Orbital Mechanics of Gravitational Slingshots 1 Introduction 2 Approach
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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 -
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. -
Asteroid Retrieval Mission
Where you can put your asteroid Nathan Strange, Damon Landau, and ARRM team NASA/JPL-CalTech © 2014 California Institute of Technology. Government sponsorship acknowledged. Distant Retrograde Orbits Works for Earth, Moon, Mars, Phobos, Deimos etc… very stable orbits Other Lunar Storage Orbit Options • Lagrange Points – Earth-Moon L1/L2 • Unstable; this instability enables many interesting low-energy transfers but vehicles require active station keeping to stay in vicinity of L1/L2 – Earth-Moon L4/L5 • Some orbits in this region is may be stable, but are difficult for MPCV to reach • Lunar Weakly Captured Orbits – These are the transition from high lunar orbits to Lagrange point orbits – They are a new and less well understood class of orbits that could be long term stable and could be easier for the MPCV to reach than DROs – More study is needed to determine if these are good options • Intermittent Capture – Weakly captured Earth orbit, escapes and is then recaptured a year later • Earth Orbit with Lunar Gravity Assists – Many options with Earth-Moon gravity assist tours Backflip Orbits • A backflip orbit is two flybys half a rev apart • Could be done with the Moon, Earth or Mars. Backflip orbit • Lunar backflips are nice plane because they could be used to “catch and release” asteroids • Earth backflips are nice orbits in which to construct things out of asteroids before sending them on to places like Earth- Earth or Moon orbit plane Mars cyclers 4 Example Mars Cyclers Two-Synodic-Period Cycler Three-Synodic-Period Cycler Possibly Ballistic Chen, et al., “Powered Earth-Mars Cycler with Three Synodic-Period Repeat Time,” Journal of Spacecraft and Rockets, Sept.-Oct. -
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. -
Future Space Launch Vehicles
Future Space Launch Vehicles S. Krishnan Professor of Aerospace Engineering Indian Institute of Technology Madras Chennnai - 600 036, India (Written in 2001) Introduction Space technology plays a very substantial role in the economical growth and the national security needs of any country. Communications, remote sensing, weather forecasting, navigation, tracking and data relay, surveillance, reconnaissance, and early warning and missile defence are its dependent user-technologies. Undoubtedly, space technology has become the backbone of global information highway. In this technology, the two most important sub-technologies correspond to spacecraft and space launch vehicles. Spacecraft The term spacecraft is a general one. While the spacecraft that undertakes a deep space mission bears this general terminology, the one that orbits around a planet is also a spacecraft but called specifically a satellite more strictly an artificial satellite, moons around their planets being natural satellites. Cassini is an example for a spacecraft. This was developed under a cooperative project of NASA, the European Space Agency, and the Italian Space Agency. Cassini spacecraft, launched in 1997, is continuing its journey to Saturn (about 1274 million km away from Earth), where it is scheduled to begin in July 2004, a four- year exploration of Saturn, its rings, atmosphere, and moons (18 in number). Cassini executed two gravity-assist flybys (or swingbys) of Venus one in April 1998 and one in June 1999 then a flyby of Earth in August 1999, and a flyby of Jupiter (about 629 million km away) in December 2000, see Fig. 1. We may note here with interest that ISRO (Indian Space Research Organisation) is thinking of a flyby mission of a spacecraft around Moon (about 0.38 million km away) by using its launch vehicle PSLV. -
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. -
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. -
Compatibility Mode
Toward the Final Frontier of Manned Space Flight Ryann Fame Luke Bruneaux Emily Russell Image: Milky Way NASA Toward the Final Frontier of Manned Space Flight Part I: How we got here: Background and challenges (Ryann) Part II: Why boldly go? Why not? (Luke) Part III: Where are we going? (Emily) Toward the Final Frontier of Manned Space Flight Part I: How we got here: Background and challenges (Ryann) Part II: Why boldly go? Why not? (Luke) Part III: Where are we going? (Emily) Challenges in Human Space Travel • Challenge 1: Leaving Earth (Space!) • Challenge 2: Can humans live safely in space? • Challenge 3: Destination Travel Curiosity and Explorative Spirit Image: NASA Curiosity and Explorative Spirit Image: NASA How did we get here? 1903 Images: Library of Congress,US Gov. Military, NASA How did we get here? 1903 1947 Images: Library of Congress,US Gov. Military, NASA How did we get here? 1903 1947 1961 Images: Library of Congress,US Gov. Military, NASA How did we get here? 1903 1947 1961 1969 Images: Library of Congress,US Gov. Military, NASA How did we get here? 1903 1947 1971 1961 1969 Images: Library of Congress,US Gov. Military, NASA How did we get here? 1903 1981-2011 1947 1971 1961 1969 Images: Library of Congress,US Gov. Military, NASA Ballistic rockets for missiles X X Images: Library of Congress,US Gov. Military, NASA Leaving Earth (space!) 1946 Image: US Gov. Military Fuel Image: Wikimedia: Matthew Bowden Chemical combustion needs lots of oxygen 2 H2 + O2 → 2 H2O(g) + Energy Chemical combustion needs lots of oxygen 2 H2 -
Interplanetary Cruising
Interplanetary Cruising 1. Foreword The following material is intended to supplement NASA-JSC's High School Aerospace Scholars (HAS) instructional reading. The author, a retired veteran of 60 Space Shuttle flights at Mission Control's Flight Dynamics Officer (FDO) Console, first served as HAS mentor in association with a June 2008 workshop. On that occasion, he noticed students were generally unfamiliar with the nature and design of spacecraft trajectories facilitating travel between Earth and Mars. Much of the specific trajectory design data appearing herein was developed during that and subsequent workshops as reference material for students integrating it into Mars mission timeline, spacecraft mass, landing site selection, and cost estimates. Interplanetary Cruising is therefore submitted to HAS for student use both before and during Mars or other interplanetary mission planning workshops. Notation in this document distinguishes vectors from scalars using bold characters in a vector's variable name. The "•" operator denotes a scalar product between two vectors. If vectors a and b are represented with scalar coordinates such that a = (a1, a2, a3) and b = (b1, b2, b3) in some 3- dimensional Cartesian system, a • b = a1 b1 + a2 b2 + a3 b3 = a b cos γ, where γ is the angle between a and b. 2. Introduction Material in this reference will help us understand the pedigree and limitations applying to paths (trajectories) followed by spacecraft about the Sun as they "cruise" through interplanetary space. Although cruise between Earth and Mars is used in specific examples, this lesson's physics would apply to travel between any two reasonably isolated objects in our solar system.