3.02 Potential Theory and Static Gravity Field of the Earth

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1 3.02 Potential Theoy and Static Gavity Field of the Eath C. Jekeli, The Ohio State Univesity, Columbus, OH, USA ª 2007 Elsevie B.V. All ights eseved Intoduction Histoical Notes Coodinate Systems Peliminay Definitions and Concepts Newton s Law of Gavitation Bounday-Value Poblems Geen s Identities Uniqueness Theoems Solutions by Integal Equation Solutions to the Spheical BVP Spheical Hamonics and Geen s Functions Invese Stokes and Hotine Integals Vening-Meinesz Integal and Its Invese Concluding Remaks Low-Degee Hamonics: Intepetation and Refeence Low-Degee Hamonics as Density Moments Nomal Ellipsoidal Field Methods of Detemination Measuement Systems and Techniques Models The Geoid and Heights 38 Refeences 4 Glossay deflection of the vetical Angle between diection of gavity and diection of nomal gavity. density moment Integal ove the volume of a body of the poduct of its density and intege powes of Catesian coodinates. distubing potential The diffeence between Eath s gavity potential and the nomal potential. eccenticity The atio of the diffeence of squaes of semimajo and semimino axes to the squae of the semimajo axis of an ellipsoid. ellipsoid Suface fomed by otating an ellipse about its mino axis. equipotential suface Suface of constant potential. flattening The atio of the diffeence between semimajo and semimino axes to the semimajo axis of an ellipsoid. geodetic efeence system Nomal ellipsoid with defined paametes adopted fo geneal geodetic and gavimetic efeencing. geoid Suface of constant gavity potential that closely appoximates mean sea level. geoid undulation Vetical distance between the geoid and the nomal ellipsoid, positive if the geoid is above the ellipsoid. geopotential numbe Diffeence between gavity potential on the geoid and gavity potential at a point. gavitation Attactive acceleation due to mass. gavitational potential Potential due to gavitational acceleation. gavity Vecto sum of gavitation and centifugal acceleation due to Eath s otation. gavity anomaly The diffeence between Eath s gavity on the geoid and nomal gavity on the

2 2 Potential Theoy and Static Gavity Field of the Eath ellipsoid, eithe as a diffeence in vectos o a diffeence in magnitudes. gavity distubance The diffeence between Eath s gavity and nomal gavity, eithe as a diffeence in vectos o a diffeence in magnitudes. gavity potential Potential due to gavity acceleation. hamonic function Function that satisfies Laplace s field equation. linea eccenticity The distance fom the cente of an ellipsoid to eithe of its foci. mean Eath ellipsoid Nomal ellipsoid with paametes closest to actual coesponding paametes fo the Eath. mean tide geoid Geoid with all time-vaying tidal effects emoved. multipoles Stokes coefficients. Newtonian potential Hamonic function that appoaches the potential of a point mass at infinity. non-tide geoid Mean tide geoid with all (diect and indiect defomation) mean tide effects emoved. nomal ellipsoid Eath-appoximating efeence ellipsoid that geneates a gavity field in which it is a suface of constant nomal gavity potential. nomal gavity Gavity associated with the nomal ellipsoid. nomal gavity potential Gavity potential associated with the nomal ellipsoid. othometic height Distance along the plumb line fom the geoid to a point. potential Potential enegy pe unit mass due to the gavitational field; always positive and zeo at infinity. sectoial hamonics Suface spheical hamonics that do not change in sign with espect to latitude. Stokes coefficients Constants in a seies expansion of the gavitational potential in tems of spheical hamonic functions. suface spheical hamonics Basis functions defined on the unit sphee, compising poducts of nomalized associated Legende functions and sinusoids. tesseal hamonics Neithe zonal no sectoial hamonics. zeo-tide geoid Mean tide geoid with just the mean diect tidal effect emoved (indiect effect due to Eath s pemanent defomation is etained). zonal hamonics Spheical hamonics that do not depend on longitude Intoduction Gavitational potential theoy has its oots in the late Renaissance peiod when the position of the Eath in the cosmos was established on moden scientific (obsevation-based) gounds. A study of Eath s gavitational field is a study of Eath s mass, its influence on nea objects, and lately its edistibuting tanspot in time. It is also fundamentally a geodetic study of Eath s shape, descibed lagely (70%) by the suface of the oceans. This initial section povides a histoical backdop to potential theoy and intoduces some concepts in physical geodesy that set the stage fo late fomulations Histoical Notes Gavitation is a physical phenomenon so pevasive and incidental that humankind geneally has taken it fo ganted with scacely a second thought. The Geek philosophe Aistotle ( BC) allowed no moe than to asset that gavitation is a natual popety of mateial things that causes them to fall (o ise, in the case of some gases), and the moe mateial the geate the tendency to do so. It was enough of a self-evident explanation that it was not yet to eceive the scutiny of the scientific method, the beginnings of which, ionically, ae cedited to Aistotle. Almost 2000 yeas late, Galileo Galilei ( ) finally took up the challenge to undestand gavitation though obsevation and scientific investigation. His expeimentally deived law of falling bodies coected the Aistotelian view and divoced the effect of gavitation fom the mass of the falling object all bodies fall with the same acceleation. This tuly monumental contibution to physics was, howeve, only a local explanation of how bodies behaved unde gavitational influence. Johannes Keple s (57 630) obsevations of planetay obits pointed to othe types of laws, pincipally an invese-squae law accoding to which bodies ae attacted by foces that vay with the invese squae of distance. The genius of Issac Newton ( ) bought it all togethe in his Philosophiae Natualis Pincipia Mathematica of 687 with a single and simple all-embacing law that in

