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Circular Orbits ANALYSIS AND MECHANIZATION OF LAUNCH WINDOW AND RENDEZVOUS COMPUTATION PART I. CIRCULAR ORBITS Research & Analysis Section Tech Memo No. 175 March 1966 BY J. L. Shady Prepared for: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION / GEORGE C. MARSHALL SPACE FLIGHT CENTER AERO-ASTRODYNAMICS LABORATORY Prepared under Contract NAS8-20082 Reviewed by: " D. L/Cooper Technical Supervisor Director Research & Analysis Section / NORTHROP SPACE LABORATORIES HUNTSVILLE DEPARTMENT HUNTSVILLE, ALABAMA FOREWORD The enclosed presents the results of work performed by Northrop Space Laboratories, Huntsville Department, while )under contract to the Aero- Astrodynamfcs Laboratory of Marshall Space Flight Center (NAS8-20082) * This task was conducted in response to the requirement of Appendix E-1, Schedule Order No. PO. Technical coordination was provided by Mr. Jesco van Puttkamer of the Technical and Scientific Staff (R-AERO-T) ABS TRACT This repart presents Khe results of an analytical study to develop the logic for a digrtal c3mp;lrer sclbrolrtine for the automatic computationof launch opportunities, or the sequence of launch times, that rill allow execution of various mades of gross ~ircuiarorbit rendezvous. The equations developed in this study are based on tho-bady orb:taf chmry. The Earch has been assumed tcr ha.:e the shape of an oblate spheroid. Oblateness of the Earth has been accounted for by assuming the circular target orbit to be space- fixed and by cosrecring che T ational raLe of the EarKh accordingly. Gross rendezvous between the rarget vehicle and a maneuverable chaser vehicle, assumed to be a standard 3+ uprated Saturn V launched from Cape Kennedy, is in this srudy aecomplfshed by: .x> direct ascent to rendezvous, or 2) rendezvous via an intermediate circG1a.r parking orbit. When operating, this subroutine Wllk determine the span or sequence of launch times in wnich a chaser vehrcie can be iauached to accomplish a pre- selected rendezvous missL;n. The Zaonch times will be restricted by such mission constraints as; 1:) Total propulsive XreiGcity change (AV 1 capability of the chaser vehicle 21 Tocai aPZ3wabPe chaser f Light time 3) Maximum time that can be spent by the chaser vehicle in a parking orbit 4) Range safety resrxiccisns on the chaser vehicle's launch azimuth. At this pQint rn time, the subroutine 1s in a state ready for initial programing and L hec kaur, 111 NORTHROP SPACE LABORATORIES TABLE OF CONTENTS Sect ion Title % NOMENCLATURE . V LIST OF ILLUSTRATIONS . xxii 1 INTRODUCTION . 1-1 2 ASSUMPTIONS AND DEFINITIONS . * 2-1 2.1 Earth Model . 2-1 2.2 Coordinate System . 2-4 2.3 Reference Time . 2-4- 2.4 Position of the Target Orbit Plane at the Reference Time . 2-8 2.5 Launch Delay Time Scale . 2-10 2.6 Target Position at Given Launch Delay Time. , . 2-12 2.7 Rendezvous Compatible Orbits . 2-15 3 ANALYSIS. 8 e 3-1 3.1 Direct Ascent to Circular Orbit Rendezvous. * . 3-1 3.2 Rendezvous Via an Intermediate Circular Parking Orbit . 3-37 4 RESULTS AND CONCLUSlONS. e * * 4-1 4.1 Subroutine Logic Flow Chart. e . 4-€ 4.2 Recommendations. 4-5 5 APPENDIX A DEFINITIONS. * A-1 APPENDIX B TRIGONOMETRIC EQUATIONS FOR THE ASCENT TRAJECTORY MODES * B-1 APPENDIX C ADDITIONAL EQUATIONS FOR COPLANAR ORBITAL TRANSFER MODES- - * C-1 APPENDIX D ADDITIONAL EQUATIONS FOR NON-COPLANAR ORBITAL TRANSFER MODES D-1 iv NORTHROP SPACE LABORATORIES NOMENCLATURE Symbo 1 Definition Units a Semi-ma jor axis m Semf-major axis transfer ellipse m a2 of 2 Chaser vehicle launch azimuth, measured AZ eastward from the launch site meridian deg, rad Maximum allowable launch azimuth deg, rad (*z)MAx Minimum allowable launch azimuth deg, rad MI* D Chord connecting points A or B and point C of the third-stage-burn-model geometry e Eccentricity e Eccentricity of transfer ellipse 2 2 Eccentric anomaly point A EA of on the variable coast e 1lip se rad Eccentric anomaly of point B on the variable EB coast el lipse rad El Eccentric anomaly of the point of orbital transfer departure (point a> rad E2B0 Eccentric anomaly of the point of chaser second stage burnout on the variable coast ellipse rad .. NQR?HROP SPACE LABORATORIES NOMENCLATURE (Continued) Symbol Definition Eccentric anomaly of the point of orbital E3 transfer conic intersection with an orbit of radius R2, ‘(point A) rad Eccentric anomaly of the point of orbital E4 transfer conic intersection with an orbit of radius R 2’ (point B) rad F Chaser vehicle third-stage thrust kg go Acceleration due to gravity at the Earth’s 2 surface, (g = 9.8045016 m/sec ) 0 GMT Greenwich mean time see GHAY Greenwich hour angle of the vernal equinox, measured westward from the prime meridian iT Target orbit inclination 3 J Earth oblateness constant, (J = 1.62345 x 16 ) K Number of completed target revolutions at the launch delay time, relative to the ascending node of the target orbit LHA Local hour angle, measured westward from the launch site meridian NOMENCLATURE (Continued) Symbo 1 Definition LHAY Local hour angle of the vernal equinox, measured from the launch site meridian m Mass