ORBITAL MECHANICS Toys in Space
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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. -
2. Orbital Mechanics MAE 342 2016
2/12/20 Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler’s Equation Position and velocity in orbit Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE342.html 1 1 Orbits 101 Satellites Escape and Capture (Comets, Meteorites) 2 2 1 2/12/20 Two-Body Orbits are Conic Sections 3 3 Classical Orbital Elements Dimension and Time a : Semi-major axis e : Eccentricity t p : Time of perigee passage Orientation Ω :Longitude of the Ascending/Descending Node i : Inclination of the Orbital Plane ω: Argument of Perigee 4 4 2 2/12/20 Orientation of an Elliptical Orbit First Point of Aries 5 5 Orbits 102 (2-Body Problem) • e.g., – Sun and Earth or – Earth and Moon or – Earth and Satellite • Circular orbit: radius and velocity are constant • Low Earth orbit: 17,000 mph = 24,000 ft/s = 7.3 km/s • Super-circular velocities – Earth to Moon: 24,550 mph = 36,000 ft/s = 11.1 km/s – Escape: 25,000 mph = 36,600 ft/s = 11.3 km/s • Near escape velocity, small changes have huge influence on apogee 6 6 3 2/12/20 Newton’s 2nd Law § Particle of fixed mass (also called a point mass) acted upon by a force changes velocity with § acceleration proportional to and in direction of force § Inertial reference frame § Ratio of force to acceleration is the mass of the particle: F = m a d dv(t) ⎣⎡mv(t)⎦⎤ = m = ma(t) = F ⎡ ⎤ dt dt vx (t) ⎡ f ⎤ ⎢ ⎥ x ⎡ ⎤ d ⎢ ⎥ fx f ⎢ ⎥ m ⎢ vy (t) ⎥ = ⎢ y ⎥ F = fy = force vector dt -
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. -
Orbital Mechanics
Orbital Mechanics Part 1 Orbital Forces Why a Sat. remains in orbit ? Bcs the centrifugal force caused by the Sat. rotation around earth is counter- balanced by the Earth's Pull. Kepler’s Laws The Satellite (Spacecraft) which orbits the earth follows the same laws that govern the motion of the planets around the sun. J. Kepler (1571-1630) was able to derive empirically three laws describing planetary motion I. Newton was able to derive Keplers laws from his own laws of mechanics [gravitation theory] Kepler’s 1st Law (Law of Orbits) The path followed by a Sat. (secondary body) orbiting around the primary body will be an ellipse. The center of mass (barycenter) of a two-body system is always centered on one of the foci (earth center). Kepler’s 1st Law (Law of Orbits) The eccentricity (abnormality) e: a 2 b2 e a b- semiminor axis , a- semimajor axis VIN: e=0 circular orbit 0<e<1 ellip. orbit Orbit Calculations Ellipse is the curve traced by a point moving in a plane such that the sum of its distances from the foci is constant. Kepler’s 2nd Law (Law of Areas) For equal time intervals, a Sat. will sweep out equal areas in its orbital plane, focused at the barycenter VIN: S1>S2 at t1=t2 V1>V2 Max(V) at Perigee & Min(V) at Apogee Kepler’s 3rd Law (Harmonic Law) The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies. a 3 n 2 n- mean motion of Sat. -
Orbital Mechanics of Gravitational Slingshots 1 Introduction 2 Approach
Orbital Mechanics of Gravitational Slingshots Final Paper 15-424: Foundations of Cyber-Physical Systems Adam Moran, [email protected] John Mann, [email protected] May 1, 2016 Abstract A gravitational slingshot is a maneuver to save fuel by using the gravity of a planet to accelerate or decelerate a spacecraft. Due to the large distances and high speeds involved, slingshots require precise accuracy to accomplish | the slightest mistake could cause the whole mission to fail. Therefore, we have developed a cyber-physical system to model the physics and prove the safety and efficiency of powered and unpowered gravitational slingshots. We present our findings and proof in this paper. 1 Introduction A gravitational slingshot is a maneuver performed to increase or decrease the speed of a spacecraft by simply approaching planetary bodies. A spacecraft's usefulness and maneuverability is basically tied to the amount of fuel it can carry, and the more fuel a spacecraft holds, the more fuel it needs to carry that fuel into orbit. Therefore, gravitational slingshots are a very appealing way to save mass, and therefore money, on deep-space missions since these maneuvers do not require any fuel. As missions conducted by national and private space programs become more frequent and ambitious, the need for these precise maneuvers will increase. Therefore, we have created a cyber-physical system that models the physics of a gravitational slingshot for a spacecraft approaching a planet. In the "Approach" section of this paper, we give a brief overview of the physics involved with orbits and gravitational slingshots. In the "Models and Properties" section of this paper, we describe what assumptions and simplifications we made to model these astrophysics in a way for us to prove our desired properties with KeYmaeraX. -
