ENGO 421: COORDINATE SYSTEMSAlexander BraunDepartment of Geomatics EngineeringSchulich School of EngineeringUniversity of Calgary2008Contents1 Introduction 41.1 What is geodesy and why coordinate systems? . . . . . . . . . . . . . . . . . 51.2 Geodetic measurements and errors . . . . . . . . . . . . . . . . . . . . . . . 92 System of Natural Coordinates 132.1 Newtons Law of Gravitation - Gravitational Acceleration and Potential . . . 132.2 Gravity Potential and Centrifugal Potential . . . . . . . . . . . . . . . . . . . 192.3 Level Surfaces and Plumb Lines . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Natural Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Celestial Coordinate Systems 303.1 The Celestial Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 The Horizon System - H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 The Right Ascension System - RA . . . . . . . . . . . . . . . . . . . . . . . . 353.4 The Hour Angle System - HA . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 The Ecliptic System - E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Reection and Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . 393.7 Transformations Between Celestial Systems . . . . . . . . . . . . . . . . . . 40Transformation between HA and RA . . . . . . . . . . . . . . . . . . . . . . 40Transformation between RA and E . . . . . . . . . . . . . . . . . . . . . . . 42Transformation between H and HA . . . . . . . . . . . . . . . . . . . . . . . 42Other transformations between H, RA, HA, and E . . . . . . . . . . . . . . . 433.8 Astronomical Triangle and Spherical Trigonometry . . . . . . . . . . . . . . 43Spherical triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Astronomical triangle connecting H and HA . . . . . . . . . . . . . . . . . . 46Time systems used in astronomic azimuth determination . . . . . . . . . . . 47Sidereal time and hour angle . . . . . . . . . . . . . . . . . . . . . . . . . . 47Determination of the astronomic azimuth . . . . . . . . . . . . . . . . . . . 483.9 Hour angle method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49The error budget for the hour angle method using Polaris: . . . . . . . . . . 50Observation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Computation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.10 Astronomic azimuth determination by the altitude method . . . . . . . . . . 52Observation procedure for azimuth determination by altitude observationsof the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53C Alexander Braun (2006-2010) 2 ENGO421: COORDINATE SYSTEMSComputation procedure for azimuth determination by altitude observationsof the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 Time Systems 554.1 Sidereal Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Solar or Universal Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Conversion between Sidereal and Solar Time . . . . . . . . . . . . . . . . . 614.4 Time Zones and Calendar Time . . . . . . . . . . . . . . . . . . . . . . . . . 62Time Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Calendar Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Atomic Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Time Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Terrestrial Coordinate Systems 665.1 The Best Fitting Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Basic Ellipsoidal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Coordinates on the Ellipsoid - Geodetic Coordinate Systems . . . . . . . . . 72Geodetic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74The Reduced Latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77The Geocentric Latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Relations Between , , , . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.4 Transformation Between Cartesian and Curvilinear Geodetic Coordinates . . 806 Coordinate Transformations 846.1 Transformation Between Systems with Different Origins and Orientations . . 856.2 Transformations Between Local and Global Systems . . . . . . . . . . . . . . 876.3 The Datum Problem Today . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.4 Summary of Coordinate Systems and Transformations . . . . . . . . . . . . 937 Basics of Map Projections 937.1 Direct and inverse geodetic problem . . . . . . . . . . . . . . . . . . . . . . 947.2 Coordinate transformations in mapping . . . . . . . . . . . . . . . . . . . . 95C Alexander Braun (2006-2010) 3 ENGO421: COORDINATE SYSTEMSPrefaceThese lecture notes have been developed during the course ENGO 421 in Fall 2006. In-dividual chapters will be subsequently distributed to the students through the Blackboardcourse management system at http://blackboard.ucalgary.ca.These lecture notes are based on Dr. K.-P. Schwarzs lecture notes on Fundamentals ofGeodesy, last published in 1999. The contents are, however, modied in order to incorpo-rate new information and knowledge as well as having a rearranged order of topics to ac-commodate for the required Astronomical Observations Lab, which now takes place in theFall term and not in the Spring term anymore. Due to the different seasonal weather con-ditions, the chapter on celestial coordinate systems has been placed before the terrestrialcoordinate systems to give students the required background to carry out the astronomicalobservations early in the term. Additional material was provided by Dr. Nico Sneeuw,Dr. Michael G. Sideris, and Dr. Rossen S. Grebenitcharsky. Their contributions are highlyappreciated.For the Fall 2008 term, the order of chapters in class was modied for scheduling reasons.The lab assignment on astronomic azimuth determination requires to cover chapters 3 and4 before chapter 2 in class, the original order is, however, kept in the notes.1 IntroductionThe Earth is a dynamic planet which changes its form, composition and location constantly.In order to quantify the shape, the deformation and the distribution of masses (rock, wa-ter, snow and ice) on Earth, which is mostly described by the gravity eld of the Earth,the discipline of geodesy is employed. One of the key aspects of geodesy is to establishcoordinate systems and reference systems which can be used to consistently describe theshape, the deformation and the gravity eld. This course is designed to make studentfamiliar with the fundamentals of coordinate systems which are frequently used in geo-matics engineering and geodesy. The student will develop an understanding of terrestrialand celestial coordinate systems, the transformation between systems, and the geodeticprinciples of map projections.In order to understand these topics, the fundamentals of geodesy and particularly thegravity eld are required. These include natural coordinates (Chapter 2), gravitationaland centrifugal acceleration and force, gravity and gravity potential, geoid, plumb linesand equipotential surfaces. As the Earth is of complex shape and mass distribution, amathematical approximation of the above parameters is thought after. These include nor-mal gravity, ellipsoid and geodetic coordinates. A description of the Earth motion in spacerequires a non-terrestrial coordinate system, hence, celestial coordinate systems (Chapter3) are introduced including the celestial sphere, Horizon system, right ascension system,hour angle system, ecliptic system, astronomical positioning and time systems (Chapter4).Terrestrial coordinate systems (Chapter 5) must be discussed including astronomic andgeodetic coordinates, the transformation of Cartesian and curvilinear coordinates, merid-ians and parallels, and the geodesic. Once the terrestrial and celestial coordinate systemshave been developed, the transformation tools are discussed which allow to transformcoordinates (Chapter 6) from one system to any other system. This concerns also thedatum problem and the sensors providing information for the establishment of referencesystems. The nal chapter will elaborate on map projections (Chapter 7) and particu-C Alexander Braun (2006-2010) 4 ENGO421: COORDINATE SYSTEMSlarly the effects involved in the projection of coordinates from the 3-dimensional to the2-dimensional space. This concerns Tissots indicatrix, map distortions, and differentialgeometry on the sphere and ellipsoid.A sketchy tour through the courseThe continuation of this course will take place in ENGO 423: Geodesy, which will be taughtin the Winter term. This course will focus on physical geodesy and more details on Earthrotation and tides, dynamic coordinate systems and a more sophisticated treatment of thegravity eld.1.1 What is geodesy and why coordinate systems?Geodetic or geomatics measurements take place in a natural environment, which showsspatio-temporal variations. The natural effects inuence the measurements and need to becorrected before an analysis can take place. After this treatment of the measurements, theuse of coordinate systems is required to represent the results of such measurements andto make them available to users. To capture the essential characteristics of a discipline,denitions are often a good starting point. The list below gives an extensive though notexhaustive collection of denitions of geodesy. The rst two, venerated by age, mark in away two extreme positions. The rst one is due to Bruns (1878), one of the most creativegeodesists of the 19th century. He states that the task of geodesy is the determination ofthe potential function W(x, y, z), i.e. of the gravity potential of the Earth. The secondone is due to Helmert (1880), one of the towering gures in geodesy around the turn ofthe 19th century. He states that geodesy is the science of measuring and mapping theEarths surface. At a rst look, these two denitions seem unrelated. At a deeper levelthough, they represent two sides of the same coin. They indicate that positioning andgravity eld determination are really not separate tasks, but need to be treated together.However, for practical purposes, we often look at them individually. A denition whichincorporates both points of view was published in 1973 by the National Research