DOCSLIB.ORG

Appendix A: Orbital Elements

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix A: Orbital Elements Appendix A: Orbital Elements Many of the items in this Appendix have been explained elsewhere in the text, but, for clarity, they have been drawn together into this one section. The size and shape of a satellite’s orbit is determined by its speed and mass, and also by the mass of the object it is orbiting. The size, shape and orientation of an orbit can be described by six orbital elements or orbital parameters, known as Keplerian ele- ments or element sets. Different sets of elements can be used for different purposes. The terms used sound complicated, but taken individually each is relatively simple. These explanations have been taken with reference to an Earth-orbiting satellite, however, they can easily be adapted for satellites orbiting other celestial bodies. If all of the elements are known, the exact location of a satellite can be calculated. The Keplerian elements are: Inclination (i) Longitude of the ascending node (W) Argument of periapsis (w) Eccentricity (e) Semi-major axis (a) Anomaly at epoch (v) Time of periapsis or perigee passage (T) may be used instead of the Anomaly at epoch (v) Inclination (i) A satellite will always travel around the centre of the body it is orbiting and therefore it will always cross the equator twice in every orbit, unless, of course, it orbits directly above the equator. It will cut it as it travels from the southern hemisphere to the northern and again when it crosses from the northern to the southern hemi- sphere. The angle it makes when it crosses the equator from the 303 304 It’s ONLY Rocket Science southern hemisphere to the northern hemisphere is called the inclination, and this defines the orientation of the orbit with respect to the equator. Therefore, an orbit that is directly above the equator has an inclination of 0° and one that goes directly over the north and south poles has an inclination of 90°. In Figure 1.2, the orbit shown in red has an inclination of 28°, which is the inclina- tion of most of the USA’s scientific satellites. The yellow orbit shows the path of the International Space Station, which has an inclination of about 52°. Longitude of the Ascending Node (W) The ascending node is where the orbit of a satellite crosses the Earth’s equator when the satellite is travelling from south to north, as shown in Figure A.1. The longitude of the ascending node is defined with reference to the terms celestial sphere, the ecliptic and the First Point of Aries, which therefore need to be defined first: The celestial sphere is the imaginary sphere encompassing the Earth, as can be seen in Figure 6.5. All objects in the sky can be thought FIGURE App A.1 Orbital Elements Appendix A: Orbital Elements 305 of as being on the sphere. The celestial equator and the celestial poles are projected from their corresponding earthly equivalents. Angles measured north or south from the line of the celestial equa- tor are called the declination and are measured in degrees. This is similar to lines of latitude. The longitude equivalent is called the Right Ascension or RA. This is measured in hours eastwards from the First point of Aries, which is given the symbol ^. The Sun does not always appear directly above the equator but instead follows a path called the ecliptic, which takes it above and below it. When its path crosses the equator, it is called an equi- nox. For the northern hemisphere, the Vernal Equinox is usually about March 20. This is when the length of daylight and darkness is exactly equal, and the Sun is directly above the equator. The direction of the Sun from the Earth on this day is known as the First point of Aries. The longitude of the ascending node is the angle from the First Point of Aries to the ascending node, taken anticlockwise if look- ing down from the north, in the plane of the Earth’s equator. Argument of Periapsis or Perigee (w) This defines where the perigee of the orbit is, in respect to the Earth’s surface. This is the angle formed, in the plane of the satel- lite’s orbit, from where the satellite crosses the Earth’s equator at the ascending node, to when it reaches perigee. If the angle is less than 180°, perigee is in the northern hemisphere, larger than 180°, it is in the southern hemisphere. Eccentricity (e) This is the shape of the orbit. Most orbits are not circular but look liked squashed circles, called ellipses. How flat the ellipse looks is called its eccentricity and it is given a value from zero to one. An ellipse with zero eccentricity is a perfect circle and a very flat ellipse has an eccentricity nearing one, as can be seen in Figure 1.3. 