Orbital Mechanics Joe Spier, K6WAO – AMSAT Director for Education ARRL 100Th Centennial Educational Forum 1 History
Orbital Mechanics Joe Spier, K6WAO – AMSAT Director for Education ARRL 100th Centennial Educational Forum 1 History Astrology » Pseudoscience based on several systems of divination based on the premise that there is a relationship between astronomical phenomena and events in the human world. » Many cultures have attached importance to astronomical events, and the Indians, Chinese, and Mayans developed elaborate systems for predicting terrestrial events from celestial observations. » In the West, astrology most often consists of a system of horoscopes purporting to explain aspects of a person's personality and predict future events in their life based on the positions of the sun, moon, and other celestial objects at the time of their birth. » The majority of professional astrologers rely on such systems. 2 History Astronomy » Astronomy is a natural science which is the study of celestial objects (such as stars, galaxies, planets, moons, and nebulae), the physics, chemistry, and evolution of such objects, and phenomena that originate outside the atmosphere of Earth, including supernovae explosions, gamma ray bursts, and cosmic microwave background radiation. » Astronomy is one of the oldest sciences. » Prehistoric cultures have left astronomical artifacts such as the Egyptian monuments and Nubian monuments, and early civilizations such as the Babylonians, Greeks, Chinese, Indians, Iranians and Maya performed methodical observations of the night sky. » The invention of the telescope was required before astronomy was able to develop into a modern science. » Historically, astronomy has included disciplines as diverse as astrometry, celestial navigation, observational astronomy and the making of calendars, but professional astronomy is nowadays often considered to be synonymous with astrophysics. -
Geosynchronous Patrol Orbit for Space Situational Awareness
Geosynchronous patrol orbit for space situational awareness Blair Thompson, Thomas Kelecy, Thomas Kubancik Applied Defense Solutions Tim Flora, Michael Chylla, Debra Rose Sierra Nevada Corporation ABSTRACT Applying eccentricity to a geosynchronous orbit produces both longitudinal and radial motion when viewed in Earth-fixed coordinates. An interesting family of orbits emerges, useful for “neighborhood patrol” space situational awareness and other missions. The basic result is a periodic (daily), quasi- elliptical, closed path around a fixed region of the geosynchronous (geo) orbit belt, keeping a sensor spacecraft in relatively close vicinity to designated geo objects. The motion is similar, in some regards, to the relative motion that may be encountered during spacecraft proximity operations, but on a much larger scale. The patrol orbit does not occupy a fixed slot in the geo belt, and the east-west motion can be combined with north-south motion caused by orbital inclination, leading to even greater versatility. Some practical uses of the geo patrol orbit include space surveillance (including catalog maintenance), and general space situational awareness. The patrol orbit offers improved, diverse observation geom- etry for angles-only sensors, resulting in faster, more accurate orbit determination compared to simple inclined geo orbits. In this paper, we analyze the requirements for putting a spacecraft in a patrol orbit, the unique station keeping requirements to compensate for perturbations, repositioning the patrol orbit to a different location along the geo belt, maneuvering into, around, and out of the volume for proximity operations with objects within the volume, and safe end-of-life disposal requirements. 1. INTRODUCTION The traditional geosynchronous (“geosynch” or “geo”) orbit is designed to keep a spacecraft in a nearly fixed position (longitude, latitude, and altitude) with respect to the rotating Earth. -
Chapter 11 Satellite Orbits
CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at rest remains at rest and a body in motion remains in motion unless acted upon by a force, (b) force equals the rate of change of momentum for a body, and (c) the law of equal but opposite forces. For a two body system comprising of the earth and a much smaller object such as a satellite, the motion of the body in the central gravitational field can be written → → dv / dt = -GM r / r3 → where v is the velocity of the body in its orbit, G is the universal gravitational constant (6.67 x 10**11 N m**2/kg**2), M is the mass of the earth, and r is position vector from the centre of mass of the system (assumed to be at the centre of the earth). The radial part of this equation has the more explicit form m d2r/dt2 - mrω2 = -GMm/r2 where ω is the angular velocity and m is the satellite mass. The second term on the left side of the equation is often referred to as the centrifugal force The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies E = 1/2 m [(dr/dt)2 + r2ω2] - mMG/r . The angular momentum is also conserved so that L = mωr2 , which allows us to rewrite the energy E = 1/2 m (dr/dt)2 + L2/(2mr2) - mMG/r . -
Orbital Mechanics Third Edition
Orbital Mechanics Third Edition Edited by Vladimir A. Chobotov EDUCATION SERIES J. S. Przemieniecki Series Editor-in-Chief Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio Publishedby AmericanInstitute of Aeronauticsand Astronautics,Inc. 1801 AlexanderBell Drive, Reston, Virginia20191-4344 American Institute of Aeronautics and Astronautics, Inc., Reston, Virginia Library of Congress Cataloging-in-Publication Data Orbital mechanics / edited by Vladimir A. Chobotov.--3rd ed. p. cm.--(AIAA education series) Includes bibliographical references and index. 1. Orbital mechanics. 2. Artificial satellites--Orbits. 3. Navigation (Astronautics). I. Chobotov, Vladimir A. II. Series. TLI050.O73 2002 629.4/113--dc21 ISBN 1-56347-537-5 (hardcover : alk. paper) 2002008309 Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, distributed, or transmitted, in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. Data and information appearing in this book are for informational purposes only. AIAA and the authors are not responsible for any injury or damage resulting from use or reliance, nor does AIAA or the authors warrant that use or reliance will be free from privately owned rights. Foreword The third edition of Orbital Mechanics edited by V. A. Chobotov complements five other space-related texts published in the Education Series of the American Institute of Aeronautics and Astronautics (AIAA): Re-Entry Vehicle Dynamics by F. J. Regan, An Introduction to the Mathematics and Methods of Astrodynamics by R. -
Exploring Space Through Math
Key Concepts It All Comes Full Circle Properties of circles, similarity Instructional Objectives The 5-E’s Instructional Model (Engage, Explore, Explain, Extend, and Problem Duration Evaluate) will be used to accomplish the following objectives. 50 minutes Students will • use geometric definitions to describe the properties of circles; Technology • find the circumference of circles; Computer with projector, • calculate arc lengths; movie player • find the measure of central angles; and Materials • use the properties of similarity, such as scale factor. - It All Comes Full Circle Student Edition Prerequisites - Real World: Calculating Shuttle Launch Windows Students should have prior knowledge of the geometric principles of circles, video circumference, central angles, arc length, and scale factor. Skills Background Identify circumference, This problem is part of a series that applies mathematical principles in central angles, arc NASA’s human spaceflight. length, scale factor The International Space Station (ISS) is an internationally developed research facility and is the largest human-built satellite in Earth’s orbit. It NCTM Standards travels around Earth at an altitude of approximately 250 miles (400 km). It - Number Operations is an incredible and complex engineering endeavor that has been - Algebra assembled while in orbit. - Geometry Construction of the ISS began in November 1998, when Russia placed the Zarya module in orbit. With the exception of the Russian Zvezda module and Zarya module, all other modules were delivered by a NASA space shuttle. Each module was installed by ISS and space shuttle crewmembers during spacewalks and using a robotic arm. The space shuttle is the only space vehicle capable of carrying the large modules that make up the ISS. -
Orbital Slots for Everyone?
