DOCSLIB.ORG

SME-SMAD Ch 10 FINAL Web 10.7.2.3.Fm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SME-SMAD Ch 10 FINAL Web 10.7.2.3.Fm 10 Orbit and Constellation Design—Selecting the Right Orbit 10.7 Design of Interplanetary Orbits compute the parameters when crossing the orbit of the target planet, at another epoch (the assumption of unper- turbed Kepler orbits allows one to do so without doing a Faster Trajectories lengthy numerical integration). One warning is appropri- Ron Noomen, Delft University of Technology ate, though; one should check whether the conditions are favorable to reach the orbit of the target planet at all. If Using the recipe given in Table 10-29, one can com- not, one should stop after Step 12 for an inbound mission pute the parameters of a Hohmann transfer between any and after Step 13 for an outbound mission; if needed, pair of planets in a straightforward manner. Attractive as proceed with a new vector V∞,1. Since we are dealing it is from an energy and computational point of view, a with circular orbits of the departure and target planets, the Hohmann transfer does have a number of drawbacks: (1) characteristics of the transfer orbit and related parameters the launch epoch is very strict, with little or no room for only depend on the relative geometry of the two planets, deviations (at least ideally; pericenter and apocenter are the absolute positions do not matter. The epochs of depar- by definition on opposite sides of the central body, with ture and arrival can be computed with the equations given consequences for the departure and arrival epochs), (2) in Sec. 10.7.2, previously, with the travel time TH re- the time-of-flight may be too long for some applications, placed by Ttr and the transfer angle π (valid for a Hohm- and (3) although it represents the most energy-efficient ann transfer) replaced by Δν = ν2–ν1 (the true anomaly of direct transfer, the total ΔV required for a direct trip to the satellite, at departure and arrival, respectively). distant planets may simply be too large to be delivered by Comparing the values given in Tables 10-29 and even the most powerful combination of launcher, upper 10web-1, it is clear that the transfer time can be reduced stage and on-board engine (an indirect trajectory, involv- significantly; in this example, the reduction is by a factor ing planetary flybys, could provide a solution; more on of 3.5. However, this fast solution does not come for free: this in Sec. 10.7.1). the amount of ΔV required increases by a factor of almost In order to compensate for the first 2 drawbacks, one 4 (with major consequences for the amount of propellant, might consider to leave the departure planet (e.g., Earth) of course). The conditions of the example in Table under conditions which are different from those of a 10web-1 are (deliberately) way off those for a Hohmann Hohmann transfer; a similar flexibility can be introduced transfer; following the same recipe one can easily study for the arrival geometry. This is the most general situa- the effect of minor variations with respect to the Hohm- tion as sketched in Fig. 10-28. Clearly, our vehicle now ann conditions (in departure epoch, in ΔV0, in launch di- follows an elliptical orbit with dimensions that exceed rection). those of a Hohmann transfer orbit. As shown in Lambert Problem Fig. 10-28, it can reach the orbit of the target planet on its In Sec. 10.7, two possibilities have been presented to way out, before reaching apocenter; in such a case we travel from planet A to planet B: a Hohmann transfer and speak of a so-called Type-I transfer, which is clearly fast- an arbitrary Kepler orbit (using the technique of forward er than a Hohmann transfer. In case the orbit of the target propagation). The first option is easy but limited in re- planet is reached after having passed the apocenter, we sults, whereas the second option leaves questions on the speak of a Type-II transfer, which has a longer transfer proper combination of the departure conditions: launch time compared to a Hohmann transfer. Both options epoch, excess velocity and direction thereof. In reality, (Type-I, Type-II) can have advantages and disadvantag- designers of interplanetary missions use a technique that es (travel time, ΔV, departure, and arrival geometry, pos- reverses the formulation of the problem: assuming that sibilities for continuation to another planet), so both the epochs of departure and arrival are known, the ques- should be considered. tions now become: what are the parameters of the Kepler In Table 10web-1, where use is made again of the orbit connecting the