THE FUNDAMENTAL CRITERION FOR ESTIMATION OF THE REFERENCE ELLIPSOID ACCURACY UDC (045)=111. Bogoljub Marjanović

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1 FACTA UNIVERSITATIS Seies: Mechnics, Auomic Conol nd Rooics Vol. 6, N o 1, 007, pp THE FUNDAMENTAL CRITERION FOR ESTIMATION OF THE REFERENCE ELLIPSOID ACCURACY UDC (045)=111 Boolju Mjnović Hihe Technicl School, Sei, Kujevc, Kosovsk 8 E-mil: oc@k.c.yu Asc. The vlue of he semi-mjo xis, nd he vlue of he semi-mino xis e he eomeicl chceisics of he efeence ellipsoid. The ohe impon chceisic is he noml viy, which epesens he mniude of he dien of he viy poenil wih he ellipsoid s n equipoenil sufce. This ppe pesens he fundmenl cieion fo esimion of ccucy of he efeence ellipsoid, which sems fom he Newon's second lw. I hs een poved, y he nlysis of he moion of picle lon meidin of he efeence ellipsoid, h he leic sum of woks of viionl cion foce nd ceniful foce, excied y oionl ellipsoid ound is xis, equls o zeo. This is n indispensle em h he diecion of viy veco is eveywhee pependicul o he ellipsoid sufce s n equipoenil sufce. This is wh mkes his cieion elile nd enles he oinin of he quniive evluion of he ellipsoid's ccucy. Applyin his cieion fo esimion of ccucy of he ellipsoid, defined ccodin o he Geodeic Refeence Sysem 1980 (GRS80), i cn e concluded h he ellipsoid's semi-mino xis conins n eo of ou 599 m. Key wods: he efeence ellipsoid, semi-mino xis, viionl cion, GRS80 1. INTRODUCTION The efeence ellipsoid is n ellipsoid of evoluion, which would e oined y oionl n ellipse ound is mino xis. I is deemined y fou consns (Moiz, 1980), nd he IUGG hs chosen he followin ones - equoil dius, semi-mjo xis, GM - eocenic viionl consn, J - dynmic fom fco nd ω - nul velociy of he Eh The ohe eomeic consn, semi-mino xis, is oined y he followin fomul = 1 e, (1) Received Mch 15, 006

2 180 B. MARJANOVIĆ whee is e fis ecceniciy. Theefoe, he efeence ellipsoid is defined s pue eomeicl fiue. The coespondin efeence poenil (he Somilin-Pizzei efeence poenil) hs een deemined fom he condiion h efeence ellipsoid should e n equipoenil sufce of he efeence viy poenil (Heisknen nd Moiz, 1967). The quesion of he ellipsoid's ccucy is ised, nd pimily of he ccucy of he vlue of is semi-mino xis. I cn' e veified y suvey lon meidin, since i is hd o esime fo evey eodesic c, o wh exen i epesens he Eh s whole (Jeffeys, 1976). The elile cieion fo esimion of he efeence ellipsoid ccucy should e oined y physics. In his ppe i hs done y Newon's second lw.. THE DEFINITION OF THE FUNDAMENTAL CRITERION Le he picle of he uni mss move in he nonesisn medium lon ficionl idel smooh oove PQE (Fi. 1) lon he meidin of he efeence ellipsoid, wih he se iniil elive velociy v 0. Applyin he Newon's second lw, he moion of he picle is descied y he equion = + () whee is he veco of viionl cion, nd F w is he oove ecion. F w P z ψ ε F d 0 Q F d c 0 d T O N Fi. 1. Relive moion of he picle lon meidin Asolue cceleion is equl o veco sum of he nsfe cceleion, elive cceleion nd he Coiolis cceleion c, so he followin equion is oined E x nd his cn e wien in he fom + + = + F c w (3) o in he fom = + F (4) w c = + F + F + F (5) w c

