1 Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften Reihe C Dissertationen Heft Nr. 67 Björn Frommknecht Integrated Sensor Analysis of the GRACE Mission München 8 Verlag der Bayerischen Akademie der Wissenschaften in Kommission beim Verlag C. H. Beck ISSN ISBN
3 Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften Reihe C Dissertationen Heft Nr. 67 Integrated Sensor Analysis of the GRACE Mission Vollständiger Abdruck der von der Fakultät für Bauingenieur- und Vermessungswesen der Technischen Universität München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigten Dissertation vorgelegt von Dipl.-Ing.Univ. Björn Frommknecht München 8 Verlag der Bayerischen Akademie der Wissenschaften in Kommission beim Verlag C. H. Beck ISSN ISBN
4 Adresse der Deutschen Geodätischen Kommission: Deutsche Geodätische Kommission Alfons-Goppel-Straße! D München Telefon ! Telefax / - Vorsitzender: Univ.-Prof. Dr.-Ing. M. Schilcher Prüfer der Dissertation:. Univ.-Prof. Dr.-Ing. Dr.-Ing.h.c. R. Rummel. Univ. Prof. Dr.-Ing. J. Müller, Leinbiz Universität Hannover Die Dissertation wurde am 3..7 bei der Technischen Universität München eingereicht und durch die Fakultät für Bauingenieur- und Vermessungswesen am 6..7 angenommen 8 Deutsche Geodätische Kommission, München Alle Rechte vorbehalten. Ohne Genehmigung der Herausgeber ist es auch nicht gestattet, die Veröffentlichung oder Teile daraus auf photomechanischem Wege (Photokopie, Mikrokopie) zu vervielfältigen ISSN ISBN
5 3 Abstract The geodetic twin satellite mission GRACE delivers gravity field models of the Earth of unprecedented accuracy. However, the originally defined mission baseline is not (yet) reached. Among others, deficiencies in the gravity field sensor system or the related signal processing could be responsible for the degraded performance. In this work the GRACE gravity field sensor system is subject to an integrated sensor analysis. First, models of the satellites environment are derived, including models for the direct gravitational accelerations caused by the Earth, the Sun and the Moon and indirect effects like Earth and ocean tides. Among the nongravitational forces air drag, solar radiation pressure and Earth albedo are investigated. The purpose of this part of the work is to model what the gravity sensor system should feel. The second part contains mathematical models of the individual elements of the gravity field sensor system, the star sensor, the accelerometer,the K-band ranging system and the GPS receiver, to derive an understanding of how the sensors detect their environment. The third part consists of the analysis of the raw instrument data. The real measurement signals are analyzed and the measurement performance in terms of a noise level is derived. They are compared to the model output and the derived noise level to the one specified by the instruments manufacturers. Finally the processing of the raw instrument data to the level that is used for the gravity field determination is investigated. A main processing step for all sensors is the application of an anti-aliasing low-pass filter. Alternative filters are tested and evaluated against the filter used for the official data processing. Zusammenfassung Die Satellitenmission GRACE liefert Schwerefeldmodelle bisher nicht verfügbarer Qualität. Aber die vor dem Start anvisierte Genauigkeit ist (noch) nicht erreicht worden. Neben anderen möglichen Ursachen könnten Fehler im Schwerefeldmesssystem und in der dazugehörigen Signalverarbeitung für die reduzierte Genauigkeit verantwortlich sein. In dieser Arbeit wird das GRACE Schwerefeldmesssystem einer integrierten Sensoranalyse unterzogen. Zunächst werden Modelle der Satellitenumgebung vorgestellt. Unter den direkten gravitativen Kräften werden die Erdanziehung sowie die Anziehung von Sonne und Mond modelliert, unter den indirekten gravitativen Kräften werden Erd- und Ozeangezeiten vorgestellt. Unter den nicht-gravitativen Kräften werden der Luftwiderstand, der Strahlungsdruck der Sonne und Erdalbedo untersucht. Diese Modelle geben Auskunft darüber, was das Schwerefeldmesssystem detektieren soll. Im zweiten Teil werden mathematische Modelle der einzelnen Sensoren des Schwerefeldmesssystems erarbeitet. Im einzelnen werden der Beschleunigungsmesser, der Sternsensoren, das K-band Abstandsmesssystem und der GPS-Empfänger modelliert, um eine Vorstellung darüber zu erlangen, wie die Satelliten ihre Umgebung wahrnehmen. Im nächsten Teil werden die rohen Messdaten der Instrumente untersucht. Die Messsignale werden analysiert und die Qualität der Sensoren, wie auch die der Messung, wird über das Fehlerniveau bestimmt. Die gemessenen Signale werden mit den Erwartungen aus den Modellen verglichen, insbesondere wird das aus den Messungen abgeleitete Rauschniveau mit den durch die Intrumentenhersteller gegebenen Spezifikationen verglichen. Schließlich wird die Prozessierung von Rohdaten zu Daten, die für die Schwerefeldbestimmung verwendet werden, untersucht. Ein Hauptschritt der Umwandlung ist die Anwendung eines Tiefpassfilters um die Datenrate ohne das Auftreten von Aliasingeffekten reduzieren zu können. Verschiedene alternative Filter werden diskutiert und miteinander verglichen.
