DOCSLIB.ORG

Aiaa 2002-4717 Application of Tisserand's Criterion to The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Aiaa 2002-4717 Application of Tisserand's Criterion to The AIAA 2002-4717 APPLICATION OF TISSERAND’S CRITERION TO THE DESIGN OF GRAVITY ASSIST TRAJECTORIES J. K. Miller* Jet Propulsion Laboratory California Institute of Technology Pasadena, California Connie J. Weekst Loyola Marymount University Los Angeles, California ABSTRACT ration of planets that otherwise would not be ac- Gravity assist involves the use of the gravita- cessible with current launch vehicle capability. tional attraction of an intermediate planet to in- Most interplanetary and planetary orbiter mis- crease the orbit energy of a spacecraft, enabling sion trajectories since the beginning of the space it to attain the target planet. The primary diffi- age have used Keplerian two-body motion in their culty in designing a gravity assist trajectory is the design. Missions requiring three-body transfers are determination of the encounter times of the launch generally limited to those involving the satellites planet, intermediate planet and target planet. of the major planets, for example missions to La- Once these times are determined, it is a relatively grange points in the Earth-Sun system, or the Voy- simple procedure to use Lambert’s theorem and nu- ager mission to the outer planets. Even though merical integration to complete the detailed design. gravity assist trajectories can be designed by re- In this paper, a procedure is described for finding peated application of two-body theory, they are in- these encounter times using Lambert’s theorem and cluded in the three-body classification because the a new criterion based on Tisserand’s criterion to gravity assist requires a simultaneous exchange of identify pairs of transfer orbits between the launch energy among three bodies. The three-body th& planet and intermediate planet and between the in- ory employed for the design of gravity assist tra- termediate planet and target planet. jectories involves the use either of vectors defin- ing the approach and departure hyperbolic asymp- INTRODUCTION totes with respect to the gravity assist planet, or of Gravity assist trajectories are an important Tisserand’s criterion, which pertains to the inter- class of trajectories that have been used by Voy- planetary Keplerian orbits connecting the launch, ager, Galileo, Cassini, and other missions to tour gravity assist and target planets. It will be shown the solar system. The design of interplanetary tra- that while both design techniques follow from the jectories involves finding an orbit that will trans- Jacobi integral, they yield significantly different re- fer a spacecraft from the vicinity of one planet to sults, since they represent different approximations the vicinity of another planet. The accessibility of of the true equations of motion. In this paper, a a target planet, particularly those beyond the or- criterion is developed that combines both of these bit of Jupiter, depends on finding a transfer orbit design techniques. with energy relative to the initial starting point, normally the Earth, within the capability of the JACOB1 INTEGRAL launch vehicle. The use of gravity assist to increase An important integral describing constraints the transfer orbit energy has opened up the explo- on energy transfer for the restricted three body problem was discovered by Carl Gustav Jacob Ja- * Member of the Navigation Systems SectionJet Propulsion cobi in the Nineteenth century. A point mass mov- Laboratory, Associate Fellow ing in the vicinity of two massive bodies in circular t Professor of Mathematics, Loyola Marymount University orbits about their barycenter will conserve a cer- Copyright 200mhe American Institute of Aeronautics and As- tain function of the state and gravitational param- tronautics Inc. All rights reserved eters of the massive bodies referred to as Jacobi’s 1 American Institute of Aeronautics and Astronautics integral. The constant of integration is called Ja- cobi's constant. The equations of motion for two massive bodies and a point mass are given by, 21 - x 22 - x X = GM1- i-GM2- rf 4 Equation 4 may be put into a form that can be in- The two massive bodies rotate around the barycen- tegrated by defining the function ter and the rotation rate is simply 27~divided by the period of the orbit, w= P3 \JGM1+GM2 and substituting into Equations 4. where p is the distance separating the two massive bodies. The geometry is illustrated on Figure 1. rY Adding Equations 6, The integral of Equation 7 given below is called the Jacobi integral. .2 .2 .2 2' +y' +z' =2u-c or Figure 1 Restricted Two Body Geometry .2 -2 .2 2' + y' + 2' = W2Xl2 + w2yt2+ GM1 GM2 The primed coordinate system (XI, y', 2') in Figure 2- + 2- -c (9) r1 r2 1 represents a rotating coordinate system in which the two massive bodies lie on the x' axis, with where C is the constant of integration. TISSERAND'S