DOCSLIB.ORG

CHARACTERISTICS of APOLLO-TYPE LUNAR ORBITS by X 10 George H

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CHARACTERISTICS of APOLLO-TYPE LUNAR ORBITS by X 10 George H GPO PRICE $ CFSTI PRICE(S) $ Hard copy (HC)—&f-)() Microfiche (MF) LS ff653 July 65 CHARACTERISTICS OF APOLLO-TYPE LUNAR ORBITS by x 10 George H. Born U TCR 7 -\ g June 6, 1966 cc o IMVPI III1IWJ • p•iZ> I' MAY 1968 r lo L'A TEXAS CENTER FOR RESEARCH P. 0. 10X 8472, UNIVERSITY STATION, AUSTIN, TEXAS 0. This report was prepared under NASA MANNED SPACE FLIGHT CENTER Contract 9-2619 under the direction of Dr. Byron D. Tapley Associate Professor of Aerospace Engineering and Engineering Mechanics CHARACTERISTICS OF APOLLO-TYPE LUNAR ORBITS by GEORGE H. BORN, B.S. THESIS Presented to the Faculty of the Graduate School of The University of Texas in Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE IN AERO-SPACE ENGINEERING THE UNIVERSITY OF TEXAS JANUARY, 1965 A B ST RA CT A computer routine for numerically integrating Lagrange's Planetary Equations for lunar satellite orbits was revised to integrate an alternate set of perturbation equations which do not have the small eccentricity restric- tion of Lagrange's Equations. This set of equations was solved numerically for the time variations of the orbit elements of circular lunar satellite orbits. Consideration was given to orbits with near equatorial inclinations and low altitudes similar to those considered for the Apollo Project. The principal perturbing forces acting on the satellite were assumed to be the triaxiality of the Moon and the mass of the Earth. The Earth was considered as a point mass revolving in an elliptical orbit about the Moon. The variations with time of the orbit elements for twelve sets of initial conditions were investigated. Data showing the results for both short time, three revolu- tions of the satellite, and long time, 80 revolutions of the satellite, are presented in both tabular and graphical form. iv PRECEDING PAGE BLANK NOT FILMED. TABLE OF CONTENTS Page PREFACE ABSTRACT ....................... iv LIST OF FIGURES ................... vi LISTOF TABLES .................. viii NOMENCLATURE ...................... ix CONSTANTS ...................... X 3. i I. INTRODUCTION .................. 1 II. ANALYSIS .................... 5 Motion of the Earth-Moon System ...... 5 Problem Definition and Assumptions ..... 6 Earth-Moon Ephemeris Equations ........ 9 Coordinate System . ........ 18 Perturbation Equations ........... 21 III. COMPUTATIONAL PROCEDURE ........... 39 IV. RESULT S ..................... 42 Initial Values of the Orbit Elements . 42 Graphical Results ............. 43 V. CONCLUSION AND RECOMMENDATIONS ......... 68 REFERENCES ...................... 71 V LIST OF FIGURES FIGURE PAGE 1. Orientation of Lunar Equatorial Plane With Respect to the Lunar Orbit Plane ..... 7 2. Geometry of Earth-Moon System .......... 11 3. Selenocentric Coordinate Systems ...... 19 4. Satellite Position With Respect to Selenocentric Inertial Coordinate Systems . 24 5. Semi-Latus Rectum and Inclination vs. Number of Revolutions for Orbit Type 3 . 51 6. Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Type 3 . 52 7. Semi-Latus Rectum and Inclination vs. Number of Revolutions for Orbit Type 4 . 53 8. Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Type 4 . 54 9. Semi-Latus Rectum and Inclination vs. Number of Revolutions for Orbit Type 9 . 55 10. Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Type 9 . 56 11. Semi-Latus Rectum and Inclination vs. Number of Revolutions for Orbit Type 10 . 57 12. Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Type 10 . 58 13. Eccentricity X Sine (Argument of Periselene) and Eccentricity X Cosine (Argument of Pen- selene) vs. Number of Revolutions for Orbit Type 10 .................... 59 vi vii FIGURE PAGE 14. Semi-Latus Rectum, Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Types 1, 3, 5 ..... 60 15. Semi-Latus Rectum, Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Types 2, 4, 6 ..... 61 16. Inclination vs. Time for Orbit Types 1 Through 6 .................. 62 17. Semi-Latus Rectum, Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Types 7, 9, 11 .... 63 18. Semi-Latus Rectum, Longitude of Ascending Node and Eccentricity vs. Number of Revolutions for Orbit Types 8, 10, 12 .... 64 19. Inclination vs. Time for Orbit Types 7 Through 12 ................. 