Deep Space Communication and Exploration of Solar System Through Inter-Lagrangian Data Relay Satellite Constellation
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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. -
General Relativity Fall 2019 Lecture 20: Geodesics of Schwarzschild
General Relativity Fall 2019 Lecture 20: Geodesics of Schwarzschild Yacine Ali-Ha¨ımoud November 7, 2019 In this lecture we study geodesics in the vacuum Schwarzschild metric, at r > 2M. Last lecture we derived the following equations for timelike geodesics in the equatorial plane (θ = π=2): d' L 1 dr 2 M L2 ML2 = ; + Veff (r) = ;Veff (r) + ; (1) dτ r2 2 dτ E ≡ − r 2r2 − r3 where (E2 1)=2 can be interpreted as a kinetic + potential energy per unit mass. The radial equation can also be rewrittenE ≡ as− d2r M 3 h i = V 0 (r) = r~2 L~2r~ + 3L~2 ; r~ r=M; L~ = L=M: (2) dτ 2 − eff − r4 − ≡ CIRCULAR ORBITS AND THE ISCO We show the effective potential in Fig. 1. In contrast to the Newtonian effective potential for orbits around a central 2 2 2 3 2 mass (i.e. Veff M=r + L =2r , without the last term ML =r ), which always has a minimum at rNewt = L =M, ≡ − − the relativistic effective potential has both a maximum and a minimun for L > p12 M, an inflection point for L = p12 M, and is strictly monotonic for L < p12 M. 0 Circular orbits (with r = constant) are such that Veff (r) = 0. Solving this equation, one finds that such orbits exist only for L > p12 M. When this condition is satisfied, the radii of circular orbits are L2 p r± = 1 1 12M 2=L2 : (3) c 2M ± − The Newtonian limit is obtained for L M, in which case r+ L2=M. -
Communication Strategies for Colonization Mission to Mars
Communication Strategies for Colonization Mission to Mars A Thesis Submitted to the Faculty of Universidad Carlos III de Madrid In Partial Fulfillment of the Requirements for the Bachelor’s Degree in Aerospace Engineering By Pablo A. Machuca Varela June 2015 Dedicated to my dear mother, Teresa, for her education and inspiration; for making me the person I am today. And to my grandparents, Teresa and Hernan,´ for their care and love; for being the strongest motivation to pursue my goals. Acknowledgments I would like to thank my advisor, Professor Manuel Sanjurjo-Rivo, for his help and guidance along the past three years, and for his advice on this thesis. Professor Sanjurjo-Rivo first accepted me as his student and helped me discover my passion for the Orbital Mechanics research area. I am very thankful for the opportunity Professor Sanjurjo-Rivo gave me to do research for the first time, which greatly helped me improve my knowledge and skills as an engineer. His valuable advice also encouraged me to take Professor Howell’s Orbital Mechanics course and Professor Longuski’s Senior Design course while at Purdue University, as an exchange student, which undoubtedly enhanced my desire, and created the opportunity, to become a graduate student at Purdue University. I would like to thank Sarag Saikia, the Mission Design Advisor of Project Aldrin-Purdue, for his exemplary passion and enthusiasm for the field. Sarag is responsible for making me realize the interest and relevance of a Mars communication network. He encouraged me to work on this thesis, and advised me along the way. -
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. -
Effective Potential
Murrary-Clay Group Notes By: John McCann Effective Potential Consider a three-body system with m3 << m2 ≤ m1, from this point narratored with m1 as a star, m2 as a planet and m3 as a small satellite. We shall use a rotating non-inertial coordinate system, which rotates about the barycenter but with the origin centered on the planet. Oriented such that the barycenter falls along the x{axis, in the x > 0 half, and the axis of rotation is parallel to the z{axis. Rewrite, m1 ≡ M∗ as the mass of the star, and m2 ≡ MP as the mass of the planet. Define ~a as the vector from center of the planet to the center of the star, ~` as the vector from the center of the planet to the barycenter, and ~r? ≡ ~ρ, as projection of the vector from the center of the planet to a given point into the plane normal to the axis of rotation (such given point denoted as ~r). ^ To be succinct, ~r? = j~rj sin(θ) sin(θ)^r + cos(θ)θ = ρρ^, where the angle is the usual spherical coordinate definition and ρ is the standard cylindrical coordinate, as used by physicist. We chose to define this last vector, since it is the relevant distance for determining the centrifugal potential, along with ` and Ω. The effective potential per unit mass, u, for a tertiary object in a planet-star system is GM GM 1 u (~r) = − P − ∗ − Ω2j~r − ~`j2: (1) eff j~rj j~a − ~rj 2 ? Respectively the terms are Newton's gravitational potential from the planet (thus defining G as Newton's gravitational constant), the gravitational potential from the star and the centrifugal potential from an object moving about the barycenter with angular frequency Ω. -
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 Toys in Space