Councilof Canada Geodesy is the discipline that deals with the measurement and representationof the Earth, including its gravity eld, in a three dimensional time varying space. Wewill use this denition to outline some of the fundamental questions that are at the coreof geodesy.Coordinates provide information about the location or position of objects, coordinatesrepresent the language in navigation, positioning, and mapping. Without coordinates,measurements do not provide sufcient information for others to fully comprehend andreproduce a position or location. Coordinates alone are not sufcient as well, as theyrequire a reference which is used to relate the coordinates to. These reference systemsrepresent the dictionary which everybody can use to look-up what a coordinate meanswith respect to a certain reference system. Finally, there are different languages and dif-ferent dictionaries, but translators allow us to communicate between different systems orto exchange coordinates between different systems. The translator in this case are thecoordinate transformation tools to be developed in this course. In principle, we learn thelanguage of geodesy in this course which enables us to communicate measurements acrosscountries and disciplines. The key points of establishing geodetic information or models isthus:1. information/measurementsC Alexander Braun (2006-2010) 5 ENGO421: COORDINATE SYSTEMS2. a reference system in space and timeC Alexander Braun (2006-2010) 6 ENGO421: COORDINATE SYSTEMSC Alexander Braun (2006-2010) 7 ENGO421: COORDINATE SYSTEMS3. measured corrections to correct information, reductions, systematic errors, bias4. models of corrections, if measurements are not available5. projection parameters to create maps6. interpretation of measurements/maps and applicationDEFINITIONS OF GEODESY1. The task of geodesy is the determination of the potential function W(x,y,z). Bruns,18782. Geodesy is the science of measuring and mapping the Earths surface. Helmert, 18803. Geodesy is a branch of science which investigates methods to accurately measureelements of the Earths surface and to determine from them geographic positions ofpoints on this surface and which studies the gure of the Earth from a theoreticalpoint of view and by evaluating results of measurements. Zakatov, 19574. Geodesy is both theoretical and practical. Its theoretical function is to determinethe size and shape of the Earth and, in conjunction with other Earth sciences, tostudy the structure of the Earth crust and of the immediately underlying layers. Itspractical function is to perform the measurements and computations that will givethe coordinates of selected control points on the Earths surface, i.e., to x theirpositions on the Earths surface. Heiskanen and Vening Meinesz, 19585. Au sens etymologique du mot, la g eod esie est la science qui a pour objet la mesuredes dimensions de la Terre. D eterminer, dune part, la forme et les dimensionspr ecises de la plante; r ealiser, dautre part, principalement au moyen de triangula-tions, la mensuration des territoires terrestres pour permettre dendresser des cartesexactes et fournir des donn ees g eom etrique pr ecises pour les diverses enterprises deling enieur, sont en effet les buts principaux, scientiques et practiques de lactivit edes g eod esiens. Dupuy and Dufour, 19696. Geodesy is a discipline that deals with measurement and representation of the Earth,including its gravity eld, in a three-dimensional time varying space. NRC, 1973(Vanicek and Krakiwsky, 1982)7. Geodesy is considered as a discipline which deals mainly with the mapping of theEarth and the monitoring of variations at its surface. From the very beginningthose tasks were connected with the gravity vector g (absolute value and direction).Groten, 19798. The problem of geodesy is to determine the gure and the external gravity eld ofthe Earth and of other heavenly bodies as functions of time; as well as to determinethe mean Earth ellipsoid from parameters observed on and exterior to the Earthssurface. Torge, 19809. Theoretical Geodesy is that part of geodesy which has as its task the solution ofscientic problems of geodesy - the determination of the gure of the Earth and itsexternal gravity eld, as well as their temporal variations - by means of geodeticmeasurements. Pellinen/Deumlich, 1981C Alexander Braun (2006-2010) 8 ENGO421: COORDINATE SYSTEMS10. Geodesy is the scientic discipline that deals with the measurement and representa-tion of the Earth, its gravitational eld and geodynamic phenomena (polar motion,earth tides, and crustal motion) in three-dimensional, time-varying space (Wikipedia,http://en.wikipedia.org/wiki/Geodesy (last accessed, Sep 2006).Today, the eld of geodesy becomes more and more interdisciplinary and touches disci-plines such as geomatics, remote sensing, geophysics, oceanography, hydrology, glaciol-ogy, environmental studies, atmospheric and space science. It represents the foundationsof geomatics engineering and provides most of its disciplines with the tools to handle andcommunicate measurements about the Earth surface, its interior, its dynamics, and thegravity eld.1.2 Geodetic measurements and errorsMeasurements take place in the physical world and include errors and uncertainties.Geodesy deals with measurements and many of its problems are related to their avail-ability, accuracy, resolution, and distribution. Measurements can never be made withouterrors. It is therefore important to distinguish between the measurement and the quantityto be measured. The rst is often called observation, the second observable. Thus, anobservation or measurement is the observable plus errors. Take as an example the mea-surement of a distance between two points A and B. The observable is in this case thedistance itself while its measurement contains errors of systematic and stochastic nature.For instance, if a tape is used, systematic errors arise from the scale factor and tape lengthvariations due to temperature or tension. These systematic errors can be modeled or cali-brated. Stochastic errors are due to the inaccuracy of reading the tape and to insufcientcontrol over the environmental conditions. Although these errors cannot be determinedexplicitly, they can be incorporated in the estimation procedure. The distinction betweensystematic and stochastic errors is a convenient way to handle the error problem but byno means a law of nature. Stochastic errors often become systematic when an increasein measurement accuracy makes it possible to identify an underlying non-stochastic phe-nomenon.The standard geodetic procedure to process measurements proceeds therefore in two steps.First, a measurement model is developed, comprising observable plus systematic effects.Second, a stochastic model is formed which characterizes the average behavior of the re-maining errors and can be used in an estimation process. Representation of the Earth andits gravity eld is the next key word in the denition. Representation is the second stage inthe modeling process. Instead of modeling an individual observation to obtain the observ-able, a set of observables is nowused to determine an adequate representation of the Earthand its gravity eld. This modeling is therefore task oriented and the same observables canbe used for different tasks. Traditionally, the two major tasks of geodesy have been denedas positioning and gravity eld determination. In the same vein, observables have beensubdivided into geometrical observables, like distances and directions, and physical ob-servables, such as gravity and its gradients. This distinction is not made anymore becausemost observables can be used either for positioning or gravity eld determination. If allobservables are used to determine positions and gravity eld components simultaneously,the terms integrated geodesy or operational geodesy are used.Starting from the measurement, a measurement model is formulated that distinguishesbetween the observable, the systematic errors or biases b, and the noise n.l = L +, (1)C Alexander Braun (2006-2010) 9 ENGO421: COORDINATE SYSTEMSwith l = measurement, L = observable, and = errors. = n +b +ge (2)with n = random errors, b = systematic errors or bias, and ge = gross errors/blunders.Neglecting the gross errors it can be seen that these three quantities (l, L, ) cannot be sep-arated on the basis of a single measurement. The observable can be estimated if redundantmeasurements are available. The bias can either be obtained by a calibration procedureor again by estimation. In some cases, it can also be eliminated by differencing betweenobservations. The random error or noise is described in a statistical manner, usually bydening its mean and variance from previous experience. It enters into the estimationprocess via a covariance matrix. Once the measurement model has been set up, the ob-servable L is parameterized in terms of the coordinates x and the gravity potential W (willbe introduced in chapter 2). The resulting equation is nonlinear. The linearization of thisequation is done by dening approximate coordinates x and a reference potential U.L = F(x, W) (3)L = f(x x0, W U) (4)By forming the difference between the actual parameters (x, W) and the reference model(x, U), by expanding into a Taylor series about (x, U) and keeping only the rst term, thelinearized model is obtained. It can take three different forms.L = Ax +BW +b (5)L b = Ax +BW (6)L b BW = Ax, (7)withA = f(x, W)x (8)B = f(x, W)W , (9)both at x0, W0. In the rst case, the coordinate corrections x, the gravity eld correctionparameter W, and the biases b are all estimated. This is the case of integrated geodesy.In the second case, the bias term is either obtained by calibration and subtracted from theobservable, or is eliminated by differencing. Thus, only x and W have to be estimated.In the third case, sufciently accurate knowledge of the gravity eld is available and thecorrections (reductions) to the observables can be made. In this case, only the coordinatecorrections x have to be estimated. Depending on the model chosen, either x