3 Potential Theoy and Static Gavity Field of the Eath 3 one bold stoke explained the dynamics of the entie univese (today thee is moe to undestanding the dynamics of the cosmos, but Newton s law emakably holds its own). The mass of a body was again an essential aspect, not as a self-attibute as Aistotle had implied, but as the souce of attaction fo othe bodies: each mateial body attacts evey othe mateial body accoding to a vey specific ule (Newton s law of gavitation; see Section ). Newton egetted that he could not explain exactly why mass has this popety (as one still yeans to know today within the standad models of paticle and quantum theoies). Even Albet Einstein ( ) in developing his geneal theoy of elativity (i.e., the theoy of gavitation) could only impove on Newton s theoy by incopoating and explaining action at a distance (gavitational foce acts with the speed of light as a fundamental tenet of the theoy). What actually mediates the gavitational attaction still intensely occupies moden physicists and cosmologists. Gavitation since its ealy scientific fomulation initially belonged to the domain of astonomes, at least as fa as the obsevable univese was concened. Theoy successfully pedicted the obseved petubations of planetay obits and even the location of peviously unknown new planets (Neptune s discovey in 846 based on calculations motivated by obseved petubations in Uanus obit was a majo tiumph fo Newton s law). Howeve, it was also discoveed that gavitational acceleation vaies on Eath s suface, with espect to altitude and latitude. Newton s law of gavitation again povided the backdop fo the vaiations obseved with pendulums. An ealy achievement fo his theoy came when he successfully pedicted the pola flattening in Eath s shape on the basis of hydostatic equilibium, which was confimed finally (afte some contovesy) with geodetic measuements of long tiangulated acs in by Piee de Maupetuis and Alexis Claiaut. Gavitation thus also played a dominant ole in geodesy, the science of detemining the size and shape of the Eath, pomulgated in lage pat by the fathe of moden geodesy, Fiedich R. Helmet (843 97). Teestial gavitation though the twentieth centuy was consideed a geodetic aea of eseach, although, of couse, its geophysical exploits should not be ovelooked. But the advancement in modeling accuacy and global application was pomoted mainly by geodesists who needed a well-defined efeence fo heights (a level suface) and whose astonomic obsevations of latitude and longitude needed to be coected fo the iegula diection of gavitation. Today, the moden view of a height efeence is changing to that of a geometic, mathematical suface (an ellipsoid) and thee-dimensional coodinates (latitude, longitude, and height) of points on the Eath s suface ae eadily obtained geometically by anging to the satellites of the Global Positioning System (GPS). The equiements of gavitation fo GPS obit detemination within an Eath-centeed coodinate system ae now lagely met with existing models. Impovements in gavitational models ae motivated in geodesy pimaily fo apid detemination of taditional heights with espect to a level suface. These heights, fo example, the othometic heights, in this sense then become deived attibutes of points, athe than thei cadinal components. Navigation and guidance exemplify a futhe specific niche whee gavitation continues to find impotant elevance. While GPS also dominates this field, the vehicles equiing completely autonomous, self-contained systems must ely on inetial instuments (acceleometes and gyoscopes). These do not sense gavitation (see Section ), yet gavitation contibutes to the total definition of the vehicle tajectoy, and thus the output of inetial navigation systems must be compensated fo the effect of gavitation. By fa the geatest emphasis in gavitation, howeve, has shifted to the Eath sciences, whee detailed knowledge of the configuation of masses (the solid, liquid, and atmospheic components) and thei tanspot and motion leads to impoved undestanding of the Eath systems (climate, hydologic cycle, tectonics) and thei inteactions with life. Oceanogaphy, in paticula, also equies a detailed knowledge of a level suface (the geoid) to model suface cuents using satellite altimety. Clealy, thee is an essential tempoal component in these studies, and, indeed, the tempoal gavitational field holds cente stage in many new investigations. Moeove, Eath s dynamic behavio influences point coodinates and Eath-fixed coodinate fames, and we come back to fundamental geodetic concens in which the gavitational field plays an essential ole! This section deals with the static gavitational field. The theoy of the potential fom the classical Newtonian standpoint povides the foundation fo modeling the field and thus deseves the focus of the exposition. The tempoal pat is a natual extension that is eadily achieved with the addition of the time vaiable (no new laws ae needed, if we neglect

4 4 Potential Theoy and Static Gavity Field of the Eath geneal elativistic effects) and will not be expounded hee. We ae pimaily concened with gavitation on and extenal to the solid and liquid Eath since this is the domain of most applications. The intenal field can also be modeled fo specialized puposes (such as submaine navigation), but intenal geophysical modeling, fo example, is done usually in tems of the souces (mass density distibution), athe than the esulting field. u z b E δ Confocal ellipsoid Sphee x Coodinate Systems Modeling the Eath s gavitational field depends on the choice of coodinate system. Customaily, owing to the Eath s geneal shape, a spheical pola coodinate system seves fo most applications, and vitually all global models use these coodinates. Howeve, the Eath is slightly flattened at the poles, and an ellipsoidal coodinate system has also been advocated fo some nea-eath applications. We note that the geodetic coodinates associated with a geodetic datum (based on an ellipsoid) ae neve used in a foundational sense to model the field since they do not admit to a sepaation-of-vaiables solution of Laplace diffeential equation (Section ). Spheical pola coodinates, descibed with the aid of Figue, compise the spheical colatitude,, the longitude,, and the adial distance,. Thei elation to Catesian coodinates is x ¼ sin cos y ¼ sin sin z ¼ cos Consideing Eath s pola flattening, a bette appoximation, than a sphee, of its (ocean) suface is an ellipsoid of evolution. Such a suface is ½Š Figue 2 Ellipsoidal coodinates. geneated by otating an ellipse about its mino axis (pola axis). The two focal points of the best-fitting, Eath-centeed ellipsoid (ellipse) ae located in the equato about E ¼ 522 km fom the cente of the Eath. A given ellipsoid, with specified semimino axis, b, and linea eccenticity, E, defines the set of ellipsoidal coodinates, as descibed in Figue 2. The longitude is the same as in the spheical case. The colatitude,, is the complement of the so-called educed latitude; and the distance coodinate, u, is the semimino axis of the confocal ellipsoid though the point in question. We call ð; ; uþ ellipsoidal coodinates; they ae also known as spheoidal coodinates, o Jacobi ellipsoidal coodinates. Thei elation to Catesian coodinates is given by p x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ E 2 sin cos p y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ E 2 sin sin ½2Š z ¼ u cos Points on the given ellipsoid all have u ¼ b; and, all sufaces, u ¼ constant, ae confocal ellipsoids (the analogy to the spheical case, when E ¼ 0, should be evident). z Peliminay Definitions and Concepts The gavitational potential, V, of the Eath is geneated by its total mass density distibution. Fo applications on the Eath s suface it is useful to include the potential,, associated with the centifugal acceleation due to Eath s otation. The sum, W ¼ V þ, is then known, in geodetic teminology, as the gavity potential, distinct fom gavitational potential. It is futhe advantageous to define a elatively simple efeence potential, o nomal potential, that accounts fo the bulk of the gavity potential Figue θ λ x Spheical pola coodinates. y