CRITERION x = x'coswt - y'sinwt Francois Felix Tisserand was a nineteenth cen- y = x' sin wt + y' cos wt tury astronomer who discovered a unique applica- tion of Jacobi's integral to identify comets. In the z = z' (3) restricted three body problem, a certain function of After differentiating Equations 3 twice, substitut- the orbit elements before and after a planetary en- ing into Equations 1 and eliminating the sine and counter is conserved. If this function is computed for two comet observations on different orbits and cosine terms, the following result is obtained as shown in Reference 1. the results are the same, one may conclude that the 2 American Institute of Aeronautics and Astronautics observations are of the same comet and the comet 14 is multiplied through by p and divided by GM1, has encountered a planet between the observations. Tisserand’s criterion in dimensionless coordinates This may be confirmed by propagating the orbits becomes forward or backward in time to see if they encoun- tered a planet. In the application of Tisserand’s criterion to gravity assist trajectory design, the procedure is If the first observation of a spacecraft or comet reversed. Transfer trajectories from the launch has orbit elements al, el and il and the second ob- planet to the intermediate planet and from the in- servation after a planetary encounter has orbit el- termediate planet to the target planet are com- ements a2, e2 and iz, then puted using Lambert’s theorem. These trajecto- ries are matched based on Tisserand’s criterion to identify viable launch and encounter opportunities. Tisserand’s criterion follows directly from Jacobi’s integral. Using Equations 3, the Jacobi integral is transformed back to inertial coordinates (the un- primed coordinates on Figure 1). GRAVITY ASSIST VECTOR DIAGRAM x2 + 7j2 + i2= 2w(xjr - yk)+ Figure 2 shows the encounter geometry in the GM1 GM2 2- +2--c vicinity of the intermediate planet that supplies r1 r2 the gravity assist energy boost to the spacecraft. For GMl much greater than GM2, the z component The incoming velocity of the spacecraft (V1)is sub- of the angular momentum vector is given by, tracted from the planet velocity (V,) to obtain the planet relative approach velocity (vi) as shown in xy - yx = h, = hcosi (11) the upper vector diagram on Figure 2. The lower vector diagram shows the same relationship for the h = dGM1 a(1- e2) outgoing velocity vectors. If the incoming and out- going velocities are computed far from the planet and from the vis viva integral the energy is given yet close enough to the planet that the heliocentric by, 21 energy may be assumed constant, the velocities k2 + y2+ t2= GMi(- - -) (12) r1 a Substituting Equations 11 and 12 into Equation 10 P gives 21 GM1(- - -) - 2wdGM1 a(1- e2)cosi = r1 a GMi GM2 2- +2--c r1 r2 Substituting Equation 2 for w and for small GM2 compared with GM1, a( 1 - e2) Cx- cosi (14) GM1a + 2GMl\/l P3 In the literature, Tisserand’s criterion is often de- veloped in dimensionless coordinates and the Ja- cobi constant modified to remove constant param- Figure 2 Gravity Assist Vector Diagram eters. If a is divided by p to define ii and Equation 3 American Institute of Aeronautics and Astronautics vi and v, are approximately the v, vectors as- The energy of the spacecraft relative to the planet, sociated with the two-body hyperbola about the the potential energy of the spacecraft relative to planet. In the limit of two-body motion assumed the Sun and the velocity of the planet relative to for patched conic trajectories, vi and v, are equal the sun may be regarded as constant. Collecting in magnitude. Since the planet velocity is also as- these “constant” terms on the left side gives, sumed to be constant during the relatively short time interval of the planet encounter, the outgoing 3GM, vector diagram may be superimposed on the incom- C=-- Vi2 ing vector diagram as shown on Figure 2. The out- P going heliocentric spacecraft velocity magnitude is N greater than the incoming velocity magnitude and N- GMs +2GMs al(l - COSil (22) a1 { P3 ea the spacecraft has acquired additional orbit energy ’ relative to the Sun. The energy acquired by the Equation 22 provides an interesting insight into spacecraft comes from the Sun and the planet. the geometrical meaning of Jacobi’s constant in the Consider the triangle formed by the spacecraft limit where one of the gravitating bodies is much and planet heliocentric velocity vectors and the in- more massive than the other. The terms in the coming velocity vector. From the law of cosines, Jacobi integral are related to the velocity vector diagram of the participating bodies and the rela- tionship to energy conservation is incidental. The orbit of the planet about the Sun may be CASSINI TRAJECTORY DESIGN approximated by a circle with velocity magnitude given by, The Cassini mission to Saturn provides an ex- ample of the application of Tisserand’s criterion to the design of a gravity assist trajectory.