7 Contents 5 Contents Abstract Zusammenfassung I Introduction Overview of the GRACE mission Definition of integrated sensor analysis Goals and topics of the work II Force models Gravitational forces Introduction Earth gravity field Gravitational forces of third bodies Direct tides Indirect tides of third bodies Ocean tides Solid Earth pole tides Non-gravitational forces Introduction Macro-model of the GRACE satellites Air drag Introduction Air density models Velocity of the satellite relative to the atmosphere Force models for the derivation of the drag coefficient Solar radiation pressure Introduction Shadow of the Earth Force model
8 6 Contents 5.5 Earth albedo Introduction Force model Comparison of the model results with real data III Sensor models Introduction Overview of the relevant sensor systems Accelerometer Introduction Logical model Mathematical model Measurement model Dynamic measurement model Star sensor Introduction Logical model Measurement model Error model K-band system Introduction Logical model Measurement model Error model GPS Receiver Introduction Measurement models Pseudorange measurements Carrier phase measurements Error model
9 Contents 7 IV Real data processing and analysis Data levels overview Data products overview Science data products Housekeeping data products Data set description Level a data analysis Analysis of the star tracker data Analysis of the accelerometer data Twangs Peaks Thruster events Accelerometer performance estimation Linear acceleration measurement performance estimation Angular accelerations Analysis of the K-band and Ka-band ranging data Level a to level b processing K-band Accelerometer Star Sensor Level b analysis GPS receiver K-band Accelerometer Disturbance effects: peaks, twangs and thruster events Performance assessment Star Sensor Combined analysis Conclusions and discussion Gravitational forces on the satellites Non-gravitational forces on the satellites Gravity field sensor system Outlook Non-gravitational forces on the satellites Level a to level b data processing Acknowledgement
10 8 Annex A Discrete Fourier transform and discrete filters A. Discrete Fourier Transform (DFT) A. Discrete filters A.3 Low-pass filter A.4 High-pass filter A.5 Differentiator A.6 Double differentiator A.7 Filter scaling Annex B Power spectral density (PSD) and standard deviation Annex C Orientation representations and coordinate transforms Annex D Coordinate frames D. Inertial reference frame (IRF) D. Orbit fixed reference frame (ORF) D.3 Earth-fixed reference frame (EFRF) D.4 Satellite fixed reference frame (SRF) D.5 Instrument frames
11 9 Part I. Introduction
12 Overview of the GRACE mission. Overview of the GRACE mission GRACE (Gravity Recovery And Climate Experiment) is a joint project of the National Aeronautics and Space Administration (NASA) and the German Space Center (DLR). The mission has been proposed in 996 jointly by the University of Texas at Austin, Center for Space Research (UTCSR), the GeoForschungsZentrum Potsdam (GFZ), the Jet Propulsion Laboratories (JPL), Space Systems/Loral (SSL), the Deutsches Zentrum für Luftund Raumfahrt e.v. (DLR), and Astrium GmbH. The GRACE mission is successor to the CHAMP mission and was selected as the second mission in NASA s Earth Science System Pathfinder project (ESSP). The Principal Investigator is Prof. Byron Tapley (UTCSR), the Co-Principal Investigator of the mission is Prof. Ch. Reigber of the GeoForschungsZentrum Potsdam. The goal of the mission is to derive highly accurate models of the Earth gravity field for a period of at least five years. As an additional goal, GPS measurements are used to recover profiles of the ionosphere and the troposphere by limb sounding. Unlike the single satellite mission CHAMP, the GRACE mission consists of two identical satellites, GRACE A and GRACE B, resp. Tom and Jerry. The satellite constellation is depicted in figure.. Figure.: Overview of the GRACE satellites configuration. Considering the satellite in the front, the GPS antenna and the star sensor baffles are visible. The K-band microwave link is indicated as a blue ray. On the back side of the satellite, the ion thruster used for orbit maintenance is visible. Each satellite is equipped with a GPS Turbo Rogue Space Receiver that allows precise orbit determination. In order to be able to take into account the effect of non-gravitational forces, both satellites are equipped with ultra-sensitive Super STAR capacitive accelerometers.
