MONITORING and MITIGATION of CLOSE PROXIMITIES in LOW EARTH ORBIT S. Aida(1)
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Low Thrust Manoeuvres to Perform Large Changes of RAAN Or Inclination in LEO
Facoltà di Ingegneria Corso di Laurea Magistrale in Ingegneria Aerospaziale Master Thesis Low Thrust Manoeuvres To Perform Large Changes of RAAN or Inclination in LEO Academic Tutor: Prof. Lorenzo CASALINO Candidate: Filippo GRISOT July 2018 “It is possible for ordinary people to choose to be extraordinary” E. Musk ii Filippo Grisot – Master Thesis iii Filippo Grisot – Master Thesis Acknowledgments I would like to address my sincere acknowledgments to my professor Lorenzo Casalino, for your huge help in these moths, for your willingness, for your professionalism and for your kindness. It was very stimulating, as well as fun, working with you. I would like to thank all my course-mates, for the time spent together inside and outside the “Poli”, for the help in passing the exams, for the fun and the desperation we shared throughout these years. I would like to especially express my gratitude to Emanuele, Gianluca, Giulia, Lorenzo and Fabio who, more than everyone, had to bear with me. I would like to also thank all my extra-Poli friends, especially Alberto, for your support and the long talks throughout these years, Zach, for being so close although the great distance between us, Bea’s family, for all the Sundays and summers spent together, and my soccer team Belfiga FC, for being the crazy lovable people you are. A huge acknowledgment needs to be address to my family: to my grandfather Luciano, for being a great friend; to my grandmother Bianca, for teaching me what “fighting” means; to my grandparents Beppe and Etta, for protecting me -
Osculating Orbit Elements and the Perturbed Kepler Problem
Osculating orbit elements and the perturbed Kepler problem Gmr a = 3 + f(r, v,t) Same − r x & v Define: r := rn ,r:= p/(1 + e cos f) ,p= a(1 e2) − osculating orbit he sin f h v := n + λ ,h:= Gmp p r n := cos ⌦cos(! + f) cos ◆ sinp ⌦sin(! + f) e − X +⇥ sin⌦cos(! + f) + cos ◆ cos ⌦sin(! + f⇤) eY actual orbit +sin⇥ ◆ sin(! + f) eZ ⇤ λ := cos ⌦sin(! + f) cos ◆ sin⌦cos(! + f) e − − X new osculating + sin ⌦sin(! + f) + cos ◆ cos ⌦cos(! + f) e orbit ⇥ − ⇤ Y +sin⇥ ◆ cos(! + f) eZ ⇤ hˆ := n λ =sin◆ sin ⌦ e sin ◆ cos ⌦ e + cos ◆ e ⇥ X − Y Z e, a, ω, Ω, i, T may be functions of time Perturbed Kepler problem Gmr a = + f(r, v,t) − r3 dh h = r v = = r f ⇥ ) dt ⇥ v h dA A = ⇥ n = Gm = f h + v (r f) Gm − ) dt ⇥ ⇥ ⇥ Decompose: f = n + λ + hˆ R S W dh = r λ + r hˆ dt − W S dA Gm =2h n h + rr˙ λ rr˙ hˆ. dt S − R S − W Example: h˙ = r S d (h cos ◆)=h˙ e dt · Z d◆ h˙ cos ◆ h sin ◆ = r cos(! + f)sin◆ + r cos ◆ − dt − W S Perturbed Kepler problem “Lagrange planetary equations” dp p3 1 =2 , dt rGm 1+e cos f S de p 2 cos f + e(1 + cos2 f) = sin f + , dt Gm R 1+e cos f S r d◆ p cos(! + f) = , dt Gm 1+e cos f W r d⌦ p sin(! + f) sin ◆ = , dt Gm 1+e cos f W r d! 1 p 2+e cos f sin(! + f) = cos f + sin f e cot ◆ dt e Gm − R 1+e cos f S− 1+e cos f W r An alternative pericenter angle: $ := ! +⌦cos ◆ d$ 1 p 2+e cos f = cos f + sin f dt e Gm − R 1+e cos f S r Perturbed Kepler problem Comments: § these six 1st-order ODEs are exactly equivalent to the original three 2nd-order ODEs § if f = 0, the orbit elements are constants § if f << Gm/r2, use perturbation theory -
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 -
Reference System Issues in Binary Star Calculations