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]
  • The Velocity Dispersion Profile of the Remote Dwarf Spheroidal Galaxy
    The Velocity Dispersion Profile of the Remote Dwarf Spheroidal Galaxy Leo I: A Tidal Hit and Run? Mario Mateo1, Edward W. Olszewski2, and Matthew G. Walker3 ABSTRACT We present new kinematic results for a sample of 387 stars located in and around the Milky Way satellite dwarf spheroidal galaxy Leo I. These spectra were obtained with the Hectochelle multi-object echelle spectrograph on the MMT, and cover the MgI/Mgb lines at about 5200A.˚ Based on 297 repeat measure- ments of 108 stars, we estimate the mean velocity error (1σ) of our sample to be 2.4 km/s, with a systematic error of 1 km/s. Combined with earlier re- ≤ sults, we identify a final sample of 328 Leo I red giant members, from which we measure a mean heliocentric radial velocity of 282.9 0.5 km/s, and a mean ± radial velocity dispersion of 9.2 0.4 km/s for Leo I. The dispersion profile of ± Leo I is essentially flat from the center of the galaxy to beyond its classical ‘tidal’ radius, a result that is unaffected by contamination from field stars or binaries within our kinematic sample. We have fit the profile to a variety of equilibrium dynamical models and can strongly rule out models where mass follows light. Two-component Sersic+NFW models with tangentially anisotropic velocity dis- tributions fit the dispersion profile well, with isotropic models ruled out at a 95% confidence level. Within the projected radius sampled by our data ( 1040 pc), ∼ the mass and V-band mass-to-light ratio of Leo I estimated from equilibrium 7 models are in the ranges 5-7 10 M⊙ and 9-14 (solar units), respectively.
    [Show full text]
  • Chapter 2: Earth in Space
    Chapter 2: Earth in Space 1. Old Ideas, New Ideas 2. Origin of the Universe 3. Stars and Planets 4. Our Solar System 5. Earth, the Sun, and the Seasons 6. The Unique Composition of Earth Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Earth in Space Concept Survey Explain how we are influenced by Earth’s position in space on a daily basis. The Good Earth, Chapter 2: Earth in Space Old Ideas, New Ideas • Why is Earth the only planet known to support life? • How have our views of Earth’s position in space changed over time? • Why is it warmer in summer and colder in winter? (or, How does Earth’s position relative to the sun control the “Earthrise” taken by astronauts aboard Apollo 8, December 1968 climate?) The Good Earth, Chapter 2: Earth in Space Old Ideas, New Ideas From a Geocentric to Heliocentric System sun • Geocentric orbit hypothesis - Ancient civilizations interpreted rising of sun in east and setting in west to indicate the sun (and other planets) revolved around Earth Earth pictured at the center of a – Remained dominant geocentric planetary system idea for more than 2,000 years The Good Earth, Chapter 2: Earth in Space Old Ideas, New Ideas From a Geocentric to Heliocentric System • Heliocentric orbit hypothesis –16th century idea suggested by Copernicus • Confirmed by Galileo’s early 17th century observations of the phases of Venus – Changes in the size and shape of Venus as observed from Earth The Good Earth, Chapter 2: Earth in Space Old Ideas, New Ideas From a Geocentric to Heliocentric System • Galileo used early telescopes to observe changes in the size and shape of Venus as it revolved around the sun The Good Earth, Chapter 2: Earth in Space Earth in Space Conceptest The moon has what type of orbit? A.
