude of the ascending node (f2) is measured from the line (0 y) to the ascending node. The ascending node lies on the line of nodes which form the intersection of the equatorial and orbital planes. orbital plane equatorial .plane Figure 2. S Orbital plane. The basis (b, , b2, b3 ) may be transformed into another orthonormal basis (B, , B2 , B3 ) by a succession of three rotations. First the basis ( b, , b,, b3 ) is rotated about the b, direction by the angle Q. The basis is then rotated about the new first coordinate vector by the angle i. The final rotation is about the new third coordinate vector by the angle 0. The position vector (r) has only one component in the rotating basis. r = rB1 3 (2) The components of r in-the fixed'basis are: r = r(cosOcosf2-sinOcosisin2) b, + r(cos0sin 2+-sin 0cosicos 2) b2 +r(sin0sini) b3 (3) Equating the components of equations (1) and (3), the following relations among the angles ( a, f ) of the spherical. coordinate system and the astronomical angles ( i, 92,0 ) can be obtained. sinf cosfl sin0sini (4) cos 0 sec(a - a) (5) = = The velocity (drdt) of the satellite is obtained by differentiating equation (2)*with respect to time (t) which results in: dr -dt - dr B, + r do (I + tan Ocot d )B 2 dt dt -dOB (6) Equations (2) and (6) represent the orbital kinematics of a satellite in the polar coordinate system. The position and velocity vector expressions describe the motion of a satellite in a particular reference system and provide the information needed to develop the equations of motion in that system. These expressions are referenced to the true, rather than mean, orbital plane and were originally formulated by Struble [Ref. 3,4,51. B. EQUATIONS OF MOTION The motion of all objects is mathematically described by the equations of motion that govern them. For an oblate planet, the expressions for the kinetic and potential energies per unit mass of an orbiting satellite in spherical coordinates are respectively: L[Or V ) ++ 2 G I-- + 4 22 (l - 3 sin'f) ds( ( (8) In the above equations, (M) is the mass of the planet, (G) is the universal gravitational constant, (R) is the equatorial radius of the planet, and (J2) is the coefficient of the second zonal harmonic of the planet's gravitational potential. The governing equations of motion can be determined by substituting equations (7) and (8) into Lagrange's equations which are represented by: d dt [(dq" 1 aq --- (T- (T-J' = 0 (9) ~dt} where: q = a, r, and fl Three equations result from Lagrange's equations which describe the motion of th . satellite. The three equations are: ( 2 ~ dt d 2, r(dfl)2 dt2 =00 cos rcos2fl dt )2 dt} dt ) = d( r2 dl+ ) ,J d \ o,. r2sin Pcosf (dh - = ar (11) (12) (12)if From the equations of motion, two integrals result which are: 2 2ol r Cos dc dt -2#-! = constant T+ V = constant (13) (14) Equation (13) results from integrating equation (10) and equation (14) simply states that the specific mechanical energy of the satellite remains constant. To change the independent variable from i to 0 , equations (1) , (2) , and (6) are used in conjunction with equation (13) and some initial conditions to form: 5 dt r2 dos i dO- Ocosi 0 II +d +tann- -cot -dO (15) 'Letting u p,!r, and using equation (15), the remaining equations of motion (11) - (12) can be rewritten as: di dO - 2 sin 0 cos 0 sin i cos 2i 2Ju )(6 + 2Ju sin 20 cos3 i 2 +U d. -3 2 , J2u co2i o2i [2 • 0 cos CiCs AU Cs 2d2U4 [ --d sin 0 (3co~-1)dO ,.u -- sin 30 Cos OCos i (3 -cos2i ) /{c a + 4Jic2 sin 20 cos4i (17) + 4u j sin 0 cos 8 i} d2 dO where: tan 0 di sin i dt (18) c = cos i0 s = sin i0 3J2 R2 2 2 po Equation (18) results from uncoupling the equations for L2 and i. The angles 0 and i can be uncoupled by applying the fact that the orbital p!ane must contain the velocity vector. The differential equations (16) - (18) are coupled by nonlinear terms and apparently cannot be solved analytically. If the right sides of equations (16) - (18) are expanded in a Taylor series expansion in powers of J, the equations simplify t': (Ii dO = dO -2.u sin 0 cos 2 3 0Ocos 0+ O(J3) c2 0 sin i cos 3 + 4Jlusc3 sin C2 ~ 6 (19) cos 2 i dd2u 02 U J cos2 i 2 -4u sin2 0 cos 4 i 2 2 c2 2 zu--sin )2-2. d 3)' •2_ si Ocos 0uco 00(d-3C + (-3co2.1 dO 2sin20Cos 2i} 4J?u-sin2O coi20' {u2 [3sin20 (I - 2 c6s2i)- 1] 4 + + 3u sin20:cos 4i C 2 + 3'cOs 2 i)1 + du2sin 0 cos 0 E4-'sin i+Isir20 a U -T (-L \2 sin2O cos 2i + 0(J3 ) dO) - dA2 L+ (20) _ 2Juc sin20 + 4j2u2c 3 sinl0 + 0(J) (21) dO Each of the neglected terms in equations-(19) - (21) are indicated by the (0) symbols. These are terms which will be multiplied by J to the third power or higher. Equations -(19) - (21) are identical to those used as a stai .ing point in the analysis of Eckstein, Shi and Kevorkian [Ref. 611. C. SOLUTION The initial breakthrough of an analytical solution of the equations of motion represented by equations (15) , (19), (20) , and (21) was obtained by Danielson arid Snider [Ref. 1, 71 . Further refinement of the solution was later formulated by Danielson, Sagovac, and Snider [Ref. 2, 81 . The thlice authors, developed the solution through the extensive use of an algebraic manipulation computer program called MACSY*1IA. Through the use of an algorithm, MACSYMA was able to solve for the variables u ( or r ),i, £2, and dildO in terms of 0. T. solution includes all terms multiplied-by J and excludes terms of order J and higher. in Z. r to maintain a solution of ord.r J when 0 1/J, the solutions also need to apprupriately include terms multiplied by J'0. The solutions which analytically dcmonstratc .,relative accuracy of order J'0 are: 7 cosy + J Po{1+eo o -i['o/S 4 eo [15(2 + e)s + 52 24-] sil - 14(4 + e"+ q." -v2- + "TE(6 + e2 (5S (3S 2)cos(2y- 20) 2)] 2-( - 2)] sin[O + wol 4) + eeo-,4(j 00 + w~o) + "-"Cos("0 - coo) 2 2 14) sin[.+3(0O so-1-6s2cos22 2 + - +--[(5eo 2) 2 - 2eo~] cos(y + Oo + co) + eos 2 cos(0 o + wo) 22 22 ) - 00Ro) +e6 "5 o s cos( eos e -- cos(y -0 0 + 3coo) + 2 0 (32 - 3 2)cosy (2 3s2) - cos(y-3 -2+ 2o 0+ 0o ) cosCy + 200) 4' 2 + -- ~)cos~y (4 - - 00- coo) + (6-Is) cosy + - 1- 3eos 8 + -4 2 (2 + 5eo) 2 cos 20 + 2 (9os 8) 2 e0 cos(y- 400 + 2coo) -T(s [2e- -1) S2-o 200 2 - [eo (2 + o20) (2- 3s) cos(2y +20) +~ + 2- - ' 20 +- (92)cs2+) osy- [ + 1)cos(y+2o) (14+ 5eo)S 2 .- cos(y - 300 + 8 2y wo)] + O(J2, , (2 0 y - 2)(0- 00) co+J{(S- 2 2 22 2 __ 2 13) -2 (5s + 96 0 2wo _1)cs 4-15'(5 20s Ce S 2 ) +w)+-- 6o( (15S -13) cos(300 [5(9eo' + 34)s + 4CU- 34)s - o(j 2,j 30,...) 56e]} + I= io+scj -cs20-+.-cosy+26)+-ycos(y-2)- cA4-- 15S) sinJ( 5s2 212(5s -1 + e-6s D a 4 sin 4s 15S4~ L~ 2 -4) sin2 -e 0 - ) JO-L 2)] (4 o[coJO-L-) osifn(300 -sino)y- 0 )sin(o-+sin)} 62 2 4 2 1so 12(s - 12(4 +2SsinJ(2o + w)] 020o+ 0 2 + 2)] sinl[2wo cos 200 w ~ o( 0 +)+-j-(7 2 -4 62 +-j0 eo4 3 s2)}c? ,)-L(7SJ3 )osI(o- 9 ) 2 5) 2 s2 + Ls ++ °( -s)cosy.-20) 2 + D. 2 (26) 20eo(3 - 4S) 's2 0 0+: 2 cosO + 20) 6 2 2 + 2 -0)]+ o(J2,3o,_...)}do 2 cos(0 0 + SIMPLIFIED SOLUTION As shown in equations (22) - ('";),if0 is restricted such that 0 1/J, the solution should be of order J and the nglected terms should be of order j2. For an Earth satellite, J<312 x 10- ', so for at least 100 revolutions the-relative error should be on the order of 10-6 . If 0 is restricted such that 0 < 1 , all of the terms of order 0 in equations (22) - (26) can be neglected if a relative e-ror of order J is desired. By neglecting terms of order JA , the solution simplifies considerably to: r = po Il+eocos 0-oo+J(0-0o)-"--2) e + JVl-"7+3s rE /&20 1 ---'" ) 1 Lr2e-(2+5eo)s21cs20 2 + 2 'e° (9s 2 - 4) cos(O + 300 - 2Co) + - (6 - 7s2) cos(O + 0o - 2coo) 2 + 22 - (6 SS) cos(0- 22 eo s_ cos(0 16 0 0 2 02w) co. + . _( 2 cos(O1, )- 30o + 2 o) 00 + 2wo) + --- (3s2 - 2) cos(O - 200 + coo) 24 -00+o) 16 cos(O - 500 + 2o) + 7 ( -2)cs - 2 ) cos(O-cvo) +o)3s -4 4 (I- 3 L ) cos(O - 200 - coo) + "" (2-3(2 10 , (27) ~0_(9S2 +1 + cos(20 -8) - 24 2(o0 ) +L-(6 IS 1-s2 cos(30 2eo'J cos(0 + 0 0) +,eoS 2 COS(0 0 1[(5eg2 - 2)s' 2o coo) - _ WO) 2 +2(2 -3s2) cos(40 + 2co) + 0_- (3S 2 -2)cs2o 3eos2 e 2 - 8- .-cos( - 40 0 + w)- -T-(S+ 1)cos((0 -L ET2eO2 - (14 + 5eo) 2 cos(0 300)+S2 cos 200 + -L-E(6-+ 5e02S2 ) - 4(1 + e02)] cos(0 + eoS 00) (300 - coo)J + O(J2, j20 io + scJ[-Lcos 20 + -L-cos(30 -wo) +eo - 1 ~cos(30 0 - w0 )- --- cos(00 + coo) _+ cos(0+ w)- 2 O(J2,Jj0,.) L + 0) eo sin(0 + wo) 2 - cos 20 0 (28) 2 !o 1 sin 200 + eo sin(00 2~ - co ) 0 0) i(0 n(36 - w0 ) (29) 2-sin(0 0 + coo)] t 10 - Jor 0 21+j[ (2-3s2) cos 20 + e(s2 _ 1) COS(0- COO) +o(O6 co(0 0) + 2 2 -e cs 20o 2 s 1- CS(Is0 (30) E. THE CRITICAL INCLINATIONS As shown in equations (22) - (26), the solution appears to be well bounded for almost all inclinations. However, two particular inclinations immediately appear that may produce a singularity. A possible singularity occurs when the inclination is equal to either 63.43 degrees or 116.57 degrees. These two inclinations have produced mountains of literature and are well known as the critical inclinations. However, if the solution is replaced by the limit as the inclination approaches either critical inclination, the solution remains finite. More specifically, the solution at either critical inclination is as follows: r= po{l+eocosy+J[ 12S2 + 7 [(6 + 5e0)s + 22 - 3+ el_5s) eo12 2~ 4(1 + eo)] cos(y - 00 + coo) + -- cos(300 - o0o) g 2 -4)coso, + 360 -oo) + ( - 7S) cosC, + 0o _ (to) + + 1[(5eo2 - 2)S2 -2 2] 2 cos(y±+0 0 +coo) + eoS cos(0+Coo) 22 eos 16 cos(y - 500 + 3oo) + - eo (3s 2 -2) cos(y - 20 0 + 2coo) - - 2 2 eos 16cos(y - 00 +3 e2 2 +24o(3s 2 2) cos0y - 300 +3vo) e°0 -3S) COSY + S2 ^S2 3s2) cosCv - 200) + - -(2- 3 Cos 200 e° (I 2 " (4- 5) .-(6 24 24 - (2- + )cos- + 200) Is2)cos(y + 20) +-!- (2- 32) cos(2y + 20) 12 2 2 -(14+ 5e)s 2 1 cos +3eos[2eo -- 8 cos0y - 400 + 2coo) - + "(s-8 cos -T 0S+1coy+2&0 C - 24 E2e02-(14+5eo) -cosO,,- 300 +oo)] 12 (31) + j 20[ " O-[15(2 + e2)s 4 - 14(4 + eo)S 2 + 241 sin(O + wo) 24 2 21 + 12 (15S2 - 14) sin 2cooj + o(J2 , j6,. y = o-co+,{(- --- 2)(O-Oo)} 2 + 2 {4 (130s 2 - 105s4 - 28) cos 2wo + -2- (15s 2 - 13) Cos 200 2 2 + 2L (15 -~-1S -- i= (32) 2 -13 _ 13)c~s(Oo+oo)+--L-(15s c(0+ )e 6 S2 E5(9eo 13) cos(30 0 - - coo) + 34)s4 + 4(9e02 - 34)s2 - 56e2j} + OJ, io + scJ cos 20 + -L- os + 20) +--cos(y-2) 2 Cos 200 - -!o - cOs(30 0 - COO) - cos(Oo + CO) (33) 2 (14 - 15s2) sin 2o o + O( +j20 +0- = 0 +-sin i 2 0-e y+ o o siny sin 200 + co sin(Oo - coo) - 2 _,eos o J0,J...) sin(y+ - coo) ___ - 2 0)--osin( 2_sin(o + oo) -20) (34) 2 eo s3s(30° - too) - 2f j + s_ 2, s 2 cos 200 + - (6 - s2)l + O(J 2, J'0, .. (2- -S2) + OS CSY 2 e°(l s2) ' s e°(3 - 4s2) ofv..2) 2 cos(, - 20) -:1--cos 20 0 + 6( - -cos + 20) 13 3) (35) + o cos(Oo + o0) + LO2cos(30o - coo) 2 2 + J20 °s (15214)sin2(o0 +O(J2,3O,...)dO Clearly, equations (31) - (35) demonstrate that the solution is indeed finite for both critical inclinations. Equations (31) - (35) are only valid for the critical inclinations and were first developed by Sagovac [Ref. 2] . The primary purpose in developing these equations was over a concern in computer programming. Some computers have major problems when a denominator approaches zero, and unlike humans, will not replace a solution with its limit. Therefore, depending upon the accuracy of a computer, equations (22) - (26) can replace equations (31) - (35) for inclinations near the critical inclinations. It should be noted, however, that the solution itself is -valid and bounded for all inclinations. It is the limitation of the computer that creates the singularity. The-simplified solution which is shown in equations (27) - (30) , is valid for all inclinations. Since all terms of order JAO have been neglected, the troublesome denominators mentioned earlier do not appear. F. SPECIFIC MECHANICAL ENERGY For all satellites under the influence of conservative forces, the specific mechanical energy remains constant. Therefore, an ideal analytical check of the solution would be to see if indeed the specific mechanical energy at any time is a constant. This simple check was performed by Danielson, Sagovac, and Snider [Ref 1, 2, 7, 8] by substituting equations (22) - (26) into equations (7)and (8). The substitution yields: T + Gjlf(l2 - eo) po G!IJR2 (l - 33sin 2flo) + 2[r(1o)] O0j2, 3 )( .. 6 (36) The first two terms on the right side of equation (36) represent the initial specific mechanical energy. All other terms multiplied by J in equations (22) - (26) combine to zero when substituted into equations (7) and (8). Equation (36) demonstrates that by neglecting all terms of order J2 and higher, the specific mechanical energ at any time 14 is precisely equalto the initial specific mechanical energy. Obviously, the solution satisfies the requirement of constant specific mechanical energy. 