Gravity and the Earth Newtonian Gravity and Earth Rotation Effects

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1 Gavity and th Eath Nwtonian Gavity and Eath Rotation Effcts Jams R. Clynch, 003 I. Nwtonian Gavity A. Nwtonian Gavitational Foc B. Nwtonian Gavitational Fild C. Nwtonian Gavitational Potntial D. Gavity Vcto and th Gadint Opato E. Sphical Eath Exampl F. Potntial Diffncs Na th Eath's Sufac II. Th Ral Gavity Fild of th Eath - Ovviw III. Rotational Effcts A. Ellipsoid Modl of Eath B. Gavity of Pfct Ellipsoid / Godtic Latitud IV. Potntial sufacs, Goid, and Hights V. Goid and Topogaphy 1

2 I. Nwtonian Gavity A. Nwtonian Gavitational Foc Nwton publishd his law of gavitational attaction in This says that all things that hav mass attach ach oth. Th attaction is popotional to th poduct of th masss and invsly popotional to th distanc btwn thm. Th valu G is a constant that was fist masud by Cavndish in 1798 using lad balls. Th two masss a assumd to b points. Howv it can b shown that if th masss a sphical symmtic th sam quation holds using th cnt of symmty as th location fo th total mass of th objct. Th quation holds fo xtndd masss in this spcial, but usful cas.

3 This can b povd by looking at th mass as bing mad up of an infinit numb of small points of volum dv and mass dm ρdv wh th dnsity is ρ. Th ffcts of all ths points is addd up. This is th intgal of calculus. (S th basic physics not fo a fw dtails.) This pocdu is quid fo th gnal cas. In th sphically symmtic cas you gt th sult abov, but only fo points outsid th mass. Fo points insid th mass, only th mass that is at a adius lss than that of th tst mass counts. So th foc dcass as you go insid a sphical mass. (Ths sults a just dpndnt on th invs squa law natu of th foc. So thy also hold fo th lctical foc.) Th cas of non-sphical Nwtonian gavity will b takn up shotly. B. Nwtonian Gavitational Fild Nwton s scond law stats that th foc and acclation a latd though th mass. Th mass occus in both th quation fo th gavitational foc and th quation fo th acclation that foc causs. Assuming that som small tst mass, m is usd, thn th foc on that mass is: 3

4 F F a g GMm ê ma GM ê a wh w hav intoducd th Nwtonian gavitational acclation g. (This is not what w call th gavitational acclation of th ath, that is slightly diffnt.) Notic that w can find th valu of th acclation of gavity fom ths fomulas, at last in th sphical appoximation: GM g. This is th valu on th sufac of a sphical ath. It can b masud, fo xampl with a falling pbbl o a pndulum. Evn if you hav an ida of th siz of th ath, and hnc, you still do not know th mass of th ath M, but only th poduct GM. This is why Cavndish said h had "wightd th ath" whn h masud G. Th is an acclation at ach point in spac du to th mass M, whth th is a tst mass th o not. W say th is a gavitational fild. This concpt is vy common in physics. Things poduc ffcts vn if th is nothing th to fl it. C. Nwtonian Gavitational Potntial Th will b a foc quid to mov a tst mass away fom ou fild souc mass, M. This mans that w hav addd potntial ngy to th tst mass. Call this potntial ngy V. It will b a function of th position lativ to th mass M. This potntial ngy must b latd to th foc though its spatial gadint. Th at of chang of this potntial ngy with spct to distanc is th sam foc as th Nwtonian gavitational foc. In mathmatical tms w say that: F V V GMm ê GMm This fom fo th potntial ngy, V has th ngy zo at an infinit distanc fom th objct M, (wh is infinit). Th ngy gts mo ngativ as th tst mass m appoachs th mass M. It bcoms infinitly lag at th oigin, but this is insid th mass and th quation dos not hold th. 4