two? What is the corresponding ΔV? patched conics approach, and where the departure condi- What are the departure and arrival geometries? What is tions are given by the position of the departure planet, the time-of-flight? This problem is generally referred to and a vectorial excess velocity V∞,1. This excess velocity as the Lambert problem. In this web article, the essential has a magnitude and a direction; the latter is specified by formulas to solve this problem for high-thrust propulsion an angle α with respect to the tangential direction (note are presented. The method also serves as a good starter that it is different from the more common flight path an- for the design of low-thrust trajectories. The theory for gle φ). By virtue of the assumption of unperturbed Kepler tackling the Lambert problem is taken from Gooding orbits, all other parameters follow unambiguously. In [1990]. general wording, this approach is called “forward propa- Referring to Fig. 10web-1, the Lambert problem is gation”: define new initial conditions (e.g., at Earth) and easily formulated: knowing an initial position r1, a final ©2011 Microcosm Inc. W10/4 Table 10web-0, Fig. 10web-0, Eq. 10web-0 W10/5 Orbit and Constellation Design—Selecting the Right Orbit 10.7 Table 10web-1. Summary of the Procedure to Compute the Characteristics of an Arbitrary Interplanetary Transfer. Live Calc The meaning of the parameters is explained in the main body of the text. The example values hold for a trip from the Earth (at 1.0 AU) to Mars (at 1.52366 AU), with circular parking orbit altitudes of 185 and 500 km, respectively. The vehicle leaves (the sphere-of-influence of) the Earth with a V∞,1 of 10 km/s which is directed at an angle α of 15 deg with respect to the tangent of the circular orbit. The “*” sign indicates that care must be taken to choose the proper quadrant (use the atan2 function; see App. D.4); the “**” sign indicates that the proper sign of V2,rad must be chosen (depending on a Type-I or Type-II transfer). Step Parameter Expression Example 1 V (heliocentric velocity of departure planet) 29.784 km/s dep Vrdep=μ( S / dep ) 2 Vtar (heliocentric velocity of target planet) Vrtar=μ()S / tar 24.129 km/s 3 V (circular velocity around departure planet) 7.793 km/s c0 Vrc0dep0=μ( / ) V (circular velocity around target planet) 4 c3 Vrc3tar3=μ()/ 3.315 km/s 5 V1,rad (radial heliocentric velocity at departure position) VV1, r a d= ∞ ,1 sinα 2.588 km/s 6 V (tangential heliocentric velocity at departure position) 39.444 km/s 1,tan VVV1, t a n=+dep ∞ ,1 cosα 7 V (heliocentric velocity at departure position) 22 39.529 km/s 1 VV1=+( 1,rad V 1,tan ) 8 H (specific angular momentum) 8 2 HrV= 11,tan 59.01 10 km /s 9 φ1 (flight path angle at departure position) (*) φ=11,rad1,tanatan(VV / ) 3.754 deg a (semi-major axis of transfer orbit) 2 8 10 tr arVtr =μμ0.5SS1 /( / − 0.5 1) 6.27 10 km 2 2 11 etr (eccentricity of transfer orbit) eHVr=+⎡12() / μ 0.5 −μ / ⎤ 0.763 tr ⎣⎢ S1S1( )⎦⎥ r (pericenter distance) 8 12 p rap =−tr(1 e tr ) 1.49 10 km r (apocenter distance) 8 13 a raatrtr=+(1 e) 11.05 10 km 14 V (heliocentric velocity at target position) 30.866 km/s 2 Vra2S()μ−()2/ 2 1/ tr 15 V (tangential heliocentric velocity at target position) 25.887 km/s 2,tan VHr2,tan= / 2 16 V (radial heliocentric velocity at target position) (**) 22 16.810 km/s 2,rad VVV2,rad=−( 2 2,tan ) 2 17 V (excess velocity at target position) ⎡ 2 ⎤ 16.901 km/s ∞,2 VVVV∞,2=+−⎢ 2,rad() 2,tan tar ⎥ ⎣ ⎦ 18 φ2 (flight path angle at target position) (*) φ=22,rad2,tanatan(VV / ) 32.997 deg 19 V (velocity at pericenter of hyperbola around departure planet) 2 14.882 km/s 0 VrV0dep0,1=μ(2/ +∞ ) 20 V (velocity at pericenter of hyperbola around target planet) 2 17.540 km/s 3 VrV3tar3,2=μ(2/ +∞ ) ΔV (maneuver at pericenter around departure planet) 21 0 Δ=VVV00 −c 0 7.089 km/s ΔV (maneuver at pericenter around target planet) 22 3 Δ=VV333c − V 14.224 km/s ΔV (total velocity change) 23 tot ΔVVVtot =Δ03 +Δ 21.313 km/s 24 ν (true anomaly at departure position) (*) ⎡ ⎤ 8.680 deg 1 νμ11,rad1,tan=−atan⎣VHVH / (S )⎦ ν (true anomaly at target position) (*) ⎡ ⎤ 25 2 νμ22,rad2,tan=−atan⎣VHVH / (S )⎦ 78.576 deg 26 E (eccentric anomaly at departure position) Eee=−+2atan⎡ (1 ) / (1 ) tanν / 2⎤ 0.056 rad 1 11⎣ ()tr tr ⎦ 27 E (eccentric anomaly at target position) Eee=−+2atan⎡ (1 ) / (1 ) tanν / 2⎤ 0.583 rad 2 22⎣ ()tr tr ⎦ 28 M1 (mean anomaly at departure position) MEe11=−sin( E 1 ) 0.013 rad M (mean anomaly at target position) 29 2 MEe22=−sin( E 2 ) 0.163 rad 6.47×106 s = 74.9 30 T ( transfer time) 3 tr TMMtr=−()21//( μ S a tr ) days = 0.205 yrs Table 10web-1, Fig.
