Appendix A: Orbital Elements
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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. -
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 -
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. -
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. -
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 -
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. -
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. -
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. -
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. -
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: [email protected] 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: [email protected] 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. -
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. -
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.