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MAE 589C Space Flight Mechanics a.k.a Astrodynamics5.55.65.75.85.9August 24, 2005 9:45 pm5.4.3 Oblateness Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.4 Odd-Zonal Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.5 Radiation pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Gauss' Form of the Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5.1 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5.2 N-Body Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Special Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Astronautics Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5-115-135-145-165-175-195-195-205-205-20Chapter 6 - Orbit Transfer and Powered Flight . . . . . . . . . . . . . . . . . . . . . . . . . .6-16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.2 Powered Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.2.1 Rocket equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.2.2 Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46.3 Impulsive Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-46.3.1 Single impulse transfers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56.3.2 Two-impulse transfer between coplanar circular orbits . . . . . . . . . . . . . . . . . 6-56.3.3 Hohmann transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-76.3.4 Bi-elliptic and bi-parabolic transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-96.3.5 Impulsive Transfers Between Inclined Orbits . . . . . . . . . . . . . . . . . . . . . . . . 6-96.3.6 Other Impulsive Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-116.4 Low Thrust Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-116.4.1 Constant Tangential Acceleration Escape Trajectories . . . . . . . . . . . . . . . . 6-116.4.2 Constant Circumferential Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-126.5 Lambert’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-136.5.1 Proof of Lambert’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-136.5.2 Euler’s equation for parabolic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-146.6 Interception and Rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-146.7 Midcourse Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-156.7.1 State transition matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-166.7.2 Constant time of arrival maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-186.7.3 Variable time of arrival maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-186.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-196.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19Chapter 7 - Special Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7-17.17.27.37.47.57.67.77.87.9Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Geosynchronous Orbits (GEO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Molniya Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Polar Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Sun-Synchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Low-Earth Orbit (LEO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Frozen Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Satellite Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Astronautics Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-v7-17-17-27-37-37-47-47-57-6

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:45 pm7.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6Appendix A - Biographical Bullets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1Appendix B - Vector and Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1Appendix C - Acronyms and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1- vi

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:42 pmList of FiguresFigureFigure 1-1.Figure 1-2.Figure 1-3.Figure 1-4.Figure 1-5.Figure 1-6.Figure 1-7.Figure 1-8.Figure 1-9.Figure 1-10.Figure 2-1.Figure 3-1.Figure 3-2.Figure 4-1.Figure 4-2.Figure 6-1.Figure 6-2.Figure 6-3.Figure 6-4.Figure 6-5.PageSpherical triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4Napier’s circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5Celestial coordinate systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6Transforming between celestial coordinate systems . . . . . . . . . . . . . . . . . . . . . . 1-7Geodetic and geocentric latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9Topocentric coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10Summary of time scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14UT1 - UTC for 1998. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14Pole location for 1998, x is along the Greenwich meridian, y is west. . . . . . . . . 1-15Local and Greenwich sidereal time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16Attraction of a homogeneous spherical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3Elliptical orbit geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4Orbit orientation elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5Zero velocity contours, µ 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4Equilateral triangle solution configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9Two impulse transfer between coplanar circular orbits . . . . . . . . . . . . . . . . . . . . 6-5Delta V requirements for transfer from rp 6700 km to geosynchronous orbit. . . 6-7Transfer time and total delta V for transfer from rp 6700 km to GEO. . . . . . . . . 6-7Bi-elliptic transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9Escape from LEO (h 300 km) using constant tangential thrust of 0.005go . . . . . . 6-12vii

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:42 pmList of TablesTableTable 1-1.Table 2-1.Table 3-1.Table 6-1.Table 7-1.PagePhysical Ephemerides for the Terrestrial Planets . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17Planetary Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9Two-body Problem Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-16Typical Rocket Motor Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2GPS Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6viii