5 Th valu of th potntial ngy p unit mass is latd to th nominal gavity on th sufac of th ath V() m GM g GM Th standad scintific units of th acclation of gavity a m/s/s. Th units of th potntial ngy p unit mass a thfo m /s, o vlocity squad. (Rmmb kintic ngy is half th mass tims th vlocity squad, so th units a consistnt with th dfinition of ngy.) D. Gavity vcto and th Gadint Opato Th diffntial opato opating on a function of position is just a vcto that xpsss th at of chang of th function in diffnt dictions: V V ê x x V + ê y V V V,, x y z y V + ê z z H th two notations fo Catsian vctos a usd. Th diction of this vcto will b along th gatst chang in V, and th magnitud is th at of chang in that diction. To obtain th at of chang in any paticula diction, just tak th dot poduct with a unit vcto in that diction. (Th is an xtnsiv tchnical not on vctos availabl. In addition th gadint opato is covd mo xtnsivly in a tchnical not on basic physics.) Th a two impotant facts about th gadint. If you tak th gadint of a potntial function V, you gt th foc. Th following is tu of this gadint and th foc: 1. Th diction of th gadint vcto is ppndicula to sufacs of constant V,. Th magnitud of th gadint is invsly popotional to th spacing of sufacs of constant potntial. If th sufacs a clos togth, th gadint is lag, if fa apat it is small. This is shown blow wh a diagam of a vtical cut of th qual potntial sufacs is shown. Th lins a spacd at qually diffncs of potntial. Th sufacs, of cous, com out of th pag and a th dimnsional. 5

6 Th distotion a gatly magnifid fo puposs of illustation. Th masud diction of gavity, and hnc th "nomal" way w dfin down, a ppndicula to ths qual potntial sufacs. If thy hav bumps (as thy do in th al wold) thn th up-down lin has vaiations. 6

7 E. Sphical Eath Exampl Th ath is vy lag compad to vyday distancs w xpinc. This is why w can assum that th ath is flat and only mak small os in most things w do. Blow is a sis of diagams of a sgmnt of th ath, with ach stp having high magnification ( small scal in th mapping tminology.) It is cla that nomal hights, such as a 100 m ( 300 ft ) 30 stoy building, a ssntially invisibl on tu scal plots of any significant sgmnt of th ath. A plot of som na ath hights and th gavitational valus fo a sphical ath a shown blow. Th valus a fo points vy na th ath. 7

8 Th valus of th potntial ngy p unit mass ( that is V/m ) a shown fo a nominal ath siz that is sphical. Notic that th potntials gt small in absolut valu vy slowly as you incas in hight. Th diffnc of ths valus, dividd by th hight diffnc is a good appoximation fo th divativ in th vtical diction, which is th gadint h. Notic that th gadint, which is th local valu of g, also dcass in magnitud as you go up, but also vy slowly. W can oftn tat th acclation of gavity na th ath sufac as a constant. (This is not tu fo satllit wok howv.) F. Potntial Diffncs Na Eath's Sufac In lmntay physics txts, w lan that th gavitational potntial ngy is givn by V V m g h m g h Th scond fom is mo coct, as w a almost always daling in diffncs of potntial ngis. This fomula can b divd fom th gnal Nwtonian Law of Gavity assuming that th hight diffnc is small compad to th adius of th ath. 8

9 Many appoximations usd fo computation a divd using som small, dimnsionlss paamt. Both bing dimnsionlss and small a impotant. H th paamt will b h/, th hight abov th ath dividd by th nominal adius of th ath. Th Nwtonian potntial is givn by: V GM m Consid this quation fo two locations, on on th sufac of th ath and th scond at a hight h abov th ath. It is th diffnc in potntials that w nd. Fo th scond w hav V(h) GMm + h Th only diffnc btwn ths two is th dnominato. W wish to comput th diffnc of ths two potntials V V(h) V(0) 1 1 [ GM m] + h Th ky to finding th most significant tm is to lt: 1 + h (1 + h / 1 (1 + α) ) wh th small paamt h/ has bn calld α. Substituting this fom into th quation fo obtains V on V GM m 1 mg α mgh α + α This is th sult dsid. In this th valu of th acclation of gavity, g, found abov has bn usd. Th appoximation 1+ α 1 was also usd. H th hight is masud fom th sufac of th ath. 9

