Computational Space Flight Mechanics

The mechanics of space flight is an old discipline. Its topic originally was the motion of planets, moons and other celestial bodies in gravitational fields. Kepler's (1571 - 1630) observations and measurements have led to probably the first mathematical description of planet's motion. Newton (1642 - 1727) gave then, with the development of his principles of mechanics, the physical explanation of these motions. Since then man has started in the second half of the 20th century to capture physically the Space in the sense that he did develop artificial celestial bodies, which he brought into Earth's orbits, like satellites or space stations, or which he did send to planets or moons of our planetary system, like probes, or by which people were brought to the moon and back, like capsules. Further he developed an advanced space transportation system, the U.S. Space Shuttle Orbiter, which is the only winged space vehicle ever in operation. Today it is no problem to solve the governing equations in the most general form using discrete numerical methods. The numerical approximation schemes, the computer power and the modern storage capacity are in such an advanced state, that solutions with high degree of accuracy can be obtained in a few seconds. Therefore the general practice in this book is to provide numerical solutions for all discussed topics and problems. This could be the orbit determination by the orbital elements, Lagrange's perturbation equations for disturbed Earth's orbits, the flight of a mass point in flight path coordinates (three degree of freedom), and the flight of a controlled space vehicle in body fixed coordinates (six degree of freedom). This book has been written not only for graduate and doctoral students but also for non-specialists who may be interested in this subject or concerned with space flight mechanics. Cover 1 Computational Space Flight Mechanics 4 364213582X 5 Preface 6 Acknowledgements 8 Table of Contents 10 1 Introduction 16 References 19 2 Coordinate Transformations 22 2.1 Basic Rotational Transformations 23 2.2 Time Derivative of Vectors in Moving Frames 26 2.2.1 The Velocity Vector 26 2.2.2 The Acceleration Vector 29 2.3 The Angular Velocity in a Body Frame: Euler Angles 32 2.4 Problems 37 References 38 3 Transformations between Often Used Coordinate Systems 40 3.1 Transformation from Geodetic to Body Frame 40 3.2 Transformation from Air Path to Body Frame 41 3.3 Transformation from Geodetic to Flight Path Frame 42 3.4 Transformation from Planetocentric to Orbital Frame 43 3.5 Problems 45 References 46 4 Kepler’s Laws of Planetary Motion and Newton’s Celestial Mechanics 48 4.1 Kepler’s 1. Law 48 4.2 Kepler’s 2. Law 49 4.3 Kepler’s 3. Law 51 4.4 Newton’s Celestial Mechanics 52 4.5 Problems 57 References 58 5 The Two-Body Problem 60 5.1 The Equation of Motion 60 5.2 The Energy Conservation 62 5.3 The Angular Momentum Conservation 63 5.4 The Orbit Equation 64 5.5 The Various Orbits 66 5.5.1 The Eccentricity e < 1 66 5.5.2 The Eccentricity e = 1 68 5.6 Test Cases for the Three Classes of Orbits 69 5.7 Time Dependency of the Orbital Variables r and ? and Kepler’s Equation 71 5.7.1 The Elliptical Orbit 74 5.7.2 Solutions of the Elliptical Test Case 1 77 5.7.3 The Hyperbolic Orbit 79 5.7.4 Solutions of the Hyperbolic Test Case 3 81 5.8 The Classical Orbital Elements 82 5.8.1 Derivation of Relations 82 5.8.2 Sample Calculations of Test Case 1 Using Orbital Elements 86 5.8.3 Sample Calculations of Test Case 1 Using the General Equations of Planetary Flight 88 5.9 Perturbations of Orbital Dynamics 90 5.9.1 Lagrange’s Planet Equations 91 5.9.2 Numerical Solutions of Lagrange’s Planet Equations 95 5.9.3 Numerical Solution of the General Equations of Planetary Flight for an Aspherical Earth 102 