ORBITAL MECHANICS Toys in Space Suggested TEKS: Grade Level: 5 Science - 5.2 5.3 Language Arts - 5.10 5.13 Suggested SCANS: Time Required: 30 minutes per toy Technology. Apply technology to task. National Science and Math Standards Science as Inquiry, Earth & Space Science, Science & Technology, Physical Science, Reasoning, Observing, Communicating Countdown: “Rat Stuff” pop-over mouse by Tomy Corp., Carson, CA 90745 Yo-Yo flight model is a yellow Duncan Imperial by Duncan Toy Co., Barbaoo, WI 53913 Wheelo flight model by Jak Pak, Inc. Milwaukee, WI 53201 “Snoopy” Top flight model by Ohio Art, Bryan, OH 43506 Slinky model #100 by James Industries, Inc., Hollidaysburg, PA 16648 Gyroscope flight model by Chandler Gyroscope Mfg., Co., Hagerstown, NJ 47346 Magnetic Marbles by Magnetic Marbles, Inc., Woodinville, WA Wind up Car by Darda Toy Company, East Brunswick, NJ Jacks flight set made by Wells Mfg. Cl., New Vienna, OH 45159 Paddleball flight model by Chemtoy, a division of Strombecker Corp, Chicago, IL 60624 Note: Special Toys in Space Collections are available from various distributors including Museum Products, the Air & Space Museum in Washington, DC, and many are on loan from your local Texas Agricultural Extension Agent Ignition: Gravity’s downward pull dominates the behavior of toys on earth. It is hard to imagine how a familiar toy would behave in weightless conditions. Discover gravity by playing with the toys that flew in space. Try the experiments described in the Toys in Space guidebook. Decide how gravity affects each toy’s performance. Then make predictions about toy space behaviors. If possible, watch the Toys in Space videotape or study a Toys in Space poster (video available through your local Texas Agricultural Extension Agent). -
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. -
Quantum Interferometric Visibility As a Witness of General Relativistic Proper Time
ARTICLE Received 13 Jun 2011 | Accepted 5 Sep 2011 | Published 18 Oct 2011 DOI: 10.1038/ncomms1498 Quantum interferometric visibility as a witness of general relativistic proper time Magdalena Zych1, Fabio Costa1, Igor Pikovski1 & Cˇaslav Brukner1,2 Current attempts to probe general relativistic effects in quantum mechanics focus on precision measurements of phase shifts in matter–wave interferometry. Yet, phase shifts can always be explained as arising because of an Aharonov–Bohm effect, where a particle in a flat space–time is subject to an effective potential. Here we propose a quantum effect that cannot be explained without the general relativistic notion of proper time. We consider interference of a ‘clock’—a particle with evolving internal degrees of freedom—that will not only display a phase shift, but also reduce the visibility of the interference pattern. According to general relativity, proper time flows at different rates in different regions of space–time. Therefore, because of quantum complementarity, the visibility will drop to the extent to which the path information becomes available from reading out the proper time from the ‘clock’. Such a gravitationally induced decoherence would provide the first test of the genuine general relativistic notion of proper time in quantum mechanics. 1 Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. 2 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Correspondence and requests for materials should be addressed to M.Z. (email: [email protected]). NATURE COMMUNICATIONS | 2:505 | DOI: 10.1038/ncomms1498 | www.nature.com/naturecommunications © 2011 Macmillan Publishers Limited. -
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. -
Assignment Week 5 Introduction
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics 8.224 Exploring Black Holes General Relativity and Astrophysics Spring 2003 ASSIGNMENT WEEK 5 NOTE: Exercises 6 through 8 are to be carried out using the GRorbits program, programmed in JAVA, available as a compressed file on the Assignments page. Download the zip file, decompress it and click on the icon GRorbits.html. If that does not work, click on the icon GRorbitsConverted.html. INTRODUCTION Now we get to the heavy lifting in general relativity! By this time we are accustomed to the surprising predictions of the Schwarzschild metric, have learned to change coordinate systems the way we change clothes, and easily use total energy as a constant of the motion to describe a stone plunging into a black hole. The topic for this week: Orbiting satellites. For orbits, there are two constants of the motion: (1) energy-measured-at-infinity, which is given by the same expression as for radial plunge, and (2) angular momentum, which is the same expression as in Newtonian mechanics, except the Newtonian universal time increment dt is replaced by the proper time increment dτ, that is, the time read on the wristwatch of the orbiting stone. What is difficult about this chapter? For many people the greatest difficulty is manipulating angular momentum, whether Newtonian or relativistic. For both theories, the angular momentum is just the radius to the particle multiplied by the particle’s component of linear momentum perpendicular to this radius. There is a different expression for linear momentum in the two cases: mds/dt for Newton, mds/dτ for Einstein.