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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 -
Trans-Neptunian Objects a Brief History, Dynamical Structure, and Characteristics of Its Inhabitants
Trans-Neptunian Objects A Brief history, Dynamical Structure, and Characteristics of its Inhabitants John Stansberry Space Telescope Science Institute IAC Winter School 2016 IAC Winter School 2016 -- Kuiper Belt Overview -- J. Stansberry 1 The Solar System beyond Neptune: History • 1930: Pluto discovered – Photographic survey for Planet X • Directed by Percival Lowell (Lowell Observatory, Flagstaff, Arizona) • Efforts from 1905 – 1929 were fruitless • discovered by Clyde Tombaugh, Feb. 1930 (33 cm refractor) – Survey continued into 1943 • Kuiper, or Edgeworth-Kuiper, Belt? – 1950’s: Pluto represented (K.E.), or had scattered (G.K.) a primordial, population of small bodies – thus KBOs or EKBOs – J. Fernandez (1980, MNRAS 192) did pretty clearly predict something similar to the trans-Neptunian belt of objects – Trans-Neptunian Objects (TNOs), trans-Neptunian region best? – See http://www2.ess.ucla.edu/~jewitt/kb/gerard.html IAC Winter School 2016 -- Kuiper Belt Overview -- J. Stansberry 2 The Solar System beyond Neptune: History • 1978: Pluto’s moon, Charon, discovered – Photographic astrometry survey • 61” (155 cm) reflector • James Christy (Naval Observatory, Flagstaff) – Technologically, discovery was possible decades earlier • Saturation of Pluto images masked the presence of Charon • 1988: Discovery of Pluto’s atmosphere – Stellar occultation • Kuiper airborne observatory (KAO: 90 cm) + 7 sites • Measured atmospheric refractivity vs. height • Spectroscopy suggested N2 would dominate P(z), T(z) • 1992: Pluto’s orbit explained • Outward migration by Neptune results in capture into 3:2 resonance • Pluto’s inclination implies Neptune migrated outward ~5 AU IAC Winter School 2016 -- Kuiper Belt Overview -- J. Stansberry 3 The Solar System beyond Neptune: History • 1992: Discovery of 2nd TNO • 1976 – 92: Multiple dedicated surveys for small (mV > 20) TNOs • Fall 1992: Jewitt and Luu discover 1992 QB1 – Orbit confirmed as ~circular, trans-Neptunian in 1993 • 1993 – 4: 5 more TNOs discovered • c. -
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 -
1 Trajectory Design for the Transiting Exoplanet
TRAJECTORY DESIGN FOR THE TRANSITING EXOPLANET SURVEY SATELLITE Donald J. Dichmann(1), Joel J.K. Parker(2), Trevor W. Williams(3), Chad R. Mendelsohn(4) (1,2,3,4)Code 595.0, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt MD 20771. 301-286-6621. [email protected] Abstract: The Transiting Exoplanet Survey Satellite (TESS) is a National Aeronautics and Space Administration (NASA) mission, scheduled to be launched in 2017. TESS will travel in a highly eccentric orbit around Earth, with initial perigee radius near 17 Earth radii (Re) and apogee radius near 59 Re. The orbit period is near 2:1 resonance with the Moon, with apogee nearly 90 degrees out-of-phase with the Moon, in a configuration that has been shown to be operationally stable. TESS will execute phasing loops followed by a lunar flyby, with a final maneuver to achieve 2:1 resonance with the Moon. The goals of a resonant orbit with long-term stability, short eclipses and limited oscillations of perigee present significant challenges to the trajectory design. To rapidly assess launch opportunities, we adapted the Schematics Window Methodology (SWM76) launch window analysis tool to assess the TESS mission constraints. To understand the long-term dynamics of such a resonant orbit in the Earth-Moon system we employed Dynamical Systems Theory in the Circular Restricted 3-Body Problem (CR3BP). For precise trajectory analysis we use a high-fidelity model and multiple shooting in the General Mission Analysis Tool (GMAT) to optimize the maneuver delta-V and meet mission constraints. Finally we describe how the techniques we have developed can be applied to missions with similar requirements. -
Secular Chaos and Its Application to Mercury, Hot Jupiters, and The