13 Overview of the GRACE mission An absolute novelty is the use of a K-band microwave link between the satellites to measure the inter-satellite distance with micrometer accuracy. To guarantee sufficient power supply and to enable operation of the K-band system, the attitude of the satellites is observed by star sensors and controlled by cold gas attitude thrusters and magnetic torque rods. The GRACE satellites were successfully launched on March 7, from Plesetsk Cosmodrome in Russia. The inclination of the satellites orbits is about 89, the semi-major axis was about 687 km, the eccentricity was almost zero:.. Since then the orbit decayed to an altitude of about 46 km. The nominal distance between the two satellites is ±5 km.
14 Definition of integrated sensor analysis. Definition of integrated sensor analysis The subject to the integrated sensor analysis is a complex system consisting of several individual sensors. The purpose of the integrated sensor analysis is to understand the functionality of each individual sensor, the interaction between the individual sensors and their operation as a sensor system. The former is achieved by deriving mathematical models of the measurement process of each sensor. A common way is to model the signal that is to be sensed and then to model the error behavior of the sensor. The error behavior of the sensor is usually given by the manufacturer. The combination of the simulated signal and simulated error is a good approximation of the real measurement behavior. The latter is reached by combining the individual models to form the integrated model: The signal flow in the sensor systems is derived, thus the dependencies can be modeled. Also the processing steps necessary to transform between the individual data levels are investigated.
15 3 3. Goals and topics of the work This work consists of three parts: The description of the forces acting on the GRACE satellites, the description of the GRACE system simulator and the description of the real data processing and analysis. The first part describes models for volume forces and surface forces that act on the GRACE satellites including models for gravitational forces like the gravity of the Earth or the effects caused by third bodies like the Sun or the Moon. Among the surface forces atmospheric drag models, models for solar radiation pressure and for Earth albedo are analyzed. Special emphasis is put on the analysis of the atmospheric drag models, as at the orbit height of GRACE air drag is the dominant surface force. The second part describes how the GRACE gravity field sensor system feels its environment. Measurement models for the sensors that constitute the gravity field sensor system, namely the star sensor, the accelerometer, the K-band ranging system and the GPS receiver are derived as mathematical models. The third part is devoted to the analysis of the real satellite data. Performance estimates for the individual instruments are derived and compared to the theoretical performance estimated in the second part. Also the transformation from raw satellite data to the data level that is used for the gravity field determination is investigated.
16 4 Part II. Force models
17 4 Gravitational forces 5 4. Gravitational forces 4.. Introduction The gravitational forces are by far the strongest forces that act on the satellites and mainly determine their orbits and relative motion. In the following sections the different kinds of gravitational forces will be briefly introduced. In terms of magnitude one expects that the gravitational forces of the Earth are strongest, followed by the forces caused by Moon, Sun and the solid Earth and ocean tides. 4.. Earth gravity field The gravitational accelerations are the first derivative of the gravitational potential of the Earth. The potential and its first order derivatives can be expressed in the following way: V i = GM r ( ) R l l λ i p (αcos(mλ) + β sin(mλ)), (4.) r l= m= l max with differentiation w.r.t λ i p α β none Plm Clm Slm r (l+) r P lm C lm S lm θ P lm C lm S lm λ m P lm S lm C lm The normalized associated Legendre functions and their derivatives are computed by recursive formulas. For convenience, the following substitutions were made: P lm = P lm (cos θ), P lm = P lm (cos θ) θ. P lm = η cos θp l,m σp l,m, (4.) P l,l = τ cos θp l,l, (4.3) P ll = ν sinθp l,l, (4.4) P lm = η(cos θp l,m sinθp l,m) σp l,m, (4.5) P l,l = τ(cos θp l,l sin θp l,l ), (4.6) P ll = ν(sin θp l,l + cos θp l,l ), (4.7) with the normalization factors: η = (l + )(l ) (l + m)(l m), σ = (l + m)(l m) (l + )(l ), τ = l + l +, ν =. (4.8) l