15 'II. A. METHOD OF ANALYSIS ORBITAL PARAMETERS 1. Argument of Latitude ( 0) Figure 2 illustrates that the position of a satellite at a particular time can be described by the argument of latitude (0) , the radius magitude (r), the inclination ('), and the longitude of the ascending node (92) . As shown in both the solution and the simplified solution, r, i, and i are only functions of J and the argument of latitude (0) Since J is a constant for all planets, a simple determination of these terms is trivial once 0 is known. However, the determination of 0 is not trivial. Although it would be ideal for all of the equations to be analytical expressions, equations (26) , (30) , and (35) contain an integral that must be evaluated in order for 0 to be determined. Herein lies the key to the solution. Given an elapsed time between observations, how can d be precisely determined? Since the initial angular momentum (h0) is known, this term can be moved to the left side of equations (26), (30), and (35) to yield equations in the form of: (t - to)ho J'r2(J, 0)(l +f(J, 0))dO (37) 00 If r was not a function of 0, an evaluation of the right side of equation (37) could easily be conducted that would yield an analytic expression. However, r is also a function of 0 and the only practical techni.ue in evaluating the integral is through numerical means. Several numerical methods could be used to evaluate the integral depending on the speed and accuracy one desires. Since accuracy and not speed is desired in this analysis, a Romberg integration routine was used to evaluate the integral. Since the right side of equations (26) , (30) , and (35) are sinusoidal in nature, the Romberg -scheme converged quickly and accurately. 16 Since 0 defines the upper limit of the integral, in order to arrive at a solution, an initial 0 must be estimated. Once 0 is estimated, the integral can be numerically integrated and the result can be compared to the left side of equation (37). If the comparison is accurate within some predetermined error, the iteration is complete and 0 has been determ-ined. If the comparison produces an error that is unacceptable, 0 can be incremented either up or down and the integral can be reevaluated. Eventually, the iteration will converge and 0 will be determined. An algorithm of the iteration procedure is as follows: 1. Estimate 0. 2. Evaluate the integral. 3. Compare the result with the left side of equation (37). 4. If outside the limit, go to (5). If within the limit, stop. 5. Increment 0 up or down as needed, go to (2). The determination of 0 involves a combination of two errors. The first error is contained in the numerical evaluation of the integral itself, while the second error involves the comparison of the result of the integration with the left side of equation. (37). Unfortunately, the errors do not linearly combine, but rather multiply since the numerical evaluation of the integral is inherently nonlinear. In order to make the comparison error meaningful, the evaluation of the integral must be made as precise as possible. In order to avoid determining whether an error is due to computing or truncation errors, the numerical technique used in this analysis did not rely on a step size constraint. Therefore, the relative error, in general, can be specifically controlled. Since in this analysis, accuracy and not speed is desired, the Romberg integration technique was utilized. The Romberg technique does not depend on any specific step size and the evaluation of the integral is determined through a converging algorithm. Also, the relative error of the integration can be specifically controlled. In general, the relative error normally demanded in the integral evaluation was on the order of 1012, and the relative error of the comparison was on the order of 10-1. Since the computer program utilized in the analysis was written for double precision accuracy, these types of relative errors presented no significant problems. The double precision accuracN enabled the computer program to calculate up to sixteen digit precision. 17 2. Radius Magnitude (r) From equations (22), (27), and (31), it can be seen that the radius magnitude (r) is a function or J and 0. Once 0 is known, r can be evaluated. From the appearance of equations (22), (27), and (31), it is not obvious how r will behave as the orbit-of a satellite -progresses. However, from observations of actual satellite motion, it is clear that the orbit should behave elliptically with r varying from a minimum value at periapsis to a maximum value-at apoapsis. The magnitude oLJ plays an important role and fortunately for most planets, oblateness effects act as a pertuibation in comparison to the main gravitational force. Therefore, a large value of J causes iargcr v riationsin r. Since equations (22), (27) , and (31) contain a number of sine and cosine terms, a sinusoidal behavior should be expected. 3. Inclination (i) The solution of the inclination is shown in equations (24) and (33) , and the inclination for the simplified solution is shown in equation (28). In general, these three solutions are quite similar. Again, once 0 is known, i can be evaluated easily. It can be seen from equations (24) , (28) , and (33) , that i will vary slightly from an initial -inclination as the orbit of a satellite progresses. Also, since a number of sine and cosine terms are present, the variation should be sinusoidal in nature. From inspection it is clear that the magnitude of the variation is dependent upon-the magnitude of J and the initial inclination. The variation of the inclination should not behave in a diverging fashion, but rather in an oscillatory fashion about some arbitrary mean inclination. This behavior is consistent with observations of actual measured satellite data. The driving factor in all inclination variations is the magnitude of J. Since for Earth, J2 is on the order of 10- , these variations should be quite small. 4. Longitude of the Ascending Node ( 2) The solution of the longitude of the ascending node (Q) is shown in equations (25) and (34) , and the longitude of the ascending node for the simplified solution is shown in equation (29). As expected, all solutions are quite similar. As with the case of r and i, 92 can easily be determined once 0 is known. Unlike the behavior of r and i, the variation of 92 is very predictable and highly meaningful. With the presence of 0 alone in equations (25), (29) , and (34) , Q possesses a linear relationship with 0 and 18 as 0 increases with time, the vaiiation of 92 from 0 should be linear. Depending upon the initial inclination, this variation will be either positive or -negative. This type of behavior is clearly consistent with the classical behavior known as nodal regression. For an oblate planet, nodal regression is a linear property whose magnitude and direction depends upon the radius magnitude and inclination of the-satellite. In equations (25), (29) , and (34) , the radius magnitude is contained in the J term. Therefore, the magnitude of the nodal regression is entirely dependent upon the magnitude of J. From the analysis of the behavior of Q as 0 increases, the nodal regression behavior should be extremely obvious. B. ROMBERG INTEGRATION TECHNIQUE The Romberg integration technique is a powerful integration method in which arbitrary accuracy can be achieved in a relatively efficient manner. The method combines any type of relatively inaccurate quadrature method -with a Richardson extrapolation in order to quickly -and accurately converge on a solution. In this analysis, a simple trapezoidal quadrature was initially used to estimate the integral and then a Richardson -extrapolation was used to improve the integration to the desired accuracy level. The trapezoidal quadrature first estimates the integral with a single interval. The estimate is then improved by using 2 intervals, 4 intervals, 8 intervals, etc. For purposes of identification, the results can be labeled I.' , Io, , and so on. These results can be arranged in column form in preparation for a Richardson extrapolation and each new member represents the technique of halving the prior interval. The length of the column is determined by the accuracy that one desires. Once the first column is arranged, a Richardson extrapolation can be performed by the following equation. 1n - I1n _ 1nh2 4 1- 1 -1 The values of 1' can be arranged in tabular form as shown in Table 1. 19 (38) Table 1. A SCHEMATIC OF ROMBERG INTEGRATION 0 12 I4 14 14 10, 1 12 13- 421 131 1616 0l- 1616 14 To test for convergence, the value represented by I," is compared with the value represented by Ifj . If these two- values are within some predetermined error, then 4" becomes the evaluation of the integral. If convergence has not been reached, then another row is calculated and the process continues. An excellent example of the Romberg integration technique is shown in Ferziger [Ref. 91 . In 'this example the following solution of the integral is desired. I = fied (39) From elementary calculus, the exact solution is: Iexac = 2.718281828 (40) The technique of Romberg integration of the integral is shown in Table 2. The relative error of I, to I is 7.81 x 10- '. The relative error of I3 to I,, is 1.97 x 10- . As can easily be seen, the integration is converging very nicely and the error found in the final solution is less than the error demanded within the Romberg integration scheme itself 20 Table 2. ROMBERG INTEGRATION 1 1.859140914 - 2 1.753931092 1.718861152 4 1.727221905 1.718318842 1.718282688 8 1.720518592 1.718284155 1.718281842 1.718281829 The advantage of the Romberg integration technique over a simple quadrature method is obvious. The number of intervals that must be evaluated is very small and the relative accuracy is very high. In order to attain the accuracy that the Romberg technique delivers, the trapezoidal method would need to divide the integral into several more intervals. This would be highly inefficient. For smooth functions, the Romberg technique is very effective and efficient. Since equation (37) is sinusoidal in nature and thus relatively smooth, the Romberg integration technique was used to evaluate the integral. If equation (37) had not been so well behaved, another integration technique might have been warranted. The Romberg integration scheme is the heart of the analysis and can be found in the computer program shown in Appendix E. 21 IV. METHOD OF COMPARISON A. NUMERICAL INTEGRATION COMPARISON In order to verify that-the theory is valid for practical application, the solution must be compared with proven numerical solutions and measured satellite data. By comparing the solution with a numerical integration of the equations of motion, theoretical accuracy can be specifically determined. As shown in equations (22) - (26), theoretically the solution is accurate to order J 20 . A numerical integration comparison will determine whether this prediction is correct. In order to verify the solution, the following parameters will be compared. 1. Delta.radius vector (ArI) 2. Earth arc angle (T) 3. Delta omega (A92) 4. Delta inclination (Ai) 5. Delta theta (AO) 6. Delta radius relative error (I Ar I/r) 7. Delta theta relative error (A00I) 8. Radial track error (RTE) 9. Along track error (,ITE) 10. Cross track error (CTE) 1. Delta Radius Vector A graphical representation of the delta radius vector (IAr[) and the Earth arc length (T) is shown in Figure 3. The delta radius vector is the magnitude of the vector separating the solution radius vector (r) from the numerical integration radius vector (r) . Mathematically, the delta radius vector can be expressed as: Ar = r-r. 22 (41) The delta radius vector describes in overall terms-the global error in the solution. Another common name for this error is the Euclidean normed difference in ephemerides. Although the delta radius vector provides ample information on the global error in the solution, this error can -also be expressed by a different parameter that will be called Earth arc angle. Ar' C C, BO Figure 3. 