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Trajectory mission design and navigation for a space weather forecast

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Trajectory mission design and navigation for a space weather forecast

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Content TRAJECTORY MISSION DESIGN AND NAVIGATION FOR A SPACE WEATHER FORECAST Copyright 2012 by Pedro J. Llanos de la Concha A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ASTRONAUTICAL ENGINEERING) May 2012 Pedro J. Llanos de la Concha Dedication A mis padres, for their undying support ll Acknowledgements I would like to acknowledge the Department of Astronautical Engineering for giving me a chance to pursue my degree at USC. In general, I am very thankful to every single member of the committee for their enthusiasm and encouragement . Specially, I am very grateful to Professor Gerald Hintz (and Mary Louise Hintz) for his trust, patience, and guidance since the very first day we started working together and for the thorough help provided in editing of my works. I am indebted to Martin Lo, not only for introducing me into the world of astrodynamics but also for challenging me in many technical ways. I would like to express my deepest gratitude to James Miller (and Connie Weeks) for his insatiable willingness to navigate my thoughts. This thesis could not have been feasible without my mentors. I will remember every single day of our meetings at Caltech and JPL filled with motiva tion, inspiration and leadership, now engraved in the course of my veins and each step of this unpredictable jou rney. Many thanks to my colleagues and hiking friends Gary Block and Vaibhav Gupta that were always there for me for their moral support and advices in and out of the technical world. I do not forget my first USC classmates and colleagues Mike Mustillo, James Horton, Patrick Copinger, Stephen De Salvo and Stefano Campagnola for sharing their knowledge and friendship since I first arrived in Los Angeles. To my marathon running friends for allowing me to sweat my griefs and pains. Remember, success lies within ones pam. I would like to thank Yen for being such supportive angle and teaching me how to be patient. To my OU colleague, Rob Pilgrim, for watching every one of the thousand steps lll of this jo urney and not letting me join the sheep, and for being such an inspirational friend. To my parents and my brother Victor Llanos for their love and undying support that made this work possible. lV Table of Contents Dedication n Acknowledgements m List Of Ta bles 1x List Of Figures x1 Abstract xx Preface xxm Chapter 1: Research Overview 1 1.1 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background and Motiva tion: L5 Lagrange Antenna Observat ory of Sun 1 1. 3 State of the Art . . . . . . . . 6 1.3.1 Lagrange's first paper 6 1.3.2 Szebehely . . . . . . . 6 1.3.3 Akioka papers 6 1.3.4 M. Gruntman, M. Lampton, J. Edelstein 7 1.3.5 R. A. Freitas, Francisco Valdes 8 1.3.6 Carles Sim6's group . . . 8 1.3.7 Masdemont Ph.D. Thesis 10 1.3.8 Benettin . . 11 1.3.9 Bertachini . . . . 11 1.3.1 0 W. Kizner 13 1.3.11 James K. Miller . 14 1.3.12 Martin W. Lo and Kathleen Howell 15 1.3.1 3 First Earth Trojan Asteroid . 16 1. 4 Missions to Study the Sun . . . . 17 1.4.1 Missions in Operation . . 17 1.4.2 Missions in Development 1Q 1.5 Research Scheme . . . . 21 1.5.1 Research Papers . . . . . 21 v 1.5.2 1.5. 3 1.5. 4 1.5.5 1.5.6 1.5.7 1.5. 8 1.5.9 1.5 .10 1.5.11 1.5.12 High Level Mission Requirements . Mission Description Mission Geometry . . . . Navigation Analysis . . . Navigation Requirements Navigation Phases . . . . Introduction to Models . Problem 1: Trajectory mission and na vigation design using CRTBP in the Sun-Earth system ...................... . Problem 2: Stability Analysis of orbits using the BCP in the Earth Moon system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 3: Elliptic RTBP vs. CRTBP in the Sun-Earth system Problem 4: Trajectory mission and na vigation design using a high fidelity model (DE421) in Sun-Earth system ............ . 22 23 26 29 30 31 33 34 35 35 35 Chapter 2: Project Outline 37 2.1 Sun-Earth £4/Ls Mission and Navigation Design 37 2.2 Mission Requirements Development . 37 2.2.1 Trajectory Mission Design . . . . . . . . . 38 2.3 Stability Analysis . . . . . . . . . . . . . . . . . . 42 2.3.1 Other Quasi-Periodic Motions around the Triangular Points . 44 Chapter 3: Mathematical Models 57 3.1 Introduction to Models . . . . . 57 3.1.1 Deriva tion of Circular RTBP 58 3.1.2 CRTBP Model . . . . . . . . 60 3.1. 3 Linearized Motion around the Triangular Points 3.1. 4 Stability Analysis . . . . . . 3.1.5 Symplectic Diagonalization 3.1.6 Variational Equations . 3.1. 7 Monodromy Matrix . . 3.2 The Bicircular Problem (BCP) 3.3 Elliptical RTBP (ER TBP) ... 3.4 Developmental Ephemeris Model (DE421) 3.5 Transition Between Models ..... . 3.6 Analytical Ephemeris Model ....... . 3.6.1 JPL DE421 Ephemeris Model .. . 3.7 Ephemeris Transf ormation from Rotating to Inertial Coordinates 3.8 Ephemeris Transf ormation from Inertial to Rotating Coordinates 3.9 Coordinate Systems ... . 3.1 0 Navigation Model ....... . 3.1 0.1 Transfer Trajectory .. . 3. 10.2 Libration Trojan Orbit . 62 69 72 74 76 76 80 83 83 85 87 89 91 91 94 98 100 Vl Chapter 4: Research Gadgets 4.1 Mission Design Tools ....................... . 4.1.1 Dynamical Systems Theory ............... . 4.1.2 Periodic Orbits (PO) and Quasi-Periodic Orbits (QPO) 4.1. 3 Invariant Manifolds 4.1. 4 Lie Series Expansions 4.1.5 Poincare Sections . . . 4.2 Navigation Design Tools ... 4.2.1 Orbit Perturba tion for Navigation Analysis 4.3 Integrated Trajectory Process ........... . 4.3.1 Ta rgeting Strategy of Departure Trajectory 4.3.2 Ta rgeting Strategy for Transfer Trajectory . 4.3.3 Ta rgeting the Trojan Orbit ..... 4.3.4 Hyperbolic Orbital Elements . . . . . . . . 4.4 Testing Integrated Trajectories with Conics . . . . 4.4.1 Hyperbolic Osculating Conic near the Earth . 4.5 Can Gravity Assist Help? 4.6 Trajectory Propagators ............... . Chapter 5: Results for Mission Design and Navigat ion Design 5.1 Trajectory Mission Design Results ...... . 5.1.1 £5 Orbits in the Sun-Earth System .. . 5.1.2 Sub-£5 Orbits in the Sun-Earth System 5.1.3 Conic and Integrated Trajectories Test . 5 .1 .4 Stability Analysis . . . . . . . . 5.2 Navigation Analysis Results ..... . 5.2.1 Trajectory Maneuver Analysis 5.2.2 Scenario 1: 1% a priori error . 5.2.3 Scenario 2: 0.5% a priori error 5.2.4 Scenario 3: 0.2% a priori error 5.2.5 Prelimin ary Maneuver Analysis Results 5 .3 Changing Maneuver Locations . . . . . . . . 5.4 Knowledge Error and Delivery Accuracy . . . . 5.5 Maneuver Analysis of Integrated Trajectories . 5.5.1 Propulsion Requirements/Mass Performance 5.6 Science at £5 . . . . . . . . . . . . . 5.7 Comm unications and Power ....... . 5.8 Sensitivity of Velocity Requirements . . . 5.9 Mission Requirements Traceability Matrix 5.10 Research Products .. 5.1 0.1 Papers ............... . 5.1 0.2 Software Tools .......... . 5.10.3 Data Base of Feasible £5 and £4 Trajectories Chapter 6: Conclusions 102 102 102 103 108 114 116 123 124 125 126 128 131 131 132 135 138 138 139 140 140 155 157 159 163 163 164 164 164 164 169 171 175 177 180 181 182 183 184 186 186 187 198 Vll Bib liography Appendices Appendix A: Acronyms and Symbols Appendix B: Different ial Correctors B.l Halo Orbits Constrained Differential Corrector B.2 Trojan Orbits Constrained Differential Corrector .... 201 207 211 211 229 Vlll List Of Tables Table 2.1: f'o. V vs. TOF companson between Ls and L4 for different periodic orbits. 40 Table 2.2 : Un stable Orbits around the Ls point ..... ........... . . . 44 Table 4. 1: Jacobi Con stant of several orbits for Stable Manifold .. ........... 113 Table 5.1: f'o.V vs. TOF comparison between trajectories launched at different times of the year, case L5( 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Table 5. 2: f'o. V vs. TOF comparison between trajectories launched at different times of the year, case L5(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Table 5.3: f'o. V vs. TOF comparison between trajectories launched at different times of the year, case L5(3) ............................. 142 Table 5.4: f'o. V vs. TOF comparison between trajectories launched at different times of the year, case L5( 4) ............................. 142 Table 5.5 : f'o.V vs. TOF comparison between trajectories launched at different times of the year, case L4 ............................... 142 Table 5.6: Relevant parameters for departure trajectory ................ 143 Table 5. 7: f'o.V vs. TOF comparison between trajectories launched at different times of the year to sub-Ls orbits in the Sun-Earth system ............ 157 Table 5.8: Relevant parameters for departure trajectory to sub-Ls in the Sun-Earth system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Table 5. 9: Environme ntal Disturbances affecting spacecraft's orbit. Table 5.10: Maneuver Analysis for Case L5( 1) Table 5.1 1: Maneuver Analysis for Case L5(2 ) Table 5.12: Maneuver Analysis for Case L5( 4) 160 166 166 167 IX Table 5.13: Maneuver Analysis for Case L5(5) ......... . 167 Table 5.1 4: Knowledge Error and Delivery Accuracy for case L5(1 ) 172 Table 5.1 5: Knowledge Error and Delivery Accuracy for case L5(2) 17 3 Table 5.16: Knowledge Error and Delivery Accuracy for case L5(4 ) 17 4 Table 5.17: Knowledge Error and Delivery Accuracy for case L5(5) 17 4 Table 5.1 8: Propellant Mass and Duration of TCM and SKM at each location during the Transfer and Station Keeping for L5(1 ) . . . . . . . . . 177 Table 5.19: Mission Programmatic Requirement Traceability . . . . . . . . . . . . . 185 X List Of Figures Figure 1. 1: Schematic of the five La grange (and Euler) Points. . . . . . . . . . . . . 2 Figure 1.2: Corotating Interaction Regions hit Ls first before arriving at the Earth. . 3 Figure 1.3: Sketch of a mission to Ls (magenta lagging orbit). The Sun and the helio centric current sheet canying solar particles are depicted in orange. The curved red arrows point in the direction of the motion of the corotating regions arriving first at Ls , then 3 to 5 days later at Earth, and finally at L4 after another 3 to 5 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Figure 1. 4: Cycler Sun-Earth. a. shows an example of a cycler from [8] in the Earth system. b. shows similar simulated cycler during the second revolution in the Sun-Earth system. The trajectory was integrated from Ls . . . 12 Figure 1.5 : First Earth Trojan asteroid 20 10 T K7 [12] in the Sun-Earth system. . 16 Figure 1.6: Departure orbit from 200 km and 28 .5° inclination parking orbit around the Earth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 1.7 : Hyperbolic departure orbit . Figure 1.8 a: Schematics of La unch Phase from a 200-km par king orbit around the Earth. b: Transfer Phase from a 200-km par king orbit to an orbit around the Triangular Point Ls. c: Schematics of a Science Phase I at Arrival to the Triangular Point Ls. d: Schematics of the Insertion Phase into the orbit around Ls. e: Schematics 23 of a Science Phase II into the orbit around Ls . . . . .. . .. ........ 25 Figure 1.9 a: Range and Range Rate profiles of the Transfer Orbit. b: Range and Range Rate profiles of the Trojan Orbit. c: Shows the radial (red) and tangential (bl ack) velocity profiles for the transfer orbit. d: Shows the radial (red) and tangential (black) velocity profiles of the Trojan orbit. e: Sun Earth Probe Angle ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 XI Figure 1. 10: a: Four-year transfer orbit from Earth 200 km circular parking orbit to Ls . b: Range and Range Rate profiles of the Transfer orbit. c: Range and Range Rate profiles of the Trojan orbit. d: Radial (red) and tangen tial (black) velocity profiles for the Transfer orbit. e: Radial (red) and tangential (bla ck) velocities at Ls f: Acceleration at Ls. . . . . . . . . . . . . . . . 36 Figure 2.1: a: Three dimensional periodic orbits around the Sun-Earth Ls for differe nt inclinations in the CRTBP. b: Two dimensional periodic orbits around the Sun-Earth Ls. c: XZ projection. d: YZ projection. . 39 Figure 2.2 : Trade between TOF and L'l. V . . . 41 Figure 2.3: Preliminary orbital velocity error analysis for 5 years: Column 1: Large orbit (0.5 2 AU amplitude); Column 2: Small orbit (598 km amplitude). Top Row: ±1% error in velocity; Middle Row: ±0 .5 % error; Bottom Row: ±0.1 % error. Initial periodic orbit is black, blue orbits are for positive errors, magenta orbits are for negative errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 43 Figure 2.4: Acceleration felt by the spacecraft at different Trojan orbits. . . . . . . . . . . 44 Figure 2.5 : Examples of linearized periodic orbits around Ls integrated for 100 years. 45 Figure 2.6: Asymmetric Orbits around Ls Earth-Moon system .. 48 Figure 2.7: Asymmetric Orbits around Ls Earth-Moon system .. 49 Figure 2.8 : Asymmetric Orbits around Ls Earth-Moon system .. 50 Figure 2.9: Asymmetric Orbits around Ls Earth-Moon system .. 51 Figure 2.10 : Fa mily of Periodic Orbits around L4(left ) and Ls (right) in the Sun-Earth system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 2.1 1: Asymmetric Orbits around Ls in the Sun-Earth system for 15 years. . 52 Figure 2.12: Other interesting trajectories in the Earth-Moon system. . . . . . . . 53 Figure 2.1 3: SHORT L4 -Ls heteroclinic connection in the Sun-Earth system. The flight time of the transfer orbit from an Earth parking orbit of 200 km to an orbit around L4 is about 300.8 days. This transfer orbit is outside the path of the Earth around the Sun. The flight time of the heteroclinic connection from L4 to Ls is about 23 0.8 days. The closest approach to the Sun of this hetero clinic connection is about 0.3 46 AU. The total flight time (transfer and hete roclinic orbits) is about 531 .6 days. Notice the relative speed of the spacecraft along the transfer trajectory being slower when it is closer to the Trojan orbits of amplitude 0. 73 AU and larger when it is closer to the Sun. The gap between two tick marks represents about 30 days of travel. ........ 54 xu Figure 2.1 4: LONG L4-Ls heteroclinic connection in the Sun-Earth system. The flight time of the transfer trajectory from Earth is about 199.1 days to the Trojan orbit of amplitude 0.7 3 AU. This transfer orbit is inside the path of the Earth around the Sun. The flight time of the heteroclinic connection from L4 -Ls is 50 5.5 days. The spacecraft depart s from an orbit around L4, then app roaches L3 and finally arrives at an orbit around Ls in the Sun-Earth system. The total flight time (transfer and heteroclinic orbits) is about 70 4.6 days... 55 Figure 2.1 5: Heteroclinic connections between the sub-Ls Trojan orbits and L3 in the Sun-Earth system. a: Shows a LONG sub-Ls -L3 Orbit of about 14 years going from a sub-Ls orbit to L3 in the Sun-Earth system. The spacecraft stays in the vicinity of L3 for about 2 years, then returns to the sub-Ls orbit after 8 years. The total flight time is about 23 .7 years. b: Shows two-SHORT Ls-L3 Orbits going from a sub-Ls orbit to L3 in the Sun- Earth system. The SHORT Ls-L3 Orbit in red has a flight time of about six years. The spacecraft orbits in the vicinity of L3 for at least 8.5 years. The SHORT Ls-L 3 Orbit in purple has a flight time of about seven years and orbits L3 for about 7.5 years. . . . . . . . . . . . . . 56 Figure 3.1: The Five La grangian Points in the Sun-Earth system. SELl, and SEL 3 are the collinear points in the Sun-Earth system. and SEL 5 are the equilateral points in the Sun-Earth system. SEL2 SEL 4 Figure 3.2 : Sketch of forces acting on the Triangular Poin ts. a g .p and a g .s are the gravitational acce lerations on Ls due to the primary and secondary bodies. ag.t is the combined gravitational accele ration on Ls as if the total mass of the system was placed at the barycenter 61 of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 3.3 : K constant as function of the mean motion. Figure 3.4: Mean motion at the Triangular points for differe nt values of�- Figure 3.5 : a: General schematics of the Bicircular Problem with the Sun and Moon movmg m circular orbits. b: La grangian Points in the Earth-Moon 70 71 System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Figure 3.6: a: Transfer and Trojan trajectories using the CRTB P and the ERTBP. b: Close-up boxed region (near the Earth) in green to see the difference between the departing transfer trajectory when using both models. The orbit in magenta is used for the CRTB P and the one in blue for the ERTB P. . . . . . . . . . . . . . . . . . . . . . . . . 81 Xlll Figure 3.7 : a: Trojan trajectories using the CRTBP and ERTBP integrated for five years. b: Close-up boxed region of the Trojan orbit shows that the Trojan orbit (red) in the ERTBP is asymmetric whereas the Trojan orbit (black) is symmetric in the CRTBP. Note the direct behaviour of the Trojan orbit in the ERTBP as it moves towards Earth. The size of the Trojan orbit is the ERTBP. . . . . . . 82 Figure 3.8 : a. Tr ajectories of the Sun, Earth and the Moon in the DE 42 1 ephemeris model for one year. b. Tr ajectory of the Sun in the DE 42 1 model for 50 years. The red-star represents the beginning of lli the trajectory (March 8 2012) and the red-circle the end of the trajectory . . . . . . . . . . . . . . ....... . 85 Figure 3.9 : Schematics of the Inertial and Rotating Frames for the 3BP.. 92 Figure 3.10: Probability Confidence Intervals. The 99% confidence level can be obtained by integrating the probability density function from zero to 2. 576a for a half normal distribution. . . . . . . . . . . . . . . . . . . 98 Figure 3.1 1: Sketch of a transfer and a Trojan orbit broken into several segments. The transfer orbit (magenta) shows several maneuvers (red star) to be per- formed at different locations after launch, midcourse, and arrival. The Trojan orbit (blue) shows several maneuvers (red star) right after insertion, during midcourse along the Trojan orbit, and before completing one Trojan revolution. . . . . . . 99 Figure 3.1 2: Left. Mapping by state transition matrix to the time of the maneuver. Right. Cov ariance mat rices mapped from the time of the maneuver to the end time, Tend . . . . . . . . . . . . . . . . . . 100 Figure 4.1: Schematics of the Tools used for the Tr ajectory Mission and Navigation Designs ..... Figure 4.2 : Halo, Lissajous and Trojan orbits obtained using a two level diffe rential corrector. The beginning of each segment is represented by a red star and the end of each segment by a red circle. Note that the less chaotic the initial guess is the more periodic the Trojan orbit becomes. Figure 4.3: Differentially corrected asymmetric orbits around Ls in the Sun-Earth system. Every loop or bounce is one Earth year long. Figures (a, c and e) show some horseshoe-like orbits with loops that do not enclose the Ls point. Figures (b, d and f) display large loops ( of the 10 3 10 5 order of half a lunar distance) which enclose the Ls point. . . . . . . . 10 6 XIV Figure 4.4: Schematics of the manifold theory. The stable manifold is in green and the unstable one in red. The stable manifold app roaches the equili brium point while the unstable manifold departs from it. d is the offset distance from a point on the orbit around the equilibrium point so in some sense these manifolds are the shadow of the real trajectories. . . . . . Figure 4.5 : Stable Manifolds Ls. These trajectories are integrated backwards m time from the Trojan orbit. Thus, the trajectory forward in time is 110 the trajectory that the spacecraft would follow in reality. The problem is that these trajectories are too long and not suitable for transfers from Earth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Figure 4.6: Error between the integrated solution obtained in Mathe matica © and the Lie series solution. Top left and right figures show the errors of x and y, respective ly, for a large Trojan orbit with amplitude of about 0.5 3 AU. Bottom left and right figures show the errors of x and y, respecti vely, for a small Tr ojan orbit with amplitude of about 100 km. These errors are lower for smaller orbits. . . . . . . . . . . . . . . . . . . . . . 115 Figure 4.7 : a: Set of trajectories integrated in the interior region of the Earth's path around the Sun from the initial points. b: Initial Poincare section. c: Initial trajectories integrated from initial points. The thick black line is the chosen plane where the trajectories intersect. . . 118 Figure 4.8 : Surface of Sections (positive side) in the Sun-Earth system. a: There are 100 130 intersections. b: There are 400628 intersections. . . . . . . 119 Figure 4.9 : Top: Magnified surface of section in the positive region from Figure 4. 8(a ). Bottom: Magnified surface of section in the negative region. Sea of islands indicating the periodi city of orbits. Empty regions indicate forbidden regions or regions where the spacecraft never passes or crosses the surface of section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure 4.10: Close-up of surface of section shown in Figure 4.8 (a). has low velocities, between 4 km/s and 7.5 km/s. . . Figure 4.1 1: Close-up of surface of section shown in Figure 4. 9(b). has large velocities, around 47 km/s. . ....... . The The spacecraft spacecraft Figure 4.1 2: L'l. V for Navigation App lications for different flight time durations of 122 123 two transfer trajectories. Each trajectory was initially perturbed either in position or veloc ity, or in both position and veloc ity. . . . . . 12 4 Figure 4.1 3: Departure B-plane at Earth shows an envelope of possible departure trajectories from a 200-km circular Earth parking orbit. . . . . 127 Figure 4.1 4: Flow chart of the integrating trajectory process . 129 XV Figure 4.1 5: Sketch depicting the patched conics approach to design trajectories. The mean motions of the Earth and spacecraft are nearly commensurable. l 34 Figure 4.1 6: B-plane targeting from L+ 10 days to desired conditions at launch. . . 136 Figure 5.1: One-year integrated trajectory to Ls. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. e: XZ zoomed in projection. f: YZ zoomed in projection. The January, Ap ril, July and October orbits are in blue, green, magenta and brown, respectively. The orbit in gold (see Figure 5.1 (b)) is the solution obtained in the CRTBP. 144 Figure 5.2: Two-years integrated trajectory to Ls. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. e: XZ zoomed in projection. f: YZ zoomed in projection. The January, Ap ril, July and October orbits are in blue, green, magenta and brown, respectively. The orbit in gold (see Figure 5. 2(b )) is the solution obtained in the CRTBP. 146 Figure 5.3: Targeting inclination by varying angle theta on the B-plane. . . . . . . 149 Figure 5.4: Leading in-lagging out two years transfer trajectory to Ls. The size of the Trojan orbit is about 0. 73 AU . . . . . . . . . . . . . . . . 150 Figure 5. 5: Two-and-one-half years transfer trajectory to Ls. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. The January, Ap ril, July and October orbits are in blue, green, magenta and brown, respectively . The orbit in gold (see Figure 5.5(b )) is the solution obta ined in the CRTBP. . . . . . . . . . . . . . . . . . . . . . 15 1 Figure 5.6: Transfer trajectories for different time of flights to Ls. a: 3D trajectory plot. b: XY projection. c: XZ projection for Ap ril, July and October launches. d: YZ projection for Ap ril, July and October launches. e: XZ projection for January launch. f: YZ projection for January launch. . . . 152 Figure 5. 7: Year-and-one-half transfer trajectories to L4. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. The January, April, July and October orbits are in blue, green, magenta and brown respectively. The orbit in gold is the solution obta ined in the CRTBP. 15 4 Figure 5.8: Different trajectories about sub-Ls in the Sun-Earth System. Green orbit has a flight time of 77 8.5 days and 777 .6 days, respectively . a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. 15 6 Figure 5. 9: Conic test. a: Comparison between an integrated trajectory and the conic trajectory. b: Close-up of Trojan orbit around Ls. c: Range and range rate for integrated and conic orbit. d: Tangential and radial velocity profiles for integrated and radial veloc ity. . . . . . . . . . . . . 15 8 XVI Figure 5.10: Displacement of Triangular Points. . . . . . . . . . . . . . . . . . . . 159 Figure 5.1 1: Sun's and Moon's orbital perturbations in the BCP. . . . . . . . . . . 162 Figure 5.12: Feasible Mission Design for different flight time durations of selected Transfer trajectories to Ls. Top left: Feasible Transfer Orbit of about 34 3 days to Trojan orbit of amplitude of 0.5 2 AU. Top right: Feasible Transfer Orbit of about 70 1 days to Trojan orbit of amplitude of 0.5 2 AU. Bottom left: Feasible Transfer Orbit of about 372 days to Trojan orbit of amplitude of about 0.0 47 AU. Bottom right: Feasible Transfer orbit of about 80 1 days to Trojan of amplitude of about 0.0 47 AU. . . . . 165 Figure 5.13: TCM and SKM L'I.V Cost. Top left: Maneuver Analysis of selected transfer trajectories for a 10% error. Top Right: Maneuver Analysis of selected transfer trajectories for a 1% error. Bottom left: Maneuver Analysis of selected Trojan orbits for a 10% error. Bottom right: Maneuver Analysis of selected Trojan orbits for a 1% error. . . . . . . 17 1 Figure 5.1 4: 99% confidence level for a fixed a priori position error. The magenta and black lines correspond to the L'l. V 99% of selected transfer and Trojan orbits, respectively. The solid lines represent CaseL5( 1) and the dash lines CaseL 5(2). The red dashed line represents the assumed upper bound confidence level of 150 m/s for both the transfer and Trojan orbits (individually). . . . . . . . . . . . . . . . . . . . . . . . 17 5 Figure 5.15: 99% confidence level for a fixed a priori velocity error. The magenta and black lines correspond to the L'l. V 99% of selected transfer and Trojan orbits, respectively. The solid lines represent CaseL5( 1) and the dash lines CaseL 5(2). The red dash line represents the assumed upper bound confidence level of 150 m/ s for both the transfer and Trojan orbits (not combined). Note that the assumed a priori velocity errors ( - 5 m/s) for the transfer orbit are very large, which is equiva- lent to about 5% error of the value of the first TCM. For the Trojan orbit, these a priori velocity errors ( - 1 m/s) are also large or about 10% of the first SKM for the studied Trojan orbits. . . . . . . . . . . 176 Figure 5.16: Mass Capability as a function of altitude. Atlas V (401) and Fa lcon 9 (Block 1) have the capability of carrying a wet mass of 35 00 kg to a parking orbit of 200 km and 28 .5° inclination. . . . . . . . . . . . . . . 179 Figure 5.17: GUI: The black trajectory represents the periodic orbit around Ls depicted by a red circle, the magenta orbit has a direct motion toward Earth and the blue trajectory has a retrograde motion away from the Earth. The dashed blue line represents the path of the Earth around the Sun. Both nominal and perturbed trajectories are 5 years long. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 XVll Figure 5. 18: TOPES (in magenta) from Earth to a Trojan orbit (in black) around Ls in the Sun-Earth system. a: One-year transfer orbit to a Trojan orbit of 0.5 2 AU amplitude. b: Two-years transfer orbit to a Trojan orbit of 0.5 2 AU amplitude. c: Two-and-one-half-y ears transfer orbit to a Trojan orbit of 0.5 2 AU amplitude. Note that the insertion bum occurs outside the path of the Earth around the Sun. d: One-year transfer orbit to a Trojan orbit of 0.0 47 AU amplitude. . . . . . . . . 18 8 Figure 5.19: TOPES (in magenta) from Earth to a Trojan orbit (in black) around Ls in the Sun-Earth system. a: Two -years and two months transfer orbit to a Trojan orbit of 0.0 47 AU amplitude. b: Four-years transfer orbit to a Trojan orbit of 0.0 47 AU amplitude. c: One year and one month - transfer orbit to a Trojan orbit of about 180,000 km amplitude. Note that the insertion burn occurs outside the path of the Earth around the Sun. d: One year and one month-transfer orbit to a Trojan orbit of 180,000 km. . . . . . . . . . . . . . . . . . . . . . . . . 189 Figure 5. 20: TOPES (in magenta) from Earth to a Trojan orbit (in black) around Ls in the Sun-Earth system. a: Two years-transfer orbit to a Trojan orbit of 28424 krn amplitude. b: Four-years transfer orbit to the equi lat eral Ls point. . . . . . . . . . . . . . . . . . . . . 190 Figure 5. 2 1: TIPES (in magenta) from Earth to a Trojan orbit (in black) around L4 in the Sun-Earth system. a: Four-years transfer orbit to a Trojan orbit of 28424 krn amplitude. b: Ten-months transfer orbit to a Trojan orbit of 28424 krn amplitude. c: Nearly nine -months transfer orbit to a Trojan orbit of 0.0 47 AU amplitude. d: One-year and eight months transfer orbit to a Trojan orbit of 0.5 2 AU amplitude. . . . . 19 1 Figure 5. 22: TIPES and TOPES (in magenta) from Earth to a Trojan orbit (in black) around L4 in the Sun-Earth system. a: Nearly seven-months TIPES to a Trojan orbit of 0.7 3 AU amplitude. b: Ten-months TOPES to a Trojan orbit of 0.7 3 AU amplitude. These are the same orbits used for the heteroclinic connections studied in this work. . . . 192 Figure 5.23: One-year integrated orbits to Ls in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in April, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equilateral point Ls is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 5 days in advance before they reach Earth. . 19 3 XVlll Figure 5. 2 4: Two -years integrated trajectories to Ls in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in Ap ril, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equila teral point Ls is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 5 days in advance before the solar events reach Earth. . . . . . . . . . . . . . . . 19 4 Figure 5. 25: Two and a half years integrated trajectories to Ls in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in Ap ril, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equila teral point Ls is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 5 days in advance before the solar events reach Earth. . . . . . . . . . . . . 19 5 Figure 5. 2 6: Several integrated trajectories to Ls in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in April, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equilateral point Ls is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 6 days in advance before the solar events reach Earth (green and blue orbits). . . . . . . 19 6 Figure 5. 27: Sub-Ls integrated orbits to Ls in J2000 coordinate frame. Space weather can be anticipated up to 7 days in advance before the solar events reach Earth (green and purple orbits) .. .......... ... 197 Figure B.1: Constrained differential corrector. a: Angle between the velocity vector L'l. Vx y and the x-direction of the rotating frame. b: Angle between the position vector of the spacecraft and the x-direction of the rotating frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Figure B.2: Constrained differential corrector. a: Angle between the position vector of the spacecraft and the x-y plane of the rotating frame. b: Angle between the velocity vector of the spacecraft and the local horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Figure B.3: Constrained differential corrector. Schematics of the inclination . 215 XIX Abstract This thesis research will be based on the trajectory mission design and na vigation design for prospective future missions to the Triangular Lagrange Points L5 and L4 in the Sun Earth and Earth -Moon systems. The research proposed here will be divided into four parts. The first problem will be devoted to studying the circular restricted three-body prob lem (CRTBP) in the Sun-Earth sys tem. With this model, we will generate potential optimized orbit solutions in the planar CRTB P and also in three-dimensional orbits in order to study the Sun above the ecliptic plane. Orbit determination analysis will also be examined using different orbit determination methods. Finally, we will analyze the stability of the trajectories and their sta tionkeeping requiremen ts. The second part of this thesis will deal with the bicircular problem (BCP) in the Earth-Moon system. As in the work on the CRTBP, we will understand and analyze the stability of the different types of periodic orbits (quasi-periodic orbits) obtained under the influence of the Moon and the Sun. The third part will describe the elliptic restricted three-body problem (ER TBP) in the Sun-Earth system. As in the work on the CRTBP, we will analyze the stability of the different types of periodic orbits (quasi-periodic orbits) obtained due to the effects of the eccentricity of the Earth around the Sun. We will partially analyze the BCP and ER TBP but the main fo cus of the research will be based on the CRTBP and the JPL Ephemeris Model. XX The last problem is the new JPL Ephemeris Model, DE421. With this ephemeris model, we will determine how accurate the models CRTBP, BCP and ER TBP are in comparison with the real one. By studying the real model, we will have a more thorough insight into why some of the orbits obtained in both the CRTBP and ER TBP lose their symmetry when adding the influence of higher order perturba tions into the dynamical model. Besides finding periodic and quasi-periodic orbits for different models, part of this trajectory mission design will be dedicated to the optimization of the trajectory, utilizing a differential corrector. Finally, we will close this section by developing some semi-analytical work based on different techniques, such as the Lie Series expa nsions. We will use these methods to have better approxi mations of the nonlinear problem in the neighborhood of the triangular points and to obtain a more accurate analysis of the stability of these orbits. Along with the trajectory mission design, part of this thesis work will be oriented towards the orbit determination analysis from the beginning of the mission at a predefined parking orbit around Earth to the end of the mission at the Libration Orbit (Trojan orbit) around the triangular points. Orbit determination will be needed to provide a more accurate estimation of the trajectory of the spacecraft at different stages: launch, mid-course and arrival . We know that after the launch phase, the spacecraft will be sensitive to large errors that make the spacecraft deviate from the nominal trajectory. The main goal will be to determine the state of the spacecraft as accurately as possible. We know that the state of the spacecraft is determined from the measuremen ts, such as range or Doppler data. Given these launch errors, we will have to perform correc tion maneuvers to adjust the perturbed trajectory to go back to the nominal trajectory or an alternate trajectory that satisfies the mission requiremen ts. Can we achieve this with a single correction maneuver? The answer is "No" for several reasons. First, the dynamical model is not perfect, even for our most realistic models. Secondly, the measurements have uncertain ties. Thirdly, the XXl spacecraft trajectory can only be estimated. Finally, each trajectory correc tion maneuver also has its own sources of execution errors. XXll Preface The main goal of the research in this thesis will be to understand and analyze the tra jectory and na vigation design from a low Earth parking orbit to the Triangular Points of the Sun-Earth as a venue for prospective fut ure space missions. Chapter 1 describes the statement of the problem. We talk about the background and main motivation of this thesis research work: Ls Lagrange Antenna Observat ory of Sun. We describe the State of the Art by giving the main literature review of this thesis work starting from the very first work done on Lagrangian mechanics back in 1772 to the most recent studies done by the Carles Sim6 group a few years ago. We also provide a literature review for Orbit Determination and Space Navigation and the missions that are currently studying the Sun. In Chapter 2, we will describe the project outline for the trajectory mission and na vigation design. We specify the main mission requirements and the mission phases. Chapter 3 introduces the dynamical models that we will study and describes each of the main problems. We also describe the mission and na vigation requirements for each problem. This chapter deals with the mathematical models that will be used, i.e., the circular restricted three-body problem (CRTBP), the bicircular problem (BCP), the elliptical restricted three-body problem (ER TBP) and the new JPL Ephemeris (DE 421), which are described in detail. The CRTBP and JPL DE421 models are used to examine the trajectory mission and na vigation design, while the other two models are studied for the stability analysis only. We then provide some reasoning for the use of the real model and how to transition from the circular model to higher fidelity models. In this sec tion, XXlll we describe the different coordinate systems, such as the rotational and inertial frames and the transition between different frames. Chapter 4 describes some of the tools that will be used in this thesis work. We start with a description of some concepts about dynamical system theory, such as periodic and quasi-periodic solutions around the equilateral points, invariant manifold theory, Poincare sections and Lie Series Expansions. Then some of the mission design tools, such as building several differential correctors to create periodic orbits and optimize the transfer trajectories, ephemeris tools and trajectory propagators are explained. This section closes with some of the visualization tools that we will use to simulate the work. Finally, several orbit determination techniques will be presen ted. One of these methods utilizes the Personal Computer Orbit Determination Program (PCODP) software which has been extended with the capability of an £5-tool as described. Chapter 5 delivers the expected results and products from this research . Some of the deliverables are trajectory design, stability analyses of periodic and quasi- periodic orbits for different models, trajectory correction maneuvers and orbit determination analyses, power, communications and mass performance analyses. Finally, we list the products of these analyses such as papers, software tools and databases for potential £5 missions. The contents of this research thesis correspond to material that I have developed at the University of Southern California, the Jet Propulsion Laboratory and in private communications with Mr. James K. Miller. At the end of the body of this thesis, we will deliver the list of references used in this work, including the ones that I have published and that are curren tly under research . Finally, this work will end with several Appendices that provide a list of acron yms, and descriptions of several differential correc tors. XXIV Chapter 1 Research Overview 1.1 Statement of Problem This thesis will consider the trajectory, mission design and na vigation for missions to the triangular Lagrange points in the Sun-Earth and Ear th-Moon systems using dynamical systems methods. 1.2 Background and Motivation: £5 Lagrange Antenna Observatory of Sun The restricted three-body problem consists of a primary body (larger mass) and a sec ondary body that revolve around their barycenter and a third body (spacecraft ) that moves in the same plane as the primaries with a negligible mass. The third body has no effect on the primary and secondary bodies under their gravitat ional fields. In the simplest case, when the primary and secondary bodies move in circular orbits, we say that this is the circular restricted three- body problem ( CRTB P). In the restricted three- body problem (R3B P), there are five equilibrium points as we can see in Figure 1.1. Three of these points are collinear lying on the same axis as the primary and secondary bodies and two are equilateral since they each form an equilateral triangle with the primaries. 1 Figure 1.1: Schematic of the five Lagrange (and Euler) Points In 2005, Akioka et al. [1] described an interesting Sun-Earth 15 mission with the Japanese Space Agency's National Space Development Agency (NASDA) and the Insti tute of Space and Astronautical Science (ISAS) as part of an international space weather observations network for in situ measurements of solar wind plasma and high-energy solar particle events. An L5 mission will be ideal for early space warnings to detect Coronal Mass Ejections (CMEs) and observe Corotating Interaction Regions (CIRs) before they hit Earth. These CIRs are aligned to the Parker spiral (see Fig.1.2) and rotate in the direction of planetary motion arriving first at L5, then 3 to 5 days later at Earth, and finally at L4 after another 3 to 5 days [22]. Hence, a mission at L5 can provide up to 5 days of advanced warning of solar storms which can wreak havoc on Earth's telecom- munications infrastructures as compared to only 1 hour warning provided by missions at L1 such as the Advanced Composition Explorer (ACE). There are other reasons why L5 is a very attractive point for in-situ scientific measurements that can clarify some of the unknown questions about CMEs and CIRs. Some of these questions are: How do CMEs and CIRs relate to solar magnetism? What is the source of the energy of these CMEs? How are particles accelerated by CMEs and what is the influence of solar flare reconnec- tion? Where and when are shocks formed in the corona and what are the mechanisms 2 that trigger their evolution? What are the processes of CMEs and CIRs that cause mag- netic storms? Therefore, in order to better understand some of these unknowns, we need to take accurate measurements from the solar interior to the atmosphere and into the heliosphere. Moreover, by placing a spacecraft at £5, we can resolve ambiguities about whether aCME is coming towards Earth or away from it. In addition, the solar wind shock and the co-rotating interaction region ( CIR) are also important goals for observation (see Figure 1.2). For this reason, non-thermal particles will be the object of study because they include pick-up ions originating within the solar system. It is believed that these pick-up ions are accelerated by the shock at the end of the heliosphere. The mechanisms that trigger such ion accelerations and generation are still under study. Therefore, if we had data from these heavy ions, non-thermal ions, and high-energy particles (HEP), we could create an entire spectrum of accelerated particles that could better explain the HEP acceleration mechanism in the bow shock and CIR. These ions, which act like thermometers in the solar wind plasma, carry significant information about the solar corona such as differences between the high and low solar winds emanating from the coronal holes. Figure 1.2: Corotating Interaction Regions hit £5 first before arriving at the Earth 3 As seen from the L15 point, there is an east limb of the sun and a west limb of the sun. The Geoffective (GEO in Earthward direction) CMEs are in front of the limb and the Solar Energetic Particles (SEP) CMEs are behind the limb. Both GEO CMEs and SEP CMEs can be measured with minimal projection effect [22] . Solar magnetic field lines of the CIRs in the interplanetary medium would rotate to Earth in approximately 3 to 5 days causing geomagnetic storms that can interrupt communications between satellites in orbit around Earth. The geometry of some of our orbits investigated in this thesis provide an additional day of solar weather prediction, yielding up to 6-7 days of advanced warnings of solar storms at Earth. Furthermore, information obtained will enable reducing radiation risks in hum an space flights. Figure 1.3: Sketch of a mission to L15 (magenta lagging orbit). The Sun and the heliocen tric current sheet carrying solar particles are depicted in orange. The curved red arrows point in the direction of the motion of the corotating regions arriving first at L15, then 3 to 5 days later at Earth, and finally at L4 after another 3 to 5 days. 4 Figure 1.3 shows the heliocentric current sheet (HCS) emanating from the Sun and separates opposite magnetic polarities in the northern and southern hemispheres within the interplanetary medium. This warping occurs because the Sun's equatorial plane is inclined ± 7° with respect to the ecliptic [2] . These current sheets carrying high speed plasma take place at around 15° and 20° latitudes above the equatorial plane. The magnetic structures drift toward the equator of the Sun causing the warps. Sun spots do not appear at random points over the surface of the Sun but are concentrated in two latitude bands on either side of the equator. These bands form at mid-latitudes, widen, and finally move toward the equator. However, these active magnetic structures get dissolved and drift towards the poles during solar minimum. The interplanetary medium transports the solar energy that is produced in the out ermost layer of the Sun or corona. The corona holes are cool, low in brightness, low in density with unipolar magnetic fields that extend along the divergent magnetic field lines reaching Earth and causing geomagnetic storm s. This energy emanating from the Sun varies over the 11 -year solar cycle. Reversal of the polar magnetic field occurs near the time of the cycle maximum. Thus, the magnetic field occurs every 22 years and it comprises two solar cycles. In 2008, the Interstellar Boundary Explorer (IBEX) mission was launched to image the region between the termination shock and the heliopause and the heliospheric sheath [23] . This mission was initially going to be launched toward one of the equilateral points but it was finally deployed in a high-apogee orbit in the Eart h-Moon system using a small launcher (Pegasus ). These points offer excellent locations to deploy a spacecraft for extreme ultraviolet (EUV) imaging because the spacecraft will be out of the geocorona and magnetosphere zones and will be able to do science for almost 100% of the time. As we can see, the Akioka et al. and Gruntman et al. studies concentrated on the science and instruments and provided relatively little information on the mission design. This observat ion persuaded us to find new promising trajectory mission designs and na vigation orbit determination analyses. 5 1.3 State of the Art 1.3.1 Lagrange's first paper Joseph Louis Lagrange (1736-1813) discovered the lib ration points £4 and £5 after Leon hard Euler (1707-1783) first discovered the collinear libration points £ 1 , £ 2 , and £3 when he was working on a Lunar Theory. Figure 1.1. shows the 5-libration points in a rotat ing frame where the two primary bodies are fixed on the x-axis with the primary mass near the origin and the secondary mass on the + x side. £4 and £5 are also called the equilateral or triangular points [36] since they each form an equilateral triangle with the primary and the secondary bodies. In 1772, Lagrange was studying the linear stability of the equilateral points in the Sun-Jupiter system and this study led him to consider the possibility of asteroids being captured around £4 (called the leading Greeks) and £5 (called the trailing Troja ns). Although, nowadays these points are referred as libration points or Lagrange points, it was Gylden in 1884 who referred these points as centers of lib ra tion. 1.3.2 Szebehely Szebehely's classic Theory of Orbits (1967) is perhaps one of the best introductions to the CRTBP, providing a detailed introduction to the problem of three bodies and fo rmulating the equations of motion in both inertial and rotating fr ames. [65] also gives an excellent discussion of the stability of the triangular points and of the periodic orbits with many classical references and important characteristics of the five Lagrangian points, such as the motion of a satellite around the equilibrium points. 