18 6 4 Gravitational forces The initial values are: P, =, P, = 3cosθ, P, = 3 sin θ, P, =, P, = 3sin θ, P, = 3cosθ. To obtain the accelerations in cartesian coordinates, the first order derivatives in spherical coordinates have to be transformed using the chain rule: ẍ e = V x ÿ e = V y z e = V z = V r r = V r = V r x + V θ θ x + V λ λ x, (4.9) r y + V θ θ y + V λ λ y, (4.) r z + V θ θ z, (4.) where the partial derivations are simply derived from geometry: r/ x = sin θ cos λ, (4.) r/ y = sin θ sin λ, (4.3) r/ z = cos θ, (4.4) θ/ x = cos θ cos λ/r, (4.5) θ/ y = cos θ sin λ/r, (4.6) θ/ z = sin θ/r, (4.7) λ/ x = sinλ/(r sinθ), (4.8) λ/ y = cos λ/(r sin θ). (4.9) For simulation purposes or for orbit integration it is necessary to transform these accelerations from the earthfixed system to the inertial system. Figure 4. shows the resulting accelerations in the Inertial Reference Frame (IRF). The magnitude is about 8.4 m/s or about 84% of the acceleration on the Earth s surface. We notice a modulation with the orbit frequency, as the distance from the satellite to the Earth and its velocity vary during a revolution. The first term of the spherical harmonic expansion of the gravity field, i.e. l=, accounts for 99% of the gravitational acceleration, a expansion up to degree, i.e. including the effect of the Earth s oblateness, represents about 99.99% of the characteristics of the gravitational acceleration of the Earth. Thus an expansion up to degree seems to be sufficient for most purposes except gravity field determination studies of course.
19 4 Gravitational forces deg deg deg 5 deg [m/s ] time [h] deg deg 5 deg [m/s ] time [h] Figure 4.: The top panel shows the gravitational accelerations induced by the Earth. The different lines show the accelerations resulting from an evaluation of the spherical harmonic series up to a certain degree. The bottom panel shows the change in acceleration from one evaluation level to the other. It seems that it is sufficient to evaluate the Legendre series up to degree for simulations, as the change for higher degrees is very small.
20 8 4 Gravitational forces 4.3. Gravitational forces of third bodies Gravitational forces caused by third bodies the gravitational forces caused by all other celestial bodies than the Earth. In terms of magnitude the tidal forces of Sun and Moon are dominating Direct tides The gravitational acceleration of a third body can be described as the acceleration of a point mass M, cf. Montenbruck and Gill (): r sat,i = GM r body r sat r body r sat 3, (4.) with r body r sat GM the geocentric position vector of the body, the geocentric position vector of the satellite, the gravitational constant times the mass of the attracting body. In an earth-fixed coordinate system the acceleration of the satellite is derived as the difference between the acceleration of the satellite caused by the body and the acceleration of the Earth caused by the body: r sat,e = r sat,i r Earth,i ( rbody r sat = GM r body r sat 3 r ) body r body 3. (4.) r-.. sat,e r-.. sat,i r-.. sat,e r-.. Earth,i r-.. sat,e Earth r-.. sat,e 3rd body Figure 4.: Tidal ellipse of relative acceleration with respect to the center of an accelerated coordinate system. In this example the resulting acceleration on a satellite caused by a third body like the Sun or the Moon in the earth-fixed coordinate system is shown. Figure 4. shows the tidal acceleration of the satellite relative to the Earth -two body system due to a third body. It has the form of a tidal ellipse. We realize that the satellite is accelerated away from the Earth if it is in front of or behind the Earth and that the resulting acceleration is directed towards the Earth if the satellite is above or below the Earth. This equation can not only be applied to a satellite orbiting the Earth, it also describes the relative acceleration of mass elements of a body with respect to its center of mass. Equation (4.) therefore also describes the solid Earth tidal deformation due to Sun and Moon.