65 20. Eccentricity X Sine (Argument of Periselene) and Eccentricity x Cosine (Argument of Pen- selene) vs. Number of Revolutions for Orbit Type 10 .................... 66 21. Change in Longitude of Ascending Node After 80 Revolutions vs. Inclination .....67 LIST OF TABLES Table 1. Initial and Final Values of the Orbit Elements . 50 viii NOMENCLATURE A 1 Principle moment of inertia about Xt axis A' L -L in s A e cos w [A] Coordinate transformation matrix a Semi-major axis of the satellite orbit B 1 Principle moment of inertia about y' axis B' L - in in B e sin w C 1 Principle moment of inertia about Z' axis C' L - S S C Perturbative acceleration in circumferential direction - D' L . in m D Elongation of Moon E Eccentric anomaly of satellite orbit e Eccentricity of satellite orbit F Perturbing force f True anomaly G Universal gravitational constant H Perturbing force potential h Angular mqmentum I Inclination of satellite orbit plane to lunar equator Inclination of Earth-Moon orbit plane to lunar equator ix x i,j,k Unit vectors in (x,y,z) direction T Unit vector in direction of ascending node of Moon's orbit about the Earth Unit vector in direction of lunar perigee Lm Geocentric mean longitude of Moon L 5 Geocentric mean longitude of Sun M Mass of the satellite Me Mass of the Earth Mm Mass of the Moon P Semi.-latus rectum B Perturbative force in radial direction re Radius of Earth res Radius from Earth to satellite t Time t Time of previous pericenter passage u Angle measured from ascending node to satellite radius vector V Velocity Total potential energy function V m Potential energy function of Moon V Potential energy function of Earth W Perturbative acceleration normal to orbital plane x,y,z Selenocentric inertial coordinates x,yt,zt Selenocentric rotating coordinates X,y,Z Coordinates of Earth xi X,Y,Z Perturbative accelerations in x,y,z direction X,Y,Z Total acceleration in x,y,z direction Latitude Unit vector in the subscript direction E ( ) Longitude e Angle between line of nodes and x' axis Angle between line e of nodes and radius vector of Earth Phase angle between x axis and periselene Lunar gravitational potential W Argument of pericenter We Angular velocity of Earth with respect to Moon w Geocentric mean longitude of Moon's perigee W Geocentric mean longitude of Sun's perigee Geocentric mean longitude of Moon's ascending node Longitude of ascending node of satellite () Vector notaticn C) First derivative of ( ) with respect to time () Second derivative of ( ) with respect to time CONSTANTS a 384422 KM 1029 KG A1 .887825 X KM B1 .888005 < 1029 KG KM2 C .888375 X 10 29 KG KM2 1 e .0549 KM/KG/sec2 G .66709998 >< 10- 19 .116384501 radians I e M .597516432 x 10 25 KG Mm .73464634 ( 10 23 KG r .173999208 X io KM .39860320 X i06 KM/ see 2 .49027779 )< 10 4 KM/sec2 radians/Sec We .266507564 )< 10 xii I. INTRODUCTION With the recommendation of the President and the approval of Congress, the United States of America has launched the scientific and technicological undertaking of manned exploration of the Moon. The program has been assigned to the National Aeronautics and Space Administra- tion and has been titled "Project Apollo." One prerequisite to lunar landing is the es- tablishment of the Apollo spacecraft in a lunar orbit. It is necessary that the variation of this orbit with time be known in order: (1) To effect a landing in a preselected area of the lunar surface, and (2) To establish stay times on the lunar sur- face in order to assure successful comple- tion of rendezvous with the command module during the return to lunar orbit. The necessity of determing the characteristics of Apollo- type lunar orbits prompted the investigation described here. The investigation of Earth satellite motion has been quite thorough, however, a relatively small amount of 1 2 work has been done in lunar satellite theory because sig- nificant interest in this area has developed comparatively recently. There are several reasons why the theory of terrestrial satellite motion is not directly applicable to lunar satellites. While usually the mass of the Moon is neglected in studies of Earth satellite motion, the greater mass of the Earth has a significant influence on the motion of a lunar satellite. Also, one of the first assumptions in most investigations of Earth satellite motion is that the Earth is aSpheroid of revolution. The