Detumbling and Nutation Canceling Maneuvers with Complete Analytic Reduction for Axially Symmetric Spacecraft
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Dynamical Adjustments in IAU 2000A Nutation Series Arising from IAU 2006 Precession A
A&A 604, A92 (2017) Astronomy DOI: 10.1051/0004-6361/201730490 & c ESO 2017 Astrophysics Dynamical adjustments in IAU 2000A nutation series arising from IAU 2006 precession A. Escapa1; 2, J. Getino3, J. M. Ferrándiz2, and T. Baenas2 1 Dept. of Aerospace Engineering, University of León, 24071 León, Spain 2 Dept. of Applied Mathematics, University of Alicante, PO Box 99, 03080 Alicante, Spain e-mail: [email protected] 3 Dept. of Applied Mathematics, University of Valladolid, 47011 Valladolid, Spain Received 20 January 2017 / Accepted 23 May 2017 ABSTRACT The adoption of International Astronomical Union (IAU) 2006 precession model, IAU 2006 precession, requires IAU 2000A nutation to be adjusted to ensure compatibility between both theories. This consists of adding small terms to some nutation amplitudes relevant at the microarcsecond level. Those contributions were derived in previously published articles and are incorporated into current astronomical standards. They are due to the estimation process of nutation amplitudes by Very Long Baseline Interferometry (VLBI) and to the changes induced by the J2 rate present in the precession theory. We focus on the second kind of those adjustments, and develop a simple model of the Earth nutation capable of determining all the changes arising in the theoretical construction of the nutation series in a dynamical consistent way. This entails the consideration of three main classes of effects: the J2 rate, the orbital coefficients rate, and the variations induced by the update of some IAU 2006 precession quantities. With this aim, we construct a first order model for the nutations of the angular momentum axis of the non-rigid Earth. -
The Spin, the Nutation and the Precession of the Earth's Axis Revisited
The spin, the nutation and the precession of the Earth’s axis revisited The spin, the nutation and the precession of the Earth’s axis revisited: A (numerical) mechanics perspective W. H. M¨uller [email protected] Abstract Mechanical models describing the motion of the Earth’s axis, i.e., its spin, nutation and its precession, have been presented for more than 400 years. Newton himself treated the problem of the precession of the Earth, a.k.a. the precession of the equinoxes, in Liber III, Propositio XXXIX of his Principia [1]. He decomposes the duration of the full precession into a part due to the Sun and another part due to the Moon, to predict a total duration of 26,918 years. This agrees fairly well with the experimentally observed value. However, Newton does not really provide a concise rational derivation of his result. This task was left to Chandrasekhar in Chapter 26 of his annotations to Newton’s book [2] starting from Euler’s equations for the gyroscope and calculating the torques due to the Sun and to the Moon on a tilted spheroidal Earth. These differential equations can be solved approximately in an analytic fashion, yielding Newton’s result. However, they can also be treated numerically by using a Runge-Kutta approach allowing for a study of their general non-linear behavior. This paper will show how and explore the intricacies of the numerical solution. When solving the Euler equations for the aforementioned case numerically it shows that besides the precessional movement of the Earth’s axis there is also a nu- tation present. -
Research on the Precession of the Equinoxes and on the Nutation of the Earth’S Axis∗
Research on the Precession of the Equinoxes and on the Nutation of the Earth’s Axis∗ Leonhard Euler† Lemma 1 1. Supposing the earth AEBF (fig. 1) to be spherical and composed of a homogenous substance, if the mass of the earth is denoted by M and its radius CA = CE = a, the moment of inertia of the earth about an arbitrary 2 axis, which passes through its center, will be = 5 Maa. ∗Leonhard Euler, Recherches sur la pr´ecession des equinoxes et sur la nutation de l’axe de la terr,inOpera Omnia, vol. II.30, p. 92-123, originally in M´emoires de l’acad´emie des sciences de Berlin 5 (1749), 1751, p. 289-325. This article is numbered E171 in Enestr¨om’s index of Euler’s work. †Translated by Steven Jones, edited by Robert E. Bradley