5.10 Problems 106 References 107 6 General Equations for Planetary Flight 108 6.1 Equations of Translational Motion 108 6.1.1 Flight without Bank Angle 108 6.1.2 Flight with Bank Angle 116 6.1.3 Equations Including Side Forces 118 6.1.4 Flight with Propulsion Force 120 6.1.5 Orbital Flight Around an Aspherical Earth 121 6.2 Equations of Rotational Motion 124 6.3 Set of Equations for Six Degree of Freedom Simulations 129 6.4 Problems 133 References 134 7 A Resume of the Aerothermodynamics of Space Flight Vehicles 136 7.1 Conventions for Aerothermodynamic Data 136 7.2 Flow Regimes and Physical Phenomena 138 7.3 Aerothermodynamic Data of the X-38 Vehicle 141 7.3.1 Data of Longitudinal Motion 143 7.3.2 Data of Lateral Motion 147 7.4 Problems 150 References 150 8 Three and Six Degree of Freedom Trajectory Simulations 152 8.1 Three Degree of Freedom Simulation for a Winged Space Vehicle 152 8.2 Three Degree of Freedom Simulation for a Non-Winged Space Vehicle 156 8.3 Six Degree of Freedom Simulations for a Winged Space Vehicle 160 8.3.1 Flight with Statically Stable Longitudinal Motion 160 8.3.2 Flight with Statically Stable Longitudinal and Yaw Motion 165 8.4 Problems 167 References 167 9 Numerical Applications of the General Equations for Planetary Flight 168 9.1 Flight in Geostationary Orbit 169 9.2 Flight in Low Earth Orbit 171 9.2.1 Circular Equatorial Orbit (Inclination Angle φ = 0) 171 9.2.2 Circular Orbit with Inclination Angle φ \neq 0 173 9.3 Elliptical Orbits 175 9.3.1 Elliptical Orbit without Aerodynamic Forces 177 9.3.2 Elliptical Orbit with Aerodynamic Forces 182 9.3.3 Elliptical Orbits with Flight in Other Directions Than West-East 187 9.4 Re-entry Flight 189 9.4.1 Deceleration of Space Vehicles and g-Loads 195 9.5 Planetary Flight and Aerocapturing Mission 201 9.6 Artillery Ballistics 208 9.6.1 Projectile’s Flight without Aerodynamic Drag 208 9.6.2 Projectile’s Flight with Aerodynamic Drag 210 9.6.3 The Principle Equation of Ballistics 214 9.6.4 Approximate Solutions of the Principle Equation of Ballistics 218 9.6.5 Shots of Shells towards the Four Cardinal Points 221 9.7 Another Illustrating Case 223 9.8 Conclusion 227 9.9 Problems 228 References 229 10 The Earth Atmosphere 230 References 236 11 Solution of Problems 238 11.1 Problems of Chapter 2 238 11.2 Problems of Chapter 3 239 11.3 Problems of Chapter 4 240 11.4 Problems of Chapter 5 243 11.5 Problems of Chapter 6 244 11.6 Problems of Chapter 7 246 11.7 Problems of Chapter 8 247 11.8 Problems of Chapter 9 247 Reference 248 Appendix A Our Planetary System 250 A.1 The First Four Planets in the Solar System 250 A.2 The First Six Planets in the Solar System 251 A.3 The Entire Solar System 252 References 253 Appendix B FORTRAN Codes 254 B.1 General Equations for Planetary Flight - Three Degree of Freedom Simulation 254 B.2 Orbit Determination with Orbital Elements 260 B.3 Lagrange’s Planet Equations 265 References 274 Appendix C MATLAB Codes 276 C.1 Kepler’s Equation for Elliptical Orbits 276 C.2 Area Approach for Elliptical Orbits 278 C.3 Area Approach for Hyperbolic Orbits 281 C.4 Six Degree of Freedom Simulation 283 References 297 Appendix D Constants, Relations, Units and Conversions 298 D.1 Constants and Relations 298 D.2 Units and Conversions 299 References 301 Appendix E Symbols 302 E.1 Latin Letters 302 E.2 Greek Letters 305 E.3 Indices 306 E.3.1 Upper Indices 306 E.3.2 Lower Indices 306 E.4 Other Symbols 307 Appendix F Glossary, Abbreviations, Acronyms 308 F.1 Glossary 308 F.2 Abbreviations, Acronyms 309 Name Index 310 Subject Index 312 364213582X,9783642135828 Springer Themechanicsofspace?ightisan olddiscipline.Itstopicoriginallywasthemotion of planets, moons and other celestial bodies in gravitational ?elds. Kepler’s (1571 - 1630) observations and measurements have led to probably