Chapter 6: the Two Body Problem
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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 -
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. -
The Celestial Mechanics of Newton
GENERAL I ARTICLE The Celestial Mechanics of Newton Dipankar Bhattacharya Newton's law of universal gravitation laid the physical foundation of celestial mechanics. This article reviews the steps towards the law of gravi tation, and highlights some applications to celes tial mechanics found in Newton's Principia. 1. Introduction Newton's Principia consists of three books; the third Dipankar Bhattacharya is at the Astrophysics Group dealing with the The System of the World puts forth of the Raman Research Newton's views on celestial mechanics. This third book Institute. His research is indeed the heart of Newton's "natural philosophy" interests cover all types of which draws heavily on the mathematical results derived cosmic explosions and in the first two books. Here he systematises his math their remnants. ematical findings and confronts them against a variety of observed phenomena culminating in a powerful and compelling development of the universal law of gravita tion. Newton lived in an era of exciting developments in Nat ural Philosophy. Some three decades before his birth J 0- hannes Kepler had announced his first two laws of plan etary motion (AD 1609), to be followed by the third law after a decade (AD 1619). These were empirical laws derived from accurate astronomical observations, and stirred the imagination of philosophers regarding their underlying cause. Mechanics of terrestrial bodies was also being developed around this time. Galileo's experiments were conducted in the early 17th century leading to the discovery of the Keywords laws of free fall and projectile motion. Galileo's Dialogue Celestial mechanics, astronomy, about the system of the world was published in 1632. -
TOPICS in CELESTIAL MECHANICS 1. the Newtonian N-Body Problem
TOPICS IN CELESTIAL MECHANICS RICHARD MOECKEL 1. The Newtonian n-body Problem Celestial mechanics can be defined as the study of the solution of Newton's differ- ential equations formulated by Isaac Newton in 1686 in his Philosophiae Naturalis Principia Mathematica. The setting for celestial mechanics is three-dimensional space: 3 R = fq = (x; y; z): x; y; z 2 Rg with the Euclidean norm: p jqj = x2 + y2 + z2: A point particle is characterized by a position q 2 R3 and a mass m 2 R+. A motion of such a particle is described by a curve q(t) where t runs over some interval in R; the mass is assumed to be constant. Some remarks will be made below about why it is reasonable to model a celestial body by a point particle. For every motion of a point particle one can define: velocity: v(t) =q _(t) momentum: p(t) = mv(t): Newton formulated the following laws of motion: Lex.I. Corpus omne perservare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare 1 Lex.II. Mutationem motus proportionem esse vi motrici impressae et fieri secundem lineam qua vis illa imprimitur. 2 Lex.III Actioni contrarium semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. 3 The first law is statement of the principle of inertia. The second law asserts the existence of a force function F : R4 ! R3 such that: p_ = F (q; t) or mq¨ = F (q; t): In celestial mechanics, the dependence of F (q; t) on t is usually indirect; the force on one body depends on the positions of the other massive bodies which in turn depend on t. -
Perturbation Theory in Celestial Mechanics
Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion. -
The Celestial Mechanics Approach: Theoretical Foundations
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library J Geod (2010) 84:605–624 DOI 10.1007/s00190-010-0401-7 ORIGINAL ARTICLE The celestial mechanics approach: theoretical foundations Gerhard Beutler · Adrian Jäggi · Leoš Mervart · Ulrich Meyer Received: 31 October 2009 / Accepted: 29 July 2010 / Published online: 24 August 2010 © Springer-Verlag 2010 Abstract Gravity field determination using the measure- of the CMA, in particular to the GRACE mission, may be ments of Global Positioning receivers onboard low Earth found in Beutler et al. (2010) and Jäggi et al. (2010b). orbiters and inter-satellite measurements in a constellation of The determination of the Earth’s global gravity field using satellites is a generalized orbit determination problem involv- the data of space missions is nowadays either based on ing all satellites of the constellation. The celestial mechanics approach (CMA) is comprehensive in the sense that it encom- 1. the observations of spaceborne Global Positioning (GPS) passes many different methods currently in use, in particular receivers onboard low Earth orbiters (LEOs) (see so-called short-arc methods, reduced-dynamic methods, and Reigber et al. 2004), pure dynamic methods. The method is very flexible because 2. or precise inter-satellite distance monitoring of a close the actual solution type may be selected just prior to the com- satellite constellation using microwave links (combined bination of the satellite-, arc- and technique-specific normal with the measurements of the GPS receivers on all space- equation systems. It is thus possible to generate ensembles crafts involved) (see Tapley et al. -