306 It’s ONLY Rocket Science Semi-major Axis (a) This defines the size of the orbit. The long axis of an ellipse is called the major axis, the shorter one, the minor axis. In a circle, both the major and minor axes are the same, and are called the diameter. Half of the major axis is called the semi-major axis. This is the equivalent of the radius in a circle. The length of the semi- major axis is used to describe the size of the ellipse. The properties of an ellipse are shown in Figure 4.4. Anomaly at Epoch (v) Anomaly in this sense is an angle. The anomaly at epoch is the angle between where the satellite is now, compared to where it was at perigee. If the satellite is at perigee, the anomaly is zero. At apogee, it is 180°, and back to perigee is 360°. As most orbits are not circular, the measurement of this angle is not easy and so the angle is given in terms of a fraction of an orbit, with one orbit being 360°. Time of Periapsis or Perigee Passage (T) This is simply the time when the satellite was at perigee. This usu- ally comprises the date and time, given in Universal Time or UT. For most purposes, this can be taken to be the same as Greenwich Mean Time (GMT). Appendix B: Coordinate Systems Geocentric Coordinate Systems Usually, the centre of the Earth is used as the origin for the coordinate system for spacecraft in orbit around the Earth. This is called a geo- centric coordinate system and is shown in Figure B.1. A geocentric equatorial coordinate system is a Cartesian system that has the fun- damental plane as the plane of the equator and the x direction towards the Vernal Equinox or First Point of Aries. The z direction is through the North Pole and the y direction makes up the right-handed set. Except for precession, the wobbling motion of the Earth’s axis of rotation, this system is fixed to the stars and the Earth rotates within it. Heliocentric Coordinate Systems For satellites travelling to other places within our solar system, a coordinate system with the origin at the centre of the Sun is used, such as the heliocentric ecliptic coordinate system. This can be seen in Figure B.2. The fundamental plane is the plane of the ecliptic, with the x direction being towards the Vernal Equinox or First Point of Aries at a particular epoch. The z direc- tion is the ecliptical North Pole, and the y direction makes the right-handed set. Perifocal Coordinate System A perifocal coordinate system is often used to describe the motion of a satellite. The fundamental plane is the plane of the satellite’s orbit, and the x-axis points towards the part of the orbit nearest to the body the satellite is orbiting, known as the periapsis. The y-axis is on the orbital plane, and rotated 90° from the x-axis in the direction of the satellite’s motion. The z-axis completes the right-handed set. 307 308 It’s ONLY Rocket Science FIGURE App B.1 Geocentric Equatorial Coordinate System FIGURE App B.2 Heliocentric Ecliptic Coordinate System Appendix C: Web Site Addresses Name Web site Comment The Radio www.amsat.org An educational organi- Amateur Satel- zation with the goal lite Corpora- of fostering amateur tion (AMSAT) radio’s participation in space research and communication. Amateur www.ariss-eu.org Common working Radio on the group of the national International amateur radio and Space Station amateur satellite radio (ARISS) societies involved in amateur radio opera- tions on board the International Space Station. Belgian Work- www.vvs.be Group that coordinate ing Group for and collate amateur’s Satellites data about the flash period of satellites. The web site is in Dutch. Celestrak www.celestrak.com Dr. T.S. Kelso’s web site, which provides a lot of satellite tracking and other information. ESA www.esa.int Web site of the Euro- pean Space Agency. HearSat www.hearsat.org Satellite radio signal monitoring web site. Heavens www.heavens-above.com Provides information Above needed to observe sat- ellites as other space- flight and astronomical information. (continued) 309 310 It’s ONLY Rocket Science Name Web site Comment International www.imo.net Ensures interna- Meteor Organ- tional cooperation ization (IMO) of meteor amateur work. Collects meteor observations by several methods from around the world. ISS Fan Club www.issfanclub.com Provides realtime information about the International Space Station. Most of the content is related to amateur radio. It is not affiliated to any official body or organization. Mike McCants www.io.com/~mmccants/ Mike McCants’ satel- lite tracking web pages.