Load more

Recommended publications
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • Spacex Rocket Data Satisfies Elementary Hohmann Transfer Formula E-Mail: [email protected] and [email protected]
    IOP Physics Education Phys. Educ. 55 P A P ER Phys. Educ. 55 (2020) 025011 (9pp) iopscience.org/ped 2020 SpaceX rocket data satisfies © 2020 IOP Publishing Ltd elementary Hohmann transfer PHEDA7 formula 025011 Michael J Ruiz and James Perkins M J Ruiz and J Perkins Department of Physics and Astronomy, University of North Carolina at Asheville, Asheville, NC 28804, United States of America SpaceX rocket data satisfies elementary Hohmann transfer formula E-mail: [email protected] and [email protected] Printed in the UK Abstract The private company SpaceX regularly launches satellites into geostationary PED orbits. SpaceX posts videos of these flights with telemetry data displaying the time from launch, altitude, and rocket speed in real time. In this paper 10.1088/1361-6552/ab5f4c this telemetry information is used to determine the velocity boost of the rocket as it leaves its circular parking orbit around the Earth to enter a Hohmann transfer orbit, an elliptical orbit on which the spacecraft reaches 1361-6552 a high altitude. A simple derivation is given for the Hohmann transfer velocity boost that introductory students can derive on their own with a little teacher guidance. They can then use the SpaceX telemetry data to verify the Published theoretical results, finding the discrepancy between observation and theory to be 3% or less. The students will love the rocket videos as the launches and 3 transfer burns are very exciting to watch. 2 Introduction Complex 39A at the NASA Kennedy Space Center in Cape Canaveral, Florida. This launch SpaceX is a company that ‘designs, manufactures and launches advanced rockets and spacecraft.
    [Show full text]
  • ORBIT MANOUVERS • Orbital Plane Change (Inclination) It Is an Orbital
    UNIVERSITY OF ANBAR ADVANCED COMMUNICATIONS SYSTEMS FOR 4th CLASS STUDENTS COLLEGE OF ENGINEERING by: Dr. Naser Al-Falahy ELECTRICAL ENGINEERING WE EK 4 ORBIT MANOUVERS Orbital plane change (inclination) It is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an orbital plane change as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector (delta v) at the orbital nodes (i.e. the point where the initial and desired orbits intersect, the line of orbital nodes is defined by the intersection of the two orbital planes). In general, inclination changes can take a very large amount of delta v to perform, and most mission planners try to avoid them whenever possible to conserve fuel. This is typically achieved by launching a spacecraft directly into the desired inclination, or as close to it as possible so as to minimize any inclination change required over the duration of the spacecraft life. When both orbits are circular (i.e. e = 0) and have the same radius the Delta-v (Δvi) required for an inclination change (Δvi) can be calculated using: where: v is the orbital velocity and has the same units as Δvi (Δi ) inclination change required. Example Calculate the velocity change required to transfer a satellite from a circular 600 km orbit with an inclination of 28 degrees to an orbit of equal size with an inclination of 20 degrees. SOLUTION, r = (6,378.14 + 600) × 1,000 = 6,978,140 m , ϑ = 28 - 20 = 8 degrees Vi = SQRT[ GM / r ] Vi = SQRT[ 3.986005×1014 / 6,978,140 ] Vi = 7,558 m/s Δvi = 2 × Vi × sin(ϑ/2) Δvi = 2 × 7,558 × sin(8/2) Δvi = 1,054 m/s 18 UNIVERSITY OF ANBAR ADVANCED COMMUNICATIONS SYSTEMS FOR 4th CLASS STUDENTS COLLEGE OF ENGINEERING by: Dr.