only, orx and W, or x, W, and b are estimated. The estimated x and W are used for therepresentation of the Earths surface and/or its gravity eld. Measurement and measure-ment space also change in time. This is true for man-made changes which usually occuron a time scale of a few years, as for instance subsidence in mining areas, as well as forchanges generated by geodynamic or large-scale climatic processes which occur on a scaleof ten thousand years and up, as for instance post glacial rebound (also know as GIA,glacial isostatic adjustment), tectonics such as plate motion or the decrease of the Earthsrotation rate. The latter group of problems is at the centre of research as this needs to beC Alexander Braun (2006-2010) 10 ENGO421: COORDINATE SYSTEMSincorporated if a stable coordinate or reference systems is required. It cannot be dealt within detail in this introductory course but some attention will be given to the effect of theseprocesses throughout the course in examples and applications. It is obviously a concernof geodesists that the coordinate system to which they refer the measurements to is stablein time or can be reduced mathematically to a stable system at a certain epoch. If a mea-surement is repeated after a certain period of time, it should show the same result exceptfor measurement errors. Changes of a local or regional scale can usually be detected byreferring all measurements to a global reference system. Changes of a global nature canonly be detected by using a reference outside the Earth, which is not part of the dynamics,such as a set of xed stars, or a selected group of quasars or a stable system of satellites.The efforts to dene and realize an inertial frame of reference are all part of this problem.Which problems have to be solved when using observables for the representation of theEarth and its gravity eld? A simple example will be used to discuss some of the majorpoints. Let us assume that a small part of the Earths surface has to be represented in formof a topographic map. Which observables are important in this case? Obviously heights,because they are represented on a topographic map. Let us assume for a moment that wehave an instrument which allows us to measure heights wherever we want. How can it beused to produce a topographic map? Obviously we have to know where the heights havebeen measured in order to plot them on the map. This means that horizontal coordinatesare needed. They will connect the discrete height observables. It also means, that we havefound a convenient three-dimensional coordinate system for the task at hand. It consistsin this case of coordinates on a reference surface to which the heights are orthogonal. Onthe reference surface, a system of horizontal coordinates can be dened in several ways. Inprinciple, the choice of an adequate coordinate system to connect discrete geodetic observ-ables is always needed to represent the Earth and its gravity eld. Different applicationsrequire different coordinate systems and the term adequate is therefore task related. Thedenition of appropriate local, global, and inertial coordinate systems and of the transfor-mations between them is therefore at the centre of this course. In the discussion so far,we have tacitly assumed that we know what a height is. Since it is a very intuitive con-cept (height is up), we seldom stop to think how it is dened. Obviously height is relatedto some reference surface and is measured as the orthogonal distance to this referencesurface. Height above a plane is an obvious example. Since a representation of heightobservables in a mapping plane is needed, a simple approach would be to dene heightwith respect to some plane tangential to the Earth in the middle of the map sheet. Thiswill obviously not work too well because the Earth has a curved surface and a large lakefor instance would show height differences in this representation. Since a lake is a levelsurface, such a representation would contradict intuition. In addition, this representationwould produce jumps in height when going from one map sheet to the next, an effectwhich is very undesirable from a practical point of view. A reference surface is thereforeneeded which represents the global shape of the Earth sufciently well. One referencesurface which is often used is that of a globally best tting ellipsoid of revolution. Heightis dened as the orthogonal distance with respect to this surface. Since it is a smoothand reasonably simple surface, its mathematical description is simple, computations areeasy, and mapping is consistent. This is one reason why the determination of a best t-ting global ellipsoid has been a central task of geodesy for many years. Today, it can beconsidered as solved with a satisfactory degree of accuracy through the use of dedicatedsatellite gravity missions. For the topographic mapping problem, the choice of an ellip-soid as reference surface for height is, however, not ideal. The ellipsoid is a mathematicalabstraction which has no physical equivalent in nature. This means that it is not possibleC Alexander Braun (2006-2010) 11 ENGO421: COORDINATE SYSTEMSto directly physically sense the ellipsoid. It is not necessarily true that water ows from aheight of 2 m above the ellipsoid to a height of 1.8 m above the ellipsoid, as the ellipsoiddoes only approximate the physical shape of the Earth. The reference surface for mostheight systems is therefore dened in a physical sense. The term height above sea levelindicates that. It is a level surface which in a rst approximation represents the idealizedsurface of the oceans and is called geoid. In land areas, this surface can only be deter-mined from a knowledge of the gravity eld of the Earth. Thus, the apparently simple taskof producing a topographic map, can in principle not be solved without a knowledge of theEarths gravity eld. This will be a recurring theme of the following chapters. Positioningand gravity eld determination are intertwined because gravity determines the structureof the space in which the measurements are taken and affects the instruments, the objectsto be measured and the measurement process.C Alexander Braun (2006-2010) 12 ENGO421: COORDINATE SYSTEMS2 System of Natural CoordinatesCoordinate systems are fundamental in science and engineering to refer observations toa reference which is unique and can be understood by others. Coordinates do not makesense without a coordinate system. For instance, an engineer gives his position relativeto the class room door, the only way to understand this position is the knowledge of thereference, here the class room door. The objective of this chapter is to establish a coor-dinate system which can be used to refer geodetic positions to and which is physicallymeaningful. In other words, the system should follow physical parameters which we knowand understand intuitively, e.g. the directions of up and down follow the direction of thegravity vector or plumbline. In geodesy, coordinate systems are a convenient way to ex-press general physical laws and to relate them to geodetic measurements. In principle,the choice of such coordinate systems is arbitrary, e.g. the reference can be the class roomdoor in the back or in the front. The engineer or scientist will, however, make a choicewhich allows to communicate the coordinates to others without complication. Also, it isadvisable to select a system with specic properties in order to simplify the representa-tion of the measurements or the computation of results. One such system is the systemof natural coordinates. Its axes are dened by directions which are physically meaningful,namely, the directions of the gravity vector and the spin axis of the Earth. The gravityvector denes the up-down direction, which is the direction orthogonal to a level surface,such as a large body of undisturbed water. The spin axis of the Earth denes the Northpole where it pierces the Earth surface and thus the north direction. Since many geodeticinstruments such as levels, theodolites, inertial survey systems are aligned to this frameduring the set up, it can be considered as a typical coordinate frame for geodetic measure-ments. It is therefore often called the system of natural coordinates outlining that naturalor physical parameters dene its orientation. It can also be considered as the coordinatesystem describing the geometry of the gravity eld as the gravity vector denes the verti-cal axis. In order to understand this concept, Newtons laws and specically Newtons lawof gravitation and its mathematical representation must be discussed. Later, the secondnatural coordinate reference, the direction of the Earth spin axis will be introduced.2.1 Newtons Law of Gravitation - Gravitational Acceleration and Po-tentialThe principle of attraction of physical bodies has been mathematically formulated by New-ton (1687) in his law of universal gravitation.1. Every body continues in its state of rest of of uniform motion in a straight line unlessit is compelled to change that state by an external impressed force.2. The rate of change of momentum of the body is proportional to the force impressedand is in the same direction in which the force acts.3. To every action there is an equal and opposite reaction.Sir Isaac Newton (1642-1727) was the rst scientist who developed a mathematical de-scription of these laws. Before him, Johannes Kepler (1571-1630) established similar lawsfor the motion of the planets and the Moon from empirical relations derived from obser-vations. As we will later see, Keplers laws can be derived from Newtons laws with certainC Alexander Braun (2006-2010) 13 ENGO421: COORDINATE SYSTEMSsimplications. Newton did not include relativistic effects in his laws as he was not awareof their existence, later in the 20th century, relativistic effects have been included in Ein-steins theory of relativity. For instance, the Newtonian momentum pN of a body is theproduct of its velocity v and mass m0: pN = m0v (10)Once the velocity increases towards the speed of light c, the rest mass m0 becomes arelativistic mass mRel.mRel = m0