1.3.3 Akioka papers As described earlier, in 2004, Akioka et al. [1] described an interesting Ls mission with the Japanese Space Agencies, the National Space Development Agency of Japan (NASDA) and the Institute of Space and Astronautical Science (ISAS) as part of an internat ional 6 space weather observa tions network for in situ measurements of solar wind plasma and high-energy solar particle even ts. This problem has been addressed recently and is focus ing a lot of attent ion on how to perform this mission. Their papers were focused on the science and instruments and provided relatively little information on the mission design. The implementa tion of the Ls mission was stated as fo llows: First the spacecraft will be launched to an approximately 200-km Earth parking orbit and then will be accelerated by approximately 3.4 km/s until escape velocity is obtained. Once in the transfer orbit, the spacecraft will take 1.2 years to reach the £5 point before decelerating by an amount of .6.. V =1. 7 km /s which would allow the spacecraft to enter a stationary orbit at the Ls point. They also believed that the instrument used may be able to measure pick up ions and reflected and accelerated particles for a 45° ± 15° view angles, electrons with an azimuthal angle distribution of 30°, and solar wind protons at 45° incident angles. As we can see, there is very little inf ormation given about the mission design such as the mission phases or mission geometry or the trajectory design. Therefore, this research thesis will be based on the analysis of libration missions in the Sun, Earth, Moon systems, including mission design, trajectory optimization, and maneuver design and na vigation analysis of nonlinear low energy trajectories for space missions to the equilateral points in the context of Three-Body Problems. 1.3.4 M. Gruntman, M. Lampton, J. Edelstein In 2005, Gruntman et al. [23] proposed deploying a spacecraft to one of the libration points £ 3 , L4 or £5 in the Ear th-Moon system. Placing an interplanetary platform at the triangular points would provide quiet observat ional conditions for the heliosphere EUV imaging. With this imaging, we will be able to establish directional and spectral properties of the glow of interstellar plasma and pick-up ions, solar wind plasmas, and galactic emissions. These new measurements will open a new window for mapping the heliopause and reveal the three-dimensional flow pattern of the solar wind, including 7 the flow over the Sun's poles. At a distance of about 384,400 km from the Earth, the spacecraft will be out of the influence of the geocorona and the magnetosphere with uninterrupted observa tions for almost 100% of the time. The implementation of the mission was stated as fo llows: "The spacecraft will be deployed to the Ear th-Moon £4 and Ls points using lunar gravity assist. The spacecraft will depart from a low Earth parking orbit and will require b. V < 3.3 km/s, including about 3.1 km/s for the launch phase, 0.1 km/s for midcourse maneuvers and 0.1 km/s for phasing into the libration-point sta tion." 1.3.5 R. A. Freitas, Francisco Valdes In 1980, Freitas et al. [17] suggested that halo orbits around the £4 and Ls points could be among the best places to start a search for evidence of ancient objects parked in the Earth-Moon system. It is known that there are objects trapped in these points in the Sun Jupiter system. Howeve r, the stability in the Earth-Moon system is very complicated due to the gravitat ional influence of the Sun and these points are no longer stable. Although these points are not stable, we could find stable orbits around the equilateral points that may serve as loci for a network of interplanetary surveillance and communication relay stations. 1.3.6 Carles Sima's group From a purely trajectory design perspective, Carles Sim6's group has presented the most extensive modern studies of orbits around the Lagrange points in their fo ur-volume opus along with many papers. Howeve r, their practical applica tions to £4 and £5 missions fo cused on the Earth-Moon and the Sun-Jupiter systems, and not on the Sun-Earth system. Extended analytical and numerical simulations of motion in the neighborhood of the triangular points in the Earth-Moon system were done using different models of motion for the solar system. According to Gomez et al. [18], there is a lack of stability of the motion 8 very close to L4 and Ls. Some of the stability studies show a strong sensitivity with respect to the model of the solar system used, the initial epoch and initial conditions. Gomez et al. concluded that when the solar gravitat ional effects are included in the Earth-Moon model, the system becomes non-autonomous and, therefore, L4 and L5 are no longer L4 and Ls points. However, the libration orbits are not fully destroyed because the satellite will still be able to orbit around the equilibrium points. However, since the satellite is not exactly at the equilibrium point, station keeping may be needed. The cost of the station keeping will decrease for larger satellite libration orbits around the equilibrium points. In order to further study the stability of these orbits, [18], [20] analyzed the normal form (expansion of the Hamiltonian around the libration point) in the CRTBP for the Earth-Moon system around the L4 and L5 points in order to generate periodic and quasi periodic orbits. Gomez et al. believe there may be orbits in the Earth-Moon system, taking into account the full solar system in its real motion, that may be in a close neighborhood of the triangular points that can be maintained with mild sta tion keeping. Whether these orbits exist and are linearly stable, we may encounter powder clouds orbiting around the triangular points. If we can find such clouds, then this means that they are mildly unstable or that the stability region is rather small. As examples of unstable orbits, they fo und that taking as a starting point one of the equilibrium points L4 and Ls, the spacecraft would spiral away from the equilibrium point until it reaches a size of the order of the mean distance between the Earth and the Moon. Then, the spacecraft continues spiraling inwards until it approaches the equilibrium point again. This behavior repeats itself until the spacecraft escapes after it finds a close encounter with either the primary or secondary body. In this thesis work, we will present similar orbit scenarios. Gomez et al. also characterized the normal form around small periodic orbits near the L5 point in the BCP and considered the quasi- periodic model including the full solar system and radiation pressure. For the BCP, they have fo und periodic orbits of moderate 9 size in the neighborhood of £4 and Ls. They classified these orbits into the well-known short period and long period families around these points. They saw that small changes using the Normal Form method produce large variat ions in the size of the periodic orbits or they tend even to disappear. These variat ions are due to the lack of convergence of any of the methods applied that use series expansions around £4 and Ls which may be caused by the fact that, at the triangular locations, the frequencies of the perturbations of the Sun, the perturba tions due to the eccentricity of the Moon, the out-of-plane oscillations and the proper short period are very close to each other. Gomez et al. concluded that regardless of the mildly unstable periodic orbits (with small components out of the plane in the Earth-Moon system ) £4 and Ls are two useful locations to place either a single satellite or two spacecraft for scientific purposes due to the stability properties. The spacecraft could be maintained in orbit by using minimum amounts of fuel. 1.3.7 Masdemont Ph.D. Thesis Masdemont [4 7] presents a study of orbital mechanics problems using invariant manifold theory. Masdemont provides an excellent study of the invariant manifolds of the triangular points and the different homoclinic and heteroclinic orbits between these points. He analyzed the shape and number of these orbits, such as the heteroclinic type I orbits between the £4 and £5 and between £5 and £4, for different mass parameters. Masdemont further analyzed some of the homoclinic type 0 orbits between £4 and £4 and between Ls and Ls. He concluded that we could find an infinite number of 0-homoclinics and 1-heteroclinics orbits for £4 and Ls, Ls and £4, £4 and £4 and Ls and Ls for certain critical mass parameters. Masdemont also applied the Floquet Theory to halo orbit fo rmation, generating in variant manifolds and analyzing different transfers between halo orbits in the circular restricted three- body problem where the influence of the Moon was also considered. 10 In this thesis work, we will investigate some interesting heteroclinic connections be tween the triangular points L4 and Ls and between Ls and the collinear point L3 in the Sun-Earth system. 1.3.8 Benettin In 1998, Benettin et al. [6] studied the Nekhoroshev-stability at L4 and Ls in the CRTBP for different mass parameters up to the Routh critical value. In order to obtain more robust conclusions about the stability, they constructed the Birkhoff normal forms of order four and eight and obtained Taylor series expan sions of the Hamiltonian. Diff erent stability analyses were derived from these expa nsions. They concluded that, depending on whether the system is directionally quasi-convex or only steep, they observed a worsening of the stability estimates near the resonanc es. According to Benettin et al., weaker stability properties might be fo und around the L4 and L5 points in the Earth-Moon system. 1.3.9 Bertachini According to Bertachini [9], libration points are very important for astronautical appli cations. L 1 is important because it can be considered a node or parking orbit where we could store fuel and supplies that a spacecraft would need to return back to Earth without needing to go to the Moon. L 2 is also important because the spacecraft would be able to see the Earth and the far side of the Moon at the same time for large periods of time. Finally, the L4 and Ls points are very interesting because we could keep a Space Station there due to the stability of the periodic orbits around the Triangular points. In 1994, Bertachini et al. [8] studied transfer orbits in the Sun-Earth planar restricted three- body problem for transferring a spacecraft from one body back to the same body after passing through several Libration points or for transferring a spacecraft from one body to the Triangular Points via a swing-by applica tion. With this trajectory applica tion, we can build a cycler transportation system between the Earth and the Triangular 11 Cycler in the Sun-Earth system (a) Berthachinni Cycler (b) Simulated Cycler Figure 1.4: Cycler Sun-Earth. a. shows an example of a cycler from [8] in the Sun-Earth system. b. shows similar simulated cycler during the second revolution in the Sun-Earth system. The trajectory was integrated backwards from Ls Points. Assuming the spacecraft starts at the Ls point with zero velocity, they found that applying a small impulsive maneuver could place the spacecraft into a transfer trajectory until it would eventually approach Earth. Then it could make a swing-by to finally return to the Ls point again with a different velocity. At this point , we have the possibility of performing another impulsive maneuver such that the spacecraft would follow the cycler again . Bertachini [8] also investigated different transfer orbits between the Earth and the Lagrangian points with minimum .6. V. Some of these transfer orbits were classified according to their time of flight s as SHORT-4-5, SHORT-5-4, LONG-4-5 and LONG-5-4. The short type orbits have smaller time of flights than the long type orbits. In some of these scenarios where the spacecraft approaches Earth after the end of the first rev- olution, the spacecraft will continue on a similar trajectory during its second revolution as we illustrate in Figure 1.4(b) such that there will be several crossing points between both trajectories for both revolutions. These crossing points are very good candidates for a one-burn maneuver to transfer the spacecraft from a trajectory it follows in the second revolution to the trajectory it followed in the first revolution. It is also possible that two impulsive maneuvers can be performed rather than a one-impulse maneuver. The advantage of this maneuver strategy is that we can keep the spacecraft parked at Ls 12 indefinitely so that we could refuel and repair the spacecraft . The disadvantage is longer flight times. Bertachini [8] used a simple trial and error technique to analyze these cy clers; that is, guessing a position for the impulse maneuver so no optimization techniques need be applied. Our simulated trajectory was generated with the same technique. In 2003, Bertachini [10] provided some comparisons of impulsive transfers to all five libration points of the Sun-Earth system. He stated that both the collinear (unstable) and triangular (stable) libration points are good for placing a space station, especially the triangular points, since they require � V for sta tion-keeping that is almost zero. Some of the results showed that we could reduce the � V for a transfer to Earth at the cost of extending the time of flight (TOF). For exam ple, we would need a � V =4.0 km/s and TOF=291 days for a transfer from the Ls point to Earth versus the �V=6.1 km/s and TOF=174 days for a transfer from £4 to Earth. 1.3.10 W. Kizner Kizner [33] describes a miss distance method for lunar interplanetary trajectories based on the components of the impact parameter. Kizner suggests constructing a system of differential corrections in order to determine the statistical errors of a spacecraft trajectory aimed to a target body. These differential corrections are the partial derivatives of the miss distance at the target body with respect to the coordinates of the spacecraft at the maneuver point. Kizner specifies the impact parameter, B, as the vector in the plane of the trajectory originating at the center of gravity and perpendicular to the incoming asymptote of the spacecraft's hyperbolic trajectory. The impact parameter has three components. S is a unit vector in the direction of the incoming asymptote, T is a unit vector lying in a fixed reference plane such as the equatorial plane and R is a unit vector fo rming the righthanded coordinate system, R = S x T. Theref ore, the two coordinates B · T and B · R will specify the direction of the miss distance of the spacecraft from the aim point in the B-plane. Kizner states that unfortunately we cannot use all three components of the 13 impact parameter vector since they depend on each other. Therefore, we should choose a third coordinate such as the linearized time of flight to describe the third component of the guidance differential correction. 1.3. 11 James K. Miller Miller [54] provides an overview of the na vigation of the Galileo mission. He defines na vigation as the process of locating the position of the spacecraft , predicting the flight path of a spacecraft (OD) and correcting the predicted flight path (m aneuver analysis or flight path control) to achieve the mission requiremen ts. Accuracy in na vigating the satellite is needed to meet the science instrument-pointing constraints while completing the mission within the propulsion requiremen ts. These tasks are definitely complex because of the trajectory sensitivity and large number of ma neuv ers. Miller suggests perf orming several propulsive error correction maneuvers (or statistical maneuvers) in order to assure the completion of the mission within mission requiremen ts. One of the main concerns of interplanetary na vigation is the removal of launch vehicle errors. Another concern is predicting the flight path when the satellite is aimed at a target. Orbit na vigation is performed using the well-known B-plane targeting coordinate system described in the previous section 1.3. 10. We can then obtain the corresponding errors in the hyperbolic miss B · R and B · T directions, the time of flight and the direction and magnitude of the hyperbolic excess velocity, V00 (C3 = V00 2 ). The Navigation system may be very challenging because of the need to trade-off between minimizing the propellant requirements and maximizing delivery accuracy. Miller [49] discusses the New Horizons Navigation to Pluto where the spacecraft nav igation during the interplanetary cruise involves estimating the trajectory based on avail able tracking data and computing the required trajectory correction maneuvers. In this specific mission, annual checkouts of 50 days duration were performed in order to provide radio metric tracking data while providing detailed science observa tions of Pluto and its 14 other moons. With these data, we can observe the motion of the spacecraft with respect to the stations that form the Deep Space Network (DSN). The DSN tracking stations transmit radio-frequency signals to the spacecraft and receive signals via the transceiver and antenna of the spacecraft . These received signals constitute Doppler and range ob servat ion data. The range data can provide us with a direct measure of the line-of-sight distance from any Earth tracking station to the spacecraft . The navigation analyst must also model the dynamics of the spacecraft motion result ing from other forces acting on the spacecraft . These fo rces are gravitat ional accelerations and non-gravitational accelerat ions, such as the solar radiation pressure, attitude control gas leaks, and dust and outgassing effects [53] . These non-gravitational effects can be considered together as a constant accelerat ion or as stochastic accelera tions. 1.3. 12 Martin W. Lo and Kathleen Howell In order to analyze an L5 mission to study the Sun, we will use dynamical systems theory. In 1991, Belbruno and Miller [4] used a low energy transfer with a ballistic capture at the Moon based on the Weak Stability Boundary (WSB) Theory. This technique allowed the Japanese spacecraft Hi ten to enter lunar orbit with a very small perturba tion and save the mission. Also, Martin W. Lo and Kathleen Howel l's Group of colleagues [44] designed the Genesis Mission using the nonlinear mission design LTo ol in 2001. NASA's Genesis Discovery Mission was the first mission to use dynamical systems theory for its designs and operations. Dynamical systems theory [27] is applied to better understand the geometry of the phase space in the three- body problem via the stable and unstable manifolds. This technique is utilized in the preliminary design of the Suess-Urey mission. These invariant manifold structures can be used to construct new spacecraft trajectories, such as a "Petit Grand Tour" of the satellites of Jupiter [19]. 15 The "InterPlanetary Superhighway" (IPS), discovered by Lo and Ross [42] is a vast network of tunnels providing ultra-low energy transport throughout the entire Solar Sys tem, generated by the Lagrange Points of all the planets and satellites. The IPS can reveal some of the profound questions about the origin of the universe. This paper mo tivated us to the new discovery of trajectories to the triangular points in the Sun-Earth system for a space weather forecast. 1.3.13 First Earth Trojan Asteroid In 2010, the Wide Infrared Survey Spacecraft (WISE) discovered [12] the first Earth Trojan Asteroid (2010 TK7) . In 2011, this asteroid was confirmed to be the first Earth Trojan Asteroid. (a) Motion of 2010 TK1 Asteroid (b) Tadpole Loops of 2010 TK1 Asteroid Figure 1.5: First Earth Trojan asteroid 2010 TK7 [12] in the Sun-Earth system. This Earth co-orbital asteroid (ECA) is in a 1:1 mean motion resonance with the Earth, that is, it goes around the Sun in the same amount of time that the Earth does. It has an approximately 390-year cycle. Due to its eccentric and inclined orbit with respect to the Earth's orbit, the asteroid seems to orbit the triangular point L4 (see Figure 1.5). Instead, it orbits in tadpole shaped loops around L4 . These orbits are rather large going as close as 20 million km from Earth (about 50 times the distance from the Earth to the Moon) and nearly as far as the opposite side of the Sun from the Earth. The motion 16 of 2010 RTK7 reverses direction and comes back to its current position. This irregular motion of this type of orbits will be described in detail in this work. 1.4 Missions to Study the Sun Last January 23, 2012, one of the strongest solar flares in the past decade blasted a stream of charged particles that headed toward Earth affe cting power grids, radio and satellite communications. Solar flares are usually fo llowed by CMEs that can reach Earth from 1 to 3 days. There are and have been many missions with one unique goal: to study and analyze the structure of the Sun and its environment so we can protect our telecommunication infrastructures at Earth. Below we mention briefly the main goals of each of these missions, both in operations and in development. 1.4. 1 Missions in Operation Advanced Composition Explorer (ACE) • How does the composition of the Sun, solar wind, solar energetic particles, inter planetary plasma clouds and cosmic rays differ and what are the main mechanisms that trigger these signatures? • How does the solar wind, energetic particles and cosmic rays affect space weather over the solar cycle? • What are the signatures that can be used as precursors to predict space weather? Hinode • Understand how energy is generated by magnetic field changes in the photosphere (lower atmosphere) and is transmitted to the corona (upper solar atmosphere). • Understand how the energy transferred within the solar atmosphere and its dynam ics affects the interplanetary space environment . Ultimately, it will help predict space weather. 17 Interstel lar Boundary Explorer (IBEX) • Image the 3-D boundary region of the heliosphere. • Understand the properties of the termination shock and the dynamics of the ener getic particles at the termination shock. • What are the properties of the solar wind flow beyond the termination shock, the heliotail and heliopause? Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) • Investigate particle accelerat ion and energy release in a solar flare. Solar and Heliospheric Observatory (SOHO) • Understand the causes and mechanisms of CME initiation and how these particles propagate through the heliosphere. • Improve our understanding of solar wind and SEPs accelera tion. Solar-Terrestrial Relations Observatory (STEREO) • Understand the causes and mechanisms that trigger CMEs and how these particles propagate through the heliosphere. • Develop a 3D magnetic field topology structure of CMEs, density, velocity and temperature of the solar wind. Transmission Region and Coronal Explorer (TRACE) • How do 3D magnetic structures emerging from the photosphere define the geometry and dynamics of the Transition Region of the Corona.? Voyager Interstellar Mission (VIM) • Examine the nature of solar wind turbulence and the behavior of the solar wind of the termination shock. 18 Wind • Measurements of interplanetary conditions to sense interplanetary disturbances and study the inner heliosphere. 1.4.2 Missions in Development Geostorm • Monitor solar activity [73] by providing a continuous data flow from sub-L 1 to allow an increase in the warning time of geomagnetic storms using solar sails. Due to the unstable nature of L 1 , station-keeping maneuvers may need to be applied but at the cost of increasing the budget of the mission. The operational mission life time is about 3 to 7 years. Space Weather Diamond • Another mission concept [13] is in development to improve space weather forecast ing from a sub-L 1 platf orm. The Space Weather Diamond (SWxO) is based on a constella tion of four platf orms that are phased into heliocentric orbits, although the spacecraft appears to orbit Earth from the perspective of a fixed Sun-Earth line. This (SW xO) concept offers a chance to make significant science return without the use of solar sails or other exotic methods of in-space propulsion. Solar Dynamics Observatory (SDO) • Understand the Sun's interior, magnetic activity and source of solar changes that affect Earth. Solar Orbiter • Determine in-situ properties and plasma dynamics of energetic particles near the Sun's heliosphere ( 48 solar radii) and evolut ion of the corona and inner heliosphere. 19 Solar Probe Plus (SP+) • Understand the structure and dynamics of the magnetic fields due to fast and slow solar wind. • Determine mechanisms that accelerate and transport energetic particles. • Perform Coronal Magnetism, Plasma and Activity Studies from Space. • Understand magnetic structure of the solar corona and the role of magnetic recon nection in the CME fo rma tion. Fine-scale Advanced Coronal Transition-region Spectograph (FACTS) • Understand flare magnetic reconnection and CMEs accelera tion. • Understand processes that accelerate particles and that cause the solar activity that affects Earth. • Develop a prediction capability of the solar activity. PERSEUS Mission Investigating heliospheric dynamics from £ 1 • Explore heliospheric material flows impacting Earth and other planets. • Understand the correlation between CMEs and CIRs. • Measure the impacts of solar events at Eart h. Solar Act ivity Farside Investigat ion (SAFARI) • Understand the source of solar variability and how magnetic fields emerge at the solar surface. JAXA SOLAR-C • Understand the internal structure of the Sun and the solar dynamo mechanism. 20 • Understand the variability of space weather in inner heliosphere from the plane of the ecliptic. Solar Polar Imager (POLARIS) • Study the helioseismology, magnetic fields of polar regions, polar view of corona, CMEs, evolution and space weather prediction. Space Weather Imaging Sentinel (SWIS) • Observe signatures, such as solar flares, CMEs and SEPs eruption. • Identify precursor signatures to fo recast CMEs, SPEs and solar flares. • Improve our ability to fo recast space weather. This work is based on how to design real trajectories for an L5 mission while provid ing analyses on the navigation, trajectory correction maneuvers and orbit determination which will help scien tists, engineers and technologists to have a unified perspective of such a demanding mission in the near future. 1.5 Research Scheme Below we present some of the published works by the author during the jo urney of the doctorate at USC. These research works are cited in the thesis and at the end of the bibliography. 1.5.1 Research Papers 1. "F easibility Study For Missions To Triangular Points PART I": Lo, Llanos, Hintz [43] 2. "Navigation Analysis for an Ls Mission in the Sun-Earth Syste m": Llanos, Miller, Hintz [39] 21 3. "3D Integrated Trajectories for an Ls Mission in the Sun-Earth Syste m" : Llanos, Miller, Hintz [38] 4. "Mission and Navigation Design of Integrated Trajectories to L4,5 in the Sun-Earth Syste m" : Llanos, Miller, Hintz, Lo [40] 5. "Trajectory Mission Design and Navigation to the Sun-Earth Sub-L5 for a Space Weather Fo recast ", Llanos [37] 1.5.2 High Level Mission Requirements 1. Optimize the transfer time and the propulsion requirement to achieve a 5-year mission in orbit about Ls. 2. Launch into a circular parking orbit at 200-km altitude with a 28.5° inclination about the Earth as displayed in Figures 1.6 and 1. 7. 3. Decouple the Ls orbit insertion maneuver (LOI) from the start of the science mission at the "arrival" to the vicinity of Ls. 4. Design periodic orbits that move 5° above and below the ecliptic plane. This value will be defined by the mission requiremen ts. 5. Limit the magnitude of the Ll V to less that 300 m/s for the sum of all TCMs and SKMs. 6. Limit the number of maneuvers to 6 TCMs for the transfer orbit and to less than 6 SKMs after insertion into the Trojan orbit. 7. Employ the optimal maneuver sequence subject to mission and operational con strain ts. 8. Provide delivery accuracy at insertion into the Trojan orbit of about 10 0rad. This parameter will be determined by the mission requiremen ts. 22 We describe each of these requirements in more detail for both trajectory mission and na vigation design. 1.5.3 Mission Description We will divide the mission into five phases: Launch or Injection Phase, Transfer Phase, Arrival Phase, Science Phase I, Insertion Phase and Science Phase II [ 43] : 1 day parking orbit ,Rp=200 krn, inc.=28.5° -150 -100 -50 0 90 50 100 150 Longitude (de g) Figure 1.6: Departure orbit from 200 km and 28.5° inclination parking orbit around the Earth Hyper bolic Departure + 4h ,Rp=200 krn, inc.=28.5° 0 x 10 4 5000 0 -5000 X(km) x 10 4 Y(km) Figure 1.7: Hyperbolic departure orbit 23 Launch Phase The launch phase (see Figure 1.8(a)) starts when the spacecraft is in a 200-km circular parking orbit about the Earth as illustrated in Figure 1.6. At the injection time, ti , an impulsive � V injects the spacecraft into the Transfer Orbit and the Transfer Phase begins. Figure 1. 7 shows the hyperbolic trajectory of the spacecraft during the first four hours after injection from the 200-km circular parking orbit with an inclination of 28.5° degrees. Transfer Phase The Time of Flight (TOF) or Transfer Time is one of the key parameters in the design of the mission (see Figure 1.8(b) ). While we wish to limit the TOF duration, the total � V is generally lower for longer TOFs. Therefore, a trade between the TOF and � V cost is the key object of study for this mission. Typically, such a transfer requires durations on the order of years or large fractions thereof. In addition, navigation correc tion maneuvers may be applied one to two weeks after leaving the parking orbit as well as midcourse trajectory maneuvers. Science Phase I We make a distinction between Arrival to the £5 vicinity from the actual Insertion into the Ls orbit because the duration between these two events is many days. Therefore, the spacecraft could take advantage of these few weeks or even months to start experiments once it has arrived in the £5 vicinity, which we will define to be approximately 0.1 AU (see Figure 1.8(c)). This value is an arbitrary parameter and may change, depending on the science requirements still to be determined. Howeve r, for orbits with amplitude > 0.1 AU, we chose 0.2 AU as the Arrival point at the Ls vicinity. Insertion Phase The duration from the Arrival to the actual insertion into the Trojan Orbit around £5 depends on the size of the orbit and TOF. In this particular exam ple, we show an ampli tude of about 0.04749 AU for the Trojan orbit. The actual duration and � V performance 24 -0.75 ., .. '· -0.95 0.4 045 (a) Science Phase I -085 -086 -0.89 � . _ _; _-/ \ hri��l�La�n �i H-29�.4 . -0.9 Intersection:at 0.1 AU 0.5 0.55 O.G X(nondim,AU) (c) � -0.86 s --0865 � � -0.87 0.65 0 48 0.7 047 Science Phase IT 049 05 051 X(nondim,AU) (e) Transfer Fhase, TOF=372.3 days 0 52 (b) Trojan Orbit Insertion 0.49 0.5 0.51 0.52 X(nondim,AU) (d) Figure 1.8: a: Schematics of Launch Phase from a 200-km parking orbit around the Earth. b: Transfer Phase from a 200-km parking orbit to an orbit around the Triangular Point £5. c: Schematics of a Science Phase I at Arrival to the Triangular Point £5. d: Schematics of the Insertion Phase into the orbit around £5. e: Schematics of a Science Phase II into the orbit around £5. 25 are described in the next chapter. The Insertion Phase (see Figure 1.8(d)) has a duration of one week to two weeks before the LOI maneuver (Libration Orbit Insertion) and one week after LOI. The extra time is reserved for na vigation analyses before and after the LOI and for additional trajectory correction maneuvers. Science Phase II After the Insertion Phase, science resumes again in Science Phase II (see Figure 1.8 (e) )until the end of the mission (EOM). Since the orbits are unstable, we may need to perform some station-keeping maneuvers during Phases I and II. We will defer the orbital stability issues to the mission performance sec tion to fo llow. 1.5.4 Mission Geometry In the mission geometry, we will analyze the range and range rate of the transfer and Trojan orbit, the velocity profiles and the Sun-Earth probe angle: Range and Range Rate Figure 1.9(a) shows the evolution of the range and range rate of the spacecraft during the transfer trajectory from Earth to the libration orbit around £5. The range varies from 1.5 to 1.83 million of kilometers, whereas the range rate excursions reach a maximum value of 11 km/s. Figure 1.9(b) illustrates the evolution of the range and range rate of the spacecraft on the Trojan orbit. Velocity Profiles Figures 1.9 (c) and 1.9( d) shows the evolu tion of the radial and tangential velocity profiles of the spacecraft for both the transfer trajectory and the periodic orbit around £5. The maximum excursions of the radial and tangential velocities are 3 km/s and 11 km/s for the transfer phase and 0.35 km/s and 0.7 km/s for the radial and tangential velocities of the Trojan orbit. 26 � : : m 1. 4 L--�-�--'---'--�-�-_j._ _ _j 0 50 1 00 150 200 250 300 350 400 time (days) Range rate for Transfer Orbit L� 0 50 1 00 150 200 250 300 350 400 time (days) (a) Rooial and Tangential velocity in function of time 12 Range for Trojan Orbit 500 1000 time (days) 1500 ""� � . . . . a o . 6 · .. .. . . .. · .. .. · · ' ·· · ·· · · ··· ··· · ") : : E o . s : ·- -- -- ---- -- --- - ---- -- --- : ------ --- __ . : -- -__ _ . Ol,) : : : : E:.:= 0 500 1000 1 500 time (days) (b) Radial and Tangential velocity in function of time 0.8 0 7 0.6 Tangentia1 I (c) 60 40 "" 20 3 � 8 -20 -40 400 Radial 02 I 0.1 .�� L- � 0- --- 1* 1- 1- 1- 1- time(days) (d) Central angle in function of time -SO L--�-�-�--:�--:�-�-�� 0 50 100 150 200 250 300 400 time( days) (e) Figure 1.9: a: Range and Range Rate profiles of the Transfer Orbit. b: Range and Range Rate profiles of the Trojan Orbit. c: Shows the radial (red) and tangential (black) velocity profiles for the transfer orbit. d: Shows the radial (red) and tangential (black) velocity profiles of the Trojan orbit e: Sun Earth Probe Angle 27 The radial and tangential velocities were calculated as follo ws: We know that, and since we know that d < i, i > di ---- =2r- dt dt d<r,r> dlrl 2 1-1-'- 2 1- 1 _,_ - -----=- - -= -- =2rr= < r r> dt dt ' The radial velocity becomes: Fr om: _, < r, r >" 1-1'" r= r= rr r :r = lrl r we take the derivative with respect to time to obtain: Then, the tangential velocity is: Sun-Earth Probe Angle -'- dlrl "' 1 _ 1 dr r = - r+ r - dt dt (1.1) (1.2) (1.3) (1.4) (1.5) (1.6) Figure 1.9(e) shows the evolution of the Sun-Earth-probe angle during the transfer trajec tory. Notice the two peaks, the first because of the spacecraft leaving the 200-km parking orbit around the Earth and the other second because of the spacecraft approaching the insertion into the Libration orbit around Ls. In our second exam ple (Figure 1.1 0), we examine a transfer orbit of the spacecraft at the actual L5 libration point. The radial and 28 tangential velocities continue decreasing as expected since the amplitude of the Trojan orbit becomes smaller. The velocity profiles fo und at the Ls point are negligible and of the order of w - 9 cm/s which is just numerical integra tion noise. This translates to the fact that some minimal station keeping might be needed to maintain the satellite at the Ls point over the time span of five years . The spacecraft departs with a velocity of about 11 km/s for the transfer orbit and it arrives with a velocity of about 0.5 km/s. Of course, since this model is theoretical in nature, the point is somewhat moot. As a check, we computed the acceleration at the £5 point over the time of five years as illustrated in Figure 1.1 0( f) and observe that the acceleration is negligible. 1.5.5 Navigation Analysis In an earlier paper [43], we studied and compared the performance of a series of planar periodic orbits around £5 suitable for such a mission. Given the feasibility of such a mission, we now study and analyze the na vigation needed for an £5 mission in the Sun Earth system. Immediately after launch of the spacecraft , orbit determination of the outgoing tra jectory will be computed and mapped to a final time. However, it will be difficult to model the accelerat ion of the spacecraft due to inconsistent errors in the state of the spacecraft as a consequence of many sources of errors. Gas vented to vacuum from a spacecraft can generate a thrust [52] that accelerates the spacecraft in the same manner as a rocket engine. There are inaccuracies in the model used to simulate the motion such as incomplete modeling of all the fo rces acting on the spacecraft and thruster model uncer tainties. Also, there are errors in tracking and orbit determination of the spacecraft trajectory during the mission. We will devote our attent ion to analyzing the magnitude of trajectory correction maneuvers (TCMs) but we will not attempt to determine the maneuver direction although this could be implem ented if desired. The resulting maneuver analysis may not be optimal 29 but is sufficient to provide a preliminary estimate for use in sizing propellant tanks, determining mass requirements and selecting the launch vehicle. When designing the trajectory, the initial state is usually considered subject to con straints that reflect mission requirements on, e.g., the altitude, flight path angle or incli nation. On the other hand, in this preliminary investigat ion of maneuver analysis, the initial state is fixed in position, velocity and time at a certain epoch Te poch [51] . We will assume that there is no correc tion maneuver at the initial sta te. Therefore, all the maneuvers will be performed at some point after the initial state. To model the spacecraft trajectory, we will use the Circular Restricted Three-Body Problem (CRTBP) with the Sun and the Earth-Moon barycenter as the primary and secondary bodies in the rotating frame. As we will mention later, the maneuver costs associated with the correc tion maneuvers during the transfer segment from Earth to the li bration orbit around £5 (Trojan orbit) and during the station keeping after insertion into the Trojan orbit will be larger when incorporating higher fidelity models, which we will analyze in a future paper. For purposes of this study, we will explain the methodology of the navigation model and the set of na vigation requirements and na vigation phases. In the next sections, we will analyze the trajectory correction maneuvers for both the transfer (TCMs) and the Trojan orbit (SKMs) for different execution errors and exam ine how the maneuver anal ysis is affected by changing the propulsion maneuvers location. The fo llowing section is devoted to the orbit determination where we will analyze the knowledge error and delivery accuracy for selected trajectories. In the final section, we will perform the corresponding propellant analysis for selected trajectories. 1.5.6 Navigation Requirements From our literature review [46, 50, 58], we gathered the following navigation requirements for an £5 mission: Requirement 1: Limit the magnitude � V to less that 300 m/s for all TCMs and SKMs. 30 Requirement 2: Limit the number of maneuvers to 6 TCMs for the transfer orbit and to less than 6 SKMs after insertion into the Trojan orbit. Requirement 3: Employ the optimal maneuver sequence subject to mission and opera tional constrain ts. Requirement 4: Provide delivery accuracy at insertion into the Trojan orbit of about 10 p,rad. This parameter will be determined by the mission requiremen ts. The locations of the TCMs and SKMs were selected to minimize the total mission propellant usage and to provide enough time between propulsion maneuvers so we can reconstruct proper correc tion maneuvers and generate an adequate sequence of upcoming maneuvers. 1.5. 7 Navigation Phases The position of a spacecraft may be affect ed by errors as a consequence of multiple fact ors such as injection errors at launch and solar radiation pressure. These factors will cause the spacecraft to deviate from its nominal trajectory, requiring execution of trajectory correc tion maneuvers (TCMs). Navigation analysis is considered in five phases: 1. Injection Phase Most of the errors are due to injection inaccuracies . We will assume the injection errors to be 10 m/s but we will examine other injection error estimates of the order of around 2 m/s for comparison. The location of these TCMs is important. The first maneuver is dominated by the injection errors while the second maneuver will be dominated by injection, first maneuver execution errors and orbit determination errors. We expect the orbit determination error to be accurate to the order of mm/s. It would be ideal to per form the first maneuver right after launch when the spacecraft is still inside the gravity well; but this is not possible because the orbit determination is not available until 5 to 6 hours later when the spacecraft has moved outside of the gravity well. As an example, the tracking data for the GSa t mission [72] was made available only after a period of five and a half hours from injection. The second maneuver (T epoch + 20 days) is done to 31 clean up accumulated errors generated after the first maneuver (T epoch+ lO days). It is very important to account for injection errors accumulated after launch. Past missions like the Galileo mission [55] performed a TCM 10 days after launch to compensate for injection errors; the Cassini mission [21] scheduled the first TCM 25 days after launch; the Messenger mission [74] planned to have the first TCM from 18 days to 58 days but it was finally performed 22 days after launch; and the Mars PathFinder mission [71] executed the first two TCMs within 60 days of launch. The Mars Science Laboratory (MSL) is a NASA rover mission [46] that will land on Mars in August of 2012. During the cruise and approach phase, the spacecraft may perform up to six TCMs. The first TCM occured at Tepoch + 15 days. 2. Midcourse Phase During the midcourse trajectory, different stochast ic errors, such as solar radiation pres sure, gas vent ing and reaction control system thruster leaks, are accumulated and we have to correct these errors by approximately (Te poch +150 days). Otherwise, canceling these dispersions will require a change of thousands of kilometers at the end of the trajectory, requiring an undesirably large correction maneuver later. The Mars Science Laboratory mission is expected to perform a second [46] TCM at Tepoch+ 120 days. 3. Pre-Insertion Phase This phase comprises two main propulsion maneuvers before the final adjustment at in sertion phase. First, the spacecraft will perform a correction maneuver at T end-30 days and then it will perform another maneuver at T end-3 days to clean up the errors from the previous maneuver . The last TCMs of the transfer trajectory, which we will refer to as an approach sequence of maneuvers, are as important as the first TCMs. This approach sequence of maneuvers was performed in past science missions. The Cassini mission [21] performed a series of three last maneuvers to 'clean-up' the execution errors from the Deep Space Maneuver (DSM) or cruise. Cassini executed a correc tion maneuver 147 days before the target (Venus swingby), then another propulsion maneuver 77 days before encounter to correct the execution errors from the previous maneuver, and finally, 32 a last maneuver 21 days before encounter to clean up execution errors incurred during the previous maneuver. Also, the Mars PathFinder mission [71] executed the last two maneuvers within 60 days of Mars arrival to clean up dispersions from execution errors. The Mars Science Laboratory mission is expected to perform three [46] TCMs at T en d-60 days, Tend-8 days and Tend-2 days. 4. Insertion Phase The last maneuver at T en d is performed to remove previous maneuver execution errors and built-up stochast ic accelerat ions during the trajectory from Earth to the Trojan Or bit. The last maneuver does not include uncertainties in the execution of the burn at T en d but it includes orbit determination errors and execution errors from previous maneuvers. These errors are of the order of a few cm/s or even mmjs. 5. Station-Keeping Phase Also, we want to perform the last maneuver as a way to have a clean interface to the next maneuver sequence for station-keeping purposes around the Trojan orbit. We will need more than one maneuver once the spacecraft inserts into the Trojan orbit, according to a strategy developed in this study. 1.5.8 Introduction to Models We propose to solve the fo llowing three problems. Our approach to solving these problems is to solve first a simplified model and then transform the solutions into the JPL ephemeris model. For the Sun-Earth problems, the simplified model is the CRTB P Model which is compared with the solution obtained in the ER TBP. For Eart h-Moon problems, the simplified model is the Bicircular Model. First, we would like to introduce the three models that we will use before we go into more detail in the section to fo llow. 1. Circul ar Restricted Three Body Problem (CR TBP) In the CRTBP, the Sun is the primary body, the Earth is the secondary body in a circular orbit around the Sun and the spacecraft is the third infinitesimal mass 33 under gravitational attrac tion of the two primary masses. The spacecraft mass is so small that it does not affect the motion of the Sun or the Eart h. 2. Bicircul ar Problem (BCP) In the BCP model, the Earth and the Moon are considered point masses that revolve around their barycenter with this center of mass revolving around the Sun (point mass) in a circular orbit. Theref ore, with this model, we pretend to study the motion of the spacecraft under the attrac tion influence of Sun, Earth and the Moon. 3. Elliptic RT BP We will use the elliptic RTBP to understand the effect of the non-circular motion of the Earth around the Sun and compare the results to the solutions obtained in the circular RTBP. 4. JPL Ephemeris Model The JPL model is considered one of the most accurate models of the Solar System. This model considers the gravitat ional attraction of the Ear th, Moon, Sun and the rest of the planets, giving the positions of the main bodies of the solar system. In this thesis, we will use the latest JPL model, DE421. 5. Navigat ion Model In our na vigation model, we will investigate the trajectory correction maneuvers after injection, midcourse maneuvers and station-keeping maneuvers, and the orbit determination for such a demanding mission. 1.5.9 Problem 1: Trajectory mission and navigation design using CRTBP in the Sun-Earth system We use the planar CRTBP to study the feasibility of an L5 mission in the Sun-Earth system. For this, we will generate periodic and quasi- periodic orbits around the equilateral 34 points and study their stability. We will analyze the mission geometry and mission performance in the Sun-Earth system. 1.5.10 Problem 2: Stability Analysis of orbits using the BCP in the Earth-Moon system We use the BCP to study quasi- periodic orbits in the Earth-Moon system to examine the effect of the eccentricity of the Moon and the Sun on a point mass satellite orbiting around L5. 1.5. 11 Problem 3: Elliptic RT BP vs. CRTBP in the Sun-Earth system We use the ER TBP to study quasi-periodic orbits in the Sun-Earth system to examine the effect of the eccentricity of the Earth orbit on a satellite orbiting around L5. Both the BCP and the ERTBP will be considered to discuss characteristics between these models and the CRTBP, but not to examine in detail the trajectory mission and na vigation analyses. 1.5. 