c 2004 1 Corollary 2. Although the earth may not be spherical, since its figure differs from that of a sphere ever so slightly, we readily understand that its moment of inertia 2 can be nonetheless expressed as 5 Maa. For this expression will not change significantly, whether we let a be its semi-axis or the radius of its equator. Remark 3. Here we should recall that the moment of inertia of an arbitrary body with respect to a given axis about which it revolves is that which results from multiplying each particle of the body by the square of its distance to the axis, and summing all these elementary products. Consequently this sum will give that which we are calling the moment of inertia of the body around this axis. -
Positional Astronomy Coordinate Systems
Positional Astronomy Observational Astronomy 2019 Part 2 Prof. S.C. Trager Coordinate systems We need to know where the astronomical objects we want to study are located in order to study them! We need a system (well, many systems!) to describe the positions of astronomical objects. The Celestial Sphere First we need the concept of the celestial sphere. It would be nice if we knew the distance to every object we’re interested in — but we don’t. And it’s actually unnecessary in order to observe them! The Celestial Sphere Instead, we assume that all astronomical sources are infinitely far away and live on the surface of a sphere at infinite distance. This is the celestial sphere. If we define a coordinate system on this sphere, we know where to point! Furthermore, stars (and galaxies) move with respect to each other. The motion normal to the line of sight — i.e., on the celestial sphere — is called proper motion (which we’ll return to shortly) Astronomical coordinate systems A bit of terminology: great circle: a circle on the surface of a sphere intercepting a plane that intersects the origin of the sphere i.e., any circle on the surface of a sphere that divides that sphere into two equal hemispheres Horizon coordinates A natural coordinate system for an Earth- bound observer is the “horizon” or “Alt-Az” coordinate system The great circle of the horizon projected on the celestial sphere is the equator of this system. Horizon coordinates Altitude (or elevation) is the angle from the horizon up to our object — the zenith, the point directly above the observer, is at +90º Horizon coordinates We need another coordinate: define a great circle perpendicular to the equator (horizon) passing through the zenith and, for convenience, due north This line of constant longitude is called a meridian Horizon coordinates The azimuth is the angle measured along the horizon from north towards east to the great circle that intercepts our object (star) and the zenith. -
Physics/0409010V1 [Physics.Gen-Ph] 1 Sep 2004 Ri Ihteclsilshr Sntfound
Physical Interpretations of Relativity Theory Conference IX London, Imperial College, September, 2004 ............... Mach’s Principle II by James G. Gilson∗ School of Mathematical Sciences Queen Mary College University of London October 29, 2018 Abstract 1 Introduction The question of the validity or otherwise of Mach’s The meaning and significance of Mach’s Principle Principle[5] has been with us for almost a century and and its dependence on ideas about relativistic rotat- has been vigorously analysed and discussed for all of ing frame theory and the celestial sphere is explained that time and is, up to date, still not resolved. One and discussed. Two new relativistic rotation trans- contentious issue is, does general[7] relativity theory formations are introduced by using a linear simula- imply that Mach’s Principle is operative in that theo- tion for the rotating disc situation. The accepted retical structure or does it not? A deduction[4] from formula for centrifugal acceleration in general rela- general relativity that the classical Newtonian cen- tivity is then analysed with the use of one of these trifugal acceleration formula should be modified by transformations. It is shown that for this general rel- an additional relativistic γ2(v) factor, see equations ativity formula to be valid throughout all space-time (2.11) and (2.27), will help us in the analysis of that there has to be everywhere a local standard of ab- question. Here I shall attempt to throw some light solutely zero rotation. It is then concluded that the on the nature and technical basis of the collection of field off all possible space-time null geodesics or pho- arXiv:physics/0409010v1 [physics.gen-ph] 1 Sep 2004 technical and philosophical problems generated from ton paths unify the absolute local non-rotation stan- Mach’s Principle and give some suggestions about dard throughout space-time. -