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: giulio.avanzini@polito.it 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]
  • Security & Defence European
    a 7.90 D 14974 E D European & Security ES & Defence 6/2019 International Security and Defence Journal COUNTRY FOCUS: AUSTRIA ISSN 1617-7983 • Heavy Lift Helicopters • Russian Nuclear Strategy • UAS for Reconnaissance and • NATO Military Engineering CoE Surveillance www.euro-sd.com • Airborne Early Warning • • Royal Norwegian Navy • Brazilian Army • UAS Detection • Cockpit Technology • Swiss “Air2030” Programme Developments • CBRN Decontamination June 2019 • CASEVAC/MEDEVAC Aircraft • Serbian Defence Exports Politics · Armed Forces · Procurement · Technology ANYTHING. In operations, the Eurofighter Typhoon is the proven choice of Air Forces. Unparalleled reliability and a continuous capability evolution across all domains mean that the Eurofighter Typhoon will play a vital role for decades to come. Air dominance. We make it fly. airbus.com Editorial Europe Needs More Pragmatism The elections to the European Parliament in May were beset with more paradoxes than they have ever been. The strongest party which will take its seats in the plenary chambers in Brus- sels (and, as an expensive anachronism, also in Strasbourg), albeit only for a brief period, is the Brexit Party, with 29 seats, whose programme is implicit in their name. Although EU institutions across the entire continent are challenged in terms of their public acceptance, in many countries the election has been fought with a very great deal of emotion, as if the day of reckoning is dawning, on which decisions will be All or Nothing. Some have raised concerns about the prosperous “European Project”, which they see as in dire need of rescue from malevolent sceptics. Others have painted an image of the decline of the West, which would inevitably come about if Brussels were to be allowed to continue on its present course.
    [Show full text]
  • Ionizing Feedback from Massive Stars in Massive Clusters III: Disruption of Partially Unbound Clouds 3
    Mon. Not. R. Astron. Soc. 000, 1–14 (2006) Printed 30 August 2021 (MN LATEX style file v2.2) Ionizing feedback from massive stars in massive clusters III: Disruption of partially unbound clouds J. E. Dale1⋆, B. Ercolano1, I. A. Bonnell2 1Excellence Cluster ‘Universe’, Boltzmannstr. 2, 85748 Garching, Germany. 2Department of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS 30 August 2021 ABSTRACT We extend our previous SPH parameter study of the effects of photoionization from O–stars on star–forming clouds to include initially unbound clouds. We generate a set 4 6 of model clouds in the mass range 10 -10 M⊙ with initial virial ratios Ekin/Epot=2.3, allow them to form stars, and study the impact of the photoionizing radiation pro- duced by the massive stars. We find that, on the 3Myr timescale before supernovae are expected to begin detonating, the fractions of mass expelled by ionizing feedback is a very strong function of the cloud escape velocities. High–mass clouds are largely unaf- fected dynamically, while lower–mass clouds have large fractions of their gas reserves expelled on this timescale. However, the fractions of stellar mass unbound are modest and significant portions of the unbound stars are so only because the clouds themselves are initially partially unbound. We find that ionization is much more able to create well–cleared bubbles in the unbound clouds, owing to their intrinsic expansion, but that the presence of such bubbles does not necessarily indicate that a given cloud has been strongly influenced by feedback.
    [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]
  • Rafael Space Propulsion
    Rafael Space Propulsion CATALOGUE A B C D E F G Proprietary Notice This document includes data proprietary to Rafael Ltd. and shall not be duplicated, used, or disclosed, in whole or in part, for any purpose without written authorization from Rafael Ltd. Rafael Space Propulsion INTRODUCTION AND OVERVIEW PART A: HERITAGE PART B: SATELLITE PROPULSION SYSTEMS PART C: PROPELLANT TANKS PART D: PROPULSION THRUSTERS Satellites Launchers PART E: PROPULSION SYSTEM VALVES PART F: SPACE PRODUCTION CAPABILITIES PART G: QUALITY MANAGEMENT CATALOGUE – Version 2 | 2019 Heritage PART A Heritage 0 Heritage PART A Rafael Introduction and Overview Rafael Advanced Defense Systems Ltd. designs, develops, manufactures and supplies a wide range of high-tech systems for air, land, sea and space applications. Rafael was established as part of the Ministry of Defense more than 70 years ago and was incorporated in 2002. Currently, 7% of its sales are re-invested in R&D. Rafael’s know-how is embedded in almost every operational Israel Defense Forces (IDF) system; the company has a special relationship with the IDF. Rafael has formed partnerships with companies with leading aerospace and defense companies worldwide to develop applications based on its proprietary technologies. Offset activities and industrial co-operations have been set-up with more than 20 countries world-wide. Over the last decade, international business activities have been steadily expanding across the globe, with Rafael acting as either prime-contractor or subcontractor, capitalizing on its strengths at both system and sub-system levels. Rafael’s highly skilled and dedicated workforce tackles complex projects, from initial development phases, through prototype, production and acceptance tests.