    [Show full text]
  • Multi-Body Trajectory Design Strategies Based on Periapsis Poincaré Maps
    MULTI-BODY TRAJECTORY DESIGN STRATEGIES BASED ON PERIAPSIS POINCARÉ MAPS A Dissertation Submitted to the Faculty of Purdue University by Diane Elizabeth Craig Davis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2011 Purdue University West Lafayette, Indiana ii To my husband and children iii ACKNOWLEDGMENTS I would like to thank my advisor, Professor Kathleen Howell, for her support and guidance. She has been an invaluable source of knowledge and ideas throughout my studies at Purdue, and I have truly enjoyed our collaborations. She is an inspiration to me. I appreciate the insight and support from my committee members, Professor James Longuski, Professor Martin Corless, and Professor Daniel DeLaurentis. I would like to thank the members of my research group, past and present, for their friendship and collaboration, including Geoff Wawrzyniak, Chris Patterson, Lindsay Millard, Dan Grebow, Marty Ozimek, Lucia Irrgang, Masaki Kakoi, Raoul Rausch, Matt Vavrina, Todd Brown, Amanda Haapala, Cody Short, Mar Vaquero, Tom Pavlak, Wayne Schlei, Aurelie Heritier, Amanda Knutson, and Jeff Stuart. I thank my parents, David and Jeanne Craig, for their encouragement and love throughout my academic career. They have cheered me on through many years of studies. I am grateful for the love and encouragement of my husband, Jonathan. His never-ending patience and friendship have been a constant source of support. Finally, I owe thanks to the organizations that have provided the funding opportunities that have supported me through my studies, including the Clare Booth Luce Foundation, Zonta International, and Purdue University and the School of Aeronautics and Astronautics through the Graduate Assistance in Areas of National Need and the Purdue Forever Fellowships.
    [Show full text]
  • Perturbation Theory in Celestial Mechanics
    Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion.
    [Show full text]
  • On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
    Noname manuscript No. (will be inserted by the editor) On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model F. Topputo Received: date / Accepted: date Abstract In this paper two-impulse Earth–Moon transfers are treated in the restricted four-body problem with the Sun, the Earth, and the Moon as primaries. The problem is formulated with mathematical means and solved through direct transcription and multiple shooting strategy. Thousands of solutions are found, which make it possible to frame known cases as special points of a more general picture. Families of solutions are defined and characterized, and their features are discussed. The methodology described in this paper is useful to perform trade-off analyses, where many solutions have to be produced and assessed. Keywords Earth–Moon transfer low-energy transfer ballistic capture trajectory optimization · · · · restricted three-body problem restricted four-body problem · 1 Introduction The search for trajectories to transfer a spacecraft from the Earth to the Moon has been the subject of countless works. The Hohmann transfer represents the easiest way to perform an Earth–Moon transfer. This requires placing the spacecraft on an ellipse having the perigee on the Earth parking orbit and the apogee on the Moon orbit. By properly switching the gravitational attractions along the orbit, the spacecraft’s motion is governed by only the Earth for most of the transfer, and by only the Moon in the final part. More generally, the patched-conics approximation relies on a Keplerian decomposition of the solar system dynamics (Battin 1987). Although from a practical point of view it is desirable to deal with analytical solutions, the two-body problem being integrable, the patched-conics approximation inherently involves hyperbolic approaches upon arrival.
    [Show full text]
  • Wobbling Stars and Planets in This Activity, You
    Activity: Wobbling Stars and Planets In this activity, you will investigate the effect of the relative masses and distances of objects that are in orbit around a star or planet. The most common understanding of how the planets orbit around the Sun is that the Sun is stationary while the planets revolve around the Sun. That is not quite true and is a little more complicated. PBS An important parameter when considering orbits is the center of mass. The concept of the center of mass is that of an average of the masses factored by their distances from one another. This is also commonly referred to as the center of gravity. For example, the balancing of a ETIC seesaw about a pivot point demonstrates the center of mass of two objects of different masses. Another important parameter to consider is the barycenter. The barycenter is the center of mass of two or more bodies that are orbiting each other, and is the point around which both of them orbit. When the two bodies have similar masses, the barycenter may be located between NASA the two bodies (outside of either of bodies) and both objects will follow an orbit around the center of mass. This is the case for a system such as the Sun and Jupiter. In the case where the two objects have very different masses, the center of mass and barycenter may be located within the more massive body. This is the case for the Earth-Moon system. scienceprojectideasforkids.com Classroom Activity Part 1 Materials • Wooden skewers 1.