5 Potential Theoy and Static Gavity Field of the Eath 5 (Section ). The nomal gavity potential, U, is defined as a gavity potential associated with a bestfitting ellipsoid, the nomal ellipsoid, which otates with the Eath and is also a suface of constant potential in this field. The diffeence between the actual and the nomal gavity potentials is known as the distubing potential: T ¼ W U; it thus excludes the centifugal potential. The nomal gavity potential accounts fo appoximately % of the total potential. The gadient of the potential is an acceleation, gavity o gavitational acceleation, depending on whethe o not it includes the centifugal acceleation. Nomal gavity,, compises % of the total gavity, g, although the diffeence in magnitudes, the gavity distubance, dg, can be as lage as seveal pats in 0 4. A special kind of diffeence, called the gavity anomaly, g, is defined as the diffeence between gavity at a point, P, and nomal gavity at a coesponding point, Q, whee W P ¼ U Q, and P and Q ae on the same pependicula to the nomal ellipsoid. The suface of constant gavity potential, W 0,that closely appoximates mean sea level is known as the geoid. If the constant nomal gavity potential, U 0,on the nomal ellipsoid is equal to the constant gavity potential of the geoid, then the gavity anomaly on the geoid is the diffeence between gavity on the geoid and nomal gavity on the ellipsoid at espective points, P 0, Q 0, shaing the same pependicula to the ellipsoid. The sepaation between the geoid and the ellipsoid is known as the geoid undulation, N, o also the geoid height (Figue 3). A simple Taylo expansion of the nomal gavity potential along the ellipsoid pependicula yields the following impotant fomula: N ¼ T This is Buns equation, which is accuate to a few millimetes in N, and which can be extended to N ¼ T= ðw 0 U 0 Þ= fo the geneal case, W 0 6¼ U 0. The gavity anomaly (on the geoid) is the gavity distubance coected fo the evaluation of nomal gavity on the ellipsoid instead of the geoid. This coection is N q=qh ¼ ðq=qhþðt=þ, whee h is height along the ellipsoid pependicula. We have g ¼ qt=qh, and hence g ¼ qt qh þ q qh T The slope of the geoid with espect to the ellipsoid is also the angle between the coesponding pependiculas to these sufaces. This angle is known as the deflection of the vetical, that is, the deflection of the plumb line (pependicula to the geoid) elative to the pependicula to the nomal ellipsoid. The deflection angle has components,,, espectively, in the noth and east diections. The spheical appoximations to the gavity distubance, anomaly, and deflection of the vetical ae given by g ¼ qt q ; ¼ qt q ; qt g ¼ q 2 T ¼ qt sin q whee the signs on the deivatives ae a matte of convention. ½3Š ½4Š ½5Š Newton s Law of Gavitation P ellipsoidal Ellipsoidal height, h h H, othometic height W = W 0 Deflection of the vetical Topogaphic suface In its oiginal fom, Newton s law of gavitation applies only to idealized point masses. It descibes the foce of attaction, F, expeienced by two such solitay masses as being popotional to the poduct of the masses, m and m 2 ; invesely popotional to the distance,,, between them; and diected along the line joining them: Figue 3 P 0 N, geoid undulation U = U g 0 Q 0 P 0 Q 0 Geoid Ellipsoid Relative geomety of geoid and ellipsoid. F ¼ G m m 2, 2 n ½6Š G is a constant, known as Newton s gavitational constant, that takes cae of the units between the left- and ight-hand sides of the equation; it can be

6 6 Potential Theoy and Static Gavity Field of the Eath detemined by expeiment and the cuent best value is (Goten, 2004): G ¼ ð6: :00030Þ0 m 3 kg s 2 ½7Š The unit vecto n in eqn [6] is diected fom eithe point mass to the othe, and thus the gavitational foce is attactive and applies equally to one mass as the othe. Newton s law of gavitation is univesal as fa as we know, equiing efomulation only in Einstein s moe compehensive theoy of geneal elativity which descibes gavitation as a chaacteistic cuvatue of the space time continuum (Newton s fomulation assumes instantaneous action and diffes significantly fom the geneal elativistic concept only when vey lage velocities o masses ae involved). We can ascibe a gavitational acceleation to the gavitational foce, which epesents the acceleation that one mass undegoes due to the gavitational attaction of the othe. Specifically, fom the law of gavitation, we have (fo point masses) the gavitational acceleation of m due to the gavitational attaction of m 2 : g ¼ G m 2, 2 n ½8Š The vecto g is independent of the mass, m, of the body being acceleated (which Galileo found by expeiment). By the law of supeposition, the gavitational foce, o the gavitational acceleation, due to many point masses is the vecto sum of the foces o acceleations geneated by the individual point masses. Manipulating vectos in this way is cetainly feasible, but fotunately a moe appopiate concept of gavitation as a scala field simplifies the teatment of abitay mass distibutions. This moe moden view of gavitation (aleady adopted by Gauss ( ) and Geen (793 84)) holds that it is a field having a gavitational potential. Lagange (736 83) fully developed the concept of a field, and the potential, V, of the gavitational field is defined in tems of the gavitational acceleation, g, that a test paticle would undego in the field accoding to the equation ÑV ¼ g whee Ñ is the gadient opeato (a vecto). Futhe elucidation of gavitation as a field gew fom Einstein s attempt to incopoate gavitation into his special theoy of elativity whee no efeence fame has special significance above all othes. It was necessay to conside that gavitational foce is not a eal ½9Š foce (i.e., it is not an applied foce, like fiction o populsion) athe, it is known as a kinematic foce, that is, one whose action is popotional to the mass on which it acts (like the centifugal foce; see Matin, 988). Unde this pecept, the geomety of space is defined intinsically by the gavitational fields contained theein. We continue with the classical Newtonian potential, but intepet gavitation as an acceleation diffeent fom the acceleation induced by eal, applied foces. This becomes especially impotant when consideing the measuement of gavitation (Section ). The gavitational potential, V, is a scala function, and, as defined hee, V is deived diectly on the basis of Newton s law of gavitation. To make it completely consistent with this law and thus declae it a Newtonian potential, we must impose the following conditions: lim,v ¼ Gm and lim V ¼ 0 ½0Š,!,! Hee, m is the attacting mass, and we say that the potential is egula at infinity. It is easy to show that the gavitational potential at any point in space due to a point mass, in ode to satisfy eqns [8] [0], must be V ¼ Gm, ½Š whee, again,, is the distance between the mass and the point at which the potential is expessed. Note that V fo, ¼ 0 does not exist in this case; that is, the field of a point mass has a singulaity. We use hee the convention that the potential is always positive (in contast to physics, whee it is usually defined to be negative, conceptually close to potential enegy). Applying the law of supeposition, the gavitational potential of many point masses is the sum of the potentials of the individual points (see Figue 4): V P ¼ G X j m j, j ½2Š And, fo infinitely many points in a closed, bounded egion with infinitesimally small masses, dm, the summation in eqn [2] changes to an integation, V P Z ¼ G mass dm, ½3Š o, changing vaiables (i.e., units), dm ¼ dv, whee epesents density (mass pe volume) and dv is a volume element, we have (Figue 5)