2. Delta radius vector and Earth arc angle. Earth Arc Angle Earth arc angle (TI) is simply the angle between the two positions if viewed from sea level on Earth. For simplicity, the position at sea level was chosen such that the arc angle from the center of the Earth was bisected. By using the law of sines and cosines, the Earth arc angle is easily determined. Since satellites are tracked b instruments on the surface of the Earth, a bet.er feel for the global error can be attained by determining the angle between the two positions. Most satellite tracking radars possess beamwidth 23 and field of view limitations; therefore, 'V will provide useful information on whether the solution is accurate enough fot satellite tracking radars. 3. Delta Omega, Delta Inclination, Delta Theta A break down of the global error can be described in the errors of delta omega (AK2) , delta inclination (Ai) , and delta theta (AO) . As mentioned previously, AQ2 will provide an insight on the motion of the line of nodes,-and specifically-if nodal regression is present. The change in inclination, will provide information on the movement and stability of the -orbital plane. The parameter in which all errors are based, AO , will provide much information on the source of the global error. It is clear that small errors in AO will contribute significantly to the accuracy of the solution. 4. Relative Errors The verification of the -solution will lie in the confirmation- of the relative errors. The delta radius relative error and the delta theta relative error will demonstrate the actual accuracy of the solution. Both parameters, Ar and AO , demonstrate a theoretical error of J2O. Therefore, the delta radius -relative error should be on the order of J 20, while the delta theta relative error should be on the order of J 2 . Comparisons of the relative errors between the sujution and- the numerical integrati.in solution will provide the evidence for theoreti.a! confirmation. 5. Track -Errors Another michod to break down the global error is in terms of track errors. Figure 4 shows a graphical representation of radial trac., error (RTE) , along track error (ATE) , and cross track error (CTE) . These errors car better be described by referring also to Figures I and 2. The radial track error is the error in the radial direction or in the B, direction. The along track error is the arc length error in the plane defined by the solution radius vector (r) and the B. direction. Together, these errors describe errors in three orthogonal directions or planes as compared with some reference position. The reference position in this case is the numerical integration solution. A mathematical derivation of these errors is as follows: Ar = r(r + Ar, 0 + AO, i + AO, 2 +A 24 )-r(,, 0, i, 2) (42) Using equation (1), Ar = (rn + Ar)(B 1 + AB1) - rnB 1 (43) Ar = rnB 1 +ArB 1 +rnAB 1 +ArAB 1 -rnB 1 (44) Neglecting-higher order terms, Ar = ArB + rnAB1 (45) AB1 = (B1 ABI)B I + (B2 . ABI)B 2 + (B3 . AB1)B3- (46) Continuing, defining AB1 , Using the rotation transformation and after performing considerable algebra, B1 • AB1 = 0 B2 .AB (47) = (AO + AQ cos i) B3 AB1 = (Ai sin 0 - A cos 0 sin1) (48) (49) Therefore, using equation (45), Ar = (Ar)B1 + rn(AO + AK2 cos i)B2 + rn(Ai sin 0 - Af cos 0 sin )B3 (50) From equation (50), the track errors can easily be defined. RTE = (51) ATE = r(A0 + Af2 cos i) (52) CTE = r(Ai sin 0 - An cos 0 sin ti (53) 25 A graphical representation of the track errors is shown in Figure 4. ATE ~CTE R TE 0 Figure 4. Track errors. Examination of equations (51) - (53) demonstrates that the track errors divide the global error into three distinct regions. Radial track errors obviously describe errors in the radial direction. Along track errors are similar in nature to Earth arc angle errors, but also include errors due to nodal regression. Cross track errors describe orbital plane errors in terms of both inclination errors and errors due to nodal regression. In general, all these parameters should give an excellent insight into the accuracy of the solution. Also included in the numerical comparison will be the simplified solution and the two-body solution. The simplified solution has been previously presented. The two-body solution can easily be determined by simply setting J = 0 . The analysis of the numerical comparison wiHl demonstrate the strengths and weaknesses of all the different solutions. 26 B. MEASURED DATA COMPARISON In order to observe how well the solution models actual satellite motion, the solution will be compared with actual measured data from operational satellites. To properly evaluate the solution, a wide range of orbit characteristics will be compared. These characteristics include orbits of various altitudes, inclinations, and eccentricities. Also included in this comparison will be the simplified solution, the two-body solution, and if available, some particular numerical solutions. The numerical solutions will consist of two forms. The first is a numerical solution that only includes perturbations involving -2 , -3 , J4 , and Js . The second numerical solution will include the following perturbation effects. 1. J2 ,J 3 ,J4 , and Js 2. Atmospheric drag. 3. Sun gravitational effects. 4. Moon gravitational effects. 5. Solar pressure effects. From the analysis, the accuracy of the solution and the simplified solution can be compared to a numerical solution as well as to actual measured data. The weakness of the two-body solution will also be demonstrated. In addition, the strengths of a well modeled numerical solution will clearly be seen. The identical error parameters described in the previous section will also be used in the measured data comparison. From this comparison, the advantages and disadvantages of the solution in regard to actual satellite motion will clearly be demonstrated. 27 V. RESULTS A. NUMERICAL INTEGRATION COMPARISON To verify that the theory is valid for practical applications, the solution was coinpared with a proven numerical solution. A numerical integration computer program called UTOPIA is currently in use at :he Colorado Center for Astrodynamic Research (CCAR) located on the campus of the University of Colorado, Boulder, Colorado. UTOPIA is primarily used to model a wide range of perturbations and can predict satellite motion with a high degree of accuracy. The UTOPIA computer program wa, developed at the University of Texas, Austin, Texas, and is currently in use at several universities and research centers. The solution was compared with the UTOPIA solution for a satellite with the following initial conditions: r = 7,3S6.IS km io = 90.03 degrees eo wo 00 no = 0.003991 = 224.3S degrees = 104.05 degrees = 322.63 degrees AD = 54,205.IS km 2Is Po = to = 7,371.29 kin 0.00 seconds In general, these initial conditions represent a slightly retrograde orbit of small eccentricity at an altitude of approximately 1000 kilometers. Essentially, it is a polar orbit at an altitude where several satellites are currently in motion. From the initial conditions, the orbit should demonstrate a slight easterly nodal regression. But, since the inclination is so close to 90 degrees, some integration routines might predict zero nodi regression. In this comparison, UTOPIA only modeled the J. perturbation; therefore, the solution should compare well if the theo*y is valid. All error parameters depicted in the comparison were calculated in the following manner. A(ErrorParanzeter) = TheoreticalSoluion - UTOPIA Nmncrical Sohion (54) 2S All solutions were compared at one hour intervals over two separate periods of time. One comparison is for a time period of one day while the second is for a time period of three days. The three day comparison was constructed to illustrate the effect of long term errors while the one day comparison allows for a more detailed analysis during the first few hours of motion. The period of rotation for the satellite is about 105.26 minutes which equates to approximately 13.68 orbits per day. The results of the numerical solution comparison are shown in Appendix A. 1. Delta Radius Vector Comparison The comparison of the delta radius vector is shown in Figures 5, 6, and 7 in Appendix A. Figure 5 includes the comparison of the solution, the simplified solution, and the two-body solution to the numerical solution. If the solution matched the numerical solution exactly, the delta radius vector would be zero. As shown in Figure 5, the solution and the simplified solution match extremely well with the numerical solution while the two-body solution contains gross errors. A more detailed plot of the delta radius vector comparison is shown in Figures 6 and 7. In these figures, the two-body solution is excluded. The difference in the solution and the simplified solution can clearly be seen. The simplified solution produces a diverging sinusoidal response about the solution. However, up to approximately four hours of motion, the solution and the simplified solution are nearly identical. The sinusoidal behavior of the simplified solution can be attributed to the fact that all terms multiplied by J20 have been neglected. As 0 grows with time, these terms become significant in the solution. As shown specifically in Figure 7, the average delta radius vector of the simplified solution clearly diverges from the solution. Figures 5, 6, and- 7 also demonstrate that for at least one day, the delta radius vector for the solution and the two-body solution are nearly linear as a function of time. Other comparisons will determine whether this relationship holds true and will be shown later. As mentioned earlier, the delta radius vector is a global error. As shown in Figures 5, 6, and 7, the solution compares well globally with the numerical solution and demonstrates a great improvement over the two-body solution. 29 2. Earth Arc Angle Comparison The comparison of the Earth arc angle is shown in Figures 8, 9, and 10 in Appendix A. From inspection, these plots are nearly identical in appearance to the delta radius vector plots. This is expected since both the delta radius vector and the Earth arc angle represent global errors. Figure 8 clearly illustrates the large error generated by the two-body solution. The two-body solution produces unsatisfactory long term satellite position prediction. After one day, a tracking radar would have a difficult time detecting a satellite with a position error of over 80 degrees. Figures 9 and 10 present much more encouraging results. Again, the simplified solution responds in a sinusoidal behavior about the solution. After one day, the position error of the solution is only approximately 0.15 degrees. Clearly, the solution and the simplified solution are superior to the two-body solution. Most tracking radars can easily handle daily position errors of 0.15 degrees. In general, the solution and the simplified solution agree very well with the numerical solution. 3. Delta Omega Comparison The comparison of the delta omega angle is shown in Figures 11 and 12 in Appendix A. At first glance, the solution and the simplified solution in Figure 11 appear not to agree well with the numerical solution. However, the scale of delta omega is multiplied by 10'. The numerical solution parallels the two-body solution nearly exactly and predicts almost no change in Q . In other words, the numerical solution predicts no nodal regression. Easterly nodal regression is represented by a positive delta omega; therefore, it is clear that the solution and the simplified solution predict greater nodal regression than the numerical solution. On a larger scale, all three solutions are essentially identical. Since the initial inclination is so close to 90 degrees, small discrepancies are not surprising. The delta omega plot does, however, invoke confidence in the solution. Although the initial inclination is very close to 90 degrees, the solution, the simplified solution, and the numerical solution predict easterly nodal regression. This result is significant. Even initial inclinations close to 90 degrees produce nodal regression in the correct direction for the solution, the simplified solution, and the numerical solution. 30 4. Delta Inclination Comparison The comparison of the delta inclination angle is shown in Figures 13 and 14 in Appendix A. On a larger scale, all of the solutions compare very well. On the scale shown in Figure 10, the two-body solution and the numerical solution are nearly identical. The solution and the simplified solution oscillate about an error of approximately -2.0 x 10- ' degrees. Obviously, this error is extremely small. In general, the solution and the numerical solution agree very well. An interesting aspect of the delta inclination comparison is the sinusoidal behavior of the solution and the simplified solution. This type of behavior is precisely what was predicted in the earlier analysis. Figure 14 demonstrates that this behavior continues for even longer periods of time. On a larger scale, this type of motion would not be detectable. 