of the chaser vehicle at second-stage burnout 2B0 M Integer number of mean solar days M Mean anomaly of the apogee a Mean anomaly of point A on the variable MA coast ellipse rad Mean anomaly of point B on the variable M% coa st e 11 ips e rad Integer number of Earth revolutions MO - Mean anomaly of the point of orbital transfer departure (point a) rad Mean anomaly of the point of chaser second-stage M2B0 burnout on the variable coast ellipse rad Mean anomaly of the point of orbital transfer M3 conic intersection with an orbit of radius RP, (point A) rad IYORTHROP SPACE LABORATORIES NOMENCLATURE (Continued) Symbo 1 Definition Units M4 Mean anomaly of the point of orbital transfer conic intersection with an orbit of radius Rp, (point B) rad N Number of target revolutions completed at the launch delay time, relative to the target's position at the reference time - N Number of target revolutions per Earth rotation - 0 P Semi- latus rectum m P Target orbita 1 period sec 0 (p)LP Limiting parabola semi-latus rectum m Q Ratio of the square of the chaser second-stage burnout velocity to the square of the circular orbit velocity at the burnout radius R1 Parking orbit radius R2 Target orbit radius Chaser second-stage burnout radius R2B0 viii NOMENCLATURE (Continued) Symbols Definition Units Apogee radius m Ra Radius to the point the chaser crossed the orbital RCOW line of nodes m R Earth equatorial radius, e 6 (Re = 6.378165 x 10 m) R Perigee radius m P SHA Sidereal hour angle deg Flight time from chaser third-stage tCC shutdown to target orbit interception sec Flight time from chaser third-stage shutdown to target orbit interception, assuming a yaw maneuver is performed s ec Launch delay time sec Launch delay time in the true mean solar day Chaser flight time from second-stage burnout to interception of the variable coast ellipse with a circular orbit of radius R 1 or R2, (point A) sec ix NORTHROP SPACE LABORATORIES NOMENCLATURE (Continued) Symbo l s Definition Units CtEC)B Chaser flight time from second-stage .burnout to interception of the variable coast ellipse with a circular orbit of radius R1 or R2, (point B) sec Total chaser ascent flight time sec (t,)a Maximum allowable chaser ascent flight time s ec [Wa] MAX Chaser flight time along orbital transfer ellipse 1 sec (%)El (%)E2 Chaser flight time along orbital transfer ellipse 2 sec Chaser flight time from second-stage burnout (tF)CC to. itarget orbit interception sec ( VTC Flight time required for the target to coast from its position at the time of chaser Sekond-stage burnout to the orbital line of nodes, (for a non-coplanar ascent trajectory mode) sec I Flight time required for the target to coast (t~)~~ from its position at the time of chaser second-stage burnout to the point of gross rendezvous, (for the coplanar ascent trajectory mode) sec X NORTHROP SPACE LABORATORIES NOMENCLATURE (Continued) Symbo Is Definition Units Chaser orbital transfer time sec (%)TRANS Flight time from perigee to point a on the tF 1 orbital transfer conic sec Flight time from perigee to point A on the tF3 orbital transfer conic s ec Flight time from perigee to point B on the orbital transfer conic s ec Coast time spend in the "fictitiousf' t~~~~ COAST target orbit sec Total chaser flight time required to accomplish ( tF 'MISS ION a given rendezvous mission sec Maximum alfowable mission flight time sec [(tF )MISSION] Chaser parking orbit time sec tPARK time (tPARK)MAx Maximum aflowabfe chaser parking orbit sec Chaser flight time from lift-off to second-stage t12B burnout sec xi NORT#ROP SPACE LABORATORIES NOMENCLATURE (Continued) Symbo 1s Definition Units Chaser third-stage-burn time t3B sec Time required for the launch site to rotate through the central angle 5 sec (t) Time requiredfor the launch site to rotate $0 through the central angle $o sec T Corrected mean solar day s ec True mean solar day (To 86,164.099 sec ) - v Chaser velocity at apogee a m/sec Chaser velocity at the apogee of transfer 1 ellipse 1 m/sec (Val* Chaser velocity at the apogee of transfer ellipse 2 m/sec Circular orbit velocity at radius R %o I1 1 mlsec Circular orbit velocity at the chaser second- (Vco)2B0 stage burnout radius RaBO mlsec xii HORTHROP SPACE LABORATORNSS NOMENCLATURE (Conr: inued) Swbo Is Definition -Units CfsreuZar orbit veloefty arb R sec 'vC~' 3 radius 2 m/ Cvei Chaser velocicy at poinc a on the orbital transfer conic mlsec Chaser velocity at A on the orbital CVe>, paint transfer conic mlsec he:)4 Chaser veisciky at point B on the orbital transfer CQXZ~G- m/sec Chaser veloafty at perigee transfer %-)2 the of ellipse 2 m/see (%om Velocity of the chaser at the time it crosses she orbital line of nsdes m/sec Chaser veiscity at pofnr: of intersection %?A the B of %hevariable coast ellipse with an orbit or radius R Rl or 2 m/sec Chaser second-stage '2BO velocity at burnout m/sec X - Y Dec ima 1 - Xiif NOMENCLATURE (Continued) Symbols Definition Units a Position of the target at the reference time, 0 relative to the ascending node deg, rad The central, angle, measured in the target orbit CY.
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