3 The Fundmenl Cieion fo Esimion of he Refeence Ellipsoid Accucy 181 whee F is he nsfe ineil foce (ceniful foce) nd F c is he Coiolis ineil foce. Thee scl equions coespond o vecoil Eq. (5); one of hem defines he elive moion of he picle in he nenil diecion, whose uni veco is T. Considein h vecos F w nd F c, nd he veco sum of vecos nd F, e ll pependicul o he nenil diecion, y pojecin Eq. (5) ono h diecion, one oins h he elive nenil cceleion T equls o zeo, h is dv = 0. (6) d whee v is he elive velociy inensiy. Tkin he do poduc of Eq. (5), y he veco of he picle's elemeny elive displcemen d, one oins he equion v d = da + da + da + da ( ) ( Fw ) ( F ) ( Fc ) whee is, on he lef-hnd-side of Eq. (7), he kineic eney of he picle's elive moion, E k = v /, nd on he ih-hnd-side e he woks of he foces, F w, F nd F c, lon he picle's elemeny displcemen. Since he elive velociy of picle is of he consn inensiy, hen hee is no chne of he elive kineic eney, nd woks of he consin ecion F w nd he Coiolis foce F c e hen equl o zeo, since hose foces e pependicul o he diecion of he picle's elemeny displcemen. Thus, usin Eq. (7), he followin equion is oined da ( ) ( ) F, (7) + da = 0. (8) The wok of he ceniful foce cn e wien in he nlyicl fom if he foce F nd elemeny elive displcemen d e expessed in ems of pojecion ono he dil diecion 0 nd he diecion c 0 which is pependicul o 0 (Fi. 1), nmely in he fom F = ω sin 0 + ω sin cos c 0 (9) nd d = d + d c (10) 0 0, whee 0 nd c 0 e he uni vecos, nd ω is he nul velociy of he Eh's oion. By he do poduc of hese vecos, he followin equion is oined (F ) da = F d = ω sin d + ω sin cos d. (11) The wok of he ceniful foce, Eq. (11), cn e wien s ol diffeenil, nd he wok of he viionl cion s diffeenil of he viionl poenil du, so h he Eq. (8) ecomes d ω sin + du = 0. (1)

4 18 B. MARJANOVIĆ The Eq. (1) consiss of only cul pefomed woks, so h i expesses he fc h he sum of ceniful nd viionl poenil is consn. Fom he definiion of wok s do poduc of he foce veco nd he veco of he elemeny displcemen, he wok cn e undesood s poduc of foce inensiy nd he pojecion of elemeny displcemen ds = d ono foce diecion (Fi. ), so he cul wok of he viionl cion cn e wien in he fom ( ) da = sin ε ds = dh (13) 1 dh 1 Q M K N 0 χ ψ ε ψ ε c 0 d S χ Fi.. Incemens of meidin The flux of he foce (Rskovic, 1956) lon QN + NS phs, is equl o he sum of cul wok lon QK ph nd woks lon KN + NS phs ( ) da = cos ψ QK cosψ KN + sin ψ NS. (14) Since he wok of he foce expessed y lon KS ph is equl o zeo, hen i cn e 0 0 (KN + NS c ) = cosψ KN + sin ψ NS = 0. (15) The ol wok of he foce lon incemen d = QN is equl o he sum of cul wok lon QK ph nd ficiious wok lon KN ph, so h he Eq. (14) cn e wien in he fom ( ) da = cos ψ d + sin ψ d (16) Incemen QN is ou doule s le s incemen QK ecuse he sme io exiss eween nles χ nd ε. Bsed on he Eq. (15), i cn e concluded h he ficiious wok occus in wo foms wih he diffeen sins u he sme vlues. The equion of n ellipse in he Cesin coodine sysem ses

5 The Fundmenl Cieion fo Esimion of he Refeence Ellipsoid Accucy 183 x z + = 1, (17) nd usin spheicl coodines (x = sin nd z = cos), he equion of he ellipse is oined in he fom =. (18) 1+ cos 1 Eq. (11) expesses flux of he foce F lon QN + NS phs. I cn e poved h is he second em, on he ih-hnd-side of Eq. (11), he ficiious wok. The Eq. (18) cn e wien in he fom sin = 1 + nd y susiuin Eq. (19) ino Eq. (11), he followin equion is oined ( F ) ω ω d da = d + ω sin cos d. By diffeeniion of Eq. (18) one oins he equion d 1 sin cos d =, 1+ 1 cos nd y susiuin Eq. (1) ino Eq. (0), he followin equion is oined (19) (0) (1) ( F ) ω da = d ω sin cos d+ ω sin cos d. () The fis em in Eq. () is he cul wok, nd nohe wo ems e ficiious woks of he foce F. The ol wok of his foce is equl o he sum of cul wok nd he ficiious wok. Considein h he Eq. (8) ives he elion eween cul woks, s well s ficiious woks, of he foces F nd, nd usin he Eqs. (16) nd (), one oins he followin wo equions nd ω sin cos d+ sin ψ d = 0 (3) ω d d + ω ψ = sin cos d cos 0. (4)