5. Delta Theta Comparison The comparison of the delta theta angle is shown in Figures 15, 16, and 17 in Appendix A. This comparison confirms the results found in the earlier comparisons. The two-body solution produces very large errors, while the solution and the simplified solution agree very well with the numerical solution. Figures 16 and 17 again illustrate the typical sinusoidal response of the simplified solution about the solution. Since the delta theta error produces all other errors, the excellent results found in the earlier comparisons are now not surprising. 6. Delta Theta Relative Error Comparison The comparison of the delta theta relative error is shown in Figures 18, 19, and 20 in Appendix A. As shown in Figure 18, the two-body solution demonstrates a relative error of 2.3J, while the relative errors of the solution and the simplified solution are much smaller. In more detail, Figures 19 and 20 indicate that the relative error of the solution is 2.8P. This result confirms the theoretical prediction that .ne delta theta relative error of the solution would be on the order of P. Figures 19 and 20 illustrate that initially the relative error of the simplified solution is also on the order of J. But as 0 increases with time, the relative error grows in a sinusoidal fashion. This result is expected since the simplified solution neglects all the terms multiplied by PA . In gen- 31 eral, the results shown in Figures 18, 19, and 20 confirm the theoretical relative accuracy of delta theta that was predicted-in the earlier analysis. 7. Delta Radius Relative' Error Comparison The comparison of the delta radius relative error is shown in-Figures 21, 22, and 23 in Appendix A. As shown in Figure 21, the two-body solution produces a relative error that is-linear in time-and proportional to 2.3.10. Again, the relative errors of the solution and the simplified- solution are-magnitudes smaller. Figures 22 and 23 present in more detail the relative errors of the solution and the simplified solution. The relative error of the solution is very linear and proportional to 2.8J'0. The relative error of the simplified solution is sinusoidal in nature and diverges from the solution. However, for up to four-hours of motion, the relative error of the solution and the simplified solution are nearly indistinguishable. Again, the- results from this comparison confirm the theoretical prediction that the delta radius relative error of the solution would be on the order of 20. 8. Radial Track Error Comparison The comparison of the radial track error is shown in Figures 24, 25, and 26 in Appendix A. As shown in Figure 24, the two-body solution oscillates about an error of approximately -11.0 kilometers, while the solution and the simplified solution both produce errors that are dramatically smaller. From inspection, the two-body solution also appears to be slowly converging as time increases. In Figures 25 and 26, the solution and the simplified solution produce contrasting behaviors. While the solution remains relatively constant, the- simplified solution slowly diverges from zero. These two different responses continue even after three days of motion. Again, the neglected J20 terms cause the significant divergence of the simplified solution. Not surprisingly, the solution and the simplified solution are clearly superior to the two-body solution. 9. Along Track Error Comparison The comparison of the along track error is shown in Figures 27, 28, and '9 in Appendix A. The results presented in this comparison parallel the results found in the earlier comparisons. Since the inclination is so close to 90 degrees, the A.Q contribution 32 is negligible and the AO contribution strongly influences the responses. As a result, the along track error comparison is practically a mirror image of the delta theta comparison. 10. Cross Track Error Comparison The comparison of the cross track error is shown in Figures 30 and 31 in Appendix A. The cross track error is strongly influenced by Ai and AQ. Since the twobody solution produced good results with these two parameters, it is not surprising that the two-body solution agrees well with the numerical solution. Fortunately, the errors produced by the solution and the simplified solution are also very small. The solution produces a maximum cross track-error of approximately ± 0.5 kilometers after one day of motion, and approximately ± -1.3 kilometers after three days of motion. Clearly in this comparison, the two-body solution is superior. However, the large errors produced by the two-body solution in the other comparisons easily overwhelm these results. In global terms, the two-body solution is no match for either the solution or the simplified solution. B. MEASURED DATA COMPARISON The solution was compared with actual measured satellite data to determine the altitude band where the theory works best. The measured satellite data was obtained from the First Satellite Control Squadron (ISCS) located at Falcon Air Force Base, Colorado. The First Satellite Control Squadron tracks several satellites for the Air Force and was able to supply measured data for three separate satellites. The three satellites are currently in motion and occupy orbits that are labeled Near Earth, Semisynchronous, and Geosynchronous, respectively. All error parameters compared in the earlier numerical comparison were also compared in this comparison using the measured data as a reference. Included in all the comparisons were the solution, the simplified solution, the twobody solution, and two numerical solutions. The two numerical solutions were also supplied by the First Satellite Control Squadmon and are labeled Spacom 1 and Spacom 2, respectively. The Spacom I solution includes all perturbation effects, while the Spacom 2 solution only includes the Earth's harmonic perturbations. All error parameters in this comparison were calculated in the following manner. 33 A(ErrorParameter) = Test Solution - Measured Data Solution (55) Unfortunately, the First Satellite Control Squadron only records measured data when an update of their numerical solution is required. Routine updates are usually conducted after about seven days of motion. Therefore, satellite data for one month usually consists of only four data points. Although more data points are needed for a more detailed analysis, a long term analysis can still be conducted. The analysis of each type of orbit will be presented separately. 1. Near Earth Orbit Comparison The near Earth orbit comparison possesses the following initial conditions. ro io eo oo 00 0o ho Po to = = = = = = 7,776.58 km 98.81 degrees 0.0003071 9.57 degrees 149.14 degrees 37.10 degrees = 53,664.37 km 2/s = 7,224.89 -km = OOOOZ 26 July 1990 The initial conditions of this satellite represent a retrograde orbit of small eccentricity at an altitude of approximately 850 kilometers. The-period of rotation for the satellite is about 101.89 minutes which equates to approximately 14.13 orbits per day. From the initial conditions, J2 should be the dominant perturbation. The orbit should demonstrate noticeable easterly nodal regression. If the theory is valid, both the solution and the simplified solution should agree well with the numerical solutions and the measured data. The results of the near Earth orbit comparison are shown in Figures 32 - 43 in Appendix B. As shown in the figur,., the solution and the simplified solution agree very well with both the Spacom I solution and the measured data. The fact that the solution and the simplified solution produce such excellent results verifies that J, is the dominant 34 - perturbation- for this satellite. Figures 32 - 43 also demonstrate the larger errors produced by-the two-body solution. In almost every comparison, both the solution and the simplified solution are far superior to the two-body solution. One surprising result is the poor comparison produced by the Spacom 2 solution. In every comparison the Spacom 2 solution either models the two-body solution exactly or produces results that are inferior to the two-body solution. It is clear that the Spacorn 2 solution does-not model the Earth's harmonic forces correctly. An explanation for the poor results cannot be determined in this analysis. A detailed analysis of the force modeling in the Spacom 2 solution must be completed in order to adequately explain the unsatisfactory results. The delta omega comparison in Figure 34 demonstrates the easterly nodal regression produced by the solution, the simplified solution, the Spacom 1 solution, and the measured data. The two-body solution represents zero nodal regression. Figure 35 presents the-delta omega comparison at a much smaller scale and excludes the two-body solution. In this figure, much more detail can be observed. There is only one comparison in which the results are mixed. The radial track error comparison in Figure 40 indicates that the solution produces a small improvement over the two-body solution while the simplified solution actually produces a greater error. In comparison with the along track errors, these errors are small. It is interesting, however, that the radial track error comparison produces such mixed results. In general, both the solution and the simplified solution produce results that are in excellent agreement with the measured data for this near Earth satellite. 35 2. Semisynchronous Orbit Comparison The semisynchronous orbit comparison possesses the following initial conditions. ro = 26,407.70 kin io = 63.66 degrees e= 0.005860 cod = 318.19 degrees 00 328.49 degrees 0= 92.13 degrees ho = 102,892.59 krn2]s Po = 26,559.96 km to = OOOOZ 22 March 1990 The initial conditions of this satellite represent a direct orbit of small eccentricity at an altitude of approximately 20,000 kilometers. The period of rotation for the satellite is about 717.96 minutes which equates to approximately 2.01 orbits per day. An important aspect -of the orbit is that the initial inclination is very close to the critical inclination of 63.43 degrees. Although the initial inclination is not exactly that of the critical inclination, an evaluation of the solution and the simplified solution near this important inclination can be made. From the initial conditions, the orbit should demonstrate substantial westerly nodal regression. Also, at this altitude, the dominance of the J2 perturbation should be diminished. Other perturbations that are not modeled should make a considerable contribution to the errors in the comparison. If the theory is valid, both the solution and the simplified solution should show a great improvement over the two-body solution. The results of the semisynchronous orbit comparison are shown in Figures 44 53 in Appendix C. As predicted earlier, the solution and simplified solution produce results that are superior to the results produced by the two-body solution. Figures 44 and 45 present the global errors of all the solutions. In global terms, the solution and the simplified solution reduce the error of the two-body solution by nearly one half. In effect, the J, perturbation accounts for approximately one half the error produced by the two-body solution. The remaining error which is represented by the solution and the simplified solution is caused by other perturbing forces. Unfortunately, the results of the Spacom 2 solution were not available. 