1 v2c2= m0 (11)Newtons momentum pN is no longer valid and becomes the relativistic momentum pRel. pRel = m0

1 v2c2v (12)It is further worth to notice that for small velocities v, the relativistic momentum becomesNewtons momentum. The equivalence of mass and velocity is a topic of theoretical physicsand will not be discussed here in more detail. However, an example using the equationsabove results in the fact that you have to reach 14% of the speed of light, or about 42 106m/s before the mass changes by 1%. Newtons second law states that the momentumchange wrt time is proportional to the force impressed:

F =

dpdt (13)In classical mechanics, the mass is considered constant, and if the momentum doesntchange or no force is impressed, the equation is equal to zero.

F =

dpdt = m0a = 0, (14)with the accelerationa. In conclusion, classical mechanics and Newtons laws are sufcientfor most geodetic applications and relativistic effects are mostly ignored with the exceptionof satellites orbiting the Earth, where these effects are accounted for already, e.g. in GPS.The previous equation leads to Newtons rst law which states that two point masses mand m

separated by a distance l, attract each other with a force F which is proportional tothe product of the two masses and inversely proportional to the square of their distance:

F = Gmm

l2 (15)The force F known as the gravitational force or gravitational attraction is directed alongthe line connecting the point masses m and m

. The constant of proportionality G, calledNewtons gravitational constant, has the valueG = 6.67108cm3g1s2= 6.671020km3kg1s2. (16)G is one of the least accurate physical constants known to a relative precision of 104while most other physical constants are determined better than 107. Consequently, G isC Alexander Braun (2006-2010) 14 ENGO421: COORDINATE SYSTEMSnot always considered a constant, however, experiments designed to determine G couldnot prove that G is not a constant. The force F has therefore the unit cmg s2= 1 dyne =105Newton. The attraction of the masses m and m

is completely symmetrical as statedby Newtons 3rd law. It is convenient, however, to consider one of them (e.g. m

) as theattracted mass and the other (m) as the attracting mass. Moreover, by dividing Newtonslaw of gravitation by m

, the attracted mass is chosen as the unity of mass. This changesthe unit to cm s2= dyne g1= 1 Gal. The unit Gal is named after the Italian physicistGalileo Galilei (1564-1642) and is frequently used in geodesy and geophysics, particu-larly to describe variations of the gravity vector at the Earth surface for exploration andsurveying. Hence, it has the unit of an acceleration, e.g. m s2.1 Gal = 102m/s2, or 1 mGal = 105m/s2(17)Assuming a Cartesian coordinate system x, y, z, the point mass m at P(x1, y1, z1) attractsthe unit mass m

= 1 at point P

(x2, y2, z2) with the force F:F = Gml2r2r1|r2r1| (18)Figure 1: Gravitational attraction between mass points.Let us dene the following quantities (Figure 1): the scalar distance l between the masspoints l = |r2r1| = |r|, the distance vector r = r2r1, and the unit vector e12 = r|r|.The transition to Cartesian coordinates is done by expressing r byr = r2r1 =

x2x1y2y1z2z1

. (19)The gravitational acceleration

b becomes:

b = Gml2e12 = Gml2rl = Gml3 (r2r1), (20)C Alexander Braun (2006-2010) 15 ENGO421: COORDINATE SYSTEMSor in Cartesian coordinates

b = G m((x2x1)2+ (x2x1)2+ (x2x1)2)32

x2x1y2y1z2z1

(21)Figure 2: Mass Point P

(x2, y2, z2) attracted by the Earth.While this relation between gravitational acceleration/attraction and mass was derivedfor a mass point, it is straightforward to consider several mass points which make up anextended body of mass by summing up the individual components (Figure 2):

b = Gni=1mil3iri (22)Let the mass element become innitesimally small, so that the ratio between the masselement and its volume at all points Q can be expressed by the density and the sumbecomes an integral. = limV 0mV (23)The resulting expression for the gravitational acceleration of an extended mass body be-comes:

b = G

V ol(Q)l3ir dV ol (24)In conclusion, if the entire mass would be concentrated in the centre of gravity,

b would beidentical, but the body must consist of concentric spheres of constant density or the entiresphere must be of homogeneous density.Now, we have developed a set of equations which describe the gravitational accelerationwith a triple of Cartesian coordinates x, y, z. Hence, three scalars are required to derive

b. The next step is to nd a physical parameter which describes the gravitational eldwith just one scalar. This is the search for the gravitational potential. There are twoC Alexander Braun (2006-2010) 16 ENGO421: COORDINATE SYSTEMSways of introducing the potential, a physical way which derives the potential from theacceleration vector, or a mathematical way, which assumes a certain function and provesthat this function describes the potential. Here, we followthe classical geodetic way, whichassumes a function and tests if this function describes the gravitational potential in relationto

b. The gravitational vector eld

b is a conservative eld, because to transport mass frompoint A to point B, the same amount of work needs to be done, no matter what path ischosen to move the mass through the eld. This can also be expressed by stating that

b iscurl-free.curl

b = rot

b =

b = 0 (25)It is also known that the curl of every gradient eld equals zero, thusrot grad V = V = 0. (26)Herein, grad V could be

b and V is a scalar eld. Then

b can be a gradient eld of a scalareld, here the gravitational potential V . V requires only one number while

b requires threenumbers to describe it entirely. What function V would fulll the relation with b? Let usassume that V takes the following form:V = Gml (27)Computing the gradient vector of the scalar function V results in:gradV =

VxVyVz

=

Vx2Vy2Vz2

(28)Deriving the partial derivative of V wrt to x2.Vx = Vx2= x2Gml = Gm x21l = Gml2lx2= Gml22(x2x1))2l = Gmx2x1l3 (29)In a similar way, differentiating the scalar function V with respect to the variables y2 andz2 gives Vy and Vz.grad V = Gml3

x2x1y2y1z2z1

(30)Comparing this last expression with the equation for

b results in

b = grad V. (31)The scalar function V (x, y, z) is called the gravitational potential. Its physical interpreta-tion is given by the work needed to bring a unit mass frominnity to the point P

(x2, y2, z2).This equation denes b as a conservative vector eld. In physical terms, a vector eld iscalled conservative if the total energy of a body moving in this eld is conserved, i.e. con-stant. In mathematical terms it means that there exists a scalar function V such that foreach point in this eld

b is equal to the gradient of the scalar function V . Typical propertiesof a conservative eld are:C Alexander Braun (2006-2010) 17 ENGO421: COORDINATE SYSTEMS1. The integral P2P1

b

dr is path independent.2.

b

dr = 0, i.e. integration over a closed path is zero.3. dV =

b

dr, i.e. dV is an exact differential.Property 1 states that in a conservative vector eld the work done in moving the bodyfrom point P1 to P2 is independent of the path taken. Property 2 is a simple consequenceof Property 1.The relation between b and V and the fact that the gravitational eld is a conservativevector eld is of basic importance. It means that the vector eld described by three scalarscan be replaced by a scalar eld consisting of only one scalar. The vector eld can then beobtained by differentiating the scalar eld with respect to the three coordinate directions.So far, a simple mathematical model consisting of two mass points attracting each otherhas been considered. Such a model is frequently applied in celestial mechanics as a rstapproximation for the solution of the two-body problem. It is possible to use this modelbecause the distances between the celestial bodies are in most cases so large that thecelestial bodies themselves can be considered as mass points. For measurements on thesurface of the Earth, this simple model is usually not applicable because the attractingmasses cannot be considered as mass points. In some cases, the potential can be modelledby a system of point masses as it was assumed earlier with b. It is occasionally used forlocal gravity eld approximation. In general, however, the dimensions of the Earth have tobe taken into account as well as the density distribution in its interior. Thus, the attractionof a mass point P by the Earth will be described as the attraction by a volume with acontinuous mass distribution. It is given that the sum of the individual contributions ofthe mass elements result in V :V = Gni=1mili(32)For innitesimal mass elements, the sumturns into an integral over the volume of the massand the gravitational potential V can be expressed by:V = G

V ol(x, y, z)l dx dy dz (33)Differentiating this equation results in the equation which was previously derived for thegravitational acceleration of an extended mass object.

b = G

V ol(x, y, z)l3 r dx dy dz (34)If the density distribution of the Earth (x, y, z) is known, then both the gravitational poten-tial and the gravitational attraction can be computed. In general, the density distributionis not known with sufcient accuracy to use this approach. These equations are, how-ever, fundamental for the denition of the relation between gravitation and mass densitydistribution.C Alexander Braun (2006-2010) 18 ENGO421: COORDINATE SYSTEMS2.2 Gravity Potential and Centrifugal PotentialUp to now, only the gravitational attraction

b and the gravitational potential V have beendiscussed. The gravitational attraction is, however, not the only force acting on a body atrest on the Earths surface. Due to the fact that the Earth is rotating about its axis of inertia,an additional force, called centrifugal force, has to be considered. Its direction is alwaysorthogonal/normal/perpendicular to the rotation axis of the Earth. It is an apparent orinertial force because it is completely dependent on the rotation of the Earth with respect toan inertial frame of reference; as soon as the attracted mass stops rotating, the centrifugalforce vanishes. Assuming again that the attracted mass is equal to unity and using it as adivisor, the total acceleration acting on a body at rest on the Earths surface is the resultantof gravitation and centrifugal acceleration and is called gravity, i.e.gravity = gravitational + centrifugal accelerationThe following will establish the equations which are required to derive the centrifugalacceleration and later also the centrifugal potential. To explain centrifugal acceleration,consider a simple example (Figure 3). Let the point P rotate about a xed origin 0 at theend of a bar which is innitely thin and without mass. Denote the distance of the rotatingpoint P from the rotation centre by p, the linear velocity by vl and the angular velocity by. From the small angle approximation, we know that the arc segment s can be related tothe radius vector p and the rotation angle :s = p (35)Two times differentiation wrt time results in an expression for the tangential acceleration:

dsdt = pddt = vl = p (36)

dvldt = pddt =at (37)It shows that for = 0 or = constant, the tangential acceleration vanishes. Moreimportant is the normal acceleration an. Again, we employ the sine law for small anglesand get two equations for the arc increment s.sp = vvl(38)st = vl (39)Substitution of s in one of the two equations results in an expression for an.vvl= vltp vt = v2lp = 2p =an (40)From the above, the centripetal (inwards directed) force Fcp can be derived.