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:42 pmChapter 1 - Coordinate Systems and Time Systems1.1 IntroductionTo develop an understanding and a basic description of any dynamical system, a physical modelof that system must be constructed which is consistent with observations. The fundamentals oforbital mechanics, as we know them today, have evolved over centuries and have continued torequire improvements in the dynamical models, coordinate systems and systems of time. Theunderlying theory for planetary motion has evolved from spheres rolling on spheres to precisionnumerical integration of the equations of motion based on general relativity. Time has evolvedfrom using the motion of the Sun to describe the fundamental unit of time to the current use ofatomic clocks to define the second. As observational accuracy has increased, models havegenerally increased in complexity to describe finer and finer detail.To apply the laws of motion to a dynamical system or orbital mechanics problem, appropriatecoordinate and time systems must first be selected. Most practical problems involve numerousreference frames and the transformations between them. For example, the equations of motion ofa satellite of Mars are normally integrated in a system where the equator of the Earth at thebeginning of year 2000 is the fundamental plane. But to include the Mars non-sphericalgravitational forces (Section 5.4) requires the satellite position in the current Mars equatorialsystem. Planetary ephemerides (Section 2.5) are usually referred to the ecliptic, so inclusion ofsolar or Jovian gravitational forces require transformations between the ecliptic and the equator.The correct development of these transformations is tedious and a prime candidate forimplementation errors.Likewise, there are usually numerous time systems in a problem. Spacecraft events might be timetagged by an on board clock or tagged with the universal time that the telemetry is received at thetracking station. In the latter case, tracking station clocks must be synchronized and the timerequired for the telemetry signal to travel from the s/c to the tracking station must be calculatedusing the s/c orbit. Depending on the precision desired, this time difference might require specialand general relativistic corrections. The independent variable for the equations of motion is calledephemeris time or dynamical time which is offset from universal time. By internationalagreement, atomic time is the basis of time and is obtained by averaging and correcting numerousatomic clocks around the world. Finally, the location of the zero or prime meridian and theequator are defined by averaging observations of specified Earth "fixed" stations. Theunderstanding of these and other coordinate systems and time systems is fundamental topracticing orbital mechanics.In this chapter only first order effects will be discussed. This book will also limit coverage to theclassical mechanics approach, i.e. special and general relativistic effects might be mentioned butwill not be included in any mathematical developments. Calculation for precise orbital mechanicsand spacecraft tracking must however include many of these neglected effects. The definitivereference for precise definitions of models and transformations is the Explanatory Supplement tothe Astronomical Almanac [Reference 1].1-1

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:42 pm1.2 Coordinate SystemsThe first issue that must be addressed in any dynamics problem is to define the relevantcoordinate systems. To specify the complete motion of a spacecraft, a coordinate system fixed inthe spacecraft at the center of mass is usually selected to specify orientation and a coordinatesystem fixed in some celestial body is used to specify the trajectory of the center of mass of thespacecraft. The interest here is primarily in the latter system.Coordinate systems are defined by specifying1. the location of the origin,2. the orientation of the fundamental plane, and3. the orientation of the fundamental direction or line in the fundamental plane.The origin is the (0,0,0) point in a rectangular coordinate system. The fundamental plane passesthrough the origin and is specified by the orientation of the positive normal vector, usually the zaxis. The fundamental direction is a directed line in the fundamental plane, usually specifyingthe x-axis. The origin, fundamental plane and fundamental line are defined either relative tosome previously defined coordinate system or in operational terms. The definitions are usuallyspecified in a seemingly clear statement like: “The origin is the center of mass of the Earth, thefundamental plane (x-y) is the Earth equator and the x-axis points to the vernal equinox.” Left asdetails are subtle issues like the fact that the center of mass of the Earth “moves” within the Earth,that the Earth is not a rigid body and the spin axis moves both in space and in the body, and thatthe vernal equinox is not a fixed direction. Some of these details are handled by specifying theepoch at which the orientation is defined, i.e. Earth mean equator of 2000.0 is frequently used.Further, it must be recognized that there is no fundamental inertial system to which all motion canbe referred. Any system fixed in a planet, the Sun, or at the center of mass of the solar system isundergoing acceleration due to gravitational attraction from bodies inside and outside the solarsystem. The extent to which these accelerations are included in the dynamical model depends onaccuracy requirements and is a decision left to the analyst.Like many other fields, conventions and definitions are often abused in the literature and thisabuse will continue in this text. So "the equator" is jargon for the more precise statement "theplane through the center of mass with positive normal along the spin axis." Likewise, anglesshould always be defined as an angular rotation about a specified axis or as the angle between twovectors. The angle between a vector and a plane (e.g. latitude) is to be interpreted as thecomplement of the angle between the vector and the positive normal to the plane. The anglebetween two planes is defined as the angle between the positive normals to each plane. The moreprecise definitions often offer computational convenience. For example, after checking theorthogonality of the direction cosines of the positive unit normal (usually z axis) and thedirection cosines of the fundamental direction in the plane (usually x), the direction cosines ofthe y axis can be obtained by a vector cross product. Thus, the entire transformation or rotationmatrix is defined by orthogonal x and z unit vectors.Exercise 1-1. Given vectors a (3,4,-6) and b (1,-3,8) in the (x,y,z) system. Define the (ξ,η,ζ)system such that the fundamental plane (normal to ζ) contains both a and b, the ζ axis such that aright hand rotation of less than π takes a into b, and the fundamental direction (ξ axis) is along a.1-2