36 The delta omega comparison in Figure 46 demonstrates the easterly nodal regression produced by the solution, the simplified solution, the Spacom 1 solution, and the measured data. Again, the two-body solution represents zero nodal regression. It is clear that at this altitude, the J2 perturbation produces -the majority of the nodal regression. The delta inclination comparison in Figure 47 indicates that the solution and the simplified solution produce results that are not much better than the results produced by the two-body solution. However, the error after 30 days of motion is extremely small. On a larger scale, the solutions would seem identical. Since the inclination is very near the critical inclination, these results produce more evidence in support of the theory. Clearly, the solution and -the simplified solution are- bounded at this inclination. The delta-theta comparison in Figure 48 demonstrates that the majority of the error produced by the solution and the simplified solution originates in the delta theta error. It is clear that the two-body solution underestimates the value of 6 while the solution and the simplified solution overestimate the value of 0. The relative error comparisons are shown in Figures 49 and 50. While the delta theta relative erroi for the solution, the simplified solution, and the two-body solution is approximately 15.0 x 10- ', the relative error produced by the Spacom I solution is far superior. This result is expected since the Spacom 1 solution models several more influential perturbations. The delta radius relative error comparison again demonstrates in global terms the amount of improvement that the solution and the simplified solution provide over that of the two-body solution. The track error comparisons in Figures 51 - 53 produce mixed results. While the two-body solution produces less radial and along track errors, the solution and the simplified solution produce much less cross track error. In comparison with the along track and cross track errors, the radial track errors are small. The poor results produced by the solution and the simplified solution in the-along track error comparison is due primarily to the large error in A0 . The very large error produced by the two-body solution in the cross track error comparison is due primarily to the very large error in AQ. In summary, although the solution and the simplified solution are superior to the two-body solution, the Spacom I solution models the satellite motion more precisely. However, the primary reason that the solution and the simplified solution are superior to the two-body solution is due exclusively to a better modeling of nodal regression or the angle 2). It is clear that the solution and the simplified solution model the J2 perturbation extremely well. The Spacom I solution is expected to perform better since it models more perturbing forces. 37 3. Geosynchronous Orbit Comparison The geosynchronous orbit comparison possesses the following initial conditions. ro = 42,156.57 kmio = 1.09 degrees eo = 0.0002341 coo = 320.06 degrees 0 = 331.32 degrees 00 = 334.85 degrees ho = 129i644.14 km 2Is Po = to = 42,166.25 km OOOOZ 21 July 1990 The initial conditions of this satellite represent a direct orbit of small eccentricity at an altitude of approximately 35,800 kilometers. The period-of rotation for the satellite is about 1436.69 minutes which-equates to approximately 1.00 orbit per day. Since the initial inclination is slightly greater than zero, the orbit should demonstrate westerly nodal regression. However, since the altitude is so large, other perturbing forces that are not modeled may influence nodal regression. At a geosynchronous altitude, the magnitude-of other perturbing forces approach that of J2 . Since at this altitude the effect of J2 is so diminished, some comparisons of the solution, the simplified solution, and the two-body solution may be nearly identical. As a result, the theory may not be any better than the two-body theory for satellites in a geosynchronous orbit. The results of the geosynchronous orbit comparison are shown in Figures 54 63 in Appendix D. The global error comparisons are shown Figures 54 and 55. In global terms, the solution and the simplified solution produce results that are surprisingly superior to -the results produced by the two-body solution. Evidently, for this satellite, the /2 perturbation is still quite dominant. However, the other comparisons may present a different picture. Once again, the Spacom 2 solution generates very poor results. The delta omega comparison in Figure 56 indicates that the actual perturbing forces produce easterly nodal regression. Conversely, the solution and the simplified solution predict westerly nodal regression. It is obvious that other perturbing forces 38 influence the nodal regression of this satellite. Although the Spacom I solution is superior, even this accurate numerical solution has trouble predicting the value of a . The solution and the simplified solution also produce poor results in the delta inclination comparison in Figure 57. All solutions, except for the Spacom 1 solution, produce identical results. Again, on a larger scale, all of the solutions would seem nearly identical. However, this detailed analysis does demonstrate a weakness in the theory. The delta theta comparison in Figure 48 indicates that the solution and the simplified solution are inferior to all solutions including the two-body solution. Clearly, other perturbing forces are at work. The relative error comparisons are shown in Figures 59 and 60. The delta theta relative error comparison simply repeats the results found in the delta theta comparison. However, the delta radius relative error comparison is much more reassuring. Again, in global terms, the solution and the simplified solution produce better results than the two-body solution. The track error comparisons in Figures 61 - 63 produce mixed results. The radial track error comparison indicates that initially the Spacom I solution is inferior to all other solutions. However, after 21 days of motion, Spacom 1 is the superior solution. Once again, the radial track errors are small when compared to the along and cross track errors. The clue to the favorable global results of the solution and the simplified solution is found in the along and cross track error comparisons. The solution and the simplified solution perform much better than the two-body solution in the along track error comparison. Although the two-body solution is superior to the solution and the simplified solution in the cross track error comparison, the difference is small. It is clear that the solution and the simplified solution are superior to the two-body solution due to a much smaller along track error. In summary, although the solution and the simplified solution are superior to the two-body solution, other perturbing forces greatly influence the satellite's motion. At this altitude, the solution and the simplified solution simply do not model the satellite's motion well. Other perturbing forces must be modeled at this altitude if proper satellite position prediction is desired. 39 VI. CONCLUSIONS AND RECOMMENDATIONS An analysis was -conducted on a perturbation solution of the main problem in artificial satellite theory. The purpose of the analysis was to compare the solution with proven numerical solutions and actual measured satellite data in order to determine if the theoretical work is valid and practical. From the analysis, the-following conclusions can be made. 1. The solution and the simplified solution are both significantly more accurate than the two-body solution. The relative error of the two-body solution is on the order of JO while the relative error of the solution and the simplified solution is on the order of AG. 2. The real physical effects of the orbit are easily distinguishable in both the solution and the simplified solution. 3. The solution and the simplified solution compare extremely well with a proven numerical solution for at least 41 revolutions with a relative error on the order of j20. 4. The solution and the simplified solution compare extremely well with actual measured satellite data for at least 297 revolutions at altitudes where the -2 perturbation dominates ( e.g., near Earth orbits ). For a satellite in orbit at an altitude of around 1000 kilometers, the solution and the simplified solution reduce the error of the two-body solution by approximately'95%. 5. The solution and the simplified solution compare less favorably with actual measured- satellite data at semisynchronous and geosynchronous altitudes. At these altitudes, however, the solution and the simplified solution reduce the error of the two-body solution by at least 50%. 6. The solution and the simplified solution are free of singularities and are valid for all orbital parameters. Clearly, the solution and the simplified solution model the J2 perturbation very well. The equations are easy to implement and can provide quick and accurate predictions of satellite motion. However, other types of analytical solutions exist that are more accurate than the solutions described here. One such solution was developed by Coffey and Alfriend [Ref. 10] through research that was conducted by Dep.'. [Ref. II, Coffcy and Deprit [Ref. 12], and Alfriend and Coffcy [Ref. 13] . The solution is called the Analitic Orbit Prediction Program generator or (AOPP). Although the program is very accurate, AOPP exten- 40 sively utilizes four different Hamiltonian transformations. As a result, the real physical effects of the orbit are not easily distinguishable. The beauty of the solution and the simplified solution is their similarity in form to the well known two-body solution and the fact that a satellite's position can be easily predicted by evaluating only one integral. Once 0 has been determined, all other orbital parameters can be calculated easily. The structure of the solution and the simplified solution is ideal for implementation with onboard spacecraft computers. Before the solutions can be adapted for practical applications, more examination and testing of the theory is required. In order to provide more confidence in the theory, the following recommendations are suggested. 1. The solution and the simplified solution need to be compared-to a numerical integration of the equations of motion for at least 100 revolutions to confirm the theoretical accuracy for long term satellite motion. 2. The solution and the simplified solution need to be compared to several more diverse sets of actual measured satellite data. 3. To increase precision, the solution needs to include the higher order zonal harmonics of the gravitational potential ( e.g., J3 , , J, etc. ). 4. For spacecraft computer implementation, the Lagrangian coefficients of the state transition matrix need to be determined. For onboard spacecraft navigation, computers make use of the state transition matrix. Currently the Lagrangian coefficients of the two-body solution are the only matrix elements that have been determined. An excellent formulation of the two-body state transition matrix is shown by Battin [Ref. 141. Once the Lagrangian coefficients of the solution are developed, onboard spacecraft navigation can be greatly improved. 41 APPENDIX A. NUMERICAL SOLUTION COMPARISON RESULTS 42 0i 0 0 II 0 ry0 00 0 f 0 C) 0 U c0M 00 < 0 0 n- LL -~ a-0 Q) -. 0 Ei E 0 00 0 -5 C) 0 - 0) 0 0 0 0 CD 0 0 ~~ 0443 C) 0O-1 00 L - C\J 0 (N 0 0 - 0 0 -06 ry 00 0 0 n0 0 -y 0 01 0~ Ei E ED 0 -5 0 0 0 -5 0 LI) 00 MWA.-' a 0.~44 i3simV c0 0 > 0(D (0 0 D 00 0 F V) < E 0 00 + 0 C/) En 0 LiU C -] E 0 aZ tj c zj INA i -4 ~ ~ O0 6to( JI3sim ~ 45 i u V - 0 C: (6 00 0) 00 < 0 C)' 00 00 -ij 0 0 0 D 5 0 DD 0 00 ET 0 0 0 0 z 05 0 0Z 0)w0 c0 0 1 0 SIRJ30 : JV o -05to 3 OV 46r O0 C4 C"J C' N 0 C6. 0 (6 0 0 20 0 C) 0( T- o V) L 0 - CDi CL 0 9D 0 00 0 -~ cv 0 00 0 -~ C-- 0 lii5 - 0 0 S 0(1OJ'TOVOJ 47. C (0 0( (0 00 < Q)0 C , 0 -j 4- Z) 0 z 00 w F c C I) Z _f 0o 0 0 S33JO 'dV' 31DNV OdV 48 01 0 0 L0 -0 00 cn 0 0 < 4-j 0 0 co Lti _0 0 Q) 0 D0 z 0 > 0 0 CD 0 =m LIJ Ii 0 0I II0 I c 0 499 00 (0~ (0 00 4-6 0( 0 0 00> 0- 00 = COf U 00_ C) 0 o 0 F ) CC Cso 0( 0 03OV3 Z3J3 0 03O 0 04' 0 0 0l CC 0 0 -0 0 C) LLJ C (1) -0 0 CD 0t M0 0 * 0 z(0 .e4 0 - F-2 cli (~) 0 0j0 00 00 0 E -(v 0- 00 0! 0 C) 03 z5 0 00 -OVMIV3 0 0 - pI ....... 