12 Problem 4: Trajectory mission and navigation design using a high fidelity model (DE421) in Sun-Earth system We use the JPL ephemeris (DE421) to analyze trajectory mission design, na vigation and orbit determination analyses in the Sun-Earth system, to generate periodic orbits around the triangular points and to study their stability. 35 Orbit Case L5(16) (a) Range for Trojan Orbit � 1 . 496f x1 _ o '_ ... " 1 496 . . ! 14 96 L_ ______ L_ ______ � ______ _L ______ -" 0 x 1 o- 1 3 � 1 �. 500 10 00 time (days) 15 00 Range rate for Trojan Orbit 2000 I :f ... ·• j 0 � 1� _ - _:- _:: _ --� "-'--c- -�- -- -- �-- ---: ·:'-:-: · ··_ ·_ ·_ · --· ---, ·-:'-:- ·· .,--- · ·_ -_ .. _·· 500 1000 1 500 time (days) (c) 7x 1 o-" Radial and Tangential velocity in function of time 6 500 10 00 time( days) (e) 15 00 Tangential I Ra dial I 2000 1 0 0 (b) Radial and Tangential velocity in function of time 500 Tangential 1 (d) Acceleration at L5 1000 time( days) (f) 1500 Figure 1.1 0: a: Fo ur-year transfer orbit from Earth 200 km circular parking orbit to £5. b: Range and Range Rate profiles of the Transfer orbit. c: Range and Range Rate profiles of the Trojan orbit. d: Radial (red) and tangential (black) velocity profiles for the Transfer orbit. e: Radial (red) and tangential (black) velocities at £5. f: Acceleration at £5. 36 Chapter 2 Project Outline 2.1 Sun-Earth L4/ £5 Mission and Navigation Design This section will consider the main Sun-Earth £4 /£5 mission and navigation require ments and some of the preliminary work in the Sun-Earth system. First, we will discuss the process of mission requirement development. In addition, we will explain how we computed the Periodic Orbits and Transfer Trajectory. Then we will talk about the mis sion design, especially the stability of some of the orbits fo und and their relevance to future missions. Finally, we will show some of the na vigation design characteristics that are relevant to the mission. 2.2 Mission Requirements Development The mission requirements will be divided into Trajectory mission requirements and nav igation requiremen ts. The first requirement for the trajectory mission design is to optimize the transfer time and the propulsion requirement to achieve a 5-year mission in orbit about £5. This will be done by automation of the transfer trajectory using a targeting strategy that we will explain in Chapter 4. 37 The second requirement is to launch the spacecraft into a 200-km circular parking orbit about the Ear th. Theref ore, we will assume that the spacecraft will be in this parking orbit before a maneuver is performed to leave it. The third requirement will be to decouple the L5 orbit insertion maneuver (LOI) from the start of the science mission at the Arrival to the vicinity of Ls. Even though we will require doing an orbit determination analysis at the beginning of the mission for possible launch correction maneuvers, this third requirement will be coupled with some of the navigation requirements (that we describe later) during the midcourse trajectory segment of the mission. During this stage, the trajectory of the spacecraft will have to be tracked via periodic orbit determination analysis. According to Mr. James K. Miller, rv 10 days will be a good time frame for doing orbit determination analyses. The last requirement in the trajectory mission design will be to design periodic orbits with ±5° about the ecliptic plane. Scientists are interested in these inclined orbits because of the advantages that these orbits yield such as the ability to obtain scien tific data from the Sun and the solar wind above the ecliptic. 2.2.1 Trajectory Mission Design In [43] we provide a study of some planar orbits around the L5 point usmg a nmve numerical continua tion method. We used a naive MAT LAB ® RKF78 integrator with a tolerance of 10 - 10 (checked against tolerance of 10 - 1 8). The simple family of periodic orbits we fo und around the Ls point is listed in Table 2.1 and shown in Figure 2.1. Some of the largest and smallest orbits fo und have amplitudes of about 0.5 AU and 100 km, respectively. Here amplitude is the apparent semimajor axes of the orbit, although they are not actually ellipses or conic orbits. All of these periodic orbits were numerically integrated for a time span of 5 years. The period of these orbits is rv 1 year. Figure 2.1 (b) shows two dimensional planar periodic orbits for different sizes in the CRTBP. The largest amplitude shown is abut 0.52 AU and the smallest amplitude is 100 km. Next we explain and analyze some of the periodic orbits fo und around the L4 or 38 � 0.05 , ,. 0 1-005 N' 3D Periodic Orbits around Sun-Earth LS -0.7 Y(nondim,AU) 0.2 X(nondim,AU) (a) 3D Periodic Orbits around Sun-Earth L5 X (c) 0.7 0.2 Periodic Orbits around Sun-Earth LS 0.4 0.5 0.6 X(nondim,AU) (b) 3D Periodic Orbits around Sun-Earth L5 -1.05 -1 -0.95 -09 -O.S5 -0.8 -075 -07 -0.65 y (d) Figure 2.1: a: Three dimensional periodic orbits around the Sun-Earth Ls for different inclinations in the CRTBP. b: Two dimensional periodic orbits around the Sun-Earth Ls. c: XZ projection. d: YZ projection Ls points. Table 2.1 provides the .6.. V required for each of the mission phases, the time of the transfer orbit arrival from which the spacecraft will start perf orming science and the TOF for each of the periodic orbits around £4 or £5. The columns of Ta ble 2.1 are described in the paragraph just after the table. We see that less .6.. V is needed at Ls than £4 because of the geometry of the vector addition of the injection orbital velocity with the Earth's velocity at launch. This is the same geometric effect as exists for planetary flybys. Since the £5 orbits are relatively stable compared with the collinear libration orbits, a sizable maneuver is required to insert into these orbits. The insertion maneuver can be reduced, but at the cost of extending the transfer time on the order of a frac tion to several years. So this is the main trade to be undertaken . In fact, if a mission can take advantage of the transfer time from Earth 39 Table 2.1: b. V vs. TOF comparison between Ls and £4 for different periodic orbits CASE Amplitude TOA TOS TOF b.Vroi b.VLoi b. V't o t a l km, au days days days kmjs kmjs kmjs £4-1 o k m 219. 3 79.8 299.1 3.427 1.993 5.420 £4 -2 o k m 1204.8 158.9 1363.7 3.258 0.541 3.709 £4-3 28424 k m 536.9 114.0 650.9 3. 288 0.907 4.195 £4-4 28424 k m 218.1 81.3 299.4 3.427 2.001 5.429 £4-5 0.04758 au 200.9 31.4 281.1 3.544 1.938 5.482 £4 -6 0.52362 au 147.6* 465.3 612.9 4.462 0.764 5.226 Ls -1 0.52362 au 312.0 * 31.4 343.4 4.622 0.995 5.617 Ls - 2 0.52362 au 540.5* 160.6 701.1 4.282 0.522 4.804 Ls - 3 0.52362 au 584.3* 329.9 914.2 3.517 0.641 4.157 Ls - 4 0.04769 au 295.4 76.9 372.3 3.440 1.573 5.013 Ls - 5 0.04769 au 701.1 100.2 801.3 3.333 0.899 4.232 Ls - 6 0.04769 au 1298.9 137.0 1435.9 3.287 0.45 1 3.738 Ls -7 0.00120 au 319.6 84.2 403.8 3.343 1.438 4.781 Ls - 8 0.00120 au 679.5 11 2.0 791.5 3.277 0.790 4.067 Ls - 9 28424 au 318.0 84.8 402.8 3.342 1.449 4.792 Ls - 10 28424 au 643.7 111.9 755.6 3.276 0.780 4.059 Ls -1 1 598 k m 321.3 89.8 411.1 3.335 1.414 4.749 Ls - 12 598 k m 643.8 118.7 762.5 3.279 0.762 4.041 Ls -1 3 w o k m 322.6 89.0 411.6 3.338 1.413 4.751 Ls -1 4 w o k m 643.1 119.7 762.8 3.275 0.761 4.036 Ls - 15 o k m 641.6 130.5 772 .1 3.277 0.795 4.072 L5 - 16 o k m 1313.0 134.9 1447.9 3.262 0.512 3.774 Arrival* for this case is 0.2 AU to L5 due to large size of orbit. to the Ls orbit, the propulsion requirements can be substant ially reduced as shown in Ta ble 2.1. Column 1 of Table 2.1 is the case identification with indications of £4 or £5 orbits. Column 2 gives the (major) amplitude of the periodic orbit around Ls. Column 3 is the TOA (Transfer Orbit Arrival) time from injection from Earth. Column 4 is the TOS (Time Of Science Phase I) between the TOA and the insertion into the Libration point. Column 5 gives the time of flight (TOF) between the injection from Earth parking orbit and the insertion into the Libration orbit. The units of columns 3, 4 and 5 are given in days. Column 6 is the Injection b. V from the 200-km Earth orbit into the transfer 40 orbit. Column 7 is the LOI b.V. Column 8 (last ) provides the main result, the total b.V needed. The units of columns 6, 7 and 8 are given in kmjs. 4.1 5.5 . '""""' � � 5 � � 4.5 4 400 600 - - - - - � - � - TOF(days) (b) 800 1000 TOF (days) (a) 1200 1400 L5 mission Transfer Orbit Injection delta V vs. Time of Science 3.46 . (c) Figure 2.2: Trade between TOF and b. V From Table 2.1, we see that a typical L5 mission will require b. V of 3. 7 to 5.6 km/sec starting from a 200-km circular Earth orbit, depending on the transfer time. These results are preliminary and assume that the Earth parking orbit and the planar libration orbit are both in the ecliptic plane. For future work, we will consider spatial orbits with a range of inclinations. In this first approach, we use the Circular Restricted Three Body model. 41 In future work, we will use the JPL Ephemeris model. Our current results characterize and size the fe asibility of a typical Ls mission. In Figure 2.2( a) we show some of the data from 2.1 the total � V required as a funct ion of the time of flight for each L5 mission and the � V needed for the LOI maneuver as a func tion of the TOS. We notice that generally, the smaller the Trojan Orbit the smaller the Transfer Orbit Injection, � Vr oi, and the larger the Trojan Orbit Insertion, � VLoi, become for similar TOFs (see orbit cases L5-1, L5-4, L5-7, L5-9, L5- 11 and L5- 13) . Also, we observe that the a same family of periodic orbits with the same size, the performance is more fa vorable but at the expense of larger TOFs. Finally, we would like to point out that for some of the Trojan Orbits, with relatively close � Vr oi values, the spacecraft will be able to do more science and therefore the TOS will be larger for larger orbits. 2.3 Stability Analysis To gain some quick insights into the stability of these orbits, we perturbed a few periodic orbits around the Sun-Earth L4 and L5. Figure 2.3 provides some indications of the orbital behavior due to the velocity errors shown in Table 2.2 A large orbit (0.52 AU amplitude, left column) and a small orbit (598 km amplitude, right column) are examined here. Velocity errors of 1.% , 0.5% and 0.1% for both orbits are examined in rows 1, 2, 3 of Figure 2.3, respectively. For negative errors, the resulting instability produced a retrograde precession around the Sun; for positive errors, the precession is direct. For the large orbit, 1% velocity error amounts to 75.79 m/s with central angle drifts around the Sun on the order of 10° after 5 years (see Table 2.2 for the exact drift). For the small orbit, 1% velocity error amounts to 0.06 em js with drifts on the order of < < 1 Jhrad for 5 years. Depending on the needs of the science mission, sta tion keeping may not be necessary as the precessions in these examples still keep the orbits in the vicinity of L5 even after five years. The mission can be designed with a velocity error biased on the positive side so that the precession is always directed towards the Eart h. This precession improves the telecom link as over time the range to Earth will decrease, even 42 Unstable orbits around the L5 point �1 �2 �3 �4 �5 �6 �7 �8 X(norrlim,AU) Unstable orbits around L5 point -0 6h-��C--.�� -065 -0.7 - --0.5%(dv) - �.5%(dv) 02 �3 �4 �5 �6 �7 X(nondim,AU) Unstable orbits around L5 point ,-------, - --O.l %(dv) -0.7 • L5 point -0.75 5' -08 ..: ,,- 1 -085 ;;:;- -09 -1 - �.l %(dv) 03 04 05 0 6 X(nondim,AU) (a) 0.52 AU Trojan Or bit 0.7 05 Unstable Orbits around L5 point �5 �5 �5 �5 X(nondim,AU) 05 ,------'U:::.:n:::st=ab::::l:::.,e orbits around L5 point -0.866 -- --0.5%(dv) -0.866 -- �.5%(dv) �5 �5 �5 �5 �5 X(nondim,AU) Unstable orbits around the L5 point 05 05 05 05 05 05 05 X(nondim,AU) (b) 598 km Trojan Or bit Figure 2.3: Preliminary orbital velocity error analysis for 5 years: Column 1: Large orbit (0.52 AU amplitude ); Column 2: Small orbit (598 km amplitude) . Top Row: ±1% error in velocity; Middle Row: ±0.5% error; Bottom Row: ±0.1% error. Initial periodic orbit is black, blue orbits are for positive errors, magenta orbits are for negative errors. as the telecom system degrades . In Table 2.2, * indicates a periodic orbit with 0.52 AU amplitude, ** indicates a periodic orbit with 598 km amplitude. 43 1.6 1 4 � 1.2 ! § 1 "' 0.8 06 Table 2.2: Unstable Orbits around the Ls point Error Vel. [km/ 8]* Drijt[0]* -% + % -% + % ±1 .0% 7.503 7.655 9.75 11. 83 ±0.5% 7.541 7.617 4.84 5.93 ±0.1% 7.571 7.587 0.96 1.19 Acceleration at Trojan Orbit = 0.52 AU 200 400 600 800 1000 1200 1400 1600 1800 t(days) (a) Vel.[km/8]** Drift[p,rad]** -% + % -% + % 5.898 6.017 0.0017 0.0022 5.927 5.987 0.0007 0.0009 5.951 5.963 0.0001 0.0002 x 10 � Acceleration at Trojan Orbit = 100 km 2 1.9 18 1.7 1.3 12 1.1 200 400 600 800 1000 1200 1400 1600 1800 t(days) (b) Figure 2.4: Acceleration felt by the spacecraft at different Trojan orbits. Using the circular RTBP, we computed the accelerat ion of two Trojan orbits (0.52 AU and 100 km) around the geometrical location of L5. During the integrated time of five years (five times around Ls), the accelerat ion is of the order of about 1.5 mm/ 8 2 and 2 x w - 6 mm/ 8 2 for both libration orbits, respectively as displayed in Figure 2.4. 2.3. 1 Other Quasi-Periodic Motions around the Triangular Points There are many other families of quasi-periodic orbits (QPOs) around the Sun-Earth triangular points. These orbits do not close on themselves. In Figure 2.5 we show a few examples for a time span of a hundred years. The linearized motion of these orbits is characterized by a combination of long- and short-period solutions (see Szebehely 1967, section 5.6), which results in the general solution of the linearized problem. We obtained some periodic solutions when both frequencies (8 1 for long-period and 8 2 for short-period motions) are present in the solution such that the ratio � = r;: is a 44 (a ) QPOl (b) QP02 (c) QP03 (d) QP04 (e ) QP05 (f) QP06 Figure 2.5: Examples of linearized periodic orbits around L5 integrated for 100 years rational with m and n positive. Some of these orbits are periodic when time approaches infinity and they do not entirely cover the inside of the region of motion. This lack of coverage is because the frequencies are commensurate to a very high order. In future work [40], we examine 3-dimensional orbits around the Triangular Points. 45 From the linear flow A, we can extract informa tion that can be applied when we use a nonlinear dynamical system, that is, we can obtain the eigenvalues A as: At 0 0 0 0 0 0 A 2 0 0 0 0 0 A= 0 A 3 0 0 0 0 0 0 A4 0 0 0 0 0 0 As 0 0 0 0 0 0 A 6 and eigenvectors V E IR 6 as: V = ( Xl X 2 X 3 X4 xs X 6 ) Then, we can integrate the linear system governed by the differential equation: to obtain the solution of the differential equation X(t) = e( A t )_X(O) where X E IR 6 A E IR 6 x 6 can be expressed as At 0 A= ( Xt X 6 ) 0 X 2 X 3 X4 xs 0 0 0 0 0 A 2 0 0 A 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A4 0 0 0 A s 0 0 0 A 6 (2.1) (2.2) (2.3) (2.4) (2.5) 46 Finally, e ) \i t 0 0 0 0 0 X 1 0 e >.. 2t 0 0 0 0 X2 A t ( X 6 ) 0 0 e >.. 3 t 0 0 0 X 3 e = x1 X2 X 3 X4 xs e >.. 4 t 0 0 0 0 0 X4 0 0 0 0 e >.. 5 t 0 xs 0 0 0 0 0 e >.. 6t X 6 (2.6) We fo und several families of asymmetric orbits in the Earth-Moon and Sun-Earth system. In this sec tion, we display a set of asymmetric orbits, tadpole orbits and horse-shoe- like orbits in both the Earth-Moon system and the Sun-Earth system. Some of the QPOs around £5 Earth-Moon system are shown in Figure 2.6 and Figure 2.7 whereas other QPOs around £4 Eart h-Moon system are shown in Figure 2.8 and Figure 2.9. All these orbits were generated using a trial and error procedure. In chapter 4, we will use a differential corrector to generate similar asymmetric orbits around Ls in the Sun-Earth system. Next, we center our attention to asymmetric orbits in the Sun-Earth system. Figure 2.10( a) and 2.10(b) display families of periodic orbits of 25 years duration around the triangular points in the Sun-Earth system. These elliptical orbits are very similar to those obtained before except that these orbits are not centered around the £4 and Ls points. Instead, the triangular points are foci of these ellipses. Figure 2.11(a) shows an asymmetric orbit around the £5 point in the Sun-Earth system using the CRTB P. This orbit was generated by trial and error guess procedure. The initial conditions used were xo = 0. 500690, Yo = -0.860785, xo = 0.01038 and Yo = -0.00141 in non-dimensional units. In Figure 2.11(b), we display another asymmetric orbit around Ls in the Sun-Earth system where the initial conditions assumed are xo = 0.50045186, Yo = -0.865666, xo = 0.00024 and Yo = -0.000275 in non-dimensional units. 47 Asymmetric Periodic Orbit around L5 Earth-Moon -0.4 X (a ) QPOl Asymmetric Periodic Orbit around L5 Earth-Moon 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 X (c) QP03 Asymmetric Periodic Orbit around L5 Earth-Moon -0.6 -0.65 -07 -0.75 -08 :>< -0.85 -11 L__�-�--�-�-�--�-� 01 0.2 0.3 0.4 0.5 0.6 0.7 0.8 X (e) QP05 Asymmetric Periodic Orbit around L5 Earth-Moon -0.4 -13 L_-�-�-�-�-�-�-�-� -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 X (b) QP02 Asymmetric Periodic Orbit around L5 Earth-Moon X (d) QP04 Asymmetric Periodic Orbit around L5 Earth-Moon -0.4 X (f) QP06 Figure 2.6: Asymmetric Orbits around Ls Earth-Moon sys tem. In Figure 2.12( a), the spacecraft starts by leaving the vicinity of Ls and ends up orbit- ing L4 in highly horseshoe-like orbits, then, it leaves the vicinity of L4. The spacecraft 's 48 Asymmetric Periodic Orbit around L5 Earth-Moon -0.4 0.4 0.6 X (a ) QP07 0.8 1.2 Asymmetric Periodic Orbit around L5 Earth-Moon (c) QP09 Asymmetric Periodic Orbit around L5 Earth-Moon -0.8 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 X (b) QPOS Asymmetric Orbit around L5 Earth-Moon -0.86 X (d) QPOlO Figure 2.7: Asymmetric Orbits around L5 Earth-Moon sys tem. trajectory depicted in Figure 2.12(b) orbits L5 for 25 Moon periods and then it leaves the vicinity of the Lagrange point. The initial velocity vector is (22.4m/ s, 4 7.9m/ s ). In Figure 2.12(c), we displayed a long tadpole-like trajectory of 250 Moon periods around Ls for an initial velocity vector of (22.4m/ s, 60.1m/ s ). The uncovered banana- shape-like area of the spacecraft in the middle of the tadpole orbit represents a fo rbidden region through which the spacecraft never trave ls. That is, the spacecraft approaches this re- gion and touches the zero-velocity surface. If we take the end points of this trajectory and reverse the velocities, the resultant trajectory would yield a trajectory that spirals inward to Ls in the Earth-Moon system. This means, that we could place a satellite on fa vorable, resonant orbits that could be retained for many years around the triangular 49 1.3 1.2 0.8 07 0.6 05 Asymmetric Orbit around L4 Earth-Moon -0.4 -0.2 0.2 0.4 0.6 0.8 X (a) QPOl Asymmetric Orbit around L4 Earth-Moon 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 X (c) QP03 Asymmetric Orbit around L4 Earth-Moon 1.3 X (b) QP02 Asymmetric Orbit around L4 Earth-Moon 013 661 X (d) QP04 Figure 2.8: Asymmetric Orbits around L4 Earth-Moon sys tem. points. Figure 2.1 2(d) displays the trajectory of a spacecraft orbiting the L4 region for a few months before it departs on an orbit of several years, then connects into a tadpole-like orbit for a few more months around the Ls in the Earth-Moon system. The trajectory event ually leaves the Ls neighborhood to reconnect again into the vicinity of the initial tadpole-like around L4. Finally, in Figure 2.12( e), we show the trajectory of a spacecraft leaving a tadpole-like around L4 and, after orbiting 2 years around the Earth, it is pushed out fo llowing a close encounter with the Moon. This final orbit may fo llow one of the trajectories that comprise the interplanetary transport network. Note the sensitivity to the initial position or velocity. Slightly changes will yield different trajectories. 50 Asymmetric Orbit around L4 Earth-Moon 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 X (a ) QP05 Asymmetric Orbit around L4 Earth-Moon 07� �-- �� L-� � � -- � � � �� 0.25 0.3 0.35 0.4 0. 45 0.5 0.55 0.6 0.65 0.7 X (c) QP07 013 665 0.8664 013 663 0 8662 013 661 >< 0.866 013 659 0 8658 013 657 0.8656 0.96 Asymmetric Orbit around L4 Earth-Moon 0.4872 0.4874 0.4876 0.4878 0.488 0.4882 0.4884 X (b) QP06 Asymmetric Orbit around L4 Earth-Moon (d) QPOS Figure 2.9: Asymmetric Orbits around L4 Earth-Moon sys tem. Heteroclinic connections between the triangular points have been investigated by [47] and [31] in the Earth-Moon system. Figure 2.13 displays a possible heteroclinic connection from L4 to Ls in the Sun-Earth sys tem. Figure 2.14 displays another possible heteroclinic connec tion from L4 to Ls in the Sun-Earth system. These trajectories may be of potential interest for fut ure missions to study the Sun and monitor other material lingering around the triangular points. In both heteroclinic connections, the gap between two tick marks is the equivalent of about one month of trav el. Figure 2.15(a) depicts a 14-year heteroclinic connection from a quasi-periodic orbit around Ls to L3. Then, the satellite orbits the vicinity of the collinear point about 2 years before going back in the clockwise direction to the equilateral point. This satellite 51 Family of Periodic Orbits around L4 Sun-Earth 0.8661 0.8661 0.866 0 866 0.8659 0.4997 0.4998 0.4998 0.4999 0.5 0.5 0.5001 X (a) -0.8658 Family of Periodic Orbits around L5 Sun-Earth 0.499 0.4992 0.4994 0.4996 0.4998 X (b) 0.5 Figure 2.10: Family of Periodic Orbits around L4 (1eft ) and L5 (right) in the Sun-Earth system. Asymmetric Orbit around L5 Sun-Earth y (a) Asymmetric Orbit around L5 Sun-Earth (b) Figure 2.11: Asymmetric Orbits around Ls in the Sun-Earth system for 15 years. 52 Homoclinic Connection between L4 and L5 Earth-Moon -25L_�� -2 -1.5 -1 -0.5 0.5 1.5 2.5 X (a) Long Orbit 250 Moon Periods around L5 Earth-Moon (c) >< Escaping a tadpole-like orbit from L5 Earth-Moon 0 -0.2 -0.6 -0.8 -1 -1.2 -1.4 -1.5 -1 -0.5 0.5 X (b) Horse-Shoe-Like Connections between L4,5 Earth-Moon 2 1.5 0.5 -0.5 -1 -1.5 -=.}L 5� --� 2 �--� 1.� 5 � - _ " - 1 � - -� o.5 = -- -�- oJ _ .5 : -- ---' �� 1.5 X (d) Leaving Horse-Shoe-Like Orbit from L4 Earth-Moon -1 -0.5 0.5 X (e) 1.5 Figure 2.12: Other interesting trajectories in the Earth -Moon system. 53 SHORT L4-L5 Heteroclinic Connection Sun-Earth 0. 8 ...... ·-· -· ··-. 0.6 0.4 ::::) <( 0.2 E "0 c 0 -S -0.2 >- -1 . -1 -0.5 0 0.5 X(nondim,AU) (a) 1.5 Figure 2.13: SHOR T £4-£5 heteroclinic connection in the Sun-Earth system. The flight time of the transfer orbit from an Earth parking orbit of 200 km to an orbit around £4 is about 300.8 days. This transfer orbit is outside the path of the Earth around the Sun. The flight time of the heteroclinic connection from £4 to L5 is about 230.8 days. The closest approach to the Sun of this heteroclinic connection is about 0.346 AU. The total flight time (transfer and heteroclinic orbits) is about 531.6 days. Notice the relative speed of the spacecraft along the transfer trajectory being slower when it is closer to the Trojan orbits of amplitude 0.73 AU and larger when it is closer to the Sun. The gap between two tick marks represents about 30 days of trav el. could also monitor the Sun's activity and the space environment at different angular resolutions [6 6]. Even though this analysis supposes long time of fligh ts, we can reduce the transfer time to a few years as seen in Figure 2.15(b). Figure 2.15(b) illustrates a possible heteroclinic connection from a different quasi-periodic orbit around £5 to another quasi- periodic orbit around £3. The transfer time is about 6 years (red) and 7 years (blue) and the test particle will remain oscillating around L 3 another 9 to 8 years . Howeve r, the test particle can be orbiting the vicinity of £3 for longer times than in the previous case (about 30 years) before escaping in the positive y direction. We may not have to perform any station-keeping maneuvers while orbiting around Sun-Earth £3. The � V 54 5 <( E -o c 0 c >::' LONG L4-L5 Heteroclinic Connection Sun-Earth 0. 8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0. 8 -1 -1 -0.5 0 0.5 X(nondim,AU) (a) Figure 2.14: LONG L4-L5 heteroclinic connection in the Sun-Earth system. The flight time of the transfer trajectory from Earth is about 199.1 days to the Trojan orbit of amplitude 0. 73 AU. This transfer orbit is inside the path of the Earth around the Sun. The flight time of the heteroclinic connection from L4-L5 is 505 .5 days. The spacecraft departs from an orbit around £4, then approaches £3 and finally arrives at an orbit around L5 in the Sun-Earth system. The total flight time (transf er and heteroclinic orbits) is about 704.6 days. needed to depart from the vicinity of £5 is about 536 m/s (blue) and 828 m/s (red) for both transfers to L 3. 55 Sun-Earth L5-L3 Heteroclinic Connection Y(mndim) (a ) Sun-Earth L5-L3 Heteroclinic Connection Y(nondim) 0.2 (b) X(nondim) Figure 2.15: Heteroclinic connections between the sub-£5 Trojan orbits and £3 in the Sun-Earth system. a: Shows a LONG sub-£5-£ 3 Orbit of about 14 years going from a sub-Ls orbit to £3 in the Sun-Earth system. The spacecraft stays in the vicinity of £3 for about 2 years, then returns to the sub-£5 orbit after 8 years. The total flight time is about 23.7 years. b: Shows two- SHOR T L5-L3 Orbits going from a sub-£5 orbit to L3 in the Sun-Earth system. The SHOR T Ls-L3 Orbit in red has a flight time of about six years. The spacecraft orbits in the vicinity of L3 for at least 8.5 years. The SH ORT Ls-L3 Orbit in purple has a flight time of about seven years and orbits £3 for about 7.5 years. 56 Chapter 3 Mathematical Models 3.1 Introduction to Models For this work, the CRTBP and the DE 421 Ephemeris model will be used. The CRTBP provides a good initial guess for the three-body problem because the CRTBP configuration space is symmetric with respect to the rotation about the x-axis where the primary and secondary bodies lie. The CRTBP will be studied first. Then, we will analyze the BCP model that provides a more realistic approach to the three-body problem since the influence of the Moon is taken into account. Finally, the Ephemeris model will be examined where we will include other body perturba tions. The rationale for using a more realistic model is that the stability properties of the orbits fo und in the CRTB can deviate slightly in some scenarios from those fo und in the real model. Below we explain each of these models. 57 3.1.1 Derivation of Circular RT BP The metric tensor components in a corotating frame are: ( G M1 G M 2 ) 2 2 2 2 900 = -2 -- + -- - W (X + y + Z ) r 1 r 2 910 = 2wy 920 = -2wx 930 = 0 9 ij = -6ij, i, j = 1, 2, 3 (3 .1) (3.2) (3.3) (3.4) (3.5) where G M 1 = 01 and G M 2 = 0 2 and w is the angular velocity of the rotating frame. Thus, the matrix tensor can be written as: 900 901 902 903 91 0 911 91 2 91 3 (3.6) 9 = 920 92 1 922 923 930 931 932 933 where r 1 and r 2 are the distances of the test particle from the primary and secondary bodies, respectively and M 1 and M 2 are the masses of the primary and secondary, re- spectively. Using this metric we can express the Lagrangian as: (3.7) The metric is derived from the mechanical energy of the system in a rotating coordinate frame. In some sense, this is akin to the "inf ormation matrix" of na vigation for a me- chanical system. Hence, using this varia tional principle by minimizing the Lagrangian, L, we obtain the equations of motion from the Euler-Lagrange equations: !!__ 8L _ 8L = O dt oqz oqt (3.8) 58 Then, we obtain: d ( 2(2wy) 1 ( 1)2 . ) 1 2 2 ( 1-t1 (x- x1) 1-t 2 (x- x 2 ) ) _ 0 - + - - X + -W X - + - dt 2 2 2 rf r� !}__ ( - 2(2wx) + � (-1)2 y ) + � w 2 2y - ( /--I- 1Y + /--I-2Y ) = 0 dt 2 2 2 rr r� !__ (�( -1)2 z )- ( �-t tz + �-t 2 z) = o dt 2 r{ r� where x1 = - �-t and x 2 = 1- I-t · Simplifying terms we obtain: ( ( 2 . ) .. 2 ( /--I-1 Y /--I- 2Y ) O - w x -y+w y--+- = rr r� z _ ( /--I-1 Z + /--I-2 Z ) = O r 3 r 3 1 2 After rearranging the terms, the differential equations can be written as: .. .,, .· 2 ( 1-t1(x + �-t) /--l- 2 (x-1+ �-t) ) X - ,:.wy = W X - + '-------- -'----- �- -'-'-- rr r� .. 2 . ( 2 /--1-1 /--1- 2 ) y + wx = w - r 3 - � y 1 2 (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) Note that the frequency w can be removed from the equation since it is unity in the rotating frame. 59 3.1.2 CRTBP Model The first key problem that we will examine is the CRTBP (see Figure 3.1). We will use the CRTBP to obtain the trajectory analysis for this study. The Sun is the pri mary body, the Earth is the secondary body and the spacecraft is the third body or infinitesimal mass in this system. To simplify the analysis we used the normalized and non-dimensionalized convention so that the mass of the secondary body is 0 < p. < 1 and the mass of the primary body is 1-p.. The distance between the primary and secondary bodies is normalized to one with the primary body located on the x-axis at -p. and the secondary body at 1-p.. The x-axis is directed from the primary body to the secondary body. The y-axis is 90° from the x-axis in the primary plane of motion. Finally, the z-axis completes the right-handed system, defining the out-of-plane direction. For this work, p. = 3.0404233 89123456e-6 is the Earth- Moon barycenter based on the combined mass of the Earth and Moon, using JPL DE405 constants (in the future we will be using the DE421 constants). Finally, time corresponds to the angle between the x-axis of the rotating frame and the x-axis of the inertial frame so that the period of the rotating frame becomes 21r. Using this convention, the motion of the infinitesimal mass in the rotating frame can be described by the governing equations: .. 2 . (1 ) x+p. x-1+p. x- y= x- -p. -- -p. r 3 r 3 1 2 .. 2 . ( 1 1 - /h /h ) y+ X= - -- -- y r 3 r 3 1 2 (3.19) where r 1 and r 2 are the distances from the spacecraft to the Sun and Earth, respectively. r 1 = V ( x + /h ) 2 + y 2 + z 2 r 2 = V ( x - 1 + /h ) 2 + y 2 + z 2 60 and The CRTBP is nondimensionalized so that the sum of the primary and secondary masses, the mean motion of the rotating frame, the distance between the primary and secondary and the gravitational constant are all unity. Adding the gravitational forces due to the Figure 3.1: The Five Lagrangian Points in the Sun-Earth system. SELl, SEL2 and SEL3 are the collinear points in the Sun-Earth system. SEL4 and SEL5 are the equilateral points in the Sun-Earth system. primary and secondary body results in a force that corresponds to the amount of the cen- trifugal force that is directed against it. The acceleration at the triangular points can be obtained through simple geometric examination as seen in Figure 3.2. Thus, the accelera tion in the Sun- Earth and Earth-Moon systems at L5 is 5.9301 mm j s2 and 2. 7143 mmj s2, respectively. The location of Ls in these systems is (7.29812336E7, -1 .295555563E8 )km and ( 1.875293E5, - 3.329001E5 )km, respectively. We assume that the libration point is an instantaneous libration point where the gravitational and centrifugal forces are in equi- librium. However, we know that these points are moving in reality around orbits with the 61 y i - -- - "' � 4 l I \ ', t I \ ', I ;' \ '\ : / \\ / Primary �� / 60o \ \ Secondar y ' I \ ' ---L-----� ��-· 1 � --------- � ��- -+ X \ , ,Barycenter Jr !, \ I ; 1 \ I r \ \ I / ' I ' \ / / ' F � \\. p /S � � '- ,, , - , ......... ... � . -- � ____ .. ...r l 5 Figure 3.2: Sketch of forces acting on the 'Ihangular Points. ag,p and ag,s are the gravita tional accelera tions on L5 due to the primary and secondary bodies. a9,t is the combined gravitational acceleration on L5 as if the total mass of the system was placed at the barycenter of the system. same eccentricity as the Earth's orbit (in the Sun-Earth system) and the Moon's orbit (in the Earth-Moon system) and coplanar with the secondary body with longitudes of 60° larger for £4 or smaller longitudes for £5 than the longitude of the secondary body. 3.1.3 Linearized Motion around the Triangular Points Recapitulating the equations of motion for the three body problem: y + 2x = Uy 62 where the potential is 1 2 2 1- �-t /h U = - ( x + y ) + - - + - 2 r 1 r 2 We can translate the rotating frame to the location of the Triangular Points £4 and Ls such that � = x + �L and ry = y + rtL where XL and YL are the locations of triangular points. The positions of the triangular points can be obtained given the fact that r 1 = 1 and r 2 = 1 Tl = v( x + �-t ? + y 2 T2 = V ( X - 1 + /h )2 + y 2 Th _1 d - ± y3 en, X L 4,5 - 2- /ha n Y L 4,5 - 2 · If we expand the potential using Taylor expansion: Therefore, we can expand the potential in the form U( x , y ) = U(�L,rJ L) + Ux(�-�L) + Uy(rt -rtL) + � [uxx(�-�L? + + Uyy(rt -rtL? + 2 UxUy(�- �L)(rt - rtL)] + H.O.T. 1 2 2 � x Ux + y Uy + 2 ( x Uxx - y Uyy + 2xyUxy) 63 We know that the potential is zero at the location of the libration points, hence, the first term on the right-hand side vanishes. Including second order terms of the Taylor expansion, the equations of motion become: X- 2 y = xUxx + yUxy + �(x 2 Uxxx- y 2 Uyyx + 2xyUxyx) jj + 2x = xUxy + yUyy + �(x 2 Uxxy - y 2 Uyyy + 2xyUxyy) (3.20) Plugging in the values for each of the partial derivatives, we obtain (let's ignore the z-component ): .. . 3 3 ;;; 1 [ 21 2 33 2 3V3 ] x - 2y = - x ± - y 3(1 - 2JJ) Y + - -(1 - 2JJ)X - -(1 - 2JJ) Y =f 2-xy 4 4 2 8 8 8 .. . 3 g 1 [ V3 2 g V3 2 33 ] y + 2x = ± - V3(1 - 2JJ)X + -y +- =f 3-x =f -y - 2-( 1 - 2JJ)xy 4 4 2 8 8 8 We should realize that the potential that accounts for higher order terms in the CRTBP has the form: (3. 21) is expressed as an expansion of the Legendre polynomials. The upper sign of "±" or " =r= " corresponds to the triangular point £4 and the lower sign corresponds to £5. If we 64 keep only the linear terms, we can describe the linearized dynamics around the triangular points by integrating the linearized system given by: X 0 0 0 1 0 0 X y 0 0 0 0 1 0 y i 0 0 0 0 0 1 z ± 3 '{3 (1 - 2p,) (3.22) X 3/4 0 0 2 0 x y ± 3 '{3 (1 - 2p,) 9/4 0 -2 0 0 y z l 0 0 -1 0 0 0 J z The characteristic equation for the triangular points can be obtained solving the determinant: Thus, yielding the fourth order polynomial characteristic equation: Solving this equation yields four pure imaginary roots: (3.23) (3.24) (3.25) (3.26) 65 where s1 = v�( -1 + J1 - 270(1- 0)) E lR and s2 = v�(-1- J1 - 270(1 - 0)) E C. The general solution to the linearized equations of motion from 3.20 in the in-plane motion can be expressed as: 4 x(t) = L Ae >. i t i =l 4 y(t) = L Bie >. i t i =l (3.27) (3.28) where Ai are the eigenvalues previously calculated and Ai and Bi are constant that are related. By plugging its correspondent derivatives: 4 x(t ) = L A>-ie >. i t i =l 4 y(t) = L Bi>.ie >. i t i =l (3.29) (3.30) obtained from equations 3.27 and 3.28 into equation 3.20 and keeping only the linear terms, we obtain that these constants are related as follows: (3.31) An alternate way of rewriting the general solution to equation 3.22 is: (3.32) 66 where ai and bi are constants that are related to Ai and Bi in the form: Now, from the first two equations of 3.33 we obtain that A 1 = (a 1 - a 2 i). Then, plugging A 1 into: (3.33) we obtain: (3.34) Similarly, we obtain the other constants: (3.35) (3.36) (3.37) 67 Since the general solution has long period 2 1r and short period 2 1r, non-commensurable S2 Sj frequencies, the solution is not periodic so that we obtain quasi- periodic orbits around the triangular points unless we eliminate some of the secular terms. In 1907, Charlier showed that there are two types of periodic orbits around the equilateral libration points: short period (the period is close to the period of the primary and secondary bodies) and long period (the period depends on the mass ratio of the primary and secondary bodies). The long-period linearized solution can be written as: xo . x(t) = xo cos(8 1t) +- sm(81 t) 8 1 y(t) = yocos(8 l t) + Yo sin(8 1 t) 8 1 and the short-period linearized solution is expressed in the form: x(t) = xo cos(8 2 t) + xo sin(8 2 t) 8 2 y(t) = yo cos(8 2 t) + Yo sin(8 2 t) 8 2 (3.38) (3.39) (3.40) (3.41) where xo, xo are yo, yo are the initial conditions which can be freely chosen or can be specified by making some assumptions. If we assume that a3 = a4 = 0, then, b3 = b4 = 0. From eq. 3.29 and 3.30, we obtain: (3.42) (3.43) (3.44) (3.45) 68 Since a 1 = 0 and h = 0, then a1 = a 2 i and 81 = � 2 x QJ..; thus: a2 · YO Ux y xo = 2 U2 + - 2 . xo 2 S! + XX 4 sf+UJY . 8t + Uxx Ux y Yo = - 2 xo + - 2 - · Yo (3.46) (3.47) which are the initial conditions for long-period orbits. If we assume that a 1 = a 2 = 0, then, b 1 = b 2 = 0. Since b3 = 0 and a3 = a 4 i, then, 8 2 = � 2 x £3., thus: a4 · YO Ux y xo = 2 + - 2 . xo 2 s2+U�x 4 s § +U £ y . 8� + Uxx Ux y YO = - 2 xo + - 2 - · YO are the initial conditions for short-period orbits. (3.48) (3.49) These are good initial guesses when searching for a periodic orbit around the equilat eral points. The time of revolution of a periodic orbit [57] can be found using T = To+ K r: 2 , where r: is a small quantity. Figure 3.3 illustrates the variation of K with w for values of wo between 0 and 1. The period of the short-period orbits around the triangular points decreases as the size of the orbit increases. For other wo values, the long-period orbits increase with increasing size of the orbit. 3.1.4 Stability Analysis The integration of the linearized equations of motion (equation 3.20) will give us infor- mation about the stability around the triangular points. To determine the stability of the linear system we need to find the eigenvectors and eigenvalues of the Jacobian, Dxf, such that the eigenvalues will exist if [D xf- >..I[ = 0 Solving numerically the Jacobi matrix using M athematica© software, we obtain the characteristic multipliers of a periodic solution, that is, the eigenvalues. These eigenvalues come in pairs. For the Earth-Moon system, the eigenvalues are >.. 1 = 1, >.. 2 = -1, A3 = 69 Constant K in function of mean motion K 40 u 20 �20 �40 Figure 3.3: K constant as function of the mean motion. 0.9545i, A4 = -0.9545i, As = 0.298211i, A 6 = -0.298211i. Some of the eigenvalues have null real part. Therefore, the equilibrium points are non-hyperb olic. If these eigenvalues are imaginary, then the triangular points are asymptotically stable, since the module of each of the eigenvalues is less than one. This means that any small perturbation will tend towards zero as time goes to infinity. So it will take infinite time for a satellite to approach the triangular points. For the Sun-Earth system, the eigenvalues come also in pairs and are At = 1, A 2 = -1, A 3 = 0.99999i, A4 = -0.999 99i, As = 0.00453026i, A5 = -0.00453026i. Therefore, the triangular points are said to have a "s addle x center x center" behavior of the linear dynamical system. The mean motions are s t = 0.004530216096014 and s 2 = 0.999989738571062 for the Sun-Earth system and s t = 0.28638514777049 and s 2 = 0.958991773568237 for the Earth- Moon system. The ratio �s = 220.7377 for the Sun-Earth system and � = 3.3486 1 S! for the Earth- Moon system. In both cases, s 2 >St . Note also that si + s� = 1 and that sis� = 2 J t-t( 1 - t-t ). 70 0 1 Mean motions at the Triangular points 0 005 0 01 0 015 0 02 0 025 0 03 0 035 0 04 11 Figure 3.4: Mean motion at the Triangular points for different values of 0· For mass parameters with 0 << 1, such that 0 2 :::::: 0 (eg. Sun-Earth system), the mean motions of the triangular points can be simplified as: (3.50) (3.51) As we can see, if 0 < 0Ruu th = � ( 1 - �) , the eigenvalues are purely imaginary so that L4 and L5 are stable as illustrated in Figure 3.4. The value of 0Ruu th can be obtained by equating the discriminant 1- 270(1 - 0) = 0, which controls the natural behavior of the roots. For 0 > 0Ruu th , the system yields all the eigenvalues to have a non-zero real part, such that L 4 and L5 are not stable. 71 3.1.5 Symplectic Diagonalization Letting 3 := 270 (1- 0) and the local diff eomorphisms 0 f---t s 1 (0) and 0 f---t s 2 (0), we can express 3 as function of s1 and s 2 as: Then, solving for 3, we obtain: � 2 27si + 27s� + 38sis� c. = (si + s�) 2 (3 .52) (3.53) We can then diagonalize the system in a manner similar to that in eq. 3.22. For simplicity, let's express the new linearization without the z-component: (3.54) where (3. 55) and lb is the 2 x 2 identity matrix. The liY' is the 2 x 2 matrix, which can be expressed as: -1 /4 ±3/4 ±3/4 ) ' -5/4 (3.56) 72 Thus, the new linearization is: 0 1 1 0 -1 0 0 1 X ' (L4,s) = (3.57) -1/ 4 ±3/4 0 1 ±3/4 -5/4 -1 0 We can obtain the eigenvalues and eigenvectors of eq. 3.57. Now we have that 3 = 3(st(0), s 2 (0)), so the eigenvectors will be more complicated. Using Mathematica © , we obtain: (3.58) (3. 59) · 1 ( s 2 + s 2 2 ) 2 ( 79s 14+ 14 2 s 1 2s 2 2 + 79s 2 4) where d 3 = v 1 . These eigenvalues are distinct and purely ( s!2 + s 2 2 ) imaginary, ±>.. 1 i, ± A. 3 i , where >.. 3 > >.. 1 . We are not showing the eigenvectors because they are very lengthy. However, given the eigenvectors � ± = �r ± i�i, such that �r (real part) and �i (ima ginary part) E JR 2 n for n=2, we can build a new basis of JR 2 n with vectors Z k , r and Z k , i for k= 1, ... ,n such that { zt,r, Z 2 , r, · · · Z k ,r, zt,i, Z 2 ,i, · · · Z k , i} is symplectic iff Z k , r] 2 nZ k , i = (] 2 n)ki . These vectors can be obtained as follows: (3.60) (3 .61) (3.62) (3.63) 73 where O:k = Jl �r · ] 2 n · �i I i- 0, k = 1, ... , n. Both Z k , r and Z k , i are vectors that span the plane IIi so that: (3.64) where each plane, IIi, is symplectically orthogonal to the others, such that IIi · ] 2 n · IIj = 0, vii- j. The matrix, M = col(zl,r, ···zn,r, Z l,i, ···Zn ,i), formed by these vectors is symplectic and satisfies the canonicity condition MT ] 2 nM = ] 2 n . Thus, the canonical transformation (q,p) f---+ (q,p) = M - 1 (q,p) conjugates the Hamiltonian P(q,p)/P T] 2 n = -] 2 n · P to: n H(q, p) = L sign ( 6, r · ] 2 n · �k,i ) s ; ( q 2 + p 2 ) k=l (3.65) We conclude that the new Hamiltonian is given by: (3.66) 3.1.6 Variational Equations Let the dynamical system i = 1, ... , 6 (3.67 ) be an autonomous nonlinear map where the initial condition is xw such that xw = xw(xl O , x 2 o, x 3 o, x4o, xso, x 6 o). We perturb the initial condition �i(O) such that x ' i(O) - xi(O) = �(0). Since �i(O) « 1 is a very small perturbation, �i(t) « 1 is also a small perturbation at a later time so that: �(t) = x ' i (t)- Xi(t) (3.68 ) 74 To first order approximation, the variational variable � satisfies the variational equations: (x -f- �) = x + � = f(x + �) = f(x) + Df(x)� + 0 ( [�[2 ) � � D f(x(t)) � (3.69) where D f ( x) is the state transition matrix. The eigenvalues of this matrix will determine the stability of the system. The variational equation 3.69 can be expressed also as: 6 � = L Pik�k i = 1, ... , 6 (3.70) k=l Then, we can linearize the variational equation around x( t). If we let the flow, ¢, associ ated with eq. 3.67 such that <1>(0) = hx 6 , ¢ E IR 6 , then: ¢ = Df(x(t))¢ (3. 71) which is our fundamental solution. Note that if: x(t) = ( � (t), rJ (t), ((t), � (t) , rj (t), ((t)) (3.72 ) is a solution of the variational equations for the RTBP, then, x(�t) = ( � (�t),rJ (�t),((�t), � (�t),rj (�t), ((�t)) (3.73 ) is also a solution. Integrating any initial conditions from a Trojan orbit around £5 backward in time will give us a final state ( eg. at Earth) so that if we integrate this state forward in time we would get the same initial conditions (at the Trojan orbit), which is the trajectory that the spacecraft would follow in the RTBP. Also, using these same initial conditions at Earth under the transformation: (x, y, z, x, y, i, t) ---+ (x, y, z, �x, y, i, t) (3.74 ) 75 we obtain symmetric transfer orbits to the other equilateral point L4. 3.1.7 Monodromy Matrix The monodromy matrix measures the periodicity of a process. Assuming that x = f(x) has a periodic solution x(t) with period T then, y(t + T) = y(t) (3.75) Let x = A(t)x be the variational equation where A(t) = Dj(x(t) ); then A(t + T) = D f(x(t+T)). Let <!>(t) = A(t)eQ t where A(t) is aT-periodic function and Q is a constant. Since <!>( t+T ) is also a solution such that <!>(t+T ) = <!>(t) ·eQT and A(t) = <!>(t)e -Q t , then, A(t + T) = <!>(t + T)e-Q ( t + T ) = <!>(t)eQ T e-Q t e-Q T = <1>(t)e -Q t = A(t). The matrix A(t) is characterized by its eigenvalues and eigenvectors. The eigenvalues give us information on how the flow either expands or contracts and the eigenvectors tells us the direction along which this expansion or contraction occurs. To calculate the period of the Trojan orbit, we let the function f(t) = [x(t)- x(to)[ such that j(t*) = 0. These Trojan orbits have a period very close to one year. To compute the exact period, we let the function be integrated in the interval [tL ow t up ] where tLow = t j -s and t up = t j + s; t j is the time at which f(t*) = 0 and sis assumed to be a small fraction of the period ( eg. s :::::: 0.1 - 0.2). 3.2 The Bicircular Problem (BCP) The second key problem that we will analyze is the Bicircular Model where the Earth and the Moon are considered point masses that revolve around the barycenter with this center of mass revolving around the Sun (point mass) in a circular orbit as we see in Figures 3.5(a) and 3.5(b). Therefore, with this model, we pretend to study the motion of the spacecraft under the attractive influence of the Sun, Earth and Moon. Even though, this model is not coherent, it is an approximation that provides a good solution before 76 applying a differential corrector to use it in the ephemeris model. The massive bodies are (a) . - EARTH EL4 ' ' ' ' ' MOON \ ' \ , ', Earth \, \ \ rbit / \ \ / \ ;!- '\- · ± : _ -:: . / LL2 /. \ , ' / ,, Moon \ \ . � \ . Sun �arth · · / Lll\ \ ------------------------�---- ---- 1 --------�---- \ -- - - . LLS L2 '- / ,' / EL2 /. , ....... , _.,;'I //LL� ,., I / I / l / �I --- - - (b) Figure 3.5: a: General schematics of the Bicircular Problem with the Sun and Moon moving in circular orbits. b: Lagrangian Points in the Earth-Moon System. not dynamically coherent. The BCP is a periodic system. Since we can look at the BCP as a periodic time-dependent perturbation of the RTBP, we can assume a parameter 77 f S = f!j for the Sun such that if we change this parameter we can generate both the r s RTBP (cs = 0) or the BCP (cs = 1): .. . X + f.J., X - 1 + f.J., X - X S x-2 y=x-(1-f.J.,)-- -f.J., -f.J.,s -E s cos B r3 r3 r3 1 2 3 .. 2 . ( 1 1 - /-" /-" ) y - Y s . 8 y+ x = -- - --y-f.J.,s -- +fs Sm r3 r3 r3 1 2 3 .. (1- /-" /-" /-"S ) z=- -- +-+- z r3 r3 r3 1 2 3 (3.76) where r 1 , r 2 and r 3 are the distances from the spacecraft to the Earth, Moon and Sun, respectively, where r 1 = J ( x + /-" ) 2 + y 2 + z 2 r 2 = J ( x - 1 + /-" ) 2 + y 2 + z 2 r 3 = J( x -xs) 2 + (y- Y s) 2 + z 2 (3. 77) xs and Y S represent the coordinates of the Sun xs = rs cos e Y s = -rssinB (3.78) and e is the angular displacement of the Sun around the Earth. (3.79) where Bo is the initial phase of the Sun and ws = 3 60��s yn = 0.92519742 is the angular speed (or synodic motion) of the Sun and where Bsyn = 360° · 3lJl25l2 5 = 29.1 06° since 78 there are 29.53 days between two consecutive new Moons. Then, The mass parameters are given by: 11 = MM = 0.012150585609411 ME +M M Ms s [LS = M M = 3.2890056 14000710 X 10 E+ M (3.80) (3.81) In 1966, Steg and De Vries [64] presented an analytical form for the equations of motion in the Sun-Earth -Moon system: x � 2i; � qx � c 2 y = c4 + C 6 cos 2¢t + c7 sin 2¢t + cs cos ¢t+ + C4 sin ¢t + cne(x COS 2¢t � y sin 2¢t)+ jj + 2:£ � c 2 x � c 3 y = c5 + c7 COS 2¢t � c 6 sin 2¢t � Cg COS ¢t+ where: + c10 sin ¢t � cnv(x sin 2¢t + y cos 2¢t)+ 3 1 q = - + - e 4 2 3 C 2 = 4 V3(1 � 2J1) 9 1 c 3 = - + - e 4 2 1 c4 = 2 exL 1 cs = 2 e YL 3 c 6 = 2 exL (3.82) 79 3 C l 2 = - - 4as (3.83) where XL and YL are the coordinates of the triangular point, as = e�r) l / 3 = 388.8111 360° 36 0o+Bs yn 2 9. 