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DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY CA 93943-5101 Approved for public release: Distribution is unlimited Orbital Maintenance of Endoatmospheric Low Earth-Orbiting Satellites by David D. Pauls Lieutenant, United States Navy B.A., Physics, Eastern Washington University, 1982 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN ASTRONAUTICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL DECEMBER 1991 Unclassified 2URITY CLASSIFICATION OF THIS PAGE FormApproved REPORT DOCUMENTATION PAGE OMBNo0704-0188 l, REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS Unclassified i SECURITY CLASSIFICATIONAUTHORITY 3 DISTRIBUTION/AVAILABILITY OF REPORT Approved for public release: ). DECLASSIFICATIONOOWNGRADING SCHEDULE Distribution is unlimited PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S) 1. NAME OF PERFORMING ORGANIZATION 6D OFFICE SYMBOL 7a NAME OF MONITORING ORGANIZATION Javal Postgraduate School (IIapplicableA)A Naval Postgraduate School : ADDRESS (City, StateandZIPCode, 7b ADDRESS (City, State, andZIPCode) Monterey, CA 93943-5000 Monterey, CA 93943-5000 ..ONRAGMAENIOZFATFIUONNDING/SPONSORING 8b (OIFIFapIpClEicaSblYeM)BOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ADDRESS (City, State, andZIPCode) 10 SOURCE OF FUNDING NUMBER PROGRAM PROJECT TASK WORK UNIT ELEMENT NO. NO. NO. ACCESSION NO TITLE (IncludeSecurityClassification) . ORBITAL MAINTENANCE OF ENDOATMOSPHERIC LOW EARTH-ORBITING SATELLITES PERSONAL AUTHORS .AVID D. PAULS i a. TYPE OF REPORT 13b TIME COVERED 14. DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT [aster' s Thesis FROM TO DECEMBER 1991 71 SUPPLEMENTARY NOTATION he views expressed are those of the author and do not reflect the official policy r position of the Department of Defense or the U.S. Government COSATI CODES 18 SUBJECTTERMS (Continueonreverseitnecessaryandidentifybyblocknumbers) orbits, satellites, low-Earth orbiting :LD GROUP SUB-GROUP satellites, fuel-minimization ABSTRACT (Continueonreverseilnecessaryandidentifybyblocknumbers) The optimization of spacecraft trajectories in vacuum has received extensive consideration ince the inception of space flight, yet, the effects of atmosphere have been largely neglected. The dvent of low Earth-orbiting, large satellites and platforms necessitates that atmosphere be included n the optimization process. A practical means of studying this topic is as a problem in minimum-fuel rbital maintenance. Optimal control theory advances the notion that orbital maintenance is optimized hrough periodic thrusting as opposed to forcing Keplarian motion by nullifying the effects of drag ith thrust. Further, this must be optimized by primer vectoring. This thesis examined the efficiency f a simple method of orbital maintenance using fixed-angle transverse thrusting. Results show that or the purpose of fuel-minimization, the width of the radial band in which the satellite is to be aintained, is dependent upon thruster size. In nearly all cases, a thrust-angle of 70 degrees aximized the fuel saved. This thesis shows that fixed-angle transverse thrusting does not improve n forced Keplarian motion and hence thrust vectoring must be optimized. XDXISUTNRCILBAUSTSIIOFNI/EAVDA/IULNALBIIMLIITTEYDOF ABSTSRAAMCETAS RPT DTIC USERS 21. AuBnSTcRlACaTsSsEiCUfRiITeYdCLASSIFICATION i. NAME OF RESPONSIBLE INDIVIDUAL ?2b.TELEPHONE (IncludeArea Code) 22c. OFFICE SYMBOL I .M. Ross (408) 646-2716 AA/RO Form 1473, JUN 86 Previous editions are obsolete. SECURITYCLASSIFICATIONOFTHISPAGE S/N 0102-LF-014-6603 Unclassified NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS ORBITAL MAINTENANCE OF ENDOATMOSPHERIC LOW EARTH-ORBITING SATELLITES by David D. Pauls DECEMBER 1991 Thesis Advisor: I M. Ross Approved for public release: Distribution is unlimited ABSTRACT The optimization of spacecraft trajectories in vacuum has received extensive consideration since the inception of space flight, yet, the effects of atmosphere have been largely neglected. The advent of low Earth-orbiting, large satellites and platforms necessitates that atmosphere be included in the optimization process. A practical means of studying this topic is as a problem in minimum-fuel orbital maintenance. Optimal control theory advances the notion that orbital maintenance is optimized through periodic thrusting as opposed to forcing Keplarian motion by nullifying the effects of drag with thrust. Further, this must be optimized by primer vectoring. This thesis examined the efficiency of a simple method of orbital maintenance using fixed-angle transverse thrusting. Results show that for the purpose of fuel-minimization, the width of the radial band in which the satellite is to be maintained, is dependent upon thruster size. In nearly all cases, a thrust-angle of 70 degrees maximized the fuel saved. This thesis shows that fixed-angle transverse thrusting does not improve on forced Keplarian motion and hence thrust vectoring must be optimized. 111 . /Jjs / TABLE OF CONTENTS I. INTRODUCTION 1 II. GENERAL FORMULATION OF THE PROBLEM 4 III. SIMPLIFIED PARAMETRIC FORMULATION 8 ... A. DEVELOPMENT OF THE EQUATIONS OF MOTION 8 .... B. DETERMINATION OF THE CONTROL VARIABLE 10 IV. DEVELOPMENT AND TESTING OF THE COMPUTER MODEL 12 . A. COMPUTER PROGRAM DEVELOPMENT 12 B. PROGRAM VALIDATION 13 1. Development of a Rule of Thumb 14 2. Development of a Control Law 15 3. Control Law design 16 a. Approximate method 16 b. Direct method 20 V. ANALYSIS AND RESULTS 22 A. USE OF A CONSTANT DENSITY MODEL 22 ... B. USE OF AN EXPONENTIAL ATMOSPHERIC MODEL 23 1 Comparison with a Forced-Keplarian Model 24 2. Further Testing 25 a. Constant Thrust, Varied Specific Impulse 25 b. Constant Specific Impulse, Varied Thrust 27 3. Final Tests 28 a. Case 1: Thrusters Fire 20 km Below R 28 c b. Case 2: Thrusters Fire 30 km Below R 30 c. Case 3: Thrusters Fire 40 km Below Ro 31 C. SUMMARY OF RESULTS 32 VI. CONCLUSIONS AND RECOMMENDATIONS 33 iv