0 0 (0 U)) 1111 00 0 zco 0 00 Q 0 0 0 S31JK o5 10NOII~nON VD3 C4 CN 0 (N6 0 0 -Fo o H-L 1-0 1- -6 0 0 0n C - D 00 0 0 0 0 4 0 LrU- cY 6 L lIJO-K QV3H ~~jj53 a-o -z( Hi r 46c V 0 0 00 V) 0 (NJ 0 f 0 - o 0 0) < 0 0o -- oo~- 0 00 Cr5U li 0 000 -R I, 0q 'Z")3 0 J93CIHiG i]HiVD0 540 0 4 00 00 0 -0 0 (0 z C) (0 L _ oii 00 (n0 03-0 00 03aO3 ([350 H0 1HiV- 0 0 0 eC*i Z) 0 0 Iy 0 y0 0 I Ii 0 4 0 11101 I~j... 0 ' _ _ _ _ _ _ _ _ 0 1j~j I F-- { I _. c~04 0 0 U)N 0 00 LjJ-4 0 0 irx - @0 0 I <L IL 0 H-4 1 H- 0 I0 I t ) 0 -57 0 03)0 CC 0 C I 00 I0 03I < (0 z~ M~ 00~ <c oE U 0 0 0 D 0 0 o6 0 09-OL K-58 r 6 SS NOIN3VIG i~d 6 4 J 6 J EIAU c 30 W ~ C04 a 0 0 6i a 0 r0) 00 0 w V) a 0 0 - an 0 oC) LL a E~ 0~ 0 CO CO ' 00 0 0 a ti 6 1n CD 0 OSNVI 00 Z 4.J 4J a da J 3 AI a9 04 c~cz4 00 Cf) 0 00 0- 0 0 o0~ 0 0 <o 0 7_5 CI) E 0 0 0 czs 5;QL zO 5 -L i 'i to CAd S3NOLiM _ u 6 t(~ &J I6 AUU C60 000 0 It 0 n,0It 0( > oy 00 E LL 0 0 eI V) K I Cf)f 0 (6" tr r) I- Cf) Diui 0 I C5 SSKOS3IG K6 ~dKJ AVI 0 0 0 0 LIL- E 00 _y 0 0 _ 000 0) a 004-0 0 00 0o 0 I-- L6 C-)0 6 os O Q) 00 62 t 0r 14 00 C5J 00 C-'0 00 0 oy 0c) 0 CCD _q 20 v ry 0 I-LID 0 t-0 0 (I) D 01 ' _ij, 4 VI C5- C) 0 C5 1 0 jlAvj 6i IV\ 630 0f - 00 0 6, rCD 4-0 00 (9) C 0. o V0 0 -6-4 to D 3AO l-VO 4 0 0 T-c.4 0 0 0 0 00 -00 ( 0 - < C.) 0 LlO 0 0 0- 0 0 Z o) -6 0 z ~ 00 a _00 09 0 (fCJu Z 0 C;. to @0 _ _ _ LO _ _ INA o 0 13X)V 3iV 65 0 iONO WJ -- 00 0 0 0 (.0 0:3 o r(' C7 0- L6 /) EI I 00 o 0z cs wq 31 vi AZMi!NM 66D jMk 00 CN 00 D( 0- (D(0 LLJC.L 0O cf) U) -J fy _1 0 (96 00 0 0 o 0 C4 0 (f) - 7co Q) 0 06 4(6 C 0l0C 0cl VIA F 67 0V 1V JONO O&E C C14 0 0\ 0-0 0 0 Cr) CO" 0 0 Li E 0~0 o) C 0 o (0 0 00 -~ C) - z >6 00 C." CCfM 0 6 INA -~68 J& AViSO 05 011 0 ') 0 c6 ((0 ry 0( 0 Cl, : 0 0 0 -1- 4-a 0 Lli zJ CD 0 0 0 0( c-f)) 0 0 Li~i 313~- DV 69~ S8 0JJ1 APPENDIX B. NEAR EARTH ORBIT COMPARISON RESULTS '"0 0 L6 1 0 ci0) 00 0 ~0< 0 ~0: ry ci F Z)W 0 0 E -6 o o 0 o Id 0§(fU 0 -(6 LU z E- o -0 LU o- o 0 Q00 0 0 0 0 0 ii ii 0 0 0 0 LO LO 00 LO LOr It rc*N VqA JO= 0 J(J 0 0 0 0 0 LO Sniv~j 0 iii0 712 CD L 0 :3 0 0 n C() V) 0 - 4 -J C 0 M 00 0 C0 0 0 00 0 C)0 6 6 0 0' ' 0 a S33 JV3 0 J)30ON 720 0 0 DO 0 0 0. U 0 0 0f 00 o0 0 c) 0 D 0 - o 2C iO L z 00> o- 0 0l ~0> cl 0 0) 00 @0 0 I 01 0 0 0 6i 00 0 ui S13djo-3K i 6 N'OG' VOIINO VD-lG 73 0 0 65 0 C4 -0 06 0 0 00 (0 0 0 0 7-0 -0 < 'J E 0) OD 00 D o) 0- 5c 0 0 CL 0 N (f 0 0 0 0 00- (fl f)U) V3qOV3 74 rd) CNJ 0 .00 0) 0 *-p 0 -5(nU Z LLJO- 0 M0 0 0 0 0 u~ oLUJ 0: <0 C6 n E 0 (0 00 S3J3 0G'NUMOIV3 cJ 0 c6 00 U)0 00 0 LO Q) 0 a)0 o C) 0 2 00U i C~ 0 0i LLJ wu 0 00 0) e0 0 U)i5 2V)U 0 0 xi04 766 0~ 0 06 04 0 cc0r0 _: _0 0 JDD liD QCU '0E0 > -0 0 0c~ 0- 11 1 E F 0 -41 0 0 0 -) Ci LLLU 0 0.. j E (5 u o * I, 0' N Lo (0 9-01* SS3-~~INj S3I( L77 0 t to -tl ' * fl) id 0, , t0 - N ) 1 t0, d AVI 0 0 0 , 0 -) 0 0 0I o - I f 0 W 0 00 o (_ o- c .f) Li 00 IfV) (o C5 LO 6 ' J~d S=NOSN~rlG 66 10& 78i 3AU 3o Ei 0 0 0 cno E 0 0 rU) 0 0 oy Li 0 C'i -oo Q0 0 _0. 0 i 0 000 0 01 0 cyi (6 c ri r--4 IAJA' JMJ )C ri '_))IJ 3 A)V' J Wc.'J 79S o C\lj 0 c6 0 EO 00 0 0 0oy c Q) 00 0 asi F - 0 0 0 - 00 cv) 0 0-I U)-- -- oo C 0 LLJ 00 cC 4 0 0 0 :3 (f0) 0 0 0080 0 )) 0 C6 0 00 q 0 Li CC < 0- V)) o o(0 ' E 0 C-)) _00 o 0i 0 0(/ CI 0 (r- 0 04 0 to LO v o o~ loc Z ok- IV\is si0 i 0 0). 0 -0 0 co 0 00 0 0( o o0o 0 Q)' ryU) V() Cfl (J)) 0Li CCf(f o) C 0 0 _ Z (0 -Q - 0 0 LLJ. 0 Cfl~)(f)a 0I NIA II II II '31:DDV JiSSM. JMRG 02 APPENDIX C. SEMISYNCHRONOUS ORBIT COMPARISON RESULTS 83 0 (6 C\J tC"4 0 I- 06 C~) <0 c LLJ c .2 - .2 IU) 0 CL C>\4 0 a 0 to 0 0 00 LO 0 oL S44 C) -6- -6 C',' 0( L(6 o ocl (. -j- 0 lit:3 0< 0 U) o 0 Tf~l-, (DC L 0 0 E-N FV) U) o5 S3 13 O-V'=VOJ 85a - 0 0 04 (00 C) -0 0 0 0 UJ M -'N 00 0 0 GDD 00 0 0< 02l 0 77 wI z 86 --- ----- C 0 0 0 0 C:N 00 c(NJ L0 0 0 0 C)C o) 0 cJ - E0~ Z0 U) 0 C 0 -;- wc -0 zI (f L.l E ZJ M~> 0 00 0 W~~ 0 0 -r o o 05 05 0 0 0 C0 0 0 00 6 65 65 6 6 00 0 Sld] 530 G 0 NOELVflNI VPI3 G 87 0 0 0 00 0 0 E 2 0 -0 ~~0 0 LLJ C' 0 c. 0 0 D 0 :3 / LU CJ40JQ0 0 liiLI I *0 0 0 00 6 0 S31JO3G HIG' VI]Hi VIDh 88 6 6 -0 0 00 Li 0 CN E (fC( EU 0= 0 0 C6 O (T)I-f) > 1--E W 0 11 11 1111 U *0 >< 0 0~ 4 6 O rv $LJ w H- a c'o 0 a0 Ld C5 C~ci SS" OIS3NIGHi~' C5 ( O&E 89i AII 3o 0 T 0 0 0 (6 LC"J nn1 C-)N <0 ry .2 0 0: :3 0 w n ~0 0 Q) 0 C,) _w (I) 0 -LW a0 00 C) o aS zKd88,33U OINM 09 C) 0 00C- C) c 0 0 -06 CJ .00 C (n 0~ 00 (/) -- 0 8 ~ l ~ I 0f c'0 C(D <)- ( < c o 291 _ 0 00 01 C 00 3-1- 00 CQ) -(D- CDl 0 0-- :3 0 Z 0 LU 0 ~ U0 0 Q EJ4 80 01 0 '4 0 0 to0 to0L' VqA~ 31V JOJd3 >13VdJi ONO1V 92 0 0 (0 0 0 0 o 0 0 0 0 (If) (-I- 00 0c. Q o 0-:3 0 . . .. ZC CIJC() .. . (f.. .. . . 000 0 0 0 0 ( U) 0 0 *0 tot Ic NA J&E 30 A)V~l 93~ S ~ 0 APPENDIX D. GEOSYNCHRONOUS ORBIT COMPARISON RESULTS 94 0 00 0 (0 c"J CJ 0 ry o 00 a Lo 0 0 - :D < <0 L 0-2 0 -C- c c.20 Lii .2~~J _00 0~~)f 0 0 a00 in i 0 S 0 0 0 C 0 0 0 0 0 0 0 0000 0 0Y) 03 Nl- to t *z:- V) Adf'MG' dJOIJ-A SfliGmvd 95 aIU 0 (0 C14 0 (5 (6 E 0 0 ai 0- 0 .2 Co o o&- o 0- (n (n p. V) 00 )- *r l r- to 1 - S33d93G' Oiv' 3J1DNV JdV 96 n i 'N 00 0( U) 00 <( tj0 0 0 2 0 -4 0-0 0t 0 UJ 0 0~~ o U)) 0 2U)U 0_______ 0T 0 00510 ~ S33 K)30 97CF-(U oNGV3 -0 C~4 0( 0~ 00 C', 0. W 00 cj <- 00 4.3 o T7c zz 0 0 >. r- (DL)) wi M C 0 0 LO l~ (f) c 0 (.5JOI wOL~ ) 0 C.\j 0 C 0 NOI~nN QLL r r~ V 08 -0 00 ul0 L- 0 00 o cs 00 0 06 U)' k20 L.L o L -0 0- 0 D:3 cCj Wi 0 0 0- C0 tO tO 0 LO 4- vi* 0D tfl LO 0 C 0 S33JOIIG ' HIG VISHi VI130 99 U 0 0 O0D 0 (0 0 00 Li wY eL 0 0 (0N Q- wC6 00 > 1 0 LU o 00 0 ~0 0 0 0 60 0 00 00 r'< c6 Li SSI1NOISN]V'IG' HI~cI 100 4 r'i (\i6 dJOdd 3AIIV13d E (0 CN E0 o 0 0 Ey w-Q 0 2 0 U)) C)) I < 0: Lio 2. 0 0 Q /)co c6 -J 0 0 00 .0 If) 0~ 0) 0~ 6600 If)o 0 SS31NOISNVJII 0 0 dJOd'K& dd3 101 AUV]B a a 0 0 CC 0~ C(n -0,-0 ry0 00 0 ry0 w 0L ) 0 (n~ 01!) aU 0 0 0 0 100 102 0 / 00 0 - 0 bJ C,) ccO* i 000"' . -5 nC)0 Li )4- 0 0N 0 103 00 (0 -5 1- 4 0 ) 1- 0 0- 01C)(n 0 0 0 00 00 co 0 (0 C4 0 U) ' 00 0 L 0 r-a uU w:0. 00 0Is) 0L Z 0 000 (f)U~)IIlelf) C) ~4 0 104 APPENDIX E.COMPUTER PRO6iRAM 105 PROGRAM COL02 C o C -C * * MAIN PROGRAM -C * C -C IMPLICIT DOUBLE PRECISION (A-I,M-Z) CHARACTER*20 LINE DIMENSION M(l00),MD(l00),E(' 00),W(1O0),WD(10O),OM(100),OMD(100) DIMENSION 1(100),ID(100),F(10),FD(O0),EC(100),ECD(100),A(100) DIMENSION R(100) ,H(100) ,N(100) ,TH( 100) ,THD( 100) DIMENSION RF(l00) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100),THF(100) DIMENSION THFD( 100) ,P(lo) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD(.100) DIMENSION DOMD(100) ,RX(100) ,RY(100),RZ(100) ,RFX(-100) ,RFY(1i00) DIMENSION RFZ(100) ,DRV(100) ,ARC(100) ,ARCD(l00),DAY(100) ,HX(100) DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100) DIMENSION NY(100),NZ(100) ,RDV(100),EX(100),EY(100) ,EZ(100) DIMENSION NDE(100),EDR(100),V(100),HT(100) ,RDRF(100) ,INTA(100) DIMENSION ROMA(iCO) ,TH~'o(100) ,ATE(l00) ,CTE(100) COMMON/OBLATE1/DAY,RXRY,RZ ,VX,VY,VZ,DT,HX,HY,HZ,Nx,NY,NZ,K,KX COMMON/OBLATE2/RDV,R,V, FX,F,\,EZ,IU,NDE,EDR,H,N,E,P,I,OM,W,F COMMON/OBLATE3/PI,EC,l,,Ht,'I,ER,TH,THD,RTD,MD,WD,OM"D,ID,ECD COMMON/OBLATE4/FD,LINE, J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM COMMON/OBLATE5/RJ,DR,DID,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER COMMON/OBLATE6/RFX, RFY ,RFZ ,ARC, ARCD , DRF,DRV,RJ2 , JN, JORBIT COMMON/OBLATE7/ INTA ,ROMA,THJO,AT, CTE C 10 C 20 PRINT--,'ENTER VERSION ( SOLUTION = 1, SIlf'LB READ* ,JVER IF(JVER. EQ. 1.OR. JVER. EQ. 2.OR.JVER. EQ. 3)THEN GOTO 20 ELSE GOTO 10 ENDIF = 2, TWO BODY PRINT*,'ENTER FIRST POINT' READ,K PRINT--e,'ENTER FINAL POINT' READ*,KK C PRINT", 'ENTER RJ2' READ*,RJ2 IF(RJ2. EQ. 1.ODO)TiHEN RJ2=0. 00108263D0 ENDIF C LINE- - - - - - - - - - PI=3. 141592653589793238462643D0 RTD=180. ODO/PI ER=6378. 137D0 HTS=1. 0iO/806. 812D0 MU=3. 986004 36D5 C OPEN(UNIT=2, STATUS='OLD', FILE='/COLO2 OUT A') 106 =3 ) OPEN(UNIT=3, STATUS=&OLD', FILE='/COLO2 DS8 A') OPENf-UNIT-=4, STATUS='OLD', FILE='/COLO2O DSS B') C CALL DATA CALL ELEMENTS CALL PINITIAL C J=1 WRITE(6,6000) WRITE(6,6000) WRITE(6,6000) WRITE(6,6000) 'POINT = , 'INTEGRATE COMPLETED' 'FORMULA COMPLETED' 'INERTIAL COMPLETED' C 30 DO 30 J = K, KK WRITE(6,6000) 'POINT =', CALL INTEGRATE WRITE(6,6000) 'INTEGRATE COMPLETED' CALL FORMULA WRITE(6,6000) 'FORMULA COMPLETED' CALL INERTIAL WRITE(6,6000) 'INERTIAL COMPLETED' CONTINUE CALL RESULTS C CLOSE(UNIT=-2) CLOSE(CUNIT=-3) CLOSEC UNIT-=4) C 6000 C C C C C C C FORMAT(3X,A,13) STOP END * SUBROUTINE DATA SUBROUTINE DATA IMPLICIT DOUBLE PRECISION (A-I,M-Z) CHARACTER*20 LINE DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(100) DIMENSION I(100),IDC100),F(100),FD(100),EC(100),ECD(100),A(100) DIMENSION R(100) ,H(100) ,N(100) ,TH(100),THD(100) DIMENSION RF(100),IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THF(100) DIMENSION ThFPD(100),P(100) ,JORBIT(100) ,DR(-100),DID(100) ,DTHID(100) DIMENSION DOMD(100) ,RX(100),RY(100) ,RZ(100) ,RFX(100) ,RFY(100) DIMENSION RFZ(100) ,DRV(100),ARC(100) ,ARCD(100) ,DAY(100),HX(100) DIMENSION HYC100),HZ(100),VX(l00),VY(100),VZ(100),DT(10o),NX(100) DIMENSION NY(100),NZC100),RDV(J.Oo),EX(100),EY(100),Ez(1o0) DIMENSION NDE(100) ,EDR(100) ,V(100),HT(100),RDRF(100) ,INTA(100) DIMENSION MONTI( 100) ,DATE( 100) ,IOUR( 100) ,MIN( 100),SEC( 100) DIMENSION ROIA( 100) ,TJ{JO( 100) ,ATiEC100) ,CTE( 100) COXMJ ON/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,IIX,H[Y,HZ,NX,NY,NZ,K,KK COMMON/OBLATE2/RDV,R,V,EX,EYv,EZ,M1U,NDE,EDR,H,N,E,P,I,OM,W,F COMMON/OBL.ATE3/PI,EC,M,A,HT,ER,Th-I,THID,RTD,M]D,WD,OMID,ID,ECD 107 COMMON/OBLATE4/FD,LINE,J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM COMMiON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER COMMON/OBLATE6/RFX,RFY,RFZ ,ARC,ARCD,RDRF,DRV,RJ2,JN,JORBIT COMMQN/OBLATE7/INTA,ROMA,THJO,ATE,CTE C o C READ IN EMPHERIS DATA OPEN(UNIT=1I, STATUS='OLD', FILE=t /COLO2 DAT', ACTION='READ') DO 10 J1= 1, KK READ(1,*) MONTH(J) ,DATE(J) ,HOUR(J) ,MIN(J) ,SEC(J) 10 C C C 20 C RX(J)=RX(J)/1000. ODO RY(J)=RY(J)/1000. 0DO RZ(J)=RZ(J)/1000. ODO READ(1,*) VX(J) ,VY(J) ,VZ(J) VX(J)=VX(J)/ 1000. 0DO VY(J)=VY(J)/ 1000. ODO VZ(J)=VZ(J)/1000. ODO CONTINUE CONVERT PARAMETERS DO 20 J = 1, KK DAY(J)=DATE(J)+((3600. ODO*HOUR(J)+ + (60. ODO*MIN(J)+SEC(J))))/86400. ODO DT(J)=(DAY(J)-DAY( 1))'*24. ODO*3600. ODO CONTINUE CLOSE(UNIT=1l) C RETURN END C C C C * SUBROUTINE ELEMENTS C SUBROUTINE ELEMENTS IMPLICIT DOUBLE PRECISION (A-I,M-Z) CHARACTER*20 LINE DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(loo) DIMENSION I(100),ID(100),F(100),FD(100),EC(l00),ECD(100),A(loo) DIMENSION R(100),H(100),N(100),TH(100),THD(100) DIMENSION RF( 100) ,IF(100),IFD( 100) ,OMF( 100) ,OMFD( 100) ,THF( 100) DIMENSION THFD(100) ,PC100) ,JORBIT(100),DR(100) ,DID(100) ,DTHD(100) DIMENSION DOMID(100),RX(100) ,RY(100),RZ(100) ,RFX(100),RFY(100) DIMENSION~ RFZ( 100) ,DRY( 100) ,ARC( 100) ,ARCD( 100) ,DAY( 100) ,HX( 100) DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(1oo),NX(100) DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(.Oo),EZ(100) DIMENSION NDE(100),EDR(100) ,V(100) ,HT(100) ,RDRF(100) ,INTA(100) DIMENSION ROMA(100) ,THJO(100) ,ATEC100) ,CTE(100) COMMON/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P, I,OM,W,F 108 COMMQN/OBLATE3/PI ,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD, ID,ECD COMMION/OBLATE4/FD,LINE,J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROI COMMON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER COMMON/OBLATE6/RFX,RFY,RFZ,ARC,ARCD,RDRF,DRV,RJ2,JN7,JORBIT COIIION/OBLATE7/INTA,ROMA,THJQ,ATE ,CTE C C C CALCULATE R,V,E,H,N,P,I,OM,W,F,EC,M,HT,TH DO 10 J = 1, KK CALL CROSS(RX(J) ,RY(J) ,RZ(J) ,VX(J) ,VY(J) ,VZ(J) ,HX(J) ,HY(J), + HZ(J)) CALL CROSS(O. ODO,O. ODO,1. ODO,HX(J),HY(J),HZ(J),NX(J),NY(J), + NZ(J)) CALL DOT(RX(J),RY(J),RZ(J),VX(J),VY(J),VZ(J),RDV(J)) R(J)=DSQRT(RX(J)*RX(J)+RYCJ)*RYCJ)+RZ(J)*RZ(J)) V(J)=DSQRT(VX(J)*VX(J)+VY(J)*VY(J)+VZ(J)*VZ(J)) EY(J)=( (V(J)*V(J)-MU/R(J))*RY(J) -RDV(J)*VYCJ) )/MIJ EZ(J)=((V(J)*V(J)-iU/R(J))*RZ(J)-RDV(J)*VZ(J))/MU CALL DO(N(J),N(J),NZ/(J),EXZ(J),EY(J)EZ(J)D)) CALL DOT(EX(J),EYZ(J),EZ(J),RX(J),RY(J),RZ(J),NER(J)) H(J)=DSQRT(HXJ)*HX(J)+HY(J)*HY(J)+HZ(J)*HZ(J)) N(J)=DSQRT(NX(J)*NX(J)+NY(J)*NY(J)+NZCJ)*NZ(J)) E(J)=DSQRT(EX(J)*EX(J)+EY(J)*EY(J)+EZ(J)*EZ(J)) P(J)=H(J)*H(J)/MU I(J)=DACOS(HZ(J)/H(J)) QM(J)=DACOS(NX(J)/N(J)) W(J)=DACQS(NDE(J)/(N(J)*E(J))) F(J)=DACOS(EDR(J)/(E(J)*R(J))) IF(NY(J). LE.0. ODO)THEN ONf(J)=2. 