On the IAU 2000 Nutation Consistency with IAU 2006 Precession Draft Note
On the IAU 2000 nutation consistency with IAU 2006 precession Draft note∗ Alberto Escapay and Nicole Capitainez Abstract The Earth precession-nutation model of the International Astronomical Union (IAU) is composed of the IAU 2006 precession and IAU 2000 nu- tation. The IAU 2006 precession, which is consistent with both dynamical theory and the IAU 2000 nutation, was adopted to replace the precession part of the IAU 2000 precession-nutation. In that process, it was noticed that very slight adjustments were required to the IAU 2000 nutation ampli- tudes in order to ensure consistency at the micro arcsecond level with the IAU 2006 precession. The formulae for these adjustments provided by Cap- itaine et al. (2005) were implanted in part of the standards and software for computing nutation (e.g., SOFA and IERS Conventions 2010). However, IAU has not adopted any resolution on that direction, so formally such an inconsistency remains. Moreover, it has been shown recently (Escapa et al. 2017) that a few additional terms should be added to the 2005 expressions. We examine the drawbacks that have arisen as a consequence of the lack of such a resolution and propose different options to address them within the framework of IAU 2006 precession and IAU 2000 nutation. They run from just supplementing current IAU resolutions to clarify the content and termi- nology of the IAU precession-nutation model, to adopting a potential new resolution that would also ensure dynamical consistency between precession and nutation. ∗This draft will be submitted to publication yDepartment of Aerospace Engineering, University of Leon,´ E-24071 Leon,´ Spain; [email protected] zSYRTE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universites,´ F-75014 Paris, France; [email protected] 1 1 Introduction Nowadays, the International Astronomical Union (IAU) precession-nutation model is based on IAU 2000A nutation (Mathews et al. -
SOFA Tools for Earth Attitude
International Astronomical Union Standards Of Fundamental Astronomy SOFA Tools for Earth Attitude Software version 18 Document revision 1.64 Version for C programming language http://www.iausofa.org 2021 April 18 MEMBERS OF THE IAU SOFA BOARD (2021) John Bangert United States Naval Observatory (retired) Steven Bell Her Majesty’s Nautical Almanac Office Nicole Capitaine Paris Observatory Maria Davis United States Naval Observatory (IERS) Micka¨el Gastineau Paris Observatory, IMCCE Catherine Hohenkerk Her Majesty’s Nautical Almanac Office (chair, retired) Li Jinling Shanghai Astronomical Observatory Zinovy Malkin Pulkovo Observatory, St Petersburg Jeffrey Percival University of Wisconsin Wendy Puatua United States Naval Observatory Scott Ransom National Radio Astronomy Observatory Nick Stamatakos United States Naval Observatory Patrick Wallace RAL Space (retired) Toni Wilmot Her Majesty’s Nautical Almanac Office (trainee) Past Members Wim Brouw University of Groningen Mark Calabretta Australia Telescope National Facility William Folkner Jet Propulsion Laboratory Anne-Marie Gontier Paris Observatory George Hobbs Australia Telescope National Facility George Kaplan United States Naval Observatory Brian Luzum United States Naval Observatory Dennis McCarthy United States Naval Observatory Skip Newhall Jet Propulsion Laboratory Jin Wen-Jing Shanghai Observatory © Copyright 2013-20 International Astronomical Union. All Rights Reserved. Reproduction, adaptation, or translation without prior written permission is prohibited, except as al- lowed under the copyright laws. CONTENTS iii Contents 1 INTRODUCTION 1 1.1 The SOFA software ................................... 1 1.2 Quick start ....................................... 1 1.3 Abbreviations ...................................... 1 2 CELESTIAL COORDINATES 3 2.1 Stellar directions .................................... 3 2.2 Precession-nutation ................................... 3 2.3 Evolution of celestial reference systems ........................ 4 2.4 The IAU 2000 changes ................................ -