Reference System Issues in Binary Star Calculations Poster for Division A Meeting DAp.1.05 George H. Kaplan Consultant to U.S. Naval Observatory Washington, D.C., U.S.A. [email protected] or [email protected] IAU General Assembly Honolulu, August 2015 1 Poster DAp.1.05 Division!A! !!!!!!Coordinate!System!Issues!in!Binary!Star!Computa3ons! George!H.!Kaplan! Consultant!to!U.S.!Naval!Observatory!!([email protected][email protected])! It!has!been!es3mated!that!half!of!all!stars!are!components!of!binary!or!mul3ple!systems.!!Yet!the!number!of!known!orbits!for!astrometric!and!spectroscopic!binary!systems!together!is! less!than!7,000!(including!redundancies),!almost!all!of!them!for!bright!stars.!!A!new!genera3on!of!deep!allLsky!surveys!such!as!PanLSTARRS,!Gaia,!and!LSST!are!expected!to!lead!to!the! discovery!of!millions!of!new!systems.!!Although!for!many!of!these!systems,!the!orbits!may!be!poorly!sampled!ini3ally,!it!is!to!be!expected!that!combina3ons!of!new!and!old!data!sources! will!eventually!lead!to!many!more!orbits!being!known.!!As!a!result,!a!revolu3on!in!the!scien3fic!understanding!of!these!systems!may!be!upon!us.! The!current!database!of!visual!(astrometric)!binary!orbits!represents!them!rela3ve!to!the!“plane!of!the!sky”,!that!is,!the!plane!orthogonal!to!the!line!of!sight.!!Although!the!line!of!sight! to!stars!constantly!changes!due!to!proper!mo3on,!aberra3on,!and!other!effects,!there!is!no!agreed!upon!standard!for!what!line!of!sight!defines!the!orbital!reference!plane.!! Furthermore,!the!computa3on!of!differen3al!coordinates!(component!B!rela3ve!to!A)!for!a!given!date!must!be!based!on!the!binary!system’s!direc3on!at!that!date.!!Thus,!a!different! -
Sentinel-1 Constellation SAR Interferometry Performance Verification
Sentinel-1 Constellation SAR Interferometry Performance Verification Dirk Geudtner1, Francisco Ceba Vega1, Pau Prats2 , Nestor Yaguee-Martinez2 , Francesco de Zan3, Helko Breit3, Andrea Monti Guarnieri4, Yngvar Larsen5, Itziar Barat1, Cecillia Mezzera1, Ian Shurmer6 and Ramon Torres1 1 ESA ESTEC 2 DLR, Microwaves and Radar Institute 3 DLR, Remote Sensing Technology Institute 4 Politecnico Di Milano 5 Northern Research Institute (Norut) 6 ESA ESOC ESA UNCLASSIFIED - For Official Use Outline • Introduction − Sentinel-1 Constellation Mission Status − Overview of SAR Imaging Modes • Results from the Sentinel-1B Commissioning Phase • Azimuth Spectral Alignment Impact on common − Burst synchronization Doppler bandwidth − Difference in mean Doppler Centroid Frequency (InSAR) • Sentinel-1 Orbital Tube and InSAR Baseline • Demonstration of cross S-1A/S-1B InSAR Capability for Surface Deformation Mapping ESA UNCLASSIFIED - For Official Use ESA Slide 2 Sentinel-1 Constellation Mission Status • Constellation of two SAR C-band (5.405 GHz) satellites (A & B units) • Sentinel-1A launched on 3 April, 2014 & Sentinel-1B on 25 April, 2016 • Near-Polar, sun-synchronous (dawn-dusk) orbit at 698 km • 12-day repeat cycle (each satellite), 6 days for the constellation • Systematic SAR data acquisition using a predefined observation scenario • More than 10 TB of data products daily (specification of 3 TB) • 3 X-band Ground stations (Svalbard, Matera, Maspalomas) + upcoming 4th core station in Inuvik, Canada • Use of European Data Relay System (EDRS) provides -
Orbit Options for an Orion-Class Spacecraft Mission to a Near-Earth Object
Orbit Options for an Orion-Class Spacecraft Mission to a Near-Earth Object by Nathan C. Shupe B.A., Swarthmore College, 2005 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Aerospace Engineering Sciences 2010 This thesis entitled: Orbit Options for an Orion-Class Spacecraft Mission to a Near-Earth Object written by Nathan C. Shupe has been approved for the Department of Aerospace Engineering Sciences Daniel Scheeres Prof. George Born Assoc. Prof. Hanspeter Schaub Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Shupe, Nathan C. (M.S., Aerospace Engineering Sciences) Orbit Options for an Orion-Class Spacecraft Mission to a Near-Earth Object