Trajectory Sensitivities for Sun-Mars Libration Point Missions
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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 -
AAS/AIAA Astrodynamics Specialist Conference
DRAFT version: 7/15/2011 11:04 AM http://www.alyeskaresort.com AAS/AIAA Astrodynamics Specialist Conference July 31 ‐ August 4, 2011 Girdwood, Alaska AAS General Chair AIAA General Chair Ryan P. Russell William Todd Cerven Georgia Institute of Technology The Aerospace Corporation AAS Technical Chair AIAA Technical Chair Hanspeter Schaub Brian C. Gunter University of Colorado Delft University of Technology DRAFT version: 7/15/2011 11:04 AM http://www.alyeskaresort.com Cover images: Top right: Conference Location: Aleyska Resort in Girdwood Alaska. Middle left: Cassini looking back at an eclipsed Saturn, Astronomy picture of the day 2006 Oct 16, credit CICLOPS, JPL, ESA, NASA; Middle right: Shuttle shadow in the sunset (in honor of the end of the Shuttle Era), Astronomy picture of the day 2010 February 16, credit: Expedition 22 Crew, NASA. Bottom right: Comet Hartley 2 Flyby, Astronomy picture of the day 2010 Nov 5, Credit: NASA, JPL-Caltech, UMD, EPOXI Mission DRAFT version: 7/15/2011 11:04 AM http://www.alyeskaresort.com Table of Contents Registration ............................................................................................................................................... 5 Schedule of Events ................................................................................................................................... 6 Conference Center Layout ........................................................................................................................ 7 Conference Location: The Hotel Alyeska ............................................................................................... -
AFSPC-CO TERMINOLOGY Revised: 12 Jan 2019
AFSPC-CO TERMINOLOGY Revised: 12 Jan 2019 Term Description AEHF Advanced Extremely High Frequency AFB / AFS Air Force Base / Air Force Station AOC Air Operations Center AOI Area of Interest The point in the orbit of a heavenly body, specifically the moon, or of a man-made satellite Apogee at which it is farthest from the earth. Even CAP rockets experience apogee. Either of two points in an eccentric orbit, one (higher apsis) farthest from the center of Apsis attraction, the other (lower apsis) nearest to the center of attraction Argument of Perigee the angle in a satellites' orbit plane that is measured from the Ascending Node to the (ω) perigee along the satellite direction of travel CGO Company Grade Officer CLV Calculated Load Value, Crew Launch Vehicle COP Common Operating Picture DCO Defensive Cyber Operations DHS Department of Homeland Security DoD Department of Defense DOP Dilution of Precision Defense Satellite Communications Systems - wideband communications spacecraft for DSCS the USAF DSP Defense Satellite Program or Defense Support Program - "Eyes in the Sky" EHF Extremely High Frequency (30-300 GHz; 1mm-1cm) ELF Extremely Low Frequency (3-30 Hz; 100,000km-10,000km) EMS Electromagnetic Spectrum Equitorial Plane the plane passing through the equator EWR Early Warning Radar and Electromagnetic Wave Resistivity GBR Ground-Based Radar and Global Broadband Roaming GBS Global Broadcast Service GEO Geosynchronous Earth Orbit or Geostationary Orbit ( ~22,300 miles above Earth) GEODSS Ground-Based Electro-Optical Deep Space Surveillance -
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. -
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. -
Aas 11-509 Artemis Lunar Orbit Insertion and Science Orbit Design Through 2013