5 3 ( note that as � A U /384, 400 ) where ns = 3 65_ 2 4 2 5 which is the distance from the Sun to the center of the system and () = l!:f = 0.005595625597 is the true anomaly of the a s Moon in its orbit around the Earth ( see Figure 3.5( a)). We will use this "restricted four-body" model to generate asymmetric orbits in the Earth-Moon system and analyze how the influence of the Sun and the Moon's orbital eccentricity affect the vicinity of the equilateral points. 3.3 Elliptical RT BP (ERTBP) The second model that we are using in this research is the elliptical restricted three- body problem ( ER TBP). The equations of motion governing the ER TBP are similar to those in the CRTBP: (3.84) 80 Transfer and Trojan Orbits in CRTBP vs. ERTBP o . * Sun . Earth �.1 .. � o 2 04 o 6 o a X(non:lim,AU) (a) 1 2 x 10� Transfer Orbit in CRTBP vs. ER TBP 1 0005 1 001 10015 X(nondim,AU) (b) 1 002 Figure 3.6: a: Transfer and Trojan trajectories using the CRTBP and ER TBP. b: Close up boxed region (near the Earth) in green to see the difference between the departing transfer trajectory when using both models. The orbit in magenta is used for the CRTBP and the one in blue for the ER TBP. where E = -e cos E and where r1 and r 2 are the distances from the spacecraft to the Sun and Earth, respectively. rt = v (x + J.t (1 + E )) 2 + y 2 + z 2 r2 = V (x-(1 + J.t )(1 + E )) 2 + y 2 + z 2 (3.85) The change w in the mean motion is a func tion of the eccentric anomaly, w = (l�)2 and w _ - 2e,ll=e2 sinE - ( 1 -ecos E )3 Figures 3.6 and 3. 7 illustrate an example of a trajectory generated using the ER TBP and compared against the CRTBP. Notice that, in this model, the orbits around the equilateral points are not periodic but quasi- periodic as shown by the direct motion (in red) of the space probe. The effect of the eccentricity of the Earth around the Sun has some effect in the vicinity of the £4,5. The black line is the periodic orbit around £5 obtained in the CRTBP and the red orbit is the quasi- periodic solution obtained in the elliptic RTBP. As for the transfer orbit, the magenta orbit is the solution obtained in the CRTBP and the blue orbit is the solution generated in the elliptic RTBP. 81 Trojan Orbit Comparison CRTBP vs. ERTBP Trojan Orbit Comparison CRTBP vs. ERTBP 03 04 0 5 0 6 07 X(nondim,AU) (a) (b) Figure 3.7: a: Trojan trajectories using the CRTBP and ER TBP integrated for five years. b: Close-up boxed region of the Trojan orbit shows that the Trojan orbit (red) in the ER TBP is asymmetric whereas the Trojan orbit (black) is symmetric in the CRTBP. Note the direct behaviour of the Trojan orbit in the ER TBP as it moves towards Earth. The size of the Trojan orbit is about 0.53 AU. The shift of this orbit tells us that the periodic orbits are not preserved in the elliptic RTBP. These weakly unstable quasi- periodic solutions were obtained numerically where the triangular points no longer subsist because of the influence of the Sun making the elliptic RTBP a nonautonomous system. This perturba tion is sufficient to cause a weak stability; yet the equilateral points are linearly stable and we can assure the existence of quasi- periodic orbits around L5 for a few years. It would take of the order of a few m/s to station-keep a point mass satellite around one of the quasi- periodic orbits. We further analyze the station-keeping maneuvers of selected orbits generated in the real solar system, the JPL model. As we anticipate, this high fidelity model yields more refined numerical solutions. In the next sec tions, we will talk about the JPL ephemeris model and how to transition between the different models. 82 3.4 Developmental Ephemeris Model (DE421) The JPL Ephemeris model [ 63] is considered one of the most accurate models of the Solar System. In our research, we used mainly the JPL Ephemeris DE421 for our high accu racy ephemeris model. Observational data are collected in order to improve the initial conditions of the planets at a given reference epoch [54]. Therefore, the equations of mo tion are numerically integrated with the new improved initial conditions to generate the planetary ephemerides. The observat ional data are obtained from different sources such as optical observa tions from the U. S. Naval Observatory, radar time-delay measurements of the planets from several telescopes and spacecraft range points from planetary encoun ters. This new model [63] gives a full precision numerical integration for the full interval 3000 BC to 3000 AD. This model contains the coefficients of a large set of Chebyshev polynomials, which are used to compute the position or velocity of a Solar System body over a determined time span well known as granule. The length of these granules pro vides a very accurate approximation with the corresponding Chebyshev polynomial. The Chebyshev coefficients have been updated after taking into account the latest planetary observa tions. The polynomials corresponding to the neighbor granules have to be equal when evaluated at the border points which guarantees that there are no discontinuities in position and velocity. 3.5 Transition Between Models Wh y transfer to the real Ep hemeris Model? We know that the JPL model cannot be studied analytically, but that it can be in tegrated numerically. We are also aware that the Lagrangian points are not equilibrium points anymore if we consider the real model [11]. The JPL model shows that the neigh borhood of the triangular points becomes unstable and nearby trajectories event ually escape. Therefore, the RTBP may not be a good model to be studied since it produces a different qualitative behavior. In addition, the BCP model does not have all the features of the real model, since relevant frequencies are omitted. 83 Also, Gomez et al. [18] have done extended simulations to study the stability of orbits around the Triangular points. They analyzed many of these orbits with the JPL ephemeris model and noted that some of them show an excellent stability character which can be considered quasi- periodic. They studied the behavior of their main frequencies for the stability region and confirmed that they are very close to the one obtained with the BCP. Furthermore, the resonances causing the instability are also the same as in the BCP. However, some of the stable orbits in the BCP become unstable when the full set of perturbations is considered. Even though the solutions obtained in the CRTBP provide similar qualitative solu tions for higher order mutli-body models [59], we would need to transfer to the Ephemeris Model to improve the accuracy of transfer trajectories and to illustrate the robustness and credibility of the CRTBP. One of the reasons to transfer to the real model is that the symmetry of the CRTBP is lost when the solution arcs are transferred to a higher order harmonic model. Hence, the libration points are no longer equilibrium points relative to the rotating frame of the primaries. For instance, we would not be able to find halo orbits as we would find in the CRTBP, but rather oscillating solutions such as quasi-periodic orbits. Therefore, we need to check the CRTBP against the new real model DE421. The CRTBP assumes that the Earth-Moon barycenter goes around the Sun in a circle but in the DE421 model we know that the real motion of the Earth, Moon and even the Sun changes with time as seen in Figure 3.8. The final state vector of the Earth shows a displacement of (1.666095E5 km, 2.934197E5 km, -3.2197 43E3 km) with respect to the initial position after one year. The final position vector displacement for the Moon is (2.875263E5 km, -4.044968E5 km, -1.0 76085E3 km). The Sun has also moved by (1.555558E5 km, -1 .424924E5 km, 1.813125E3 km) in the x, y and z directions, respec tively. 84 1-Year Tra je<:tories of Sun Earth Moon in Ecliptic 12000 Sun Motion of 50 years in 12000 frame 4 4 � N 0 ! I 0 -2 -I -2 -2 X 10 ° x w' -3 2 X (km) Y(km ) X I 0 ° 0 -I -2 -2 X (km) Y(km ) (a) (b) Figure 3.8: a. Trajectories of the Sun, Earth and the Moon in the DE421 ephemeris model for one year. b. Trajectory of the Sun in the DE421 model for 50 years. The red- star represents the beginning of the trajectory (March 8th 2012) and the red- circle the end of the trajectory. In this research, we used the DE406 constants in our CRTBP to analyze the trajectory mission design and navigation analysis for an L5 mission. However, we use the JPL DE 421 model to generate more refined trajectories. 3.6 Analytical Ephemeris Model In this section, we constructed an Ephemeris model that includes Pluto and the Earth- Moon barycenter. This ephemeris was checked against the real Ephemeris DE421. To know how close this analytical ephemeris model is with respect to the real model, we calculated the position of the Earth-Moon barycenter (EMB) with respect to the Sun and obtained an error of -1467.15 km, 317.55 km and -289.81 km in the x, y and z directions, respectively. The error in velocity was obtained to be 12.56 cm/s, 6.3 mm/s and 5.16 cm/s in the x, y and z directions, respectively. This analytical method will use the planetary elements and their rates with respect to the mean ecliptic and equinox of J2000 valid for the interval 1800 AD-2050 AD [63] . This procedure comprises six steps: 85 1. Obtain each planet's orbital elements using the planetary elements {a, e, i, 0, w, L} and their rates {a, e, i, 0, w, L} according to q = qo + qdT where q is any of the planetary T h - 2 45 1545 .0 . . elements and where dT = ep oc 3 6 5. 2 5 , where the Juhan date at J2000 lS 2451545.0 2. Get the argument of perihelion, w, and the mean anomaly, M, using: w=w-0 M=L -w (3.86) (3.87) 3. Compute the eccentric anomaly, E, using the mean anomaly, M, and applying Newton's method to second order: E = M esin(M) + 1- sin(M +e)+ sin(M) f f !" E ne w = Eold- J ' ( 1 + 21 , 2 ); (3.88) (3.89) where f = E- ( M + 1 - sin(�i��s in ( M ) ), and f' and f" are the first and second derivatives of f with respect to the mean anomaly. 4. Obtain the planet's heliocentric coordinates in its own orbital plane, fhel, from: x� el = a(cos E - e) Y � el =a � sin E I O· 2hel = ' (3.90) (3.91) (3.92) 5. Get the ecliptic coordinates, r ee l, in the J2000 ecliptic plane, with the x-axis aligned towards the equinox, such that: Xecl = ( cosw cos 0- sinw sin 0 cos I)xhel + (- sinw cos 0- cosw sin 0 cos I)Yhel (3.93) Yecl = (cos w sin 0 + sin w cos 0 cos I)x hel + (- sin w sin 0 + cos w cos 0 cos I)Yhel (3. 94) Zecl = (sin w sin I)xecl + (cos w sin I)Yhel (3. 95) 86 6. Compute the equatorial coordinates in the J2000 frame: Xeq = Xecl Yeq = (cos c:)Y ecl - (sin c:) zecl Zeq = (sin c: )Yecl + (cos c: )zecl (3.96) (3. 97) (3.98) where the obliquity of the Earth equatorial plane with respect to the ecliptic is 23.43928° . 3.6.1 JPL DE421 Ephemeris Model After having used the CRTBP as our initial model to generate trajectories, we proceed to use a higher fidelity model based on the JPL planetary ephemeris DE421 model [16] This model includes the gravitat ional influence of all the planets (and Pluto) and the Moon from 1900 to 2050 [63]. The JPL DE421 model represents the current best estimates of the trajectories of the planets and the Moon. While the lunar orbit is known to the sub-meter accuracy, the orbits of the inner planets (Venu s, Earth, and Mars) are known to the sub-kilometer accuracy and the outer planets (Jupiter and Saturn) are known to tens of kilometers. Both, the new JPL DE421 ephemeris or the JPL DE405 ephemeris are sufficiently accurate to use when generating integrated trajectories in the real model. The lunar position is separating with respect to the Earth. The Moon separated 6 meters in 2008, to 8 meters in 2012. The Moon will continue separating to 11 meters in 2015 and 16 meters in 2020. This separation occurs along the radial direction [75]. From [56] , we loaded the public SPICE kernels that were used in our high fidelity model. These kernels are de-4 03-mass es.tpc which is based on the same constants from DE403, de421. bsp which is the binary planetary and lunar ephemeris, naif0009.tls which is the leap second kernel file, and the pck00009. tpc which is the kernel file for the general planetary constan ts. 87 In order to generate the ecliptic positions of the Earth and the Moon, we will use the table of planetary elements and their rates. These tables provide the classical orbital ele ments of the Ear th-Moon barycenter (EMB ) but it does not provide the orbital elements of the Earth and Moon independently. Therefore, we can compute the position of the EMB in the ecliptic with respect to the Sun but we are interested in knowing the position of the Earth and the Moon in the ecliptic with respect to the Sun. How do we calculate these vectors? We can calculate them as fo llows: First, from the JPL ephemeris, we can calculate the state vectors of the Earth and the Moon with respect to the EMB in the ecliptic J2 000 coordinates at the reference epoch J2000. This is done with the following expressions: [EMB E ] = cspice_spke zr( ' EARTH ' , et, ' ECLIPJ2000 1 , 1 NONE ' , ' EMB ' ) [EMBM] = cspice_spke zr( ' MOON ' ,et, ' ECLIPJ2000 1 , 1 NONE ' , ' EMB ' ) where et2 = cspice_ str2et( ' 2451545.0J D ' ). Given these state vectors in the ecliptic then, we can obtain the classical orbital elements for both the Earth and the Moon. With these orbital elements and neglecting their classical rates, we can generate the state vectors of the Earth and the Moon in the ecliptic coordinates at a given time. Then, we will have to rotate these ecliptic vectors in position to generate their corre sponding position vectors in the ecliptic J2000 coordinates. The transformation will be of 23.43928° around the x-ax is. Finally, the position vectors of the Earth and the Moon with respect to the Sun will be the sum of their corresponding position vectors in the ecliptic with respect to the EMB and the position vectors in the ecliptic J2 000 coordinates. 88 3.7 Ephemeris Transformation from Rotating to Inertial Coordinates 1. Find the position and velocity of the Sun and the Eart h-Moon barycenter and their relative distance, using the J PL DE 421 (com pared against my own developed Ephemeris) (3.99) The relative state vector of the EMB with respect to the Sun can be computed using the SPICE command: [ X EMBd = cspice_spke zr( ' EME ' ,et, 1 ECLIPJ2000 1 , 1 NONE ' , ' S UN ' ) Similarly, we could obtain the relative state vector of the EMB and the Sun with respect to the Solar System Barycenter (SSB ) as fo llows: [ X EMB, SS B] = cspice_spke zr( ' EM E ' , et, ' ECLIP J2000 ' , ' NONE ' , ' SSE ' ) [ X 0,ssB] = cspice_spke zr( ' SUN 1 ,et, 1 ECLIPJ2000 1 , 1 NONE ' , ' SSE ' ) where et is the ephemeris time. Hence: (3.100) 2. Calculate the position and velocity of the center of mass of the Ear th-Moon and rotation rate of the synodic frame. (3.101) 89 3. Calculate the position and velocity in synodic coordinates i'R =II r8,EMB II (rR,non - (1 - fi,, 0, 0) ) VR =II i'0,EMB II llw RIIvR,no n (3.102) (3.103) 4. Compute the rotation matrix to transform the rotating coordinates to the inertial coordinates. where i'EMB,G A wR A e3 x e 1 e1 = ll rEMBdl ; e3 = llwRII ; e2 = lle 3 x et ll 5. Compute inertial frame coordinates i'J,syn = I [C] R · i'R (3.104) (3.105) (3.106) where the synodic notation represents that the center of the synodic coordinates is at the center of mass of the Sun-Earth-Moon barycent er. 6. Compute the inertial position and velocity of spacecraft with respect to the center of mass of the Sun. i'I,8 = i'I,syn + i'EMB,8 Vf,8 = VJ,syn + w X i'I,syn + VEMB,8 (3.107) (3.108) 90 3.8 Ephemeris Transformation from Inertial to Rotating Coordinates We fo llow a similar approach as above assuming we know the state vector in the inertial frame with respect to the center of the Sun. 1. Calculate the inertial frame coordinates - - - rl,syn = rl,0 - rEMB,0 i.h,syn = ih,0 - VEMB,0 - WR X TJ,syn (3.109) (3.11 0) 2. Obtain the position and velocity vectors in the inertial coordinates using the inverse of the matrix transf ormation. 3. Compute the position and and velocity vectors in the synodic coordinates. TR,non = TR II - II + (1 - fh, o, o) rEMB,0 VR VR,non = ,-,--- -- -----,- -,-,---- ---,-,- llr EMBd[ [[wsyn[[ 3.9 Coordinate Systems Rotational frame (3.111) (3.112) (3.11 3) The barycenter of the primary and secondary bodies is used as the origin of a three- body system. This frame is defined such that the x-direction unites the primary and secondary bodies and is always directed from the larger to the smaller body. The y-direction is 90 degrees from the x-direction in the plane of motion of the primary and secondary bodies. Finally, the z-direction is defined completing a right-handed coordinate sys tem. In ertial fra me with diffe rent centers A very common reference frame is the Earth Centered Inertial (ECI). The origin of this frame is the Eart h's center of mass. X is the direction of the vernal equinox (intersection 91 of the equatorial and ecliptic planes) . Z is the instantaneous direction of the rotation axis of the Earth and Y completes the right-handed coordinate system . This frame is not exactly inertial because the origin is accelerating as the Earth travels around the Sun and the rotation axis precesses and nutates . The Earth's precession is about a 23 .43928 degrees cone with a period of about 25,7 70 years and the nutation has a period of about 18.6 years . The motion of this frame is small enough that it can be neglected and fixed to a particular date like the standard J2000 frame. Tra nsition between fra mes The transformation between the rotating and inertial frames ( see Figure 3.9) is crucial for tackling the 3BP and is given by: (3.11 4) where w is the angular velocity, r is the position vector in inertial coordinates and () is the angle between the rotating and inertial frames . The transformation between the y y A i SATELLITE wm /, , _ __ WJIE i � /:/ · · · · · - - E : . . __ ... 7 X k?p x, : Secondary /! 1 \ Mass (M2} /,/ .. ;/ ............ J. e ... ... ..... .... ........... ...... ··· ·· ··· ····· -.. - > .X I Primar y Mass (M1} X Figure 3.9: Schematics of the Inertial and Rotating Frames for the 3BP. 92 rotational and inertial frames can be expressed in the form cos() sin() [ r f = [: : ] [ r rl d 0 (3.115) sin() cos() 0 where d is the distance between the barycenter and the primary body, R is the rotation matrix for transforming the position state from inertial frame to a rotating fra me, and R is the derivative of this rotation with respect to time. No ndimensional fra me cos () sin () 0 R = - sin () cos () 0 0 0 1 (3.116) To simplify things, we will use the normalized and non-dimensionalized convent ion for length, mass and time quantities. Therefore, we can define the reference characteristic length x * = x2 - x1 (normalized distance between primary and secondary body is equal to 1), that is, x1 = -p,, x2 = 1- p,. The reference characteristic mass is defined as m * = m 1 + m 2 where m1 = 1 - p,, m2 = p,, such that m1 + m 2 = 1 is also normalized. Finally, using Kepler's third law we can obtain the third quantity that we want to normalize, time: (3.1 17) 93 where G is the gravitat ional constant also normalized and equal to 1. In addition, the mean motion will also be normalized and equal to 1. Therefore, the three normalized distances and normalized time (Figure 3.9) will be: (3.118) Coordinate translation to the Tr iangular Points We will define the coordinate system of the equilibrium points by using the fo llowing translation (positive sign for £4 and negative sign for Ls). 1 X £= X-X E, XE =± 2 (3.119) V3 Y L = Y - YE , Y E = ± 2 (3.12 0) We also introduce the synodic coordinates with respect to the Sun-Earth (or Earth Moon) system, using the fo llowing transforma tion. ( x ) ( cos t sin t ) ( X ) y - sin t cos t Y (3.121) By applying this transf ormation, we obtain the equations of motion of the BCP that we introduced in section 3.2 3.10 Navigation Model In this analysis, we will start with a diagonal a priori covar iance (6 x 6) matrix, Po , where the first three elements of the matrix correspond to the variance of a priori errors in position and the last three elements of the diagonal correspond to the variance of 94 the a priori errors in velocity. Then, we obtain the covar iance at each of the maneuver locations: (3.122) When i = 0, we start with the corresponding injection error associated with launch and the error due to the Trojan insertion. The covariance can be expressed as: P f i = (3.12 3) where Ppv and Pv p are the correlations of the position with respect to velocity. Now use the state transition matrix, <I>, to obtain the error in velocities. which can be displayed as: <I>s = O x f (8x s ) - l 8xo 8xo oP 1 oP 1 OF; OV'i <I> i = ov 1 ov 1 oF; oV; (3.12 4) (3.125) 95 Calculate the a priori error in velocity, ay, from the covariance error in velocity, Pv; = (8Pt)- l (8Pt)- T Pv; oV£ P p ; oV£ a-� = Tr(Pv; ) = o- � i + a- � + o- � i Finally, we can compute the a priori covar iance matrix at the end: where Pv P; = 0.01Pv; (3.126) (3.127) (3.12 8) (3.129) where we assumed a conservatively large execution error (10%) which will be compared against a 1% execution error in a later section. This execution error represents the percentage of the planned maneuver that is being executed and it will depend on the mission requiremen ts. This fact or is conservatively large since we know that the spacecraft will encounter much smaller execution errors. Finally, the a priori error at the next burn will be: (3.130) Similarly, we can do the same for the rest of the maneuvers. We are interested in calcu- lating the 99% cumulative probability distribution [48, 61] b. V 99 % = p, + 2.5 76o- (3.131) In this na vigation analysis, we will obtain the standard deviation a-y1 , o- v 2 pv 3 pv 4 , o- v 5 and o- v 6 for the error made by each of the maneuvers, and their corresponding a priori error 96 covar iances given by Co, C1, C 2 , C 3 , C4, Cs , C6. We obtain the absolute value of O" V r ota t for each maneuver where CT�, = :2 1 �_ 1 CTy 2 .. For a single maneuver, we obtain a half YT ota l - J normal distribu tion. Adding all the maneuvers yields a normal distribution according to the Central Limit Theorem. For each trajectory, we have a series of maneuvers and each of these maneuvers have a half normal distribution. The velocity for each maneuver in any given direction is normally distributed. However, since the fuel consumption is related to the absolute value of the velocity, the fuel consumption is distributed half normally. Each half normal distribution (see Figure 3.10) may require different fuel consumption. If we add these half normal distributions statistically, we obtain a normal curve which is displaced by the mean. If one of the maneuvers dominates, the total will not be exactly a normal distribu tion. If we know the mean and the standard devia tion, we can calculate the 99% value [61]. Therefore, if we perform 100 missions, we know that for each mission the spacecraft will need a different .6. V but 99% of the missions will be successful and only one mission will fail due to lack of fuel. The mean and standard deviation were obtained using the following expressions [67] The probability density function for a half normal distribution, f ( x), is: x ElR x < O,x E lR The mean or expected value is obtained by integrat ing the first moment from 0 to +oo as: E[ x] = /"" = l eo xf(x) dx = f!,o < x < +oo (3.132) The variance can be computed by integrating the second moment as: CT(x) 2 = 1 00 (x- E[ x]) 2 f(x)d x = E[ x 2 ]- (E[ x]) 2 (3.133) CT(x) = J1- � (3.134) 97 Assuming the spacecraft will perf orm several maneuvers, then the variance, mean and standard deviation for a half normal distribution can be expressed [48] as: CT� = Tr(Pv;), i = 1, ... , Nman where N man is the number of maneuvers. f(x Prob(-l <x<+!) � 68.27% Prob(-2<x<+2) � 95.45% Prob (-2.57<x<+2.57) � 99% -I t-, I ' ' \ I ' I \ I \ / Half- Normal I \ I , \ I ',, ___ _ 0 +I +2 +2.576 x(a) (3. 135) (3.136) (3. 137) Figure 3.10: Probability Confidence Intervals. The 99% confidence level can be obtained by integrating the probability density function from zero to 2.576cr for a half normal distribution. 3.10.1 Transfer Trajectory The transfer trajectory is divided into five segments or legs as illustrated in Figure 3.11 where we will perform propulsion maneuvers. Therefore, the spacecraft will perform six TCMs after injection, during cruise and arrival at the Trojan orbit. By performing multiple TCMs instead of a single maneuver, the mission objectives can be accomplished with higher accuracy and less propellant. The orbit determination software provides the 98 Transfer Trajectory T.poch +lOd "� +20d \ I +150d � I / / Trojan Orbit -- ........ / ' , / \ +20 � I +to� · I T. �� j( I Fr ,, . - 3 d I / I J · L 5 I f T.," - 3 0d 1 / � +150 d / I ·.\ / / , , / . ........ ___ / ' - ·· - · · --...,.,. Figure 3.11: Sketch of a transfer and a Trojan orbit broken into several segments. The transfer orbit (magenta) shows several maneuvers (red star) to be performed at different locations after launch, midcourse, and arrival. The Trojan orbit (blue) shows several maneuvers (red star) right after insert ion, during midcourse along the Trojan orbit, and before completing one Trojan revo lution. mapping matrices and cov ariances as shown in Figure 3.12, that is, the state transition matrix (Mi, i = 1, ... , Nman) maps state deviations from a defined epoch, Tepoch, to the time of the maneuver . The maneuver analysis requires the mapping and covariance matrices (see Figure 3.12). Let C1,C2,C3,C4,C 5 and C6 denote the covar iances and M1, M2, M3, M4, M5 and M6 the mapping matrices at each point where the maneuvers will be performed. The covar iances at the time of each maneuver will be mapped to the end time, Tend· 99 3.10.2 Libration Trojan Orbit Similarly, we will divide the Trojan orbit into several segments as we process the tra- jectory maneuvers one at a time. The time of arrival of the spacecraft on the transfer trajectory will be defined as the epoch time or start time for the Trojan orbit . The pe- riod of any of these orbits around Ls is about one year. The spacecraft will perform six � ax, J:fXJCh �ml 'lr - '1- -- -'f. -------- -'f. ------- '1- -------� - -'f. +2d +lOd +20d C0 C1 C2 M 1 M 2 +l50d c, M, T,,, - 30d T, .. J - 3d C4 C5 C6 M4 M5 M6 � ax, a x1 a x �� \\ ( �� a x,� �nd .,. - Y.. --_.,. --------_.,. -------Y.. -------'f.. - -'f. +2d +lOd +20d C0 C1 C, M 1 M, +l50d c, M, T,.," - 30d T'"" - 3d c. c, c6 M4 M5 M6 Figure 3.12: Left . Mapping by state transition matrix to the time of the maneuver. Right . Covariance matrices mapped from the time of the maneuver to the end time, T end. correction maneuvers after insertion into the Trojan orbit during the first year. Then, it will execute 4 maneuvers during the second year. The first maneuver after insertion into the Trojan orbit will be performed 10 days after insertion to compensate primarily for errors generated at the insertion maneuver. The second maneuver will take place 20 days after insertions to clean up dispersions from execution errors produced from the previous maneuver. A third maneuver will be performed at about 150 days after insertion to com- pensate for different stochastic errors and additional built up execution errors incurred when orbiting Ls . The spacecraft will perform a maneuver about 30 days before complet ing its first orbit around Ls (assuming the spacecraft follows the nominal trajectory) to clean up dispersions from execution errors built up during previous maneuvers. Finally, the last two maneuvers will be executed, one at 3 days before the completion of the first orbit and the last one about one year later after insertion into the Trojan orbit in order to remove previous maneuver execution errors and built-up stochastic accelerations. We 100 will assume that the spacecraft orbits Ls for about two years. After the spacecraft has completed its first orbit around Ls, it will not be required to perform the first two maneu vers again since the purpose of those maneuvers was to remove errors due to the insertion maneuver . Theref ore, only the last four correction maneuvers will suffice to maintain the spacecraft in orbit for station keeping purposes during the second year around the Trojan orbit. 101 Chapter 4 Research Gadgets 4.1 Mission Design Tools 4. 1.1 Dynamical Systems Theory Below we show a flow chart (see Figure 4.1) of some of the tools and theory behind the dynamical systems theory that we will be using in this research . Starting with some initial conditions, we can generate some periodic orbits around the equilateral points by a trial and error method. We have used a naive continuation method and saw that it can give us a pretty good initial guess to create some of these orbits. Howeve r, the initial parameters need to be tuned to obtain some of the periodic orbits that we fo und. Therefore, no optimization method was used to produce these orbits although this will be the next step to produce more accurate periodic orbits. Once we have periodic orbits fo rmed, we will proceed to study possible invariant manifolds by slightly perturbing the periodic orbit conditions. So far we have obtained different .6.. V performances by trial and error method. However, this method may not provide us with the optimal trajectory, so we will need to build a differential corrector to automate the trajectory optimization. 102 Moreover , we will perform an orbit determination study needed for trajectory cor- rection maneuvers for launch corrections, cruise, and insertion in order to generate a stochastic analysis of the trajectory. The rest of this section will descr ibe some of tools that we will use in this proposed thesis: Periodic and Quasi-Periodic Orbits, Invariant Manifolds and Poincare sections. INITIAL GUESS: OPTIMIZ ATION: ----------------- ----,---------------------, Model: Initial Conditions JPL DE42 Models: � � 0 � � L5-tool in PCODP : Mission Navigation/O.D. Design Strategy Strategy PO/QPO Orbits � J, Stability Trajectory O.D. Trial and Lindstedt Diff. Orbits Analysis Analysis Error Poincare Corrector l � Mission Requirements Manifolds Tracea bility Matrix Invariant J -------------------- �l--------------------- Figure 4.1: Schematics of the Tools used for the Trajectory Mission and Navigation Designs 4.1.2 Periodic Orbits (PO) and Quasi-Periodic Orbits (QPO) Finding periodic orbits (PO) is a key point to understan ding the properties and stability behavior of a dynamical system . For several dynamical systems applications, the mon odromy matrix must be known. The monodromy matrix is a particular fundamental 103 solution of the variational equation along a periodic orbit evaluated after one period of the orbit. From the monodromy matrix, we can obtain the eigenvalues and eigenvectors which can be used to estimate the local geometry of the phase space in the neighborhood of a fixed point. The eigenvalues determine properties of the orbit, whereas the eigenvectors give the directions of the stable and unstable manifolds. If the eigenvalues are inside the unit circle (real part smaller than unity), they identify stable manifolds, while eigenvalues outside of the unit circle (real part larger than unity) identify unstable manifolds. Eigenvectors associated with stable eigenvalues yield stable manifolds which approach the periodic orbits asymptotically as time goes to infinity. Simila ry, eigenvectors associ ated with unstable eigenvalues yield unstable manifolds which approach the orbit asymp totically as time approaches negative infinity. Finally, pure imaginary eigenvalues are associated with periodic and quasi-periodic orbits (central manifo ld). We will explain the stable and unstable manifolds more in detail in the section to fo llow. In the CRTBP, there are six eigenvalues associated with the monodromy matrix that appears in recip rocal complex conj ugate pairs (Lyapunovs Theorem) and two of them have the value of one. Quasi-periodic orbits (QPOs) about the libration points are extremely practical and useful [41] for space missions modeled in the three-body problem to a level of significance for the exploration of the solar system that has only been glimpsed. Many studies have sought QPOs using different techniques. Some of these techniques are the Poincare sec tions method [34] in 2007, the Lindstedt-Poincare method [18] up to order 35 providing very accurate solutions in 2001, a parallel shooting method [28] in 1988, the Lindstedt Poincare method up to third order [60] as an initial guess for the Lissajous orbits in 1980. We analyze and describe some QPOs around the triangular points in the CRTBP model. In the vicinity of the equilibrium points, we can find two types of oscillations. The first type takes place in the plane of motion of the primary and secondary body and is called the Planar Lyapunov Periodic Orbit (PLPO) or in-plane motion as illustrated in Figure 104 � 1 x ,- ' Lyapunov Orbit around L2 Sun-Earth L2 �- 0 -Starting point � _1 'Endpoint ""· -2 -3 -0.65 -07 -0.75 -0.8 � -085 � -0.9 c g -0.95 :.:; -1.1 -1.15 1.0085 1.009 1.0095 1.01 1.0105 1.011 X(not>:lim,AU) (a) Differ entially Correct&:! Trojan Orbit Sun-Earth 0.2 0.4 0.5 0.6 X(not>:lim,AU) (c) 0.7 -4 _, -8 x 1 o-' Lissajous Orbit around L2 Sun-Earth 1.004 1.005 1.006 1.007 1.008 1.009 1.01 1.011 1.012 X(nondim,AU) (b) Differ entially Correct&:! Trojan Orbit Sun-Earth (d) Figure 4.2: Halo, Lissajous and Trojan orbits obtained using a two level different ial correc tor. The beginning of each segment is represented by a red star and the end of each segment by a red circle. Note that the less chaotic the initial guess is the more periodic the Trojan orbit becomes. 4.2(a). The second one is perpendicular to this plane and is called the Vertical Lyapunov Periodic Orbit (VLPO) or out-of -plane motion. We say that when both frequencies are commensurable, then a Halo Orbit (HO) is fo rmed as explained in [25] . Howeve r, when oscillations (planar and vertical) are not commensurable so that non-linear terms are significant , the motion is not periodic and other types of orbits are fo rmed. For small amplitudes, the orbits are called Lissajous Orbits or Quasi- Periodic Orbits ( QPOs) as we see in Figure 4.2(b ). The Lissajous orbit is comprised of 10 revo lutions. Both the Lyapunov orbit and the Lissajous orbits were differen tially corrected using the two level 105 -0.8657 -0.8658 5' -0.8659 � � -0.866 § s -0.8661 ;.. -0.8662 -0.8663 -0.8664 -0 8652 -0.8654 -0.8656 5' -0 8658 � ,,- -0 866 1 -0 8662 � -0.8664 -0.8666 -0.8668 Asymmetric Orbit around L5 Sun-Earth 0.4 996 0.4997 0.4998 0.4999 0.5 0.5001 0.5002 0.5003 0 .5004 0.5005 X(nondi m,AU) (a) Horse-Shoe-Like Orbit around L5 Sun-Earth - O . SSlc_ _ c- 0.4 � ,c-, -- -0 . � 4 , c:- 95 = -- - - o � .5 -- ---c- 0.5 � 0--c 05 = -- - -oc -:. 5�01_J X(nondi m,AU) -0.8652 -0 8654 -0 8656 5' -0.8658 � -0 866 =g -0 8662 � -0.8664 (c) Horse-Shoe-Like Orbit around L5 Sun-Earth 0.4995 0.5 0.5005 X(nondi m,AU) ( e ) -0.8658 -0.8659 � s -0.866 'i3 § � -08661 -0.8655 -0.8656 -0.8657 � -0.8658 � -0.8659 § � -0.866 -08661 -08662 -08663 -08657 -08658 -0.8659 � s -0866 'i3 § � -08661 Asymmetric Orbit around L5 Sun-Earth 0.4996 0.4997 0.4998 0.4999 0.5 0.5001 0.5002 0.5003 0.5004 X(nondim,AU) (b) Asymmetric Orbit around L5 Sun-Earth 0.5 0.5002 X(nondim,AU) (d) 0.5004 Asymmetric Orbit around L5 Sun-Earth Starting point 0.5006 0.4996 0.4997 0.4998 0.4999 0.5 0.5001 0. 5002 0.5003 0.5004 X(nondim,AU) (f) Figure 4.3: Diff erent ially corrected asymmetric orbits around Ls in the Sun-Earth system. Every loop or bounce is one Earth year long. Figures (a, c and e) show some horseshoe like orbits with loops that do not enclose the £5 point. Figures (b, d and f) display large loops (of the order of half a lunar distance) which enclose the Ls point. 106 method. This method that corrects for both position and velocity until a certain tolerance for position and velocity is reached is explained in more detail in [28]. Given an orbit around L2, we can break it into several segments or legs of different sizes. The red star indicates the beginning and the red circle is the end point of each segment. In the case of the Lyapunov orbit, a perturba tion of 0.0001 AU and 95 segments was assumed. In Figures 4.2(c) and 4.2(d), we display an exam ple of a Trojan orbit with an am plitude of about 0.52 AU differen tially corrected using the same two-level different ial corrector method. The initial guess has a perturbation of 0.01 AU and 0.05 AU to the solution obtained in the CRTBP in Figures 4.2( c) and 4.2( d), respectively. Each orbit case has 16 segments. In the first case, the differen tially-corrected Trojan orbit is almost periodic and in the second case the Trojan orbit starts to show indications of quasi periodicity. Unlike halo orbits, the initial guess assumed for the Trojan orbits can be more chaotic and still obtain a quasi- periodic orbit around the triangular point. We further analyzed smaller QPOs in the Sun-Earth system. The next exam ple is an orbit of 0.00120 AU (about half the Lunar distance). For smaller Trojan orbits, it is convenient to use a larger number of segments. In Figure 4.3(a), 4.3(d) and 4.3(e), we used 270 segments and in Figures 4.3(b), 4.3(c) and 4.3(f), we segmented the Trojan orbit into 135 legs. For all the cases, a perturbation of 0.01 AU was used. The first three plots of Figure 4.3 were obtained by integrating over five years and the last three orbits were obtained by integrating over the time period of ten years. The left column of Figure 4.3 shows examples of horse-shoe-like orbits centered at L5 in the Sun-Earth system whereas the right column displays other types of asymmetric orbits. The top of these asymmetric orbits shows a direct motion towards the Earth as explained in the stability section when using the CRTBP. The middle one starts spiraling inward and then outwa rd. The bottom one displays a QPO where the start and end points are 205 km apart, which is very close to being periodic. 107 4. 1.3 Invariant Manifolds "No matter where it goes, a particle that starts on that surface will remain on that surface forever unless you give it a knock" (Martin W. Lo). In other words, the stable and unstable manifolds asymptotically approach and depart from the fundamental solution [26]. Here we will introduce the stable, unstable and center manifolds. The local stable manifold associated with an equilibrium point is the set of all initial conditions, in the vicinity of this point such that the flow approaches asymptotically this point as time goes to infinity. The local unstable manifold associated with an equilibrium point is the set of all initial conditions in the neighborhood of this point such that the flow approaches the equilibrium point asymptotically as the time goes to negative infinity. Here flow is the solution to the different ial equation of motion and it tells us how far from the initial point we are. This is the real flow, which gives us the actual trajectory. Our goal will be to mimic this trajectory. For simplicity, we will use the state transition matrix that is obtained when we linearize the differential equation of the dynamical system: x = f(x) ( 4.1) where f : 0 <;;; IR. n ----+ IR. n . The state transition matrix is the linearized flow, '-P / '-P : JR. x 0 ----+ 0, generated by the different ial equation. Assume that xo is a hyperbolic equilibrium point (does not have a center manifold) of eq. 4.1. The charac terization of the stable and unstable invariant subspaces depends only on local behavior solutions near the equilibrium point. Thus, we define the stable manifold and unstable manifold of xo as the sets: W8(xo) := {x E 0/ lim t_p(t, x) = xo} t--+co ( 4.2) (4.3) 108 Besides the approaching (stable) and departing (unstable) manifolds, there is another manifold called a center manifold that is associated with the fo rmation of both periodic and quasi- periodic orbits around the equilibrium point. We cannot characterize the cen ter manifold because the invariant center subspace depends on the behavior of solutions possibly far from the equilibrium point. Thus, ( 4.4) which means that the center manifold is connected to both stable and unstable manifolds. Let's assume now that Y is a neighborhood of xo, then the stable manifold and unstable manifold of xo can be described as: ws,loc(xo) := {x E y I lim <p(t, x) = xo } t--700 (4.5) Wu,loc (xo) := {xEY / lim <p(t, x) =xo} t--7-oo (4.6) where ws,loc(xo) <;;;;; ws(xo) and wu,loc(xo) <;;;;; wu(xo) are nonempty relative invariant sets because they contain the equilibrium point. In order to obtain a complete picture of the manifold subspaces, we need to compute the global stable and unstable manifolds associated with the equilibrium point. To do this computation, we start with an estimate of the initial conditions that depends on a perturba tion parameter, d, as displayed in Figure 4.4. This parameter requires special atten tion. This perturbation (offset ) has to be small enough to maintain the linear accuracy of the state. In other words, if d is too small, a spacecraft will take a long time to leave the periodic orbit and, if d is too large, it will leave the periodic orbit without preserving the manifold structure. Proper values of d [59] are 30 km - 70 km for the Earth-Moon system and 150 km-200 km in the Sun-Earth system. Choosing different values will affect the trajectory selected along the approaching manifold. It is usually convenient to normalize the eigenvectors relative to the components of the position. This ensures that the perturbation d along the eigenvector will be unif orm. 109 ' Periodic Orbit "d" is the offset Manifold Shadow ' 1>. \ \ I j t j ; r direction Unstable direction Figure 4.4: Schematics of the manifold theory. The stable manifold is in green and the unstable one in red. The stable manifold approaches the equilibrium point while the unstable manifold departs from it. d is the offset distance from a point on the orbit around the equilibrium point so in some sense these manifolds are the shadow of the real trajectories . There are several ways [59] to compute the manifolds. First) we can calculate the mon odromy matrix for a chosen fixed point and obtain the corresponding eigenvectors. By choosing a different fixed point) we can compute a different monodromy that would give us different eigenvectors and so on. A quicker and more accurate way is to use the state transiti on matrix to shift the eigenvector along the orbit. We will use the second method. Once we have determined the magnitude of d and normalized the eigenvectors) we can integrate the non-linear equations of motion from the perturbed initial conditions backward or forward in time to generate the stable or unstable manif olds) respectively. 1. Find a 'Ito jan orbit using a trial and error process or via a differen tial corrector. 110 2. Find the monodromy matrix, M, by solving the state transition matrix ¢(T, To). 3. Find the eigenvalues and eigenvectors of the monodromy matrix. We will denote v s the eigenvector of the stable manifold and vu the eigenvector of the unstable manifold which can be computed from ¢(XL, t 0 ). Then, we can compute the final monodromy matrix, M J , that is, ¢(XL, T). Finally, we compute the stable and unstable manifolds as fo llows: Wi = M 1 Vi i = 1 2 ' ' 4. Approximate the initial conditions of the manifolds: Xa = XoL + d · W s X0 = XoL + d · Wu (4.7) (4.8) where X0 and X0 are the points on the stable and unstable manifolds of the Xo . In eq. 4.8, d is a constant which is suggested to be small enough to keep linerarity (linear approxi mation). The Barcelona group suggested using an offset of d = 200 km. This value works very well when generating manifolds that depart from or arrive to the halo orbits around the unstable collinear points. This offset does not yield any realistic manifolds around the stable triangular points due in part with the stability properties associated with the Trojan orbits around Ls. Therefore, we used different offsets yielding the manifolds depicted in Figure 4.5. - - 5. Integrate the above initial conditions X0 or X0 forward or backward in time under the nonlinear equations of motion to plot the stable or unstable manifolds. If we were able to use one of these trajectories that com prise the manifold, it would take 10 years for the spacecraft to reach this specific Trojan orbit (0.52 AU in ampli tude) when using such a large perturba tion. Using smaller perturbations would make 111 Stable Manifolds (10 years) from Earth, d of f se t =3E6 k m -0.5 0 0.5 X(not>:lim,AU) (a) 1.5 Stable Manifolds (1 0 years) from Earth, d offse, =384400 k m (c) Stable Manifold s (10 years) from Earth, d offse t 1E6 k m -0.5 0 0.5 X(nondim,AU) (b) Stable Manifolds (10 years) from Earth, d offset =1E4 k m (d) Figure 4.5: Stable Manifolds £5. These trajectories are integrated backwards in time from the Trojan orbit. Thus, the trajectory forward in time is the trajectory that the spacecraft would fo llow in reality. The problem is that these trajectories are too long and not suitable for transfers from Earth. the spacecraft reach the £5 vicinity in much longer time of flights which are unrealistic compared with the mission requiremen ts. From Table 4.1, we observe that for large offsets the structure of the manifold is not very well preserved. As we make these perturba tions smaller, the structure of the manifold is better preserved as shown by the Jacobi constant . In our simulation, each manifold comprises 10 orbits. Offsets of about 200 km yields a good manifold structure but it would take about 10000 years to reach it. 112 Table 4.1: Jacobi Constant of several orbits for Stable Manifold. Orbit dojf s et = 3E6km dojf s et = 1E6km dojf s et = 384400km dojf s et = 1E4km 1 2.96 982044 2. 97877902 2.98126 884 2.98272154 2 2.97045977 2.97 897434 2.98134177 2. 98272341 3 2.97139811 2.97926214 2. 98144939 2.98272616 4 2.972 32491 2.97954768 2.9815 5633 2. 98272890 5 2.97323013 2. 97982774 2.981661 39 2.98273159 6 2.97410677 2. 98010005 2.98176368 2. 98273421 7 2.97495021 2. 98036301 2.98186259 2. 98273675 8 2.97575754 2.98061557 2.98195771 2.98273919 9 2.97652717 2. 98085709 2. 98204876 2.98274153 10 2.977 25838 2.9 8108720 2.98213560 2. 98274377 Our invest igation of some of these manifolds suggest that we cannot use these invariant manifolds or ultra-low-energy trajectories for a space mission. Therefore, we concluded from this analysis that unlike halo orbits, the invariant manifolds of Earth Trojan orbits are unsuitable for transfers from the Eart h. The Trojan orbit has a period of one year. The monodromy matrix over one year is characterized by its eigenvalues and eigenvectors. The eigenvalues are classified as stable, unstable and center. The stable eigenvalues are sn (1) = -187.767611 4623 and sn(2) = -0.0053257321; the unstable eigenvalues are un(1) = 3.1239463988 and un(2) = 0.3201079251; and the center eigenvalues are cn(1) = 0.9999999547 + 0.0003009356i and cn(2) = 0.9999999547-0.00030 09356i. Note that sn(1) · sn(2) = 1; un(1) · un(2) = 1 and that the center eigenvalues are complex of modulus one and come in pairs. The stable eigenvalues have a negative real part and the unstable eigenvalues have a positive real part. The real part of both center eigenvalues is basically unity. The correspondent eigenvector with eigenvalue equal to one is the tangent vector to the Trojan orbit. Note also that un(1) is the dominant unstable eigenvalue (un(1) > 1) and so its eigenvector gives the most expanding direction of the variational flow whereas un(2) < 1 is the smallest unstable eigenvalue. This initial eigenvector can be propagated in time using 113 the flow. This eigenvector with the vector tangent to the Trojan orbit spans a plane which is tangent to the local unstable manifold wu,loc. Similarly, we can obtain the stable manifold. 