Orbital Mechanics Course Notes David J. Westpfahl Professor of Astrophysics, New Mexico Institute of Mining and Technology March 31, 2011 2 These are notes for a course in orbital mechanics catalogued as Aerospace Engineering 313 at New Mexico Tech and Aerospace Engineering 362 at New Mexico State University. This course uses the text “Fundamentals of Astrodynamics” by R.R. Bate, D. D. Muller, and J. E. White, published by Dover Publications, New York, copyright 1971. The notes do not follow the book exclusively. Additional material is included when I believe that it is needed for clarity, understanding, historical perspective, or personal whim. We will cover the material recommended by the authors for a one-semester course: all of Chapter 1, sections 2.1 to 2.7 and 2.13 to 2.15 of Chapter 2, all of Chapter 3, sections 4.1 to 4.5 of Chapter 4, and as much of Chapters 6, 7, and 8 as time allows. Purpose The purpose of this course is to provide an introduction to orbital me- chanics. Students who complete the course successfully will be prepared to participate in basic space mission planning. By basic mission planning I mean the planning done with closed-form calculations and a calculator. Stu- dents will have to master additional material on numerical orbit calculation before they will be able to participate in detailed mission planning. There is a lot of unfamiliar material to be mastered in this course. This is one field of human endeavor where engineering meets astronomy and ce- lestial mechanics, two fields not usually included in an engineering curricu- lum. Much of the material that is familiar to students of those disciplines will be unfamiliar to engineers. Students are probably already familiar with Newton’s Laws and Newtonian gravity. These will be used to develop the particular applications needed to describe orbits and orbital maneuvers. Space missions are expensive and risky, especially if people or living ani- mals are sent into space. Thus, it is important to check and recheck calcula- tions and assumptions. Computer programs are subject to the imperfections of the humans who write them. This, it becomes necessary to develop phys- ical insight into orbit calculations to have a sense of when a programming bug is leading to inaccurate answers. We will spend time developing physical intuition and understanding what it is. Notation Well I remember being a student and being frustrated by notation used by printers of textbooks that was impossible to write by hand at note-taking speed, and not used by the professor, anyway. Thus, I have tried to use 3 notation that is consistent with the text, but also within my abilities to write on the board and within the abilities of students to write in their notes. Some examples follow. A scalar is written as an ordinary math symbol, as in a. A vector is written with an arrow above, as in ~r. A unit vector is written with a hat, as inr ˆ. If the unit vector is a basis vector of a coordinate set it’s symbol is usually capitalized, as in Iˆ. A matrix is written in a boldfaced capital letter and covered by a tilde, as in D˜ . This is a compromise. The tilde is easy enough to write in notes or on the board, but boldface is not. The boldface is used to make the matrix instantly recognizable in the notes, at the cost of inconsistency. 1 The inverse of the same matrix is written as D˜ − . The transpose of a matrix is written in boldface with a tilde and a trailing superscript capital T, as in D˜ T. A triangle with vertices A, B, and C is named ABC. Its line segments are named AB, BC, and CA. Order does not matter, so AB and BA describe the same line segment. The angle between segments AB and BC is labeled ABC[ . These notes were made using the LaTeX math symbols of AMS TeX. Anyone who has posted hundreds of pages of LaTeX notes has probably discovered hundreds of typos and left undiscovered scores of others. I am no exception. If you discover typos please report them to me by email at [email protected]. D. J. W. Albuquerque, NM January, 2011 4 Contents 1 Two-Body Orbital Mechanics 1 1.1 Kepler’sLaws ........................... 1 1.2 Newton’sLaws .......................... 2 1.2.1 AnExample........................ 4 1.2.2 History........................... 8 1.3 The Equation of Motion for Two Orbiting Bodies . 9 1.3.1 ChoosingaGoal ..................... 9 1.3.2 Making Things Complicated . 10 1.3.3 Making Things Simple Again . 10 1.4 PartialSolutionsoftheEquationofMotion . 14 1.4.1 ConstantsoftheMotion . 14 1.4.2 TheTrajectoryEquation . 17 1.5 ConicSections........................... 18 1.5.1 PolarEquationsofConics . 19 1.6 PropertiesofConic-SectionOrbits. 24 1.6.1 Relating the Constants of the Motion to the Geometry oftheOrbit ........................ 