6 184 B. MARJANOVIĆ The Eq. (4) epesens he sum of cul woks nd ficiious woks of ceniful foce nd viionl cion foce lon incemens of meidin. By ineion of Eq. (4) wihin limis fom posiion P o posiion Q (Fi. 1) whee nle chnes fom zeo o, nd dius chnes fom o, one oins he equion ( Q) ( Q) ω +ω sin cos d cosψ d = 0. (5) ( P) ( P) The fis inel in Eq. (5) cn e solved usin Eq. (18), o oin ( Q) cos d(cos ) 1 sin cos d ln = = ( P) cos cos nd usin Eq. (19), hen Eq. (5) cn e wien in he fom (6) ( Q) ω 1 ω ( P) sin + ln cosψ d = 0. (7) cos The fis em in Eq. (7) epesens he ceniful poenil in posiion Q. I could e ken h is cos ψ 1 ecuse he hihes vlue of he nle ψ is less hn If he picle is movin fom he poin P he pole o he poin E he equo, hen in Eq. (7) he susiuion should e mde = π / nd =, o ive ( E) ω 1 ω + ln d 0. (8) ( P) 1 The Eq. (8) should e sisfied if he efeence ellipsoid is ccue. If i is no, hen he sum of he ol woks of he ceniful foce nd he viionl cion foce shll no e equl o zeo nd shll epesen he eo of he ellipsoid A, h is ( E) ω 1 ω A = + ln d (9) ( P) 1 The Eq. (9) epesens he fundmenl cieion fo esimion of he efeence ellipsoid ccucy.

7 The Fundmenl Cieion fo Esimion of he Refeence Ellipsoid Accucy THE APPLICATION OF THE FUNDAMENTAL CRITERION The fundmenl cieion will e pplied fo he esimion of he ccucy of GRS80 ellipsoid whose semi-xes e = m, = m nd he viy is deemined ccodin o he Gviy Fomul 1980, which eds ( ) = [ sin φ sin φ ] m/ s, (30) whee φ is he eodeic liude. The vlue of inel in Eq. (9) will e compued usin he elionship eween viionl cion nd heih h. In h sense, he nle POE = π / will e divided o n equl ps, whee one will oin n ppoximely equl semens on he poion PQE of he ellipse (Fi. 3). The posiions of hese semens deemine y he (n+1) poins hei ends. Fo ech poin he mniude of he dius veco cn e compued sed on Eq. (18), nd hen he heih of he picle's ise duin he elive moion, s he diffeence of nd of he semi-mino xis h. (31) P Q χ F T O φ χ N φ E Fi. 3. Relevn viles fo deeminion of viionl cion The nle χ (Fi.) eween he veco nd dil diecion cn e compued y he equion 1 d χ=, (3) d nd hen sed on Eq. (1), he equion is oined sin cos cos χ= Fo ech of he poins whose posiion is deemined y nle, he eodeic liude cn e oined ccodin o fomul (33) φ =π +χ (34)