4. 1.4 Lie Series Expansions We apply the Lie Series technique to the CRTBP in order to obtain the motion of the third infinitesimal body in the rota ting fr ame. We will use a recurrence fo rmula [14] which will enable us to obtain high order terms very rapidly in M athematica © . With this expa nsion, we intend to obtain analytical expansions to obtain periodic orbits around the L5 point. We would also like to test how good this method is if compared with the integrated solution obtained with M athematica © . Below we provide some terms of the solution of the Lie Series expansion up to 10 t h _order. As an exam ple, we analyze one Trojan orbit (0.52 AU) and obtain its Lie Series expan sion. Due to the local behavior of the Lie Series, we break the Trojan orbit into segments. For each segment, we perform the Lie Series expansion and obtain an analytic algebraic expression. This expression is then compared with the integrated solution generated also with M athematica © . In Figure 4.6, we show the trajectories of one of the segments of two Trojan orbits (0.52 AU and 100 km) and the correspondent errors when comparing the Lie Series solution up to 10 t h - order with the numerically integrated solution. Both the Lie Series expansion and the numerically integrated trajectories were evaluated at �t = 60.8737 days. This is sufficient to show that the Lie Series solution fits the integrated solution obtained numerically for short periods of times yielding errors of about 6 km and 2 km in the XLie and YLie , respectively. As we observe, the Lie Series expansion behaves well locally. Larger �t values ( eg. �t = Q 1.31 days) provide a less accurate solution with about 300 km and 105 km errors with respect to the integrated solution. The errors in the segment of the second Trojan orbit of 100 km were about 4 km and less than 1 km in the XLie and YLie , respectively for �t = 60.8737 days. When increasing the time, we notice that the errors were almost the same with values of about 4 km and 1.5 km, respectively. 114 Absolute Error in X between Integrated and Lie Series Trajectories Absolute Error in Y between Integrated and Lie Series Trajectories Abs Errorx[t] Abs Error,[!] 4. X 10-8 3. X 10-S 1. X 1W8 0.2 0.4 0.6 0.8 Absolute Error in X between Integrated and Lie Series Trajectories Abs Errorx[t] 3. X lQ-8 25 X 10-8 2. X 10-8 15 X lQ-8 0.2 0.4 0.6 0.8 1.0 1. X lQ-8 0.2 0.4 0.6 0.8 Absolute Error in Y between Integrated and Lie Series Trajectories Abs Error,[!] 4.x1 o - • 0.2 0.4 0.6 0.8 1.0 Figure 4.6: Error between the integrated solution obtained in Mathematica © and the Lie series solution. Top left and right figures show the errors of x and y, respectively, for a large Trojan orbit with amplitude of about 0.53 AU. Bottom left and right figures show the errors of x and y, respectively, for a small Trojan orbit with amplitude of about 100 km. These errors are lower for smaller orbits. (4.9) 11 6 (4.10) 4. 1.5 Poincare Sections Jules-Henri Poincare (1854-1912) was studying the Three-Body Problem in the 1890s when he noticed the periodicity of some orbits. We will study Poincare sections to analyze the stability of the computed periodic solutions for different models. Poincare invented a general technique based on reducing a continuous time system to a discrete time system. This method can be used to classify orbits by plotting position and velocity in phase space as described in [70] and [69], for instance. To find a Poincare section or surface of section (SOS), we first need to define a crossing plane. We need to ensure [34] that the velocity vector is as close to perpendicular to the transversal plane as possible. In other words, Poincare sections are the set of all points of successive crossing with the hyperplane or surface of section in JR n - 1 , perpendicular to the flow. A trajectory that intersects the SOS is integrated until it crosses the SOS again. For the PCRTBP in JR 4 , the SOS is specified by fixing one of the coordinates (in our case y=O) in order to generate a surface in JR3 . 116 Initially, we set up a grid (NI, N J ), where N1 and NJ are the number of grid points of the phase space we will be integrating ov er. The grid, which is specified by the user, is represented with lower and upper bounds where XL and xu are the lower and upper bounds of the first coordinate of phase space and where X L and xu are the lower and upper bounds of the second coordinate of phase space. The variable XL corresponds to the left end-point of the interval of points that will be integra ted. Remember that the integration process should be generated away from the singularities - JJ and 1 - IJ · The variable xu representing the right end-point gives us an idea how far to the right of XL we can integra te. Then, we can determine the step size between successive initial conditions for both coordinates of phase space, specified by (xL-xu)/(NI -1) and After we have set the initial conditions, we integrate them forward in time using an eighth-seventh order Runge- K utta integra tor. When integrat ing the spacecraft trajecto- ries, we will obtain the intersection of each trajectory with the plane that we have defined and neglect the first intersection to allow the dynamics to remove the effect of the grid. (4.11) Poincare surfaces of sections (Figures 4.8( a) and 4.8(b)) are generated in the phase space [ 68] to com pare the regions of periodic, quasi- periodic, and stochast ic motion around the stability of the Sun-Earth system. The motion is restricted to be on a set or surface of sec tion: (4.12) which defines the hyperplane or two dimensional surface y = 0 when the motion is restricted to only those orbits that cross this plane with y. In eq. 4.11, y is positive 117 04 0.3 0.2 0.1 .;; 0 -0.1 -0.2 -0.3 -0.4 Trajectory Overview of Poincare Section 08 0 6 -0 6 -08 -1 L - __ J- __ _L __ �-- � �� ----�--�--� -1 -0 8 -0 6 -04 -02 0 02 04 06 08 X(nondim,AU) (a) Trajectory Overview Poincare points on Surface of Section Poincare section L-�--�--�--�--�--�--�---- os oss o_s o_ss o 7 o 75 o_a X (b) Poincare Points -1 x 10-• Initial Conditions of Poincare Section I 3:2 Orbit/ �Initial 1\oints / Surface of Sectim \ 0 5567 0 5567 0 5568 0 5568 0 5569 0.5569 0_557 X(nondim,AU) (c ) Trajectories integrate d from initial points Figure 4.7: a: Set of trajectories integrated in the interior region of the Eart h's path around the Sun from the initial points. b: Initial Poincare sec tion. c: Initial trajec tories integrated from initial points. The thick black line is the chosen plane where the trajectories intersect . because the intersection is trav ersed. We will use this surface of section to represent the stable motion in the phase plot. On this surface, I;, the motion can only take place on (4.13) 118 Poincare Section L5 orbit Sun-Earth, C=2.98275971091 1038 0.4 0.3 0.2 0.1 :< > 0 -0.1 -0.2 -0.3 -0.4 0.5 0.6 0.7 0.8 0.9 X (a) Poincare Section L5 orbit Sun-Earth, C=2.98275971091 1038 0.4 0.5 0.6 0.7 0.8 0.9 1. 1 X (b) Figure 4.8: Surface of Sections (positive side) in the Sun-Earth system. a: There are 100130 intersections. b: There are 400628 intersections. 119 Surf aces of Section L5 Earth- Sun system 0.51 0.515 0.52 0.525 0.53 0.535 X (a) Smfaces of Section L5 Earth-Sun system (b) Figure 4.9: Top: Magnified surface of section in the positive region from Figure 4.8(a). Bottom: Magnified surface of section in the negative region. Sea of islands indicating the periodicity of orbits. Empty regions indicate forbidden regions or regions where the spacecra ft never passes or crosses the surface of section. such that Figure 4. 7(a) shows an example of interior intregrated trajectories in the CRTBP from the initial points labeled. The Poincare section intersecting the surface of 120 section takes place on the positive side of the x-axis as illustrated in Figure 4. 7(b ). For these interior orbits, the velocity of the spacecraft is larger than the velocity of the Earth around the Sun since the period is shorter. The spacecraft velocity is even faster when approaching periapse. These orbits are on a 3:2 resonance with the Earth. That is, the Earth goes around the Sun twice in the same time that the spacecraft goes around the Sun three times. Similarly, we can obtain a set of integrated trajectories on the nega tive side of the x-axi s. The corresponding trajectories fall outside the path of the Earth around the Sun. For these trajectories, the velocity of the spacecraft is lower than the velocity of the Earth around the Sun since the period of the spacecraft is larger. But the velocity of the spacecraft is larger when it approaches periapse. This lower rate of the spacecraft translates into a reverse motion in the rotating fram e. The technique explained above can be used to generate the Poincare sections in order to search for islands that correspond to different resonances . Figure 4.8( a) shows a section of the Poincare surface of section mapping fo und for the Ear th-Sun system and different dimensionless velocities representing each of these surfa ces. In Fig 4.8(a), there appears to be a sea of various islands. We can see this sea of islands in a chain because the trajectory of the closed orbit associated with this chain cuts the SOS before closing on itself. At the center of each island sits the point in which the phase-space trajectory of a closed orbit strikes the SOS. The central closed orbit of a chain of islands is always stable. This stable closed orbit gives rise to a chain of 10 islands as seen in Figure 4. 10. Each of these islands contain a set of invariant curves surrounding one of the fixed stable orbits. Figure 4.10 also shows a magnification for some of the concentric islands in Figure 4.9 (a) and their corresponding dimensionless velocities. Figure 4.9 (b) shows another section of the Poincare surface of section mapping fo und for the Eart h-Sun system and different dimensionless velocities representing each of these surfac es. A magnified portion of Figure 4.9(b) is better shown in Figure 4.11. Figure 4.11 shows invariant curves that form resonance islands around the fixed points of stable periodic motion. This motion is described as a regular motion. Around these islands and 121 Surf aces of Section L5 Earth- Sun system . . . . . . . . :. . . . :::- � t;;. ;: .3:jt : ·· . . . • • • . •• .. · # · .. : # ' -: .,.·, •i · :;. :-: · � · .-ot ' · . • 0.08 • • • • • • .. . . . . . . �: : : · · •• , • • • • ·'• , · • • • • 0 . 1 . 69396� ' i J .,' . . .. . · . .. • • · · �. I 0.06 • • � • ... .. .'2-P • ; .... � · :/.; 0.131291 . 0.04 . : · ... . . :..it� .. ,. .. . : f . } ' · · · � \ · � ' � . 0.02 . - :� �0.1 63946 0 �� 0.133758 . · . · ·: • t • ll; \f ... ... . . .· . · . . . .. , . . . . -0.02 \ • ' • • , . : . -0.04 • • -0.06 -0.08 0.512 0.513 · · . ! · · · .. · . 0.514 0.515 0.516 X Surf aces of Section L5 Earth- Sun system 0.517 0.529 0.53 0.531 0.532 0.533 0.534 0.535 0.536 X Figure 4. 10: Close-up of surface of section shown in Figure 4.8(a). The spacecraft has low velocities, between 4 km/s and 7.5 km/s. between regions of finite area, we see chaotic motion around unstable periodic orbits. Notice also how the regions of the regular motion and chaotic motion are mixed together with a third type of motion or unbounded escaping motion as shown in Figure 4.8(a). Here, we see that the hyperbolic points appear to be joined by a separatrix that divides the stable from the unstable motion. 122 � Surfaces of Section L5 Earth-Sun system -197 -1 965 -196 -1 .955 -195 -1945 -1 94 -1 935 -1.93 -1 925 0 04 0.03 0 02 0 01 • 0 -0 01 · . -0 02 . . -0.03 -0 04 -2.08 X Surfaces of Section L5 Earth-Sun system -· '· ' · . . . . ::, . · ' . 1.685465. . 1.�6 \ 87 ' 6 . 1.695�� t : . \/:::. . . . . /) . . . : ;.: � .706880 . . \ .···, : . . . : . .... j 00 ' • '• 4'1 � I t ' . :. Y: .. ·; :. � ·: . / f . . . ·: . 1 ' .. 1 . . . . ; ; . 1 r • '1 .. . • • , r ; • ; ; • l i . t, ... . .: \�; ·, . � : .::· -2.07 \ I •''1 •: · : � � . ' � ! \j . . ' . . '\ :· . }. i· . '·\ \: � •• r • \ \ . �- -2.06 . · .: . ·· ·.J . •:;·*' · .• -2.05 X -2.04 -2.03 -2.02 Figure 4.11: Close-up of surface of section shown in Figure 4.9(b). The spacecraft has large velocities, around 4 7 km /s. 4.2 Navigation Design Tools Trajectory design for current m1ss1ons are very challenging to perform because of the increasing number of mission constrain ts. Nowadays, different softwares, such as the the NASA Goddard software or the JPL LTo ol software, are used to design real trajectories. In this research, an L5- Tool was designed to generate integrated trajectories using the 123 JPL DE421 ephemeris which is part of the software PCODP built by Mr. James K. Miller. This tool offers a decent insight for understanding the natural dynamics of the multi-body problem and can be used to make real trajectories for future foreseeable missions to the triangular points. 4. 2.1 Orbit Perturbation for Navigation Analysis In this section, we explore the effect of different perturbations in phase space for the Trojan Orbits and for the transfer trajectories. For each test orbit, we introduced an initial perturbation in position and velocity. 50 40 30 � :;; -10 -20 -30 -40 -50 15 10 \ 1 km and 0.1 nun/ s e rr ors 6 • Time( days) 10 12 (a) Transfer Or bit of about 343 days 11 V for Navigation Applications of Trojan Orbit 0.1 km and0.01 nun/s errors 5 6 7 Time( days) \ ( c ) One year Trojan Or bit 10 11 14 40 -40 1.5 11 V for Navigation Applications of Transfer Orbit 10 km and 1 nun/ s e rr or s \ ! ! 0.1 km and 0.01 nun/s errors 1 km <Kid 0 .1 nun/ s e rr ors 6 Time( days) 10 (b) Transfer Or bit of about 372 days 11 V for Navigation Applications of Tro jan Orbit 1 km and 0.1 nun/ s e rr or s � 0.1 kmand0.01 nun/s errors� 6 Time( days) (d) One year Trojan Or bit 10 12 12 Figure 4.12: D. V for Navigation Applica tions for different flight time durations of two transfer trajectories. Each trajectory was initially perturbed either in position or velocity, or in both position and velocity. 124 The motivation behind this is that during the time of injection, the booster produces an uncertainty in position of about 2588 km [64] and the injection accuracy must be better than 16 km in position and 0.152 m/s in velocity. We found that for these two particular exam ples, the .6.. V penalty can be as much as tens of meters per second for the transfer orbit during the first two weeks after injection (see Figures 4.12(a) and 4.1 2(b)) and of the order of a few meters per second for the Trojan orbit as illustrated in Figures 4. 12(c) and 4. 12(d). Then, we integrated forward and obtained the difference in position at the end of the time. Primarily, we are interested in the trajectory correction maneuver that the spacecraft would have to perform to compensate for this difference in position with respect to the nominal trajectory. However, this trajectory correction maneuver may be very large if it is not taken care of within the first few days after launch, particularly during the first 10 days. Similarly, we will mainly fo cus on the maneuvers needed a few days after the spacecraft has inserted into the libration orbit around Ls. We ran several orbit cases for different perturbations given the fact that similar perturbations can have varying effects along the trajectory of the spacecraft. These perturbations will provide us with an estimate of the size of the maneuver .6.. V (in m/s) required so we can reduce the na vigation uncertainties especially when the effects of the perturba tions on the spacecraft are large. Initially, we will perform three fictitious perturbations in position and velocity: large perturba tions (10 km and 1mm/s); medium perturbations (1 km, 0.1 mm/s) and small perturba tions (0.1 km and 0.01 mm/s) as we illustrate in Figures 4. 12(a) and 4.12(c) 4.3 Integrated Trajectory Process The orbit selection is a complex process that involves trade offs between different pa rameters, such as the time of flight, total mission cost, environment, viewing geometry and payload perf ormance. One of the most common trade offs consists of the velocity requirements to achieve the orbit and the TOF. 125 At injection, we will interprete the orbit-related mission requirements in terms of orbit parameters. The two most important parameters are the altitude and inclination. The spacecraft will start at a 200-km altitude circular parking orbit with an inclination of 28.5 degrees. We used the CRTBP in the non-dimensional rotating frame to obtain a first initial guess of the trajectory from Earth to an orbit around £5 as described in an earlier paper [43]. In this frame, the spacecraft is moving without any real date assigned. Howeve r, we would like to express the position and velocity of the spacecraft in the Earth Mean Equator J2000 coordinate frame where we can freely input a particular date. From now on, the state vector of the spacecraft is expressed in the J2000 coordinate frame. Having the state vector in this frame allows the user to input it into the JPL flight software to generate real time trajectories. For the targeting trajectory process, we used the corresponding initial state vector of the trajectory of the spacecraft at two days after launch. At this time, the spacecraft is settling into a more stable dynamics [39] after leaving the influence of the Eart h's gravi tational field. Then, we shepherd the trajectory from To + 2 days, where To is the initial time of epoch, back to the exact conditions at launch using hyperbolic osculating elements in an iterative process. This process converges rapidly providing all the informa tion we need by obtaining both sets of classical and hyperbolic elements. Next, we present the optimal targeting scheme described in [38]. 4.3.1 Targeting Strategy of Departure Trajectory The methodology that we implem ented in this research is applied to target both the departing trajectory near the Earth and the arrival trajectory at the Trojan orbit. At departure, the orientation of the hyperbolic departure plane (see Figure 4. 13) must contain the center of mass of the Earth and the relative velocity vector V 00. Consider an imaginary line parallel to the V 00 that passes through the center of the Earth. Then, when rotating the hyperbola about this line, we obtain an envel ope of possible departure 126 trajectories. All these trajectories have one point in common from which all the trajec tories depart. That is, by making a small change in the position of the V 00 vector, we can generate a different trajectory that will target the desired Trojan orbit. The 200-km altitude and inclination of 28.5° corresponds to the parking orbit that is on the locus of all the parking orbits that have the same V 00 vector. Targeting these two constraints can be achieved by refining the orientation and periapsis altitude which defines the direction of the V 00 vector. Departure B-plane at Earth . I Locus of 1 1 H yperbolic ' � -= = ='= = = I � - A - B-Piane • -' S 1 V I j,' I QQ Angle v, I I I I I I I Hyperbola 1 200 km 1 I Envelope 1 . � /I ! T0 + 2 days I I I \ \ . • � B-plane Imaginary line through center of Earth V ro1 (I njectio n ) I ! Asymptote I Figure 4.13: Departure B-plane at Eruth shows an envelope of possible departure trajec tories from a 200-km circular Eru·th parking orbit. To avoid the gravitational influence of the Earth, we selected the input state vector (position and velocity) of the spacecraft in the J2000 frame at Tepoch = To + 2 days from injection and targeted back to the desired conditions defined before, where To is the initial time of epoch. This state vector conesponds to a certain time of epoch, Tepoch· Then, we convert this state vector to classical elements to obtain the set {a, e, T P , i, 0, w} and to hyperbolic elements to obtain the set {B,B,Tp, V00 ,a00,0 00}. Usually, convergence to 127 the desired inclination occurs after a few iterations. If the desired conditions at launch are met, then we stop the iteration process. Otherwise, we use f) as a free parameter to update the injection state vector until the correct orbit inclination, corresponding to the latitude of launch site, is achieved. 4.3.2 Targeting Strategy for Transfer Trajectory This targeting strategy contains the option of targeting either the position or velocity at £5. We used both options to target different cases that we will present in Chapter 5. The first option is to target the position on the insertion orbit. In some scenari os, this option may be a harder (more lengthy) task than targe ting velocity only (but it depends on the analyzed case) since we are targeting in all three directions at the parking orbit. We can compute the velocity correction required to correct the position at the insertion orbit point: (4.14) where � is the change in the position state vector at insertion with respect to the change in the velocity state vector at injection, obtained by numerical integra tion of the var iational equations, and where ft a rg et ( t5) is the initial guess of the position obtained from the CRTBP. Fixing the position of the spacecraft is like having a cord between two fixed sides (Earth parking orbit and Trojan orbit) and letting the cord oscillate freely while keeping one extreme of the cord at a certain angle. By fixing the position, the trajectory of the spacecraft bows up or down, depending on the imparted velocity at injec tion. The second option is targeting the velocity at £5. We use the upper right partition (3 x 3) of the state transition matrix to update the velocity at launch. This matrix can be inverted to obtain the changes of the velocity with respect to the position. The velocity vector at injection is then updated and so is the state vector in the J2 000 frame. We iteratively continue the process until the inclination and altitude at injection converge 128 Continue iteration Process Gel Classical and Hyperbolic Elcmcllls: E, = {a,, e,:z;,,,, i,, Q,, w), E, = e,(t,,X,(t,)) H, = {B, , B,, T,, ,, , Voc ,, i>oc, a00J, H, = h,(t,,X,(t,)) .---- -- <frl �lsroP lnclination,i - 28 S __j V � � 1 Hyper bolic B�Planc Angle: Impact Parameter: 80 = B, - (i, - ¢,) B 0 = f(V.,rr,,u) I Get New Classical and Hyperbolic Elements: E(t0) = {t0p,,e,, T,, ,, ,i ,, Q,,w,,t) H(t0) = {t0,B0 , 80, T,, ,, , Vx,; , D00 , i, ax,;,t ) ! Get New Cartesian State Vector: X(t11) = h;;,, {t11, H(t 0)} X(t0) = e�;P {10, £{10) Figure 4. 14: Flow chart of the integrating traj ectory process to the desired values at the parking orbit . Targeting the velocity yields a faster and smoother convergence. Since we are not forcing the position of the spacecraft to be fix ed, the traj ectory is more or less free to librate around Ls with a certain shift with respect to Ls . The orbits around Ls are displaced either to the right or the left of Ls in Figure 1.3. In Figure 4. 14, we present a flow chart that describes the integrating process for integrating and targeting traj ectories . The state vector of the spacecraft at time ti can be represented by Xi (ti)=[fi iii ] in to the position and velocity state vectors of the spacecraft at To+2 days (or To + 10 days) . Let Hi=[Bi,ei ,T p , i,V oo , i,aood , i/5ood , i,ti] the set of hyperbolic elements, where B is the miss 129 distance orB-plane magnitude, () is the B-plane angle, Tp is the time from periapsis, V00 is the magnitude of the V 00 vector along the hyperbolic asymptote, a00 is the right ascension of the hyperbolic trajectory, and 500 is the declination of the hyperbolic trajectory. Let semimajor axis of the hyperbolic path, e is the eccentricity of the hyperbola, Tp is the time from periapsis, 0 is the right ascension, i is the inclination and w is the argument of periapsis. That is, Hi = hd ep[ti , Xi (ti)] and Ei = e[ti, Xi (ti)]. The aiming (target) radius (magnitude of B-vector) [33] is a function of the periapse radius and the hyperbolic excess velocity when departing the planet with a mass parameter !J, that is, E 0 = f(V 00, rp, !J) . Then, we redefine the new epoch time at periapse as To = Tp,i · Thus, we can compute the new hyperbolic angle (), which represents the orientation on the B-plane: (4 .15) where c/JL is the latitude of the launch site. In our simulations, we are launching from Cape Canaveral so that ¢L = 28 .5°. Hence, the new set of hyperbolic elements is H(to) = [to, Eo, eo, Tp,i, Voo , i, aood , i, 5ood , i, ti] at this new epoch time. Thus, we can obtain the new state vector at peripase as X(t o) = h"d e;(to, Ho). We also know that the state vector at £5 is X t 5 = f(to, X o, qd y n), where qd y n are the dynamic parameters and this state is determined by numerical integra tion. Finally, we can compute the velocity correction required to correct the altitude at the parking orbit: - - [ av] -1 Vo i = Vo i_ 1 - aV o 3x /'hoi(t5) - :Vt arg et (t 5)) (4 .16) where � is the change in the velocity state vector at insertion with respect to the change in the velocity state vector at injection, obtained by numerical integra tion of the var iational equations, and where Vt arg et ( t5) is the initial guess of the velocity obtained from the CRTBP. This targeting method converges quite fa st, achieving the conditions at the parking orbit within a few iterations. 13 0 4.3.3 Targeting the Trojan Orbit The final position state vector of the transfer trajectory must be the same as the initial position state vector of the Trojan orbit. At the end time of the transfer trajectory, we perform a maneuver so that the spacecraft inserts into the Trojan orbit. When the spacecraft inserts into the Trojan orbit, we assume we have a fictitious planet with very small mass parameter, J-l, at L5. The use of this imaginary planet allows us to simulate the dynamics of the spacecraft near the equilateral point. The gravity is assumed for the purpose of calculating the target parameters and it does not affect the dynamics of the trajectory. It is not included in the equations of motion. We can generate a set of orbital elements (as for the departure leg) so we can know the time of closest approach where we will perform the insertion maneuver . When including a higher fidelity model like the DE421 ephemerides, these trajectories show different behavior than those obtained with the CRTBP [43] , yielding various trade offs between the TOF and the L)..Vro T A£=.6..Vroi +.6.V LoJ . Moreover, the sizes of these Trojan orbits are also different from the sizes of the solutions generated in the CRTBP. 4.3.4 Hyperbolic Orbital Elements Given the state vector of the spacecraft as Xi (ti)=[ri vi], the hyperbolic set of orbital elements can be obtained as follo ws: 1. Obtain the pole vector, P = r x v where P = J Pd + PJ + PJ . 2. Get magnitude of position and velocity vectors, R = J x 2 + y 2 + z 2 and V vx 2 + iP + i 2 . 3. Find hyperbolic excess velocity, V00 = J v 2 - '*· 4. Compute the semilatus rec tum, p = � 2 • y 2 5. Obtain energy, s = 2 · 6. Get the eccentricity, e = J 1 + P( �s ) . 131 7. Get semimajor axis, a = � · 00 8. Obtain the impact parameter, B = y'rijJ. 9. Compute time of periapsis from epoch time, Tp = t � /¥: ( e sinh F � F ) where sinh F = r e v JI;, coshF = a ;:;, r so that F = ln(sinh F + cosh F). 10. Find the S = sin(77 � rn)r x P � cos771r along the hyperbolic asymptote where 7] l = cos - 1 ( � 1 ), sin7] = r e v if;, COS 7] = P;;i- so that 7] = tan - 1 (sin 7], COS7J). 11. Get the right ascension and declination of the asymptote, a00 = tan - 1 (S y, Sx), £ · - 1 ( S ) Uoo = Sln � . y Sx+Sy 12. GetT= �( Sy,�Sx,o ) andR = (�TySz,SzTx,TySx�TxSy). These expres sions will be proved in sec tion 4.4.1. 13. Compute the hyperbolic true anomaly, () = tan - 1 (sin(), cos()) where cos() = �R · P and sin () = T · P. 14. Assemble the six hyperbolic elements (B, (), T P , V00 , a00, 500) that map one to one into position and velocity. Note that () and 77 are determined in four quadrants. 4.4 Testing Integrated Trajectories with Conics The targeting strategy explained in section 4.3.1 is then tested with a conics approach that we explain in this section. 1. Read Ephemerides 2. Convert the state vector of the Earth from J2000 to orbital elements (p, e, Tp, 0, i, w). 3. Compute the semimajor axis for the Earth, a$ = p/(1 � e 2 ). 4. Get the nominal period for the Earth using Kepler's third law, FEB = 21r jf- . 132 5. Obtain the mean motion of the Ear th, n$ = �· 6. Find the mean motion of the spacecraft , nsc = (n$ · TOF - X SL5 · 6071)180- (/r)/TOF where XSL5 = ±1, +1 if chosen an orbit around L5 and -1 for an orbit around L 4 . The time of flight, TOF, and the size of the Trojan orbit, (, are also desired inputs. 7. Calculate the period of the spacecraft , Psc = 21r /nsc 8. Get the semimajor axis of the spacecraft, asc = ( ��c �) 1/ 3 . 9. Obtain the difference in the semimajor axis, Lla = 2(a$ - asc), between the Earth and spacecraft orbits. 10. Compute the radius of periapsis of the spacecraft , Rp =a $ - Lla · f where f is the fraction of the orbit (offset ) from periapsis and determines the eccentricity of the orbit. This offset parameter is another input that can be chosen as desired. 11. Obtain the radius of apoapsis, Ra = 2asc - Rp. 12. Get the eccentricity of the spacecraft orbit, e = 1 -�. 13. Find the semilatus rectum, p = Rp(1 + e). 14. Calculate the angle from periapsis to the desired crossing node, rJ = cos � 1 ( � � 1 ). 15. Obtain the time of flight from periapsis to the desired crossing node for both Earth and spacecraft . Note that the angle between the line of periapsis to this crossing node is the same for both the Earth and the spacecraft. We add 10 days to this new time of flight from periapsis so the spacecraft is in a more stable regime. Then, we target the state of the spacecraft back to the desired conditions at the parking orbit. 16. Compute the new state vectors of the Earth, x$ and spacecraft Xsc at the corre sponding crossing node for this new flight time. 133 17. Update the new ephemeris time. 18. Get the state vector of the Earth centered inertial frame, Xeq = Xsc - XEf!· Rotate this vector to rotating frame. Finally, use this Earth state vector in the rotating frame to obtain the state vector of the spacecraft in the Earth centered inertial frame. In Figure 4.15, we present a sketch that shows how the trajectory generation process works. The blue dashed line corresponds to the orbit of the Earth around the Sun with a mean motion, nEf!, and the black dashed line to the orbit of the spacecraft around the Sun with a mean motion, nsc· For simplicity, we assume eccentric coplanar orbits such that both the Earth and the satellite orbit a central mass (Sun). Figure 4.15: Sketch depicting the patched conics approach to design trajectories. The mean motions of the Earth and spacecraft are nearly commensurable. There are two points where both orbits intersect, giving us two locations where we can launch the spacecraft . The first point corresponds to the Ascending Crossing Node (ACN) and the second point corresponds to the Descending Crossing Node (DCN). If the spacecraft is ahead of the Earth, then it will move slower because the Earth acting from behind brakes the spacecraft. Instead, if the spacecraft lags the Earth then, the Earth 134 will accelerate the spacecraft . The gravitat ional effect by the Earth on the spacecraft will be stronger when the spacecraft lags the Earth. When this happens, the spacecraft looses some angular momentum and the spacecraft falls closer to the Earth increasing its mean motion. In other words, the relative distance and relative velocity are both smaller. Since P o: a 3 1 2 , we can obtain the perturbation in the orbital period, Ps c, of the spacecraft � = .:i 2 6- a . Since we know the difference in period of the spacecraft with rsc a respect to the period of the Earth around the Sun, we can obtain the limiting semimajor axis, aR. For a semimajor axis values a > aR, resonances do not exist. For example, an offset of 0.05 AU translates into aR = 1.09575AU and the limiting period P R = 1.147. 4.4.1 Hyperbolic Osculating Conic near the Earth We can specify the orientation of the satellite conic [33] by using the fo llowing set of - - vectors: � which is parallel to the apse line of the departure hyperbola, ( is the normal vector to the orbital plane defined as �� · Let's define another vector z = ( x � = R 2 v - (RV ) R 2 can be obtained as follo ws: R -h "' A R 2 . v- (R . V)R R 2 . v z- -��-� - fRI · h - fRffR X V f R. ( R V cos a: ) R V = + -=��--- fRffR x V f fRff V f sin a: R COS O: v = - � · -- + �-- fRI sin a: [V f sin a: R(R · f!) fRffR x V f ( 4.17) (4.18) (4 .19) where 0: = cos - 1 ( � ( 1I - 1)) is the polar angle or true anomaly. If R points in the negative direction normally with respect to the Earth equatorial plane, then we can replace R by - R. Also, multiplying the previous expression by sin a:, we finally obtain an expression for { € = R cos 0: - z sin 0: (4.20) 135 The vector fi can be easily obtained given the fact that fi is perpendicular to { Thus, fi = R sin Ot + z cas Ot (4.21) Fl:om Figure 4.16, we can obtain the three unit vectors that defined the B-plane, S, T and R. S is the vector in the direction of the hyperbolic asymptote. T is the vector (perpendicular to S) parallel to the line of intersection between the B-plane and the J2000 Earth Mean Ecliptic plane being positive in the direction of decreasing right ascension. R completes the set of orthogonal unit vect ors. Also, we will use B defined as the impact parameter which is the vector from the center of the target (Earth) perpendicular to the asymptote of the hyperbola. The physical meaning of the B-plane vector represents the miss distance by the satellite if the Earth was massl ess. e II 'g T Figure 4.16: B-plane targeting from 1+10 days to desired conditions at launch. 136 S = COSete + sin a(( X e) [3 = b(S X () (4.22) (4.23) Knowing that the eccentricity vector, e, is parallel to � ' then, S and R can be expressed in the fo llowing form: (4.24) (4.25) where b is the magnitude of B and can be obtained from lE I = b = YPfaT. Therefore, if we know S, we know T and R. Note that S can be also computed as a funct ion of the declination and right ascension of the asymptote of the hyperbola in the fo rm: s = (cos 5oo cos Ctoo, cos 5oo sin Ctoo, sin 5oo ). We also know that T and S are perpendicular to each other, thus, T x = J 1 - tz = S y and T y = -� = -S x and Tz = 0. By normalizing the T vector, we obtain: (4.26) Finally, we can obtain the R vector as R = T x S, then: (4.27) where R is pointing in the negative direction normally with respect to the Earth equatorial plane, �. 137 4.5 Can Gravity Assist Help? Sun-Earth triangular Lagrange point insertion using a lunar gravity assist has never been attempted [5] . In fact, no trajectory optimization to these specific Lagrange points had been published in astrodynamics conference proceedings until 2007. Benavides et al. [5] designed optimal trajectories to £4 and Ls using a lunar gravity assist that will require smaller Ll V s than those predicted by co-orbital rendezvous. Their approach produced for the best L5 rendezvous a trajectory with a total LlV of 5.057 km/s and a time of flight of 410.9 days. This solution was 8.7% greater than the direct transfer to L5 not using any lunar gravity assist. Thus, a lunar flyby is not helpful for this mission and we will not be able to take advantage of it. 4.6 Trajectory Propagators We use different Trajectory Propagators in this research : i) the Runge- Kutta seventh eighth-order integrator (RK78), ii) RK87. The RK78 is used for the CRTB P and the ERTBP, rather than the convent ional MATLAB® ode45 integra tor. Howeve r, we could also use the MATLAB® ode113 integra tor, which was tested against the RK78 integra tor, providing almost the exact an swer . The RK78 method comes from Classical Fifth-, Sixth-, Seven th-, and Eighth-order Runge- Kutta Formulas with Stepsize Control, Erwin Fehlberg, NASA TR R-287 [35] . A good integrator is determined by how many times the integrator calls the derivative func tion. More refined Trojan orbits can be obtained using an RK87 integrator instead of an RK78. When using the PCODP, we will use RK56 as a trajectory propaga tor. 138 Chapter 5 Results for Mission Design and Navigation Design This last section will deliver the results obtained from this research, such as the trajectory design optimization, navigation analysis, mission trade studies and fe asibility analysis of the mission. In the trajectory design optimization part, we will present plots and tables of the cost performance and stability analysis. For the na vigation analysis, we will also show the corresponding plots and tables for the maneuver design and orbit determination. In the mission parameter trade studies, results for power, communication and propulsion mass performance will be shown. Finally, we will provide a section explaining the feasibility of the mission for each of the orbits found based on the trajectory mission design and navigation analyses. Last, we will provide final products such as papers, software tools and a database of orbits. Our objective in this part of the work will be to determine the spacecraft nominal trajectory while satisfying a set of predefined mission constrain ts. Some of these con straints are the predefined altitude for the parking orbit of 200-km at 28.5° inclination, 139 the number of trajectory correction maneuvers (maneuver frequ ency), the time of flight and fuel usage (thruster performance). For our purpose where we will consider an interplanetary trajectory; we will have to account for launch errors from the 200-km parking orbit around the Earth. This means that maneuvers will be needed to correct for the deviated trajectory. Moreover, because of the uncertainties of the state of the spacecraft , additional midcourse and arrival maneuvers will also be needed. We expect that after a sequence of correction maneuvers the spacecraft will be deployed at the aim point with sufficient accuracy. We will pay special atten tion once the spacecraft has arrived in the vicinity of the triangular point and it is orbiting around this point. We know that these orbits are mildly stable, that is, they drift away from the Earth (retrograde) or towards the Earth (direct) as seen in the stability section. The fact that we are using a linear dynamical model could result in additional trajectory errors so more correction maneuvers may be needed when orbiting around the triangular point. 5.1 Trajectory Mission Design Results This section presents the results for the trajectory mission design that we present in different tables. Each table shows the comparison between the performance of the mission and the corresponding TOF. These results were obtained using the Ephemeris model, DE421. 5.1.1 L5 Orbits in the Sun-Earth System Recap itulating, we used the CRTBP model to obtain our first initial guess for the tra jectory. For this, we chose a point on the Trojan orbit and integrated backwards in time until we target a 200-km Earth parking orbit. In a way, we are forcing the transfer trajectory to have a unique TOF given the fact that no other perturbations affect the dynamics of the spacecraft in this model. When including a higher fidelity model, like the JPL DE421 ephemerides, these trajectories show different behaviors yielding variant 140 trade offs between the TOF and the �V Toi and the �V Lo J. Moreover, the size of these Trojan orbits are also distinct with respect to the solution generated in the CRTBP [38]. Next we examine several trajectories previously analyzed in the CRTBP [39, 43]. Each trajectory is shown in Table 5.1 to Table 5.5 with six columns. The first column gives the time of the year at which the trajectory was launched. We analyzed four different epochs four months apart: January, April, July and October. The second column provides the Transfer Orbit Insertion � V, � V roi , at injection from the Earth parking orbit. The third column shows the Libration Orbit Insertion (LOI) � V, � V Loi . The fourth column gives the total �V, �V ToTAL = �V Toi + �V LO I· The fifth column gives the total time of flight (TOF). The sixth column shows the targeted inclination of the the trajectory at injec tion. The seventh column provides the targeted radius of the trajectory at injec tion. Ta ble 5.1: �V vs. TOF comparison between trajectories launched at different times of the year, case L5 ( 1) Tep och �V Toi �V Loi �V TOTAL TOF me. rp (km/s) (km/s) (km/s) (days) (deg) (km) Jan 4.388 1.050 5.438 366.9 28.22 6578.135978 Apr 4.249 1.050 5.2 99 361.8 28.67 6578. 136035 Jul 4.141 1.12 8 5.2 69 358.9 28.57 6578.135989 Oct 4.265 1.12 5 5.390 363.4 28.41 6578.135987 CRTBP 4.622 0.995 5.617 343.4 na 6578.136 000 Ta ble 5.2: � V vs. TOF comparison between trajectories launched at different times of the year, case L5(2) Tep och �V Toi �V Loi �V TOTAL TOF me. rp (km/s) (km/s) (km/s) (days) (deg) (km) Jan 4.219 0.529 4.748 718.5 28.47 6578.135989 Apr 4.113 0.529 4.642 713.5 28.45 6578.135967 Jul 3.998 0.564 4.562 709.2 28.43 65 78.135993 Oct 4.094 0.570 4.664 713.4 28.40 6578. 136028 CRTBP 4.326 0.522 4.848 701.1 na 6578.136 000 141 Ta ble 5.3: Ll V vs. TOF comparison between trajectories launched at different times of the year, case L5(3) Tep och LlV roi LlV Loi LlV roTAL TOF inc. rp (km/s) (km/s) (km/s) (days) (deg) (km) Jan 3.505 0.621 4. 126 877.6 28.48 6578.136 000 Apr 3.405 0.661 4.066 888.8 28.52 6578.136017 Jul 3.566 0.596 4.162 897.3 28.50 6578.136 000 Oct 3.748 0.544 4.292 897.2 28.50 6578. 136007 CRTBP 3.562 0.641 4. 203 914.2 na 6578.136 000 Ta ble 5.4: Ll V vs. TOF comparison between trajectories launched at different times of the year, case L5 ( 4) Tep och LlV roi LlV Loi LlV roTAL TOF inc. rp (km/s) (km/s) (km/s) (days) (deg) (km) Jan 3.509 0.744 4.253 806.5 28.40 6578.135996 Apr 3.377 0.753 4. 130 805.2 28.49 6578.136 048 Jul 3.434 0.724 4. 158 753.2 28.50 6578.136016 Oct(2) 3.336 0.775 4.111 756.1 28.53 6578.135215 CRTBP 3.514 0.789 4.303 702.0 na 6578.136 000 Ta ble 5.5: Ll V vs. TOF comparison between trajectories launched at different times of the year, case L4 Tep och LlV roi LlV Loi LlV roTAL TOF me. rp (km/s) (km/s) (km/s) (days) (deg) (km) Jan 4.468 0.744 5.212 560.3 28.38 65 78.135966 Apr 4.039 0.830 4.869 559.3 28.60 6578.136 034 Jul 3.875 0.903 4.778 559.2 28.51 6578. 136005 Oct 3.452 0.710 4.162 573.4 28.42 65 78.13597 4 CRTBP 4.462 0.764 5.226 612.9 na 6578.136 000 In Table 5.6, we show the relevant parameters for the departure trajectory for each case that we have examined. Column 1 gives launch month for each trajectory. Column 2 is the C3 which is the main performance parameter that represents the minimum energy 142 Ta ble 5.6: Relevant parameters for departure trajectory Case C3 V oo 8oo O:oo b /3h y p (km 2 /s 2 ) (km/s) (deg ) (deg ) (km) (deg ) L51A Jan 26.97 5.193 -18.46 44.58 15419.0 46.21 Apr 23.60 4.858 -24.89 112.91 16293.2 43.97 Jul 21.03 4.585 -17.96 20.27 17107.8 42.07 Oct 24.00 4.899 -25.6 304.2 16179.7 44.25 L51B Jan 22.91 4.786 -21.99 33.22 16498.9 43.47 Apr 20.36 4.512 -26.60 106.00 17346.1 41.5 3 Jul 17.63 4.198 -13.95 190.54 18461.8 39.22 Oct 19.89 4.460 -19.01 188.02 17518.7 41.16 L51C Jan 6.26 2.503 -28.42 359.82 29671.8 25.00 Apr 4.01 2.003 -4.92 70.19 36754.5 20.29 Jul 7.64 2.763 -4. 11 172.60 27019.4 27.37 Oct 11.80 3.434 -28.49 274.03 22087.6 33.17 L527 Jan 6.34 2.519 -14.06 349.98 29491.0 25.15 Apr 3.38 1.838 -1.11 74.37 39938.9 18.71 Jul 4.67 2.161 -1.51 171.2 6 34154.0 21.80 Oct 2.47 1.572 -26.83 257.22 46527.5 16.09 L417 Jan 28.92 5.379 -22.37 217.73 14983.9 47.40 Apr 18.62 4.315 -26.4 7 302.58 18025.6 40.10 Jul 14.74 3.840 -26.23 30.22 19972.7 36.46 Oct 24.57 4.957 -24.01 117.01 16021.5 44.64 requirement to accomplish the mission. Column 3 is the V00 of the magnitude of the vector difference between the departure velocity of the spacecraft along the hyperbolic trajectory and the orbital velocity of the Earth. Columns 4 and 5 are the declination and right ascension of the departure hyperbolic asymptote. Column 6 is the semiminor axis of the departure hyperbola. Column 7 is the angle of the asymptote of the departure hyperbolic trajectory from the line of apsides. 143 s ..;_ � 0 0 � -1 X 10-4 10 . ... s 4 ..;_ s 2· ] 0 � -2 . - 4 . - 6 . Orbit Case L5(1) . . . . . . . . . . . - · .. · . .. . . _ . .. : 'S UN . . . . . � . ,. . Y(nondim,AU) (a ) Orbit Case L5(1) ... .. . . ···· ·· -1 O L 2 ---- �� -- � 0 � 6 ---- - 0 �8� -- - - L--- � 12 X(nondim,AU) (c ) Orbit Case L5(1) (e) 0 . SUN -0. 2 : . 1. 5 -0 B -1 Orbit Case L5(1) EARTH ! , , , t: I I .......... ...... .. .. '/' . / '�!:... .... _, Earth ru:hit. r (b) Orbit Case L5(1) -0 6 -04 -0 2 Y(nondim,AU) (d) Orbit Case L5(1) (f) Figure 5.1: One-year integrated trajectory to Ls. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. e: XZ zoomed in projection. f: YZ zoomed in projection. The January, April, July and October orbits are in blue, green, magenta and brown, respectively. The orbit in gold (see Figure 5.1(b)) is the solution obtained in the CRTBP. 144 Numerically, we can obtain the osculating classical elements and the hyperbolic ele- ments for each trajectory as fo llows: 1. From the eccentricity and semimajor axis of the departure asymptote, we can obtain the point of injection which represents the location at which the spacecraft reaches the velocity required to enter the hyperbolic departure trajectory rp = a (e- 1). 2. Obtain the circular velocity, Vc = fi!iii of the spacecraft. V r p 3. Determine the departure velocity at injection, V p = V fJ., ( r 2 p + �) . 5. b. VL oi is solved numerically as described in subsection 4.3.3. 6. TOF is obtained numerically as explained in subsection 4.3.3. 7. The hyperbolic excess velocity, VH E can be determined since it is the same as VeX) on the departure hyperbola, VH E = Voo = J v; - ve� C ) where Ve sc = �· 8. Get C3 from the hyperbolic excess velocity, C3 = VJ g 9. Calculate the semiminor axis, b, of the departure hyperbola, b = r p 10. Determine the angle of asymptote, f3hy p = tan - 1 ( b�!E). 11. 600 and o:00 are obtained as explained in subsec tion 4.3.4. In this chapt er, we describe the behavior of each of these integrated trajectories 1 . For each trajectory, we showed four different times of the year: January, April, July and October. The trajectory in blue depicts a January launch, the trajectory in green displays an April launch, the trajectory in magenta represents a launch in July and the brown trajectory corresponds to a launch in October. In Figure 5.1, we illustrate an integrated trajectory from Earth to £5 using the JPL ephemeris model, DE421. The b. V roi at injection is 4.622 km/s (compared with b. V r OJ = 1 EARTH denotes location of Earth at injection in the plots. 145 Orbit Case LS (2) Y(nondim,AU) -1 (a) Orbit Case L5(2) (c) Orbit Case L5(2) (e) Orbit Case L5(2) 0.4 0.6 0 .B X(nondim,AU) (b) Orbit Case L5 (2) (d) Orbit Case L5(2) - 0 .2 (f) 1.2 Figure 5.2: Two-years integrated trajectory to Ls. a: 3D trajectory plot . b: XY projection. c: XZ projection. d: YZ projection. e: XZ zoomed in projection. f: YZ zoomed in projection. The January, April, July and October orbits are in blue, green, magenta and brown, respectively. The orbit in gold (see Figure 5 .2(b)) is the solution obtained in the CRTBP. 