Coordinate Systems in Geodesy
COORDINATE SYSTEMS IN GEODESY E. J. KRAKIWSKY D. E. WELLS May 1971 TECHNICALLECTURE NOTES REPORT NO.NO. 21716 COORDINATE SYSTElVIS IN GEODESY E.J. Krakiwsky D.E. \Vells Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton, N .B. Canada E3B 5A3 May 1971 Latest Reprinting January 1998 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. TABLE OF CONTENTS page LIST OF ILLUSTRATIONS iv LIST OF TABLES . vi l. INTRODUCTION l 1.1 Poles~ Planes and -~es 4 1.2 Universal and Sidereal Time 6 1.3 Coordinate Systems in Geodesy . 7 2. TERRESTRIAL COORDINATE SYSTEMS 9 2.1 Terrestrial Geocentric Systems • . 9 2.1.1 Polar Motion and Irregular Rotation of the Earth • . • • . • • • • . 10 2.1.2 Average and Instantaneous Terrestrial Systems • 12 2.1. 3 Geodetic Systems • • • • • • • • • • . 1 17 2.2 Relationship between Cartesian and Curvilinear Coordinates • • • • • • • . • • 19 2.2.1 Cartesian and Curvilinear Coordinates of a Point on the Reference Ellipsoid • • • • • 19 2.2.2 The Position Vector in Terms of the Geodetic Latitude • • • • • • • • • • • • • • • • • • • 22 2.2.3 Th~ Position Vector in Terms of the Geocentric and Reduced Latitudes . • • • • • • • • • • • 27 2.2.4 Relationships between Geodetic, Geocentric and Reduced Latitudes • . • • • • • • • • • • 28 2.2.5 The Position Vector of a Point Above the Reference Ellipsoid . • • . • • • • • • . .• 28 2.2.6 Transformation from Average Terrestrial Cartesian to Geodetic Coordinates • 31 2.3 Geodetic Datums 33 2.3.1 Datum Position Parameters . -
Classical and Celestial Mechanics, the Recife Lectures, Hildeberto Cabral and Florin Diacu (Editors), Princeton Univ
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, Pages 121{125 S 0273-0979(03)00997-2 Article electronically published on October 2, 2003 Classical and celestial mechanics, the Recife lectures, Hildeberto Cabral and Florin Diacu (Editors), Princeton Univ. Press, Princeton, NJ, 2002, xviii+385 pp., $49.50, ISBN 0-691-05022-8 The photographs of Recife which are scattered throughout this book reveal an eclectic mix of old colonial buildings and sleek, modern towers. A clear hot sun shines alike on sixteenth century churches and glistening yachts riding the tides of the harbor. The city of 1.5 million inhabitants is the capital of the Pernambuco state in northeastern Brazil and home of the Federal University of Pernambuco, where the lectures which comprise the body of the book were delivered. Each lecturer presented a focused mini-course on some aspect of contemporary classical mechanics research at a level accessible to graduate students and later provided a written version for the book. The lectures are as varied as their authors. Taken together they constitute an album of snapshots of an old but beautiful subject. Perhaps the mathematical study of mechanics should also be dated to the six- teenth century, when Galileo discovered the principle of inertia and the laws gov- erning the motion of falling bodies. It took the genius of Newton to provide a mathematical formulation of general principles of mechanics valid for systems as diverse as spinning tops, tidal waves and planets. Subsequently, the attempt to work out the consequences of these principles in specific examples served as a catalyst for the development of the modern theory of differential equations and dynamical systems. -
A New Celestial Mechanics Dynamics of Accelerated Systems
Journal of Applied Mathematics and Physics, 2019, 7, 1732-1754 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 A New Celestial Mechanics Dynamics of Accelerated Systems Gabriel Barceló Dinamica Fundación, Madrid, Spain How to cite this paper: Barceló, G. (2019) Abstract A New Celestial Mechanics Dynamics of Accelerated Systems. Journal of Applied We present in this text the research carried out on the dynamic behavior of Mathematics and Physics, 7, 1732-1754. non-inertial systems, proposing new keys to better understand the mechanics https://doi.org/10.4236/jamp.2019.78119 of the universe. Applying the field theory to the dynamic magnitudes cir- Received: July 2, 2019 cumscribed to a body, our research has achieved a new conception of the Accepted: August 13, 2019 coupling of these magnitudes, to better understand the behavior of solid rigid Published: August 16, 2019 bodies, when subjected to multiple simultaneous, non-coaxial rotations. The results of the research are consistent with Einstein’s theories on rotation; Copyright © 2019 by author(s) and Scientific Research Publishing Inc. however, we propose a different mechanics and complementary to classical This work is licensed under the Creative mechanics, specifically for systems accelerated by rotations. These new con- Commons Attribution International cepts define the Theory of Dynamic Interactions (TDI), a new dynamic mod- License (CC BY 4.0). el for non-inertial systems with axial symmetry, which is based on the prin- http://creativecommons.org/licenses/by/4.0/ Open Access ciples of conservation of measurable quantities: the notion of quantity, total mass and total energy. -
Relativistic Acceleration of Planetary Orbiters