Fcp = man = m2l (41)In case of the Earth, Figure 3 represents a plane orthogonal to the rotation axis of theEarth, i.e. a section through the parallel of latitude = constant. The point P is a pointC Alexander Braun (2006-2010) 19 ENGO421: COORDINATE SYSTEMSFigure 3: Circular motion and centrifugal acceleration.on the Earths surface, a distance p away from the nearest point on the rotation axis, andE is the angular velocity of the Earth, considered to be constant in this example. Notethat the rotation period of a planet and the length of day are different quantities as theyhave different references, i.e. xed stars or the Sun, respectively. Figure 4 shows thesituation, where a Cartesian coordinate system has been chosen in such a way that its z-axis coincides with the spin axis. The centrifugal acceleration fc at point P(x, y, z) in thiscase is equal to the normal acceleration an and can thus be expressed as

fc =an = 2 p (42)By writing the vector p as p =

xy0

(43)one obtains

fc =

2Ex2Ey0

, (44)withp = | p| =

x2+y2. (45)Since the centrifugal acceleration is proportional to the distance p normal to the rotationaxis, it becomes zero at the rotation poles,

fPolec = 0, (46)and will reach its maximum at the equator with an equatorial radius RE,

fEqc = RE2E. (47)Assuming the following function for the centrifugal potential Vc,Vc = 122Ep2= 122E(x2+y2), (48)C Alexander Braun (2006-2010) 20 ENGO421: COORDINATE SYSTEMSFigure 4: The Earth centrifugal force.it can be shown that the gradient of this expression equals the centrifugal acceleration fc.gradVc =

VcxVcyVcz

=

2Ex2Ey0

= fc (49)As both the gravitational and centrifugal potentials are scalar and only a function of space,both terms can be added and the sum represents the gravity potential W.W = V +Vc (50)Considering the previous equations for the individual potentials, W becomes,W = G

V ol(x, y, z)l dV ol + 122E(x2+y2) (51)The gradient of the gravity potential W is dened as the gravity vector g with the compo-nents

b and fc,g =

b + fc = grad W = G

V olrl3 dV ol +2E p (52)Note that gravitation decreases with the squared distance from the attracting masses whilethe centrifugal acceleration increases with distance p from the rotation axis. The gravita-tional vector

b points inward while the centrifugal vector points outward. Figure 5 depictsthis situation in graphical form. The magnitude of g is called gravity and is measured inC Alexander Braun (2006-2010) 21 ENGO421: COORDINATE SYSTEMSGal or more commonly in mGal. Its value at the poles is about 983 Gal and it decreasessystematically to about 978 Gal at the equator. This is due to differences of centrifugaland gravitational acceleration at the equator and at the poles. The centrifugal accelerationis about 3.4 Gal at the equator pointing outward and zero at the poles. The magnitudeof gravity is therefore reduced at the equator. The attening of the Earth at the polesdecreases the distance to the centre of mass of the Earth by about 22 km, which is about3%o of the Earths radius. This, in turn, would increase the magnitude of the gravitationalattraction

b at the poles.Figure 5: Interaction of gravitational and centrifugal accelerations

b and fc.The combined effect of the change in centrifugal acceleration and the change in gravita-tional acceleration due to the attening of the Earth results in the gravity difference ofabout 5 Gal between the equator and the poles. 35% are due to attening, 65% are dueto Earth rotation. If this change was completely systematic and symmetric, a simple globalmodel for the change of gravity could be derived. However, due to the inhomogeneousdensity distribution and related mass irregularities in the interior of the Earth, the actualglobal gravity model is much more complicated. The simplied model is often used as arst approximation and is then called the normal gravity model.2.3 Level Surfaces and Plumb LinesSo far, only the magnitude of the gravity vector g has been considered. In this section, thedirection of the gravity vector and the characteristics of the surface orthogonal to it will bediscussed. Remember, the objective of this chapter is to introduce a physically meaningfulreference system. Let us start by settingW(x, y, z) = WP = constant. (53)The surfaces dened in this way are surfaces of constant potential, called equipotentialsurfaces or in the case of the gravity potential, level surfaces. The levelling bubble of atheodolite orients itself to lie in this level surface. Equipotential surfaces coincide with theC Alexander Braun (2006-2010) 22 ENGO421: COORDINATE SYSTEMSsurface of a homogeneous uid in equilibrium with no external forces except the gravityeld, which explains the term level surface. In a rst approximation, the idealized surfacesof lakes can be considered as such level surfaces. They approximate W = constant for aspecic value WP. Differentiating W = W(x, y, z) with respect to (x, y, z) givesdW = Wx dx + Wy dy + Wz dz = gradW dr, (54)where drT= (dx, dy, dz) is the displacement vector. Let dr lie in the equipotential surfaceW(x, y, z) = WP, then dW = 0 andgradW dr = 0 = g dr (55)on WP. If the dot product of two non-zero vectors (both g and dr are non-zero) is equalto zero, then the vectors are orthogonal to each other, so that the gravity vector must beorthogonal to dr. In addition, this means that the gravity vector is normal to the equipo-tential surface passing through the point P. It is therefore simple to nd the directionof the gravity vector on the surface of the Earth. It is orthogonal to the surface estab-lished by a level bubble, or, in other words, the bubble represents the level surface in thatspecic point. This fundamental principle is used extensively in the levelling of geodeticinstruments.Figure 6: Level surfaces, plumblines, geoid, orthometric height H.The lines which intersect all level surfaces of the Earth orthogonally are called plumblines. They are curved lines and the gravity vector is obviously tangent to the plumb lineat the points of intersection. A good approximation of such a tangent, and therefore ofthe direction of gravity, is the string holding a plumb bob. Each specic WP = constantdenes a different equipotential surface (Figure 6). The particular equipotential surfaceC Alexander Braun (2006-2010) 23 ENGO421: COORDINATE SYSTEMSwhich coincides with the idealized surface of the oceans is called the geoid. It is assumedthat there are no forces acting on the ocean such as ocean dynamics, tides, wind, waves etc.Then, the approximation of the mean sea level can be used to describe the geoid, however,this is only a rst order approximation and when considering the effects mentioned above,the mean sea level is not equal to the geoid surface. The difference between them is alsoreferred to as sea surface topography. The name geoid was proposed by Listing to describethe gure of the Earth. The geoid is used as a reference surface for the orthometric heightsystem which will be discussed later in this chapter. It denes the height H of a point atthe physical surface of the Earth by its distance from the geoid measured along the plumbline. The following properties of the Earths equipotential surfaces are of importance ingeodesy.Equipotential surfaces are continuous surfaces, never cross each other, are not necessarily parallel to one another, change their curvature smoothly from point to point.2.4 Natural CoordinatesThe objective of this chapter is to develop a system of natural coordinates using axesdened by directions which are physically meaningful in the terrestrial space, e.g. thedirections of the gravity vector and the spin axis of the Earth. How can these spatialdirections be related to positions in an Earth-xed coordinate system? It is obvious fromthe preceding sections that the gravity potential and its gradients are important in thiscontext. They dene the direction of the gravity vector by gradients of W in an Earth-xedCartesian coordinate system x, y, z. The relationship between grad W and x, y, z will bebriey discussed in this section. Natural coordinates will be used to dene the three axesof orthogonal coordinate systems; together with the origin, this denes a reference frame.The simplest representation of the gravity vector is obtained in the Local Astronomic Frame(LA) which is dened by: Local Astronomic Frame (LA) Origin: At observers point P Primary axis (z): Orthogonal to level surface at P(WP = constant) Secondary axis (x): Tangent to astronomic meridian pointing north Tertiary axis (y): Orthogonal to complete a left-handed systemThe LA is the frame which is used to take measurements in the eld. The word localindicates that the frame is used in the local measurement environment (Figure 7). TheC Alexander Braun (2006-2010) 24 ENGO421: COORDINATE SYSTEMSword astronomic indicates that the natural coordinates are used which have a physicalmeaning. In this system, the gravity vector has the coordinatesgLA = grad WLA =