21 4 Gravitational forces Indirect tides of third bodies As mentioned above, the tidal forces of the other planets of our solar system deform the Earth. The deformation depends on the local properties of the Earth s crust. The so called Love numbers reflect the elasticity of the Earth s crust. The deformation leads to a change of the mass distribution of the Earth in an earth-fixed coordinate system and therefore has to be taken into account when modeling the forces acting on a satellite. As this effect is not caused directly by the gravitating body, it is called indirect. In principle not only the Earth is deformed by this effect, but also all the other planets. For satellites orbiting the Earth this effect is neglected, as the other planets are assumed to be point masses. The change in the mass distribution of the Earth causes a change of the geopotential. It is expanded in a series of spherical harmonic coefficients and described in more detail e.g. in McCarthy (6). We will give here only a brief outline: C nm i S nm = k nm 3 ( ) GM j RE P nm(sinφ j )e imλ j, (4.) n + GM j= E R j where k nm is the nominal Love number for degree n and order m, R E is the equatorial radius of the Earth, GM j is the gravitational parameter for the Moon (j = ) and the Sun (j=3), GM E is the gravitational parameter for the Earth, R j is the distance from geocenter to Moon or Sun, Φ j is the body fixed geocentric latitude of the Moon or the Sun, λ j is the body fixed geocentric longitude of the Moon or the Sun, and P nm are the normalized associated Legendre functions. The redistribution of the masses depends also on the frequency band affected by the different tidal effects, therefore a frequency dependent correction to the equation above has to be derived: C nm i S nm = f(n,m) A(f(n, m)) e iθ f(n,m), (4.3) where f(n, m) A(f(n, m)) θ f(n,m) is the frequency of the excitation, is the amplitude of the excitation, accounts for the Earth rotation and the type of the tide. The needed amplitudes and frequencies are tabulated in McCarthy (6) Ocean tides The oceans react to the tidal forces as well. There are two effects:. the change in the mass distribution of the Earth due to the movement of the water,
22 4 Gravitational forces. the change in the mass distribution of the Earth due to the changed load on the Earth s crust caused by the changing water load. The effect on the geopotential is again formulated as a change in the spherical harmonic coefficients: C nm i S nm = F nm s(n,m) (C snm ± S snm)e ± ±iθsnm ), (4.4) + where F nm accounts for the changed load on the crust, C snm, ± Ssnm pm are the ocean tide coefficients for the tide type s, θ snm is the argument of the tide type s. The summation over + and - denotes the respective addition of the retrograde and prograde waves. The interested reader may consult McCarthy (6) for a more detailed description Solid Earth pole tides The pole tides are generated by the centrifugal effect of the polar motion. They affect only the geopotential coefficients C and S : C = C (m C m ), (4.5) S = C (m + C m ), (4.6) where C =.333 9, C =.5, m, m are the wobble variables directly related the polar motion variables (x p, y p). A detailed description is to be found in McCarthy (6). Figure 4.3 shows the tidal accelerations on a GRACE satellite caused by Sun and Moon in an earth-fixed inertial reference frame over one day. We notice a strong signal on twice per revolution for all constituents. The direct tidal acceleration caused by the Moon is strongest at a level of 7 7 m/s followed by the direct tidal acceleration caused by the Sun at a level of about 3 7 m/s, i.e. the tidal force of the Moon is about three times stronger than the tidal force caused by the Sun. The ratio between the forces of Sun and Moon depends on the orientation of the orbit with respect to the positions of Sun and Moon. Here the orbit normal points towards the Sun, i.e. the distance between the satellite and the Sun is almost constant, therefore the force is also almost constant. The orbit plane is aligned with the direction to the Moon, therefore the exerted force by the Moon varies strongly. The magnitude of the indirect tidal accelerations caused by the Moon are at level of about 7 m/s, those caused by the Sun are at level of about 8 8 m/s. The accelerations caused by the ocean tides vary between
23 4 Gravitational forces 6 acceleration [m/s ] time [h] sun moon indirect moon indirect sun freq. dep. ocean pole tide Figure 4.3: Tidal accelerations due to Sun and Moon over one day for a GRACE satellite. acceleration [m/s /srqt(hz)] sun moon indirect moon indirect sun freq. dep. ocean pole tide frequency [Hz] Figure 4.4: Power spectral density of the tidal accelerations caused to Sun and Moon for a GRACE satellite. 8 m/s and 7 m/s. The magnitude of the frequency dependent corrections is about.5 8 m/s, the pole tide varies between 3 9 m/s and 8 m/s and shows also a semidiurnal period. All effects can be detected by the GRACE sensor system, as its sensitivity is about 3 m/s.