7 Potential Theoy and Static Gavity Field of the Eath 7 m l z l 2 P l j spheically symmetic density we may choose the integation coodinate system so that the pola axis passes though P. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, ¼ 9 2 þ 2 29 cos 9; d, ¼ 9 sin 9d9, m 2 l3 m j ½5Š y It is easy to show that with this change of vaiables (fom to,) the integal [4] becomes simply x Figue 5 n V P m 3 Figue 4 Discete set of mass points (supeposition pinciple). dv ρ Continuous density distibution. Z ¼ G volume dv ½4Š, In eqn [4],, is the distance between the evaluation point, P, and the point of integation. In spheical pola coodinates (Section ), these points ae ð; ; Þ and ð9; 9; 9Þ, espectively. The volume element in this case is given by dv ¼ 9 2 sin 9 d9 d9 d9. V and its fist deivatives ae continuous eveywhee even in the case that P is on the bounding suface o inside the mass distibution, whee thee is the appaent singulaity at, ¼ 0. In this case, by changing to a coodinate system whose oigin is at P, the volume element becomes dv ¼, 2 sin c d dc d, (fo some diffeent colatitude and longitude c and ); and, clealy, the singulaity disappeas the integal is said to be weakly singula. Suppose the density distibution ove the volume depends only on adial distance (fom the cente of mass): ¼ ð9þ, and that P is an exteio evaluation point. The suface bounding the masses necessaily is a sphee (say, of adius, R) and because of the l s P V ð; ; Þ ¼ GM ; R ½6Š whee M is the total mass bounded by the sphee. This shows that to a vey good appoximation the extenal gavitational potential of a planet such as the Eath (with concentically layeed density) is the same as that of a point mass. Besides volumetic mass (density) distibutions, it is of inteest to conside suface distibutions. Imagine an infinitesimally thin laye of mass on a suface, s, whee the units of density in this case ae those of mass pe aea. Then, analogous to eqn [4], the potential is V P ¼ G s ds ½7Š, In this case, V is a continuous function eveywhee, but its fist deivatives ae discontinuous at the suface. O, one can imagine two infinitesimally close density layes (double laye, o laye of mass dipoles), whee the units of density ae now those of mass pe aea times length. It tuns out that the potential in this case is given by (see Heiskanen and Moitz, 967, p. 8) V P ¼ G q ds ½8Š s qn, whee q=qn is the diectional deivative along the pependicula to the suface (Figue 5). Now, V itself is discontinuous at the suface, as ae all its deivatives. In all cases, V is a Newtonian potential, being deived fom the basic fomula [] fo a point mass that follows fom Newton s law of gavitation (eqn [6]). The following popeties of the gavitational potential ae useful fo subsequent expositions. Fist, conside Stokes s theoem, fo a vecto function, f, defined on a suface, s: I ðf Þ? n ds ¼ f? d ½9Š s whee p is any closed path in the suface, n is the unit vecto pependicula to the suface, and d is a p

