DOCSLIB.ORG

Orbital Mechanics Course Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Orbital Mechanics Course Notes 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.
Load more

Recommended publications
  • Appendix a Orbits
    Appendix A Orbits As discussed in the Introduction, a good ¯rst approximation for satellite motion is obtained by assuming the spacecraft is a point mass or spherical body moving in the gravitational ¯eld of a spherical planet. This leads to the classical two-body problem. Since we use the term body to refer to a spacecraft of ¯nite size (as in rigid body), it may be more appropriate to call this the two-particle problem, but I will use the term two-body problem in its classical sense. The basic elements of orbital dynamics are captured in Kepler's three laws which he published in the 17th century. His laws were for the orbital motion of the planets about the Sun, but are also applicable to the motion of satellites about planets. The three laws are: 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. The line joining the planet to the Sun sweeps out equal areas in equal times. 3. The square of the period of a planet is proportional to the cube of its mean distance to the sun. The ¯rst law applies to most spacecraft, but it is also possible for spacecraft to travel in parabolic and hyperbolic orbits, in which case the period is in¯nite and the 3rd law does not apply. However, the 2nd law applies to all two-body motion. Newton's 2nd law and his law of universal gravitation provide the tools for generalizing Kepler's laws to non-elliptical orbits, as well as for proving Kepler's laws.
    [Show full text]
  • Astrodynamics
    Politecnico di Torino SEEDS SpacE Exploration and Development Systems Astrodynamics II Edition 2006 - 07 - Ver. 2.0.1 Author: Guido Colasurdo Dipartimento di Energetica Teacher: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Two–Body Orbital Mechanics 1 1.1 BirthofAstrodynamics: Kepler’sLaws. ......... 1 1.2 Newton’sLawsofMotion ............................ ... 2 1.3 Newton’s Law of Universal Gravitation . ......... 3 1.4 The n–BodyProblem ................................. 4 1.5 Equation of Motion in the Two-Body Problem . ....... 5 1.6 PotentialEnergy ................................. ... 6 1.7 ConstantsoftheMotion . .. .. .. .. .. .. .. .. .... 7 1.8 TrajectoryEquation .............................. .... 8 1.9 ConicSections ................................... 8 1.10 Relating Energy and Semi-major Axis . ........ 9 2 Two-Dimensional Analysis of Motion 11 2.1 ReferenceFrames................................. 11 2.2 Velocity and acceleration components . ......... 12 2.3 First-Order Scalar Equations of Motion . ......... 12 2.4 PerifocalReferenceFrame . ...... 13 2.5 FlightPathAngle ................................. 14 2.6 EllipticalOrbits................................ ..... 15 2.6.1 Geometry of an Elliptical Orbit . ..... 15 2.6.2 Period of an Elliptical Orbit . ..... 16 2.7 Time–of–Flight on the Elliptical Orbit . .......... 16 2.8 Extensiontohyperbolaandparabola. ........ 18 2.9 Circular and Escape Velocity, Hyperbolic Excess Speed . .............. 18 2.10 CosmicVelocities
    [Show full text]
  • Maneuver Detection of Space Object for Space Surveillance
    MANEUVER DETECTION OF SPACE OBJECT FOR SPACE SURVEILLANCE Jian Huang*, Weidong Hu, Lefeng Zhang $756WDWH.H\/DE1DWLRQDO8QLYHUVLW\RI'HIHQVH7HFKQRORJ\&KDQJVKD+XQDQ3URYLQFH&KLQD (PDLOKXDQJMLDQ#QXGWHGXFQ ABSTRACT on the technique of a moving window curve fit. Holzinger & Scheeres presented an object correlation Maneuver detection is a very important task for and maneuver detection method using optimal control maintaining the catalog of the orbital objects and space performance metrics [5]. Kelecy & Jah focused on the situational awareness. This paper mainly focuses on the detection and reconstruction of single low thrust in- typical maneuver scenario where space objects only track maneuvers by using the orbit determination perform tangential orbital maneuvers during a relative strategies based on the batch least-squares and long gap. Particularly, only two classical and extended Kalman filter (EKF) [6]. However, these commonly applied orbital transition manners are methods are highly relevant to the presupposed considered, i.e. the twice tangential maneuvers at the maneuvering mode and a relative short observation gap. apogee and the perigee or one tangential maneuver at In this paper, to address the real observed data, we only an arbitrary time. Based on this, we preliminarily consider the common maneuvering modes and estimate the maneuvering mode and parameters by observation scenarios in practical. For an orbital analyzing the change of semi-major axis and maneuver, minimization of fuel consumption is eccentricity. Furthermore, if only one tangential essential because the weight of a payload that can be maneuver happened, we can formulate the estimation carried to the desired orbit depends on this problem of maneuvering parameters as a non-linear minimization.