Thesis directed by Prof. Daniel Scheeres Based on the recommendations of the Augustine Commission, President Obama has pro- posed a vision for U.S. human spaceflight in the post-Shuttle era which includes a manned mission to a Near-Earth Object (NEO). A 2006-2007 study commissioned by the Constellation Program Advanced Projects Office investigated the feasibility of sending a crewed Orion spacecraft to a NEO using different combinations of elements from the latest launch system architecture at that time. The study found a number of suitable mission targets in the database of known NEOs, and pre- dicted that the number of candidate NEOs will continue to increase as more advanced observatories come online and execute more detailed surveys of the NEO population. -
Circular Orbits
ANALYSIS AND MECHANIZATION OF LAUNCH WINDOW AND RENDEZVOUS COMPUTATION PART I. CIRCULAR ORBITS Research & Analysis Section Tech Memo No. 175 March 1966 BY J. L. Shady Prepared for: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION / GEORGE C. MARSHALL SPACE FLIGHT CENTER AERO-ASTRODYNAMICS LABORATORY Prepared under Contract NAS8-20082 Reviewed by: " D. L/Cooper Technical Supervisor Director Research & Analysis Section / NORTHROP SPACE LABORATORIES HUNTSVILLE DEPARTMENT HUNTSVILLE, ALABAMA FOREWORD The enclosed presents the results of work performed by Northrop Space Laboratories, Huntsville Department, while )under contract to the Aero- Astrodynamfcs Laboratory of Marshall Space Flight Center (NAS8-20082) * This task was conducted in response to the requirement of Appendix E-1, Schedule Order No. PO. Technical coordination was provided by Mr. Jesco van Puttkamer of the Technical and Scientific Staff (R-AERO-T) ABS TRACT This repart presents Khe results of an analytical study to develop the logic for a digrtal c3mp;lrer sclbrolrtine for the automatic computationof launch opportunities, or the sequence of launch times, that rill allow execution of various mades of gross ~ircuiarorbit rendezvous. The equations developed in this study are based on tho-bady orb:taf chmry. The Earch has been assumed tcr ha.:e the shape of an oblate spheroid. Oblateness of the Earth has been accounted for by assuming the circular target orbit to be space- fixed and by cosrecring che T ational raLe of the EarKh accordingly. Gross rendezvous between the rarget vehicle and a maneuverable chaser vehicle, assumed to be a standard 3+ uprated Saturn V launched from Cape Kennedy, is in this srudy aecomplfshed by: .x> direct ascent to rendezvous, or 2) rendezvous via an intermediate circG1a.r parking orbit. -
Some of the Scientific Miracles in Brief ﻹﻋﺠﺎز اﻟﻌﻠ� ﻲﻓ ﺳﻄﻮر English [ إ�ﻠ�ي - ]
Some of the Scientific Miracles in Brief ﻟﻋﺠﺎز ﻟﻌﻠ� ﻓ ﺳﻄﻮر English [ إ���ي - ] www.onereason.org website مﻮﻗﻊ ﺳﺒﺐ وﻟﺣﺪ www.islamreligion.com website مﻮﻗﻊ دﻳﻦ ﻟﻋﺳﻼم 2013 - 1434 Some of the Scientific Miracles in Brief Ever since the dawn of mankind, we have sought to understand nature and our place in it. In this quest for the purpose of life many people have turned to religion. Most religions are based on books claimed by their followers to be divinely inspired, without any proof. Islam is different because it is based upon reason and proof. There are clear signs that the book of Islam, the Quran, is the word of God and we have many reasons to support this claim: - There are scientific and historical facts found in the Qu- ran which were unknown to the people at the time, and have only been discovered recently by contemporary science. - The Quran is in a unique style of language that cannot be replicated, this is known as the ‘Inimitability of the Qu- ran.’ - There are prophecies made in the Quran and by the Prophet Muhammad, may God praise him, which have come to be pass. This article lays out and explains the scientific facts that are found in the Quran, centuries before they were ‘discov- ered’ in contemporary science. It is important to note that the Quran is not a book of science but a book of ‘signs’. These signs are there for people to recognise God’s existence and 2 affirm His revelation. As we know, science sometimes takes a ‘U-turn’ where what once scientifically correct is false a few years later. -