AAS 11-509 ARTEMIS LUNAR ORBIT INSERTION AND SCIENCE ORBIT DESIGN THROUGH 2013 Stephen B. Broschart,∗ Theodore H. Sweetser,y Vassilis Angelopoulos,z David C. Folta,x and Mark A. Woodard{ As of late-July 2011, the ARTEMIS mission is transferring two spacecraft from Lissajous orbits around Earth-Moon Lagrange Point #1 into highly-eccentric lunar science orbits. This paper presents the trajectory design for the transfer from Lissajous orbit to lunar orbit in- sertion, the period reduction maneuvers, and the science orbits through 2013. The design accommodates large perturbations from Earth’s gravity and restrictive spacecraft capabili- ties to enable opportunities for a range of heliophysics and planetary science measurements. The process used to design the highly-eccentric ARTEMIS science orbits is outlined. The approach may inform the design of future eccentric orbiter missions at planetary moons. INTRODUCTION The Acceleration, Reconnection, Turbulence and Electrodynamics of the Moons Interaction with the Sun (ARTEMIS) mission is currently operating two spacecraft in lunar orbit under funding from the Heliophysics and Planetary Science Divisions with NASA’s Science Missions Directorate. ARTEMIS is an extension to the successful Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission [1] that has relocated two of the THEMIS spacecraft from Earth orbit to the Moon [2, 3, 4]. ARTEMIS plans to conduct a variety of scientific studies at the Moon using the on-board particle and fields instrument package [5, 6]. The final portion of the ARTEMIS transfers involves moving the two ARTEMIS spacecraft, known as P1 and P2, from Lissajous orbits around the Earth-Moon Lagrange Point #1 (EML1) to long-lived, eccentric lunar science orbits with periods of roughly 28 hr. -
An Examination of the Mass Limit for Stability at the Triangular Lagrange
San Jose State University SJSU ScholarWorks Master's Theses Master's Theses and Graduate Research Spring 2015 An Examination of the Mass Limit for Stability at the Triangular Lagrange Points for a Three-Body System and a Special Case of the Four-Body Problem Sean Kemp San Jose State University Follow this and additional works at: https://scholarworks.sjsu.edu/etd_theses Recommended Citation Kemp, Sean, "An Examination of the Mass Limit for Stability at the Triangular Lagrange Points for a Three-Body System and a Special Case of the Four-Body Problem" (2015). Master's Theses. 4546. DOI: https://doi.org/10.31979/etd.4tf8-hnqx https://scholarworks.sjsu.edu/etd_theses/4546 This Thesis is brought to you for free and open access by the Master's Theses and Graduate Research at SJSU ScholarWorks. It has been accepted for inclusion in Master's Theses by an authorized administrator of SJSU ScholarWorks. For more information, please contact [email protected]. AN EXAMINATION OF THE MASS LIMIT FOR STABILITY AT THE TRIANGULAR LAGRANGE POINTS FOR A THREE-BODY SYSTEM AND A SPECIAL CASE OF THE FOUR-BODY PROBLEM A Thesis Presented to The Faculty of the Department of Physics and Astronomy San José State University In Partial Fulllment of the Requirements for the Degree Master of Science by Sean P. Kemp May 2015 c 2015 Sean P. Kemp ALL RIGHTS RESERVED The Designated Thesis Committee Approves the Thesis Titled AN EXAMINATION OF THE MASS LIMIT FOR STABILITY AT THE TRIANGULAR LAGRANGE POINTS FOR A THREE-BODY SYSTEM AND A SPECIAL CASE OF THE FOUR-BODY PROBLEM by Sean P. -
Design of Small Circular Halo Orbit Around L2
Design of Small Circular Halo Orbit around L2 Kohta Tarao* and Jun’ichiro Kawaguchi** *Graduate Student, Department of Aeronautics and Astronautics, School of Engineering, The University of Tokyo ** The Institute of Space and Astronautics Science / Japan Aerospace Exploration Agency (ISAS/JAXA) 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, JAPAN. Abstract A Lagrange point of Sun-Earth system, L1 or L2 is conceived one of the best point for astronomy and connection ports for outer planets. This paper describes a new useful control method for realizing and keeping small circular Halo orbits around the L2 point. First, a control law is mathematically derived from the restricted three body problem formulation and the associated properties are discussed. And next, some useful cases for the space mission are verified numerically using real ephemeris. The resulted Halo orbit is compact and circular realized with satisfactorily small thrust. The paper concludes the strategy is applied to many kinds of missions. L2 点周りの小円ハロー軌道設計 多羅尾康太(東大・工・院)、川口淳一郎(ISAS/JAXA) 摘要 太陽―月・地球系におけるラグランジュ点の中で、L2 点は太陽と地球との距離がほぼ一定で、太 陽、地球との幾何学的関係が保たれ、またダウンリンク通信が太陽の影響を受けない点で L1 点よ りも有利である。