00g

(56)To relate this representation to an Earth-xed Cartesian system, the Conventional Terres-trial Frame (CT) is dened: Conventional Terrestrial Frame (CT) Origin: Earth centre of mass Primary axis (z): Conventional (or mean) spin axis of the Earth Secondary axis (x): Intersection of the conventional (or mean) equator plane andthe mean Greenwich meridian plane Tertiary axis (y): Orthogonal to complete a right-handed systemThis coordinate frame is of fundamental importance in geodesy. It will be more rigorouslydened in ENGO423: Geodesy, where the terms mean or conventional get a proper deni-tion based on Earth rotation, precession and nutation. The word conventional indicatesthat there is a denition involved, a convention or average position of the physical quanti-ties. The word terrestrial describes the origin as the Earth centre of mass as opposed tolocal, where the local point is the origin. The plane orthogonal to the conventional/meanspin axis is called conventional equator plane. The direction of g in the CT-frame is givenby two angles, the astronomic latitude and the astronomic longitude . The denitionof is tied to the denition of the astronomic meridian plane, i.e the mean Greenwichmeridian plane.The astronomic meridian plane of a point P is the plane containing the gravity vector atP and the parallel to the conventional rotation axis of the Earth through P. Thus, it isorthogonal to the conventional equator plane. The astronomic longitude is the anglebetween the astronomic meridian planes of two points. The convention is that the angle is counted counterclockwise from the mean astronomic meridian plane of Greenwich.The astronomic latitude of a point P is the smallest angle between the conventionalequator plane and the vector normal to the level surface in P measured in the meridianplane of P. The normal vector is opposite in direction to the gravity vector in P. The angle is conventionally counted from the mean equator plane positive towards the north poleand negative towards the south pole. Since each point on the surface of the geoid can beexpressed by a pair of coordinates ,, these coordinates can be mapped onto the surfaceof a unit sphere. By connecting a specic coordinate pair to the centre of the unit sphere,we obtain a unique spatial unit vector n for each pair. This unit vector gives the directionof the vector normal to the geoid at P expressed as a function of and . This directionis shown in Figure 8.The vector p in Figure 8 is the projection of the normal vector n onto the conventionalequator plane. The angle between the vectors n and p is the astronomic latitude and theC Alexander Braun (2006-2010) 25 ENGO421: COORDINATE SYSTEMSFigure 7: Local astronomic frame and astronomic coordinates.angle between p and the mean Greenwich meridian plane is the astronomic longitude .Since |n| = 1 and | p| = cos, the vector p has the coordinates p =

cos coscos sin0

(57)Based on this, the vector n can be expressed because it only differs from p in the z-component.nCT=

cos coscos sinsin sin

(58)Using this denition of the normal vector n, the gravity vector g can be expressed asgCT= g n (59)Finally, the gravity potential can be substituted and we obtain,gCT= gradW =

WxWyWz

=

WxWyWz

=

cos coscos sinsin sin

. (60)C Alexander Braun (2006-2010) 26 ENGO421: COORDINATE SYSTEMSFigure 8: Direction of normal vector in terms of and .The last equation shows an interesting connection between physics and geometry. Startingfrom physics (gravity potential), the astronomic coordinates , can be derived and thegeometry (e.g. the curvature of the Earth) can be determined. It denes the gradients ofthe gravity potential in terms of astronomic coordinates. The reverse formulas expressing and as gradients of the gravity potential, can also be obtained.W2x +W2y = g2cos2 (cos2 +sin2) = g2cos2 (61)Wz

W2x +W2y= g sing cos = tan (62)WyWx= tan (63)Solving for the astronomical latitude and longitude, a function of the potential gradientscan be derived. = arctan Wz

W2x +W2y(64) = arctanWyWx(65)This shows that, if the gravity potential W(x, y, z) is given, the coordinates and canalways be determined. To describe the position of a point in three-dimensional space,three coordinates are needed. They can be Cartesian x, y, z, curvilinear , , H, or anyother coordinate triple. It has been shown that and give the position of a point onan equipotential surface. It makes sense, therefore, to dene the third coordinate as beingorthogonal to this surface. It has been mentioned before that this coordinate is called theorthometric height H if the reference surface used is the geoid (Figure 9). To dene HC Alexander Braun (2006-2010) 27 ENGO421: COORDINATE SYSTEMSmathematically, let us use the equation again, which relates a displacement vector with gand dW.dW = g dr (66)Earlier, the displacement vector dr lay in the equipotential surface, and we have shownthat g is normal on W. This time, let dr point upward along the plumb line, i.e. |

dr| = dH.Figure 9: Denition of the orthometric height H.As the angle between g and dr then becomes 180 deg, this yieldsdW = g dr = |g| |

dr|cos(g, dr) = g dH cos(180) = g dH (67)This is the fundamental equation for the denition of heights. It relates a potential dif-ference to a height difference. The proportionality factor is the magnitude of the gravityvector. It can also be rewritten in the formdH = dWg (68)The equation denes the height in terms of potential differences and gravity. Since gravitycannot be easily measured inside the Earth, different approximations for g inside the Earthmust be made. This leads to different height systems and denitions. A height wouldonly be accurately determined, if we would know g along the entire plumbline betweenthe point P at the Earth surface and the geoid. Different height systems solve this prob-lem in different ways, but always use an approximation of g, sometimes obtained usingmeasurements at the surface or sometimes by assuming normal gravity .Geopotential Numbers: A height difference can be derived by knowing the potentialdifference dW between two points and g. Let us dene two potential values at the surfaceWP and on the geoid W0.dW = W0WP = C (69)Herein, C is called geopotential number. The height can now be dened as:Height = C g (70)In this case, g is an approximate magnitude of g derived from gP and g0, but dependingon the choice of the approximation, different heights can be derived, e.g. orthometricC Alexander Braun (2006-2010) 28 ENGO421: COORDINATE SYSTEMSheights, dynamic heights, normal heights or Helmert heights can be dened. Details aboutthe different height systems will be discussed in ENGO423-Geodesy. Geopotential num-bers are expressed in terms of geopotential units (g.p.u.), i.e. 1 g.p.u. = 1 kGal meter. Asa rule of thumb, C in g.p.u is always close to the height above the geoid in meters, as g isapproximately 0.98 kGal. The system of natural coordinates can therefore be expressed byeither one of the two coordinate triples , , H or , , C. The rst has a simple geomet-rical explanation but introduces some assumptions in the denition of H. The second ismore precise in terms of dening the coordinates but lacks the intuitive geometrical mean-ing. In both cases, the coordinates can be uniquely expressed by potential differences andgradients of W, thus explaining Bruns (1878) concise statement: The task of geodesyis the determination of the potential function W(x,y,z). It is important to note that thedetermination of heights always involves the gravity eld of the Earth as the gravity eldaffects the measurement device, e.g. a theodolite. Once the observer changes location, thevalue of g may change and consequently the height changes, and the levelling bubble willbe in a different orientation. This effect leads to the misclosure of leveling loops, if grav-ity is not accounted for. A closed loop of geometric leveling leads to the misclosure anddepends on the path. The misclosure disappears once geopotential differences are takeninto account. This statement indicates that gravimetry is mandatory for high precision sur-veying, however, the knowledge of the potential gradient in the surveyed area allows foran estimation of the effect on the height determination. In area of small geoid gradients,e.g. the Prairies, gravimetry is not required for most surveying applications. However, inmountainous regions, where gravity changes signicantly, it becomes important and mustbe considered in order to achieve high accuracy.As the geoid is not known perfectly, geodesists have developed alternative surfaces tobe used as a height reference. One example is the ellipsoid, or rotationally symmetricellipsoid. The task is to approximate the gravity eld of the Earth or the equipotentialsurface called geoid with a mathematical surface which can be derived exactly at everylocation. In a rst step, this surface could be a sphere, but the height differences wouldbe up to 11 km at the poles and the equator, as the Earth attening causes a difference inradius of about 22 km between the poles and the equator. The next and signicantly betterapproximation is an ellipsoid with semi-major axis a and semi-minor axis b. Once this tis made, the differences between the ellipsoid and the geoid are less than 100 meters. Theellipsoid will be discussed in more detail in chapter 5. As terrestrial coordinate frameshave been dened, the denition of celestial coordinate frames is the next step whichallows also to observe the motion of the Earth from a non-terrestrial point of view.C Alexander Braun (2006-2010) 29 ENGO421: COORDINATE SYSTEMS3 Celestial Coordinate SystemsIn the previous chapters, the coordinate systems were xed to the Earth, which implies thatthe coordinate systems move as the Earth moves with respect to other planetary objectsor the Sun. In order to identify coordinate systems, which allow us to observe the Earthmotion, celestial coordinate systems will be discussed in this chapter. In other words, wesearch for a connection between local, terrestrial, and inertial systems which allow us torelate coordinates across different coordinate systems. In addition, the denition of thenatural system of coordinates (, , H) is tied to the knowledge of the potential functionW and its gradients. This function is not known with sufcient accuracy on Earth to actu-ally determine positions on the surface by using formulas such as (64, 65). It is possible,however, to establish the direction of the local gravity vector with good accuracy at eachpoint on the Earths surface. Is it possible to use this direction in space to determine and? The answer is yes, if is direction can be related by measurements to the conventionalequator and the mean Greenwich meridian which are the reference for and , respec-tively. Previous to the GPS era, the classical way of doing this is by way of astronomicalobservations. (Note: Knowledge in Astronomical observations is still mandatory as theavailability of GPS is not guaranteed and the accreditation of our geomatics program re-quires this.) This means that an operational inertial frame of reference must be implicitlydened by making certain assumptions about the average direction of the spin axis andthe average position of the Greenwich observatory. It also means that Earth rotation withrespect to celestial objects has to be known with high accuracy. This chapter thereforediscusses celestial coordinate systems used in observational astronomy and their relationto the terrestrial systems, assuming Earth rotation as known.3.1 The Celestial SphereThe celestial sphere is based on a simple concept frequently used in geodetic astronomy.The celestial sphere is a 2-dimensional system where a point is fully described by 2 coor-dinates only. Consider that all stars are projected onto a sphere with an innite radius andthe spheres centre being the Earths centre. Further assume that the Earths movementabout the Sun can be neglected. The direction to each star is then given by the direction ofthe unit vector from the Earths centre of mass. It pierces the celestial sphere at a uniquepoint which can be denoted by only two spherical coordinates. If these coordinates arecalled and , the unit vector can be written in the general forme =