    [Show full text]
  • Last Class Commercial C Rew S a N D Private Astronauts Will Boost International S P a C E • Where Do Objects Get Their Energy?
    9/9/2020 Today’s Class: Gravity & Spacecraft Trajectories CU Astronomy Club • Read about Explorer 1 at • Need some Space? http://en.wikipedia.org/wiki/Explorer_1 • Read about Van Allen Radiation Belts at • Escape the light pollution, join us on our Dark Sky http://en.wikipedia.org/wiki/Van_Allen_radiation_belt Trips • Complete Daily Health Form – Stargazing, astrophotography, and just hanging out – First trip: September 11 • Open to all majors! • The stars respect social distancing and so do we • Head to our website to join our email list: https://www.colorado.edu/aps/our- department/outreach/cu-astronomy-club • Email Address: [email protected] Astronomy 2020 – Space Astronomy & Exploration Astronomy 2020 – Space Astronomy & Exploration 1 2 Last Class Commercial C rew s a n d Private Astronauts will boost International S p a c e • Where do objects get their energy? Article by Elizabeth Howell Station's Sci enc e Presentation by Henry Universe Today/SpaceX –Conservation of energy: energy Larson cannot be created or destroyed but ● SpaceX’s Demo 2 flight showed what human spaceflight Question: can look like under private enterprise only transformed from one type to ● NASArelied on Russia’s Soyuz spacecraft to send astronauts to the ISS in the past another. ● "We're going to have more people on the International Space Station than we've had in a long time,” NASA –Energy comes in three basic types: Administrator Jim Bridenstine ● NASAis primarily using SpaceX and Boeing for their kinetic, potential, radiative. “commercial crew”program Astronomy 2020 – Space Astronomy & Exploration 3 4 Class Exercise: Which of the following Today’s Learning Goals processes violates a conservation law? • What are range of common Earth a) Mass is converted directly into energy.
    [Show full text]
  • Fundamentals of Orbital Mechanics
    Chapter 7 Fundamentals of Orbital Mechanics Celestial mechanics began as the study of the motions of natural celestial bodies, such as the moon and planets. The field has been under study for more than 400 years and is documented in great detail. A major study of the Earth-Moon-Sun system, for example, undertaken by Charles-Eugene Delaunay and published in 1860 and 1867 occupied two volumes of 900 pages each. There are textbooks and journals that treat every imaginable aspect of the field in abundance. Orbital mechanics is a more modern treatment of celestial mechanics to include the study the motions of artificial satellites and other space vehicles moving un- der the influences of gravity, motor thrusts, atmospheric drag, solar winds, and any other effects that may be present. The engineering applications of this field include launch ascent trajectories, reentry and landing, rendezvous computations, orbital design, and lunar and planetary trajectories. The basic principles are grounded in rather simple physical laws. The paths of spacecraft and other objects in the solar system are essentially governed by New- ton’s laws, but are perturbed by the effects of general relativity (GR). These per- turbations may seem relatively small to the layman, but can have sizable effects on metric predictions, such as the two-way round trip Doppler. The implementa- tion of post-Newtonian theories of orbital mechanics is therefore required in or- der to meet the accuracy specifications of MPG applic ations. Because it had the need for very accurate trajectories of spacecraft, moon, and planets, dating back to the 1950s, JPL organized an effort that soon became the world leader in the field of orbital mechanics and space navigation.
    [Show full text]
  • 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.