    [Show full text]
  • Reduction of Saturn Orbit Insertion Impulse Using Deep-Space Low Thrust
    Reduction of Saturn Orbit Insertion Impulse using Deep-Space Low Thrust Elena Fantino ∗† Khalifa University of Science and Technology, Abu Dhabi, United Arab Emirates. Roberto Flores‡ International Center for Numerical Methods in Engineering, 08034 Barcelona, Spain. Jesús Peláez§ and Virginia Raposo-Pulido¶ Technical University of Madrid, 28040 Madrid, Spain. Orbit insertion at Saturn requires a large impulsive manoeuver due to the velocity difference between the spacecraft and the planet. This paper presents a strategy to reduce dramatically the hyperbolic excess speed at Saturn by means of deep-space electric propulsion. The inter- planetary trajectory includes a gravity assist at Jupiter, combined with low-thrust maneuvers. The thrust arc from Earth to Jupiter lowers the launch energy requirement, while an ad hoc steering law applied after the Jupiter flyby reduces the hyperbolic excess speed upon arrival at Saturn. This lowers the orbit insertion impulse to the point where capture is possible even with a gravity assist with Titan. The control-law algorithm, the benefits to the mass budget and the main technological aspects are presented and discussed. The simple steering law is compared with a trajectory optimizer to evaluate the quality of the results and possibilities for improvement. I. Introduction he giant planets have a special place in our quest for learning about the origins of our planetary system and Tour search for life, and robotic missions are essential tools for this scientific goal. Missions to the outer planets arXiv:2001.04357v1 [astro-ph.EP] 8 Jan 2020 have been prioritized by NASA and ESA, and this has resulted in important space projects for the exploration of the Jupiter system (NASA’s Europa Clipper [1] and ESA’s Jupiter Icy Moons Explorer [2]), and studies are underway to launch a follow-up of Cassini/Huygens called Titan Saturn System Mission (TSSM) [3], a joint ESA-NASA project.
    [Show full text]
  • Orbital Mechanics
    Orbital Mechanics These notes provide an alternative and elegant derivation of Kepler's three laws for the motion of two bodies resulting from their gravitational force on each other. Orbit Equation and Kepler I Consider the equation of motion of one of the particles (say, the one with mass m) with respect to the other (with mass M), i.e. the relative motion of m with respect to M: r r = −µ ; (1) r3 with µ given by µ = G(M + m): (2) Let h be the specific angular momentum (i.e. the angular momentum per unit mass) of m, h = r × r:_ (3) The × sign indicates the cross product. Taking the derivative of h with respect to time, t, we can write d (r × r_) = r_ × r_ + r × ¨r dt = 0 + 0 = 0 (4) The first term of the right hand side is zero for obvious reasons; the second term is zero because of Eqn. 1: the vectors r and ¨r are antiparallel. We conclude that h is a constant vector, and its magnitude, h, is constant as well. The vector h is perpendicular to both r and the velocity r_, hence to the plane defined by these two vectors. This plane is the orbital plane. Let us now carry out the cross product of ¨r, given by Eqn. 1, and h, and make use of the vector algebra identity A × (B × C) = (A · C)B − (A · B)C (5) to write µ ¨r × h = − (r · r_)r − r2r_ : (6) r3 { 2 { The r · r_ in this equation can be replaced by rr_ since r · r = r2; and after taking the time derivative of both sides, d d (r · r) = (r2); dt dt 2r · r_ = 2rr;_ r · r_ = rr:_ (7) Substituting Eqn.