8 8 Potential Theoy and Static Gavity Field of the Eath diffeential displacement along the path. Fom eqn [9], we find g ¼ 0 ½20Š since ¼ 0; hence, applying Stokes s theoem, we find with F ¼ mg that I w ¼ F? ds ¼ 0 ½2Š That is, the gavitational field is consevative: the wok, w, expended in moving a mass aound a closed path in this field vanishes. In contast, dissipating foces (eal foces!), like fiction, expend wok o enegy, which shows again the special natue of the gavitational foce. It can be shown (Kellogg, 953, p. 56) that the second patial deivatives of a Newtonian potential, V, satisfy the following diffeential equation, known as Poisson s equation: 2 V ¼ 4G ½22Š whee 2 ¼ Ñ? Ñ fomally is the scala poduct of two gadient opeatos and is called the Laplacian opeato. In Catesian coodinates, it is given by 2 ¼ q2 qx 2 þ q2 qy 2 þ q2 qz 2 ½23Š Note that the Laplacian is a scala opeato. Eqn [22] is a local chaacteization of the potential field, as opposed to the global chaacteization given by eqn [4]. Poisson s equation holds wheeve the mass density,, satisfies cetain conditions simila to continuity (Hölde conditions; see Kellogg, 953, pp ). A special case of eqn [22] applies fo those points whee the density vanishes (i.e., in fee space); then Poisson s equation tuns into Laplace equation, 2 V ¼ 0 ½24Š It is easily veified that the point mass potential satisfies eqn [24], that is, 2 ¼ 0 ½25Š, whee, ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x 0 Þ 2 þðy y 0 Þ 2 þðz z 0 Þ 2 and the mass point is at ðx9; y9; z9þ. The solutions to Laplace equation [24] (that is, functions that satisfy Laplace equations) ae known as hamonic functions (hee, we also impose the conditions [0] on the solution, if it is a Newtonian potential and if the mass-fee egion includes infinity). Hence, evey Newtonian potential is a hamonic function in fee space. The convese is also tue: evey hamonic function can be epesented as a Newtonian potential of a mass distibution (Section ). Whethe as a volume o a laye density distibution, the coesponding potential is the sum o integal of the souce value multiplied by the invese distance function (o its nomal deivative fo the dipole laye). This function depends on both the souce points and the computation point and is known as a Geen s function. It is also known as the kenel function of the integal epesentation of the potential. Functions of this type also play a dominant ole in epesenting the potential as solutions to bounday-value poblems (BVPs), as shown in subsequent sections Bounday-Value Poblems If the density distibution of the Eath s inteio and the bounday of the volume wee known, then the poblem of detemining the Eath s gavitational potential field is solved by the volume integal of eqn [4]. In eality, of couse, we do not have access to this infomation, at least not the density, with sufficient detail. (The Peliminay Refeence Eath Model (PREM), of Dziewonsky and Andeson (98), still in use today by geophysicists, epesents a good pofile of Eath s adial density, but does not attempt to model in detail the lateal density heteogeneities.) In this section, we see how the poblem of detemining the exteio gavitational potential can be solved in tems of suface integals, thus making exclusive use of accessible measuements on the suface Geen s Identities Fomally, eqn [24] epesents a patial diffeential equation fo V. Solving this equation is the essence of the detemination of the Eath s extenal gavitational potential though potential theoy. Like any diffeential equation, a complete solution is obtained only with the application of bounday conditions, that is, imposing values on the solution that it must assume at a bounday of the egion in which it is valid. In ou case, the bounday is the Eath s suface and the exteio space is whee eqn [24] holds (the atmosphee and othe celestial bodies ae neglected fo the moment). In ode to study the solutions to these BVPs (to show that solutions exist and ae

9 Potential Theoy and Static Gavity Field of the Eath 9 unique), we take advantage of some vey impotant theoems and identities. It is noted that only a athe elementay intoduction to BVPs is offeed hee with no attempt to addess the much lage field of solutions to patial diffeential equations. The fist, seminal esult is Gauss divegence theoem (analogous to Stokes theoem, eqn [9]), Z? f dv ¼ f n ds ½26Š v whee f is an abitay (diffeentiable) vecto function and f n ¼ n? f is the component of f along the outwad unit nomal vecto, n (see Figue 5). The suface, s, encloses the volume, v. Equation [26] says that the sum of how much f changes thoughout the volume, that is, the net effect, ultimately, is equivalent to the sum of its values pojected othogonally with espect to the suface. Conceptually, a volume integal thus can be eplaced by a suface integal, which is impotant since the gavitational potential is due to a volume density distibution that we do not know, but we do have access to gavitational quantities on a suface by way of measuements. Equation [26] applies to geneal vecto functions that have continuous fist deivatives. In paticula, let U and V be two scala functions having continuous second deivatives, and conside the vecto function f ¼ UÑV. Then, since n? Ñ ¼ q=qn, and? ðuv Þ ¼U? V þ U 2 V ½27Š we can apply Gauss divegence theoem to get Geen s fist identity, Z v U? V þ U 2 V dv ¼ s s U qv qn ds ½28Š Intechanging the oles of U and V in eqn [28], one obtains a simila fomula, which, when subtacted fom eqn [28], yields Geen s second identity, Z v U 2 V V 2 U dv ¼ s U qv qn V qu ds qn ½29Š This is valid fo any U and V with continuous deivatives up to second ode. Now let U ¼ =,, whee, is the usual distance between an integation point and an evaluation point. And, suppose that the volume, v, is the space exteio to the Eath (i.e., Gauss divegence theoem applies to any volume, not just volumes containing a mass distibution). With efeence to Figue 6, conside the evaluation point, P, to be inside the volume (fee Figue 6 identity. space) that is bounded by the suface, s; P is thus outside the Eath s suface. Let V be a solution to eqn [24], that is, it is the gavitational potential of the Eath. Fom the volume, v, exclude the volume bounded by a small sphee,, centeed at P. This sphee becomes pat of the suface that bounds the volume, v. Then, since U, by ou definition, is a point mass potential, 2 U ¼ 0 eveywhee in v (which excludes the inteio of the small sphee aound P); and, the second identity [29] gives qv s, qn V q ds qn, qv þ, qn V q d ¼ 0 qn, ½30Š The unit vecto, n, epesents the pependicula pointing away fom v. On the small sphee, n is opposite in diection to, ¼, and the second integal becomes n Eath qv q þ V q q d ¼ V d ¼ qv q 2 d qv q d 4 V ½3Š whee d ¼ 2 d, is the solid angle, 4, and V is an aveage value of V on. Now, in the limit as the adius of the small sphee shinks to zeo, the ighthand side of eqn [3] appoaches 0 4V P. Hence, eqn [30] becomes (Kellogg, 953, p. 29) V P ¼ 4 s ds Geomety fo special case of Geen s thid qv, qn V q qn l s σ n, ds l P dv ½32Š

