Orbital Mechanics
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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 -
Launch and Deployment Analysis for a Small, MEO, Technology Demonstration Satellite
46th AIAA Aerospace Sciences Meeting and Exhibit AIAA 2008-1131 7 – 10 January 20006, Reno, Nevada Launch and Deployment Analysis for a Small, MEO, Technology Demonstration Satellite Stephen A. Whitmore* and Tyson K. Smith† Utah State University, Logan, UT, 84322-4130 A trade study investigating the economics, mass budget, and concept of operations for delivery of a small technology-demonstration satellite to a medium-altitude earth orbit is presented. The mission requires payload deployment at a 19,000 km orbit altitude and an inclination of 55o. Because the payload is a technology demonstrator and not part of an operational mission, launch and deployment costs are a paramount consideration. The payload includes classified technologies; consequently a USA licensed launch system is mandated. A preliminary trade analysis is performed where all available options for FAA-licensed US launch systems are considered. The preliminary trade study selects the Orbital Sciences Minotaur V launch vehicle, derived from the decommissioned Peacekeeper missile system, as the most favorable option for payload delivery. To meet mission objectives the Minotaur V configuration is modified, replacing the baseline 5th stage ATK-37FM motor with the significantly smaller ATK Star 27. The proposed design change enables payload delivery to the required orbit without using a 6th stage kick motor. End-to-end mass budgets are calculated, and a concept of operations is presented. Monte-Carlo simulations are used to characterize the expected accuracy of the final orbit. -
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 -
AFSPC-CO TERMINOLOGY Revised: 12 Jan 2019
AFSPC-CO TERMINOLOGY Revised: 12 Jan 2019 Term Description AEHF Advanced Extremely High Frequency AFB / AFS Air Force Base / Air Force Station AOC Air Operations Center AOI Area of Interest The point in the orbit of a heavenly body, specifically the moon, or of a man-made satellite Apogee at which it is farthest from the earth. Even CAP rockets experience apogee. Either of two points in an eccentric orbit, one (higher apsis) farthest from the center of Apsis attraction, the other (lower apsis) nearest to the center of attraction Argument of Perigee the angle in a satellites' orbit plane that is measured from the Ascending Node to the (ω) perigee along the satellite direction of travel CGO Company Grade Officer CLV Calculated Load Value, Crew Launch Vehicle COP Common Operating Picture DCO Defensive Cyber Operations DHS Department of Homeland Security DoD Department of Defense DOP Dilution of Precision Defense Satellite Communications Systems - wideband communications spacecraft for DSCS the USAF DSP Defense Satellite Program or Defense Support Program - "Eyes in the Sky" EHF Extremely High Frequency (30-300 GHz; 1mm-1cm) ELF Extremely Low Frequency (3-30 Hz; 100,000km-10,000km) EMS Electromagnetic Spectrum Equitorial Plane the plane passing through the equator EWR Early Warning Radar and Electromagnetic Wave Resistivity GBR Ground-Based Radar and Global Broadband Roaming GBS Global Broadcast Service GEO Geosynchronous Earth Orbit or Geostationary Orbit ( ~22,300 miles above Earth) GEODSS Ground-Based Electro-Optical Deep Space Surveillance -
The Velocity Dispersion Profile of the Remote Dwarf Spheroidal Galaxy