    [Show full text]
  • Electric Propulsion System Scaling for Asteroid Capture-And-Return Missions
    Electric propulsion system scaling for asteroid capture-and-return missions Justin M. Little⇤ and Edgar Y. Choueiri† Electric Propulsion and Plasma Dynamics Laboratory, Princeton University, Princeton, NJ, 08544 The requirements for an electric propulsion system needed to maximize the return mass of asteroid capture-and-return (ACR) missions are investigated in detail. An analytical model is presented for the mission time and mass balance of an ACR mission based on the propellant requirements of each mission phase. Edelbaum’s approximation is used for the Earth-escape phase. The asteroid rendezvous and return phases of the mission are modeled as a low-thrust optimal control problem with a lunar assist. The numerical solution to this problem is used to derive scaling laws for the propellant requirements based on the maneuver time, asteroid orbit, and propulsion system parameters. Constraining the rendezvous and return phases by the synodic period of the target asteroid, a semi- empirical equation is obtained for the optimum specific impulse and power supply. It was found analytically that the optimum power supply is one such that the mass of the propulsion system and power supply are approximately equal to the total mass of propellant used during the entire mission. Finally, it is shown that ACR missions, in general, are optimized using propulsion systems capable of processing 100 kW – 1 MW of power with specific impulses in the range 5,000 – 10,000 s, and have the potential to return asteroids on the order of 103 104 tons. − Nomenclature
    [Show full text]
  • Exploding the Ellipse Arnold Good
    Exploding the Ellipse Arnold Good Mathematics Teacher, March 1999, Volume 92, Number 3, pp. 186–188 Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). More than 200 books, videos, software, posters, and research reports are available through NCTM’S publication program. Individual members receive a 20% reduction off the list price. For more information on membership in the NCTM, please call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright May 1991 by the National Council of Teachers of Mathematics. All rights reserved. Arnold Good, Framingham State College, Framingham, MA 01701, is experimenting with a new approach to teaching second-year calculus that stresses sequences and series over integration techniques. eaders are advised to proceed with caution. Those with a weak heart may wish to consult a physician first. What we are about to do is explode an ellipse. This Rrisky business is not often undertaken by the professional mathematician, whose polytechnic endeavors are usually limited to encounters with administrators. Ellipses of the standard form of x2 y2 1 5 1, a2 b2 where a > b, are not suitable for exploding because they just move out of view as they explode. Hence, before the ellipse explodes, we must secure it in the neighborhood of the origin by translating the left vertex to the origin and anchoring the left focus to a point on the x-axis.
    [Show full text]
  • 2.3 Conic Sections: Ellipse
    2.3 Conic Sections: Ellipse Ellipse: (locus definition) set of all points (x, y) in the plane such that the sum of each of the distances from F1 and F2 is d. Standard Form of an Ellipse: Horizontal Ellipse Vertical Ellipse 22 22 (xh−−) ( yk) (xh−−) ( yk) +=1 +=1 ab22 ba22 center = (hk, ) 2a = length of major axis 2b = length of minor axis c = distance from center to focus cab222=− c eccentricity e = ( 01<<e the closer to 0 the more circular) a 22 (xy+12) ( −) Ex. Graph +=1 925 Center: (−1, 2 ) Endpoints of Major Axis: (-1, 7) & ( -1, -3) Endpoints of Minor Axis: (-4, -2) & (2, 2) Foci: (-1, 6) & (-1, -2) Eccentricity: 4/5 Ex. Graph xyxy22+4224330−++= x2 + 4y2 − 2x + 24y + 33 = 0 x2 − 2x + 4y2 + 24y = −33 x2 − 2x + 4( y2 + 6y) = −33 x2 − 2x +12 + 4( y2 + 6y + 32 ) = −33+1+ 4(9) (x −1)2 + 4( y + 3)2 = 4 (x −1)2 + 4( y + 3)2 4 = 4 4 (x −1)2 ( y + 3)2 + = 1 4 1 Homework: In Exercises 1-8, graph the ellipse. Find the center, the lines that contain the major and minor axes, the vertices, the endpoints of the minor axis, the foci, and the eccentricity. x2 y2 x2 y2 1. + = 1 2. + = 1 225 16 36 49 (x − 4)2 ( y + 5)2 (x +11)2 ( y + 7)2 3. + = 1 4. + = 1 25 64 1 25 (x − 2)2 ( y − 7)2 (x +1)2 ( y + 9)2 5. + = 1 6. + = 1 14 7 16 81 (x + 8)2 ( y −1)2 (x − 6)2 ( y − 8)2 7.