10 20 Potential Theoy and Static Gavity Field of the Eath with n pointing down (away fom the space outside the Eath). This is a special case of Geen s thid identity. A change in sign of the ight-hand side tansfoms n to a nomal unit vecto pointing into v, away fom the masses, which confoms moe to an Eath-centeed coodinate system. The ight-hand side of eqn [32] is the sum of single- and double-laye potentials and thus shows that evey hamonic function (i.e., a function that satisfies Laplace equation) can be witten as a Newtonian potential. Equation [32] is also a solution to a BVP; in this case, the bounday values ae independent values of V and of its nomal deivative, both on s (Cauchy poblem). Below and in Section , we encounte anothe BVP in which the potential and its nomal deivative ae given in a specified linea combination on s. Using a simila pocedue and with some exta cae, it can be shown (see also Couant and Hilbet, 962, vol. II, p. 256 (footnote)) that if P is on the suface, then V P ¼ 2 s qv, qn V q qn, ds ½33Š whee n points into the masses. Compaing this to eqn [32], we see that V is discontinuous as one appoaches the suface; this is due to the doublelaye pat (see eqn [8]). Equation [32] demonstates that a solution to a paticula BVP exists. Specifically, we ae able to measue the potential (up to a constant) and its deivatives (the gavitational acceleation) on the suface and thus have a fomula to compute the potential anywhee in exteio space, povided we also know the suface, s. Othe BVPs also have solutions unde appopiate conditions; a discussion of existence theoems is beyond the pesent scope and may be found in (Kellogg, 953). Equation [33] has deep geodetic significance. One objective of geodesy is to detemine the shape of the Eath s suface. If we have measuements of gavitational quantities on the Eath s suface, then conceptually we ae able to detemine its shape fom eqn [33], whee it would be the only unknown quantity. This is the basis behind the wok of Molodensky et al. (962), to which we etun biefly at the end of this section Uniqueness Theoems Often the existence of a solution is poved simply by finding one (as illustated above). Whethe such as solution is the only one depends on a coesponding uniqueness theoem. That is, we wish to know if a cetain set of bounday values will yield just one potential in space. Befoe consideing such theoems, we classify the BVPs that ae typically encounteed when detemining the exteio potential fom measuements on a bounday. In all cases, it is an exteio BVP; that is, the gavitational potential, V, is hamonic ( 2 V ¼ 0) in the space exteio to a closed suface that contains all the masses. The exteio space thus contains infinity. Inteio BVPs can be constucted, as well, but ae not applicable to ou objectives. Diichlet poblem (o, BVP of the fist kind ). Solve fo V in the exteio space, given its values eveywhee on the bounday. Neumann poblem (o, BVP of the second kind ). Solve fo V in the exteio space, given values of its nomal deivative eveywhee on the bounday. Robin poblem (mixed BVP, o BVP of the thid kind ). Solve fo V in the exteio space, given a linea combination of it and its nomal deivative on the bounday. Using Geen s identities, we pove the following theoems fo these exteio poblems; simila esults hold fo the inteio poblems. Theoem. If V is hamonic (hence continuously diffeentiable) in a closed egion, v, and if V vanishes eveywhee on the bounday, s, then V also vanishes eveywhee in the egion, v. Poof. Since V ¼ 0 on s, Geen s fist identity (eqn [28]) with U ¼ V gives Z ðv Þ 2 dv ¼ s V qv qn ds ¼ 0 ½34Š The integal on the left side is theefoe always zeo, and the integand is always non-negative. Hence, ÑV ¼ 0 eveywhee in v, which implies that V ¼ constant in v. Since V is continuous in v and V ¼ 0ons, that constant must be zeo; and so V ¼ 0inv. This theoem solves the Diichlet poblem fo the tivial case of zeo bounday values and it enables the following uniqueness theoem fo the geneal Diichlet poblem. Theoem 2 (Stokes theoem). If V is hamonic (hence continuously diffeentiable) in a closed egion, v, then V is uniquely detemined in v by its values on the bounday, s. Poof. Suppose the detemination is not unique: that is, suppose thee ae V and V 2, both hamonic

11 Potential Theoy and Static Gavity Field of the Eath 2 in v and both having the same bounday values on s. Then the function V ¼ V 2 V is hamonic in v with all bounday values equal to zeo. Hence, by Theoem, V 2 V ¼ 0 identically in v, ov 2 ¼ V eveywhee, which implies that any detemination is unique based on the bounday values. Theoem 3. IfV is hamonic (hence continuously diffeentiable) in the exteio egion, v, with closed bounday, s, then V is uniquely detemined by the values of its nomal deivative on s. Poof. We begin with Geen s fist identity, eqn [28], as in the poof of Theoem to show that if the nomal deivative vanishes eveywhee on s, then V is a constant in v. Now, suppose thee ae two hamonic functions in v: V and V 2, with the same nomal deivative values on s. Then the nomal deivative values of thei diffeence ae zeo; and, by the above demonstation, V ¼ V 2 V ¼ constant in v. Since V is a Newtonian potential in the exteio space, that constant is zeo, since by eqn [0], lim,! V ¼ 0. Thus, V 2 ¼ V, and the bounday values detemine the potential uniquely. This is a uniqueness theoem fo the exteio Neumann BVP. Solutions to the inteio poblem ae unique only up to an abitay constant. Theoem 4. Suppose V is hamonic (hence continuously diffeentiable) in the closed egion, v, with bounday, s; and, suppose the bounday values ae given by g ¼ V j s þ qv ½35Š qn s Then V is uniquely detemined by these values if = > 0. Poof. Suppose thee ae two hamonic functions, V and V 2, with the same bounday values, g, ons. Then V ¼ V 2 V is hamonic with bounday values ðv 2 V Þj s þ qv 2 qn qv qn ¼ 0 ½36Š s With U ¼ V ¼ V 2 V, Geen s fist identity, eqn [28], gives Z ððv 2 V ÞÞ 2 dv ¼ ðv 2 V Þ ð V 2 V Þds v Then Z ððv 2 V ÞÞ 2 dv þ s s ½37Š ðv 2 V Þ 2 ds ¼ 0 ½38Š Since = > 0, eqn [38] implies that ðv 2 V Þ ¼ 0 in v; and V 2 V ¼ 0ons. Hence V 2 V ¼ constant in v; and V 2 ¼ V on s. By the continuity of V and V 2, the constant must be zeo, and the uniqueness is poved. The solution to the Robin poblem is unique only in cetain cases. The most famous poblem in physical geodesy is the detemination of the distubing potential, T, fom gavity anomalies, g, on the geoid (Section ). Suppose T is hamonic outside the geoid; the second of eqns [5] povides an appoximate fom of bounday condition, showing that this is a type of Robin poblem. We find that ¼ 2=, and, ecalling that when v is the exteio space the unit vecto n points inwad towad the masses, that is, q=qn ¼ q=q, weget ¼. Thus, the condition in Theoem 4 on = is not fulfilled and the uniqueness is not guaanteed. In fact, we will see that the solution obtained fo the spheical bounday is abitay with espect to the coodinate oigin (Section ) Solutions by Integal Equation Geen s identities show how a solution to Laplace s equation can be tansfomed fom a volume integal, that is, an integal of souce points, to a suface integal of BVPs, as demonstated by eqn [32]. The uniqueness theoems fo the BVPs suggest that the potential due to a volume density distibution can also be epesented as due to a genealized density laye on the bounding suface, as long as the esult is hamonic in exteio space, satisfies the bounday conditions, and is egula at infinity like a Newtonian potential. Molodensky et al. (962) supposed that the distubing potential is expessible as T ¼ ds ½39Š, whee is a suface density to be solved using the bounday condition. With the spheical appoximation fo the gavity anomaly, eqn [5], one aives at the following integal equation 2 cos s q q s, þ 2, ds ¼ g ½40Š The fist tem accounts fo the discontinuity at the suface of the deivative of the potential of a density laye, whee is the deflection angle between the nomal to the suface and the diection of the (adial) deivative (Heiskanen and Moitz, 967, p. 6; Günte, 967, p. 69). This Fedholm integal equation of the second kind can be simplified with futhe appoximations, and, a solution fo the density,, ultimately