    [Show full text]
  • Optimal Triangular Lagrange Point Insertion Using Lunar Gravity Assist
    Optimal Triangular Lagrange Point Insertion Using Lunar Gravity Assist Julio César Benavides Elizabeth Davis Brian Wadsley Aerospace Engineering Aerospace Engineering Aerospace Engineering The Pennsylvania State University The Pennsylvania State University The Pennsylvania State University 306 George Building 306 George Building 306 George Building (814) 865-7396 (814) 865-7396 (814) 865-7396 [email protected] [email protected] [email protected] ABSTRACT 1.1 Lagrange Point Geometry In this paper, we design optimal trajectories that transfer satellites Given two massive bodies in circular orbits around their common from Earth orbit to the Sun-Earth triangular Lagrange points (L4 center of mass, there are five positions in space where a third and L5) using a lunar gravity assist during the year 2010. An body of comparatively negligible mass will maintain its position optimal trajectory is defined as one that minimizes total mission relative to the two primary bodies. These five points, known as velocity change (∆v), which is directly proportional to fuel Lagrange points, are the stationary solutions of the circular, consumption and operational cost. These optimal trajectories are restricted three-body problem [1]. Figure 1.1 shows the location determined using two heuristic algorithms: Differential Evolution of the five Lagrange points relative to the positions of two and Covariance Matrix Adaptation. The three parameters being primary masses in the rotating coordinate frame. varied are the mission commencement time, Hohmann transfer target radius above the center of the moon, and the midcourse correction maneuver time of flight. L4 Optimal trajectory parameters predicted by Differential Evolution 60° and Covariance Matrix Adaptation agree closely.
    [Show full text]
  • USAAAO First Round 2015
    USAAAO First Round 2015 This round consists of 30 multiple-choice problems to be completed in 75 minutes. You may only use a scientific calculator and a table of constants during the test. The top 50% will qualify for the Second Round. 1. At arms length, the width of a fist typically subtends how many degrees of arc? a. 1​o b. 5​o c. 10​o d. 15​o e. 20​o 2. To have a lunar eclipse, the line of nodes must be pointing at the sun. The moon must also be in what phase? a. New b. First Quarter c. Waxing Gibbous d. Full e. Waning Crescent 3. Mars orbits the sun once every 687 days. Suppose Mars is currently in the constellation Virgo. What constellation will it most likely be in a year from now? a. Virgo b. Scorpius c. Aquarius d. Taurus e. Cancer 4. To calculate the field of view of a telescope, you measure the time it takes Capella (RA:5.27h, dec:45.98​o)​ to pass across the eyepiece. If the measured time is 2 minutes and 30 seconds, what is the field of view in arcseconds? a. 11.6’ b. 26.5’ c. 37.5’ d. 52.5’ e. 66.8 5. A telescope with focal length of 20 mm and aperture of 10 mm is connected to your smartphone, which has a CCD that measures 4.0mm by 4.0mm. The CCD is 1024 by 1024 pixels. Which is closest to the field of view of the telescope? a.
    [Show full text]
  • Orbit Transfer Optimization of Spacecraft with Impulsive Thrusts Using Genetic Algorithm
    ORBIT TRANSFER OPTIMIZATION OF SPACECRAFT WITH IMPULSIVE THRUSTS USING GENETIC ALGORITHM A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY AHMET YILMAZ IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING SEPTEMBER 2012 Approval of the thesis: ORBIT TRANSFER OPTIMIZATION OF A SPACECRAFT WITH IMPULSIVE THRUST USING GENETIC ALGORITHM Submitted by AHMET YILMAZ in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Süha Oral Head of Department, Mechanical Engineering Prof. Dr. M. Kemal Özgören Supervisor, Mechanical Engineering Dept., METU Examining Committee Members Prof. Dr. S. Kemal İder Mechanical Engineering Dept., METU Prof. Dr.M. Kemal Özgören Mechanical Engineering Dept., METU Prof. Dr. Ozan Tekinalp Aerospace Engineering Dept., METU Asst. Prof. Dr. Yiğit Yazıcıoğlu Mechanical Engineering Dept., METU Başar SEÇKİN, M.Sc. Roketsan Missile Industries Inc. Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name: Ahmet Yılmaz Signature : iii ABSTRACT ORBIT TRANSFER OPTIMIZATION OF A SPACECRAFT WITH IMPULSIVE THRUST USING GENETIC ALGORITHM Yılmaz, Ahmet M.S., Department of Mechanical Engineering Supervisor: Prof. Dr. M.Kemal Özgören September 2012, 127 pages This thesis addresses the orbit transfer optimization problem of a spacecraft.
    [Show full text]