Milos Pick Geophysical Institute of the Czech Academy of Sciences, Czech Republic.

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1 GRAVIMETRY Mils Pick Gephysical Institute f the Czech Academy f Sciences, Czech Republic. Keywds:Gavimety, gavity ptential, gavity fce, geid, gavity field, eductin f gavity measuements, efeence suface, equiptential suface, deflectin f the vetical, cuvatue f the level suface, quasigeid, heights, leveling. Cntents 1. Intductin. Newtn s they f ptential 3. Ptential f sme simple fmatins, appximate in shape t the figue f the Eath 4. Ptential f the centifugal fce, gavity fce 5. Buns fmula, Claiaut pblem 6. Gavitatinal cnstant 7. Geid 8. Chaacteistics f the Eath s gavity field 9. Gavity measuements 10. Reductin f gavity measuements, Faye anmaly, Bugue anmaly 11. Othe types f anmalies 1. Refeence suface 13. Pjectin methd f tansfming ne tatinal ellipsid int anthe ne 14. Equiptential (level) suface 15. Deflectins f the vetical, Laplace equatin 16. The cuvatue f level suface (the cnvegence f level sufaces) 17. Bunday cnditin f the gavity distubing ptential 18. Stkes fmula 19. Slutin f the inne Stkes pblem f a sphee 0. Vening-Meinesz fmulae 1.Mldensky quasigeid. Heights abve the sea level 3.Genealizatin f the system f nmal heights 4. Ellipsidal heights, astnmic leveling 5. Deteminatin f the tignmetic heights 6. Ast-gavimetic leveling Glssay Bibligaphy Bigaphical Sketch Summay Afte a bief suvey f the Newtn s they f the ptential and f the gavitatinal and gavity fces, a discussin f facts f the main pblems f the gavimety fllws. It is the questin f the they f the geid, actually quasigeid, level sufaces and deviatins fm the cnsequent deflectins f the vetical. The basic methds f the measuements f the gavity fces ae examined as well as thei eductins. Then the

2 applicatins f the gavimety in the pactice ae intduced, f example the heights cmputatin and diffeent methds f leveling. 1. Intductin Gavimety as an independent scientific banch began t evlve nly at the end f the 19th centuy. Hweve, its ts g back t the times f the celebated pinees f classical mechanics, G. Galile and I. Newtn (16th and 18th centuies). A numbe f utstanding 18th and 19th centuy scientists (e.g. A.C. Claiaut, G. Stkes, H. Buns, F.R. Helmet) wked n vaius fundamental gavimetic pblems cncened in paticula with the applicatin f the they f ptential t eseach int the Eath s bdy. Nevetheless, the tem gavimety (fm Latin gavis=heavy and the Geek metein=t measue) nly became established in science in the 0th centuy. The name des nt fully eflect the natue f the subject. A gavimetician nt nly must measue gavity, seach f new methds f measuements and cnstuct gavimetic instuments, but als has t slve many fundamental gavimetic pblems theetically and pactically. Amng the mst imptant pblems f cntempay gavimety it is the study f the figue and the dimensins f the Eath s bdy (the geid and its extenal gavity field). Teatment f the imptant pblem f the equilibium f the Eath s cust (the pblem f isstasy) was anthe significant task f gavimeticians. Hweve, the fist bsevatins f the atificial satellites aleady shwed that isstasy des nt exist. The Eath s cust is als acted upn by attactive fces due t the masses f celestial bdies, f which the mn and the sun have the lagest effects. These distubing fces esult in peidical mvements in the Eath s cust called the tides f the slid cust. Similaly, the same fces act n the seas and ceans, ceating the sea tides (geneally knwn as tides and ebbs), and n the gaseus shell f the Eath causing atmspheic tides, caused mstly by the influence f the themal adiatin f the Sun. Of geat ecnmic imptance is the pblem f applying gavimety t suveying and investigating depsits f useful mineals and aw mateials. Fm this list f sme f me imptant gavimetic pblems it can be seen that gavimety is clsely elated t gedesy and gelgy. As cmpaed with gedesy which uses gemetic methds t investigate the figue f the Eath, gavimety emplys physical methds f the same pblem. Gelgy uses applied gavimety in the investigatin f the gelgical stuctue f the uppe pats f the Eath s cust, and gavimetic methds ae emplyed in pspecting. Gavimety is als intimately cnnected with physics, mathematics, astnmy and the elated sciences. The fist athe inaccuate measuements f the acceleatin f gavity wee made by Galile Galilei ( ), wh bseved the path f feely falling bdies. Accding t the law f fee fall, which he discveed in abut 1590, the length f the path f a bdy falling fm its initial est psitin in the fist secnd is equal t half f the value f acceleatin f gavity at the pint f bsevatin.