    [Show full text]
  • PHY140Y 3 Properties of Gravitational Orbits
    PHY140Y 3 Properties of Gravitational Orbits 3.1 Overview • Orbits and Total Energy • Elliptical, Parabolic and Hyperbolic Orbits 3.2 Characterizing Orbits With Energy Two point particles attracting each other via gravity will cause them to move, as dictated by Newton’s laws of motion. If they start from rest, this motion will simply be along the line separating the two points. It is a one-dimensional problem, and easily solved. The two objects collide. If the two objects are given some relative motion to start with, then it is easy to convince yourself that a collision is no longer possible (so long as these particles are point-like). In our discussion of orbits, we will only consider the special case where one object of mass M is much heavier than the smaller object of mass m. In this case, although the larger object is accelerated toward the smaller one, we will ignore that acceleration and assume that the heavier object is fixed in space. There are three general categories of motion, when there is some relative motion, which we can characterize by the total energy of the object, i.e., E ≡ K + U (1) 1 GMm = mv2 − , (2) 2 r2 where K and U are the kinetic and gravitational energy, respectively. 3.3 Energy < 0 In this case, since the total energy is negative, it is not possible for the object to get too far from the mass M, since E = K + U<0 (3) 1 GMm ⇒ mv2 = + E (4) 2 r GM ⇒ r = (5) 1 2 − E 2 v m GM ⇒ r< , (6) −E since r will take on its maximum value when the denominator is minimized (or when v is the smallest value possible, ie.
    [Show full text]
  • Hyperbolic'type Orbits in the Schwarzschild Metric
    Hyperbolic-Type Orbits in the Schwarzschild Metric F.T. Hioe* and David Kuebel Department of Physics, St. John Fisher College, Rochester, NY 14618 and Department of Physics & Astronomy, University of Rochester, Rochester, NY 14627, USA August 11, 2010 Abstract Exact analytic expressions for various characteristics of the hyperbolic- type orbits of a particle in the Schwarzschild geometry are presented. A useful simple approximation formula is given for the case when the deviation from the Newtonian hyperbolic path is very small. PACS numbers: 04.20.Jb, 02.90.+p 1 Introduction In a recent paper [1], we presented explicit analytic expressions for all possible trajectories of a particle with a non-zero mass and for light in the Schwarzschild metric. They include bound, unbound and terminating orbits. We showed that all possible trajectories of a non-zero mass particle can be conveniently characterized and placed on a "map" in a dimensionless parameter space (e; s), where we have called e the energy parameter and s the …eld parameter. One region in our parameter space (e; s) for which we did not present extensive data and analysis is the region for e > 1. The case for e > 1 includes unbound orbits that we call hyperbolic-type as well as terminating orbits. In this paper, we consider the hyperbolic-type orbits in greater detail. In Section II, we …rst recall the explicit analytic solution for the elliptic- (0 e < 1), parabolic- (e = 1), and hyperbolic-type (e > 1) orbits in what we called Region I (0 s s1), where the upper boundary of Region I characterized by s1 varies from s1 = 0:272166 for e = 0 to s1 = 0:250000 for e = 1 to s1 0 as e .
    [Show full text]
  • Study of a Crew Transfer Vehicle Using Aerocapture for Cycler Based Exploration of Mars by Larissa Balestrero Machado a Thesis S
    Study of a Crew Transfer Vehicle Using Aerocapture for Cycler Based Exploration of Mars by Larissa Balestrero Machado A thesis submitted to the College of Engineering and Science of Florida Institute of Technology in partial fulfillment of the requirements for the degree of Master of Science in Aerospace Engineering Melbourne, Florida May, 2019 © Copyright 2019 Larissa Balestrero Machado. All Rights Reserved The author grants permission to make single copies ____________________ We the undersigned committee hereby approve the attached thesis, “Study of a Crew Transfer Vehicle Using Aerocapture for Cycler Based Exploration of Mars,” by Larissa Balestrero Machado. _________________________________________________ Markus Wilde, PhD Assistant Professor Department of Aerospace, Physics and Space Sciences _________________________________________________ Andrew Aldrin, PhD Associate Professor School of Arts and Communication _________________________________________________ Brian Kaplinger, PhD Assistant Professor Department of Aerospace, Physics and Space Sciences _________________________________________________ Daniel Batcheldor Professor and Head Department of Aerospace, Physics and Space Sciences Abstract Title: Study of a Crew Transfer Vehicle Using Aerocapture for Cycler Based Exploration of Mars Author: Larissa Balestrero Machado Advisor: Markus Wilde, PhD This thesis presents the results of a conceptual design and aerocapture analysis for a Crew Transfer Vehicle (CTV) designed to carry humans between Earth or Mars and a spacecraft on an Earth-Mars cycler trajectory. The thesis outlines a parametric design model for the Crew Transfer Vehicle and presents concepts for the integration of aerocapture maneuvers within a sustainable cycler architecture. The parametric design study is focused on reducing propellant demand and thus the overall mass of the system and cost of the mission. This is accomplished by using a combination of propulsive and aerodynamic braking for insertion into a low Mars orbit and into a low Earth orbit.
