3.02 Potential Theory and Static Gravity Field of the Earth C. Jekeli, The Ohio State University, Columbus, OH, USA ª 2007 Elsevier B.V. All rights reserved. 3.02.1 Introduction 12 3.02.1.1 Historical Notes 12 3.02.1.2 Coordinate Systems 14 3.02.1.3 Preliminary Definitions and Concepts 14 3.02.2 Newton’s Law of Gravitation 15 3.02.3 Boundary-Value Problems 18 3.02.3.1 Green’s Identities 18 3.02.3.2 Uniqueness Theorems 20 3.02.3.3 Solutions by Integral Equation 21 3.02.4 Solutions to the Spherical BVP 22 3.02.4.1 Spherical Harmonics and Green’s Functions 22 3.02.4.2 Inverse Stokes and Hotine Integrals 25 3.02.4.3 Vening-Meinesz Integral and Its Inverse 26 3.02.4.4 Concluding Remarks 27 3.02.5 Low-Degree Harmonics: Interpretation and Reference 28 3.02.5.1 Low-Degree Harmonics as Density Moments 28 3.02.5.2 Normal Ellipsoidal Field 30 3.02.6 Methods of Determination 32 3.02.6.1 Measurement Systems and Techniques 33 3.02.6.2 Models 37 3.02.7 The Geoid and Heights 38 References 41 Glossary geodetic reference system Normal ellipsoid with deflection of the vertical Angle between direction defined parameters adopted for general geodetic of gravity and direction of normal gravity. and gravimetric referencing. density moment Integral over the volume of a geoid Surface of constant gravity potential that body of the product of its density and integer closely approximates mean sea level. powers of Cartesian coordinates. geoid undulation Vertical distance between the disturbing potential The difference between geoid and the normal ellipsoid, positive if the geoid Earth’s gravity potential and the normal potential. is above the ellipsoid. eccentricity The ratio of the difference of squares geopotential number Difference between gravity of semimajor and semiminor axes to the square of potential on the geoid and gravity potential at a the semimajor axis of an ellipsoid. point. ellipsoid Surface formed by rotating an ellipse gravitation Attractive acceleration due to mass. about its minor axis. gravitational potential Potential due to gravita- equipotential surface Surface of constant tional acceleration. potential. gravity Vector sum of gravitation and centrifugal flattening The ratio of the difference between acceleration due to Earth’s rotation. semimajor and semiminor axes to the semimajor gravity anomaly The difference between Earth’s axis of an ellipsoid. gravity on the geoid and normal gravity on the 11 12 Potential Theory and Static Gravity Field of the Earth ellipsoid, either as a difference in vectors or a dif- normal gravity Gravity associated with the normal ference in magnitudes. ellipsoid. gravity disturbance The difference between normal gravity potential Gravity potential asso- Earth’s gravity and normal gravity, either as a dif- ciated with the normal ellipsoid. ference in vectors or a difference in magnitudes. orthometric height Distance along the plumb line gravity potential Potential due to gravity from the geoid to a point. acceleration. potential Potential energy per unit mass due to the harmonic function Function that satisfies gravitational field; always positive and zero at Laplace’s field equation. infinity. linear eccentricity The distance from the center of sectorial harmonics Surface spherical harmonics an ellipsoid to either of its foci. that do not change in sign with respect to latitude. mean Earth ellipsoid Normal ellipsoid with para- Stokes coefficients Constants in a series expan- meters closest to actual corresponding parameters sion of the gravitational potential in terms of for the Earth. spherical harmonic functions. mean tide geoid Geoid with all time-varying tidal surface spherical harmonics Basis functions effects removed. defined on the unit sphere, comprising products of multipoles Stokes coefficients. normalized associated Legendre functions and Newtonian potential Harmonic function that sinusoids. approaches the potential of a point mass at infinity. tesseral harmonics Neither zonal nor sectorial non-tide geoid Mean tide geoid with all (direct harmonics. and indirect deformation) mean tide effects zero-tide geoid Mean tide geoid with just the removed. mean direct tidal effect removed (indirect effect due normal ellipsoid Earth-approximating reference to Earth’s permanent deformation is retained). ellipsoid that generates a gravity field in which it is a zonal harmonics Spherical harmonics that do not surface of constant normal gravity potential. depend on longitude. 