Moon, however, is best approximated as a triaxial ellipsoid; i.e. the Moon is not a body of revolution. Therefore, it has no plane of symmetry. Consequently, the lunar gravitational field is more complex than the gravitational field of the Earth. Moreover, its orientation with respect to an in- ertial coordinate system is changing with time due to the rotation of the Moon about its axis. Due to these factors, the problem of describing the motion of a near lunar sat- ellite is in general a different and more complex problem than that of describing the motion of a near Earth sat- ellite. It should be noted that the presence of atmospheric drag, which greatly complicates the motion of near Earth satellites, is of no consequence in the motion of lunar satellites. 3 A survey of the literature concerned with this problem reveals that the primary effort, thus far, has been the development of approximate closed form solutions to Lagrange's Planetary Equations, which are a set of first order, nonlinear, differential equations for the time rates of change of the orbit elements.
Load more

Recommended publications
  • A Study of Ancient Khmer Ephemerides
    A study of ancient Khmer ephemerides François Vernotte∗ and Satyanad Kichenassamy** November 5, 2018 Abstract – We study ancient Khmer ephemerides described in 1910 by the French engineer Faraut, in order to determine whether they rely on observations carried out in Cambodia. These ephemerides were found to be of Indian origin and have been adapted for another longitude, most likely in Burma. A method for estimating the date and place where the ephemerides were developed or adapted is described and applied. 1 Introduction Our colleague Prof. Olivier de Bernon, from the École Française d’Extrême Orient in Paris, pointed out to us the need to understand astronomical systems in Cambo- dia, as he surmised that astronomical and mathematical ideas from India may have developed there in unexpected ways.1 A proper discussion of this problem requires an interdisciplinary approach where history, philology and archeology must be sup- plemented, as we shall see, by an understanding of the evolution of Astronomy and Mathematics up to modern times. This line of thought meets other recent lines of research, on the conceptual evolution of Mathematics, and on the definition and measurement of time, the latter being the main motivation of Indian Astronomy. In 1910 [1], the French engineer Félix Gaspard Faraut (1846–1911) described with great care the method of computing ephemerides in Cambodia used by the horas, i.e., the Khmer astronomers/astrologers.2 The names for the astronomical luminaries as well as the astronomical quantities [1] clearly show the Indian origin ∗F. Vernotte is with UTINAM, Observatory THETA of Franche Comté-Bourgogne, University of Franche Comté/UBFC/CNRS, 41 bis avenue de l’observatoire - B.P.
    [Show full text]
  • Elliptical Orbits
    1 Ellipse-geometry 1.1 Parameterization • Functional characterization:(a: semi major axis, b ≤ a: semi minor axis) x2 y 2 b p + = 1 ⇐⇒ y(x) = · ± a2 − x2 (1) a b a • Parameterization in cartesian coordinates, which follows directly from Eq. (1): x a · cos t = with 0 ≤ t < 2π (2) y b · sin t – The origin (0, 0) is the center of the ellipse and the auxilliary circle with radius a. √ – The focal points are located at (±a · e, 0) with the eccentricity e = a2 − b2/a. • Parameterization in polar coordinates:(p: parameter, 0 ≤ < 1: eccentricity) p r(ϕ) = (3) 1 + e cos ϕ – The origin (0, 0) is the right focal point of the ellipse. – The major axis is given by 2a = r(0) − r(π), thus a = p/(1 − e2), the center is therefore at − pe/(1 − e2), 0. – ϕ = 0 corresponds to the periapsis (the point closest to the focal point; which is also called perigee/perihelion/periastron in case of an orbit around the Earth/sun/star). The relation between t and ϕ of the parameterizations in Eqs. (2) and (3) is the following: t r1 − e ϕ tan = · tan (4) 2 1 + e 2 1.2 Area of an elliptic sector As an ellipse is a circle with radius a scaled by a factor b/a in y-direction (Eq. 1), the area of an elliptic sector PFS (Fig. ??) is just this fraction of the area PFQ in the auxiliary circle. b t 2 1 APFS = · · πa − · ae · a sin t a 2π 2 (5) 1 = (t − e sin t) · a b 2 The area of the full ellipse (t = 2π) is then, of course, Aellipse = π a b (6) Figure 1: Ellipse and auxilliary circle.