3D Rigid Body Dynamics: Tops and Gyroscopes
J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L30 - 3D Rigid Body Dynamics: Tops and Gyroscopes 3D Rigid Body Dynamics: Euler Equations in Euler Angles In lecture 29, we introduced the Euler angles as a framework for formulating and solving the equations for conservation of angular momentum. We applied this framework to the free-body motion of a symmetrical body whose angular momentum vector was not aligned with a principal axis. The angular moment was however constant. We now apply Euler angles and Euler’s equations to a slightly more general case, a top or gyroscope in the presence of gravity. We consider a top rotating about a fixed point O on a flat plane in the presence of gravity. Unlike our previous example of free-body motion, the angular momentum vector is not aligned with the Z axis, but precesses about the Z axis due to the applied moment. Whether we take the origin at the center of mass G or the fixed point O, the applied moment about the x axis is Mx = MgzGsinθ, where zG is the distance to the center of mass.. Initially, we shall not assume steady motion, but will develop Euler’s equations in the Euler angle variables ψ (spin), φ (precession) and θ (nutation). 1 Referring to the figure showing the Euler angles, and referring to our study of free-body motion, we have the following relationships between the angular velocities along the x, y, z axes and the time rate of change of the Euler angles. The angular velocity vectors for θ˙, φ˙ and ψ˙ are shown in the figure. -
General Relativity and Spatial Flows: I
1 GENERAL RELATIVITY AND SPATIAL FLOWS: I. ABSOLUTE RELATIVISTIC DYNAMICS* Tom Martin Gravity Research Institute Boulder, Colorado 80306-1258 [email protected] Abstract Two complementary and equally important approaches to relativistic physics are explained. One is the standard approach, and the other is based on a study of the flows of an underlying physical substratum. Previous results concerning the substratum flow approach are reviewed, expanded, and more closely related to the formalism of General Relativity. An absolute relativistic dynamics is derived in which energy and momentum take on absolute significance with respect to the substratum. Possible new effects on satellites are described. 1. Introduction There are two fundamentally different ways to approach relativistic physics. The first approach, which was Einstein's way [1], and which is the standard way it has been practiced in modern times, recognizes the measurement reality of the impossibility of detecting the absolute translational motion of physical systems through the underlying physical substratum and the measurement reality of the limitations imposed by the finite speed of light with respect to clock synchronization procedures. The second approach, which was Lorentz's way [2] (at least for Special Relativity), recognizes the conceptual superiority of retaining the physical substratum as an important element of the physical theory and of using conceptually useful frames of reference for the understanding of underlying physical principles. Whether one does relativistic physics the Einsteinian way or the Lorentzian way really depends on one's motives. The Einsteinian approach is primarily concerned with * http://xxx.lanl.gov/ftp/gr-qc/papers/0006/0006029.pdf 2 being able to carry out practical space-time experiments and to relate the results of these experiments among variously moving observers in as efficient and uncomplicated manner as possible. -
The Precession and Nutation of Deformable Bodies
The Precession and Nutation of Deformable Bodies Zdengk Kopal - -. ..I > / IPAQES) (NASA CR OR TUX OR AD NUMBER) MATH E MATI C S R E S E A R C H DECEMBER 1966 D1-82-0590 THE PRECESSION AND NUTATION OF DEFORMABLE BODIES by Zdengk Kopal I This research was supported in part by the National Aeronautics and Space Administration under Contract No. NASW-1470. Mathematical Note No. 496 Mathematics Research Laboratory E BOEING SCIENTIFIC RESEARCH LABORATORIES December 1966 ABSTRACT The aim of the present communication has been to set up the Eulerian system of equations which governs the motion of a self-gravitating de- formable body (regarded as a compressible fluid of arbitrarily high viscosity) about its own center of gravity in an arbitrary external field of force. If the latter were