the ?rst mathematical description of planet’s motion. Newton (1642 - 1727) gave then, with the devel- ment of his principles of mechanics, the physical explanation of these motions. Since then man has started in the second half of the 20th centuryto capture ph- ically the Space in the sense that he did develop arti?cial celestial bodies, which he brought into Earth’s orbits, like satellites or space stations, or which he did send to planets or moons of our planetary system, like probes, or by which p- ple were brought to the moon and back, like capsules. Further he developed an advanced space transportation system, the U.S. Space Shuttle Orbiter, which is the only winged space vehicle ever in operation. In the last two and a half decades there were several activities in the world in order to succeed the U.S. Orbiter, like the HERMES project in Europe, the HOPE project in Japan, the X-33, X-34 and X-37 studies and demonstrators in the United States and the joint U.S. - European project X-38. However, all these projects were cancelled. The motion of these vehicles can be described by Newton’s equation of motion. The mechanics of space flight is an old discipline. Its topic originally was the motion of planets, moons and other celestial bodies in gravitational fields. Kepler's (1571-1630) observations and measurements have led to probably the first mathematical description of planet's motion. Newton (1642-1727) gave then, with the development of his principles of mechanics, the physicalexplanation of these motions. Since then man has started in the second half of the 10th century to capture physically the Space in the sense that he did develop artificial celestial bodies, which he brought into Earth's orbits, like satellites or space stations, or which he did send to planets or moons of our planetary system, like probes, or by which people were brought to the moon and back, like capsules. Further he developed an advanced space transportation system, the U.S. Space Shuttle Orbiter, which is the only winged space vehicle ever in operation. Today it is no problem to solve the governing Equations in the most general form using discrete numerical methods. The numerical approximation schemes, the computer power and the modern storage capacity are in such an advanced state, that solutions with high degree of accuracy can be obtained in a few seconds. Therefore the general practice in this book is to provide numerical solutions for all discussed topics and problems. This could be the orbit determination by the orbital elements, Lagrange's perturbation equations for disturbed Earth's orbits, the flight of a mass point in flight path coordinates (three degree of freedom), and the flight of a controlled space vehicle in body fixed coordinates (six degree of freedom). This book has been written not only for graduate and doctoral students but also for nonspecialists who may be interested in this subject or concerned with space flight mechanics. --Book Jacket Front Matter....Pages - Introduction....Pages 1-5 Coordinate Transformations....Pages 7-23 Transformations between Often Used Coordinate Systems....Pages 25-31 Kepler’s Laws of Planetary Motion and Newton’s Celestial Mechanics....Pages 33-43 The Two-Body Problem....Pages 45-92 General Equations for Planetary Flight....Pages 93-120 A Resumé of the Aerothermodynamics of Space Flight Vehicles....Pages 121-136 Three and Six Degree of Freedom Trajectory Simulations....Pages 137-152 Numerical Applications of the General Equations for Planetary Flight....Pages 153-214 The Earth Atmosphere....Pages 215-221 Solution of Problems....Pages 223-233 Our Planetary System....Pages 235-238 FORTRAN Codes....Pages 239-259 MATLAB Codes....Pages 261-282 Constants, Relations, Units and Conversions....Pages 283-286 Back Matter....Pages -