Lecture 7 Launch Trajectories
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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 -
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 -
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. -
Flight and Orbital Mechanics
Flight and Orbital Mechanics Lecture slides Challenge the future 1 Flight and Orbital Mechanics AE2-104, lecture hours 21-24: Interplanetary flight Ron Noomen October 25, 2012 AE2104 Flight and Orbital Mechanics 1 | Example: Galileo VEEGA trajectory Questions: • what is the purpose of this mission? • what propulsion technique(s) are used? • why this Venus- Earth-Earth sequence? • …. [NASA, 2010] AE2104 Flight and Orbital Mechanics 2 | Overview • Solar System • Hohmann transfer orbits • Synodic period • Launch, arrival dates • Fast transfer orbits • Round trip travel times • Gravity Assists AE2104 Flight and Orbital Mechanics 3 | Learning goals The student should be able to: • describe and explain the concept of an interplanetary transfer, including that of patched conics; • compute the main parameters of a Hohmann transfer between arbitrary planets (including the required ΔV); • compute the main parameters of a fast transfer between arbitrary planets (including the required ΔV); • derive the equation for the synodic period of an arbitrary pair of planets, and compute its numerical value; • derive the equations for launch and arrival epochs, for a Hohmann transfer between arbitrary planets; • derive the equations for the length of the main mission phases of a round trip mission, using Hohmann transfers; and • describe the mechanics of a Gravity Assist, and compute the changes in velocity and energy. Lecture material: • these slides (incl. footnotes) AE2104 Flight and Orbital Mechanics 4 | Introduction The Solar System (not to scale): [Aerospace -
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. -
Coplanar Air Launch with Gravity-Turn Launch Trajectories
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AFTI Scholar (Air Force Institute of Technology) Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-23-2004 Coplanar Air Launch with Gravity-Turn Launch Trajectories David W. Callaway Follow this and additional works at: https://scholar.afit.edu/etd Part of the Aerospace Engineering Commons Recommended Citation Callaway, David W., "Coplanar Air Launch with Gravity-Turn Launch Trajectories" (2004). Theses and Dissertations. 3922. https://scholar.afit.edu/etd/3922 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. COPLANAR AIR LAUNCH WITH GRAVITY-TURN LAUNCH TRAJECTORIES THESIS David W. Callaway, 1st Lieutenant, USAF AFIT/GAE/ENY/04-M04 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. AFIT/GAE/ENY/04-M04 COPLANAR AIR LAUNCH WITH GRAVITY-TURN LAUNCH TRAJECTORIES THESIS Presented to the Faculty Department of Aeronautics and Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautical Engineeering David W. -
Mscthesis Joseangelgutier ... Humada.Pdf
Targeting a Mars science orbit from Earth using Dual Chemical-Electric Propulsion and Ballistic Capture Jose Angel Gutierrez Ahumada Delft University of Technology TARGETING A MARS SCIENCE ORBIT FROM EARTH USING DUAL CHEMICAL-ELECTRIC PROPULSION AND BALLISTIC CAPTURE by Jose Angel Gutierrez Ahumada MSc Thesis in partial fulfillment of the requirements for the degree of Master of Science in Aerospace Engineering at the Delft University of Technology, to be defended publicly on Wednesday May 8, 2019. Supervisor: Dr. Francesco Topputo, TU Delft/Politecnico di Milano Dr. Ryan Russell, The University of Texas at Austin Thesis committee: Dr. Francesco Topputo, TU Delft/Politecnico di Milano Prof. dr. ir. Pieter N.A.M. Visser, TU Delft ir. Ron Noomen, TU Delft Dr. Angelo Cervone, TU Delft This thesis is confidential and cannot be made public until December 31, 2020. An electronic version of this thesis is available at http://repository.tudelft.nl/. Cover picture adapted from https://steemitimages.com/p/2gs...QbQvi EXECUTIVE SUMMARY Ballistic capture is a relatively novel concept in interplanetary mission design with the potential to make Mars and other targets in the Solar System more accessible. A complete end-to-end interplanetary mission from an Earth-bound orbit to a stable science orbit around Mars (in this case, an areostationary orbit) has been conducted using this concept. Sets of initial conditions leading to ballistic capture are generated for different epochs. The influence of the dynamical model on the capture is also explored briefly. Specific capture trajectories are then selected based on a study of their stabilization into an areostationary orbit. -
Spacecraft Trajectories in a Sun, Earth, and Moon Ephemeris Model