8 186 B. MARJANOVIĆ The mniude of viy cn e compued ccodin o Eq. (30), nd of ceniful foce ccodin o fomul F = sin ω (35) whee is Eh's nul velociy ω = d / s. The mniude of viionl cion is deemined ccodin o he cosine heoem = + F + F cos φ. (36) The ph of viionl cion ins heih h hs he shpe shown in Fiue 4. p e. dh O dh h Fi. 4. Gviionl cion vesus heih The vlue of inel in Eq. (9) is popoionl o he e unde he ph, which mouns Nm. I is ppoxime equl o he e of peze, nmely ( E) p + e dh ( ) = Nm, (37) ( P) whee p is he viionl cion he pole, nd e he equo. These vlues e compued ccodin o Eq. (36) nd hey moun o p = N nd e = N. The eminin p he ih-hnd-side of Eq. (9) epesens he ceniful foce wok nd i mouns o Nm. The leic sum of hese woks ccodin o Eq. (9) epesens he eo of he ellipsoid which is Nm. The ellipsoid eo wouldn' e sinificnly less even unde he consideion of he viionl cion o e unchnele wih he mximum vlue p = N, in which cse i would e 5697 Nm. The ceniful foce wok wekly depends on he possile eo in he vlue of semi-mino xis, considein he Eq. (9), so i cn e ken h he clculed vlue of his wok cn e consideed s ppoximely ccue, while he vlue of he viionl cion foce wok is diecly dependen on ccucy of he semi-xis. In ode fo ellipsoid o e ccue, vlues of hese woks mus e equl, nmely

9 The Fundmenl Cieion fo Esimion of he Refeence Ellipsoid Accucy 187 p + e ( ) Nm. (38) If one ssumes h vlues of, p nd e e ccue, hen fom Eq. (38) one oins h vlue of semi-mino xis should moun o m, wh is smlle hn he vlue oined ccodin o he model GRS80 of ou 599 m. 4. CONCLUSION The fundmenl cieion fo he esimion of ccucy of he efeence ellipsoid is elile cieion ecuse i ppes fom Newon's second lw. This is n indispensle em h he diecion of viy is eveywhee pependicul o he ellipsoid sufce s n equipoenil sufce. Applyin his cieion fo esimion of ccucy of he ellipsoid defined ccodin o he Geodeic Refeence Sysem 1980, i cn e concluded h he ellipsoid's semi-mino xis conins n eo of ou 599 m. REFERENCES 1. Moiz, H., (1980). Geodeic Refeence Sysem 1980, Bull Geod 54: [hp://eodesy.en.ohiose.edu/couse/efppes/ pdf]. Heisknen, W.A. nd Moiz, H., (1967). Physicl Geodesy, Feemn, Sn Fncisco, pp Jeffeys, S.H., (1976). The Eh is oiin hisoy nd physicl consiuion, 6 h ediion, Univesiy pess Cmide, Cmide, pp Rsković, D., (1956). Mechnics III Dynmics, he second ediion, scienific ook, Belde, pp. 389 (in Sein). OSNOVNI KRITERIJUM ZA OCENU TAČNOSTI REFERENTNOG ELIPSOIDA Boolju Mjnović Veličin veće poluose, i veličin mnje poluose su eomeijske kkeisike efeenno elipsoid. Du in kkeisik je nomln vicij, koj pesvlj inenzie dijen viciono poencijl n ekvipoencijlnoj povšini elipsoid. U ovom du je pesvljen osnovni kieijum z odedjivnje čnosi efeenno elipsoid koji poisiče is duo Njunovo zkon. Anlizom kenj meijlne čke, duž meidijn efeenno elipsoid, je dokzno d je leski zi dov sile viciono pivlčenj i cenifulne sile, nsle onjem elipsoid oko njeove ose, jednk nuli. Ovo je neophodn uslov d pvc veko vicije ude svud upvn n povšinu elipsoid ko ekvipoencijlnu povšinu. Zo je ovj kieijum pouzdn i omoućv d se doije kvniivn ocen čnosi elipsoid. Pimenjujući ovj kieijum z ocenu čnosi elipsoid definisno pem he Geodeic Refeence Sysem 1980 (GRS80), može se zključii d mnj poluos elipsoid sdži ešku od oko 599 m. Ključne eči: efeenni elipsoid, mnj poluos, viciono pivlčenje, GRS80

LECTURE 5. is defined by the position vectors r, 1. and. The displacement vector (from P 1 to P 2 ) is defined through r and 1.

LECTURE 5. is defined by the position vectors r, 1. and. The displacement vector (from P 1 to P 2 ) is defined through r and 1. LECTURE 5 ] DESCRIPTION OF PARTICLE MOTION IN SPACE -The displcemen, veloci nd cceleion in -D moion evel hei veco nue (diecion) houh he cuion h one mus p o hei sin. Thei full veco menin ppes when he picle