    [Show full text]
  • Interplanetary Trajectories in STK in a Few Hundred Easy Steps*
    Interplanetary Trajectories in STK in a Few Hundred Easy Steps* (*and to think that last year’s students thought some guidance would be helpful!) Satellite ToolKit Interplanetary Tutorial STK Version 9 INITIAL SETUP 1) Open STK. Choose the “Create a New Scenario” button. 2) Name your scenario and, if you would like, enter a description for it. The scenario time is not too critical – it will be updated automatically as we add segments to our mission. 3) STK will give you the opportunity to insert a satellite. (If it does not, or you would like to add another satellite later, you can click on the Insert menu at the top and choose New…) The Orbit Wizard is an easy way to add satellites, but we will choose Define Properties instead. We choose Define Properties directly because we want to use a maneuver-based tool called the Astrogator, which will undo any initial orbit set using the Orbit Wizard. Make sure Satellite is selected in the left pane of the Insert window, then choose Define Properties in the right-hand pane and click the Insert…button. 4) The Properties window for the Satellite appears. You can access this window later by right-clicking or double-clicking on the satellite’s name in the Object Browser (the left side of the STK window). When you open the Properties window, it will default to the Basic Orbit screen, which happens to be where we want to be. The Basic Orbit screen allows you to choose what kind of numerical propagator STK should use to move the satellite.
    [Show full text]
  • 2 Coordinate Systems
    2 Coordinate systems In order to find something one needs a system of coordinates. For determining the positions of the stars and planets where the distance to the object often is unknown it usually suffices to use two coordinates. On the other hand, since the Earth rotates around it’s own axis as well as around the Sun the positions of stars and planets is continually changing, and the measurment of when an object is in a certain place is as important as deciding where it is. Our first task is to decide on a coordinate system and the position of 1. The origin. E.g. one’s own location, the center of the Earth, the, the center of the Solar System, the Galaxy, etc. 2. The fundamental plan (x−y plane). This is often a plane of some physical significance such as the horizon, the equator, or the ecliptic. 3. Decide on the direction of the positive x-axis, also known as the “reference direction”. 4. And, finally, on a convention of signs of the y− and z− axes, i.e whether to use a left-handed or right-handed coordinate system. For example Eratosthenes of Cyrene (c. 276 BC c. 195 BC) was a Greek mathematician, elegiac poet, athlete, geographer, astronomer, and music theo- rist who invented a system of latitude and longitude. (According to Wikipedia he was also the first person to use the word geography and invented the disci- pline of geography as we understand it.). The origin of this coordinate system was the center of the Earth and the fundamental plane was the equator, which location Eratosthenes calculated relative to the parts of the Earth known to him.