10 II. Th Ral Gavity Fild of th Eath - Ovviw Th a sval diffncs btwn th al gavity fild of th ath and th abov simplification. Th two majo physical diffncs a: 1. Th ath is otating. This causs an additional ffctiv foc to b flt fo all objcts otating with th ath. This includs appls, pndulums, and popl. It dos not includ ath satllits. This lads to ffcts on th od of 1/300 of th Nwtonian gavitation valus.. Th ath is not homognous. Ths ffcts a 1/10,000 o lss than th main ffcts. But thy can b masud and hav som pactical consquncs. Th otational ffcts a not insignificant. Not only do thy modify th valu of th acclation of gavity, thy sult in a chang in th shap of th ath. Th ath is appoximatly an llips volvd about th pola axis. This shap chang lads to significant changs is th valus of latitud. It also indictly lads to many of th map us poblms as diffnt popl and oganizations us diffnt appoximations fo th tu figu of th ath. This lads to most of th diffnc in datums. (Th is an xtnsiv tchnical not on datums.) In addition th a som tminology diffncs btwn what Godsy uss and physics. In th fild of Godsy, th a som tms and concpts that look and fl lik things studid in lmntay physics, but a slightly diffnt. This can caus som confusion, spcially if on taks quations fom both physics books and godsy books. Physics Godsy Gavity Nwtonian Gavitation Nwtonian Gavitation Plus Rotational Effcts Foc Foc, Fma Body Foc f F/m Foc p unit mass, Acclation, g, As sn on Rotating Eath Potntial Foc - Gadint (Potntial) Acclation + Gadint (Potntial) F V g + V Coodinats Pola Angl Masud fom Pol (0 to 180 dgs) Masud fom Equato ( φ ' ) (-90 to 90 dgs) Th main souc of diffnc is th inclusion of th otational ffcts in th tms usd in godsy. This is don to flct th masumnts sn by an obsv otating on th ath s sufac, that is masumnts mad on th al wold. Th dfinitions of sval itms w mad bfo many of th concpts usd in today s lmntay physics w discovd. Nwton o his contmpoais mad many dfinitions. Th concpts of potntial ngy, kintic ngy, and th consvation of ngy followd this wok by 100 yas. Th scond diffnc is th lation btwn th foc o acclation and a potntial function. Fist godsy dals in acclations, not focs. This is bcaus th ffcts of th mass of an objct in Nwton s scond law (F ma) and his law of gavity ( F GMm/ ) cancl out. 10

11 In godsy th acclation is takn as th gadint 1 of th potntial. In physics th foc is th ngativ gadint of th potntial ngy. In godsy th ngativ sign is omittd. In physics tms, th potntial ngy of somthing sitting on th ath is ngativ. You hav to add ngy to mov it off th ath. In godsy th gavity potntial is a positiv numb. It is th absolut valu of th physics valu of potntial ngy p unit mass. Th two potntials a latd by a minus sign. W will hav th diffnt sts of quantitis usd in godsy, 1. Quantitis basd on a sphical, non-otating ath,. Quantitis basd on a otating, llipsoidal ath without dnsity vaiations, 3. Quantitis basd on th al ath. Th fist st is usd in lmntay physics books. Thy a not usd in godsy. Th scond st is usd as a baslin o fnc st in godsy. Diffncs fom th smooth llipsoidal modl a potd. Eath Gavity Fild Quantitis Potntial Gavity Acclation Sphical Eath V Ellipsoidal Eath UV+Φ γ Ral Eath WU+T g H, V is th Nwtonian Potntial fo th modl ath ( sphical o llipsoidal) U is th gavity potntial fo th modl ath - with otational ffcts includd, Φ is th potntial fo otational ffcts. Φ ω p /, wh p and ω a discussd blow, W is th gavity potntial fo th al ath, T is th ffct of th inhomognitis in th ath, γ is th acclation du to th modl potntial, U, and g is th al gavity acclation of a body co-otating with th ath. 1 Th gadint is a masu of how fast a function changs. In this cas th changs a with spct to position, a at of chang with distanc. Th gadint is a divativ. It is also a vcto. It points in th diction of maximum positiv chang. Fo a sphical ath, th gadint of th gavity potntial points to th cnt of th ath. Fo th al ath it points in th local down diction. (Invs of th local vtical). 11