3.02.1 Introduction (or rise, in the case of some gases), and the more material the greater the tendency to do so. It was Gravitational potential theory has its roots in the late enough of a self-evident explanation that it was not Renaissance period when the position of the Earth in yet to receive the scrutiny of the scientific method, the cosmos was established on modern scientific the beginnings of which, ironically, are credited to (observation-based) grounds. A study of Earth’s grav- Aristotle. Almost 2000 years later, Galileo Galilei itational field is a study of Earth’s mass, its influence (1564–1642) finally took up the challenge to under- on near objects, and lately its redistributing transport stand gravitation through observation and scientific in time. It is also fundamentally a geodetic study of investigation. His experimentally derived law of fall- Earth’s shape, described largely (70%) by the surface ing bodies corrected the Aristotelian view and of the oceans. This initial section provides a historical divorced the effect of gravitation from the mass of backdrop to potential theory and introduces some the falling object – all bodies fall with the same concepts in physical geodesy that set the stage for acceleration. This truly monumental contribution to later formulations. physics was, however, only a local explanation of how bodies behaved under gravitational influence. Johannes Kepler’s (1571–1630) observations of pla- 3.02.1.1 Historical Notes netary orbits pointed to other types of laws, Gravitation is a physical phenomenon so pervasive principally an inverse-square law according to and incidental that humankind generally has taken it which bodies are attracted by forces that vary with for granted with scarcely a second thought. The the inverse square of distance. The genius of Issac Greek philosopher Aristotle (384–322 BC) allowed Newton (1642–1727) brought it all together in his no more than to assert that gravitation is a natural Philosophiae Naturalis Principia Mathematica of 1687 property of material things that causes them to fall with a single and simple all-embracing law that in Potential Theory and Static Gravity Field of the Earth 13 one bold stroke explained the dynamics of the entire needed to be corrected for the irregular direction of universe (today there is more to understanding the gravitation. Today, the modern view of a height dynamics of the cosmos, but Newton’s law remark- reference is changing to that of a geometric, mathe- ably holds its own). The mass of a body was again an matical surface (an ellipsoid) and three-dimensional essential aspect, not as a self-attribute as Aristotle had coordinates (latitude, longitude, and height) of points implied, but as the source of attraction for other on the Earth’s surface are readily obtained geometri- bodies: each material body attracts every other mate- cally by ranging to the satellites of the Global rial body according to a very specific rule (Newton’s Positioning System (GPS). The requirements of law of gravitation; see Section 3.02.2). Newton gravitation for GPS orbit determination within an regretted that he could not explain exactly why Earth-centered coordinate system are now largely mass has this property (as one still yearns to know met with existing models. Improvements in gravita- today within the standard models of particle and tional models are motivated in geodesy primarily for quantum theories). Even Albert Einstein (1879– rapid determination of traditional heights with 1955) in developing his general theory of relativity respect to a level surface. These heights, for example, (i.e., the theory of gravitation) could only improve on the orthometric heights, in this sense then become Newton’s theory by incorporating and explaining derived attributes of points, rather than their cardinal action at a distance (gravitational force acts with the components. speed of light as a fundamental tenet of the theory). Navigation and guidance exemplify a further spe- What actually mediates the gravitational attraction cific niche where gravitation continues to find still intensely occupies modern physicists and important relevance. While GPS also dominates cosmologists. this field, the vehicles requiring completely autono- Gravitation since its early scientific formulation mous, self-contained systems must rely on inertial initially belonged to the domain of astronomers, at instruments (accelerometers and gyroscopes). These least as far as the observable universe was concerned. do not sense gravitation (see Section 3.02.6.1), yet Theory successfully predicted the observed perturba- gravitation contributes to the total definition of the tions of planetary orbits and even the location of vehicle trajectory, and thus the output of inertial previously unknown new planets (Neptune’s discov- navigation systems must be compensated for the ery in 1846 based on calculations motivated by effect of gravitation. By far the greatest emphasis in observed perturbations in Uranus’ orbit was a major gravitation, however, has shifted to the Earth triumph for Newton’s law). However, it was also dis- sciences, where detailed knowledge of the configura- covered that gravitational acceleration varies on tion of masses (the solid, liquid, and atmospheric Earth’s surface, with respect to altitude and latitude. components) and their transport and motion leads Newton’s law of gravitation again provided the back- to improved understanding of the Earth systems (cli- drop for the variations observed with pendulums. An mate, hydrologic cycle, tectonics) and their early achievement for his theory came when he suc- interactions with life.