    [Show full text]
  • New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion
    New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion Michelle Chernick∗ and Simone D'Amicoy Aeronautics and Astronautics, Stanford University, Stanford, California, 94305, USA This paper addresses the fuel-optimal guidance and control of the relative motion for formation-flying and rendezvous using impulsive maneuvers. To meet the requirements of future multi-satellite missions, closed-form solutions of the inverse relative dynamics are sought in arbitrary orbits. Time constraints dictated by mission operations and relevant perturbations acting on the formation are taken into account by splitting the optimal recon- figuration in a guidance (long-term) and control (short-term) layer. Both problems are cast in relative orbit element space which allows the simple inclusion of secular and long-periodic perturbations through a state transition matrix and the translation of the fuel-optimal optimization into a minimum-length path-planning problem. Due to the proper choice of state variables, both guidance and control problems can be solved (semi-)analytically leading to optimal, predictable maneuvering schemes for simple on-board implementation. Besides generalizing previous work, this paper finds four new in-plane and out-of-plane (semi-)analytical solutions to the optimal control problem in the cases of unperturbed ec- centric and perturbed near-circular orbits. A general delta-v lower bound is formulated which provides insight into the optimality of the control solutions, and a strong analogy between elliptic Hohmann transfers and formation-flying control is established. Finally, the functionality, performance, and benefits of the new impulsive maneuvering schemes are rigorously assessed through numerical integration of the equations of motion and a systematic comparison with primer vector optimal control.
    [Show full text]
  • Impulsive Maneuvers for Formation Reconfiguration Using Relative Orbital Elements
    JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 38, No. 6, June 2015 Impulsive Maneuvers for Formation Reconfiguration Using Relative Orbital Elements G. Gaias∗ and S. D’Amico† DLR, German Aerospace Center, 82234 Wessling, Germany DOI: 10.2514/1.G000189 Advanced multisatellite missions based on formation-flying and on-orbit servicing concepts require the capability to arbitrarily reconfigure the relative motion in an autonomous, fuel efficient, and flexible manner. Realistic flight scenarios impose maneuvering time constraints driven by the satellite bus, by the payload, or by collision avoidance needs. In addition, mission control center planning and operations tasks demand determinism and predictability of the propulsion system activities. Based on these considerations and on the experience gained from the most recent autonomous formation-flying demonstrations in near-circular orbit, this paper addresses and reviews multi- impulsive solution schemes for formation reconfiguration in the relative orbit elements space. In contrast to the available literature, which focuses on case-by-case or problem-specific solutions, this work seeks the systematic search and characterization of impulsive maneuvers of operational relevance. The inversion of the equations of relative motion parameterized using relative orbital elements enables the straightforward computation of analytical or numerical solutions and provides direct insight into the delta-v cost and the most convenient maneuver locations. The resulting general methodology is not only able to refind and requalify all particular solutions known in literature or flown in space, but enables the identification of novel fuel-efficient maneuvering schemes for future onboard implementation. Nomenclature missions. Realistic operational scenarios ask for accomplishing such a = semimajor axis actions in a safe way, within certain levels of accuracy, and in a fuel- B = control input matrix of the relative dynamics efficient manner.
    [Show full text]
  • NOAA Technical Memorandum ERL ARL-94
    NOAA Technical Memorandum ERL ARL-94 THE NOAA SOLAR EPHEMERIS PROGRAM Albion D. Taylor Air Resources Laboratories Silver Spring, Maryland January 1981 NOAA 'Technical Memorandum ERL ARL-94 THE NOAA SOLAR EPHEMERlS PROGRAM Albion D. Taylor Air Resources Laboratories Silver Spring, Maryland January 1981 NOTICE The Environmental Research Laboratories do not approve, recommend, or endorse any proprietary product or proprietary material mentioned in this publication. No reference shall be made to the Environmental Research Laboratories or to this publication furnished by the Environmental Research Laboratories in any advertising or sales promotion which would indicate or imply that the Environmental Research Laboratories approve, recommend, or endorse any proprietary product or proprietary material mentioned herein, or which has as its purpose an intent to cause directly or indirectly the advertised product to be used or purchased because of this Environmental Research Laboratories publication. Abstract A system of FORTRAN language computer programs is presented which have the ability to locate the sun at arbitrary times. On demand, the programs will return the distance and direction to the sun, either as seen by an observer at an arbitrary location on the Earth, or in a stan- dard astronomic coordinate system. For one century before or after the year 1960, the program is expected to have an accuracy of 30 seconds 5 of arc (2 seconds of time) in angular position, and 7 10 A.U. in distance. A non-standard algorithm is used which minimizes the number of trigonometric evaluations involved in the computations. 1 The NOAA Solar Ephemeris Program Albion D. Taylor National Oceanic and Atmospheric Administration Air Resources Laboratories Silver Spring, MD January 1981 Contents 1 Introduction 3 2 Use of the Solar Ephemeris Subroutines 3 3 Astronomical Terminology and Coordinate Systems 5 4 Computation Methods for the NOAA Solar Ephemeris 11 5 References 16 A Program Listings 17 A.1 SOLEFM .