Secular chaos and its application to Mercury, hot SPECIAL FEATURE Jupiters, and the organization of planetary systems Yoram Lithwicka,b,1 and Yanqin Wuc aDepartment of Physics and Astronomy and bCenter for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, IL 60208; and cDepartment of Astronomy and Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H4 Edited by Adam S. Burrows, Princeton University, Princeton, NJ, and accepted by the Editorial Board October 31, 2013 (received for review May 2, 2013) In the inner solar system, the planets’ orbits evolve chaotically, by including the most important MMRs (11).] It is the cause, for driven primarily by secular chaos. Mercury has a particularly cha- example, of Earth’s eccentricity-driven Milankovitch cycle. How- otic orbit and is in danger of being lost within a few billion years. ever, on timescales ≳107 years, the evolution is chaotic (e.g., Fig. Just as secular chaos is reorganizing the solar system today, so it 1), in sharp contrast to the prediction of linear secular theory. has likely helped organize it in the past. We suggest that extra- That appears to be puzzling, given the small eccentricities and solar planetary systems are also organized to a large extent by inclinations in the solar system. secular chaos. A hot Jupiter could be the end state of a secularly However, despite its importance, there has been little theo- chaotic planetary system reminiscent of the solar system. How- retical understanding of how secular chaos works. Conversely, ever, in the case of the hot Jupiter, the innermost planet was chaos driven by MMRs is much better understood. -
Abstracts of the 50Th DDA Meeting (Boulder, CO)
Abstracts of the 50th DDA Meeting (Boulder, CO) American Astronomical Society June, 2019 100 — Dynamics on Asteroids break-up event around a Lagrange point. 100.01 — Simulations of a Synthetic Eurybates 100.02 — High-Fidelity Testing of Binary Asteroid Collisional Family Formation with Applications to 1999 KW4 Timothy Holt1; David Nesvorny2; Jonathan Horner1; Alex B. Davis1; Daniel Scheeres1 Rachel King1; Brad Carter1; Leigh Brookshaw1 1 Aerospace Engineering Sciences, University of Colorado Boulder 1 Centre for Astrophysics, University of Southern Queensland (Boulder, Colorado, United States) (Longmont, Colorado, United States) 2 Southwest Research Institute (Boulder, Connecticut, United The commonly accepted formation process for asym- States) metric binary asteroids is the spin up and eventual fission of rubble pile asteroids as proposed by Walsh, Of the six recognized collisional families in the Jo- Richardson and Michel (Walsh et al., Nature 2008) vian Trojan swarms, the Eurybates family is the and Scheeres (Scheeres, Icarus 2007). In this theory largest, with over 200 recognized members. Located a rubble pile asteroid is spun up by YORP until it around the Jovian L4 Lagrange point, librations of reaches a critical spin rate and experiences a mass the members make this family an interesting study shedding event forming a close, low-eccentricity in orbital dynamics. The Jovian Trojans are thought satellite. Further work by Jacobson and Scheeres to have been captured during an early period of in- used a planar, two-ellipsoid model to analyze the stability in the Solar system. The parent body of the evolutionary pathways of such a formation event family, 3548 Eurybates is one of the targets for the from the moment the bodies initially fission (Jacob- LUCY spacecraft, and our work will provide a dy- son and Scheeres, Icarus 2011). -
Estabilidade De Veículos Espaciais Em Ressonância
sid.inpe.br/mtc-m21c/2018/03.12.13.05-TDI ESTABILIDADE DE VEÍCULOS ESPACIAIS EM RESSONÂNCIA Rubens Antonio Condeles Júnior Tese de Doutorado do Curso de Pós-Graduação em Engenharia e Tecnologia Espaciais/Mecânica Espacial e Controle, orientada pelos Drs. Antonio Fernando Bertachini de Almeida Prado, e Tadashi Yokoyama, aprovada em 06 de abril de 2018. URL do documento original: <http://urlib.net/8JMKD3MGP3W34R/3QMPS3E> INPE São José dos Campos 2018 PUBLICADO POR: Instituto Nacional de Pesquisas Espaciais - INPE Gabinete do Diretor (GBDIR) Serviço de Informação e Documentação (SESID) Caixa Postal 515 - CEP 12.245-970 São José dos Campos - SP - Brasil Tel.:(012) 3208-6923/6921 E-mail: [email protected] COMISSÃO DO CONSELHO DE EDITORAÇÃO E PRESERVAÇÃO DA PRODUÇÃO INTELECTUAL DO INPE (DE/DIR-544): Presidente: Maria do Carmo de Andrade Nono - Conselho de Pós-Graduação (CPG) Membros: Dr. Plínio Carlos Alvalá - Centro de Ciência do Sistema Terrestre (COCST) Dr. André de Castro Milone - Coordenação-Geral de Ciências Espaciais e Atmosféricas (CGCEA) Dra. Carina de Barros Melo - Coordenação de Laboratórios Associados (COCTE) Dr. Evandro Marconi Rocco - Coordenação-Geral de Engenharia e Tecnologia Espacial (CGETE) Dr. Hermann Johann Heinrich Kux - Coordenação-Geral de Observação da Terra (CGOBT) Dr. Marley Cavalcante de Lima Moscati - Centro de Previsão de Tempo e Estudos Climáticos (CGCPT) Silvia Castro Marcelino - Serviço de Informação e Documentação (SESID) BIBLIOTECA DIGITAL: Dr. Gerald Jean Francis Banon Clayton Martins Pereira - Serviço de Informação -