3 The funde f the science f the attactin f masses (gavitatin) is Si Isaac Newtn ( ), wh was the fist t publish the law f univesal gavitatin in Apat f these tw geat men, many the imptant scientists have a place in the histy f gavimety. A cntempay f Newtn, Ch. Huygens ( ) dealt with the pblem f the Eath s shape as well. He als elabated the they f the physical pendulum in The disageements between the theetical cnsideatins f Newtn and Huygens wee explained by A.C. Claiaut ( ), wh als shwed hw the flattening f the Eath culd be cmputed fm gavimetic bsevatins. C. Maclauin, P.S. Laplace, K. Jacbi, A.M. Lyapunv, A.M. Legende, G. Stkes, G. Geen, F.R. Helmet, J. Buns, and many thes dealt with vaius theetical pblems f gavimety in the 17th, 18th, 19th and 0th centuies. Stkes study ( ) was especially imptant. He indicated that the figue f the Eath (the geid) culd be deived if the distibutin f the acceleatin f gavity was knwn ve the whle suface f the Eath. His theetical esults have, in fact, nly been explited in the 1950s as sufficient gavity data fm the whle wld became available. Many scientists f the 19th and 0th centuy dealt with the figue f the geid: L. Tanni, M.S. Mldenskii, P. Pizzetti, and many thes. In the 17th and 18th centuies, Jacques, Jean and Daniel Benulli elabated the they f the physical pendulum. Vaius mdificatins f evesible pendulums wee used f measuements f the acceleatin f gavity. The fist pendulum measuements f the acceleatin f gavity n seas and ceans wee caied ut in 198 by F.A. Meinesz.. Newtn s They f Ptential Attactive fces called gavitatinal fces peate between the Eath and an abitay bdy. They cause evey bdy t have weight, and if a bdy has an pptunity t mve, it falls twads the Eath with a given acceleatin. These fces gven the mtins f celestial bdies. Fm the deed mvements f the planets aund the sun, Keple deived his thee laws. Fm the latte, Newtn in tun fmulated the law f univesal gavitatin. This law is as fllws: The fce with which tw bdies f masses m 1 and m attact each the is diectly pptinal t the pduct f bth masses and invesely pptinal t the squae f the distance between them: mm 1. F = G, (.1) whee the gavitatinal cnstant G = kg m s.

4 Theefe, the bdies fm gavitatinal field in space which can be descibed by means f the intensity f the field by means f the gavitatinal ptential. The intensity f the gavitatinal field is undestd t be the fce which acts n a unit mass at a given pint. The gavitatinal ptential at a pint is the amunt f wk equied t be dne t mve a unit mass fm infinity (pint f ze ptential) t that pint. In this case the ptential is defined as a fce functin in mechanics. It shuld nt be mistaken f the ptential enegy. The gemetic lcus f pints with the same ptential is an equiptential suface. If a celestial bdy is sufficiently small elative t the distance fm the Eath, we can cnside the gavitatinal effect f this bdy as effect f a pint mass. Let us have a set f pint masses. The cdinates f the ple P ae ( ξ, ηζ, ). The cdinates f the vaiable pint P i ae ( xi, yi, z i). The gavitatinal effect f this set is given as a sum effects f individual mass pints detemined by Eq. (.1). We usually put m P = 1. F G = 1 we have n m F = i= 1 Ai 3 i. i (.) The fllwing expessin hlds f the cmpnents f fce F in the diectin f the cdinate axes: n n n Ai i i Ai i i Ai i i i= 1 i= 1 i= 1 (.3) X = m ( x ξ ), Y = m ( y η), Z = m ( z ζ). The cmpnents X, Y, Z may be cnsideed as patial deivatives f the ptential f the pint masses, U, i.e., X = U ξ, Y = U η, Z = U ζ. The ptential U is detemined fm the equatin n i= 1 1 i U =. m. (.4) Ai The ptential U is called vlume ptential. Eq.(.4) can be witten in the fm U dτ = ρ, (.5) τ whee ρ is the density f the bdy and τ is the egin f integatin. The fist deivatives f the functin U can be cnsideed in the fm