    [Show full text]
  • Mission Concept for a Satellite Mission to Test Special Relativity
    Mission Concept for a Satellite Mission to Test Special Relativity VOLKAN ANADOL Space Engineering, masters level 2016 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering LULEÅ UNIVERSITY of TECHNOLOGY Master Thesis SpaceMaster Mission Concept for a Satellite Mission to Test Special Relativity Supervisors: Author : Dr. Thilo Schuldt Volkan Anadol Dr. Norman Gürlebeck Examiner : Assoc. Prof. Thomas Kuhn September 29, 2016 Abstract In 1905 Albert Einstein developed the theory of Special Relativity. This theory describes the relation between space and time and revolutionized the understanding of the universe. While the concept is generally accepted new experimental setups are constantly being developed to challenge the theory, but so far no contradictions have been found. One of the postulates Einsteins theory of Relativity is based on states that the speed of light in vac- uum is the highest possible velocity. Furthermore, it is demanded that the speed of light is indepen- dent of any chosen frame of reference. If an experiment would find a contradiction of these demands, the theory as such would have to be revised. To challenge the constancy of the speed of light the so- called Kennedy Thorndike experiment has been developed. A possible setup to conduct a Kennedy Thorndike experiment consists of comparing two independent clocks. Likewise experiments have been executed in laboratory environments. Within the scope of this work, the orbital requirements for the first space-based Kennedy Thorndike experiment called BOOST will be investigated. BOOST consists of an iodine clock, which serves as a time reference, and an optical cavity, which serves as a length reference.
    [Show full text]
  • 1 CHAPTER 10 COMPUTATION of an EPHEMERIS 10.1 Introduction
    1 CHAPTER 10 COMPUTATION OF AN EPHEMERIS 10.1 Introduction The entire enterprise of determining the orbits of planets, asteroids and comets is quite a large one, involving several stages. New asteroids and comets have to be searched for and discovered. Known bodies have to be found, which may be relatively easy if they have been frequently observed, or rather more difficult if they have not been observed for several years. Once located, images have to be obtained, and these have to be measured and the measurements converted to usable data, namely right ascension and declination. From the available observations, the orbit of the body has to be determined; in particular we have to determine the orbital elements , a set of parameters that describe the orbit. For a new body, one determines preliminary elements from the initial few observations that have been obtained. As more observations are accumulated, so will the calculated preliminary elements. After all observations (at least for a single opposition) have been obtained and no further observations are expected at that opposition, a definitive orbit can be computed. Whether one uses the preliminary orbit or the definitive orbit, one then has to compute an ephemeris (plural: ephemerides ); that is to say a day-to-day prediction of its position (right ascension and declination) in the sky. Calculating an ephemeris from the orbital elements is the subject of this chapter. Determining the orbital elements from the observations is a rather more difficult calculation, and will be the subject of a later chapter. 10.2 Elements of an Elliptic Orbit Six numbers are necessary and sufficient to describe an elliptic orbit in three dimensions.