24 4 Gravitational forces In order to compare the variations of the different effects, we consider figure 4.4, where the tidal accelerations are displayed in the frequency domain as a power spectral density. We notice that the dominant frequencies are harmonics of the orbit frequency. The dominant amplitudes are found on twice per revolution. Concerning the magnitude of the different effects, the direct tides of the Moon are strongest, followed by the ocean tides, the indirect tides of the Moon, the direct tides of the Sun, the indirect tides of the Sun, the pole tides and the frequency dependent corrections. The ocean tides show a less strong correlation with harmonics of the orbit frequency, they seem to be rather independent from the satellite s position.
25 5 Non-gravitational forces 3 5. Non-gravitational forces 5.. Introduction The non-gravitational forces acting on the satellites are significantly smaller than the major gravitational ones. Depending on the orbit height, either the solar radiation pressure (as for high orbiting GPS satellites) or the atmospheric drag and earth albedo (as for low orbiting satellites) have significant influence on the forces acting on the satellites. As the GRACE satellites are low orbiting satellites, we expect dominant influence of the atmospheric drag, followed by the earth albedo and the solar radiation pressure. We did not investigate the effects of anisotropic thermal emission, the recoil from the radiowave beams used for the data downlink, electrostatic effects and others that are considered negligible. The main emphasis was put on the analysis of different air drag models, as the air drag is the dominant nongravitational force. A general idea was to assess if the accelerometer measurements can be replaced by model outputs. In the following sections we will first present a macro model of the GRACE satellites, that is a prerequisite for all force models, and then present the individual models. 5.. Macro-model of the GRACE satellites Figure 5. shows an engineering drawing of the view from the front and figure 5. shows the view from the side of a GRACE satellite. Top Port outer Starboard outer Port inner Bottom Starboard inner Boom Figure 5.: Engineering drawing of the front of a GRACE satellite. Image from Bettadpur (7). Table 5. shows the properties of the individual surface elements of the used macro-model of the GRACE satellites. The surface element location, area, normal vector in the SRF, the coefficients describing the interaction with visible light and infra-red radiation and the type of material is given.
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
Examination Space Missions and Applications I AE2103 Faculty of Aerospace Engineering Delft University of Technology SAMPLE EXAM Please read these instructions first: This are a series of multiple-choice
ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there
Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital
Exercises 131 The Falling Apple (page 233) 1 Describe the legend of Newton s discovery that gravity extends throughout the universe According to legend, Newton saw an apple fall from a tree and realized
Solar System 1. The diagram below represents a simple geocentric model. Which object is represented by the letter X? A) Earth B) Sun C) Moon D) Polaris 2. Which object orbits Earth in both the Earth-centered
Sunlight and its Properties EE 495/695 Y. Baghzouz The sun is a hot sphere of gas whose internal temperatures reach over 20 million deg. K. Nuclear fusion reaction at the sun's core converts hydrogen to
Solar System Fundamentals What is a Planet? Planetary orbits Planetary temperatures Planetary Atmospheres Origin of the Solar System Properties of Planets What is a planet? Defined finally in August 2006!
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,
5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
Lecture 5: Satellite Orbits Jan Johansson email@example.com Chalmers University of Technology, 2013 Geometry Satellite Plasma Posi+oning physics Antenna theory Geophysics Time and Frequency GNSS
RS platforms Platform vs. instrument Sensor Platform Instrument The remote sensor can be ideally represented as an instrument carried by a platform Platforms Remote Sensing: Ground-based air-borne space-borne
ESCI 107/109 The Atmosphere Lesson 2 Solar and Terrestrial Radiation Reading: Meteorology Today, Chapters 2 and 3 EARTH-SUN GEOMETRY The Earth has an elliptical orbit around the sun The average Earth-Sun
1 MCQ - ENERGY and CLIMATE 1. The volume of a given mass of water at a temperature of T 1 is V 1. The volume increases to V 2 at temperature T 2. The coefficient of volume expansion of water may be calculated
Günter Seeber Satellite Geodesy 2nd completely revised and extended edition Walter de Gruyter Berlin New York 2003 Contents Preface Abbreviations vii xvii 1 Introduction 1 1.1 Subject of Satellite Geodesy...