5 U x ξ ρ = ρ dτ = 3 cs(, x ) dτ ξ. (.6) τ τ The same applies t the cmpnents Y and Z. If the density ρ is capable f integatin and limited, then the vlume ptential U and its deivatives ae cntinuus functins in the whle egin and they ae limited, t. The secnd deivatives f the vlume ptential U U U, Δ U = + + ξ η ζ whee Δ Ui = 4 πρ( P), P within τ, (Pissn equatin), Δ U = 0, P utside τ, (Laplace equatin), and e (.7) ΔU Δ U = 4 πρ. (.8) e i i Futhe (Gauss fmula) 1 U M = d, 4π (.9) n whee M is the ttal mass f the bdy, limited by the suface, n is the ute nmal t the suface. At imppe (infinitely distant) pints lim U = M. The functin U is egula at infinity. Fm the they f the electmagnetic field was adpted the cncept f the suface ptentials, the ptential f the single laye and the ptential f the duble laye. The ptential f the duble laye W is defined by the equatins csϕ 1 W = d = d ν ν ( ), (.10) n whee n is the ute nmal t the suface, ϕ is the angle between the adius vect and the extenal nmal at the vaiable pint.

6 The ptential f the duble laye is a discntinuus functin, if the ple P passes thugh the suface. If the ple P appaches in infinity, fm inside utside, we btain the limiting values W and W espectively i e W ( P) = W( P ) + πν ( P ), i W ( P) = W( P ) πν ( P ), e csϕ W( P) = ν d, (.11) whee ϕ is the angle between the extenal nmal at pint A t the suface and adius vect, = PA. Ptential f a single laye V is defined by the equatin μ V = d (.1) f the density μ. We can als wite the expessins f the nmal deivatives f the single laye ptential. If the pint P appximates the pint P n the suface fm inside utside, we get Vi = V ni n + πμ( P), Ve = V ne n πμ( P), V csψ = ν n d, ο (.13) whee the angle ψ ο is the angle between the ute nmal n at the pint P n the suface and the adius vect = P A. The cnsideable imptance f equatins (.10) - (.13) was pinted ut by Fedhlm. They can be applied in slving the bunday pblems f the ptential they by means f integal equatins. Hithet, we have assumed that the vlume and suface ptentials ae elated t bdies and sufaces which ae statinay. If the bdy is tating aund a fixed axis f tatin with a cnstant angula velcity ω, then the fce F = ω ρ and the ptential 1 U = ω ρ appea.

7 The gavitatinal ptential V is due t the actin f the gavitatinal fces, and the ptential U is due t the centifugal fce. The sum f these ptentials W = U + V (.14) is ptential f the gavity, the gavity ptential. 3. Ptential f Sme Simple Fmatins, Appximate in Shape t the Figue f the Eath Let us cnside a sphee, cented at O (0, 0, 0), with adius R. Pint P is utside the sphee, its cdinate ae P( ρ, 0, 0). F the extenal ple, the gavitatinal ptential is given by the elatin 3 4 R Ve = πgκ, ρ R, (3.1) 3 ρ and f intenal pint by the equatin Vi = πgκ(3 R ρ ), ρ R. (3.) 3 Equiptential tatinal ellipsid is usually put f the mathematical mdel f the Eath. Accding t Pizzetti, its gavity ptential, that is the sum f the gavitatinal ptential and ptential f the centifugal fce, is given by the elatin b b W = ( α + β) A α{[ B B C+ C ] }, + b C a a whee α, β ae cnstants, A, BC, ae Jacbi s integals. (3.3) The gavity γ f this bdy can be detemined by taking the deivative f this equatin in the diectin f the ute nmal: W W γ = ( ) + ( ). h u h s u s (3.4) The influence f the secnd tem can be mstly neglected. Thus, we can wite W γ, h u u b α β γ = [( B B)cs u+ C ( C C ) cs u], (3.5) Q a a b Q whee u is the educed latitude.