0D0*PI-OM(J) ENDIF IF(EZ(J). LE. 0. ODO)THEN ENDIF IF(RDV(J). LE. 0.ODO)TIIEN F(J)=2. ODO*PI-F(J) ENDIF EC(J)=DACOS((E(J)+DCOS(F(J)))/(1. ODO+E(J)*DCOS(F(J)))) IF(F(J). GE. PI)THEN EC(J)=2. ODO*PI-EC(J) ENDIF M(J)=EC(J) -E(J)*-'DSIN(EC(J)) 20 AT=(MTJU/(N(1)*N( 1)) )**(1. ODOI 3.ODO) HT(J)=R(J)-ER. RJ=3. OD0*RJ2*ER*-ER/( 2.ODO*P( 1:P( 1)) TH( 5)=F( J)+W( J) THD( J)=TH( J)*RTD IF(THD(J).GT. 360. ODO)THEN TIID(J)=THD(J) -360. ODO GOTO 20 END IF THC J)=THD(5) /RTD C 109 C C CONVERT ORBITAL ELEMENTS TO DEGREES MD(J)=M(J)"RTD WD(J)=W(J)"RTD OMD( J)=OM(J)*RTD ID(J)=I(J)*RTD ECD(J)=EC(J)*RTD FD(J)=F(J)RTD THD(J)--TH(J)*RTD 10 CONTINUE RETURN END C C C C C C C C C C A * * SUBROUTINE CROSS * SUBROUTINE CROSS(AX,AY,AZ,BX,BY,BZ,CX,CY,CZ) IhPLICIT DOUBLE PRECISION (A-I,M-Z) CALCULATE THE CROSS PRODUCT OF 'WO VECTORS A AND B CX=AY-*BZ-AZ*BY CY=AZ*BX-AX*:BZ CZ=AX*BY-AY*BX C RETURN END C C C C * SUBROUTINE DOT C SUBROUTINE DOT(AX,AY,AZ,BX,BY,BZ,ADB) IMPLICIT DOUBLE PRECISION (A-I,M-Z) C C C CALCULATE THE DOT PRODUCT OF TWO VECTORS A AND B ADB=AXC*BX+AY--BY+AZ*BZ C RETURN END C C C - C C SUBROUTINE PINITIAL ** C C SUBROUTINE PINITIAL IMPLICIT DOUBLE PRECISION (A-I,M-Z) CIIARACTER*:20 LINE 110 DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(100) DIMENSION I(100),ID(100),F(100),FD(100),EC(100),ECD(100),A(100) DIMENSION R(100),H(100),N(100),TH(100),THD(100) DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THF(100) DIMENSION THFD(100) ,P(100) ,JORBIT(100) ,DR(100) ,DID(100),-DTHD(100) DIMENSION DOMID(100),RX(100),RY(100),RZ(100),RFX(100),RFY(100) DIMENSION RFZ(100),DRV( 100) ,ARC( 100) ,ARCD( 100) ,DAY( 100) ,HX( 100) DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100) DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100) DIMENSION NDE(100) ,EDR(100) ,V(100) ,HT(100) ,RDRF(100) ,INTA(100) DIMENSION ROMA( 100) ,THJO( 100) ,ATE( 100) ,CTE( 100) COMMION/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK COMMfON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P, I,OM1,W,F COMMON/OBLATE3/PI ,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD,ID,ECD COMMON/OBLATE4/FD ,LINE, J ,THF ,THFD,IF,IFD ,OMF ,OMFD ,RF, INT ,ROM COMMON/OBLATE5/RJ ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER COMMON/OBLATE6/RFX,RFY,RFZ ,ARC ,ARCD,RDRF,DRV,RJ2,JN,JORBIT COMMON/OBLATE7/INTA,ROMA ,THJO,ATE ,GTE C C C PRINT INITIAL ORBITAL ELEMENTS + + + + + + + + C 6000 6100 6200 C WRITE(6,'(/)') WRITE(2,'(/)') WRITE(6,6000) 'ORBITAL ELEMENTS' WRITE(2,6000) 'ORBITAL ELEMENTS' WRITE(6,6100) LINE WRITE(2,6100) LINE WRITE(6,6200) :M = ',MD(1),'N = ',N(1),'EI ',= ) 'w = 'WD(l), OM = 'OMD(1),' , D=) 'EC = ,ECD(l),'A =,IA(1), IR = Rl) 'HT = 'HT(l),'H = ',H(1),'F = 'F~) 'TH = ,TIID(1) WRITE(2,6200) 'M = ',MD(1),'N = ',N(1),'E = ,l) 1w = IWD(l),lOM = 'OMD(l),' =i II~) 'EC = ',ECD(l),'A =',A(l), 'R = ,~) 'HT = ,HT(l),'H = ',H(.1),'F FDl) 'TH = 'THD(l) FORMAT(3X,A) FORMAT(3X,A20/) FORMAT(13(3X,A5,D18. 10/)!) RETURN END C C C C o C C * * * SUBROUTINE INTEGRATE SUBROUTINE INTEGRATE IMPLICIT DOUBLE PRECISION (A-I,M-Z) CHARACTER*20 LINE DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(100) DIMENSION I(100),ID(100),F(100),FD(100),EC(100),ECD(100),A(100) III DIMENSION R(100) ,H(100) ,N(100) ,TH( 100) ,THD( 100) DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMfF(100),OMFD(100) ,THF(100) DIMENSION THFD(.'0),P(100),JORBIT(100),DR(100),DID(100),DTHD(100) DIMENSION DOME '0),RX( 100) ,RY( 100) ,RZ( 100) ,RFX( 100) ,RFY( 100) DIMENSION RFZ(O ),DRV(100),ARC(100),ARCD(100),DAY(100),HX(100) DIMENSION HY(10,j),HZ(100),VX(100),VY(100),VZ(10Q),DT(100),NX(100) DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100) DIMENSION NDE(100),EDR(100),V(100),HT(100),RDRF(100),INTA(100) DIMENSION ROMAC 100) ,THJO( 100) ,ATE( 100) ,CTE( 100) COIMON/OBLATE1/DAY,RX,RY,RZ,'YX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K5 KK COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P, I,0MW,F COMMON/OBLATE3/PI,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD,ID,ECD COMMON/OBLATE4/FD,LINE,J,THF,THFD,IF,IFD,OMIF,OMFD,RF,INT,ROM COMMION/OBLATE5/RJ,DR,DID ,DTHD ,DOMD ,ESTERR ,ACTERR ,TERROR, JVER COMMION/OBLATE6/RFX ,RFY ,RFZ ,ARC ,ARCD ,RD,DRV,RJ2,JN, JORB IT COMMON/OBLATE7/INTA,ROMA,THJO,ATE,CTE C C C EQUATE INITIAL VALUES THF( 1)=TH( 1) THFD( 1)=THD( 1) IF(1)=I( 1) IFD( 1)=ID( 1) OMF( 1)=OM( 1) OMFD(1)=OMD(1) RF(1)=R(1) C C C ESTIMATE UPPER BOUND OF THETA THF(J)=TH(J)+(J-1)*0. 57D0*2. ODO*PI C C C C C C 10 C 30 20 ROM00080 INITIALIZE (1X1O)-12 ESTERR=0. 00000000000 iDO INT=DT(J)*H( 1) DTHF=0. 1745329251994 (1X1O)-10 TERROR=0. 000000000 iDO CALL ROMBERG ACTERR=-INT-ROM IF(ACTERR. LT.0. ODO)THEN THF(J)=THF(J) -DTHF GOTO 10 END IF TEMPTHF=-THF( J) GOTO 20 ROM00190 CALL ROMBERG ACTERR=INT-ROM IF(ACTERR. GE. 0. ODO)THEN IF((ACTERR/INT). LE. ESTERR)THEN GOTO 40 ELSE TEMPTHF=-THF( J) 112 THF( J) THF( J)+DTHF GOTO 30 ENDIF ELSE DTHF=DTHF/2. ODO THF( J)TEMPTHF+DTHF GOTO 30 END IF C 40 INTA(J)=INT ROMA(-J)=ROM RETURN END C o o C o SUBROUTINE ROMBERG* C * C C SUBROUTINE ROMBERG ROM00190 IMPLICIT DOUBLE PRECISION (A-I,Mi-Z) CHARACTER*20 LINE DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM1(100),OMD(100) DIMENSION I(100),ID(100),F(100),,FD(100),EC(100),ECD(100),A(100) DIMENSION R(1OO),H(1O0),N(100),T{ 100),THD(10O) DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THF(100) DIMENSION THFD( 100),P( 100) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD( 100) DIMENSION DOMD(100),RX(100),RY(100),RZ(100),RFX(100),RFY(100) DIMENSION RFZ( 100) ,DRV( 100) ,ARC( 100) ,ARCD( 100),DAY( 100) ,HX( 100) DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100) DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100) DIMENSION NDE( 100) ,EDR( 100)-,V(100) ,HT( 100) ,RDRF( 100) ,INTA( 100) DIMENSION ROMAC 100) ,THJO( 100) ,ATE( 100) ,CTE( 100) COMMION/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK COMMON/OBLATE2/RDV,R,V,EX,EY.,EZ,MU,NDE,EDR,H,N,E,P,I,OM,Wi,F COMMON/OBLATE3/PI ,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD, ID,ECD COMMON/OBLATE4/FD,LINE,J,THF,THFD,IF,IFD,OMIF,OMFD,RF,INT,ROM COMMON/OBLATE5 /RJ ,DR ,DID, DTHD ,DOMD ,ESTERR ,ACTERR ,TERROR, JVER COMIMON/OBLATE6/RFX ,RFY,RFZ ,ARC, AROD ,RDRF ,DRV ,RJ2 ,JN, JORB IT COMMON/OBLATE7/INTA,ROMA,THJO,ATE ,CTE C C C C C C C EXTERNAL FUNC R0M00460 INITIALIZE VARIABLES R0M00560 HS=THF(J) -THF( 1) FUNCA=FUNC(RJ,A( 1),1(1) ,E(1) ,W(1) ,TH( 1),THF(1) ,JVER) FUNCB=FUNC(RJ,A( 1),1(1) ,E(1) ,W(1) ,TH(1) ,THF(J) ,JVER) P(1)=HS*(FUNCA+FUNCB)/2. ODO JM=1 ROM005 70 BEGIN THE ROMBERG LOOP. ROM005 80 R0M00590 ROM00610 ROM00620 DO 10 JN = 1, 100 OLDINT=P( 1) ROM00630 ROM00640 113 20 30 10 40 C C C* C HH=HS SS=O.ODO TT=-THF( 1)+HH/2. ODO DO 20 L = 1, JM T=Tr SS=SS+FUNC(RJ,A(l),I(1),E(i),W(l),TH(1),T,JVER) fTT-HHCONTINUE SUM=HH*SS P(JN-I1)=(P(JN)+SUM)/2. 0DO D=1. 0Db DO 30 JK = JN, 1, -1 D=4. 0D0*D P(JK)=P(JK+1)+(P(JK+1)-P(JK))/(D-.-ODO) CONTINUE -ERROR=(PC1) -OLDINT) IF(JN. GE. 10)THEN IF (DABS(ERROR/OLDINT). LB.TERROR)THEN GOTO 40 ENDIF ENDIF HS=HS/2. ODO JM=JM*2 CONTINUE ROM=P(1): RETURN END R0M00670 R0M00680 R0M00690 R0M00720 R0M00730 R0M00740 ROM00 750 R0M00760 R0M00780 ROM00800 R0M00830 R0M00840* R0M00850 ROIIO0860 R0M00870 ROM00900 R0M00930 R01100940 R0M00950 ROM009 60 R0M00970 FUNCTION FUNC C * C C FUNCTION FUNC(RJ,A1,11,E1,W1,TH1l,THFJ,JVER) IMPLICIT DOUBLE PRECISION (A-I,O-Z) C EXTERNAL RADIUS C S=DSIN( II) S2=S*S S4=52*52 56=S4*52 E2=E 1*E 1 C RAD=RADIUS(RJ,A1,11,E1,Wl,THI,THFJ,JVER) C C C F ( SOLUTION) IF(JVER. EQ. 1)THEN C Y1=112. ODO-75. ODO*S6+260. OD0*S4-296. ODO*S2 Y2=RJ*THFJ*(2. 5D0*S2-2. 0Db) Y3=2.ODO*Wl-Y2 Y4=24. ODO*(5. 0D0*S2-4.ODO)*(5. 0DO*S2-4. 0Db) Y5=E2*S2*( 14.0Db- 15. ODO*S2)*( 15. OD0*S2- 13. 0Db) 114 ROM00030 Y6=9. ODO*E2+34. ODO Y9=12. ODO*(5. ODO*S2-4. ODO) Y12=9. ODO*E2-34. ODO C YF=(2. 5DO*S2-2. 0DO)*(THFJ-TH1)+E2*Y1*DSIN(Y2)*DC0S(Y3)/Y4 C + + + + YS=Y5*DCOS( 2.ODO*W1)/(2. ODO*Y9)+ E1*S2*( 15. ODO*S2-13. ODO)*DCOS(TH1+W1)/2. ODO+ E1*S2*( 15. ODO*S2-13. ODO)*DCOS(3. ODO*TH1-W1)/6. ODO+ S2*( 15. ODO*S2-13. ODO)*DCOS(2. ODO*TH1)/2. ODO+ (5.ODO*Y6*S4+4. ODO*Y12*S2-56. ODO*E2)/96. ODO C Y=THFJ-Wl+RJ*YF+RJ*RJ*THFJ*YS C F=(2. ODO-3. ODO*S2)*DCOS(2. ODO*THFJ),/2. ODO+ El*(S2-1)*DCOS(Y)+E1*(3. QDO-4. ODO*S2)*DCOS(Y+2. ODO*THFJ)/6. ODO+ E1'(1. ODO-2. ODO*S2)*DCOS(Y-2. ODO*THFJ)/2. ODO+S2-1. ODO+ E2*S2*( 15. ODO*S2-14. ODO)*DSIN(Y2)*DSIN(Y3)/Y9+ S2*DCOS( 2.ODO*TH1)/2. ODO+E1*S2*DCOS(3. ODO*TH1-W1)/6. ODO+ + E1*S2*DCOS(TH1+W1)/2. ODO + + + + C END IF C o C F ( SIMPLIFIED SOLUTION) IF(JVER. EQ. 2)THEN CF=(2. ODO-3. ODO*S2)*DCOS(2. ODO*THFJ)/2. ODO+ +- E1*(S2-1)*DCOS(THFJ-W1)+ E1*(3. ODQ-4. ODO*S2)*DCOS(3. ODO*THFJ-Wl)/6. ODO+ + + El*(1. ODO-2. ODO*~S2)*DCOS(THFJ+W1)/2. ODO+S2-1. ODO+ S2*DCOS(2. ODO*TH1)/2. ODO+E1*S2*DCOS( 3.ODO*TH1-Wl)/6. ODO+ + +- E1*S2*DCOS(TH1+1?1)/2. ODO C ENDIF C C C F ( TWO BODY SOLUTION) IF(JVER. EQ. 3)THEN F=O. ODO ENDIF C C C FUNCTION FUNC=RAD*RAD*(1. ODO+RJ*F)- C RETURN END C C C o C C C FUNCTION RADIUS FUNCTION RADIUS(RJ,A1,I1,E1,W1,TI1,TFJ,JVER) 115 IMPLICIT DOUBLE PRECISION (A-I,O-Z) C C C CALCULATE E, SINE, AND COSINE FUNCTIONS S=DSIN(I1) S2=S*S S4=S2*S2 S6=S4*S2 C=DCOS( Il) C2=C*C SC=DSIN( I1)*DCOS( Il) E2=El*El PO=A1*(1. ODO-E2) C C C RADIUS BOTTOM ( SOLUTION) IF(JVER. EQ. 1)THEN C Y1=112. ODO-75. ODO*S6+260. ODO*S4-296. ODO*S2 Y2=RJ*TFJ*c(2:-5D0*52-2. ODO) Y3=2. ODO*W1-Y2 Y4=24. ODO*(5. ODO*S2-4. ODO)*(5. ODO*S2-4. ODO) Y5=E2*S2*( 14. ODO-iS. ODO*S2)*( 15. ODO*S2-13. ODO) Y6=9. ODO*E2+34. ODO Y9=12. ODO*(5. ODO*S2-4. ODO) C Y11=15. ODO*(2. ODO+E2)*S4-14. ODO*(4. ODO+E2)*S2+24. ODO Y12=9. ODO*E2-34. ODO YF=(2. 5DO*S2-2. ODO)*(TFJ-TH1)+E2*Y1*DSIN(Y2)*DCOS(Y3)/Y4 C + + + + yS=y5*DCOS( 2.ODO*W1)/( 2.ODO*Y9)+ E1*S2*( 15. ODO*S2-13. ODO)'*DCOS(TH1+W1)/2. ODO+ E1*S2*( 15. ODO*S2-13. ODO)*DCOS(3. QDO*THI-W1)/6. ODO+ S2*( 15. ODO*S2-13. ODO)*DCOS(2. ODQ*TH1)/2. ODO+ (5.ODO*Y6*S4+4.-ODO*Y12*S2-56. ODO*E2)/96. ODO C Y=TFJ-W1+RJ*YF+RJ*rRJ*TFJ*YS C + + + + + + RF1=1. ODO-1. 5D0*S2+E2*(1. ODO-1. 25D0*S2)((2. ODO+5. ODO*E2)*S2-2. ODO*E2)*DCOS(2. ODO*TFJ)/12. ODO+ E2*(9. ODO*S2-8. ODO)*DCOS(2. ODO*Y)/12. ODO+ E1*(-11. ODO*S2+6. ODO)*DCOS(Y+2. ODO*TFJ)/24. ODO+ E2*( -3. ODO*S2+2. ODO)*DCOS(2. ODO*Y+2. ODO*TFJ)/24. ODO+ E2*(3. ODO*S2-2. ODO)*DCOS(2. ODO*Y-2. ODO*TFJ)/8. ODO+ E1*Yl1*DSIN(Y2)*DSIN(TFJ+Wl)/Y9 + + + + + + RF2=E2*S2*( 15. ODO*S2-14. ODO)*DSIN(Y2)*DSIN(Y3)/(O. 5D0*Y9)E2*S2*DCOS(Y-TH1+3. ODO*W1)/16. ODO+ E2*(3. ODO*S2-2. ODO)*DCOS(Y-3. ODO*TH1+3. ODO*W1)/24. DOE2*S2*DCOS( Y-5. ODOI-TH1+3. ,ODO*W1) /16. ODO+ El*(3. ODO*S2-2. ODO)*DCOS(Y-2. ODO*TH1+2. ODO*W1)/4. ODO3. ODO*E1*S2*DCOS(Y-4. ODO*TH1+2. ODO*W)/8. ODOE1*(S2+1. ODO)*DCOS(Y+2. ODO*'W1)/4. ODO -C C 116 ROM00030 + + + + + + RF3=((5. ODO*E2-2. ODO)*S2-2. ODO*E2)*DCOS(Y+TH1+W1)/8. ODO+ ((5. ODO*E2+6. ODO)*S2-4. ODO*(E2+1. ODO))*DCOS(Y-THI1W1)/4. ODO+ (2.ODQ*E2-S2*(5. ODO*E2+14. ODO))*DCOS(Y-3. ODO*TH1+W1)f 24. ODO+ E2*(9. ODO*S2-4. ODO)*DCOS(Y+3. ODO*TH1-W1)/48. DO+ E2*(6. ODO-7. ODO*S2)*DCOS(Y+TH1-W1)/8.ODO+ E2*(4..ODO-5. ODO*S2)*DCOS(Y-TH1-W1)/16. ODO+ E1*(2. ODO*S2-1. ODO)*DCOS(Y+2. ODO*THI)14. ODO + + + RF4=El*(1. ODQ-3.'ODO*S2):*DCOS(Y-2. ODO*TH1)14. ODO+ E1*1(2. ODQ-3. ODO*S2)*DCOS(Y)/4. ODO+ E1*S2*DCOS(TH1+Wl)+S2i"DCOS(2. ODO*TH1)+ E1*S2*DCOS(k3. ODO*THI-W1)/3. ODO c c RFB=1. ODO+E 1*DCOS( Y)+RJ*(RF1+R2+P3+RF4) END IF C o C RADIUS BOTTOM ( SIMPLIFIED SOLUTION) IF(JVER. EQ. 2)THEN c + + + + + + + + + + + + RF1=1. ODO-1. 5D0*S2+E2*(1. ODQ-1. 25D0*S2)((2. ODO+5. ODO*E2)*S2-2. ODO*E2)*DCOS(2. ODO*TFJ)/12. ODO+ E2*(9. ODO*S2-8. ODO)*DCOS(2. ODO*(TFJ-Wi))/12. ODO+ E1*( -11. ODO*S2±6. ODO )*DCOS( 3. ODO*TFJ-W1 ) /24. ODO+ E2*( -3. ODO*S2+2. ODO)*DCOS(4. ODO*TFJ-2. ODO*W)/24. ODO+ E2*(3. ODQ*S2-2. ODO)*DCOS(2. QDO*W1)/8. ODOE2*S2*DCOS(TFJ-TH1+2. ODO*W1)/16. ODO RF2=E2*(3. ODO*S2-2. ODO)*DCOS(TFJ-3. ODO*TH1+2. ODO*W1)/24. DOE2*S2*DCOS(TFJ-5. ODO*TH1+2. ODO*W1)/16. ODO+ El*(3. ODO*S2-2. ODO)*DCOS(TFJ-2. ODO*TH1+W1l)/4. ODO3.ODO*E1*S2*DCOS(TFJ-4. ODO*TH1+Wl)/8. aDOE1*(S2+1. ODO)*DCOS(TFJ+W1)/4. ODO+ ((5. ODO*E2-2. ODO)*S2-2. ODO*E2)*DCOS(TFJ+TH1)/8. ODO+ ((5. ODO*E2+6. ODO)*S2-4. ODO*(E2+1. ODO))*DCOS(TFJ-THI)/4. DO C RF3=(2. ODO*E2-S2*(5. ODO*E2+14. ODO))*DCOS(TFJ-3. DO*TH1)/24. DO+ E2*(9. ODO*S2-4. ODO)*DCOS(TFJ+3. ODO*TH1-2. ODO*W1)/48. DO+ E2*(6. ODO-7. ODO*S2)*DCOS(TFJ+THI-2. ODO*Wl)/8. ODO+ E2'*(4. ODO-5. ODO*S2)*DCOS(TFJ-TH1-2. ODO*Wl)/16. ODOI El*(2. ODO*S2-1. ODO)*DCOS(TFJ+2. ODO*THI-W1)/4. ODO+ E1*(1. ODO-3. ODO*S2)*DCOS(TFJ-2. ODO*TH1-W1)/4. ODO+ + El*(2. ODO-3. ODO*S2)*DCOS(TFJ-Wl)/4. ODO + + + + + C + RF4=E1*S2*DCOS(TH1+Wl)+S2*DCOS(2. ODO*TH1)+ El*S2*DCOS(3. ODO*THl-W1)/3. ODO + RFB=E1*DCOS(TFJ-W1+RJ*(TFJ-TH1)*( 2. 5DO*S2-2. ODO) )+ 1. ODO+RJ*(RFI+RF2+RF3+RF4) C * C ENDIF C C C RADIUS BOTTOM ( TWO BODY SOLUTION) IF(JVER. EQ. 3)TIIEN 117 RFBl.ODO+E1*DCOS(TFJ-W1) ENDIF C C RADIUS=PO/RFB C RETURN END R0M00410 R0M00420 C o SUBROUTINE FORMULA o C' C SUBROUTINE FORMULA IMPLICIT DOUBLE PRECISION (A-I,M-Z) CHARACTER*20 LINE DIMENSION M(1OIMD(100),E(100),W(100),WD(1O0),OM(100),OMID(1o0) DIMENSION I(100)_ ID(100),F(100),FD(JO0),EC(100),ECD(100),A(100) DIMENSION R(100),H(100),N(100),TH(100),THD(100) DIMENSION RF(100)4TF(100),IFD(100),OMF(100),OMFD(100),THF(100) DIMENSION THFD(1OQ) ,P(100) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD( 100) DIMENSION DOMD( 100) ,RX( 100) ,RY( 100) ,RZ( 100) ,RFX( 100) ,RFY( 100) DIMENSION RFZ(100) ,DRV(100) ,ARC(100) ,ARCD(100) ,DAY(100) ,HX(100) DIMENSION HY(100),-HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100) DIMENSION NY(100) ,NZ( 100) ,RDV(100) ,EX(100) ,EY(100) ,EZ(100) DIMENSION NDE(100) ,EDR(100) ,V(100) ,HT(i0),RDRF(100) ,INTA(100) DIMENSION ROMA( 100) ,THJO( 100) ,ATE( 100) ,CTE( 100) COMMON/OBLATE1/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MIU,NDE,EDR-,H,N,E,P, I,OMI,W,F COMMON/OBLATE3/PI,EC,M1,A,HT,ER,TH,THD,RTD,MD,WD,OMD,ID,ECD COMMION/OBLATE4/FD,LINE,J,THF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM COMMON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR,ACTERR,TERROR,JVER COMMON/OBLATE6/RFX, RFY ,RFZ ,ARC, AROD ,RDRF ,DRV ,RJ2 ,JN, JORB IT COMMON/OBLATE7/INTA,ROMA,THJO ,ATE , TE C C C C -EXTERNAL RADIUS CALCULATE E, SINE, AND COSINE FUNCTIONS S=DSIN(I(1)) S2=S*S S4=S 2*S2 S6=S4*S2 C=DCOS(I(1)) C2=C*C SC=DSIN(1(l) )*DCOS(I( 1)) E2=E( 1)*E( 1) C C FORMULA ( SOLUTION) IF(JVER. EQ. 1)THEN C Yl112. ODO-75. ODO*S6+260. 