    [Show full text]
  • Brief Introduction to Orbital Mechanics Page 1
    Brief Introduction to Orbital Mechanics Page 1 Brief Introduction to Orbital Mechanics We wish to work out the specifics of the orbital geometry of satellites. We begin by employing Newton's laws of motion to determine the orbital period of a satellite. The first equation of motion is F = ma (1) where m is mass, in kg, and a is acceleration, in m/s2. The Earth produces a gravitational field equal to Gm g(r) = − E r^ (2) r2 24 −11 2 2 where mE = 5:972 × 10 kg is the mass of the Earth and G = 6:674 × 10 N · m =kg is the gravitational constant. The gravitational force acting on the satellite is then equal to Gm mr^ Gm mr F = − E = − E : (3) in r2 r3 This force is inward (towards the Earth), and as such is defined as a centripetal force acting on the satellite. Since the product of mE and G is a constant, we can define µ = mEG = 3:986 × 1014 N · m2=kg which is known as Kepler's constant. Then, µmr F = − : (4) in r3 This force is illustrated in Figure 1(a). An equal and opposite force acts on the satellite called the centrifugal force, also shown. This force keeps the satellite moving in a circular path with linear speed, away from the axis of rotation. Hence we can define a centrifugal acceleration as the change in velocity produced by the satellite moving in a circular path with respect to time, which keeps the satellite moving in a circular path without falling into the centre.
    [Show full text]
  • Elliptical Orbits
    1 Ellipse-geometry 1.1 Parameterization • Functional characterization:(a: semi major axis, b ≤ a: semi minor axis) x2 y 2 b p + = 1 ⇐⇒ y(x) = · ± a2 − x2 (1) a b a • Parameterization in cartesian coordinates, which follows directly from Eq. (1): x a · cos t = with 0 ≤ t < 2π (2) y b · sin t – The origin (0, 0) is the center of the ellipse and the auxilliary circle with radius a. √ – The focal points are located at (±a · e, 0) with the eccentricity e = a2 − b2/a. • Parameterization in polar coordinates:(p: parameter, 0 ≤ < 1: eccentricity) p r(ϕ) = (3) 1 + e cos ϕ – The origin (0, 0) is the right focal point of the ellipse. – The major axis is given by 2a = r(0) − r(π), thus a = p/(1 − e2), the center is therefore at − pe/(1 − e2), 0. – ϕ = 0 corresponds to the periapsis (the point closest to the focal point; which is also called perigee/perihelion/periastron in case of an orbit around the Earth/sun/star). The relation between t and ϕ of the parameterizations in Eqs. (2) and (3) is the following: t r1 − e ϕ tan = · tan (4) 2 1 + e 2 1.2 Area of an elliptic sector As an ellipse is a circle with radius a scaled by a factor b/a in y-direction (Eq. 1), the area of an elliptic sector PFS (Fig. ??) is just this fraction of the area PFQ in the auxiliary circle. b t 2 1 APFS = · · πa − · ae · a sin t a 2π 2 (5) 1 = (t − e sin t) · a b 2 The area of the full ellipse (t = 2π) is then, of course, Aellipse = π a b (6) Figure 1: Ellipse and auxilliary circle.
    [Show full text]
  • Equation of Time — Problem in Astronomy M
    This paper was awarded in the II International Competition (1993/94) "First Step to Nobel Prize in Physics" and published in the competition proceedings (Acta Phys. Pol. A 88 Supplement, S-49 (1995)). The paper is reproduced here due to kind agreement of the Editorial Board of "Acta Physica Polonica A". EQUATION OF TIME | PROBLEM IN ASTRONOMY M. Muller¨ Gymnasium M¨unchenstein, Grellingerstrasse 5, 4142 M¨unchenstein, Switzerland Abstract The apparent solar motion is not uniform and the length of a solar day is not constant throughout a year. The difference between apparent solar time and mean (regular) solar time is called the equation of time. Two well-known features of our solar system lie at the basis of the periodic irregularities in the solar motion. The angular velocity of the earth relative to the sun varies periodically in the course of a year. The plane of the orbit of the earth is inclined with respect to the equatorial plane. Therefore, the angular velocity of the relative motion has to be projected from the ecliptic onto the equatorial plane before incorporating it into the measurement of time. The math- ematical expression of the projection factor for ecliptic angular velocities yields an oscillating function with two periods per year. The difference between the extreme values of the equation of time is about half an hour. The response of the equation of time to a variation of its key parameters is analyzed. In order to visualize factors contributing to the equation of time a model has been constructed which accounts for the elliptical orbit of the earth, the periodically changing angular velocity, and the inclined axis of the earth.
    [Show full text]
  • Moon-Earth-Sun: the Oldest Three-Body Problem
    Moon-Earth-Sun: The oldest three-body problem Martin C. Gutzwiller IBM Research Center, Yorktown Heights, New York 10598 The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler’s laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today’s many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem.
    [Show full text]