1 Resonant Kuiper Belt Objects
Resonant Kuiper Belt Objects - a Review Renu Malhotra Lunar and Planetary Laboratory, The University of Arizona, Tucson, AZ, USA Email: [email protected] Abstract Our understanding of the history of the solar system has undergone a revolution in recent years, owing to new theoretical insights into the origin of Pluto and the discovery of the Kuiper belt and its rich dynamical structure. The emerging picture of dramatic orbital migration of the planets driven by interaction with the primordial Kuiper belt is thought to have produced the final solar system architecture that we live in today. This paper gives a brief summary of this new view of our solar system's history, and reviews the astronomical evidence in the resonant populations of the Kuiper belt. Introduction Lying at the edge of the visible solar system, observational confirmation of the existence of the Kuiper belt came approximately a quarter-century ago with the discovery of the distant minor planet (15760) Albion (formerly 1992 QB1, Jewitt & Luu 1993). With the clarity of hindsight, we now recognize that Pluto was the first discovered member of the Kuiper belt. The current census of the Kuiper belt includes more than 2000 minor planets at heliocentric distances between ~30 au and ~50 au. Their orbital distribution reveals a rich dynamical structure shaped by the gravitational perturbations of the giant planets, particularly Neptune. Theoretical analysis of these structures has revealed a remarkable dynamic history of the solar system. The story is as follows (see Fernandez & Ip 1984, Malhotra 1993, Malhotra 1995, Fernandez & Ip 1996, and many subsequent works). -
The Long-Term Dynamical Evolution of Planetary Systems
The Long-Term Dynamical Evolution of Planetary Systems Melvyn B. Davies Lund University Fred C. Adams University of Michigan Philip Armitage University of Colorado, Boulder John Chambers Carnegie Institution of Washington Eric Ford The Pennsylvania State University, University of Florida Alessandro Morbidelli University of Nice Sean N. Raymond University of Bordeaux Dimitri Veras University of Cambridge This chapter concerns the long-term dynamical evolution of planetary systems from both theoretical and observational perspectives. We begin by discussing the planet-planet interactions that take place within our own Solar System. We then describe such interactions in more tightly-packed planetary systems. As planet-planet interactions build up, some systems become dynamically unstable, leading to strong encounters and ultimately either ejections or collisions of planets. After discussing the basic physical processes involved, we consider how these interactions apply to extrasolar planetary systems and explore the constraints provided by observed systems. The presence of a residual planetesimal disc can lead to planetary migration and hence cause instabilities induced by resonance crossing; however, such discs can also stabilise planetary systems. The crowded birth environment of a planetary system can have a significant impact: close encounters and binary companions can act to destabilise systems, or sculpt their properties. In the case of binaries, the Kozai mechanism can place planets on extremely eccentric orbits which may later circularise to produce hot Jupiters. 1. INTRODUCTION views the application of theoretical models to observations of the solar system and extrasolar planetary systems. Currently observed planetary systems have typically Planetary systems evolve due to the exchange of angu- evolved between the time when the last gas in the protoplan- lar momentum and / or energy among multiple planets, be- etary disc was dispersed, and today. -
Orbital Resonances and Chaos in the Solar System
Solar system Formation and Evolution ASP Conference Series, Vol. 149, 1998 D. Lazzaro et al., eds. ORBITAL RESONANCES AND CHAOS IN THE SOLAR SYSTEM Renu Malhotra Lunar and Planetary Institute 3600 Bay Area Blvd, Houston, TX 77058, USA E-mail: [email protected] Abstract. Long term solar system dynamics is a tale of orbital resonance phe- nomena. Orbital resonances can be the source of both instability and long term stability. This lecture provides an overview, with simple models that elucidate our understanding of orbital resonance phenomena. 