12 III. Rotational Effcts A. Ellipsoid Modl of Eath What w call "gavity" in godsy is th sum of th Nwtonian gavitational attaction and otational ffcts. Th otation ffct is calld cntifugal acclation. All things co-otating with th ath fl it. This causs th ath to hav a bulg at th quato. Fo a unifom ath, th shap is an llipsoid of volution. (This is also calld a sphoid). This is th godtic modl w will discuss in this sction. Th complxity of th bumps in th al wold will b discussd in th lat sctions. On of th ways to dfin th amount th llips diffs fom a cicl is calld th flattning, f. Th flattning, f, is givn by th quation f a b b 1, a a wh a is th long quatoial adius and b is th shot pola adius. Th flattning of th ath is about 1/300. If th ath w a fluid it would b about 1/30. Th ath acts almost as a fluid ov long tim fams. Nwton was th fist to pdict th flattning of th ath. Anoth common way to dscib th shap of th llips is with th ccnticity,. This is dfind as: a 1 b, o a 1. b Th obsvd acclation of gavity is th sum of th Nwtonian gavitational attaction and th otational foc. Th shap of th ath minimizs th potntial ngy du to th two focs. This maks 1

13 th diction of th obsvd acclation of gavity ppndicula to th llipsoid. This lin dos not go to th cnt of th ath xcpt fo th pols and points on th quato. Th Nwtonian acclations is givn by g n GM ê wh G is th univsal gavitational constant, M is th mass of th ath, is th adius of th ath (o distanc fom ath cnt of mass to obsv), and ê is a unit vcto that points outwad fom th cnt of th ath to th obsvation point. This quation is only valid outsid th ath. Insid th ath only th blow th obsv contibuts to th Nwtonian acclation. Th cntifugal acclation is givn by wh a c ω pê p ω π / T is th otation at of th ath (T is 1 sidal day ), p is th distanc fom th axis of otation to th obsv shown in th figu, and ê is a unit vcto along th outwad diction of p. p This applis only to objcts that otat with th ath. Satllits in spac do not fl this acclation. 13

14 B. Gavity of Pfct Ellipsoid / Godtic Latitud Th figu blow is a summay of th acclation vctos and th up-down lins fo a point on th otating ath. Th up lin is dfind by th obsvd gavity acclation, g, fo an objct co-otating with th ath. This is th sum of th otational and Nwtonian gavitation ffcts. Ths togth caus th shap of th ath to budg at quato and fom an llipsoid in th pfct ath cas. Th Nwtonian gavitational attaction, which w will dnot g n, dos point at th cnt of mass of th ath. But th sum of focs w fl dos not. Th otational acclation, which w will call a c, acts outwad fom th axis of otation. Ths acclations a shown in th diagam. Not that th vctos a not dawn to scal. Th lin fom th obsvd up-down lin gnats th usual latitud w us. This is calld Godtic (φ ). It is th kind found on maps. A latitud fom th lin to th cnt of th ath also can b dfind. This is gocntic (φ ) latitud. It would b th obsvd latitud on a sphical ath. In th al wold, it is usd mostly in satllit wok. Th obsvd acclation, g, is dfind by th vcto sum of th g n and a c. Th total is th ath's acclation of "gavity" as dfind in godsy. Th otational acclation tm is maximum at th quato and zo at th pols bcaus th momnt am p vais with latitud. In th al wold, th maximum a c is small than g n by a facto of 300. Thfo th ffcts on g, and angls, a small. But th ath is lag and vn small ffcts on angls can caus changs of 10 s of km. 14