    [Show full text]
  • Drift-Free Solar Sail Formations in Elliptical Sun-Synchronous Orbits
    Acta Astronautica 139 (2017) 201–212 Contents lists available at ScienceDirect Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro Drift-free solar sail formations in elliptical Sun-synchronous orbits Khashayar Parsay *, Hanspeter Schaub Aerospace Engineering Sciences Department, 431 UCB, Colorado Center for Astrodynamics Research, University of Colorado, Boulder, CO, 80309-0431, USA ARTICLE INFO ABSTRACT Keywords: To study the spatial and temporal variations of plasma in the highly dynamic environment of the magnetosphere, Solar sail multiple spacecraft must fly in a formation. The objective for this study is to investigate the feasibility of solar sail fl Formation ying formation flying in the Earth-centered, Sun-synchronous orbit regime. The focus of this effort is to enable for- Magnetosphere mission mation flying for a group of solar sails that maintain a nominally fixed Sun-pointing attitude during formation flight, solely for the purpose of precessing their orbit apse lines Sun-synchronously. A fixed-attitude solar sail formation is motivated by the difficulties in the simultaneous control of orbit and attitude in flying solar sails. First, the secular rates of the orbital elements resulting from the effects of solar radiation pressure (SRP) are determined using averaging theory for a Sun-pointing attitude sail. These averaged rates are used to analytically derive the first-order necessary conditions for a drift-free solar sail formation in Sun-synchronous orbits, assuming a fixed Sun-pointing orientation for each sail in formation. The validity of the first-order necessary conditions are illustrated by designing quasi-periodic relative motions. Next, nonlinear programming is applied to design truly drift-free two-craft solar sail formations.
    [Show full text]
  • Transformation Between the Celestial and Terrestrial Systems
    IERS Technical Note No. 32 5 Transformation Between the Celestial and Terrestrial Systems The coordinate transformation to be used to transform from the terres- trial reference system (TRS) to the celestial reference system (CRS) at the epoch t of the observation can be written as: [CRS] = Q(t)R(t)W (t) [TRS], (1) where Q(t), R(t) and W (t) are the transformation matrices arising from the motion of the celestial pole in the celestial system, from the rotation of the Earth around the axis of the pole, and from polar motion respec- tively. The frame as realized from the [TRS] by applying the transfor- mations W (t) and then R(t) will be called “the intermediate reference frame of epoch t.” 5.1 The Framework of IAU 2000 Resolutions Several resolutions were adopted by the XXIVth General Assembly of the International Astronomical Union (Manchester, August 2000) that concern the transformation between the celestial and terrestrial reference systems and are therefore to be implemented in the IERS procedures. Such a transformation being also required for computing directions of celestial objects in intermediate systems, the process to transform among these systems consistent with the IAU resolutions is also provided at the end of this chapter. Resolution B1.3 specifies that the systems of space-time coordinates as defined by IAU Resolution A4 (1991) for the solar system and the Earth within the framework of General Relativity are now named the Barycentric Celestial Reference System (BCRS) and the Geocentric Ce- lestial Reference System (GCRS) respectively. It also provides a gen- eral framework for expressing the metric tensor and defining coordinate transformations at the first post-Newtonian level.
    [Show full text]
  • Analytical Representation for Ephemeride with Short Time Spans Application to the Longitude of Titan
    A&A 635, A91 (2020) Astronomy https://doi.org/10.1051/0004-6361/201937148 & c X. J. Xi and A. Vienne 2020 Astrophysics Analytical representation for ephemeride with short time spans Application to the longitude of Titan X. J. Xi1,2,3 and A. Vienne2,3 1 National Time Service Centre (NTSC), Chinese Academy of Sciences, PO Box 18, Lintong, Shaanxi 710600, PR China 2 IMCCE, Observatoire de Paris, PSL Research University, CNRS, UPMC Univ. Paris 06, Paris, France e-mail: [email protected] 3 Université de Lille, Observatoire de Lille, 59000 Lille, France e-mail: [email protected] Received 20 November 2019 / Accepted 15 January 2020 ABSTRACT Context. The ephemerides of natural satellites resulting from numerical integration have a very good precision on the fitting to recent observations, in a limited interval. Meanwhile, synthetic ephemerides like the Théorie Analytique des Satellites de Saturne (TASS) by Vienne and Duriez describe in detail the dynamical system by a representation based on the combinations of the proper frequencies. Some theoretical studies need to have both advantages. For example, to study the rotation of Titan, one needs to know the representa- tion of its longitude. Aims. We aim to use these two types of ephemerides in order to rebuild a long-lasting and high-precision ephemeris with proper frequencies based on the numerical integration ephemeris. The aim is to describe the numerical ephemerides with formulas similar to analytical ones. Methods. We used the representation of the orbital elements from the TASS ephemeris analysed over 10 000 years as a reference template.
    [Show full text]