SPACECRAFT TRAJECTORIES IN A SUN, EARTH, AND MOON EPHEMERIS MODEL A Project Presented to The Faculty of the Department of Aerospace Engineering San José State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Aerospace Engineering by Romalyn Mirador i ABSTRACT SPACECRAFT TRAJECTORIES IN A SUN, EARTH, AND MOON EPHEMERIS MODEL by Romalyn Mirador This project details the process of building, testing, and comparing a tool to simulate spacecraft trajectories using an ephemeris N-Body model. Different trajectory models and methods of solving are reviewed. Using the Ephemeris positions of the Earth, Moon and Sun, a code for higher-fidelity numerical modeling is built and tested using MATLAB. Resulting trajectories are compared to NASA’s GMAT for accuracy. Results reveal that the N-Body model can be used to find complex trajectories but would need to include other perturbations like gravity harmonics to model more accurate trajectories. i ACKNOWLEDGEMENTS I would like to thank my family and friends for their continuous encouragement and support throughout all these years. A special thank you to my advisor, Dr. Capdevila, and my friend, Dhathri, for mentoring me as I work on this project. The knowledge and guidance from the both of you has helped me tremendously and I appreciate everything you both have done to help me get here. ii Table of Contents List of Symbols ............................................................................................................................... v 1.0 INTRODUCTION -
Elliptical Orbits
1 Ellipse-geometry 1.1 Parameterization • Functional characterization:(a: semi major axis, b ≤ a: semi minor axis) x2 y 2 b p + = 1 ⇐⇒ y(x) = · ± a2 − x2 (1) a b a • Parameterization in cartesian coordinates, which follows directly from Eq. (1): x a · cos t = with 0 ≤ t < 2π (2) y b · sin t – The origin (0, 0) is the center of the ellipse and the auxilliary circle with radius a. √ – The focal points are located at (±a · e, 0) with the eccentricity e = a2 − b2/a. • Parameterization in polar coordinates:(p: parameter, 0 ≤ < 1: eccentricity) p r(ϕ) = (3) 1 + e cos ϕ – The origin (0, 0) is the right focal point of the ellipse. – The major axis is given by 2a = r(0) − r(π), thus a = p/(1 − e2), the center is therefore at − pe/(1 − e2), 0. – ϕ = 0 corresponds to the periapsis (the point closest to the focal point; which is also called perigee/perihelion/periastron in case of an orbit around the Earth/sun/star). The relation between t and ϕ of the parameterizations in Eqs. (2) and (3) is the following: t r1 − e ϕ tan = · tan (4) 2 1 + e 2 1.2 Area of an elliptic sector As an ellipse is a circle with radius a scaled by a factor b/a in y-direction (Eq. 1), the area of an elliptic sector PFS (Fig. ??) is just this fraction of the area PFQ in the auxiliary circle. b t 2 1 APFS = · · πa − · ae · a sin t a 2π 2 (5) 1 = (t − e sin t) · a b 2 The area of the full ellipse (t = 2π) is then, of course, Aellipse = π a b (6) Figure 1: Ellipse and auxilliary circle. -
A Tool for Preliminary Design of Rockets Aerospace Engineering
A Tool for Preliminary Design of Rockets Diogo Marques Gaspar Thesis to obtain the Master of Science Degree in Aerospace Engineering Supervisor : Professor Paulo Jorge Soares Gil Examination Committee Chairperson: Professor Fernando José Parracho Lau Supervisor: Professor Paulo Jorge Soares Gil Members of the Committee: Professor João Manuel Gonçalves de Sousa Oliveira July 2014 ii Dedicated to my Mother iii iv Acknowledgments To my supervisor Professor Paulo Gil for the opportunity to work on this interesting subject and for all his support and patience. To my family, in particular to my parents and brothers for all the support and affection since ever. To my friends: from IST for all the companionship in all this years and from Coimbra for the fellowship since I remember. To my teammates for all the victories and good moments. v vi Resumo A unica´ forma que a humanidade ate´ agora conseguiu encontrar para explorar o espac¸o e´ atraves´ do uso de rockets, vulgarmente conhecidos como foguetoes,˜ responsaveis´ por transportar cargas da Terra para o Espac¸o. O principal objectivo no design de rockets e´ diminuir o peso na descolagem e maximizar o payload ratio i.e. aumentar a capacidade de carga util´ ao seu alcance. A latitude e o local de lanc¸amento, a orbita´ desejada, as caracter´ısticas de propulsao˜ e estruturais sao˜ constrangimentos ao projecto do foguetao.˜ As trajectorias´ dos foguetoes˜ estao˜ permanentemente a ser optimizadas, devido a necessidade de aumento da carga util´ transportada e reduc¸ao˜ do combust´ıvel consumido. E´ um processo utilizado nas fases iniciais do design de uma missao,˜ que afecta partes cruciais do planeamento, desde a concepc¸ao˜ do ve´ıculo ate´ aos seus objectivos globais. -
Kepler's Equation—C.E. Mungan, Fall 2004 a Satellite Is Orbiting a Body
Kepler’s Equation—C.E. Mungan, Fall 2004 A satellite is orbiting a body on an ellipse. Denote the position of the satellite relative to the body by plane polar coordinates (r,) . Find the time t that it requires the satellite to move from periapsis (position of closest approach) to angular position . Express your answer in terms of the eccentricity e of the orbit and the period T for a full revolution. According to Kepler’s second law, the ratio of the shaded area A to the total area ab of the ellipse (with semi-major axis a and semi-minor axis b) is the respective ratio of transit times, A t = . (1) ab T But we can divide the shaded area into differential triangles, of base r and height rd , so that A = 1 r2d (2) 2 0 where the polar equation of a Keplerian orbit is c r = (3) 1+ ecos with c the semilatus rectum (distance vertically from the origin on the diagram above to the ellipse). It is not hard to show from the geometrical properties of an ellipse that b = a 1 e2 and c = a(1 e2 ) . (4) Substituting (3) into (2), and then (2) and (4) into (1) gives T (1 e2 )3/2 t = M where M d . (5) 2 2 0 (1 + ecos) In the literature, M is called the mean anomaly. This is an integral solution to the problem. It turns out however that this integral can be evaluated analytically using the following clever change of variables, e + cos cos E = .