15 Th nominal Nwtonian acclation is about 980 cm/s and th maximum otational tm about 3 cm/s. In godsy acclations of gavity a calld gal's, shot fo Galilo. On gal is on cm/s. This is a lag unit fo masuing vaiations, so th milligal, mgal o 1/1000th gal, commonly occus. Th gavity valu at ach godtic latitud fo an llipsoid is givn by: g g cos φ + g p 1 1 sin φ sin φ wh g is th acclation at th quato, g p, is th acclation at th pols, and is th ccnticity. This is not as complicatd as it looks. It is asy to show that b a 1. This quation is oftn wittn using th symbol gamma, γ, instad of g. Oftn gamma is usd fo thotical gavity fom a pfct llipsoid and g fo th al gavity fom th lumpy, non-unifom al ath. Anoth fom of this quation is with th auxiliay valu k givn by 1+ k sin φ γ g, 1+ sin φ k b g p 1. a g This valu is gaphd in th nxt figu along with th two componnts. 15

16 Th two componnts and th total acclation of gavity on th llipsoid a a function of latitud as show in th abov figu. Th total acclation is obtaind by adding th Nwtonian tm and th cntifugal tm as vctos. Th Nwtonian acclation is opposd by th otation tms vywh xcpt at th pols. It is thfo lag than th total valu. Th Nwtonian tm incass at th pols bcaus th sufac is clos to th cnt of mass. Th gavity at th pols is about 5 gal ( 5 cm/s/s ) lag than at th quato. About half of this ffct is bing clos to th cnt and th st is th diffnc in th otation tm, th cntifugal acclation. Th diction Up is also dfind fom th sum of th focs. It is ppndicula to th llipsoid sufac. It s invs, Down, dos not point to th cnt of th ath. W histoically dfind latitud fom obsvations of stas, with spct to th local up calld th local vtical. Th histoical dfinition of latitud cosponds to th angl mad th lin ppndicula to th llipsoid, not th on to th cnt of th ath. Th local vtical is ppndicula to th llipsoid in th absnc of inhomognats. 16

17 Th latitud w s on maps, calld godtic latitud, coms fom this ppndicula to th llipsoid. This latitud is commonly calld gogaphic latitud, but this tm is not wll dfind in th scintific litatu. Officially it is godtic latitud. Usually whn you s a tm calld godtic somthing, it fs to dfinitions basd on th llipsoid. Th coodinats using th vtical snsd by a bubbl lvl, which sponds to th al vaiation in th gavity fild, a discussd in sction VII. Th angl mad by th lin to th cnt of th ath is calld gocntic latitud. This is th only latitud in th sphical ath modl. Today gocntic latitud is usd mainly in th fild of atificial ath satllits. 17

18 IV. Potntial Sufacs, Goid, and Hights Th sufacs of constant W a sufacs of constant potntial in th godsy nomnclatu. Ths a also sufacs of constant potntial ngy. Th minus sign dosn't chang th fom of a constant sufac. Wat would assum on of ths sufacs in a bowl, lak o ocan. Ths sufacs a calld Lvl Sufacs. Man sa lvl is on paticula sufac of constant W. This paticula sufac can b xtndd ov land. This paticula lvl sufac is calld th goid. To this point, th ath has bn idalizd. But th al ath has ocans, mountains and oth dnsity vaiations. Ths vaiations a small, but caus th al gavity valus to diff fom th simpl llipsoid modl. In th llipsoid modl, th goid is an llipsoid. In th al wold th goid has bumps. Now w will tun to th ffcts of th inhomognats of th al wold. Th lagst ffcts will b on hights. Th fom of th sufacs of W must b masud; thy cannot b computd fom a fw constants and a modl. Thy dpnd on th al wold mass vaiations, which vais on many distanc scals. Computing th distanc fom th goid to th cnt of th ath is not possibl without lag numbs of al wold masumnts. Th llipsoid is dfind mathmatically with spct to th cnt of th ath and its adius is asily computd. Knowing th distanc btwn th llipsoid and goid thfo would dfin th goid. Th fom of th goid ov lag distancs was not wll known until satllits w launchd. Stating with a "stak in th sand" that dfins man sa lvl at th sho, th hights of th land can b masud. With classical suvy tchniqus, this is don with a sis of masumnts mad fom point to point. Ths hights masud by classical mthods a masud fom th goid. This happns bcaus of th way hights a masud. Tansits (tlscops on a tipod) a usd, with angls masud with spct to th local vtical. Th local up vcto masud with a plumb bob o spiit lvl is th ppndicula to th al goid. Lvl Sufacs a th th-dimnsional analogu of contous in two dimnsions. In on dimnsion if you find th locations wh th function has a paticula valu, you gt a st of points. If you hav a function of two dimnsions, such as hight as a function of latitud and longitud, th st of points of a paticula valu is a lin. This is on of th contous. If you hav a function of latitud, longitud and hight th st of points of a paticula valu is a sufac. Picking on potntial dfins a sufac. On paticula valu dfins th goid. 18