The Velocity Dispersion Profile of the Remote Dwarf Spheroidal Galaxy Leo I: A Tidal Hit and Run? Mario Mateo1, Edward W. Olszewski2, and Matthew G. Walker3 ABSTRACT We present new kinematic results for a sample of 387 stars located in and around the Milky Way satellite dwarf spheroidal galaxy Leo I. These spectra were obtained with the Hectochelle multi-object echelle spectrograph on the MMT, and cover the MgI/Mgb lines at about 5200A.˚ Based on 297 repeat measure- ments of 108 stars, we estimate the mean velocity error (1σ) of our sample to be 2.4 km/s, with a systematic error of 1 km/s. Combined with earlier re- ≤ sults, we identify a final sample of 328 Leo I red giant members, from which we measure a mean heliocentric radial velocity of 282.9 0.5 km/s, and a mean ± radial velocity dispersion of 9.2 0.4 km/s for Leo I. The dispersion profile of ± Leo I is essentially flat from the center of the galaxy to beyond its classical ‘tidal’ radius, a result that is unaffected by contamination from field stars or binaries within our kinematic sample. We have fit the profile to a variety of equilibrium dynamical models and can strongly rule out models where mass follows light. Two-component Sersic+NFW models with tangentially anisotropic velocity dis- tributions fit the dispersion profile well, with isotropic models ruled out at a 95% confidence level. Within the projected radius sampled by our data ( 1040 pc), ∼ the mass and V-band mass-to-light ratio of Leo I estimated from equilibrium 7 models are in the ranges 5-7 10 M⊙ and 9-14 (solar units), respectively. -
The Speed of a Geosynchronous Satellite Is ___
Physics 106 Lecture 9 Newton’s Law of Gravitation SJ 7th Ed.: Chap 13.1 to 2, 13.4 to 5 • Historical overview • N’Newton’s inverse-square law of graviiitation Force Gravitational acceleration “g” • Superposition • Gravitation near the Earth’s surface • Gravitation inside the Earth (concentric shells) • Gravitational potential energy Related to the force by integration A conservative force means it is path independent Escape velocity Example A geosynchronous satellite circles the earth once every 24 hours. If the mass of the earth is 5.98x10^24 kg; and the radius of the earth is 6.37x10^6 m., how far above the surface of the earth does a geosynchronous satellite orbit the earth? G=6.67x10-11 Nm2/kg2 The speed of a geosynchronous satellite is ______. 1 Goal Gravitational potential energy for universal gravitational force Gravitational Potential Energy WUgravity= −Δ gravity Near surface of Earth: Gravitational force of magnitude of mg, pointing down (constant force) Æ U = mgh Generally, gravit. potential energy for a system of m1 & m2 G Gmm12 mm F = Attractive force Ur()=− G12 12 r 2 g 12 12 r12 Zero potential energy is chosen for infinite distance between m1 and m2. Urg ()012 = ∞= Æ Gravitational potential energy is always negative. 2 mm12 Urg ()12 =− G r12 r r Ug=0 1 U(r1) Gmm U =− 12 g r Mechanical energy 11 mM EKUrmvMVG=+ ( ) =22 + − mech 22 r m V r v M E_mech is conserved, if gravity is the only force that is doing work. 1 2 MV is almost unchanged. If M >>> m, 2 1 2 mM ÆWe can define EKUrmvG=+ ( ) = − mech 2 r 3 Example: A stone is thrown vertically up at certain speed from the surface of the Moon by Superman. -
Optimisation of Propellant Consumption for Power Limited Rockets
Delft University of Technology Faculty Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Optimisation of Propellant Consumption for Power Limited Rockets. What Role do Power Limited Rockets have in Future Spaceflight Missions? (Dutch title: Optimaliseren van brandstofverbruik voor vermogen gelimiteerde raketten. De rol van deze raketten in toekomstige ruimtevlucht missies. ) A thesis submitted to the Delft Institute of Applied Mathematics as part to obtain the degree of BACHELOR OF SCIENCE in APPLIED MATHEMATICS by NATHALIE OUDHOF Delft, the Netherlands December 2017 Copyright c 2017 by Nathalie Oudhof. All rights reserved. BSc thesis APPLIED MATHEMATICS \ Optimisation of Propellant Consumption for Power Limited Rockets What Role do Power Limite Rockets have in Future Spaceflight Missions?" (Dutch title: \Optimaliseren van brandstofverbruik voor vermogen gelimiteerde raketten De rol van deze raketten in toekomstige ruimtevlucht missies.)" NATHALIE OUDHOF Delft University of Technology Supervisor Dr. P.M. Visser Other members of the committee Dr.ir. W.G.M. Groenevelt Drs. E.M. van Elderen 21 December, 2017 Delft Abstract In this thesis we look at the most cost-effective trajectory for power limited rockets, i.e. the trajectory which costs the least amount of propellant. First some background information as well as the differences between thrust limited and power limited rockets will be discussed. Then the optimal trajectory for thrust limited rockets, the Hohmann Transfer Orbit, will be explained. Using Optimal Control Theory, the optimal trajectory for power limited rockets can be found. Three trajectories will be discussed: Low Earth Orbit to Geostationary Earth Orbit, Earth to Mars and Earth to Saturn. After this we compare the propellant use of the thrust limited rockets for these trajectories with the power limited rockets. -