RELATIVISTIC ACCELERATION OF PLANETARY ORBITERS Bernard Godard(1), Frank Budnik(2), Trevor Morley(2), and Alejandro Lopez Lozano(3) (1)Telespazio VEGA Deutschland GmbH, located at ESOC* (2)European Space Agency, located at ESOC* (3)Logica Deutschland GmbH & Co. KG, located at ESOC* *Robert-Bosch-Str. 5, 64293 Darmstadt, Germany, +49 6151 900, < firstname > . < lastname > [. < lastname2 >]@esa.int Abstract: This paper deals with the relativistic contributions to the gravitational acceleration of a planetary orbiter. The formulation for the relativistic corrections is different in the solar-system barycentric relativistic system and in the local planetocentric relativistic system. The ratio of this correction to the total acceleration is usually orders of magnitude larger in the barycentric system than in the planetocentric system. However, an appropriate Lorentz transformation of the total acceleration from one system to the other shows that both systems are equivalent to a very good accuracy. This paper discusses the steps that were taken at ESOC to validate the implementations of the relativistic correction to the gravitational acceleration in the interplanetary orbit determination software. It compares numerically both systems in the cases of a spacecraft in the vicinity of the Earth (Rosetta during its first swing-by) and a Mercury orbiter (BepiColombo). For the Mercury orbiter, the orbit is propagated in both systems and with an appropriate adjustment of the time argument and a Lorentz correction of the position vector, the resulting orbits are made to match very closely. Finally, the effects on the radiometric observables of neglecting the relativistic corrections to the acceleration in each system and of not performing the space-time transformations from the Mercury system to the barycentric system are presented. -
SATELLITES ORBIT ELEMENTS : EPHEMERIS, Keplerian ELEMENTS, STATE VECTORS
www.myreaders.info www.myreaders.info Return to Website SATELLITES ORBIT ELEMENTS : EPHEMERIS, Keplerian ELEMENTS, STATE VECTORS RC Chakraborty (Retd), Former Director, DRDO, Delhi & Visiting Professor, JUET, Guna, www.myreaders.info, [email protected], www.myreaders.info/html/orbital_mechanics.html, Revised Dec. 16, 2015 (This is Sec. 5, pp 164 - 192, of Orbital Mechanics - Model & Simulation Software (OM-MSS), Sec 1 to 10, pp 1 - 402.) OM-MSS Page 164 OM-MSS Section - 5 -------------------------------------------------------------------------------------------------------43 www.myreaders.info SATELLITES ORBIT ELEMENTS : EPHEMERIS, Keplerian ELEMENTS, STATE VECTORS Satellite Ephemeris is Expressed either by 'Keplerian elements' or by 'State Vectors', that uniquely identify a specific orbit. A satellite is an object that moves around a larger object. Thousands of Satellites launched into orbit around Earth. First, look into the Preliminaries about 'Satellite Orbit', before moving to Satellite Ephemeris data and conversion utilities of the OM-MSS software. (a) Satellite : An artificial object, intentionally placed into orbit. Thousands of Satellites have been launched into orbit around Earth. A few Satellites called Space Probes have been placed into orbit around Moon, Mercury, Venus, Mars, Jupiter, Saturn, etc. The Motion of a Satellite is a direct consequence of the Gravity of a body (earth), around which the satellite travels without any propulsion. The Moon is the Earth's only natural Satellite, moves around Earth in the same kind of orbit. (b) Earth Gravity and Satellite Motion : As satellite move around Earth, it is pulled in by the gravitational force (centripetal) of the Earth. Contrary to this pull, the rotating motion of satellite around Earth has an associated force (centrifugal) which pushes it away from the Earth. -
Problem Set #6 Motivation Problem Statement
PROBLEM SET #6 TO: PROF. DAVID MILLER, PROF. JOHN KEESEE, AND MS. MARILYN GOOD FROM: NAMES WITHELD SUBJECT: PROBLEM SET #6 (LIFE SUPPORT, PROPULSION, AND POWER FOR AN EARTH-TO-MARS HUMAN TRANSPORTATION VEHICLE) DATE: 12/3/2003 MOTIVATION Mars is of great scientific interest given the potential evidence of past or present life. Recent evidence indicating the past existence of water deposits underscore its scientific value. Other motivations to go to Mars include studying its climate history through exploration of the polar layers. This information could be correlated with similar data from Antarctica to characterize the evolution of the Solar System and its geological history. Long-term goals might include the colonization of Mars. As the closest planet with a relatively mild environment, there exists a unique opportunity to explore Mars with humans. Although we have used robotic spacecraft successfully in the past to study Mars, humans offer a more efficient and robust exploration capability. However, human spaceflight adds both complexity and mass to the space vehicle and has a significant impact on the mission design. Humans require an advanced environmental control and life support system, and this subsystem has high power requirements thus directly affecting the power subsystem design. A Mars mission capability is likely to be a factor in NASA’s new launch architecture design since the resulting launch architecture will need take into account the estimated spacecraft mass required for such a mission. To design a Mars mission, various propulsion system options must be evaluated and compared for their efficiency and adaptability to the required mission duration.