Prospects for an Improved Lense-Thirring Test with SLR and the GRACE Gravity Mission J. C. Ries, R. J. Eanes, B. D. Tapley Center for Space Research The University of Texas at Austin Austin, TX G. E. Peterson
Chapter 2 Mission Analysis As noted in Chapter 1, orbital and attitude dynamics must be considered as coupled. That is to say, the orbital motion of a spacecraft affects the attitude motion, and the attitude
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
Exam # 1 Thu 10/06/2010 Astronomy 100/190Y Exploring the Universe Fall 11 Instructor: Daniela Calzetti INSTRUCTIONS: Please, use the `bubble sheet and a pencil # 2 to answer the exam questions, by marking
5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
Attitude and Orbit Dynamics of High Area-to-Mass Ratio (HAMR) Objects and Carolin Früh National Research Council Postdoctoral Fellow, AFRL, firstname.lastname@example.org Orbital Evolution of Space Debris Objects Main
Milan Bursa Karel Pec Gravity Field and Dynamics of the Earth With 89 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Preface v Introduction 1 1 Fundamentals
Lesson 11 Physics 168 1 Oscillations and Waves 2 Simple harmonic motion If an object vibrates or oscillates back and forth over same path each cycle taking same amount of time motion is called periodic
UNIT HEAT. KINETIC THEORY OF GASES.. Introduction Molecules have a diameter of the order of Å and the distance between them in a gas is 0 Å while the interaction distance in solids is very small. R. Clausius
GRAVITATION CONCEPTS Kepler's law of planetry motion (a) Kepler's first law (law of orbit): Every planet revolves around the sun in an elliptical orbit with the sun is situated at one focus of the ellipse.
Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007 Name: Directions: Listed below are twenty (20) multiple-choice questions based on the material covered by the lectures this past week. Choose
APPENDIX D: SOLAR RADIATION The sun is the source of most energy on the earth and is a primary factor in determining the thermal environment of a locality. It is important for engineers to have a working
Sound Power Measurement A sound source will radiate different sound powers in different environments, especially at low frequencies when the wavelength is comparable to the size of the room 1. Fortunately
2 Absorbing Solar Energy 2.1 Air Mass and the Solar Spectrum Now that we have introduced the solar cell, it is time to introduce the source of the energy the sun. The sun has many properties that could
Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall Gravitational Potential Energy In space, an object or gas cloud has
1 ESCI-61 Introduction to Photovoltaic Technology Sun Earth Relationships Ridha Hamidi, Ph.D. Spring (sun aims directly at equator) Winter (northern hemisphere tilts away from sun) 23.5 2 Solar radiation
A2 Physics Notes OCR Unit 4: The Newtonian World Momentum: - An object s linear momentum is defined as the product of its mass and its velocity. Linear momentum is a vector quantity, measured in kgms -1
Can Hubble be Moved to the International Space Station? 1 On January 16, NASA Administrator Sean O Keefe informed scientists and engineers at the Goddard Space Flight Center (GSFC) that plans to service
IB PHYSICS: Gravitational Forces Review 1. This question is about gravitation and ocean tides. (b) State Newton s law of universal gravitation. Use the following information to deduce that the gravitational
Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled
APN-064 IMU Errors and Their Effects Rev A Introduction An Inertial Navigation System (INS) uses the output from an Inertial Measurement Unit (IMU), and combines the information on acceleration and rotation
HybridSail Hybrid Solar Sails for Active Debris Removal Final Report Authors: Lourens Visagie (1), Theodoros Theodorou (1) Affiliation: 1. Surrey Space Centre - University of Surrey ACT Researchers: Leopold
SF2A 2011 G. Alecian, K. Belkacem, R. Samadi and D. Valls-Gabaud (eds) IN-FLIGHT CALIBRATION OF THE MICROSCOPE SPACE MISSION INSTRUMENT: DEVELOPMENT OF THE SIMULATOR E. Hardy 1, A. Levy 1, G. Métris 2,
The Kinetic Theory of Gases Sections Covered in the Text: Chapter 18 In Note 15 we reviewed macroscopic properties of matter, in particular, temperature and pressure. Here we see how the temperature and
Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged
Physics A Unit: G484: The Newtonian World 1(a) State Newton s second law of motion. The resultant force on an object is proportional to the rate of change of momentum of the object In part (a) the candidate
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
The Sun Solar Factoids (I) The sun, a medium-size star in the milky way galaxy, consisting of about 300 billion stars. (Sun Earth-Relationships) A gaseous sphere of radius about 695 500 km (about 109 times
Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 20 Conservation Equations in Fluid Flow Part VIII Good morning. I welcome you all
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L17 - Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to
Chapter 6: SOLAR GEOMETRY Full credit for this chapter to Prof. Leonard Bachman of the University of Houston SOLAR GEOMETRY AS A DETERMINING FACTOR OF HEAT GAIN, SHADING AND THE POTENTIAL OF DAYLIGHT PENETRATION...