cos coscos sinsin

. (71)Since the length of each vector is innite, it does not give the position of the star in 3Dspace but only its direction from the Earths centre of mass. Figure 10 shows the celestialsphere in equatorial orientation from the viewpoint of the Earth. The Earths spin axispierces the celestial sphere at the North Celestial Pole (NCP) and the antipodal SouthCelestial Pole (SCP). The plane perpendicular to the Earths spin axis and containing thecentre of the celestial sphere is the celestial equator. A great circle through the star andthe celestial poles is called an hour circle. Each hour circle is orthogonal to the celestialequator. A small circle parallel to the celestial equator, is called celestial parallel. TheC Alexander Braun (2006-2010) 30 ENGO421: COORDINATE SYSTEMSplane containing the celestial poles and the zenith is called celestial meridian or observershour circle.Figure 10: Top: Celestial Sphere from a Terrestrial Point of View. Bottom: Celestial Spherefrom the Viewpoint of the Sun-Earth/Moon SystemC Alexander Braun (2006-2010) 31 ENGO421: COORDINATE SYSTEMSFigure 10 shows the celestial sphere in an ecliptic orientation, i.e. from the viewpointof the Sun-Earth/Moon system. The Earth/Moon system, or in fact its barycentre (thebarycentre is the centre of mass of two bodies), orbits the Sun in a plane which is inclinedwith respect to the celestial equator plane by an angle = 23.5o. The orbital plane is calledthe plane of the ecliptic and can also be considered as the apparent path of the Sun as seenfrom the Earth/Moon barycentre. The ecliptic is dened by two vectors. The rst one is thevector between the centre of the Sun and the barycentre of Earth/Moon system, the otherone is the inertial velocity vector of the barycentre orbiting about the Sun. The eclipticdened in this way does not contain perturbations from the planets of the solar system,where Venus and Jupiter cause the largest effects, however, they are small and within 2of the Suns apparent path. The northern ecliptic pole (NEP) and the southern ecliptic pole(SEP) are dened by the points where a normal to the ecliptic plane and through the spherecentre pierces the celestial sphere. The vernal equinox is the point where the apparent pathof the Sun crosses the celestial equator in spring coming from the southern and moving tothe northern hemisphere. In fall, the apparent path of the Sun crosses the celestial equatoragain, but in this case passing from the northern to the southern hemisphere; this point isknown as the autumnal equinox. Vernal and autumnal points are thus the points on thecelestial sphere which dene the intersection of the ecliptic with the celestial equator. Theyare used to dene the direction of one coordinate axis in most celestial coordinate systems.Summer and winter solstices are points on the ecliptic which have an angular separationof 90 from either equinox. They mark the largest distance of the Earths orbit (or Sunsapparent path) from the celestial equator. The equinoctial (solstitial) colure is the greatcircle through the celestial poles and the equinoxes (solstices) or, in other words, the hourcircles of the equinoxes (solstices). Equinoctial and solstitial colures are orthogonal to eachother. Further, we dene the nadir as the point on the celestial sphere, where the extensionof the local gravity vector pierces the sphere and we dene the zenith antipodal, where theextension of the gravity vector in the opposite direction pierces the sphere. The zenith canbe generally described as directly above the observer, with reference to the gravity vectorin the observers point. The following list summarizes the astronomical quantities used inthis chapter: Celestial poles NCP, SCP: Earth spin axis extension piercing celestial sphere Celestial equator: extension of Earth equator plane to celestial sphere Hour circle: great circle with NCP, SCP and normal to celestial equator Celestial parallel: Any circle parallel to celestial equator plane Observer related quantities: Zenith (Z): Direction of g intersecting celestial sphere Nadir (N): Antipodal to Zenith on celestial sphere Celestial horizon: Great circle orthogonal to Z, N Celestial vertical circle: Greta circle with Z, N, and g and orthogonal to celestialhorizon Celestial meridian: Great circle with NCP, SCP, and Z Observers hour circle: identical with celestial meridianC Alexander Braun (2006-2010) 32 ENGO421: COORDINATE SYSTEMS Prime vertical: Celestial vertical circle orthogonal to celestial meridian Ecliptic: Plane stretched out by the apparent path of the Sun about Earth/Moonbarycentre North/South ecliptic poles (NEP, SEP): Axis pierces celestial sphere and goes throughcentre of sphere Angel of the ecliptic: 23.5oangle between celestial equator and ecliptic Vernal equinox: Point where ecliptic intersects celestial equator from the South Autumnal equinox: Point where ecliptic intersects celestial equator from the North Winter solstice: 90oseparation from equinox on ecliptic, Southern hemisphere wrtcelestial equator Summer solstice: 90oseparation from equinox on ecliptic, Northern hemisphere wrtcelestial equator Equinoctial colure: Hour circle with both equinoxes Solstitial colure: Hour circle with both solstices and orthogonal to equinoctial colureIn order to imagine the three different types of reference systems, consider that there arethree equatorial planes and three types of poles orthogonal to these planes. The reason forthat is that these systems have a different point-of-view or purpose. While we know thetilt between the ecliptic and the celestial equator, the tilt wrt the celestial horizon changeswith the observers location and depends on the changing gravity vector direction. Hence,there is no constant geometric relation between the celestial horizon and the other twoequatorial planes.1. Observer related: Z and N as poles, celestial horizon as equatorial plane2. Terrestrial: NCP and SCP as poles, celestial equator as equatorial plane3. Sun-Earth/Moon: NEP, SEP as poles, ecliptic as equatorial planeIn order to dene the following four celestial coordinate systems (H, RA, HA, E), threebasic assumptions will be made: Stars have xed positions on the celestial sphere The celestial sphere has an innite radius The celestial sphere has its origin at the Earths centre of massThese assumptions simplify the denition of celestial coordinate systems because i) starpositions can be used as a uniform reference system for all observations from the Earth,and ii) star positions can be cataloged by using two coordinates only. Note that the as-sumption about the geocentric coordinate system is specic to geodetic astronomy, whilein most other cases, heliocentric systems are used with the Sun in the celestial spherescentre. The above assumptions give a useful model of physical reality. The corrections toC Alexander Braun (2006-2010) 33 ENGO421: COORDINATE SYSTEMSthe coordinates due to the movement of the spin axis do not exceed 50 per year. Thecorrections for the centre/origin shift from heliocentre to observers position (topocentric)are below 1. All corrections are time dependent and can be computed from parametersgiven in star catalogs. In accordance with the third assumption, all celestial coordinatessystems discussed here have a geocentric origin. In each celestial coordinate system, thepositions are xed by two spherical coordinates: the rst is dened in the primary plane,the second in the secondary plane. Depending on the denition of the primary plane andthe purpose of the coordinate systems, we distinguish the following celestial coordinatesystems: Horizon System (H) Right Ascension System (RA) Hour Angle System (HA) Ecliptic System (E).The different systems will be dened by the direction of their primary, secondary andtertiary axes. Thus, the primary axis denes the primary plane, the secondary axis thesecondary plane, etc. In all gures, S indicates the position of the star on the celestialsphere.3.2 The Horizon System - HThe horizon system is a typical observational system, it is comparable to the local astro-nomical frame (LA) as the observers gravity vector denes the primary axis/plane. Itscoordinates are the azimuth A and zenith distance z or the altitude angle a, which is thecorresponding angle to z, e.g. a = 90o z. a, z correspond directly to the readings ofa leveled theodolite. The horizon system is an earth-xed system and thus rotates withrespect to an inertial frame of reference. Horizon System (H) Origin: At observers position P (topocentric in geodetic astronomy) Primary axis (z): Zenith at P (normal toWp = constant) Secondary axis (x): Tangent to the celestial meridian at P (Z - NCP) pointing north Tertiary axis (y): Pointing east in the celestial horizon plane to complete a left-handed system (LHS)From Figure 11 we can derive, with pHbeing the horizontal component and zHthe verticalcomponent of the distance vectorpH= sinz = cosa (72)zH= cosz = sina (73)C Alexander Braun (2006-2010) 34 ENGO421: COORDINATE SYSTEMSFigure 11: The Horizon system - Hand, sincex = p cosA (74)y = p sinA (75)the unit vector becomeseH=