146 4.265 km/s obtained from the CR3BP solu tion). The �V LOI at insertion is 1.125 km/s (compared with � VLoJ= 0.995 km/s obtained from the CR3BP solut ion). The total � V of the mission is greater than the one obtained with the CRTBP solution. The time of flight is TOF= 363.4 days (20 days longer than TOF in CR3B P). In the first example (see Table 5.1), we observe that launching in July gives the lowest � V required for the mission and also the shortest time of flight . Even though, this trajectory has the largest insertion maneuver � VLoi = 1.128 which is about 75 m/s greater than for a January or April launch, the main savings ( �100 m/s for April launch to about 250 m/s for January launch) take place at injection when performing a �V Toi = 4.141 km/s at the 200-km altitude parking orbit. If we combine the TOI and LOI maneuvers, we see that the July launch gives the lowest � V r oi ( 481 m /s less than the solution obtained in CRTBP) of the mission, then April launch with less than 373 m/s, fo llowed by a launch in October with less than 357 m/s and finally the January launch with less than 234 m/s, if compared with the solution obtained in the CRTBP. Note that the injection � V is lower when the spacecraft is launched in July which corresponds to when the Earth is at its furthest position with respect to the Sun (aphe lion), that is, when it has the lowest velocity. In Figure 5.1, we illustrate a one-year transfer trajectory from Earth at the 200-km altitude parking orbit and 28.5° inclination to a Trojan orbit of 0.52 km amplitude around £5. The spacecraft orbits £5 for 3 years for each trajectory launched at a different time of epoch. Looking at the XZ and YZ projections of Figure 5.1, we see after the data are converted to dimensional units that the transfer trajectories for missions launched in April and January are off the ecliptic plane by about 1.35 x 10 5 km and 1.2 x 10 5 km, respectively. These excursions off the ecliptic plane are less pronounced for the transfer trajectories launched in October or July with about 2 x 10 4 km and 7500 km, respectively. Although, a July launch requires slightly less � V than an April launch by about 30 m /s with almost the same TOF (see Table 5.1), a mission to £5 launched in April will provide more interesting science results since the space probe will be able to obtain data from the Sun 147 from higher inclinations with respect to the ecliptic. Ideally, the spacecraft could start science as soon as leaving the Earth. Once it arrives at the Trojan orbit, the spacecraft can continue taking data for a few more years, while orbiting L5. In our case, we illustrate only a 2-year Trojan orbit in Figure 5.1 but the time of science of the spacecraft could be extended for a few more years. In the second case (see Table 5. 2), launching in July provides also the lowest b. V of the mission, fo llowed by a launch in April, July and October. Launching in July implies 328 m/s savings in the b. V roi with respect to the solution obtained in the CRTBP, 213 mjs savings in April, 232 m/s savings in October and 107 mjs savings in January. The lowest b. V at injection corresponds to when launching in July for the same reason as in the first case analyzed. For both cases analyzed, launching in January and April yields the same b. VL oi at insertion whereas launching in July and October yields also about the same insertion b.VLoJ. The next example tabulated (see Table 5.3) corresponds to a two-and-one-half-year transfer orbit to a Trojan orbit. In this case, the insertion burn occurs outside the path of the Earth around the Sun. At this point, the spacecraft has an orbital velocity that is in the opposite direction from that of the Earth around the Sun. This insertion maneuver yields the lowest b.V when launching in April (instead of July as in the other two cases ). The total b. V of the April launch mission is 4.066 km js which is 137 mjs lower than the solution we obtained in the CRTBP, 89 m/s lower than launching in October, 77 mjs lower for a January launch, 41 m/s for a launch in July. Figure 5.2, shows a two-years transfer orbit from Earth at the 200-km altitude parking orbit and 28.5° inclination to a Trojan orbit of 0.52 km amplitude around L5. The spacecraft orbits L5 for 2 years for each trajectory launched at a different time of epoch. Our targeting strategy for the altitude at 200 km shows very accurate results with less than 4 em error for the worst scenario. Finally, the target orbit inclination (for the one- year transfer orbit) is also reached with very good results with less than 0. 28° error 148 for January and 0.07° for July from the actual desired value at 28 .5°. For the other two cases, we observed that the error in the target orbit inclination is between 0° and 0.05° with respect to the desired value and the target orbit altitude is also improved from an error of a few centimeters down to an error of a few millimeters. Targeting inclination by varying angle theta 33 32 � LS(la)-July 31 30 \ \ LS(l c)-July 26 LS(lc)-October 25 24 23�--�--��--�--�----�--�----�--� 152 153 154 155 156 157 e(deg) (a) 158 159 160 Figure 5.3: Targeting inclination by varying angle theta on the B-plane. The errors produced in the targeting of the inclination are negligible or zero for the cases studied. A 0.1° error in the inclination translates by equation 4.15 to an error of about 0.05°-0.1° in the angle theta (angle on the B-plane ). Our targeting strategy generates very good results converging to the desired values within a few iterations for most scenarios as seen in Figure 5.3. Some of the departure trajectories (see Figure 5.4) begin moving in the direction of the Earth's motion around the Sun crossing the Earth orbit soon after launch. The spacecraft crosses the Earth orbit doing an inner excursion (inside the Earth orbit) after leaving the parking orbit a few days after launch. Then, it does an outer excursion (outside the Earth orbit) after a few days or weeks. Therefore, when targeting these trajectories, we see that the spacecraft enters into a very sensitive, non-linear regime when targeted from L+2 149 X 10 7 4 2 0 -2 -4 ]' -6 � -8 -10 -12 -14 -16 0 Leading In-Lagging Out Transfer 2 4 6 8 10 12 14 16 18 X(km) X 10 7 (a) Figure 5.4: Leading in-lagging out two years transfer trajectory to £5. The size of the Trojan orbit is about 0.73 AU days backward in time to the exact condition at launch. Targeting these trajectories can be tricky and lengthy because of the complex dynamics when the spacecraft is near the Earth or in the influence of the Moon. In these more challenging circumstances, we can alleviate some of these targeting issues (and therefore, convergence issues) by shepherding the trajectory from L+ lO days (instead of L+2 days) or by launching at a different time of the year. For all the orbits investigated in this work, we observe that the spacecraft motion shows resonant natural frequen cies. Right after departing from the Earth parking orbit at 200-km, the spacecraft undergoes smooth oscillations but they grow in time and become even more accentuated when the spacecraft is close to Earth approximately one year later. The Earth is oscillating, and, therefore, the satellite orbit is oscillating around the Earth-Moon barycenter because of the influence of the Moon with respect to the center of mass. These fluctua tions show a chaotic growth during the rest of the transfer phase before inserting into the Trojan orbit where the oscillations are less pronounced. 15 0 Orbit Case L5(3) � s ] 0 � -1 -2 -I X(nondim.AUl (a) Orbit Case L5 (3) (c) Y(nondim,AU) Orbit Case L5(3) 0.4 O.G 0.8 X(nondim,AU) (b) Orbit Case L5(3) (d) Figure 5.5: Two- and-one-half years transfer trajectory to L5. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. The January, April, July and October orbits are in blue, green, magenta and brown, respectively. The orbit in gold (see Figure 5.5(b)) is the solution obtained in the CRTBP. Figure 5.2 illustrates another example that analyzed how the different times of launch affect the mission perf ormance. As in the first case analyzed, launching a space probe in July gives the lowest total � V for the mission fo llowed by an April launch ( rv80 m/s more expensive). The total � V of the mission is cheaper than the one obtained with the CRTBP solution. This trajectory has a TOF of a little less than 2 years (see Ta ble 5.2 for exact times). The target orbit inclination was targeted with less than 0.1 o difference with respect to the desired value of 28.5°. Similarly, the target orbit altitude was targeted with less than 3 em in position. For this case, the April trajectory reaches excursions of about 2.2 x 10 6 km while the July trajectory reaches excursions of about 6 x 10 4 km off the 151 � 2 s 1 N' -2 X 10 - 4 SUN ¥ • : Orbit Case L5(27) ... · · · · ;-- · ::::·:·., . . • Eartl) o�bit . ' • . 'f • . '··<· . . : - . __ : . : · :.. : J - - 1. 2 ·-. : . . � X(nondim,AU) Y(nondi m,AU) � s · '6 ! -0 5 � � -1 001 § -001 . s N -002 -0 03 . -004 . (a) Orbit Case L5(27) (c) Orbit Case L5(27) ( e ) -0 1 05 -0 8 -0.9 0.2 � l N Orbit Case L5(27) 0.4 0.6 0. 8 X(nondim,AU) (b) Orbit Case L5(27) (d) Orbit Case L5(27) (f) TH Figure 5.6: Transfer trajectories for different time of flights to £5. a: 3D trajectory plot. b: XY projection. c: XZ projection for April, July and October launches. d: YZ projection for April, July and October launches . e: XZ projection for January launch. f: YZ projection for January launch. 152 ecliptic plane. Either trajectory will provide good data collection, but picking one or the other trajectory will depend on the scientific mission requiremen ts. A two-years transfer trajectory yields not only a slightly lower � V roi than a one-year transfer trajectory but also the � VLoi is reduced to half as shown in Table 5.2 at the expense of increasing the TOF one more year. In Figure 5.5, we show a trajectory with a TOF � 2.5 years with different times of epoch: January, April, July and October. In the previous cases analyzed, a July launch seemed to give the lowest �V of the mission. Howeve r, in this case (see Ta ble 5.3), April yields the lowest total � V = 4.066 km/s for the mission fo llowed by a January launch with �V = 4.12 6 km/s, a July launch with �V = 4.162 km/s and a October launch with � V = 4.292 kmjs. These trajectories were targeted to the desired altitude and inclination (see Table 5.3). In this case, we observe that the time of flight is about half a year longer than in the previous case analyzed for comparable � V Loi and about 430 m/s cheaper � Vr oi at injection if the launch occurs in July. For this case, the spacecraft will be about 1.1 x 10 4 km above and 9 x 10 3 km below the ecliptic. By launching the space probe in April, the excursions above and below the ecliptic are of the order of 3.4 x 10 4 km. Similarly, excursions of about 2.5 x 10 4 km off the ecliptic plane are obtained for the January case. Notice that the insertion burn into the Trojan orbit occurs outside the path of the Earth (see Figure 5.5) whereas for the other two cases (see Figures 5.1 and 5.2) previously analyzed, the libration insertion occurs inside the path of the Earth. This fact is not conclusive but indicates that launching at different times of epoch may be fa vored by the location of the insertion maneuver into the Trojan orbit. Figure 5.6 illustrates a series of interesting trajectories from the science point of view. In previous examples, we showed several transfer trajectories with varying TOFs from one year to about 2.5 years. Then, the spacecraft describes concentric orbits around £5 for several years . In these other cases, we show transfer trajectories of about 2 to 2.2 years and different sizes of Trojan orbits. The purpose of showing these orbits is not to compare different times of launch for the same TOFs but to show that by changing 153 Orbit Case L4(1) X 10 8 Y(km ) 5 X (km) X 10 7 0 (a) X 10 5 Orbit Case L4(1) (c) X 10 7 6 ,. 4 , ... 2 ···· suN· · I 0 + · · ··· 0 x w' Orbit Case L4(1) 10 X (km) (b) Orbit Case L4(1) Y(km ) (d) 15 X 10 7 X 10 7 Figure 5.7: Year- and-one-half transfer trajectories to L4. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. The January, April, July and October orbits are in blue, green, magenta and brown, respectively. The orbit in gold is the solution obtained in the CRTBP. slightly the TOF (see Table 5.4), the spacecraft can orbit a Trojan orbit which is actually shifted from the L5 location. Trojan orbits shifted behind L5 in the Earth orbit would have a great advantage to anticipate space weather warnings because the space probe could collect data from the west limb of the Sun. We know that space weather could nominally be predicted 3-5 days in advanced from L5; thus, these orbits (April launch) could anticipate space weather at least one day before the nominal prediction time. We also exam ined a few trajectories to orbits around L4. In Figure 5.7, we observe different trajectories launched at different times from Earth parking orbit having a 200-km 154 altitude and 28.5° inclination to the other triangular point £4. From Table 5.5, we notice that, in this case, launching the spacecraft in October yields the lowest total .6. V for the mission with 4.162 km/s, which is about 1.064 km/s cheaper than the solution obtained in the CRTB P. The TOF is about 13 days longer for a launch in October than launching at a different time of epoch. The excursions above and below the ecliptic plane are of the order of 10 5 km for July and October launches. These excursions are of the order of 5 x 10 4 km above the ecliptic for April and January launches but the spacecraft spends very little time below the ecliptic. Therefore, the October launch is a more attractive trajectory for science purposes since we can achieve higher excursions above and below the ecliptic. The July launch would yield slightly lower excursions than the October launch with a TOF of only 13 days longer but with a penalty of 616 m/s compared to the October case, that is, 423 m/s more at injection and 193 more at insertion. Mission analysis to £4 is not further analyzed in this research. 5.1.2 Sub-£5 Orbits in the Sun-Earth System Improved science-dat a-collection [37] opportunities are studied for a space weather fore cast mission to the sub-£5 point, which is shifted from £5, in the Sun-Earth system. These sub-£5 orbits (see Figure 5.8) provide excellent locations to monitor solar activity, such as solar flares and coronal mass ejections before they reach Earth, enhancing the warning time of geomagnetic storms. The green and purple orbits could anticipate space weather up to seven days before these solar storms arrive at Eart h. These orbits have comparable flight times of the order of 778 days (see Tabe 5. 7 for exact values) while the total .6. V is 105 m/s larger for the orbit in purple. On the other hand, the trajectory in red corresponds to a much shorter flight time about a sub- £5 orbit which is also smaller than the other two orbits. As we studied in section 2.2.1, smaller orbits around £5 tend to be more inclined in the z-direction than Trojan orbits of larger size. Although, the sub-£5 orbits are not orbiting £5, their proximity to the triangular point makes the orbits very inclined. The orbit in red has the largest excursions of the order of about 10 5 km 155 whereas the green and purple orbits have excursions of about 52000 km and 10000 km, respectively. Sub-L5 Orbits in Sun-Earth System 0 2 -I Y(nondim,AU) X(nondim,AU) (a) 8 x 1 0 ... Sub-LS Orbits in Sun-Earth System 4 � a 1 N' - 2 - 4 - 6 . : - 8 � 0 -- �0 � 2 -- �0�4 -- - 706��0 � 8 --�-- �1�2 X(norrlim,AU) (c) -I 8 Sub-L5 Orbits in Sun-Earth System o .4 o .6 o.8 X(nondim,AU) (b) x 1 0 ... Sub-LS Orbits in Sun-Earth System 1. 2 \ 'EARTH - 8 L__ __ -"- 1 -- � '::--- -- _ 7- 0 6� ----: - 0:>-: 4- - ----:" "::" -- -- 7 0- Y(nondim,AU) (d) Figure 5.8: Different trajectories about sub-L5 in the Sun-Earth System. Green orbit has a flight time of 778.5 days and 777.6 days, respectively. a: 3D trajectory plot. b: XY projection. c: XZ projection. d: YZ projection. These solar events rotate in the direction of planetary motion arriving first at Ls, then 3-7 days later at Earth, and finally at L4 after another few days. Hence, this L5 mission can provide up to 7 days of advanced warning of solar storms, which can wreak havoc on Earth's telecommunication infrastructures, as compared to only 1 hours warning provided by missions at Lt such as the Advanced Composition Explorer. Diff erent flight times from Earth to sub-L5 orbits can be accommodated as displayed in Table 5.7 by selecting science experiments to be performed in the cruise phase. Because 156 Ta ble 5. 7: Ll V vs. TOF comparison between trajectories launched at different times of the year to sub-L5 orbits in the Sun-Earth system Tep och LlV roi Ll V Loi LlV roTAL TOF me. rp (km/s) (km/s) (km/s) (days) (deg) (km) Oct 3.452 0.710 4.162 573.4 28.50 6578.136019 Oct 3.608 0.785 4.393 778.5 28.52 6578.136015 Oct 3.724 0.774 4.498 777.6 28.51 6578. 136041 the sub-L5 orbits are only mildly stable, we may need to apply station-keeping maneuvers to conduct the desired science without affect ing the instrument's perf ormance. Thus, this new geometry may serve as a more appealing observat ory to study the Sun than placing a spacecraft in an orbit around L5. These orbits were very well targeted to the desired parking orbit conditions for both altitude of 200 km and inclination of 28.5°. Ta ble 5.8: Relevant parameters for departure trajectory to sub-L5 in the Sun-Earth system Sub-L5 Orbits C3 V oo 800 O:oo b f3hy p (km 2 /s 2 ) (km/s) (deg) (deg) (km) (deg) 1. Red- orbit 5.06 2.249 -2.1 0 171.39 32855.9 22.64 2. Green-orbit 8.60 2.932 -28.51 280.99 25561.1 28.86 3. Purple-orbit 11.2 5 3.354 -28.29 286 .61 22572.8 32.49 Using the JPL ephemeris DE421 model, we generate integrated transfer and sub-L5 orbits that will provide further key parameters of the trajectory design and na vigation analyses for such a demanding mission. In Ta ble 5.8, we depict the orbit parameters for the departure transfer trajectory of selected sub-L5 orbits. 5.1.3 Conic and Integrated Trajectories Test We tested our integrated trajectories by using a conic method in the planar case. In Figure 5.9( a), we illustrate that both the integrated and the conic trajectories are very 157 similar. We assume that the spacecraft is at L+ 10 days so we are not trying to target at launch, as done in previous analyzed cases. Instead, our goal is to compare how close the conic is from the actual real trajectory. The TOF for the conic orbit is about 745.4 days and the actual �V Loi = 652.2m/ s. (a) x 1 o ' Range for Integrated and Conic Trajectories � t\••••• f\ •••••••;.. :•••••• ;.. ••••••!•;.. •••• :• jjj� 0 500 10 00 1500 time (days) Range Rate for Integrated and Conic Trajectories �lnt£gra� . 500 10 00 time (days) (c) -1 . 55� ....L... �� .......... ��7�� ..L.... �� ......._. X(km) (b) Radial and Tangential velocity in function oftime - - - Radial (Coole) --- Tangential (Cooic) - Radial (Integrated) - Tangential (Integrated) tOOO time( days) (d) 2000 Figure 5.9: Conic test. a: Comparison between an integrated trajectory and the conic trajectory. b: Close-up of Trojan orbit around Ls. c: Range and range rate for integrated and conic orbit. d: Tangential and radial velocity profiles for integrated and radial velocity. As for the integrated trajectory, the insertion is about �V Loi = 649.7m/ s but we can see that a correc tion maneuver would have to be performed in order to correct for the retrograde motion of the spacecraft depicted in red in Figure 5.9( b) once it arrives at the Trojan orbit. In Figure 5.9(b), we also observe that the actual position of the Ls 158 point suffers a slight displacement outwards as indicated (in blue) by the real trajectory of the Earth. This is caused by the eccentricity effect of the Earth's orbit. The maximum difference in range between the integrated and the conic trajectories is about 5 x 105 km as illustrated in top of Figure 5.9(b ). The maximum differ ences in the range rate between the integrated trajectory and the conic orbit is about 150 mjs for both the transfer orbit and the Trojan orbit, respectively as shown in bottom of Figure 5.9( c). Figure 5.9( d) shows the tangential and radial velocity profiles of both the integrated and conic trajectories. The difference in radial velocity is about 0.15 km/s for the transfer orbit and 0.05 km/s for the Trojan orbit. 5.1.4 Stability Analysis The Triangular points are locations of especial interest for astrophysical and astronautical purposes [24]. These natural sites could be very attractive for future manned space sta tions. The material displaced from such natural sites could be used to capture (artificial) satellites but this will not be investigated further. rAU -------------------- r;-- , . --T;---- ' ' , ' : / / ' / ' . ' . / / ' I ', - / / rL ,', rL ' / ./ 5 ' ' 5 I I \' /, / / Displaced /).() _;, ;��:�-- { Neighborhood L, _' .... '! ', • 5 (/ll /1y) ,.. .. < , .. .., ' •"" " .... - ·"" Figure 5.10: Displacement of Triangular Points. In the CRT BP, the position of the Triangular points are unchanged but their actual positions depend on other secondary perturbing bodies (V enus, Jupiter, Saturn) when 159 including the ephemeris model. Thus, the actual positions of these points as illustrated in Figure 5.10 slightly shift in or out from the original position. Additional perturbing bodies do not strongly influence the triangular points for the inner planets (Venu s, Earth and Mars) but for the outer planets (Saturn, Uranus and Neptune), the triangular points are mainly influenced by Jupiter. Even though Jupiter's perturba tion may be strong enough to slightly alter the Earth-Venus resonance, it is restored in a short time. This resonance stabilizes the planet orbits, because if one planet lags behind, it is attracted by the other planet. Besides the resonances and the effect of other plane ts, the eccentricity of the Earth around the Sun can generate periodic changes that affect the Trojan orbits in the long run. Ta ble 5.9: Environmental Disturbances affe cting spacecraft's orbit Disturbance Force (N) Ll V (m/ s ) per year Lld (km ) per year Gravitational Fo rces Venus 2x w- 7 m/s 2 6 258 Jupiter 3x w- 7 m/s 2 10 437 Saturn 2x1 0- 8 m/s 2 ::; 1 30 Solar Wind effect 2x 10- 7 6 258 Radiation Pressure 7x w- 4 60 9x 10 5 Galactic Cosmic Rays 8x w- 8 2 110 In Table 5.9, we observe that Jupiter produces the largest gra vitational disturbance to Earth's orbit of about 437 km per year fo llowed by Venus with about 258 km per year. The total Ll V requirement including the main gravitat ional bodies is about 17 m/s per year. Note that the accelerat ion due to these perturbations is comparable to the acceleration due to the solar wind effect . The most dominant force comes from the solar radiation pressure with about 7 x 10- 4 N. The smallest contribution comes from the galactic cosmic rays with about 8 x 10- 8 N. We assumed that the effective area of the spacecraft is about 80 m 2 . For the solar wind model [24 ], we used the analytical 160 expression Fs win d = Pw · Ae ff V';- 2 where the momentum flux at 1 AU is 2.3 x 10- 9 kg jms 2 and for the solar radiation pressure, FsR P = A specul a r c A eff F A u . Our preliminary analysis on the stability of the triangular points shows that the Triangular points are quite stable for long periods of time. In this particular case, the integration time was 150 years. We can obtain the ranges and angular displacements from � R = ri, � :: A u and � f) = () �5 - () L5. Displacements range from about 10 4 -10 5 km to 10 6 km in the x and y directions, respectively. These range displacements translate into angular displacements between 0.16° and 0.58° but they can yield larger angular displacemetns. These results suggest clearly that £5 is quite a stable point for long periods of time. We know that the Lagrange point Ls oscillates. Thus, the motion of Ls can be represented [24] to first order with longitudinal and radial simple harmonic motions as: (5.1) (5.2) where AL5 is the maximum longitudinal displacement of the Lagrange point, TL5 is the period of Ls, WL5 is the maximum radial width of the equilateral point and eo is the offset from £5: (5.3) (5 .4) In our case, the mean distance T£5 from the Sun is one AU. In the Sun-Earth system, L5 librates with a period of about 222 years. Assuming one of the largest displaced am plitudes of about 10 6 km, the corresponding longitudinal and radial displacements are 0.053958 AU and 0.049152 AU, respectively. 161 Stability Analysis BCP In Figure 5.11, we illustrate the influences of the Sun's gravity and the Moon's orbital eccentricity on the Earth-Moon libration point L5 as the four body perturba tion. For this particular example on an asymmetric orbit as shown in Figure 5.11 (a), the Sun perturba tion and the Moon's orbital eccentricity are quite important for station-keeping purposes. These perturbative accelerat ions will cause the space probe to move in irregular orbits around the libration point. As we can see from Figure 5.11 (b), these combined perturba tive effects will tend to shift the satellite by about 0.5 km over the first 5 years, then, this effect increases up to 1.5 km over the time span of 75 years. Note also that the effect of Asymmetric Orbit around LS Earth-Moon Y( km) (a ) Sun and Moon's orbital eccentricity Pertur bation (km) Sun 4 Moon (b) T (years) Figure 5.11: Sun's and Moon's orbital perturbations in the BCP. 162 the eccentricity of the Moon is comparable to the effect of the Sun at early times (about 50 years). The effect of the Sun is more dominant at later times between about 75 and 225 years. Then, both perturbations are comparable again beyond 225 years (period of £5) for about another 50 years and the modular cycle repeats itself. A space probe around the Earth-Moon triangular points will tend to move away from these points due to the influence of the Sun and the eccentricity of the Moon causing the spacecraft to librate in irregular orbits around these points. Theref ore, station-keeping maneuvers are required so that the mission is not compromised. 5.2 Navigation Analysis Results In a prevwus paper [43], we charac terized a typical m1sswn m a planar orbit around £5 to observe the Sun and indicated that such a mission is in fact fe asible. Next, we provide a maneuver and orbit determination analysis for an £5 mission. This study will be performed for both the transfer trajectory and the Trojan orbit around £5. Orbit determination will be needed to have a more accurate estimation of the trajectory of the spacecraft at different stages: launch, midcourse, arrival and Trojan orbit. Therefore, we will analyze the required propulsion maneuvers needed to compensate primarily for injection errors at launch and station keeping maneuvers around £5 and £4. 5.2.1 Trajectory Maneuver Analysis We will examine several orbit cases [43] that will be tested for different execution errors to see how the effects of the total injection and insertion maneuvers may vary. All of these orbits can be classified [39] as Trajectories Outside the Path of the Earth around the Sun (TOPES). We will assume that the errors are spherical, that is, the error is the same in all directions. To get the exact injection correction maneuver will require a more detailed analysis. Since the exact launch date is uncertain and we may encounter attitude control issues, we may not know the exact direction that we need to thrust in advance. Therefore, our approach will give an upper bound for the errors even though we know 163 that the actual injection error will be smaller. In order words, we are considering large margins that will lead to a relatively small increase in the size of the propellant tank. In [39], we analyze several cases that represent anomalous a priori errors for different execution errors. Generally, we will assume a priori errors of 1%, 0.5% and 0.2%. We examine three different scenarios for each orbit case. For example, for orbit Case 1, the a pnon errors are: 5.2.2 Scenario 1: 1% a priori error For the transfer orbit, the a priori errors in position and velocity are about 7,987 km and 46 m js. For the Tto jan orbit, the a priori errors in position and velocity are about 1, 719 km and 10 m/s, respectively. 5.2.3 Scenario 2: 0.5% a priori error For the transfer orbit, the a priori errors in position and velocity are about 3,993 km and 23 mjs. For the Trojan orbit, the a priori errors in position and velocity are about 860 km and 5 mjs, respectively. 5.2.4 Scenario 3: 0.2% a priori error For the transfer orbit, the a priori errors in position and velocity are about 1597 km and 9 m/s. For the Trojan orbit, the a priori errors in position and velocity are about 344 km and 2 m/s, respectively. Similarly, we can obtain the corresponding a priori errors in position and velocity for both the transfer and Trojan orbit for each of the orbits analyzed. 5.2.5 Preliminary Maneuver Analysis Results Ta bles 5.10 to 5.1 3 show the maneuver analysis for orbits outside of the Earth's path around the sun. Column 1 gives the a priori errors of the transfer and Trojan orbit for each of the new L5 orbits fo und; column 2 the correction maneuvers in m/s (10 days 164 after launch) for both the transfer trajectory �V rcM�l and the station keeping for the Trojan orbit � VsKM� l; column 3 the � V r cM� 2 and � VsKM� 2 20 days after launch; column 4 the � V r cM� 3 and � VsKM� 3 150 days after launch; column 5 the � V r cM� 4 and �V sKM� 4 30 days before Tend; column 6 the �V r cM� 5 and �V sKM� 5 3 days before Tend and column 7 provides the �V rcM� 6 and �V sKM� 6 at Tend · Column 8 gives the 99% probability level �V 99% . 0.8 . 0.6 . Orbit Case L5(1) -0.5 0 0.5 X(norrlim.AU) (a) Feasible Transfer Or bit of about 343 days Orbit Case L5( 4) -0.1 (c) Feasible Transfer Or bit of about 372 days Orbit Case L5(2) (b) Feasib le Transfer Or bit of about 701 days ···· +·· -01 .. .. \ SUN -0.2 Orbit Case L5(5) (d) Feasib le Transfer Or bit of about 801 days Figure 5.12: Fe asible Mission Design for different flight time durations of selected transfer trajectories to Ls. Top left: Fe asible Transfer Orbit of about 343 days to Trojan orbit of amplitude of 0.52 AU. Top right: Fe asible Transfer Orbit of about 701 days to Trojan orbit of amplitude of 0.52 AU. Bottom left: Feasible Transfer Orbit of about 372 days to an Trojan orbit of amplitude of about 0.047 AU. Bottom right: Feasible Transfer Orbit of about 801 days to Trojan orbit of amplitude of about 0.047 AU. 165 Table 5.10: Maneuver Analysis for Case 15(1) Pri.Err. To + 10d To + 20d To + 150d T t - 30d T t -3 d T t �v 99% Transfer �V rcM (m/s) 1% 78.972 8.093 0.872 0.598 3.111 0.311 196.7 0.5% 39.486 4.053 0.462 0.535 1.867 0.1 87 92.99 0.2% 15.7 96 1.639 0.246 0.516 1.323 0.1 32 38.1 Troj an �V s KM (m/s) 1% 10.505 1.341 0.677 0.538 1.719 0.203 28.7 0.5% 5.341 0.963 0.588 0.321 1.664 0.198 15. 3 0.2% 2.370 0.828 0.561 0.225 1.6 48 0.196 9.0 where To denotes the Tepoc h in days, T t represents Tend in days and 99% confidence level indicates that 99% of the time, the sum of the maneuvers will be less than the value � V99% listed. Table 5.11: Maneuver Analysis for Case 15(2) Pri.Err. To + 10d To + 20d To + 150d T t - 30d T t -3 d T t �v 99% Transfer �V rcM (m/s) 1% 75.068 7.836 26.774 4.212 4.544 0.454 207.7 0.5% 37.534 3.922 13.389 2.107 2.356 0.235 103.9 0.2% 15.015 1.582 5.361 0.847 1.151 0.115 41. 8 Troj an �V s KM (m/s) 1% 5.588 0.979 0.614 0.303 2.025 0.236 16. 3 0.5% 2.939 0.847 0.587 0.230 2.009 0.234 10.7 0.2% 1. 522 1.166 0.580 0.205 2.001 0.233 8.1 The first set of analyzed orbits consists of TOPES trajectories [43] . Figure 5.12(a) displays an example of a transfer trajectory in the rotating frame from a 200-km parking low Earth orbit to a Trojan orbit. The amplitude of the Trojan orbit (maximum radius measured from center of the libration orbit) is 0.52 AU. The time of flight (TOF) between the injection from Earth parking orbit and the insertion into the libration orbit (10I) is 343.4 days. The injection � V from Earth orbit into the transfer orbit is 4.622 km/s and the �V required for 10I is 0.995 kmjs. The total �V is 5.617 km/s. Figure 5.12(b) shows an exam ple of a transfer trajectory in the rotating frame from 200-km parking low Earth orbit to a Trojan orbit. The amplitude of the Trojan orbit is 166 Table 5.12: Maneuver Analysis for Case 15( 4) Pri.Err. To + lOd To + 20d To + 150d T t - 30d T t -3 d T t �v99% Transfer � V rcM (m/s) 1% 58.348 5.869 135 .9 1.915 1.917 0.192 370.8 0.5% 29.174 2.935 69.085 0.958 0.993 0.099 187.8 0.2% 11.6 59 1.174 30.585 0.384 0.482 0.049 81. 4 Pri.Err. To + 10d To + 20d To + 60d T t - 30d T t -3 d T t �v99% Transfer � V rcM (m/s) 1% 50.315 4.687 0.586 1.622 0. 240 0.050 124.4 0.5% 25.258 2.345 0.294 0.812 0. 125 0.047 62.2 0.2% 10.107 0.946 0.123 0.328 0.062 0.045 25.0 Troj an � V sKM (m/s) 1% 16.397 1.847 0.693 0.666 1.849 0.212 40.6 0.5% 8.252 1.175 0.514 0.394 1.7 83 0.206 21. 8 0.2% 3.448 0.901 0.45 1 0.273 1.7 64 0.204 11 .3 Table 5.13: Maneuver Analysis for Case 15(5) Pri.Err. To + lOd To + 20d To + 150d T t - 30d T t -3 d T t �v99% Transfer � V rcM (m/s) 0.2% 123.535 1.017 0.495 0.748 0.631 0.064 274.4 0.1% 61 .765 0.509 0.247 0.374 0.317 0.034 137.2 0.15% 92.650 0.763 0.371 0.561 0.474 0.049 205.8 Trojan � V sKM (m/s) 1% 9.492 1.252 0.538 0.425 1.995 0.224 24.9 0.5% 4.722 0.946 0.467 0.298 1.970 0.222 14.2 1.5% 14.173 1.635 0.638 0.578 2.036 0.228 35.7 0.52 AU. The TOF between the injection from Earth parking orbit and the insertion into the 101 is 701.1 days. The injection � V from Earth orbit into the transfer orbit is 4.326 km/s and the � V required for 101 is 0.522 km/s. The total �V is 4.848 kmjs. Figure 5.1 2( c) corresponds to an example of a transfer trajectory in the rotating frame from a 200-km parking low Earth orbit to a Trojan orbit. The amplitude of the Trojan orbit is 0.04769 AU. The TOF between the injection from Earth parking orbit and the insertion into the 101 is 372.3 days. The injection � V from Earth orbit into the transfer 167 orbit is 3.483 km/s and the b. V required for LOI is 1.573 kmjs. The total b. V is 5.056 kmjs. Figure 5.1 2( d) presents an exam ple of a transfer trajectory in the rotating frame from a 200-km parking low Earth orbit to a Trojan orbit. The amplitude of the Trojan orbit is 0.04769 AU. The TOF between the injection from Earth parking orbit and the insertion into the LOI is 801.3 days. The injection b. V from Earth orbit into the transfer orbit is 3.377 km/s and the b. V required for LOI is 0.899 kmjs. The total b. V is 4.276 km/s. For these particular cases, we performed the maneuver trajectory analysis (see Tables 5.1 0 to 5. 13) as discussed earlier. We analyze the maneuver analysis of Case 1 displayed in Ta ble 5.10. For a 1% error, the spacecraft performs a TCM-1 of 79 m/s at + 10 days after launch to compensate primarily for injection errors. TCM-1 accounts for about 86% of the total budget needed for correc tion maneuver analysis of the transfer orbit. A second maneuver TCM-2, of about 8.1 m/s, was performed at +20 days after launch primarily to clean up the execution errors due to the first maneuver . TCM-3 was performed at + 150 days after launch and applied about 0.87 mjs in order to compensate for errors in the execution of previous maneuvers. TCM-4 and TCM-5 were performed to counterbalance built-up stochast ic accelerations and remove previous maneuver execution errors. TCM-4 and TCM-5 will be performed to improve the delivery accuracy at the entry interface into the Trojan orbit. These maneuvers were executed 30 days and 3 days before insertion into the Trojan orbit and required about 0.6 mjs and 3.1 mjs, respectively (see Table 5.1 0 for exact values). Finally, a last maneuver, TCM-6, at insertion required another small maneuver of about 31.1 cm/s and decreases to about 18 cm/s and 13 cm/s for smaller errors of 0.5% and 0.2%, respectively. This last maneuver will be available in order to correct any unplanned late anomalies. For TOPES cases, the maximum errors assumed were 7,987 km and 46 m/s and the minimum errors considered were 1597 km and 9 mjs. Although these large errors provide informa tion about the station keeping strategy to follow, the spacecraft is likely to encounter much smaller errors. 168 For the Trojan orbit of Case 1, the sum of the two first correction maneuvers after insertion needed for station keeping was about four times smaller than the sum of those maneuvers needed after injection for the transfer leg. In fact, the first two SKMs after insertion are smaller for all the TOPES cases. The maximum value correction maneuver for station keeping was 16.4 m/s for 1% error. SKM-1 is the dominant maneuver ac counting for about 70% of the total cost for the Trojan orbit over one year to correct for errors from the LOI maneuvers. The errors in the position and velocity for Case 1 were of the order of about 1719 km and 10 mjs. The errors in position and velocity were as large as 2718 km and 1. 57 m/s for Case 3. In the rest of the TOPES cases, the errors were smaller. 5.3 Changing Maneuver Locations By changing the locations [39] of some of the correction maneuvers, we could significantly reduce the propellant used. In Case 2, we changed the maneuver location of TCM-3. The propulsion maneuver at this location To+60 days was significantly reduced with respect to the maneuver performed at To + 150 days. For a 1% error, TCM-3 decreased from 26.774 m/s to 1.2 29 m/s; for a 0.5% error, TCM-3 was reduced from 13.389 m/s to 0.636 m/s and, for a 0.2% error, TCM-3 decreased from 5.361 m/s to 0.307 mjs. The subsequent approach sequence of maneuvers would change with only a penalty of 3 mm/s and 2 mm/s at Tend-30 days and Tend-3 days, respectively. However, by moving TCM-5 from Tend-3 days to Tend-20 days which is closer to the propulsion maneuver executed at Tend-30 days, we reduced the correction maneuvers notably. For a 1% error, TCM-5 was cut back from 4.544 m/s to 0.623 m/s so that the last maneuver TCM-6 at Tend was decreased from 0.45 m/s to 0.057 mjs. TCM-5 was reduced from 2.356 m/s to 0.323 m/s for an error of0.5% and TCM-6 was cut down from 23.5 cm/s to 2.9 cmjs. Finally, for an error of 0.2%, TCM-5 was reduced from 1.151 m/s to 0.15 8 m/s and TCM-6 was lowered from 11.5 cm/s to 1.5 cm/s . 169 In Case 3 for a 1% a priori error, TCM-3 was decreased from 135 .9 m/s to 0.586 m/s when changing the location of the maneuver from To+150 days to To+60 days. TCM-5 was reduced from 1. 917 m/s to 0.240 when performing the maneuver at Tend-20 days instead of Tend-3 days. The final propulsion maneuver, TCM-6, was cut down from 19.2 cm/s to 5 cmjs. Simila rly, for a 0.5% error, TCM-3 at To+150 days was reduced from 69.085 m/s to 0.294 m/s at To+60 days. Also, TCM-5 was decreased from 0.993 m/s at Tend-3 days to 12.5 cm/s at Tend-20 days; and TCM-6 was finally cut back from 9.9 cm/s to 4.7 cm/s at Tend· Finally, for a 0.2% error, TCM-3 was reduced from 30.585 m/s at T 0+150 days to 12.3 cm/s at T0+60 days, TCM-5 was brought down from 0.482 m/s at Tend-3 days to 6.2 cm/s at Tend-20 days; and TCM-6 was further cut down from 4.9 cm/s to 4.5 cm/s at Tend· The .6.V at a 99% confidence level was reduced from 370.8 m/s down to 124.4 m/s for 1% execution error. The .6.V budget at a 99% confidence level for all the cases investigated are of the order of 200 mjs. This value is comparable with the one of about 180 m/s obtained for generic halo orbits [32] around the collinear point L 2 . As we mentioned at the beginning of the paper, all these results were obtained for a very large execution error of 10%. Maneuver errors of the order of 5% were obtained in previous studies [29] . Although the spacecraft may not encounter such large errors, in this study, we are being very conservative. Howeve r, execution errors of 10% were analyzed and com pared against execution errors of 1% as we display in Figure 5.13. From this analysis, we observe that the total maneuver cost difference between these execution errors range from about 3% to about 17% for the selected transfer orbits, and from about 0.5% to about 3.5% for their corresponding Trojan orbits. Why is the total cost of all trajectory correction maneuvers low when compared to other missions? First, the spacecraft is not far from the Earth so the orbit determination is very accura te. Secondly, the time of flight is very long and so a change in miss distance at the Trojan Orbit will require very small impulsive correc tion maneuvers. Thirdly, we are not trying to hit a specific target such as Mars. If the spacecraft were required to fly by Mars at a specific date, we would have to ensure that the spacecraft is within a few 170 01L_ __ �2t= ==� 3 �� 4 � � == s � � � SKM number Figure 5.1 3: TCM and SKM 6. V Cost. Top left: Maneuver Analysis of selected transfer trajectories for a 10% error. Top Right : Maneuver Analysis of selected transfer trajecto ries for a 1% error. Bottom left: Maneuver Analysis of selected Trojan orbits for a 10% error. Bottom right: Maneuver Analysis of selected Trojan orbits for a 1% error. minutes or seconds of the target arrival time so we do not miss the interception at Mars. However, in our case, we are only trying to get to the vicinity of Ls. Because of stability properties [43], we do not need to have very accurate maneuvers since the spacecraft will still be orbiting in the vicinity of L5. 5.4 Knowledge Error and Delivery Accuracy In this section, we will provide the orbit determination covar iance analyses (knowledge error) and the mapping orbit determination and maneuver execution analyses (delivery accuracy) for some of the orbits [39] . All statistical maneuvers are 1CT. 171 With the covar iance analysis, we can characterize the na vigation performance given the stat istical properties and measurements of the system. By covar iance analysis, we refer to the computation of the covar iance matrix that gives us informa tion about the accuracy level at which the spacecraft orbit can be estimated while providing predictions of the delivery accuracy. Once we know a more accurate uncertainty, we can propagate it to simulate more realistic estimates of the state of the spacecraft . In addition, we perform a sensitivity analysis to evaluate which fa ctors, such as the execution errors and the a priori errors, contribute the most to the improvement or degra- dation of the na vigation analysis. Therefore, different execution errors and a priori errors for both position and velocity will be studied as a way to simulate the spacecraft being exposed to unexpected environmental conditions or any other unpredicted anomalies. Ta ble 5.14: Knowledge Error and Delivery Accuracy for case 15(1) CASE 1 Location Transfer Orbit To + lOd To + 20d To + 30d T J + 150d T J -3 d T J Knowledge Error Err.P os.(km) 206.7 255.4 272.8 282.1 175.7 174.4 Err. Vel. (em Is) 23.06 13.83 10.53 2.17 0.60 0.59 Delivery Accuracy Err.P os.(km) 1013 .1 628.7 461.7 197.1 174.5 174.4 Err. Vel. (em Is) 117.2 5 36.38 17.81 1.52 0.59 0.59 Trojan Orbit To + lOd To + 20d To + 150d T J - 30d T J -3 d T J Knowledge Error Err.P os.(km) 991.0 970.5 458.5 28 3.1 277.4 278 Err. Vel. (em Is) 3.50 1.2 6 8.23 8.24 7.66 7.59 Delivery Accuracy Err.P os.(km) 1063.0 987.7 373.7 288.9 278.4 278 Err. Vel. (em Is) 36.05 23.29 10.10 8.12 7.64 7.59 In this particular example (see Table 5.14), we see that the errors in position for both the knowledge and the delivery accuracy are of the order of a few kilometers. These errors are smallest for the delivery accuracy after the last two correc tion maneuvers . The errors in velocity are of the order of a few cmls or mmls. A similar orbit determination 172 analysis can be extracted from Table 5.15, Table 5.16 and Ta ble 5.17, which correspond to orbit cases 15(2), 15(4) and 15(5), respectively. Ta ble 5.15: Knowledge Error and Delivery Accuracy for case 15(2) CASE 2 Location Transfer Orbit To + 10d T0 + 20d T0 + 60d T 1 - 30d T t -3 d T t Knowledge Error Err.Pos. (km) 159.9 205.80 254 .1 96. 3 108.2 118.3 Err.Vel.(cm/ s ) 18.51 11.91 4.90 0.74 0.18 0.20 Delivery Accuracy Err.Pos. (km) 853.5 540.3 248.7 118.3 118.3 118.3 Err.Vel.(cm/ s ) 98.79 31.27 4.80 0.02 0.20 0.20 Trojan Orbit To + 10d T0 + 20d T0 + 150d T 1 - 30d T t -3 d T t Knowledge Error Err.Pos. (km) 995.4 983.6 519.1 331.4 307.6 306.4 Err.Vel.(cm/ s ) 115.21 56.92 4.00 1.15 0.98 0.97 Delivery Accuracy Err.Pos. (km) 1089.4 992.8 419.8 327.8 307.6 306.4 Err.Vel.(cm/ s ) 126.08 57.45 3.24 1.13 0.98 0.97 How is the orbit determination at every orbit time compared with the pointing ac- curacy? We assume that the most accurate pointing requirement is about 10 0rad per pixel (typical imaging camera), that is, the ratio of the orbit determination error to the distance to the Sun will set the minimum instrument resolution. For exam ple, the New Horizons Navigation to Pluto [50] used two cameras with different resolutions (5 0rad and 20 wad per pixel ). Once the spacecraft has inserted into the Trojan orbit, it will continue doing science for two years. The orbit determination error 10 days after inser- tion is about 1063 km, therefore, the orbit determination to pointing accuracy will be about 8.6 0rad per pixel. After one year, the orbit determination drops to 278 km with a pointing accuracy of about 2.2 0rad per pixel. For the rest of the cases, the resolution was between 2 0rad per pixel and 2.5 0rad per pixel. This resolution can be achieved using a heliospheric and magnetic imager 173 Ta ble 5.16: Knowledge Error and Delivery Accuracy for case L5( 4) CASE 4 Location Transfer Orbit To + 10d To + 20d To + 60d T 1- 30d T 1 -3 d T t Knowledge Error Err.Pos.(km) 156.3 206 .1 251.7 117.5 96.7 44.9 Err.V el.(cm/ s) 18.09 11.93 4.86 0.40 0.30 0.10 Delivery Accuracy Err.Pos.(km) 288.6 174.2 80.1 46.2 46.1 44.9 Err.V el.(cm/ s) 33.40 10.08 1.55 0.16 0.14 0.14 Trojan Orbit To + 10d To + 20d To + 150d T 1- 30d T 1 -3 d T t Knowledge Error Err.Pos.(km) 174.0 207.1 214.1 125.4 67.3 58.8 Err.Vel.(cmjs) 20.14 11.99 1.65 0.43 0.22 0.19 Delivery Accuracy Err.Pos.(km) 308.5 208.3 71.7 60.0 59.1 58.8 Err.Vel.(cmjs) 35.71 12.05 0.55 0.21 0.19 0.19 Ta ble 5.17: Knowledge Error and Delivery Accuracy for case L5(5) CASE 5 Location Transfer Orbit To + 10d To + 20d To + 150d T t - 30d T t -3 d T t Knowledge Error Err.Pos. (km) 20.4 30.9 42.7 12.1 6.8 8.9 Err.Vel.(cm/ s) 2.37 1.79 0.33 0.02 0.01 0.01 Delivery Accuracy Err.Pos. (km) 111. 3 45.32 27.8 8.9 8.9 8.9 Err.Vel.(cm/ s) 12.88 2.62 0.21 0.01 0.01 0.01 Trojan Orbit To + 10d To + 20d To + 150d T t - 30d T t -3 d T t Knowledge Error Err.Pos. (km) 1038.7 1073.7 488.1 336 .1 326.1 326.0 Err. Vel. (em/ s) 120.2 62.1 3 3.77 1.16 1.04 1.03 Delivery Accuracy Err.Pos. (km) 1053.3 994.6 473.6 333.0 326.2 326.0 Err.Vel.(cm/ s) 121.91 57.56 3.65 1.15 1.04 1.03 with a 0.5 ± 0.01 arc sec per pixel. As a point of reference, the Hubble Space Telescope resolution [50] of the Wide Field Planetary Camera is 0.043 arc sec per pixel. 174 The Deep Space Network (DSN) can monitor the spacecraft position and velocity when orbiting the Trojan orbit about £5. Even though, the position and velocity will not be exactly known due to errors in the orbit determination process, the SKMs will maintain the spacecraft within its nominal trajectory. 5.5 Maneuver Analysis of Integrated Trajectories We limit the magnitude of the TCMs to less than 150 m /s for the transfer trajectory. This value is the value obtained from the pre-launch thruster performance estimates for the Genesis mission. Our maneuver strategy [40] also contains another 150 m/s for SKMs after insertion into the Trojan orbit. We studied two scena rios. For each scenario, we examined the effect of the a priori position and velocity errors on the 99% confidence level assuming a 1% execution error. 99 % Confidence Level (A Priori Position fixed) 0 2 3 4 5 6 7 8 9 10 Error in A Priori Velocity (m/s) Figure 5.14: 99% confidence level for a fixed a priori position error. The magenta and black lines correspond to the Ll V 99% of selected transfer and Trojan orbits, respectively. The solid lines represent CaseL5(1) and the dash lines Case L5(2). The red dashed line represents the assumed upper bound confidence level of 150 m/s for both the transfer and Trojan orbits (individually). 