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical
Interaction of Energy and Matter Gravity Measurement: Using Doppler Shifts to Measure Mass Concentration TEACHER GUIDE EMR and the Dawn Mission Electromagnetic radiation (EMR) will play a major role in
References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that
14 CHAPTER 2 ORBITAL DYNAMICS 2.1 INTRODUCTION This chapter presents definitions of coordinate systems that are used in the satellite, brief description about satellite equations of motion and relative
www.led-professional.com ISSN 1993-890X Trends & Technologies for Future Lighting Solutions ReviewJan/Feb 2015 Issue LpR 47 International Year of Light 2015 Tech-Talks BREGENZ: Mehmet Arik Well-Being in
CLASSICAL CONCEPT REVIEW 8 Kinetic Theory Information concerning the initial motions of each of the atoms of macroscopic systems is not accessible, nor do we have the computational capability even with
% *UDYLW\)LHOG7XWRULDO The formulae and derivations in the following Chapters 1 to 3 are based on Heiskanen and Moritz (1967) and Lambeck (1990). ([SDQVLRQRIWKHJUDYLWDWLRQDOSRWHQWLDOLQWRVSKHULFDOKDUPRQLFV
.1.1 Measure the motion of objects to understand.1.1 Develop graphical, the relationships among distance, velocity and mathematical, and pictorial acceleration. Develop deeper understanding through representations
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
Name Period 4 th Six Weeks Notes 2015 Weather Radiation Convection Currents Winds Jet Streams Energy from the Sun reaches Earth as electromagnetic waves This energy fuels all life on Earth including the
Hyperspectral Satellite Imaging Planning a Mission Victor Gardner University of Maryland 2007 AIAA Region 1 Mid-Atlantic Student Conference National Institute of Aerospace, Langley, VA Outline Objective
Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be
Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws
1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: email@example.com Abstract: There are many longstanding
Chapter 5. Gravitation Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 5.1 Newton s Law of Gravitation We have already studied the effects of gravity through the
Simulation, prediction and analysis of Earth rotation parameters with a dynamic Earth system model Florian Seitz Earth Oriented Space Science and Technology (ESPACE) 20.9.2011 Earth rotation parameters
entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 214 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Wednesday
USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney firstname.lastname@example.org Introduction The numerical calculations performed by scientists and engineers
Presentation of problem T1 (9 points): The Maribo Meteorite Definitions Meteoroid. A small particle (typically smaller than 1 m) from a comet or an asteroid. Meteorite: A meteoroid that impacts the ground
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125
Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost
Chapter 5: Circular Motion, the Planets, and Gravity 1. Earth s gravity attracts a person with a force of 120 lbs. The force with which the Earth is attracted towards the person is A. Zero. B. Small but
entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 1 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Friday 18
Homework #8 203-1-1721 Physics 2 for Students of Mechanical Engineering Part A 1. Four particles follow the paths shown in Fig. 32-33 below as they pass through the magnetic field there. What can one conclude
Coordinate Systems Orbits and Rotation Earth orbit. The earth s orbit around the sun is nearly circular but not quite. It s actually an ellipse whose average distance from the sun is one AU (150 million
Coverage Characteristics of Earth Satellites This document describes two MATLAB scripts that can be used to determine coverage characteristics of single satellites, and Walker and user-defined satellite
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
The Four Seasons A Warm Up Exercise What fraction of the Moon s surface is illuminated by the Sun (except during a lunar eclipse)? a) Between zero and one-half b) The whole surface c) Always half d) Depends
Artificial Satellites Earth & Sky Name: Introduction In this lab, you will have the opportunity to find out when satellites may be visible from the RPI campus, and if any are visible during the activity,
Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular
The Universe is thought to consist of trillions of galaxies. Our galaxy, the Milky Way, has billions of stars. One of those stars is our Sun. Our solar system consists of the Sun at the center, and all
Newton s Law of Universal Gravitation The greatest moments in science are when two phenomena that were considered completely separate suddenly are seen as just two different versions of the same thing.