    [Show full text]
  • THE BINARY COLLISION APPROXIMATION: Iliiilms BACKGROUND and INTRODUCTION Mark T
    O O i, j / O0NF-9208146--1 DE92 040887 Invited paper for Proceedings of the International Conference on Computer Simulations of Radiation Effects in Solids, Hahn-Meitner Institute Berlin Berlin, Germany August 23-28 1992 The submitted manuscript has been authored by a contractor of the U.S. Government under contract No. DE-AC05-84OR214O0. Accordingly, the U S. Government retains a nonexclusive, royalty* free licc-nc to publish or reproduce the published form of uiii contribution, or allow othen to do jo, for U.S. Government purposes." THE BINARY COLLISION APPROXIMATION: IliiilMS BACKGROUND AND INTRODUCTION Mark T. Robinson >._« « " _>i 8 « g i. § 8 " E 1.2 .§ c « •o'c'lf.S8 ° °% as SOLID STATE DIVISION OAK RIDGE NATIONAL LABORATORY Managed by MARTIN MARIETTA ENERGY SYSTEMS, INC. Under Contract No. DE-AC05-84OR21400 § g With the ••g ^ U. S. DEPARTMENT OF ENERGY OAK RIDGE, TENNESSEE August 1992 'g | .11 DISTRIBUTION OP THIS COOL- ::.•:•! :•' !S Ui^l J^ THE BINARY COLLISION APPROXIMATION: BACKGROUND AND INTRODUCTION Mark T. Robinson Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032, U. S. A. ABSTRACT The binary collision approximation (BCA) has long been used in computer simulations of the interactions of energetic atoms with solid targets, as well as being the basis of most analytical theory in this area. While mainly a high-energy approximation, the BCA retains qualitative significance at low energies and, with proper formulation, gives useful quantitative information as well. Moreover, computer simulations based on the BCA can achieve good statistics in many situations where those based on full classical dynamical models require the most advanced computer hardware or are even impracticable.
    [Show full text]
  • Exploring the Orbital Evolution of Planetary Systems
    Exploring the orbital evolution of planetary systems Tabar´e Gallardo∗ Instituto de F´ısica, Facultad de Ciencias, Udelar. (Dated: January 22, 2017) The aim of this paper is to encourage the use of orbital integrators in the classroom to discover and understand the long term dynamical evolution of systems of orbiting bodies. We show how to perform numerical simulations and how to handle the output data in order to reveal the dynamical mechanisms that dominate the evolution of arbitrary planetary systems in timescales of millions of years using a simple but efficient numerical integrator. Trough some examples we reveal the fundamental properties of the planetary systems: the time evolution of the orbital elements, the free and forced modes that drive the oscillations in eccentricity and inclination, the fundamental frequencies of the system, the role of the angular momenta, the invariable plane, orbital resonances and the Kozai-Lidov mechanism. To be published in European Journal of Physics. 2 I. INTRODUCTION With few exceptions astronomers cannot make experiments, they are limited to observe the universe. The laboratory for the astronomer usually takes the form of computer simulations. This is the most important instrument for the study of the dynamical behavior of gravitationally interacting bodies. A planetary system, for example, evolves mostly due to gravitation acting over very long timescales generating what is called a secular evolution. This secular evolution can be deduced analytically by means of the theory of perturbations, but can also be explored in the classroom by means of precise numerical integrators. Some facilities exist to visualize and experiment with the gravitational interactions between massive bodies1–3.
    [Show full text]