Principles and Techniques of Remote Sensing
BASIC ORBIT MECHANICS 1-1 Example Mission Requirements: Spatial and Temporal Scales of Hydrologic Processes 1.E+05 Lateral Redistribution 1.E+04 Year Evapotranspiration 1.E+03 Month 1.E+02 Week Percolation Streamflow Day 1.E+01 Time Scale (hours) Scale Time 1.E+00 Precipitation Runoff Intensity 1.E-01 Infiltration 1.E-02 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Length Scale (meters) 1-2 BASIC ORBITS • Circular Orbits – Used most often for earth orbiting remote sensing satellites – Nadir trace resembles a sinusoid on planet surface for general case – Geosynchronous orbit has a period equal to the siderial day – Geostationary orbits are equatorial geosynchronous orbits – Sun synchronous orbits provide constant node-to-sun angle • Elliptical Orbits: – Used most often for planetary remote sensing – Can also be used to increase observation time of certain region on Earth 1-3 CIRCULAR ORBITS • Circular orbits balance inward gravitational force and outward centrifugal force: R 2 F mg g s r mv2 F c r g R2 F F v s g c r 2r r T 2r 2 v gs R • The rate of change of the nodal longitude is approximated by: d 3 cos I J R3 g dt 2 2 s r 7 2 1-4 Orbital Velocities 9 8 7 6 5 Earth Moon 4 Mars 3 Linear Velocity in km/sec in Linear Velocity 2 1 0 200 400 600 800 1000 1200 1400 Orbit Altitude in km 1-5 Orbital Periods 300 250 200 Earth Moon Mars 150 Orbital Period in MinutesPeriodOrbitalin 100 50 200 400 600 800 1000 1200 1400 Orbit Altitude in km 1-6 ORBIT INCLINATION EQUATORIAL I PLANE EARTH ORBITAL PLANE 1-7 ORBITAL NODE LONGITUDE SUN ORBITAL PLANE EARTH VERNAL EQUINOX 1-8 SATELLITE ORBIT PRECESSION 1-9 CIRCULAR GEOSYNCHRONOUS ORBIT TRACE 1-10 ORBIT COVERAGE • The orbit step S is the longitudinal difference between two consecutive equatorial crossings • If S is such that N S 360 ; N, L integers L then the orbit is repetitive. -
Orbit Determination Using Modern Filters/Smoothers and Continuous Thrust Modeling
Orbit Determination Using Modern Filters/Smoothers and Continuous Thrust Modeling by Zachary James Folcik B.S. Computer Science Michigan Technological University, 2000 SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE 2008 © 2008 Massachusetts Institute of Technology. All rights reserved. Signature of Author:_______________________________________________________ Department of Aeronautics and Astronautics May 23, 2008 Certified by:_____________________________________________________________ Dr. Paul J. Cefola Lecturer, Department of Aeronautics and Astronautics Thesis Supervisor Certified by:_____________________________________________________________ Professor Jonathan P. How Professor, Department of Aeronautics and Astronautics Thesis Advisor Accepted by:_____________________________________________________________ Professor David L. Darmofal Associate Department Head Chair, Committee on Graduate Students 1 [This page intentionally left blank.] 2 Orbit Determination Using Modern Filters/Smoothers and Continuous Thrust Modeling by Zachary James Folcik Submitted to the Department of Aeronautics and Astronautics on May 23, 2008 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautics and Astronautics ABSTRACT The development of electric propulsion technology for spacecraft has led to reduced costs and longer lifespans for certain -