19 As th tansit is movd fom plac to plac, th local vtical will follow th hills and vallys of W. Thfo th goid is th fnc sufac fo classical hight masumnts. As on chif of th US National Godtic Suvy onc said, w a in th position of knowing hights vy accuatly with spct to a sufac w know pooly. Hights dtmind in this mano a calld othomtic hights o man sa lvl (msl) hights. (Othomtic and msl hight a slightly diffnt, but th two tms a commonly usd to man th sam thing.) 19

20 Hights dtmind in this mano a calld othomtic hights o man sa lvl (msl) hights. Excpt in godtic scinc litatu, wh you s msl hight, it usually mans othomtic hight. Tchnically othomtic and msl hight a slightly diffnt. Ocan cunts and oth ffcts slightly modify th al sufac hight of th sa. All th diffnc btwn al msl and othomtic hights can b classifid as ocanogaphy. Hights masud fom th llipsoid a calld llipsoidal hights o godtic hights. Satllit systms masu llipsoidal hights. This is not what w s on maps o in databass. Ths contain othomtic hight. Bcaus satllit basd positioning is now vy impotant, th is a nd fo a good masumnt of th goid location. This is don by masuing th vtical distanc btwn th llipsoid and th al wold goid as a function of location. This diffnc is calld undulation of th vtical o spaation of th goid. Th symbol N is usd fo th undulation. Whil th symbols h and H a commonly usd fo th two hights, th is no standad fo which mans othomtic o llipsoidal. H w will us th DoD standad, h H fo llipsoidal, and fo othomtic hight. Not that in all cass th undulation, N is th othomtic minus th llipsoidal hights. It is th distanc fom th llipsoid to th goid. 0

21 h N Thus H H + N h, wh th valus a takn at th sam point. N is a function of location. Finding N as a function of latitud and longitud is how w dtmin th goid. Th abov figu givs a woldwid viw of th goid sufac. Th valus ang fom about -100 m to m. Wold wid llipsoids a chosn so that th avag undulation is zo ov th wold. This mans that th a as many placs with th goid abov th llipsoid as th a with th goid blow th llipsoid. Th a many small aa vaiations in th goid that do not show up in this figu. Bcaus othomtic hights (msl) w th only ons histoically masuabl, thy a th hights shown on maps and in databass. All hights on maps a othomtic. Howv position masumnts mad using satllit systms, such as GPS, a inhntly don in an ath cntd, ath fixd Catsian X-Y-Z systm. Ths can asily b convtd to latitud, longitud, and llipsoidal hight. (Also calld godtic hight). Th goid undulation, N, is ndd to convt ths to map typ (othomtic o msl) hights. This valu is only know at th mt lvl in most placs. 1

22 V. Goid and Topogaphy Th goid has vaiations, o componnts, at many spatial scals. At long lngths it fls th influncs of most of th ath and its vaiations in dnsity. Howv on a shot scal, small vaiations a dominatd by local ffcts such as na by mountains tc. This is illustatd in th abov figu. Th goid sms to follow th mountain ang up and down. This can b sn fom th condition that th goid is ppndicula to th local gavity vcto. Th absolut valu of th goid lvl can vay a lot bcaus it is influncd by mass vaiations ov a vy long ang, but th small bumps look lik local topogaphy - at last on th km ang.