0D0*S4-296. ODO*S2 118 -Y2=RJ*T~i(J)*(2. rDO*S2-2. ODO) Y3=2.ODO*W(1) -Y2Y4=24. ODO*(5. qDO*S2- 4. ODO)*(5. ODO*S2-4. ODO) Y.5 =E2*S2*( 14. ODO- 15. ODO*'S2)*( 15. ODO*S2-13. ODO) Y6=9.0OD*E2+34.0OD Y77i5. ODO*S4-45. ODO*S2+28. ODO Y8=6.ODO*(5. ODO*S2-4. ODO)*(5. ODO*S2-4. ODO) Y9=12. ODO*(5. ODO*S2-4. ODO) 'C + Y1O=(6. ODO-S2)/12. ODO-E(1)*S2*DCOS(3. ODO*TH(1)-W(1))/3. ODO-S2*DCOS(2.ODO*TH(l))+E2*(7. ODO*S2-4. ODO)/24. ODO 1!12=9. ODO*E2-34. ODO C YF=(2. 5D0*S2-2. ODO)*(THF(J)-TH(l))+E2*Y1*DSIN(Y2)*DCOS(Y3)/Y4 YS=Y5*DCO S(2:ODO*W( 1))/(2..ODO*Y9)+ + E(1)*S2*(15. ODO*S2-13. ODO)*DCOS(TH(1)+W(l))/2. ODO+ + E(1)*S2*(15. ODO*S2.13. ODO)*DCOS(3. ODO*TH(1)-W(1))/6. ODOI + S2*(15. ODO*S2-13. ODO)*DCOS(2. ODO*TH(1l))/2. ODO+ 4- '(5. ODO*Y6*S4+4. ODO*YL12*S2-56. ODO*E2)/96. ODO ,Y=TfHF( J) -W(1) +RJ*rYF+RJ*RJ*THF( J) *YS C C CALCULATE INCLINATION ( SOLUTION) IF(J)=I( 1)+SC*~RJr,(DCOS(2. ODO*THF(J))/2. ODO+ E(1)*DCOS(Y+2. ODO*THF(J))/6.DO+E(1)*DCOS(Y-2.DO*THF(J))/2.DO+ E2*( 14. DO-15.-DO*S2)*DSIN(Y2)*DSIN(Y3)/(12. DQ*(5. DO*S2-4. DO))DCOS(2. ODO*TH(1))/2. ODO-E(1)*DCOS(3. ODO*TH(1)-W(1))/6. ODO+ E(1)*DCOS(TH(i)+W(1))/2. ODO) + + + C C C CALCULATE LONGITUDE OF THE ASCENDING NODE ( SOLUTION) RJ2=RJ*RJ C OMF(J)=O1( 1)+C*RJ*(TH(1)-TIIF(J)+DSIN(2. ODO*THF(J))/2. ODOE(1l)*DSIN(Y)+E(1)*DSIN(Y+2. ODO*THF(J))/6. ODOE(1)*DSIN(Y-2. ODO*THF(J))/2. ODO-DSIN(2. ODO*TH(l))/2. ODO+ E(1y*DSIN(TII(1)-Wi(l))-E(1)*DSIN(3. ODO*TH(l)-W(1))/6. ODOE(1)*DSIN(TH(1)+W (1))/2. ODO+E2*Y7*DSIN(Y2)*DCOS(Y3)/Y8)+ + C*RJ2*THF(J)*(E2*S2*( 15. ODO*S2-14. ODQ)*DCOS( 2.DO*W( 1)) /Y9+ E(1y*S2*DCOS(TH(1)+Ii(1))+YlO) + + + + C ENDIF C C C FORMULA ( SIMPLIFIED SOLUTION) IF(JVER. EQ. 2)THEN C C C CALCULATE INCLINATION ( SIMPLIFIED SOLUTION) IF(J)I( 1)+SC*RJ*c(DCOS(2. ODO*THF(J))/2. ODO+ E(1)*DCOS(3. ODO*THF(J)-Wi(1))/6. ODO+ E(1)*DCOS(THIF(J)+W(1))/2. ODO-DCOS(2. ODO*TH(1))/2. ODO+ E(1)*DCOS(3. ODQ*Ti(1)-W?(1))/6. ODO+ + 119 + C C E(:1,)*DCOS(TH(1)+W(l))/2. ODO) CALCULATE LONGITUDE OF TlHE ASCENDING NODE-( SIMPLIFIED SOLUTION) O1M(J)=OM(1)+ C*RJ*(THCI)'-THF(J)+DSIN(2. ODO*T}{F(J))/2. ODOE(1)*DSIN -THF(J)-W(1l))+E(1)*DSIN(3.OD0*THF(J)-W(l))/6. 0D0+ E(1)*DSIN(THF(J)+W(l))f". ODO-DSIN(2. ODO*TH(l))/2. 0D0+ + E(1)*DSIN(TH(1)-W(l))-E(1y*DSIN(3. ODO*TH(1)-W(1l))/6. ODO+ E(1)*DSIN(TH(1)+W(l))/2. ODO) + C END IF C C c FORMULA ( TWMO BODY SOLUTION) IF(JVER. EQ. 3,)THEN C CALCULATE INCLINATION TW l~O BODY SOLUTION) C IF(J)1( 1) C C C CALCULATE LONGITUDE OF THE ASCENDING NODE CTWO BODY SOLUTION) OMF(J)0OM( 1) C ENDIF -C C C CALCULATE RADIUS ( SOLUTION, SIMPLIFIED, OR TWO0 BODY) RF(j)=RADIUS(RJ,A(1) ,I(1) ,E(1) ,W(1) ,TH(1) ,THF(J) ,JVER) C C C 10 20 C C C CONVERT ANGLES TO DEGREES OLIFD( J)=OMF(J)*RTD IFD( J)1F( J)*RTD THFD( J)=THF( J)*RTD JORBIT( J)=O IF(THFDCJ). GT. 360. 0D0)THEN THFD(J)=-THFD(j)36O. ODO JORBIT( J)=JORBI'T.(J)+1 GOTO 10 END IF THJO(J)=JORBIT(J)*2. OD0'*PI+TH(J) -TiI( 1) IF(OMFD(J). GT. 360. 0D0)T[EN OMFD(J)=OMFD(J) -360. ODO GOTO 20 END IF THF(J)=THFDCJ) /RTD OMF( J)=OMFD(J) /RTD CALCULATE DELTAS DRCJ)=RFCJ) -R(J) DID(J)=IFD(J) -ID(J) DTHD(J)=TIIFD(J) -TIID(J) IF(DABS(DTHD(j)). GE. 180. ODO)THEN IF(DTHD(J). LT, 0. ODO)THEN 120 DTHD(J)=DTHD(J)+360. ODOELSE DTHD(J)=DTHD(J) -360. ODO ENDIF ENDIF DOMD(J)OMFD(J) -OMD(J-) C, RETURN END C C o o o C C C SUBROUTINE INERTIAL * C SUBROUTINE INERTIAL IMPLICIT DOUBLE PRECISION (A-I ,M-Z) CHARACTER*20 LINE DIMENSION M(100),M D(100),E(100),W(100),WD(1OO),OM(100),OMD(100) DIMENSION I(100),ID(100),FX100),FD(100),EC(100),ECD(100),A(100) DIMENSION R(100),H(100),-.;N(100) ,TH(100) ,THD(100) DIMENSION RF(100) ,IF(100) ,IFD(100) ,OMF(100) ,OMFD(100) ,THIF(100) DIMENSION THFD( 100) ,PGLOO) ,JORBIT( 100) ,DR( 100) ,DID( 100) ,DTHD( 100) DIMENSION DOMD(100) ,RX(100) ,RY(100) ,RZ(100) ,RFX(100) ,RFY(100) DIMENSION RFZ(100) ,DRV(100) ,ARC(100),ARCD(100) ,DAY(100) ,HX(100) DIMENSION HY(100),HZ(100),VX(100),VY(100),VZ(100),DT(100),,NX(100) DIMENSION NY(100),NZ(100),RDV(100),EX(100),EY(100),EZ(100) DIMENSION NDE(100) ,EDR(100),V(100) ,HT(100) ,RDRF(100) ,INTA(100) DIMENSION EARC( 100) ,EARCD( 100) ,PDR( 100) DIMEN-SION ROMiA(QO) ,THJO(100) ,ATE(-100) ,CTE(100) COMMION/OBLATE1/DAY,RX,RY,RZ,VX,VIY,VZ,DT,HX,HY,HZ,N.,NY,NZ,K,KK COMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P,I,OM,W,F COMMION/OBLATE3/PI,EC,M,A,HT,ER,TH,THD,RTD,MD,WD,OMD, ID,ECD COMMON/OBLATE4/FD,LINE,J,THF,THFD, IF,'IFD,OMF,OMIFD,RF, INT,ROM COMMON/OBLATE5/RJ,DR,DID ,DTHD,DOMD,ESTERR ,ACTERR,TERROR,JVER COMMON/OBLATE6/RFX ,RFY,RFZ ,ARC, ARCD ,RDRF,DR,RJ2,J,JORBIT COMMfON/OBLATE7/INTA,ROMA,THJO ,ATE ,CTE COMMON/SPECIAL/EARC ,EARCD,PDR,ENG,ENGF C C C CALCULATE INITIAL ENERGY ENG=V( 1)*V( 1)/2. ODO-MU/R( 1) ENGF=V( 1)*V(1)/2. ODO-IIU/RF( 1) C C C CALCULATE INERTIAL COORDINATES RFX(J)=RF(J)*(DCOS(THF(J) )*DCOS(OMF(J)-) DSIN(THF(J))*DCOS(IF(J))*DSIN(OMF(J))) RFY(J)=RF(J)*(DCOS(THF(J) )*DSIN(ON1FCJ) )+ + DSIN(THF(J) )*DCOS( IF(J) )*DCOS(OMF(J))) RFZ(J)=RF(J)*(DSIN(THF(J) )*DSIN( IF(J))) + C C C CALCULATE DR VECTOR DRV(J)=DSQRT( (RFX(J) -RX(J) )*(RFX(J) -RX(J) )+ 121 + + (RFY(J)--RY(J))*(RFY(J)-RY(J))+ (RPZ(J)-RZ(J))*(RFZ(J)-RZ(J))) ,PDR(J)=DRV(J)/R(J) C C CALCULATE ANGLE ERROR C CALL DOT(RX(J),RY(J),RZ(J),RFX(J),RFY(J),RFZ(J),RDRF(J)) ARC(J)F=DACOS(RDRF(J)/(R(J)*RF(J))) CC=RF( J) CCP=R( J) BB=ER AA=DSQRT(BB*BB+CC*CC-2. ODO*BB*CC*DCOS(ARC(J)/2. ODO)) AAP=DSQRT(BB*BB+CCP*CCP-2. ODO*BB*CCP*VDCOS(ARC(J)/2. Oj)0)), CCA=PI-DASIN(CC*DSIN(ARC(J) /2.ODO)/AA) CCPA=PI-DASIN(CCP*DSIN(ARC(J)/2. ODO) fAAP) EARC(J)=2. ODO*PI-CCA-CCPA ARCD(J)=ARC(J)*RTD EARCD(J)=EARC( J)*RTD C C C CALCULATE DOWNRANGE AND CROSSRANGE ERRORS ATE(J)=R(J)'*(DTHD(-J)/RTD+DCOS( 1(J) )*DOMD(J)/RTD) CTE(J)=R(J)*(DSIN(TH(J) )*DID(J)/RTD+ DCOS(TH(J)Y)*DSIN(I(J))*DOMD(J)/RTD) C RETURN END C C C* C C C * SUBROUTINE RESULTS JJJJJ..L.J....J4JJJJ.*......J C SUBROUTINE RESULTS IMPLICIT DOUBLE PRECISION (A-I,M-Z) CHARACTER*20 LINE CHARACTER-*11 VERS ION DIMENSION M(100),MD(100),E(100),W(100),WD(100),OM(100),OMD(loO) DIMENSION I(100),ID(100),F(100),FD(100),EC(1oo),ECD(1oo),A(100) DIMENSION R(100) ,H(100) ,N(100) ,TH(100) ,THD(100) DIMENSION RF(100) ,IF(100),IFD(100) ,OMF(100) ,OMFD(100) ,THF(100) DIMENSION THFD(100) ,P(100) ,JORBIT(100) ,DR(100) ,DID(100) ,DTHD(100) DIMENSION DOMD(100) ,RX(100) ,RY(100),RZ(100) ,RFX(100) ,RFY(100) DIMENSION RFZ( 100) ,DRV( 100) ,ARC( 100) ,ARCD( 100) ,DAY( 100) ,HX( 100) DIMENSION HYC100),HZ(100),VX(100),VY(100),VZ(100),DT(100),NX(100) DIMENSION NY(100),NZ(1CO),RDV(100),EX(100),EY(.Oo),EZ(100) DIMENSION NDE(100),EDR(100) ,V(100),HT(100) ,RDRF(100),INTA(100) DIMENSION EAROC 100) ,EARCD( 100) ,PDR( 100)DIMENSION ROMA(100),THJO(100),ATE(100) ,CTE(100) COMMON/OBLATEI/DAY,RX,RY,RZ,VX,VY,VZ,DT,HX,HY,HZ,NX,NY,NZ,K,KK COMMON/OBLATE2/RDV,R,V,EX,EY,EZ,MU,NDE,EDR,H,N,E,P,I ,OM,W,F COMMION/OBLATE3/PI,EC,M,A,IIT,ER,TH,TIID,RTD,MID,WD,OMD,ID,ECD COMMON/OBLATE4/FD,LINE,J,THIF,THFD, IF,IFD,OMF,OMFD,RF, INT,ROM,, COMMON/OBLATE5/RJ,DR,DID,DTHD, DOMD,ESTERR,ACTERR,TERROR,JVER 122 COMMON/OBLATE6/RFX,RFY,RFZ,ARC,ARCD,RDRF,DRV,RJ2-,JN,jORBITT COMMON/OBLATE7/INTA,ROMA,THJO,ATE,CTE COMMON/SPECIAL/EARO ,EARCD, PDR,ENG,ENGF C DR( 1)O.ODO DID(1)=0.ODO DTHD( 1)=0. ODO DOMD( 1)=0. ODO DRV( 1)0. ODO ARCD( 1)=O. ODO EARCD( 1)=0. ODO PDR( 1)=0. ODO THJO( 1)0. ODO ATE(1)=0. ODO CTE(1)=0.ODO C IF(JVER.EQ. 1)THEN VERSION' SOLUTION ELSE IF(JVER. EQ. 2)THEN VERSION='SIMPLIFIED' ELSEIF(JVER. EQ. 3)THEN VERSION=' SECULAR ENDIF IF(RJ. EQ.0. ODO)THEN VERSION='TWO BODY ENDIF C C OUTPUT RESULTS FOR DISSPLA VC IF(JVER. EQ. 1)THEN WRITE(3,3000) K WRITE(3,3100) RJ ENDIF If C 10 C C C DO 10 J = 1, KK WRITE(3,3100) WRITE(3,3100) WRITE(3,3100) WRITE(4,3100) CONTINUE DR(J) ,DID(J) ,DTHD(J) DOiM-D(J),DRV(J) ,EARCD(J) PDR(J),ATE(J),CTE(J) THJO(J) PRINT RESULTS WRITE(6,(/)') WRITE(2,'(/)') WRITE(6,6000) 'RESULTS' WRITE(2,6000) 'RESULTS' WRITE(6,6100) LINE WRITE(2,6100) LINE WRITE(6,6200) 'J = IRJ WRITE(2,6200) 'J = ',RJ WRITE(6,6300) 'VERSION = 'VERSION WRITE(2,6300) 'VERSION = 'VERSION WRITE(6,6100) LINE WRITE(2,6100) LINE C 123 DO 20 J = K, KK C + + WRITE(6,6400) 'POINT = 'ROMBERG WRIITE(2,6400) 'POINT = 'ROMBERG ',J,'ORBIT = ',JORBIT(J), ITERATIONS = ,JN IPj,'ORBIT = ,JORBIT(J), ITERATIONS = 'JN C WRITE(6,6500) 'R WRITE(2,6500) 'R = = WRITE(6,6500) 'I WRITE(2,6500) 'I T ( ,5 0 T WRITE(6,6500) 'ITH = WRITE(6,6500) '0OM WRITE(2,6500) '014 = = ',R(J),'RF 'R(J): RF 'RF(J),:DR ',RF(J),'DR = = = = ',DR(J) 'pDR(J) C CR 'ID(J),:IF ',IFD(J).,'DI = ',DID(J) ,IFD(J),'DI = ',DID(J) , H ( ) : HF = : T F ( ) : T TD J = ',THD(J),'THF = 'THFD(J), DITH = ',DTHDCJ) ='ID(J),'IF = = C ',OMD(J),'OMF ',OMD(J),'OMF = 'OMFD(J),'D014f= =,OMFD(J),'DOM = ',DOMD(J'pr ',DOMD(J,% C W'RITE(6,6500) 'RX = ',RX(J),'RY WRITE(2,6500) 'RX = 'RX(J),'RY = = 'RYCJ),'RZ = ',RZ(J) ,RY(J),'RZ = ',RZ(J) C WRITE(6,6500) 'RFX = ',RFX(J),'RFY W'RITE(2,6500) 'RFX = ',RFX(J),'RFY =',RFY(J),'-RFZ W'RITE(6,6500) 'DRy = 'EARC= WRITE(2,6500) 'DRV = 'EARC= ',DRV(J),'PDR ',EARCD(J) ',DRV(J),'PDR ',EARCD(J) =',PDR(J), WRITE(6,6500) 'RITE 'CITE WRITE(2,6500) 'RITE 'CITE 'DR(J),'ATE = ',ATE(J), ',CTE(J) 'DR(J),'ATE = ',ATE(J), ',CTE(J) =',RFY(J), C + + =',PDR(J), C + + = = = = C WRITE(6,6600) 'INT = 'INTA(J),'ROM = ',ROMA(J) WRITE(2,6600) 'INT = 'INTA(J),'ROM = ',ROMA(J) C 20 C CONTINUE WRITE(6,6500) 'EG = ',ENG,'EGF = 'ENGF WRITE(2,6500) 'EG = ',ENG,'EGF = ,ENGF C UWRITE(6,'(/)') WRITEC2, '(I)') C 3000 3100 C 6000 6100 6200 6300 6400 6500 FOR14AT(3X,13) FOR.%AT(3(3X,D18. 10)) FORKA4T(3X,A) FORM.AT(3X,A20//) FOR14AT(3X,A,F8. 6) FORNAT(3X,A,Al1//) FORMAC2(3X,AB,13/) ,3X,A21,I3//) FORMAT(3(3X,A6,F23. 15/)) 124 = ',RFZ(J) RFZ = ',RFZ(J) 6600 C FORMAT(3(3X,A6,F23. 8/)) RETURN END C 125 LIST OF REFERENCES 1. Snider, J.R., Satellite Motion Around An Oblate Planet: A PerturbationSolution for All Orbital Parameters, Ph.D. Dissertation, Naval Postgraduate School, Monterey, California, June, 1989. 2. Sagovac, C. P., A PerturbationSolution of the Main Problem in Artificial Satellite Theory, Master's Thesis, Naval Postgraduate School, Monterey, California, June, 1990. 3. Struble, R. A., "A Geometrical Derivation of the: Satellite Equations," Journal of Mathematical Analysis and Applications, Volume 1, 1960, pp. 300-307. 4. Struble, R. A., "The Geometry of the Orbits of Artificial Satellites," Architectural Rational and Mechanical Analysis, Volume 7, 1961, pp. 87-104. 5. Struble, R. A., "An Application of the Method of Averaging in the Theory of Satellite Motion," Journal of Mathematics and Mechanics, Volume 10, 1961, pp. 691-704. 6. Eckstein, M. C., Shi, Y. Y., and Kevorkian, J., "Satellite Motion for All' Inclinations Around an Oblate Planet," Proceedings of Symposium No. 25, International Astronomical Union, Academic Press, 1966, pp. 291-322, equations 61. 7. Danielson, D. A. and Snider, J.R., "Satellite Motion Around an Oblate Earth: A Perturbation Solution for All Orbital Parameters: Part I - Equatorial and Polar Orbits," Proceedingsof the AAS/AIAA Astrodynamics Conference, Stowe, Vermont, August, 1989. 8. Danielson, D. A., Sagovac, C. P., and Snider, J. R., "Satellite Motion Around an Oblate Earth: A Perturbation Solution for All, Orbital Parameters: Part II - Orbits for All Inclinations," Proceedings of the AAS/AIAA Astrodynamics Conference, Portland, Oregon, August, 1990. 9. Ferziger, J. H., Numerical Methods for Engineering Application, John Wiley & Sons, New York, New York, 1981, pp. 32-37. 10. Coffey, S. L. and Alfriend, K. T., "An Analytic Orbit Prediction Program Generator," Journal of Guidance, Control, and Dynamics, Volume 7, September-October, 1984, pp. 575-581. 11. Deprit, A., "The Elimination of Parallax in Satellite Theory," Celestial Mechanics, Volume 24, June, 1981, pp. 111-153. 12. Coffey, S. L. and Deprit, A., "A Third Order Solution to the Main Problem in Satellite Theory," Journal of Guidance, Control, and Dynamics, Volume 5, JulyAugust, 1982, pp. 366-371. 13. Alfriend, K. T. and Coffey, S. L., "Elimination of the Perigee in the Satellite Problem," Celestial Mechanics, Volume 32, February, 1984, pp. 163-172. 126 14. -Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics, American Institute of Aeronautics and Astronautics, New York, New York, 1987, pp. 128-130, pp. 450-470. 127

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