Preparation of Papers for AIAA Technical Conferences
DUKSUP: A Computer Program for High Thrust Launch Vehicle Trajectory Design & Optimization Spurlock, O.F.I and Williams, C. H.II NASA Glenn Research Center, Cleveland, OH, 44135 From the late 1960’s through 1997, the leadership of NASA’s Intermediate and Large class unmanned expendable launch vehicle projects resided at the NASA Lewis (now Glenn) Research Center (LeRC). One of LeRC’s primary responsibilities --- trajectory design and performance analysis --- was accomplished by an internally-developed analytic three dimensional computer program called DUKSUP. Because of its Calculus of Variations-based optimization routine, this code was generally more capable of finding optimal solutions than its contemporaries. A derivation of optimal control using the Calculus of Variations is summarized including transversality, intermediate, and final conditions. The two point boundary value problem is explained. A brief summary of the code’s operation is provided, including iteration via the Newton-Raphson scheme and integration of variational and motion equations via a 4th order Runge-Kutta scheme. Main subroutines are discussed. The history of the LeRC trajectory design efforts in the early 1960’s is explained within the context of supporting the Centaur upper stage program. How the code was constructed based on the operation of the Atlas/Centaur launch vehicle, the limits of the computers of that era, the limits of the computer programming languages, and the missions it supported are discussed. The vehicles DUKSUP supported (Atlas/Centaur, Titan/Centaur, and Shuttle/Centaur) are briefly described. The types of missions, including Earth orbital and interplanetary, are described. The roles of flight constraints and their impact on launch operations are detailed (such as jettisoning hardware on heating, Range Safety, ground station tracking, and elliptical parking orbits). -
Flowers and Satellites
Flowers and Satellites G. Lucarelli, R. Santamaria, S. Troisi and L. Turturici (Naval University, Naples) In this paper some interesting properties of circular orbiting satellites' ground traces are pointed out. It will be shown how such properties are typical of some spherical and plane curves that look like flowers both in their name and shape. In addition, the choice of the best satellite constellations satisfying specific requirements is strongly facilitated by exploiting these properties. i. INTRODUCTION. Since the dawning of mathematical sciences, many scientists have attempted to emulate mathematically the beauty of nature, with particular care to floral forms. The mathematician Guido Grandi gave in the Age of Enlightenment a definitive contribution; some spherical and plane curves reproducing flowers have his name: Guido Grandi's Clelie and Rodonee. It will be deduced that the ground traces of circular orbiting satellites are described by the same equations, simple and therefore easily reproducible, and hence employable in satellite constellation designing and in other similar problems. 2. GROUND TRACE GEOMETRY. The parametric equations describing the ground trace of any satellite in circular orbit are: . ... 24(t sin <p = sin i sin tan (A-A0 + £-T0) = cos i tan — — where i = inclination of the orbital plane; <j>, A = geographic coordinates of the subsatellite point; t = time parameter, defined as angular measure of Earth rotation starting from T0 ; T0 , Ao = time and geographic longitude relative to the crossing of the satellite at the ascending node; T = orbital satellite period as expressed in hours. In the specific case of a polar-orbiting satellite, equations (i) become tan(AA + t) = where T0 = o; AA = A — Ao ; /i = 24/T.