particularized to represent the tidal attraction of the Sun and the Moon, this motion would represent the luni-solar precession and nutation of a fluid Earth; if, on the other hand, the external field of force were governed by the Earth (or the Sun), the motion would define the physical librations of the Moon regarded as a deformable body. All these (and other) cases arising in the solar system will be treated in due course. The specific aim of this first of a series of reports in which these problems will be discussed will be to establish the explicit form of the system of differential equations which are basic to our problem. One specific aspect of their solution--namely, dynamical tides on deformable bodies and the consequent dissipation of energy--will be deferred to a second report of this series; while reports 111 and IV will be concerned with particular cases of the precession and librations of the Earth and the Moon. -
Arxiv:1007.1861V1 [Gr-Qc] 12 Jul 2010 Feulms N Hp.Temaueeto Uhagravito-Mag a Such of Measurement 10 the Is Shape
A laser gyroscope system to detect the Gravito-Magnetic effect on Earth A. Di Virgilio∗ INFN Sez. di Pisa, Pisa, Italy K. U. Schreiber and A. Gebauer† Technische Universitaet Muenchen, Forschungseinrichtung Satellitengeodaesie Fundamentalstation Wettzell, 93444 Bad K¨otzting, Germany J-P. R. Wells‡ Department of Physics and Astronomy, University of Canterbury, PB4800, Christchurch 8020, New Zealand A. Tartaglia§ Polit. of Torino and INFN, Torino, Italy J. Belfi and N. Beverini¶ Univ. of Pisa and CNISM, Pisa, Italy A.Ortolan∗∗ Laboratori Nazionali di Legnaro, INFN Legnaro (Padova), Italy Large scale square ring laser gyros with a length of four meters on each side are approaching a sensitivity of 1 10−11rad/s/√Hz. This is about the regime required to measure the gravito- magnetic effect (Lense× Thirring) of the Earth. For an ensemble of linearly independent gyros each measurement signal depends upon the orientation of each single axis gyro with respect to the rotational axis of the Earth. Therefore at least 3 gyros are necessary to reconstruct the complete angular orientation of the apparatus. In general, the setup consists of several laser gyroscopes (we would prefer more than 3 for sufficient redundancy), rigidly referenced to each other. Adding more gyros for one plane of observation provides a cross-check against intra-system biases and furthermore has the advantage of improving the signal to noise ratio by the square root of the number of gyros. In this paper we analyze a system of two pairs of identical gyros (twins) with a slightly different orientation with respect to the Earth axis. The twin gyro configuration has several interesting properties. -
An Essay on the Coriolis Force
James F. Price Woods Hole Oceanographic Institution Woods Hole, Massachusetts, 02543 http:// www.whoi.edu/science/PO/people/jprice [email protected] Version 3.3 January 10, 2006 Summary: An Earth-attached and thus rotating reference frame is almost always used for the analysis of geophysical flows. The equation of motion transformed into a steadily rotating reference frame includes two terms that involve the rotation vector; a centrifugal term and a Coriolis term. In the special case of an Earth-attached reference frame, the centrifugal term is exactly canceled by gravitational mass attraction and drops out of the equation of motion. When we solve for the acceleration seen from an Earth-attached frame, the Coriolis term is interpreted as a force. The rotating frame perspective gives up the properties of global momentum conservation and invariance to Galilean transformation. Nevertheless, it leads to a greatly simplified analysis of geophysical flows since only the comparatively small relative velocity, i.e., winds and currents, need be considered. The Coriolis force has a simple mathematical form, 2˝ V 0M , where ˝ is Earth’s rotation vector, V 0 is the velocity observed from the rotating frame and M is the particle mass. The Coriolis force is perpendicular to the velocity and can do no work. It tends to cause a deflection of velocity, and gives rise to two important modes of motion: (1) If the Coriolis force is the only force acting on a moving particle, then the velocity vector of the particle will be continually deflected and rotate clockwise in the northern hemisphere and anticlockwise in the southern hemisphere.