    [Show full text]
  • An Overview of New Worlds, New Horizons in Astronomy and Astrophysics About the National Academies
    2020 VISION An Overview of New Worlds, New Horizons in Astronomy and Astrophysics About the National Academies The National Academies—comprising the National Academy of Sciences, the National Academy of Engineering, the Institute of Medicine, and the National Research Council—work together to enlist the nation’s top scientists, engineers, health professionals, and other experts to study specific issues in science, technology, and medicine that underlie many questions of national importance. The results of their deliberations have inspired some of the nation’s most significant and lasting efforts to improve the health, education, and welfare of the United States and have provided independent advice on issues that affect people’s lives worldwide. To learn more about the Academies’ activities, check the website at www.nationalacademies.org. Copyright 2011 by the National Academy of Sciences. All rights reserved. Printed in the United States of America This study was supported by Contract NNX08AN97G between the National Academy of Sciences and the National Aeronautics and Space Administration, Contract AST-0743899 between the National Academy of Sciences and the National Science Foundation, and Contract DE-FG02-08ER41542 between the National Academy of Sciences and the U.S. Department of Energy. Support for this study was also provided by the Vesto Slipher Fund. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the agencies that provided support for the project. 2020 VISION An Overview of New Worlds, New Horizons in Astronomy and Astrophysics Committee for a Decadal Survey of Astronomy and Astrophysics ROGER D.
    [Show full text]
  • Satellite Systems
    Chapter 18 REST-OF-WORLD (ROW) SATELLITE SYSTEMS For the longest time, space exploration was an exclusive club comprised of only two members, the United States and the Former Soviet Union. That has now changed due to a number of factors, among the more dominant being economics, advanced and improved technologies and national imperatives. Today, the number of nations with space programs has risen to over 40 and will continue to grow as the costs of spacelift and technology continue to decrease. RUSSIAN SATELLITE SYSTEMS The satellite section of the Russian In the post-Soviet era, Russia contin- space program continues to be predomi- ues its efforts to improve both its military nantly government in character, with and commercial space capabilities. most satellites dedicated either to civil/ These enhancements encompass both military applications (such as communi- orbital assets and ground-based space cations and meteorology) or exclusive support facilities. Russia has done some military missions (such as reconnaissance restructuring of its operating principles and targeting). A large portion of the regarding space. While these efforts have Russian space program is kept running by attempted not to detract from space-based launch services, boosters and launch support to military missions, economic sites, paid for by foreign commercial issues and costs have lead to a lowering companies. of Russian space-based capabilities in The most obvious change in Russian both orbital assets and ground station space activity in recent years has been the capabilities. decrease in space launches and corre- The influence of Glasnost on Russia's sponding payloads. Many of these space programs has been significant, but launches are for foreign payloads, not public announcements regarding space Russian.
    [Show full text]
  • Up, Up, and Away by James J
    www.astrosociety.org/uitc No. 34 - Spring 1996 © 1996, Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, CA 94112. Up, Up, and Away by James J. Secosky, Bloomfield Central School and George Musser, Astronomical Society of the Pacific Want to take a tour of space? Then just flip around the channels on cable TV. Weather Channel forecasts, CNN newscasts, ESPN sportscasts: They all depend on satellites in Earth orbit. Or call your friends on Mauritius, Madagascar, or Maui: A satellite will relay your voice. Worried about the ozone hole over Antarctica or mass graves in Bosnia? Orbital outposts are keeping watch. The challenge these days is finding something that doesn't involve satellites in one way or other. And satellites are just one perk of the Space Age. Farther afield, robotic space probes have examined all the planets except Pluto, leading to a revolution in the Earth sciences -- from studies of plate tectonics to models of global warming -- now that scientists can compare our world to its planetary siblings. Over 300 people from 26 countries have gone into space, including the 24 astronauts who went on or near the Moon. Who knows how many will go in the next hundred years? In short, space travel has become a part of our lives. But what goes on behind the scenes? It turns out that satellites and spaceships depend on some of the most basic concepts of physics. So space travel isn't just fun to think about; it is a firm grounding in many of the principles that govern our world and our universe.