xyz

H=

sinz cosAsinz sinAcosz

=

cosa cosAcosa sinAsina

. (76)3.3 The Right Ascension System - RAThe Right Ascension System (RA) is a close approximation of an inertial frame of reference(Figure 12). It is thus xed with respect to distant galaxies and its two coordinates, rightascension and declination are used to catalog stars. It is also used as a referencein satellite geodesy providing the inertial frame of reference for Newtons equations ofmotion. The star catalog can be used in positioning and is updated regularly with newobservations of the two coordinates and , which fully describe the position of a star onthe celestial sphere. the right ascension angle is measured in easterly direction from thex-axis and has the units of 0-24 hours.C Alexander Braun (2006-2010) 35 ENGO421: COORDINATE SYSTEMS Right Ascension System (RA) Origin: Earths centre of mass Primary axis (z): Spin axis of the Earth, NCP, SCP Secondary axis (x): Direction to vernal equinox Tertiary axis (y): Completing a right-handed system (RHS)Using z = sin and p = cos, the unit vector takes the formeRA=

xyz

RA=

cos coscos sinsin

. (77)Figure 12: The Right Ascension System - RA3.4 The Hour Angle System - HAThe hour angle system is a transformational system and connects the observational system(H) to the catalog system (RA) (Figure 13). Its coordinates are the hour angle h and theC Alexander Braun (2006-2010) 36 ENGO421: COORDINATE SYSTEMSdeclination . The hour angle is dened as Sidereal time (ST) of the vernal equinox minusthe right ascension of the star. Thus, time is one of the variables dening this coordinatesystem. Note the equivalence of time and horizontal angles. The hour angle is measuredin westerly direction in units of 0-24 hours. Hour Angle System (HA) Origin: Earths centre of mass Primary axis (z): Spin axis of the Earth, NCP, SCP Secondary axis (x): Intersection of celestial meridian/observers hour circle and ce-lestial equator plane Tertiary axis (y): Completing LHSThe unit vector takes the formeHA=

xyz

HA=

cos coshcos sinhsin

. (78)Figure 13: The Hour Angle System - HAC Alexander Braun (2006-2010) 37 ENGO421: COORDINATE SYSTEMS3.5 The Ecliptic System - EThe ecliptic system is considered the most stable reference system and thus represents thebest realization of an inertial frame of reference (no acceleration) (Figure 14). It is usedparticularly for Sun catalogs. Its coordinates are ecliptic longitude and ecliptic latitude, and both are referred to the ecliptic plane. Ecliptic System (E) Origin: Earths centre of mass Primary axis (z): Orthogonal to the ecliptic plane (apparent path of the Sun) andthrough the centre of the celestial sphere Secondary axis (x): Pointing towards vernal equinox Tertiary axis (y): Completing a RHSThe unit vector takes the formeE=

xyz

E=

cos coscos sinsin

. (79)Figure 14: The Ecliptic System - EC Alexander Braun (2006-2010) 38 ENGO421: COORDINATE SYSTEMS3.6 Reection and Rotation MatricesIn order to transform between different coordinate systems, the required mathematicaloperations are the reection and the rotation of coordinate axes. The tools to performthese operations can be generally expressed by reection and rotation matrices. Considerthe following example: Two coordinate systems (x, y) and (x

, y

) have the same origin,but their orientation in space is different (see Figure 15). In order to express coordinatesobtained in one system by coordinates in the other system, let us write down the relationbetween the coordinate pairs and the angles and in Figure 15.Figure 15: Rotation of a Coordinate Systemx

= s cos( ) (80)y

= s sin( ) (81)x = s cos (82)y = s sin (83)By solving the argument of the sin and cos term and substituting s cos by x, and s sinby y,x

= s cos cos +s sin sin = x cos +y sin (84)y

= s sin cos s cos sin = y cos x sin, (85)or in matrix notation,

x

y

=

cos sinsin cos

xy

. (86)While the above rotation was only in 2-d space, it is straightforward to realize that therotation actually was performed about a third axis z which is orthogonal to x and y. theC Alexander Braun (2006-2010) 39 ENGO421: COORDINATE SYSTEMSthird component can thus be introduced to the matrix.

x

y

z

=

cos sin 0sin cos 00 0 1

xyz

= R3()

xyz

(87)The rotation matrix R3() describes a rotation by the angle about the z-axis. In an analogway, the rotation matrices for rotations about the other two axis can be derived.R1() =

1 0 00 cos sin0 sin cos

(88)R2() =

cos 0 sin0 1 0sin 0 cos

(89)Reection matrices are even simpler. Applying a reection matrix to a coordinate systemor coordinates results in changing the direction of one axis, or simply a sign change forone component of the coordinates. Reection matrices are denoted by the axis which isreected, e.g. P2 reects the y-axis. The three refection matrices are as follows:P1 =

1 0 00 1 00 0 1

(90)P2 =

1 0 00 1 00 0 1

(91)P3 =

1 0 00 1 00 0 1

(92)A combination of rotation and reection matrix operations is often required to transformfrom one coordinate system into another. It is therefore useful to investigate the propertiesof these operators. The product of a reection and rotation matrix is commutative, if theirindex is identical.Pi Ri() = Ri() Pi, for i = 1, 2, 3 (93)For different indices, the sign of the argument changes.Pi Rj() = Rj() Pi = R1j () Pi = RTj () Pi (94)These mathematical tools will be extensively used in the next section for transformingcelestial coordinate systems, but also in chapter 6 - Coordinate transformations.3.7 Transformations Between Celestial SystemsTransformation between HA and RAThe HA and RA systems are shown in Figure 16. Both systems have the same primaryplane, the celestial equator, and also share the declination as one coordinate. TheirC Alexander Braun (2006-2010) 40 ENGO421: COORDINATE SYSTEMScoordinates in the equatorial plane are related byST = +h, (95)where ST is the hour angle of the vernal equinox which is also known as sidereal time.However, the two systems differ in their handedness. The transformation consists of twosteps: Left-handed into right-handed system: apply P2 Rotation about z-axis by ST: apply R3(ST) = R3( h)Expressed through a unit vector, th