12 22 Potential Theoy and Static Gavity Field of the Eath leads to the solution fo the distubing potential (Moitz, 980). Othe foms of the initial epesentation have also been investigated, whee Geen s functions othe than =, lead to simplifications of the integal equation (e.g., Petovskaya, 979). Nevetheless, most pactical solutions ely on appoximations, such as the spheical appoximation fo the bounday condition, and even the fomulated solutions ae not stictly guaanteed to convege to the tue solutions (Moitz, 980). Futhe teatments of the BVP in a geodetic/mathematical setting may be found in the volume edited by Sansò and Rummel (997). In the next section, we conside solutions fo T as suface integals of bounday values with appopiate Geen s functions. In othe wods, the bounday values (whethe of the fist, second, o thid kind) may be thought of as souces, and the consequent potential is again the sum (integal) of the poduct of a bounday value and an appopiate Geen s function (i.e., a function that depends on both the souce point and the computation point in some fom of invese distance in accodance with Newtonian potential theoy). Such solutions ae eadily obtained if the bounday is a sphee. appoximation based on an ellipsoid of evolution is biefly examined in Section fo the nomal potential. In spheical pola coodinates, ð; ; Þ,the Laplacian opeato is given by Hobson (965, p. 9) 2 ¼ q 2 2 q þ q q q 2 sin q sin q q þ q 2 2 sin 2 q 2 ½4Š A solution to 2 V ¼ 0 in the space outside a sphee of adius, R, with cente at the coodinate oigin can be found by the method of sepaation of vaiables, wheeby one postulates the fom of the solution, V,as V ð; ; Þ ¼ f ðþg ðþh ðþ ½42Š Substituting this and the Laplacian above into eqn [24], the multivaiate patial diffeential equation sepaates into thee univaiate odinay diffeential equations (Hobson, 965, p. 9; Mose and Feshbach, 953, p. 264). Thei solutions ae well-known functions, fo example, o V ð; ; Þ ¼ P nm ðcos Þ sin m nþ ½43aŠ Solutions to the Spheical BVP V ð; ; Þ ¼ P nm ðcos Þcos m nþ ½43bŠ This section develops two types of solutions to standad BVPs when the bounday is a sphee: the spheical hamonic seies and an integal with a Geen s function. All thee types of poblems ae solved, but emphasis is put on the thid BVP since gavity anomalies ae the most pevalent bounday values (on land, at least). In addition, it is shown how the Geen s function integals can be inveted to obtain, fo example, gavity anomalies fom values of the potential, now consideed as bounday values. Not all possible invese elationships ae given, but it should be clea at the end that, in pinciple, vitually any gavitational quantity can be obtained in space fom any othe quantity on the spheical bounday Spheical Hamonics and Geen s Functions Fo simple boundaies, Laplace s equation [24] is elatively easy to solve povided thee is an appopiate coodinate system. Fo the Eath, the solutions commonlyelyonappoximatingtheboundaybyasphee. This case is descibed in detail and a moe accuate whee P nm ðþis t the associated Legende function of the fist kind and n, m ae integes such that 0 m n, n 0. Othe solutions ae also possible (e.g., gðþ ¼ e a ðaprþ and hðþ ¼ n ), but only eqns [43] ae consistent with the poblem at hand: to find a eal-valued Newtonian potential fo the exteio space of the Eath (egula at infinity and 2-peiodic in longitude). The geneal solution is a linea combination of solutions of the foms given by eqns [43] fo all possible integes, n and m, and can be witten compactly as V ð; ; Þ ¼ X n¼0 X n m¼ n R nþ v nm Y nm ð; Þ ½44Š whee the Y nm ae suface spheical hamonic functions defined as ( cos m; m 0 Y nm ð; Þ ¼ P nm j j ðcos Þ sinjmj; m < 0 ½45Š and P nm is a nomalization of P nm so that the othogonality of the spheical hamonics is simply