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:42 pmDevelop, symbolically and numerically, the 3 by 3 transformation matrix Φ from the (ξ,η,ζ)asystem to the (x,y,z) system. For example, the first column of Φ is ----- a0.38410.5151– 0.7682Common origins for coordinate systems of interest in astrodynamics include:1. Topocentric: at an observer fixed to the surface of a planet,2. Heliocentric, Geocentric, Areocentric, Selenocentric, etc.: at the center of mass of theSun, Earth, Mars, Moon, etc.3. Barycentric: at the center of mass of a system of bodies, i.e. the solar system, EarthMoon system, etc.Astronomical observations were traditionally referred to topocentric coordinates since the localvertical and the direction of the spin axis could be readily measured at the site. For dynamicsproblems, topocentric coordinates might be used for calculating the trajectory of a baseball or alaunch vehicle. For the former case, the rotation of the Earth and the variation in gravity withaltitude can be ignored because these effects are small compared the errors introduced by theuncertainty in the aerodynamic forces acting on a spinning, rough sphere. For the latter case, theseeffects cannot be ignored; but, gravitational attraction of the Sun and Moon might be ignored forapproximate launch trajectory calculations. The decision is left to the analyst and is usually baseon "back of the envelope" calculations of the order of magnitude of the effect compared to thedesired accuracy.Heliocentric, areocentric, etc. coordinates are traditionally used for calculating and specifying theorbits of both natural and artificial satellites when the major gravitational attraction is due to thebody at the origin. During calculation of lunar or interplanetary trajectories, the origin is shiftedfrom one massive body to another as the relative gravitational importance changes; however, thefundamental plane is often kept as the Earth equator at some epoch. Often in what follows onlyEarth geocentric systems are discussed, but the definitions and descriptions generally apply toplanets and moons. Geocentric systems are either terrestrial or celestial. Terrestrial systems arefixed to the rotating Earth and can be topocentric, geocentric, or geodetic. Celestial systems haveeither the equator or the ecliptic as the fundamental plane and the vernal equinox as thefundamental direction.1.2.1Spherical trigonometryTransformations of position and velocity vectors between coordinate systems are represented inmatrix notation and developed by vector outer and inner products as mentioned above. However,the understanding of the basic concepts of spherical trigonometry is also a necessity when dealingwith orbital mechanics. It is convenient to introduce the concept of the celestial sphere. Thecelestial sphere is a spherical surface of infinite radius. The location of the center of the celestialsphere is therefore unimportant. For example, one can think of the center as being simultaneouslyat the center of the Earth and Sun and observer. Any unit vector or direction can thus berepresented as a point on the sphere and vice versa. For example, the Earth to Sun line and Sun to1-3

MAE 589C Space Flight Mechanics a.k.a AstrodynamicsAugust 24, 2005 9:42 pmEarth lines could be represented by two points 180 degrees apart. Two distinct points on thesphere can be connected by a great circle formed by the intersection on the sphere of the planeformed by the two points and the center of the sphere. If the points are not coincident or 180 apart, the great circle is unique.The distance or length between two points onthe surface is the central angle subt

practicing orbital mechanics. In this chapter only first order effects will be discussed. This book will also limit coverage to the classical mechanics approach, i.e. special and general relativistic effects might be mentioned but will not be included in any mathematical developments. Calculation for precise orbital mechanicsFile Size: 2MB