26 1.6.2 Some Important Properties of Individual Conic Orbits 27 1.6.3 Relationships Among the Conics . 30 1.6.4 What is B~ ? ........................ 32 1.6.5 The Eccentricity Vector . 34 1.7 CanonicalUnits.......................... 34 2 Orbit Determination from Observations 39 2.1 CoordinateSystems. 39 2.1.1 Heliocentric-Ecliptic Coordinates . 40 2.1.2 Geocentric-Equatorial Coordinates . 41 2.1.3 Right Ascension-Declination Coordinates . 41 5 6 CONTENTS 2.1.4 PerifocalCoordinates. 42 2.2 ClassicalOrbitalElements . 42 2.3 Determining the Orbital Elements from ~r and ~v ........ 44 2.3.1 Three Fundamental Vectors . 44 2.3.2 Solving for the Orbital Elements . 45 2.4 Determining ~r and ~v fromtheOrbitalElements . 49 2.5 CoordinateTransformations . 51 2.5.1 Transformations Change the Basis Vectors . 51 2.5.2 SimpleTransformations . 52 2.5.3 More Challenging Transformations . 55 2.6 MechanicsontheRotatingEarth . 63 2.6.1 AnIntroductiontoTime . 63 2.6.2 Position, Velocity, and Acceleration . 67 2.7 TheEllipsoidalEarth. 72 2.7.1 The Measurement of Latitude . 73 2.7.2 StationCoordinates. 73 2.8 TheGroundTrackofaSatellite . 85 2.8.1 Launch Site, Launch Azimuth, and Orbital Inclination 85 3 Real Orbits and Orbital Maneuvers 89 3.1 SomeTypesofOrbits. .. .. 89 3.1.1 Classification of Earth Orbit by Altitude . 89 3.1.2 Classification of Orbits by Inclination . 91 3.1.3 Sun-SynchronousOrbits . 92 3.1.4 SpecialOrbits. .. .. 92 3.2 TheEarth’sEquatorialBulge . 93 3.3 In-PlaneOrbitChanges . 94 3.3.1 Launching a Satellite and Adjusting its Orbit . 94 3.3.2 HohmannTransfer . 96 3.3.3 GeneralCoplanarTransfer . 100 3.4 Bi-ellipticTransfer . .106 3.5 Out-Of-PlaneOrbitChanges. 111 4 ~r and ~v as Functions of Time 113 4.1 WhatweHaveDone .......................113 4.2 Elliptical Time of Flight as a Function of E ..........114 4.2.1 Two Approaches: Geometric and Analytical . 114 4.2.2 Why is This Integral so Difficult? . 116 CONTENTS 7 4.2.3 Kepler’s Geometric Method - Developing Kepler’s Equa- tion ............................117 4.2.4 Time of Flight Between Arbitrary Points . 121 4.2.5 AnalyticalMethod . .123 4.3 Parabolic Time of Flight as a Function of D ..........125 4.4 Hyperbolic Time of Flight as a Function of F .........127 4.4.1 AreaasaMeasureofAngle . .127 4.4.2 HyperbolicTimeofFlight . 131 5 Orbit Determination from Two Positions and Time 137 6 Ballistic Missile Trajectories 139 6.1 History...............................139 6.2 Purpose ..............................140 6.3 Basics of Ballistic Missiles . 140 6.3.1 The Non-dimensional Parameter, Q . 141 6.3.2 TheFree-FlightRangeEquation. 141 6.3.3 TheFlight-PathAngleequation . 142 7 Lunar Trajectories 145 7.1 SphereofInfluence . .145 7.2 The Patched Conic Approximation . 146 8 Interplanetary Trajectories 147 8 CONTENTS Chapter 1 Two-Body Orbital Mechanics A story has to start somewhere. Our story starts with Kepler’s Laws. 1.1 Kepler’s Laws Following our text, Fundamentals of Astrodynamics by Bate, Mueller, and White, we start with Kepler’s Laws of Planetary Motion, which are general- izations derived from the planetary position data of Tycho Brahe. According to our text, Kepler published the first two laws in 1609, and the third in 1619. Kepler’s Laws Kepler’s First Law - The orbit of each planet is an ellipse with the Sun at one focus. Kepler’s Second Law - The line joining any planet to the Sun sweeps out equal areas in equal times. Kepler’s Third Law - The squares of the periods of any two planets are in the same proportion as the cubes of their mean distances from the Sun. These laws explain what the orbits are. Their shapes are ellipses, and the local, instantaneous speeds within the ellipses change so that the second and third laws are true. These laws do not explain why the shapes and speeds of the orbits behave in this way. Our first goal will be to apply the work of Newton to understand why. 1 2 CHAPTER 1. TWO-BODY ORBITAL MECHANICS 1.2 Newton’s Laws Newton did his fundamental work in the 1660’s, but did not publish until the Principia appeared in 1687. Without the help of Edmund Halley the Principia might never have seen the light of day. The Principia contains Newton’s three Laws of Motion and the Law of Universal Gravitation (more on that later). You will find variations in the statements of the Laws of Motion because the Principia is in Latin, and there is more than one English translation. Newton’s Laws of Motion Newton’s First Law - A body at rest stays at rest, and a body in motion stays in uniform, straight-line motion, unless acted upon by a net force. Newton’s Second Law - The time rate of change of momentum is propor- tional to the impressed force and is in the same direction as that force, or d~p ~ dt = Ftot.