13 Potential Theoy and Static Gavity Field of the Eath 23 4 ( ¼ Y nm ð; ÞY n9m9 ð; Þd ; n ¼ n9 and m ¼ m9 0; n 6¼ n9 o m 6¼ m9 ½46Š and whee ¼ fð; Þj0 ; 0 2g epesents the unit sphee, with d ¼ sin d d. Fo a complete mathematical teatment of spheical hamonics, one may efe to Mülle (966). The bounding spheical adius, R, is intoduced so that all the constant coefficients, v nm, also known as Stokes constants, have identical units of measue. Applying the othogonality to the geneal solution [44], these coefficients can be detemined if the function, V, is known on the bounding sphee (bounday condition): v nm ¼ V ð; ; RÞY nm ð; Þd ½47Š 4 Equation [44] is known as a spheical hamonic expansion of V and with eqn [47] it epesents a solution to the Diichlet BVP if the bounday is a sphee. The solution thus exists and is unique in the sense that these bounday values geneate no othe potential. We will, howeve, find anothe equivalent fom of the solution. In a moe fomal mathematical setting, the solution [46] is an infinite linea combination of othogonal basis functions (eigenfunctions) and the coefficients, v nm,ae the coesponding eigenvalues. One may also intepet the set of coefficients as the spectum (Legende spectum) of the potential on the sphee of adius, R (analogous to the Fouie spectum of a function on the plane o line). The integes, n, m, coespond to wave numbes, and ae called degee (n) and ode (m), espectively. The spheical hamonics ae futhe classified as zonal (m ¼ 0), meaning that the zeos of Y n0 divide the sphee into latitudinal zones; sectoial (m ¼ n), whee the zeos of Y nn divide the sphee into longitudinal sectos; and tesseal (the zeos of Y nm tessellate the sphee) (Figue 7). While the spheical hamonic seies has its advantages in global epesentations and spectal intepetations of the field, a Geen s function epesentation povides a moe local chaacteization of the field. Changing a bounday value anywhee on the globe changes all coefficients, v nm, accoding to eqn [47], which poses both a numeical challenge in applications, as well as in keeping a standad model up to date. Howeve, since the Geen s function essentially depends on the invese distance (o highe powe theeof), a emote change in bounday value geneally does not appeciably affect the local detemination of the field. When the bounday is a sphee, the solutions to the BVPs using a Geen s function ae easily deived fom the spheical hamonic seies epesentation. Moeove, it is possible to deive additional integal elationships (with appopiate Geen s functions) among all the deivatives of the potential. To fomalize and simultaneously simplify these deivations, conside hamonic functions, f and h, whee h depends only on and, and function g, defined on the sphee of adius, R. Thus let f ð; ; Þ ¼ X n¼0 X n m¼ n R nþ f nm Y nm ð; Þ ½48Š hð; Þ ¼ X ð2n þ Þ R nþ h n P n ðcos Þ ½49Š n¼0 gð; ; RÞ ¼ X n¼0 X n m¼ n g nm Y nm ð; Þ ½50Š p whee P n ðcosþ ¼ P n0 ðcosþ= ffiffiffiffiffiffiffiffiffiffiffiffiffi 2n þ is the nth degee Legende polynomial. Constants f nm and g nm ae the espective hamonic coefficients of f and g when these functions ae esticted to the sphee of adius, R. Then, using the decomposition fomula fo Legende polynomials, P 80 (cos θ) P 88 (cos θ) cos 8λ P 87 (cos θ) cos 7λ Figue 7 Examples of zonal, sectoial, and tesseal hamonics on the sphee.

14 24 Potential Theoy and Static Gavity Field of the Eath P n ðcos cþ ¼ 2n þ whee X n m¼ n Y nm ð; ÞY nm ð9; 9Þ ½5Š cos c ¼ cos cos 9 þ sin sin 9cosð 9Þ ½52Š it is easy to pove the following theoem. Theoem (convolution theoem in spectal analysis on the sphee). f ð; ; Þ ¼ gð9; 9; RÞhðc; Þd 4 ½53Š if and only if f nm ¼ g nm h n Hee, and in the following, d ¼ sin 9d9d9. The angle, c, is the distance on the unit sphee between points ð; Þ and ð9; 9Þ. Poof. The fowad statement [53] follows diectly by substituting eqns [5] and [49] into the fist equation [53], togethe with the spheical hamonic expansion [50] fo g. A compaison with the spheical hamonic expansion fo f yields the esult. All steps in this poof ae evesible, and so the evese statement also holds. Conside now f to be the potential, V, outsidethe sphee of adius, R, and its estiction to the sphee to be the function, g : gð; Þ ¼ V ð; ; RÞ. Then, clealy, h n ¼, fo all n;bythetheoemabove,wehave V ð; ; Þ ¼ 4 V ð9; 9; RÞUðc; Þd ½54Š whee Uðc; Þ ¼ X ð2n þ Þ R nþ P n ðcos cþ ½55Š n¼0 Fo the distance p, ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ R 2 2R cos c ½56Š between points ð; ; Þ and ð9; 9; RÞ, with R, the identity (the Coulomb expansion; Cushing, 975, p. 55),, ¼ X R nþ P n ðcos cþ ½57Š R n¼0 yields, afte some aithmetic (based on taking the deivative on both sides with espect to ), Uðc; Þ ¼ R2 ð R 2 Þ, 3 ½58Š Solutions [44] and [54] to the Diichlet BVP fo a spheical bounday ae identical (in view of the convolution theoem [53]). The integal in [54] is known as the Poisson integal and the function U is the coesponding Geen s function, also known as Poisson s kenel. Fo convenience, one sepaates Eath s gavitational potential into a efeence potential (Section ) and the distubing potential, T. The distubing potential is hamonic in fee space and satisfies the Poisson integal if the bounday is a sphee. In defeence to physical geodesy whee elationships between the distubing potential and its deivatives ae outinely applied, the following deivations ae developed in tems of T, but hold equally fo any exteio Newtonian potential. Let Tð; ; Þ ¼ GM R X X n n¼0 m¼ n R nþ C nm Y nm ; ð Þ ½59Š whee M is the total mass (including the atmosphee) of the Eath and the dc nm ae unitless hamonic coefficients, being also the diffeence between coefficients fo the total and efeence gavitational potentials (Section ). The coefficient, C 00,is zeo unde the assumption that the efeence field accounts completely fo the cental pat of the total field. Also note that these coefficients specifically efe to the sphee of adius, R. The gavity distubance is defined (in spheical appoximation) to be the negative adial deivative of T, the fist of eqns [5]. Fom eqn [59], we have gð; ; Þ ¼ q q ð ¼ GM X R 2 T ; ; Þ X n R nþ2 ðn þ n¼0 m¼ n ÞC nm Y nm ð; Þ ½60Š and, applying the convolution theoem [53], we obtain Tð; ; Þ ¼ R gð9; 9; RÞHðc; Þd ½6Š 4 whee with g nm ¼ ðn þ ÞdC nm =R and f nm ¼ dc nm,we have h n ¼ f nm =g nm ¼ R= ðn þ Þ, and hence (taking cae to keep the Geen s function unitless) Hðc; Þ ¼ X n¼0 2n þ n þ R nþ P n cos c ð Þ ½62Š The integal in [6] is known as the Hotine integal, the Geen s function, H, is called the Hotine kenel, and with a deivation based on equation [57], it is given by (Hotine 969, p. 3) Hðc; Þ ¼ 2R, ln þ, 2R sin 2 ½63Š c=2

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