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. -
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. -
Orbit and Spin
Orbit and Spin Overview: A whole-body activity that explores the relative sizes, distances, orbit, and spin of the Sun, Earth, and Moon. Target Grade Level: 3-5 Estimated Duration: 2 40-minute sessions Learning Goals: Students will be able to… • compare the relative sizes of the Earth, Moon, and Sun. • contrast the distance between the Earth and Moon to the distance between the Earth and Sun. • differentiate between the motions of orbit and spin. • demonstrate the spins of the Earth and the Moon, as well as the orbits of the Earth around the Sun, and the Moon around the Earth. Standards Addressed: Benchmarks (AAAS, 1993) The Physical Setting, 4A: The Universe, 4B: The Earth National Science Education Standards (NRC, 1996) Physical Science, Standard B: Position and motion of objects Earth and Space Science, Standard D: Objects in the sky, Changes in Earth and sky Table of Contents: Background Page 1 Materials and Procedure 5 What I Learned… Science Journal Page 14 Earth Picture 15 Sun Picture 16 Moon Picture 17 Earth Spin Demonstration 18 Moon Orbit Demonstration 19 Extensions and Adaptations 21 Standards Addressed, detailed 22 Background: Sun The Sun is the center of our Solar System, both literally—as all of the planets orbit around it, and figuratively—as its rays warm our planet and sustain life as we know it. The Sun is very hot compared to temperatures we usually encounter. Its mean surface temperature is about 9980° Fahrenheit (5800 Kelvin) and its interior temperature is as high as about 28 million° F (15,500,000 Kelvin). -
Curriculum Overview Physics/Pre-AP 2018-2019 1St Nine Weeks
Curriculum Overview Physics/Pre-AP 2018-2019 1st Nine Weeks RESOURCES: Essential Physics (Ergopedia – online book) Physics Classroom http://www.physicsclassroom.com/ PHET Simulations https://phet.colorado.edu/ ONGOING TEKS: 1A, 1B, 2A, 2B, 2C, 2D, 2F, 2G, 2H, 2I, 2J,3E 1) SAFETY TEKS 1A, 1B Vocabulary Fume hood, fire blanket, fire extinguisher, goggle sanitizer, eye wash, safety shower, impact goggles, chemical safety goggles, fire exit, electrical safety cut off, apron, broken glass container, disposal alert, biological hazard, open flame alert, thermal safety, sharp object safety, fume safety, electrical safety, plant safety, animal safety, radioactive safety, clothing protection safety, fire safety, explosion safety, eye safety, poison safety, chemical safety Key Concepts The student will be able to determine if a situation in the physics lab is a safe practice and what appropriate safety equipment and safety warning signs may be needed in a physics lab. The student will be able to determine the proper disposal or recycling of materials in the physics lab. Essential Questions 1. How are safe practices in school, home or job applied? 2. What are the consequences for not using safety equipment or following safe practices? 2) SCIENCE OF PHYSICS: Glossary, Pages 35, 39 TEKS 2B, 2C Vocabulary Matter, energy, hypothesis, theory, objectivity, reproducibility, experiment, qualitative, quantitative, engineering, technology, science, pseudo-science, non-science Key Concepts The student will know that scientific hypotheses are tentative and testable statements that must be capable of being supported or not supported by observational evidence. The student will know that scientific theories are based on natural and physical phenomena and are capable of being tested by multiple independent researchers. -
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.