1. INTRODUCTION The phenomenon of resonance is a familiar one to everybody from childhood. A very young child is delighted in a playground swing when an older companion drives the swing at its natural frequency and rapidly increases the swing amplitude; the older child accomplishes the same on her own without outside assistance by driving the swing at a frequency twice that of its natural frequency. Resonance phenomena in the Solar system are essentially similar – the driving of a dynamical system by a periodic force at a frequency which is a rational multiple of the natural frequency. In fact, there are many mathematical similarities with the playground analogy, including the fact of nonlinearity of the oscillations, which plays a fundamental role in the long term evolution of orbits in the planetary system. But there is also an important difference: in the playground, the child adjusts her driving frequency to remain in tune – hence in resonance – with the natural frequency which changes with the amplitude of the swing. Such self-tuning is sometimes realized in the Solar system; but it is more often and more generally the case that resonances come-and-go. -
The Fate of Debris in the Pluto-Charon System
MNRAS 000,1{13 (2016) Preprint 6 August 2018 Compiled using MNRAS LATEX style file v3.0 The Fate of Debris in the Pluto-Charon System Rachel A. Smullen,1? Kaitlin M. Kratter,1 1Steward Observatory, University of Arizona, Tucson, AZ 85721, USA Accepted XXX. Received YYY; in original form ZZZ ABSTRACT The Pluto-Charon system has come into sharper focus following the fly by of New Horizons. We use N-body simulations to probe the unique dynamical history of this binary dwarf planet system. We follow the evolution of the debris disc that might have formed during the Charon-forming giant impact. First, we note that in-situ formation of the four circumbinary moons is extremely difficult if Charon undergoes eccentric tidal evolution. We track collisions of disc debris with Charon, estimating that hundreds to hundreds of thousands of visible craters might arise from 0.3{5 km radius bodies. New Horizons data suggesting a dearth of these small craters may place constraints on the disc properties. While tidal heating will erase some of the cratering history, both tidal and radiogenic heating may also make it possible to differentiate disc debris craters from Kuiper belt object craters. We also track the debris ejected from the Pluto-Charon system into the Solar System; while most of this debris is ultimately lost from the Solar System, a few tens of 10{30 km radius bodies could survive as a Pluto-Charon collisional family. Most are plutinos in the 3:2 resonance with Neptune, while a small number populate nearby resonances. We show that migration of the giant planets early in the Solar System's history would not destroy this collisional family. -
1 on the Origin of the Pluto System Robin M. Canup Southwest Research Institute Kaitlin M. Kratter University of Arizona Marc Ne
On the Origin of the Pluto System Robin M. Canup Southwest Research Institute Kaitlin M. Kratter University of Arizona Marc Neveu NASA Goddard Space Flight Center / University of Maryland The goal of this chapter is to review hypotheses for the origin of the Pluto system in light of observational constraints that have been considerably refined over the 85-year interval between the discovery of Pluto and its exploration by spacecraft. We focus on the giant impact hypothesis currently understood as the likeliest origin for the Pluto-Charon binary, and devote particular attention to new models of planet formation and migration in the outer Solar System. We discuss the origins conundrum posed by the system’s four small moons. We also elaborate on implications of these scenarios for the dynamical environment of the early transneptunian disk, the likelihood of finding a Pluto collisional family, and the origin of other binary systems in the Kuiper belt. Finally, we highlight outstanding open issues regarding the origin of the Pluto system and suggest areas of future progress. 1. INTRODUCTION For six decades following its discovery, Pluto was the only known Sun-orbiting world in the dynamical vicinity of Neptune. An early origin concept postulated that Neptune originally had two large moons – Pluto and Neptune’s current moon, Triton – and that a dynamical event had both reversed the sense of Triton’s orbit relative to Neptune’s rotation and ejected Pluto onto its current heliocentric orbit (Lyttleton, 1936). This scenario remained in contention following the discovery of Charon, as it was then established that Pluto’s mass was similar to that of a large giant planet moon (Christy and Harrington, 1978).