そのため L2 点周辺の軌道は、赤外線天文衛星や外惑星探査のための起・終点と しての宇宙港の軌道として利用が考えられている。本論文では、新しい制御法により L2 点回りに 小さな円形の Halo 軌道が作れることを示す。まず初めに、円制限三体問題における線形化された 方程式から理論的に制御則を導く。その後、実軌道要素を用いて軌道制御方法を検証する。軌道制 御に必要な推力は十分小さく大変実用的な制御方法である。 INTRODUCTION million kilometers and also due to its highly alternating geometry. A Lissajous orbit also exists A Lagrange point is the point where the gravity of around the L2 point naturally, however, the orbital two planets such as Sun and Earth is equal to plane rotates and the solar power availability is centrifugal force by their circular motions. L1 and often restricted due to the shadow by the Earth. -
Stability of the Lagrange Points, L4 and L5
Stability of the Lagrange Points, L4 and L5 Thomas Greenspan January 7, 2014 Abstract A proof of the stability of the non collinear Lagrange Points, L4 and L5. We will start by covering the basics of stability, stating a theorem (without proof) with a few corollaries, and then turn to the Lagrange points, proving first the stability of all Lagrange points in the z-direction and then restricting our attention to the the points, L4 and L5. Introduction The three body problem is one that has been studied for many centuries. It consists of considering 3 bodies, subject to only to mutually attracting forces (determined by the inverse square force, gravitation) and to solve for the motions of the three bodies. This task can be extremely complicated and, following Poincar´e's example, most mathematicians and physicists have interested themselves in the finding periodic solutions to the problem, that is, where x(t) = x(t + T ) where T is the period [3]. Indeed not only does this offer a useful framework from within which to work, but also describes states that are most applicable and useful to real world situations. Finding these periodic solutions is by no means trivial. It is a problem that has been worked on for centuries and still is in modern times, one of the latest publications dealing with it as recent as March of 2013 [9]. Among the first to interest themselves in and find solutions to the 3 body problem were Leonhard Euler in 1765 and Joseph Lagrange in 1772 [3]. In his publication, Essai sur le probl`emedes trois corps (Essay on the 3-Body Problem), Lagrange proposed a method that had never been used until then, that of considering only the distances between the three bodies rather than their absolute positions [2]. -
Applications of Multi-Body Dynamical Environments: the ARTEMIS Transfer Trajectory Design
Paper ID: 7461 61st International Astronautical Congress 2010 Applications of Multi-Body Dynamical Environments: The ARTEMIS Transfer Trajectory Design ASTRODYNAMICS SYMPOSIUM (CI) Mission Design, Operations and Optimization (2) (9) Mr. David C. Folta and Mr. Mark Woodard National Aeronautics and Space Administration (NASA)/Goddard Space Flight Center, Greenbelt, MD, United States, david.c.folta<@nasa.gov Prof. Kathleen Howell, Chris Patterson, Wayne Schlei Purdue University, West Lafayette, IN, United States, [email protected] Abstract The application of forces in multi-body dynamical environments to pennit the transfer of spacecraft from Earth orbit to Sun-Earth weak stability regions and then return to the Earth-Moon libration (L1 and L2) orbits has been successfully accomplished for the first time. This demonstrated transfer is a positive step in the realization of a design process that can be used to transfer spacecraft with minimal Delta-V expenditures. Initialized using gravity assists to overcome fuel constraints; the ARTEMIS trajectory design has successfully placed two spacecraft into Earth Moon libration orbits by means of these applications. INTRODUCTION The exploitation of multi-body dynamical environments to pennit the transfer of spacecraft from Earth to Sun-Earth weak stability regions and then return to the Earth-Moon libration (L 1 and L2) orbits has been successfully accomplished. This demonstrated transfer is a positive step in the realization of a design process that can be used to transfer spacecraft with minimal Delta-Velocity (L.\V) expenditure. Initialized using gravity assists to overcome fuel constraints, the Acceleration Reconnection and Turbulence and Electrodynamics of the Moon's Interaction with the Sun (ARTEMIS) mission design has successfully placed two spacecraft into Earth-Moon libration orbits by means of this application of forces from multiple gravity fields. -