175 First, we fixed the a priori position error while varying the error in a priori velocity for both the transfer (magenta line) and the Trojan (black line) as illustrated in Figure 5.1 4. For a fixed a priori error in position of 15 km on the transfer orbit (L5a), the b. V 99% is between 83 mjs and 105 m/s when assuming variat ions in the a priori velocity errors of 10 cm/s and 10 mjs, respectively (see Figure 5.1 4). b.V 99% can be reduced for lower errors in the a priori position. For the correspondent Trojan orbit, the b. V99% ranges between 94 m/s and 150 m/s assuming an a priori position error of 3 km. By decreasing this a priori error to 1 km, we notably reduce the b. Vg 9% down to values between 31 m/s and 136 mjs. For the second orbit (L5b ), the b. V99% of the transfer segment is between 99 % Confidence Level (A priori Velocity fixed) 200 . 150 � I? >g:, 100 <J (;)' 0 . 0 Transfer L5a t ransfer LSb · · Apri\i � i ,;; 5 riV s� · · � ¥ ApriVel = 5 rn/s I I 5 10 15 20 25 30 35 Error in A Priori Position (km) 40 Figure 5.15: 99% confidence level for a fixed a priori velocity error. The magenta and black lines correspond to the b. V99% of selected transfer and Trojan orbits, respectively. The solid lines represent CaseL5(1) and the dash lines CaseL5(2). The red dash line represents the assumed upper bound confidence level of 150 m/s for both the transfer and Trojan orbits (not combined). Note that the assumed a priori velocity errors ( ""' 5 m/s) for the transfer orbit are very large, which is equivalent to about 5% error of the value of the first TCM. For the Trojan orbit, these a priori velocity errors ( rv 1 m/s) are also large or about 10% of the first SKM for the studied Trojan orbits. 33 mjs and 133 m/s for an a priori position error of 1 km whereas for the Trojan leg, the b. V99% fa lls between 18 m/s and 46 m/s when keeping constant an a priori error in 176 position of 5 km. The transfer leg is depicted by the magenta-dash line and the Trojan segment by the black-dash line. Second, we vary the a priori error in position while maintaining constant the a priori velocity error as displayed in Figure 5.15. As in the first scenario, the dot-dashed red line denotes the limit requirement for the Ll V 99% . Our analysis shows that the correction maneuvers (TCMs and SKMs) for an Ls mission can be achieved with less than 300 mjs. 5.5. 1 Propulsion Requiremen ts/Mass Perf ormance The total mission propellant of the spacecraft is the sum of the propellant used to execute TCMs and SKMs and the propellant used by the Attitude Control System (ACS) system. The total propellant consumption, for a given confidence level (99%), will have to be below the total propellant stored in the tanks to ensure a successful mission as displayed in Table 5.1 8. The total propellant computation accounts for the initial injection dispersion, orbit determination and the maneuver execution errors. Ta ble 5.1 8: Propellant Mass and Duration of TCM and SKM at each location during the Transfer and Station Keeping for 15(1) Leg To + 10d To + 20d To + 150d Tt - 30d Tt -3 d Tt Mt o t al Transfer M rcMs Mass(kg) 18.871 1.9 04 0.205 0.140 0. 730 0.073 21.92 Time( min) 14 3.11 14.443 1.55 4 1.065 5.538 33 sec Troj an(1 yr) M sKMs Mass( kg) 1.7 33 0.221 0.111 0.089 0. 283 0.034 2.47 Time( min) 13.15 1.67 51 sec 40.32 sec 2.72 15 sec Trojan(2 yr) M sKMs Mass(kg) n/a n/a 0.111 0.089 0. 283 0.034 0.52 Time( min) n/a n/a 51 sec 40.32 sec 2.72 15 sec LlM 99% (kg) 24.91 ACS(kg) 5 Mission( kg) 44.86 Margin( kg) 14.95 where n/a means that a TCM is not applicable 177 The maneuver analysis in this study enable us to find the fuel expenditure for the whole mission and for individual TCMs and SKMs. The relationship between the fuel requirement of the total number of maneuvers is expressed as: where M0 and Mp are the initial and final mass after each correction maneuver . We assumed that the injected mass of the spacecraft from a 200-km parking orbit is 3500 kg and the payload mass is 465 kg. Akioka et.al [1] suggested that a small spacecraft of about 450 kg with instruments will be sufficient for an £5 mission to be accomplished using a medium size launch vehicle or with a dual launching on H2-A class rocke ts. Also, using the Launch Vehicle Perf ormance Web Site [7], we display the availability of some launch vehicles. For example, we could use the Atlas V (401) (see Figure 5.1 6) to bring a spacecraft of about 465 kg to a 200-km parking low Earth orbit and 28.5° inclina tion. This payload mass is comparable to the payload mass in The New Horizons to Pluto mission with about 4 78 kg. The results from Table 5.1 8 confirm that the total propellant required adding all the TCMs and SKMs is about 44.86 kg for the entire mission. This includes a propellant margin of about 10 kg (or 33 % of the mission propellant allocation) and extra 5 kg of propellant for the attitude control system (ACS). We assumed that the thrust of the thruster was 5.0 N. In the Case 1, TCM-1 was performed 10 days after launch for a burn duration of about 143 minutes using about 19 kg of propellant. TCM-2, performed 20 days after launch, used about 1.9 kg of propellant during 14.4 minutes. The rest of the TCMs used about 1.1 kg combined and their duration was between 33 seconds and 5.5 minutes (see Ta ble 5.1 8 for exact values). Once the satellite has inserted into the Trojan orbit, it will perform a SKM-1 using less than 1.7 kg for about 13.15 minutes. SKM-2 will use about 0.22 kg for a burning time of 100 seconds. As mentioned earlier, these two maneuvers will only be required after 178 9,000 8,500 '81 8,000 � Ill Ill � 7,500 7,000 6,SQQ NASA ELY Performance Estimation Curve(s) lEO Circular with inclination 28.5 Please note ground rules and assumptions be low. , ...... Falcon 9 (Block 1l -+- A tl as V (401) I I I I I I I I I I I I I I I I I I I -+-----+-----+-----+-----+-----�-----�-----�-----� I I I I I I I I I I I I I I I I I I I I I I t-----�----:-----�---- -� -----:-----�-----�-----� I I I I i �� I I I T-----r-----r-- -r-----r-----r-----r- � - -----r-----r I I I I I I I I I I I I I I I I I I I I I I I I I I �-----�-----�-----�----- -----�-----�-----�-----�---- 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I _T _____ T _____ T _____ T -----�-----�-- - -----�-----�-----� I I I I I I I I I I I I I I I I I I I I I I I I I I I I +-----+-----+-----+-----+-----+-----�-----�- --�-----r I I I I I I I I I I I I I I I I I I I I I I I I I I 200 400 600 800 1,000 1,200 Altitude (,l<m) 1,400 1,600 1,800 2,000 Figure 5.16: Mass Capability as a functi on of altitude. Atlas V (401) and Falcon 9 (Block 1) have the capability of carrying a wet mass of 3500 kg to a parking orbit of 200 km and 28.5° inclinati on. insertion into the 'Itojan orbit. After the satellite has completed one year around the Trojan orbit) these two first SKMs will not be required and only the last four maneuvers will suffice to maintain the spacecraft around the 'Ito jan orbit for another year. The last maneuver will burn 34 grams for about 15 seconds. As mentioned earlier) the TOMs during the cruise and approach phase were reduced significantly when performed at different locations. Some of these TOMs are very small compared to the TOMS required to compensate primarily for injection errors) such that the total mass of propellant )) savings )) was solely a few kilograms) and it does not have a critical impact on the final prop ellant performance. However) there are other TOMs that resulted in a much higher penalty if maneuver relocation was not considered as in the Case 2 with about 30 % more fuel used. 179 The propellant analysis was obtained assuming a specific impulse of 232 seconds (monomethyl hydrazyne thrusters ). With this engine parameter, the spacecraft was left off with about 2 kg for Case 1 after orbiting the Trojan orbit for two years. We could use this extra fuel if further anomalies occur. 5.6 Science at L5 In order to detect particles emerging from the Sun, we need a detector that is able to sweep a certain region of the field of view angle. The detector can be pointing at specific regions (solar flares or CME) near the Sun and is not required to sweep the celestial hemisphere. Depending on the angular distribution that the detector can achieve, the satellite will be able to monitor data from different particles. For exam ple, a detector with a 4° angular resolution [1] can detect the energy of particles (coming in at 45°) within a small percentage ( � 10%) of their initial energy. Thus, by pointing the detector (assuming the same angular resolution) at different regions of the Sun, the instrument will be able to collect other particles with larger or smaller energies. Diff erent part icles can be monitored from different angular distributions from the Sun: Thermal protons at ±10° , heavy ions at ±5° and thermal and non-thermal electrons at ±3° angular distribu tion. Solar wind electrons can be monitored easier since they come from all directions. The angular resolution of the detector to perform science and analyze data from different particle collection can take advantage of the 3D orbits around L5 that lay above and below the ecliptic as discussed in this research . From our simulations, the largest angular inclinations of the Trojan orbits around L5 occur for January and April launches whereas the lowest angular inclinations occur during July and October launches. In particular, case L5(2) shows inclined Trojan orbits about 2° and 4° for January and April launches, respectively. Also, case L5( 4) has the maximum inclined Trojan orbit with about 9.1 ° and 0.1 ° for January and April launches, respectively. Case L5 (1) gives inclinations of 0.2° and 0.25° for the Trojan orbit for January and April launches . For 180 other launches at different times of the year, the inclinations of the Trojan orbits result in less than O.F in general. The geometry of these orbits will help monitoring different particles so we can accu rately measure the speed of CMEs in order to predict their arrival time at Earth. Diff erent CMEs travel at different speeds [22]. For instance, the average speed of CMEs is about 450 km/s which would take about 3.8 days to arrive at Eart h, but SEP CMEs travel at about 1500 km/s taking slightly over a day to get to Earth. Ground Level Enhancement (GLE) CMEs associated with SEP have high energies to be detected by ground-based neutron monitors. These CMEs are sudden and short lived so they travel to the Earth in less than a day . There are also other types of CMEs such as Geoeffective CMEs, Halo CMEs, Magnetic Clouds CMEs and Coronal type II bursts which have different velocity distributions and therefore different flight times to the Eart h. Understanding how these particles travel is crucial for space weather prediction. 5.7 Communications and Power Reference [22] suggests that the space probe should have three antennas: 1) High Gain Antenna (HGA) in dual frequency (Ka and X-band ), 2) Medium Gain Antenna (MGA) and 3) Omni antennas. Each antenna has its own funct ionality. A HGA can be used for general mission support at 2 kbps. A MGA can be operated at 2 kbps in X-band during the early transfer phase when the HGA is not pointing at Earth. These two antennas (X-band up and down and a K-band antenna down) can be used to determine the range between the spacecraft and Earth. Omni antennas are usually located 180° apart on the spacecraft to provide support at launch and during the early phase after launch. These omni antennas can operate at 2 kbps when the spacecraft is in the vicinity of Earth or at about 100 bps when it is at L5. For a given transmitter power of an antenna, the data rate from a satellite antenna to an Earth antenna is proportional to the size of the antenna and the square of the frequency and inversely proportional to the square of the distance. Thus, the data rate transmitted 181 in the X-band frequency is the same as for S-band at four times the distance. Assuming that the instruments are body mounted then, we can turn the spacecraft and point the antenna at Ear th. We can start science as soon as it leaves Earth; thus, the spacecraft can point the instruments at the Sun to collect data when cruising. The spacecraft can collect data into a tape recorder or other storage device every week. Then, the spacecraft can be turned around to point the antenna at Earth and download the collected data. The power system of the EASCO mission would need of the order of 194 W to operate the instruments that will send data to Earth ground stations at the rate of 90 kbps. The New Horizons to Pluto is a current mission that generates of the order of 240 W from radioisotope thermal generators (R TGs). RT Gs can be used to keep the instruments warm so they can perform smoothly. The spacecraft can be thoroughly protected with thermal blankets to retain heat, and louvres to release excess of heat accumulated internally in the spacecraft when needed. Thus, a comparable electric power system can be used for this Ls mission. 5.8 Sensitivity of Velocity Requirements If we miss the launch window by a few days because of other schedule or operations issues, there will be a delay in the mission before another opportunity occurs. During this time, the spacecraft is orbiting Earth at an altitude of 200-km and inclination of 28.5° inclination and the trajectory is affected by some of the spherical harmonics produced by the Earth. The effects of orbit inclination, the J 2 oblateness of the Earth and the time that the spacecraft stays in the parking orbit before the departure along the hyperbola trajectory needs to be considered because they can change the rate of regression of nodes of the departing orbit. This regression rate affects the departure vector that lies in the orbit 182 plane, thus impacting the b. Vat injec tion. In other words, the first-order secular pertur- bation because of the oblateness of the Earth causes the orbit plane to regress according to the fo llowing average rate: . (R$)2 !.1 = -37r h -----;;:- cos i (5.5) We can obtain the error in the predicted location of the ascending node for a given stayt ime in orbit at 200 km and inclina tion. Then, using the var iations of parameters, we can obtain the change of nodes with respect to time to calculate the penalty in b. V. (5.6) (5.7) For a duration of 10 days and inclination of 28.5°, an error in the ascending node of about 5° would affect the predicted date of departure and introduce a b. Vo penalty of about 246 mjs. 5.9 Mission Requirements Traceability Matrix The L5 Lagrange Antenna Observat ory of Science seems a very promising mission to study the Sun. The spacecraft will be able to monitor CMEs and CIRs from the SEL5 with uninterrupted observa tions for 100% of the time. It will observe magnetic properties emanating from sunspots, active regions and coronal holes as well as the solar regions associated with the production of SEPs. The continuous in-situ monitoring of the various solar events will be anticipated from 3-6 days in advance. However, science observa tions can start a few months before insertion into the Trojan orbit. Thus, additional science is possible but it will be dictated by the type of instruments on board given the fact that some instruments may be affected by the TCMs or SKMs. For exam ple, this mission should have the capability of a Magnetic and Doppler Imager on board which will take measurements of the photospheric, magnetic and velocity fields 183 in order to map the photosphere of the Sun and be able to extract the physical conditions of the tachocline (region between the convect ion and conduction zones of the Sun). The requirement for this to be possible is that the radial velocity of the satellite while orbiting £5 should not be more than V ra d = V$ · 0.0167 = 0.5 km/s greater [22] than that of the Earth. This is important so we can accurately measure the CMEs speed to predict the travel time to Eart h. In this research, we also fo und other irregular (asymmetric) orbits around £5. Placing the spacecraft in quasi- periodic orbits around £5 with a less chaotic nature is fa vorable since range rate variat ions will be reduced as much as possible and be able to operate the instruments on board of the spacecraft . Unlike the SOHO mission, a mission to £5 will be able to measure the strength and direction of the magnetic field to study the CMEs and CIRs. Also, the STEREO mission did not have a magnetograph so it could not measure the magnetic field locally. A mission to £5 could be launched around 2022-2023. The spacecraft would arrive 2-3 years later around the time of the next solar minimum cycle 2025-2026 with sufficient payload capability to conduct the science required at £5. Ta ble 5.19 presents the compliance matrix with respect to the mission requirements of the trajectory and navigation analyses. The Trojan geometry has a great flexibility of selecting certain orbit parameters besides the period of the libration orbit for a reasonable amount of delta-V. The location of the burn into the insertion libration orbit plays a critical role in the reduction of the final delta- V budget . Our simulations suggest that the final delta-V budget can be larger if the insertion burn is performed when the Earth is at apoapse instead of at periapse. 5.10 Research Products The products of this research are papers, software tools, and extensive data base of orbits for both transfers trajectories to £5 and Trojan orbits around £5. 184 Table 5.19: Mission Programmatic Requirement Traceability Req. Description Specifications Compl iance (Y /N) Trajectory Mission Design 1 Transfer time TOF� 1-3 years y 2 Parking orbit (200 km, 28.5°) !:1rp < 1m, !:1i < 0.2° y 3 Libration orbit decoupling 2-3 years around Ls y 4 "Periodic orbits" ±5° off ecliptic Orbits with 2° - g o y Orbits with 0.1° - 0.3° (few orbits) Navigation Mission Design 5 !:1 V(TCMs + SKM s) < 300m/s !:1 Vgg< r., � 200 - 300 m /s y 6 Frequency TCMs/SKMs 6 TCMs and 6 SKMs y 7 Optimal maneuver frequency Relocate maneuvers y 8 Delivery accuracy 2-10 t-trad y Science at Ls g Payload/ scientific measurements 465 kg payload capability y 450 kg 10 Science observa tions �1-4 months y months before LOI 11 Radial velocity does not Magnetic Doppler Imager y exceed 0.5 km/s of V $ Trojan size< 0.05 - 0.06AU (most orbits) 12 Space weather prediction Continuous monitoring y 3-5 days in advanced at L5 3-7 days in advance at L5 185 5.10. 1 Papers This work has been the seed of several papers that have been published and submitted to confer ences: 1. (LLH10) Lo, M., Llanos P., Hintz G., " An L5 Mission to Observe the Sun and Space Weather, Part I", AAS/ AIAA Astrodynamicist Specialist Conference, San Diego, Fe bruary, 2010. 2. (LMHll) Llanos P., Miller J., Gerald R. Hintz, "Navigation Analysis for an Ls Mis sion in the Sun-Earth Syste m", AAS / AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, August 2, 2011. 3. (LMH12) Llanos P., Miller J., Hintz G., "3D Integrated Trajectories for an L5 Mission in the Sun-Earth Syste m", (to appear). 4. (LMHL12) Llanos P., Miller J ., Hintz G., Lo M., "Mission and Navigation Design of Integrated Trajectories to L4,5 in the Sun-Earth Syste m" , AAS/ AIAA Astrody namics Specialist Conference (accepted). 5. (Lla12) Llanos Pedro, "Trajectory Mission Design and Navigation to the Sun-Earth Sub-L5 for a Space Weather Fo recast ", (to appear). 5.10.2 Software Tools In this work, we include a MATLAB® GUI simulations for some selected orbits as shown in Figure 5.1 7. This representation tool could be used to generate and interact trajectories so the user can have a sense of the stability behavior of these orbits. 186 Grid xo - � d.XO • � Type of PIOI €) XY2Dpl>t Q XZ2Dptot Q YZ2Dptot 0 XYZ30p101 ( RUWI>LOT J ( CLOSE ) -0 . 2 CR.T BP Model : : : : . . . - .. P. .E!r:tl1 r!>�9 . .!r<Jj ?. r1. Qr.�i.t�.Ar.<J .ut:Jd, .. L5 .. ........ ; ..... . .. Di r�(;L Figure 5.17: G Ul: The black trajectory represents the periodic orbit around Lt depicted by a red circle, the magenta orbit has a direct motion toward Earth and the blue trajectory has a retrograde motion away from the Earth. The dashed blue line represents the path of the Earth around the Sun. Both nominal and perturbed trajectories are 5 years long. 5.10.3 Data Base of Feasible Lt and £4 Traj ectories Below we illustrate some of the feasible orbits from Earth to the triangular points £4,5 in the Sun-Earth system. Some of these orbits have been analyzed using the ephemeris model and described in Chapter 5. There are other orbits that are not shown because they look almost the same. Note the different types of orbit: trajectories outside the path of the Earth around the Sun (TOPES) and tr aj ectori es inside the path of the Earth around the Sun (TIPES). 187 -0.1 Orbit Case L5(1) (a) Orbit Case L5(3) 0.4 0.6 0.8 X(not>:lim,AU) (c) -0_1 0.2 Orbit Case L5(2) (b) Orbit Case L5( 4) 0.4 0.6 0.8 X(nondim,AU) (d) 1.2 Figure 5.18: TOPES (in magenta) from Earth to a Trojan orbit (in black) around Ls in the Sun-Earth system. a: One-year transfer orbit to a Trojan orbit of 0.52 AU amplitude. b: Two-years transfer orbit to a Trojan orbit of 0.52 AU amplitude. c: Two-and-one half-years transfer orbit to a Trojan orbit of 0.52 AU amplitude. Note that the insertion burn occurs outside the path of the Earth around the Sun. d: One-year transfer orbit to a Trojan orbit of 0.047 AU amplitude. 188 -0.1 Orbit Case L5(5) 0.4 0.6 X(norrlim,AU) (a) Orbit Case L5(7) 0.4 0.6 0.8 X(norrlim,AU) (c) -0.2 . 5' -0.3 , .. ...: . s -0.4 'i3 6 -0.5 s >< -0.6 Orbit Case L5(6) (b) Orbit Case L5(8) 0.4 0.6 0.8 X(nondim,AU) (d) Figure 5.19: TOPES (in magenta) from Earth to a Trojan orbit (in black) around Ls in the Sun-Earth system. a: Two-years and two months transfer orbit to a Trojan orbit of 0.047 AU amplitude. b: Fo ur-years transfer orbit to a Trojan orbit of 0.047 AU amplitude. c: One year and one month-transfer orbit to a Trojan orbit of about 180,000 km amplitude. Note that the insertion burn occurs outside the path of the Earth around the Sun. d: One year and one month-transfer orbit to a Trojan orbit of 180,000 km. 189 Orbit Case L5( 1 0) -0 1 5 -0 .3 ... · ..... ....... · .. .......... : ........ .. : . --< . § -0 . 4 ........ """"" "d § -0 5 .... .... � :;:;- -0 6 -0 . 9 . 0 02 -02 ........ ........ .... . 5' -0.3 --< . § -0 . 4 "d � 8 -0 . 5 ;:.-. -0 .6 -0 . 7 -08 04 0 6 X(nondim,AU) (a ) Orbit Case L5(16) 08 0 0 1 02 0 3 04 0 5 0 6 0 7 08 0 9 X(nondim,AU) (b) Figure 5.20: TOPES (in magenta) from Earth to a Trojan orbit (in black) around £5 in the Sun-Earth system. a: Two years-transfer orbit to a Trojan orbit of 28424 km amplitude. b: Fo ur-years transfer orbit to the equilateral £5 point. 190 Orbit Case L4(2) o.4 o_s o_a X(norrlim,AU) (a) Orbit Case L4( 4) (c) '2 SUN I Orbit Case L4(3) 0.4 0.6 0.8 X(nondim,AU) (b) Orbit Case L4(6) 02 0.4 0.6 X(nondim,AU) (d) Figure 5.21: TIPES (in magenta) from Earth to a Trojan orbit (in black) around £ 4 in the Sun-Earth system. a: Fo ur-years transfer orbit to a Trojan orbit of 28424 km amplitude. b: Ten-months transfer orbit to a Trojan orbit of 28424 km amplitude. c: Nearly nine-months transfer orbit to a Trojan orbit of 0.047 AU amplitude. d: One-year and eight months transfer orbit to a Trojan orbit of 0.52 AU amplitude. 191 1.2 ' .. � 0 . 8. ::J . � E '5 0 6 : c . 0 c :;::- 0 . 4 ; 0 0 . 2 Orbit Case L4(7) 0 . 4 0 . 6 0 . 8 X(nondim,AU) (a) Orbit Case L4(8) 02 0 . 4 0 . 6 0 . 8 X(nondim,AU) (b) 1.2 Figure 5.22: TIPES and TOPES (in magenta) from Earth to a Trojan orbit (in black) around £ 4 in the Sun-Earth sys tem. a: Nearly seve n-months TIPES to a Trojan orbit of 0.73 AU amplitude. b: Ten-months TOPES to a Trojan orbit of 0.73 AU amplitude. These are the same orbits used for the heteroclinic connections studied in this work. 192 /""'. 0 * :s uN -0 02 � -0.4 <r: 8 o ,....; "C) § -0 06 � '--' >-< -0 08 October -1 ' --- - � :. 0 0 02 Orbit Case L5 (1) :E ARTH I ' '· 0 0 0 ....... 0 • 0 0 0 ••• 0 •••• : •• 0 0. 0 0 ••• 1: .. 0 ••• 0. 0 J : I I I /1 \ .1 Earth orbit / 004 006 0 08 X(nondim,AU) 1 1 02 Figure 5.23: One-year integrated orbits to £5 in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in April, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBPO The dash blue line is the path of the Earth around the Sun. The equilateral point £5 is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 5 days in advance before they reach Earth. 193 _,...__ 0 * ' - -- - - · SUN -0.2 � <r: 8 -0. 4 . ,....; "C) r::: 0 5-0.6 >-< -0.8 - October -1 0 0.2 Orbit Case L5(2) a : I · EARTH f I _I Earth orbit� / : 0. 4 0.6 0.8 X(nondim,AU) 1 1. 2 Figure 5.24: Two-years integrated trajectories to Ls in J2000 coordinate fram e. The blue orbit represents a launch in January, the green orbit corresponds to a launch in April, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equilateral point L5 is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 5 days in advance before the solar events reach Earth. 194 0 . * SUN -0.1 Orbit Case L5(3) -0.2 ··--······· -·······--·······- ·······--······· ·-· -0. 7 -0.8 · · · · · · CRTBP · -0. 9 . Ea rth orbit � ' � - -1 - - - - ...... ... ·. 0 0.2 0.4 0.6 X(nondim,AU) 0.8 ' 1 Figure 5.25: Two and a half years integrated orbits to Ls in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in April, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equilateral point £5 is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 5 days in advance before the solar events reach Ear th. 195 0 SUN -0.1 -0.2 . . -0. 3 , _ __......_ � � -0.4 � -0.5 � 0 5 -0.6 . . � -0. 7 -- Orbit Case L5(27) EARTH 1 ., ·· -' · I . - Earth Qrbit� I I ······················ . ············: ·····" / ···· .. . . • J U ly Janua ry -0.8 -0. 9 . . ··'::4. . . _ _ : _ -1 �-- · -� - -· · 0 0.2 0.4 0.6 X(nondim,AU) 0.8 1 Figure 5.26: Several integrated trajectories to Ls in J2000 coordinate frame. The blue orbit represents a launch in January, the green orbit corresponds to a launch in April, the magenta orbit depicts a launch in July and the brown orbit symbolizes a launch is October. The orange orbit in gold is the solution obtained in the CRTBP. The dash blue line is the path of the Earth around the Sun. The equilateral point £5 is denoted by the solid black dot. The red star represents the Sun. Space weather can be anticipated up to 6 days in advance before the solar events reach Earth (green and blue orbits) 196 Sub-L5 Orbits in Sun-Earth Sy stem ' 0 - * -------···· -0.2 8 <r: -0.4 8 . ,....; "0 � 0 -0.6 � '--' ;>-< -0.8 :SUN -1 -- - 0 0.2 :E ARTH i - - ' r Earth orbit / : : : � , 0.4 0.6 0.8 X(nondim,AU) I I 1 1. 2 Figure 5.27: Sub-L5 integrated orbits to L5 in J2000 coordinate frame. Space weather can be anticipated up to 7 days in advance before the solar events reach Earth (green and purple orbits). 197 Chapter 6 Conclusions This work characterizes a space weather fo recast mission to analyze solar events that may send ejecta that will arrive at Eart h. The trajectory, mission design and navigation are analyzed. To begin with, we studied the feas ibility for such a mission in the planar CRTBP where a space probe in a planar periodic orbit around Ls can obtain in-situ measurements and observe the Sun. Our simulations indicate that the insertion maneuver into periodic orbits around £5 can be reduced at the cost of extending the transfer time on the order of a fraction to several years. Therefore, if a mission can take advantage of the transfer time from Earth to the Ls orbit, the propulsion requirements could be substant ially reduced. We further analyzed these periodic orbits by studying their stability and concluded that station-keeping may not be necessary as the precessions in the orbits analyzed still keep the orbits in the vicinity of £5. Small perturbations added on the velocity error generate a drift in the orbit that if biased can improve the telecom link as the science phase continues since the range will decrease due to the prograde precession towards the Earth. We attempted to use the invariant manifolds of the Trojan orbits to see if we could develop a better transfer algorit hm. Our invariant manifold analysis suggests that these manifolds are not suitable for a mission to Ls due to the long flight times. 198 Our simulations using more refined models, such as the BCP and ER TBP, show that the effect of the gravity of the Sun, the Moon and the eccentricity of the Earth around the Sun have an important effect on the periodicity of the orbits. The periodicity of the Trojan orbits disintegrates and is replaced by quasi- periodic orbits around L5. Integrated trajectories at different times of launch were investigated using a high fidelity model (DE 421). These refined trajectories indicate that space weather can be monitored from 3-dimensional orbits around L5 to have a better view of the Sun above and below the ecliptic and improve the science data collected. The total b. V budget of the integrated trajectories is greater than the one obtained in the RTBP as a consequence of other body perturba tions and the fact that the Earth is oscillating due to the influence of the Moon with respect to their barycen ter. These perturbations are the cause of natural frequencies during the transfer phase and Trojan orbit. In this L5 mission, a na vigation strategy for orbit determination and maneuver anal ysis was studied. From the maneuver analysis results, we conclude that we can reduce the propellant budget considerably by relocating the third TCM. The TCMs during the approach phase will have little effect on the total propellant budget since these maneuvers are smaller than the first TCMs after launch. The delta- V at a 99% confidence level for TCMs of most of the transfer orbits analyzed is about 200 m/s; for one case, it exceeds 300 mjs but maneuver relocation reduced the delta-V of TCMs to less than 125 mjs. The delta- V at the 99% confidence level for SKMs ranges from about 16 m/s to about 40 m/s per year for the worst scenario of execution errors analyzed on the Trojan orbits. Thus, we can decrease the cost of a series of maneuvers by reducing the a priori errors in position and velocity of the maneuver or by perf orming the maneuver at the correct lo cation. This can be accomplished by using an adequate attitude control system designed to the highest accuracy. Even though in our research, we have not fully optimized the placement of the maneuvers, we were able to reduce significantly the statistical maneuver cost of individual maneuvers by changing the maneuver location and therefore reducing the final delta-V budget to an acceptable amount. 199 Trajectory correc tion maneuver analysis of selected integrated orbits showed that about 136 m/s were required for the transfer phase and between 18 m/s and 46 m/s for maintenance of the Trojan orbit. Overa ll, the total (TCMs and SKMs) budget for an £5 mission can be achieved with less than the assumed upper bound of 300 m/s. The spacecraft can start science observa tions months before arriving at the libration orbit insertion location. Moreover, it can be orbiting £5 in certain orbits providing a continuous monitoring of the solar events and be able to anticipate space weather m some scenarios from 3 to 6-7 days before the arrival of ejecta at Eart h. A mission to £5 could be launched around 2022-2023. The space probe will arrive 2-3 years later around the time of the next solar maximum 2025-2026 in the solar cycle with sufficient payload ( rv 465 kg) to conduct the desired science at £5. For example, New Horizons is spending about 9 years in cruise, so long flight times can be accommodated by selecting science experiments to be performed in the cruise phase or putting the spacecraft into hibernation (as is being done for New Horizons) to reduce cost of operations. 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AAS jA IA A Astrodynamics Specialists Conference, Copper Mountain, Colorado, Jan 23-27, 2005. [75] J. G. Williams, D. H. Boggs, and W. M. Fo lkner. DE421 Lunar Orbit, Physi cal Librations, and Surface Coordinates. Interoffice Memorandum 335-JW,DB, WF- 200803 14-001, Jet Propulsion Laboratory, Pasadena, Califo rnia, 14-March 2008. 206 Appendix A Acronyms and Symbols 3BP ACE AU /3hy p b J3 BCP Three- Body Problem Right ascension of departure hyperbolic asymptote, deg Advanced Composition Explorer Astronomical Unit, km Asymptote angle, deg Semiminor axis of departure hyperbolic asymptote, km Impact parameter, km Bicircular Problem C3 Mission design parameter= v �' km 2 I s 2 C IRs Corotating Interaction Regions C ME Coronal Mass Ejection CRT BP Circular Restricted Three-Body Problem L}.99% L}. V Loi L}. V roi L}. V rot al Declination of departure asymptote, deg Maneuver delta-V, km Is 99% confidence level for total delta-V, mls Delta- V for Libration Orbit Insertion Maneuver, km Is Delta-V for Transfer Orbit Insertion (Injection) Maneuver, kmls Total Maneuver delta- V, km Is 207 r5 i j DE421 1} E[x] Kronecker delta function Developmental Ephemeris 421 True anomaly, deg Expected value of x EM B Earth Moon Barycenter ERT BP Elliptical Restricted Three-Body Problem EUV Extreme Ultraviolet 9 e 9 i,j G GUI H HCS HEP H.O.T. fsp L £ 4, 5 LM3A LM3C M o M 1 M 2 Mp 1-L State transition matrix Flow Matrix metric tensor Gravity at Earth Metric tensor components, i, j = 1, 2, 3 Gravitational constant Graphical User Intergace Hamiltonian Heliocentric Current Sheet High Energy Particles High Order Te rms Specific impulse, sec Lagrangian Triangular points Long March Vehicle Long March Vehicle Initial mass Mass of primary body Mass of secondary body Burnout (final) mass Gravitational mass parameter, GM 208 PO ·j q QPO R R3BP s SEP SKM sos SRP e T t T TCM Periodic Orbit Generalized coordinates Generalized velocity coordinates Quasi- Periodic Orbit B-plane vector completes orthogonal set with S and T Restricted Three-Body Problem (or RTBP) Standard deviation B-plane vector in the direction of the incoming hyperbolic asymptote Solar Energetic Part icle Station-Keeping Maneuver Surface of Section Solar Radiation Pressure True anomaly along hyperbola, deg True anomaly of out-going hyperbolic asymptote, deg Generalized time coordinate Time unit B-plane vector perpendicular to S and R Trajectory Correction Maneuver TIP ES Trajectory Inside the Path of the Earth around the Sun TOA TOF Time of Arrival, days Time of Flight, days TOPES Trajectory Outside the Path of the Earth around the Sun TOS Time of Science, days au aq; au aq;q; au oq; au oq ;iJ; Circular velocity 209 Vp Spacecraft 's departure (periapsis) velocity, kmjs V esc Escape velocity, kmj s V HE Hyperbolic excess velocity (V 00), kmjs w Angular velocity X State vector (position and veloci ty), km / s Subscripts � Variable number j Variable number Symbols 0 Sun EB Earth sc Spacecraft 210 Appendix B Diff erential Correctors The constraint differential correc tor, briefly explained in [45] and [3] can be used in the computation of Earth-to-halo transfer orbits. This appendix provides a more detailed mathematical description of this differential correc tor. This different ial corrector is then used to describe a constraint differential corrector that can be used for Trojan orbits. B .1 Halo Orbits Constrained Differential Corrector A constrained different ial corrector will eliminate random guessing and decrease signifi- cantly the computational time rather than using trial and error approaches. The State Transition Matrix (STM) will be used to estimate the changes needed in the initial state to arrive to the final state. Next, we show the entire procedure although we will only attempt to constrain the altitude parameter. Let i h and xo denote the state vector on the halo orbit and the state vector during the insertion, respectively. xo ( xo YO zo xo YO z o ) (B.l) X h ( Xh Y h Zh Xh Y h zh ) (B.2) 211 A y A X EARTH '\a A X (a) (b) Figure B.l: Constrained differential corrector. a: Angle between the velocity vector 6. Vx y and the x-direction of the rotating frame. b: Angle between the position vector of the spacecraft and the x-direction of the rotating frame. The initial state vector is defined to be close to the nominal halo orbit. In order to have a better insight of the halo insertion maneuvers, we introduce a new set of variables such that the new state vector is expressed as: xo ( xo Yo zo 6.V x y f3 6. Vz ) (B.3) where xo , Yo and zo are the components of the position state vector of the spacecraf t on the transfer path along the manifold and 6. Vx y is given by (B.4) We will define 6. Vz = io - ih and j3 denote the angle between the velocity vector 6. Vx y and the rotating x-axis such that (B.5) The final state vector is represented by new parameters (B.6) 212 where h alt is the altitude of the spacecraft above the surface of the Earth, o: is the angle between the projection of the vector r on the x-y plane and the rotating x-axis, A r is the angle between the r vector and the x-y plane, 'Y is the flight path angle or angle between the local horizon and the velocity vector v f , and � is the inclination or angle between the x-y plane and the final plane of motion of the spacecraft . Position of spacecraft at final state where Xf,$ = Xf - 1 + fh· Altitude Angle between r projection and x-y plane, o: tan o: = J!.L --+ o: = arctan ( J!.L ) Xh$ Xh$ Angle between r and x-y plane, A r tan A r = ZJ --+ Ar =arctan ( ZJ ) J xJ ,$ + YJ J xJ ,$ + YJ (B.7) (B.8) (B.9) (B.10) 213 A z _ ) A �! EARTH vJ EARTH / A A ' ' y / y X .. ,� 'It X (a) (b) Figure B.2: Constrained differential corrector. a: Angle between the position vector of the spacecraft and the x-y plane of the rotating frame . b: Angle between the velocity vector of the spacecraft and the local horizon. Fli ght Path Angle The angular momentum is defined as, L = r f x v f . . YJZJ - ZJY! (B.ll ) From the magnitude of the angular momentum L = rfVf cos /, we obtain that the flight path angle can be expressed as (B. 12) where the magnitude of the final velocity of the spacecraft is given by Vj = x} +if }+ z}. 214 Angle between the final plane and the x-y plane The inclination angle is expressed as: (B. 13) When considering this constraint, we would have to account for the fact that the Earth's equatorial plane is tilted 23.44 degrees with respect to the ecliptic plane. However, we will not attempt to constrain the inclination. z / A y X Figure B.3: Constrained differen tial corrector. Schematics of the inclination 215 Computation of variations Applying the chain rule we can compute the var iations of the var iables of the final state as a funct ion of the var iations of the initial states (B.14) (B.l5) (B.16) (B.17) where (B.18) 8h = h alt,d - h alt where h alt,d , cxd, 'Yd and � d are the desired values for the altitude, right ascension, flight path angle and inclination, respectively. Assuming that Ll V z = 0 and that the desired flight path angle is zero at injection ('Yd = 0), then the var iation of the flight path angle, 8"(, is zero. Hence, equations B.14, 216 B.15 and B.16 can be simplified to obtain (B.19) (B.20) (B.21) (B.22) From B.20 equation, we can solve for 6tr (B.23) Arranging equations B.19 and B.21, we obtain: 6h ah otj + (B.24) 8a 6(3 Hence, after plugging B.23 into B.24, the expression becomes: 8h (B.25) 8(3 217 which we can express as: ( ;: ) ( (B.26) The terms that we need to correct , 6 Ll Vx y and 6 j3, due to var iations in altitude and right ascension can be obtained as fo llows: T 6h 1 (B.27) 6j3 -B 12 Bu Similarly, if we want to constrain the inclination we can arrange equation B.22 to obtain: a� __ky___ a� 8-r l [ · : ; • Y 6 E a� 8f'>V xy a� � (B.28) �-Nf � �- Nf .§J_ 8 t 1 8 t 1 Thus, __ky___ lb. ___j}J!__ _ J!j._ 8f'> 8 Vx y ah ah a(3 6h at:.. Vxy at! ..£.]'_ a;3 - at1 � 6LlV xy 8 t f a t 1 (B.29) a E __ky___ a� a� � 5E a� 8f'>V xy 8j3 �-Nf � Oi3- Nf � a t 1 a t 1 which can be expressed in the new form: ( :; ) ( (B.30) Bu 5 j3 218 and the corrected terms, 6L)..Vx y and 6(3, due to var iations in altitude and inclination are: T 15.6. V xy c 22 � c 21 15h 1 (B.31) En C22 � E12C 21 15(3 � E 12 En 15� Computation of partial derivatives To compute the varia tion in altitude we need to obtain four partial derivatives. The first one 1s Proof ah ( oh oh 8 .6.. V xy = TfXi <JYi which can be expressed as: = ( ah ) az; ¢14 ¢1 5 ¢16 ¢24 ¢2 5 ¢26 </Y34 </Y35 </Y36 oh ) O Z f (B.32) (B.33) (B.34) 219 where: (B.35) or oxo � OXJ =( � OXj OXj ) � (B.36) all Vxy oxo o ifo oio a,6.Vx y � oil. Vx y which can be expressed in terms of known state transition matrix elements: (B.37) Then, a�Vx y and &��x y can be obtained in the same way: (B.38) 220 (B.39) Hence, equation B.32 has been proved. We can obtain the partial derivatives for the var iation of altitude. The first partial derivatives of the altitude with respect to each of the final components of the state position vector of the spacecraft are: 8h Xj ,$ (B.40) 8xf Jx },$ + YJ + z ] 8h Y! (B.41) 8yt j x },$ + YJ + z ] 8h ZJ (B.42) 8z t Jx },$ + YJ + z ] To calculate the partial derivatives af't9x y , o � x y and aft9x y we know that .6. V x y = .6. V x y (xo, Yo, io), therefore, we can compute the change of .6. V x y with respect to xo, Yo and io. Hence, the change of xo, yo and i o with respect to .6. Vx y will be the inverse of 8f1Jx y , aft?x y and o io . l � , respective y. 8xo 8.6. V x y ((xo- xh ) 2 + (yo- Yh? ) 1 1 2 XO - Xh 8yo 8.6. Vx y ((xo- xh? +(yo - Yh? ) 1 1 2 YO- Yh 8i o = 0 8.6. V x y (B.43) (B.44) (B.45) 221 The second partial derivative (used to compute the B 1 2 -term) for the var iation in altitude lS: �� = ( ah ah Oxf OYj where ¢1 4 ¢1 5 ah ) Fif ¢24 ¢2 5 ¢ 34 ¢ 35 8x0 Llv ;Y af3 YO- Yh oiJo Ll v x 2 y 8{3 xo - xh 8i o = 0 8f3 � a/3 ¢16 ¢26 ayo (B.46) O(j ¢ 3 6 aio O(j (B.4 7) (B.48) (B.49) Finally, the third and fo urth partial derivatives for the variat ion of the altitude are ob- tained as fo llows: � =0 allV z (B. 50) OX f 7ft Xj :t� = ( ah ah ah ) OX f = ( ah ah ah ) Y! (B. 51) OX f Oyj OZj 7ft OX j a y, OZj OX f Zj Cit 222 Computation of the B-matrix terms En Te rm The En-term can be computed as fo llows: (B. 52) where X f Y ! ZJ :z = ( a, Ox a, Oy a, Oz a, ox a, uy a, liZ ) (B. 53) X f Y ! Zf and ¢14 ¢1 5 ¢16 � ¢24 ¢2 5 ¢26 o.6. Vxy 0 "( - ( a, a, a, a, a, a, ) ¢34 ¢3 5 ¢36 o ifo (B.54) ollV xy - ox Oy oz ox o if az ¢44 ¢45 ¢46 0.6. Vxy ¢ 54 ¢ss ¢ 56 a· ¢64 ¢65 ¢66 &.6. t9x y 223 Finally, we compute (using Mathematica) a�;,i where and where (x Nf!XJ + Yt il t + z1z1) ( -y ] x1 + x1,$Y!Yt + z1 ( -z1x1 + x1,$it ) ) R} Vt1 2 14 (xf,(f)Xf + Y!Y! + ZJZJ ) ( ( x } ,$ + z ]) Y! - Y! (xf,(f)Xf + ZJZJ ) ) R} Vt1 2 14 ( - Zf (xf,(f)Xf + Y!Y! ) + ( x } ,$ + YJ) i t ) (xf,(f)Xf + Y!Y! + ZJZJ ) R} Vt1 2 14 (xf,(f)Xf + Y!Y! + ZJZJ ) ( -y fXJY! - ZJXJZJ + Xf,(f) (i!J + z ])) RtV /1 2 14 (xj,(f)Xf + Y!Y! + ZJZJ ) (il t (xj,(f)Xf + ZJZJ )- Y! ( x ] + i J)) RtV /1 2 14 (xf,(f)Xf + Y!Y! + ZJZJ ) ( - Zf ( x} + Y J) + (xf,(f)Xf + Y!Y! ) ZJ ) RtV }12 14 (xf,(f)Xf + Y!Y ! + ZJZJ ) 2 Rt 2 V f 2 Xf,(f) = -1 + fh + Xf B 12 Te rm (B. 55) (B. 56) (B. 57) (B.58) (B. 59) The B 12 -term can be computed using some of the partial derivatives computed previously: C! h B _ ah _ ot; a'Y 12 - 8 j3 a , 8 j3 otj (B.60) 224 where ¢14 ¢15 ¢16 axo ¢24 ¢25 ¢26 � �; = ( � � � � � 'b_ ) ¢34 ¢35 ¢36 0& (B.61) ax ay az ax ay az ¢44 ¢45 ¢46 af3 ¢ 5 4 ¢ 55 ¢ 56 (§_ ¢64 ¢6 5 ¢66 af3 B 21 Te rm Using some of the previous computed partial derivatives we can calculate the B 21 -term: where where oa ah a, Tftj B 21 = 8 .6..Vx y - a 1 o�V x y at 1 00: ( a a a a ) ot t = ax , ay f 00: - yf OXf - x} , EB + YJ 00: X j - 1 + /.t oy f x} , EB + YJ X j Y ! (B.62) (B.63) (B.64) (B.65) 225 and (B.66) B 22 Te rm Given the previous computed partial derivatives we can calculate the last term, B 22 (B.67) where oxo ap �; = ( oa oa ) ( 1H ¢1 5 ¢16 ) !}jjQ OXj o y f ¢2 5 ¢26 of3 ¢24 (B.68) (tiQ of3 C 21 Ter m The C 21 term has the form (B.69) This term requires the computation of new partial derivatives (B. 70) 226 a� a� a� aE · . where .,..,-- ,.,...,-- , .,...,-- and ..,--- t are g1ven by. uxo uyo uzo u f (h + R e ) v f 3/ 2 -(YfXf-Xf,$Yj )2 +(h+Re) 2 V j (h+Re) 2 V 1 . 2 . . · 2 Xf,ff! X j + YfX!Yf + Xf,ff!Zj (h + Re) v f 3/ 2 -(YfXf-Xj,$Y! )2 +(h+Re) 2 V j (h+Re) 2 V 1 (yrx1 - x1,ff! ii t ) 2 1 Xj Y! Y! Zj and the rest of the partial derivatives have been already computed. (B.71) (B.72) (B.73) (B. 74) 227 C 22 Ter m To obtain the C 22 term, we also need to compute where � C _ 8� 8� af3 22 - 8 f3 - 8t a::r f at; 8� 8� 8xo 8� 8yo 8� 8io -= -- + -- + -- 8 /3 8xo 8 /3 8iJo 8 /3 8io 8f3 (B.75) (B.76) that is, C 22 can be expressed in fun ction of partial derivatives, previously computed. Compute new velocity components Squaring B .5 expression and since Lj,.Vx y 2 = L�.Vx 2 + Lj, V y 2 then, Similarly, 2 Lj, V y 2 tan f3 tan f3 = 2 2 ---+ Lj, V y = ( 2 )1/2 Lj, Vx y Lj, Vx y - Lj, V y 1 + tan f3 (B. 77) (B. 78) Since Lj, V x = xo- Xh and Lj, V y = iJo - Yh , then we can obtain the next velocity components expressed as: xo , new = ( \ )1/2 Lj, V x y + xh 1 + tan f3 tan f3 . YO , new = (1 + tan 2 /3 )1/2 Lj, V x y + Yh (B.79) (B.80) 228 These expressions can be furt her simplified to obtain: where XO,new = Ll V xy,ne w COS f3new + Xh YO ,ne w = Ll V xy,ne w sin f3new + Yh f3 ne w = f3 + 8 ;3 and where 8Ll V xy and 8 ;3 can be computed using equation B.31 B.2 Trojan Orbits Constrained Differential Corrector (B.81) (B.82) (B.83) (B.84) Fo llowing the constrained different ial corrector obtained for halo orbits, we derive a sim- ilar constraint different ial corrector for Trojan orbits. Computation of variations Applying the chain rule we can compute the var iations of the var iables of the final state B. 6 as a function of the var iations of the initial states (B.85) (B.86) (B.87) (B.88) 229 where 15"1 = 'Yd - 'Y (B.89) 15h = h alt,d - h alt 15o: = 0:d - 0: where h azt,d , o:d, 'Yd and � d are the desired values for the altitude, right ascension, flight path angle and inclination, respectively. We will assume that the desired flight path angle is zero at injection ('Yd = 0), then the variat ion of the flight path angle, 15"(, is zero. Hence, equations B.85, B.86, B.87 and B.88 can be simplified to obtain (B.90) (B.91) (B.92) (B.93) From B.91 equation, we can solve for 15tt : (B.94) 230 Substituting B.94 equation into B.90, B.92 and B.93, we obtain: ah oh ai:�xy ah ah 8 ? &p 6h �- Tffj � 8 t t (f(j- Tffj � 8 t t __!b____ lb._ � _ oa fJ""e Vxy oa oa 8!3 ()/::,. Vxy Ot f 5!:1.. a/3 - at 1 � f) t f f) t ! B E B E aJ:?-x y B E B E 8 ? &p �- Tffj � (f(j- (i[j� 8 t t f) t f which we can express as: 6h En B 12 Bt3 6o: B 21 B 22 B23 6� B3 1 B32 B33 ah __§l_ oh 8D. Vz a;sv; - Tffj 3:1_ 8 t t __§l_ � _ !}g_ 8�Vz 8/::,. V z Otj 5!:1.. a t 1 B E __§l_ BE f) D. Vz a;sv; - Tffj 3:1_ 8 t t 6Ll Vx y 6{3 6LlV z 6{3 (B.95) (B.96) The terms that we need to correct , 6Ll Vxy, 6{3 and 6Ll Vz due to variat ions in altitude, right ascension and inclination can be obtained as fo llows: 231 which can be simplified as 6(3 Computation of partial derivatives En , B 12, B 21 , B 22 , B3 1 = C 21 and B32 = C 22 were computed in the previous constrained differential corrector (see equations B.69 and B.75). The new partial derivatives are now computed (B.99) (B. 100) (B.101) (B.102) 232 We need the fo llowing partial derivatives � aV z ¢1 4 ¢1 5 ¢16 ah ( ah ah ah ) ayo ailV z = ox; Oyf OZj ¢24 ¢2 5 ¢26 ov; (B. 103) ¢ 34 </1 35 </1 3 6 az o ov; Equation B.103 can be easily simplified since � = � = 0 and � = 1. Thus, and a , ( a, a, !b_ a, all Vz = ax, a y f az f ax· ! which can be further simplified as ah ah all Vz = ¢36 az1 ¢1 4 ¢24 !b_ a, ) ¢ 34 a y f a i f ¢ 44 ¢ 54 ¢6 4 ¢1 5 ¢16 ¢2 5 ¢26 </13 5 </13 6 ¢ 45 ¢ 4 6 ¢ 55 ¢ 5 6 ¢6 5 ¢66 (B. 104) ayo (B.l05) a�V z (B.106) 233 The following partial derivatives are required � oVz oa ( ah ah ) ( ¢1 4 (h s ¢16 ) o iJo oilV z = ox; Oiij ¢2 5 ¢26 ov; ¢2 4 (B.l07) ozo ov; which can be simplified as: (B. 108) and oa ) OZj Y ! (B.109) ZJ B33Term Finally, we will need (B.ll O) but it can be simplified as (B.lll) 234