    [Show full text]
  • The Space-Based Global Observing System in 2010 (GOS-2010)
    WMO Space Programme SP-7 The Space-based Global Observing For more information, please contact: System in 2010 (GOS-2010) World Meteorological Organization 7 bis, avenue de la Paix – P.O. Box 2300 – CH 1211 Geneva 2 – Switzerland www.wmo.int WMO Space Programme Office Tel.: +41 (0) 22 730 85 19 – Fax: +41 (0) 22 730 84 74 E-mail: SAT-Help-Desk@wmo.int Website: www.wmo.int/pages/prog/sat/ WMO-TD No. 1513 WMO Space Programme SP-7 The Space-based Global Observing System in 2010 (GOS-2010) WMO/TD-No. 1513 2010 © World Meteorological Organization, 2010 The right of publication in print, electronic and any other form and in any language is reserved by WMO. Short extracts from WMO publications may be reproduced without authorization, provided that the complete source is clearly indicated. Editorial correspondence and requests to publish, reproduce or translate these publication in part or in whole should be addressed to: Chairperson, Publications Board World Meteorological Organization (WMO) 7 bis, avenue de la Paix Tel.: +41 (0)22 730 84 03 P.O. Box No. 2300 Fax: +41 (0)22 730 80 40 CH-1211 Geneva 2, Switzerland E-mail: Publications@wmo.int FOREWORD The launching of the world's first artificial satellite on 4 October 1957 ushered a new era of unprecedented scientific and technological achievements. And it was indeed a fortunate coincidence that the ninth session of the WMO Executive Committee – known today as the WMO Executive Council (EC) – was in progress precisely at this moment, for the EC members were very quick to realize that satellite technology held the promise to expand the volume of meteorological data and to fill the notable gaps where land-based observations were not readily available.
    [Show full text]
  • Archons (Commanders) [NOTICE: They Are NOT Anlien Parasites], and Then, in a Mirror Image of the Great Emanations of the Pleroma, Hundreds of Lesser Angels
    A R C H O N S HIDDEN RULERS THROUGH THE AGES A R C H O N S HIDDEN RULERS THROUGH THE AGES WATCH THIS IMPORTANT VIDEO UFOs, Aliens, and the Question of Contact MUST-SEE THE OCCULT REASON FOR PSYCHOPATHY Organic Portals: Aliens and Psychopaths KNOWLEDGE THROUGH GNOSIS Boris Mouravieff - GNOSIS IN THE BEGINNING ...1 The Gnostic core belief was a strong dualism: that the world of matter was deadening and inferior to a remote nonphysical home, to which an interior divine spark in most humans aspired to return after death. This led them to an absorption with the Jewish creation myths in Genesis, which they obsessively reinterpreted to formulate allegorical explanations of how humans ended up trapped in the world of matter. The basic Gnostic story, which varied in details from teacher to teacher, was this: In the beginning there was an unknowable, immaterial, and invisible God, sometimes called the Father of All and sometimes by other names. “He” was neither male nor female, and was composed of an implicitly finite amount of a living nonphysical substance. Surrounding this God was a great empty region called the Pleroma (the fullness). Beyond the Pleroma lay empty space. The God acted to fill the Pleroma through a series of emanations, a squeezing off of small portions of his/its nonphysical energetic divine material. In most accounts there are thirty emanations in fifteen complementary pairs, each getting slightly less of the divine material and therefore being slightly weaker. The emanations are called Aeons (eternities) and are mostly named personifications in Greek of abstract ideas.
    [Show full text]
  • 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) (celletti@mat.uniroma2.it) 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.
    [Show full text]