8 Oiginally, accding t the ecmmendatin f the Intenatinal Unin f Gedesy and Gephysics f Luzen 1967, the physical and gemetical ppeties f the level tatinal ellipsid wee fully detemined by fu paametes: maj semiaxis a, the pduct f the Eath mass M and gavitatinal cnstant G, Stke s cefficient J and the angula velcity f the Eath s tatin ω. Late, accding t Bwas ppsal, instead f the value a, the mean value f the ptential f the wld seas W was accepted TO ACCESS ALL THE 46 PAGES OF THIS CHAPTER, Visit: Bibligaphy M.Pick, J.Pícha, V.Vyskčil (1973) They f the Eath Gavity Field. Elsevie. [Textbk f the mden gavimety with paticula efeence t Mldensky s they.] M.S.Mldenskij (1945) Osnvnyje vpsy gedězičeskj gavimetii. Tudy CNIIGAiK, Vyp. 4, Mscw. [Basic pblems f gedetic gavimety. The English vesin by the Office f Technical Sevices U.S. Washingtn D.C.] M.S.Mldenskij, V.F.Eemeev, M.I.Yukina (---- ) Methds f Study f the Extenal Gavitatinal Field and F???. The suvey f the mden Mldensky s schl. P.Vaníček, E.Kakiwsky (1986) Gedesy the cncepts. Elsevie Amstedam. [ Mden textbk f the gavimety. A gd suvey f the whle mden gedesy.] W.A.Heiskanen, H.Mitz (1967) Physical gedesy. Feeman. [Classic mngaph f the physical gedesy. The mden ways f slutin ae hinted nly.] Bigaphical Sketch Milš Pick, bn in Luže, Czech Republic. He is a pfess f Highe Gedesy, Czech Technical Univesity, Pague and cnsultant, Gephysical Institute, Czech Academy f Sciences, Pague : student f gedesy, Czech Technical Univesity, Pague : eseach fellw, Militay Tpgaphic Institute, dpt. f mathematical catgaphy, Dbuška : eseachwke, Gephysical Institute, Czech Academy f Sciences, Pague degee CSc. (candidate f science PhD equivalent), thesis Investigatin f the Eath Gavity Field in Muntains 1963 degee DSc. (dct f sciences), thesis Geid and Oute Eath Gavity Field :diect, Gephysical Institute, Czechslvak Academy f Sciences, Pague :seni eseachwke, Gephysical Institute, Czechslvak Academy f Sciences, Pague 1967 invited seies f lectues f Newtnian Ptential They, Chales Univesity, Pague 1991 cnsultant, Gephysical Institute, Czech Academy f Sciences, Pague 1994 pfess f Highe Gedesy, Czech Technical Univesity, Pague

9 C-auth f the mngaph They f the Eath Gavity Field (c-auths J. Pícha, V. Vyskčil); Elsevie, Amstedam 1973, als in Czech as a text-bk.auth f the 1 th Chapte The Geid and tectnic fces f the multiauths mngaph Geid and its gephysical intepetatin, edited by P. Vaníček and N. T. Chistu, CRC Pess, Bca Ratn, Ann Ab, Lndn, Tky 1994.Auth f 7 undegaduate textbks, and cca 00 iginal scientific papes.included in sme Wh s Wh publicatins (f example Maquis Wh s Wh in the Wd ).Membe f the Editial Bad f the junal Studia Gephysica and Gedaetica.F many yeas was chaiman f the cmmissins f awading the title f candidate f sciences (Phd. equivalent) and dct f sciences in gedesy and gephysics.oganized tw intenatinal sympsia in the scpe f the Intenatinal Assciatin f Gedesy.Membe f the Special Study Gup Testing Aeas, headed the Subgup Testing Aeas Atificial Mdels in the scpe f IAG.