CENTER FOR SPACE POLICY AND STRATEGY MARCH 2017 ORBITAL SLOTS FOR EVERYONE? JOSEPH W. GANGESTAD THE AEROSPACE CORPORATION © 2017 The Aerospace Corporation. All trademarks, service marks, and trade names contained herein are the property of their respective owners. Approved for public release; distribution unlimited. OTR201700473 JOSEPH W. GANGESTAD Joseph W. Gangestad, Ph.D., is an expert in orbital mechanics and space-system architectures at The Aerospace Corporation. In addition to publishing several scholarly journal articles in the field of astrodynamics, he authored the articles on “Celestial Mechanics” and “Orbital Motion” for the McGraw-Hill Encyclopedia of Science and Technology. He is also the author of Good Grad! A Practical Guide to Graduate School in the Sciences and Engineering. ABOUT THE CENTER FOR SPACE POLICY AND STRATEGY The Center for Space Policy and Strategy is a specialized research branch within The Aerospace Corporation, a nonprofit organization that operates a federally funded research and development center providing objective technical analysis for programs of national significance. Established in 2000 as a Center of Excellence for civil, commercial, and national security space and technology policy, the Center examines issues at the intersection of technology and policy and provides nonpartisan research for national decisionmakers. Contact us at www.aerospace.org/policy or [email protected] Foreword “New Space” entrepreneurs propose constellations that number in the thousands of satellites, and in the rapidly evolving space industry, they may very well succeed. But constellations of this size bring greater risk for collisions and the creation of debris, and no organization is responsible for assessing how they may impact the broader space community. -
Enabling Deep Space Exploration with an In-Space Propellant Depot Supplied from Lunar Ice
AIAA SPACE Forum 10.2514/6.2017-5376 12 - 14 Sep 2017, Orlando, FL AIAA SPACE and Astronautics Forum and Exposition Enabling Deep Space Exploration with an In-Space Propellant Depot Supplied from Lunar Ice Sophia Casanova1, Jack Henry de Frahan2, Vinicius Guimaraes Goecks3, Sumudu Herath4, Mercedes Herreras Martinez5, Nicholas Jamieson6, Therese Jones7, Sung Wha Kang8, Sydney Katz9, Gary Li10, Donal O’Sullivan11, Daniel Pastor12, Nathan Sharifrazi13, Bryan Sinkovec14, Joseph Sparta15, Matthew Vernacchia16 Deep-space missions are heavily constrained by the amount of payload mass the launch vehicle can carry. Furthermore, the amount of payload mass the launch vehicle can carry is limited by the delta-V losses of escaping both Earth’s gravity well and its atmosphere. Instead of launching the propellant mass to be used for trajectories to deep- space, if the propellant can be delivered in-space, the vehicle may carry a significantly larger payload from the surface of the Earth to the destination. Such an architecture is a paradigm shift for space exploration, enabling spacecraft to fly to the furthest reaches of the Solar System with more mass and/or in less time. An international team of sixteen students met at the 2017 Caltech Space Challenge to design Lunarport: a station which provides vehicles traveling to destinations around the solar system with propellant created from water ice extracted at the lunar south pole. A complete system concept design and architecture was produced, entitled ‘Ice Rush’, which leverages mostly TRL 6+ technology and is capable of refueling crewed Mars missions by 2032 at a total cost of $17B.