Abstract (if available)

Abstract This thesis research will be based on the trajectory mission design and navigation design for prospective future missions to the Triangular Lagrange Points L5 and L4 in the Sun-Earth and Earth-Moon systems. The research proposed here will be divided into four parts. ❧ The first problem will be devoted to studying the circular restricted three-body problem (CRTBP) in the Sun-Earth system. With this model, we will generate potential optimized orbit solutions in the planar CRTBP and also in three-dimensional orbits in order to study the Sun above the ecliptic plane. Orbit determination analysis will also be examined using different orbit determination methods. Finally, we will analyze the stability of the trajectories and their stationkeeping requirements. ❧ The second part of this thesis will deal with the bicircular problem (BCP) in the Earth-Moon system. As in the work on the CRTBP, we will understand and analyze the stability of the different types of periodic orbits (quasi-periodic orbits) obtained under the influence of the Moon and the Sun. ❧ The third part will describe the elliptic restricted three-body problem (ERTBP) in the Sun-Earth system. As in the work on the CRTBP, we will analyze the stability of the different types of periodic orbits (quasi-periodic orbits) obtained due to the effects of the eccentricity of the Earth around the Sun. We will partially analyze the BCP and ERTBP but the main focus of the research will be based on the CRTBP and the JPL Ephemeris Model. ❧ The last problem is the new JPL Ephemeris Model, DE421. With this ephemeris model, we will determine how accurate the models CRTBP, BCP and ERTBP are in comparison with the real one. By studying the real model, we will have a more thorough insight into why some of the orbits obtained in both the CRTBP and ERTBP lose their symmetry when adding the influence of higher order perturbations into the dynamical model. ❧ Besides finding periodic and quasi-periodic orbits for different models, part of this trajectory mission design will be dedicated to the optimization of the trajectory, utilizing a differential corrector. Finally, we will close this section by developing some semi-analytical work based on different techniques, such as the Lie Series expansions. We will use these methods to have better approximations of the nonlinear problem in the neighborhood of the triangular points and to obtain a more accurate analysis of the stability of these orbits. ❧ Along with the trajectory mission design, part of this thesis work will be oriented towards the orbit determination analysis from the beginning of the mission at a predefined parking orbit around Earth to the end of the mission at the Libration Orbit (Trojan Orbit) around the triangular points. Orbit determination will be needed to provide a more accurate estimation of the trajectory of the spacecraft at different stages: launch, mid-course and arrival. ❧ We know that after the launch phase, the spacecraft will be sensitive to large errors that make the spacecraft deviate from the nominal trajectory. The main goal will be to determine the state of the spacecraft as accurately as possible. We know that the state of the spacecraft is determined from the measurements, such as range or Doppler data. Given these launch errors, we will have to perform correction maneuvers to adjust the perturbed trajectory to go back to the nominal trajectory or an alternate trajectory that satisfies the mission requirements. Can we achieve this with a single correction maneuver? The answer is "No" for several reasons. First, the dynamical model is not perfect, even for our most realistic models. Secondly, the measurements have uncertainties. Thirdly, the spacecraft trajectory can only be estimated. Finally, each trajectory correction maneuver also has its own sources of execution errors.

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Asset Metadata

Creator Llanos de la Concha, Pedro J. (author)

Core Title Trajectory mission design and navigation for a space weather forecast

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Astronautical Engineering

Publication Date 05/04/2012

Defense Date 03/02/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag astronautics,maneuver analysis,navigation,OAI-PMH Harvest,orbit determination,space physics,Sun-Earth triangular points,trajectory mission design

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Erwin, Daniel A. (committee chair), Gruntman, Michael (committee member), Hintz, Gerald (committee member), Lo, Martin W. (committee member), Rhodes, Edward J., Jr. (committee member), Wang, Joseph (committee member)

Creator Email llanos@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-30564

Unique identifier UC11289845

Identifier usctheses-c3-30564 (legacy record id)

Legacy Identifier etd-Llanosdela-765.pdf

Dmrecord 30564

Document Type Dissertation

Rights Llanos de la Concha, Pedro J.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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