Rideshare and the Orbital Maneuvering Vehicle: the Key to Low-Cost Lagrange-Point Missions
SSC15-II-5 Rideshare and the Orbital Maneuvering Vehicle: the Key to Low-cost Lagrange-point Missions Chris Pearson, Marissa Stender, Christopher Loghry, Joe Maly, Valentin Ivanitski Moog Integrated Systems 1113 Washington Avenue, Suite 300, Golden, CO, 80401; 303 216 9777, extension 204 [email protected] Mina Cappuccio, Darin Foreman, Ken Galal, David Mauro NASA Ames Research Center PO Box 1000, M/S 213-4, Moffett Field, CA 94035-1000; 650 604 1313 [email protected] Keats Wilkie, Paul Speth, Trevor Jackson, Will Scott NASA Langley Research Center 4 West Taylor Street, Mail Stop 230, Hampton, VA, 23681; 757 864 420 [email protected] ABSTRACT Rideshare is a well proven approach, in both LEO and GEO, enabling low-cost space access through splitting of launch charges between multiple passengers. Demand exists from users to operate payloads at Lagrange points, but a lack of regular rides results in a deficiency in rideshare opportunities. As a result, such mission architectures currently rely on a costly dedicated launch. NASA and Moog have jointly studied the technical feasibility, risk and cost of using an Orbital Maneuvering Vehicle (OMV) to offer Lagrange point rideshare opportunities. This OMV would be launched as a secondary passenger on a commercial rocket into Geostationary Transfer Orbit (GTO) and utilize the Moog ESPA secondary launch adapter. The OMV is effectively a free flying spacecraft comprising a full suite of avionics and a propulsion system capable of performing GTO to Lagrange point transfer via a weak stability boundary orbit. In addition to traditional OMV ’tug’ functionality, scenarios using the OMV to host payloads for operation at the Lagrange points have also been analyzed. -
Libration Orbit Mission Design: Applications of Numerical and Dynamical Methods
Libration Orbit Mission Design: Applications Of Numerical And Dynamical Methods David Folta and Mark Beckman NASA - Goddard Space Flight Center Libration Point Orbits and Applications June 10-14, 2002, Girona, Spain Agenda • NASA Enterprises • Challenges • Historical and future missions • Libration mission design via numerical and dynamical methods • Applications ¾Direct transfer ¾Low thrust ¾Servicing ¾Formations: Constellation-X and Stellar Imager ¾Conceptual studies • Improved tools • Conclusions NASA Themes and Libration Orbits NASA Enterprises of Space Sciences (SSE) and Earth Sciences (ESE) are a combination of several programs and themes SSE ESE SEU SEC Origins • Recent SEC missions include ACE, SOHO, and the L1/L2 WIND mission. The Living With a Star (LWS) portion of SEC may require libration orbits at the L1 and L3 Sun-Earth libration points. • Structure and Evolution of the Universe (SEU) currently has MAP and the future Micro Arc-second X-ray Imaging Mission (MAXIM) and Constellation-X missions. •Space Sciences’ Origins libration missions are the Next Generation Space Telescope (NGST) and The Terrestrial Planet Finder (TPF). • The Triana mission is the lone ESE mission not orbiting the Earth. • A major challenge is formation flying components of Constellation-X, MAXIM, TPF, and Stellar Imager. Future Libration Mission SSE and ESE Challenges ¾ Orbit ¾Biased orbits when using large sun shades ¾Shadow restrictions ¾Very small amplitudes ¾Reorientation to different planes and Lissajous classes ¾Rendezvous and formation flying ¾Low