LECTURE 3: EARTH S FIGURE, GRAVITY, AND GEOID. Earth s shape, tides, sea level, internal structure, and internal dynamics, are all

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1 GG612 Letue 3 2/1/11 1 LECTURE 3: EARTH S FIGURE, GRAVITY, AND GEOID Eath s shape, tides, sea level, intenal stutue, and intenal dynamis, ae all ontolled by gavitational foes. To undestand gavitation and how it affets Eath, we stat with Newton s laws: Gavitational Potential Fo a point mass: Newton s law of gavitation: Then the aeleation due to gavity is: F = m a = "G mm 2 g = "G M 2 The wok by a foe F on an objet moving a distane d in the dietion of the foe is: The hange in potential enegy is: dw = Fd de p = -dw = -Fd The gavitational potential is the potential enegy pe unit mass in a gavitational field. Thus: Then the gavitational aeleation is: The gavitational potential is given by: md = "Fd = "mgd g = "# % U = " $ $, $ $y, $ ( ' * U & $z ) = "G M Fo a distibution of mass: If a mass is distibuted within a body of volume V, then we an integate to find the total potential: = "G # ( $ ) dv Fo the speial ase of a spheial shell of thikness t, applying this integal V yields: = " GM as if the sphee wee onentated at the ente. Thus, eveywhee outside a sphee of mass M: = " GM Clint Conad 3-1 Univesity of Hawaii

2 GG612 Letue 3 2/1/11 2 Centifugal Potential Fo a otating body suh as Eath, a potion of gavitational self-attation dives a entipetal aeleation towad the ente of the Eath. When viewed in the fame of the otating body, the body epeienes a entifugal aeleation away fom the Eath s ais of otation. Angula veloity: " = d# dt = v whee = sin" Centifugal aeleation: a = " 2 = v 2 But a = "# U, so we an alulate the entifugal potential by integating: U = " 1 2 #2 2 = " 1 2 #2 2 sin 2 $ θ ω a Figue of the Eath Eath s atual sufae is an equipotential sufae (sea level), a sufae fo whih + U = onstant. The figue of the Eath a smooth sufae that appoimates this shape and upon whih moe ompliated topogaphy an be epesented. The eath appoimates an oblate spheoid, whih means it is elliptially-shaped with a longe equatoial adius than a pola adius. The flattening (o oblateness) is the atio of the diffeene in adii to the equatoial adius: f = a " b a Fo eath, f= , o 1/ , and the diffeene in the pola and equatoial adii is about 21 km. Clint Conad 3-2 Univesity of Hawaii

3 GG612 Letue 3 2/1/11 3 The Intenational Refeene Ellipsoid is an ellipsoid with dimensions: Equatoial Radius: Pola Radius Radius of Equivalent Sphee: a = km = km R = km Flattening f = 1/ Aeleation Ratio Moment of Inetia Ratio m = a C a G = "2 a 3 GM E = 1/ H = C " A C = 1/ Hydostati equilibium pedits that the flattening should be: f=1/299.7 This is smalle than the obseved flattening by about 113 m [see Chambat et al., Flattening of the Eath: futhe fom hydostatiity than peviously estimated, Geophys. J. Int., 183, , 2010]. The gavitational potential of an ellipsoid is given by: = "G M ( E " G C " A ) ( 3os 2 # " 1) = "G M ( E " G C " A ) P 2 ( os# ) whee A and C ae the moments of inetia about the equatoial and pola aes. Moe geneally: = "G M # ) 2 # E R & & % 1" *% ( J n P n ( os+ )( % n=2$ $ ' ( ' O θ z P y Whee P n ae the Legende polynomials and the oeffiients J n ae measued fo Eath. The most impotant is the dynamial fom fato: J 2 = C " A = #10"6 2 M E R The net tem, J 3, desibes pea-shaped vaiations: a ~17 m bulge at Noth pole and ~7 m bulges at mid-southen latitudes (~1000 times smalle than J 2 ) Clint Conad 3-3 Univesity of Hawaii

4 GG612 Letue 3 2/1/11 4 The gavitational potential of the Eath (the geopotential) is given by: U g = " 1 2 #2 2 sin 2 $ = " GM + G (C " A) % 3os2 $ " 1( ' * " 1 3 & 2 ) 2 #2 2 sin 2 $ The geopotential is a onstant (U 0 ) eveywhee on the efeene ellipsoid. Then: At the equato: U 0 = " GM a + G 2a 3 (C " A) " 1 2 #2 a 2 Then: At the pole: U 0 = " GM + G (C " A) 3 (C " A) # f = a " = a 2 % M E a 2 $ a & ( + 1 ' 2 a 2 ) 2 GM E * 3 2 J m Whee we have appoimated a~ on the ight hand side. Gavity on the Refeene Ellipsoid ( ) To fist ode: = a 1" f sin 2 # Geoenti latitude = λ (measued fom ente of mass) λ λ g g a Geogaphi latitude = λ g (in ommon use) To fist ode: sin 2 " # sin 2 " g $ f sin 2 2" g The aeleation of gavity on the efeene ellipsoid is given by: g = "# U g Pefoming this diffeentiation gives: g = GM " 3GM E a2 J 2 3sin 2 # " 1 " $ 2 os 2 # * # Rewiting and simplifying gives: g = g e 1+ 2m " 3 2 J & -, % 2 ( sin 2 )/ + $ '. * # Witing in tems of λ g gives: g = g e 1+ % 5 2 m " f " 17 $ 14 mf & # ( sin 2 ) g + f 2 % ' 8 " 5 $ 8 mf & -, ( sin 2 2) g / +, './ Equatoial gavity is: [ ] g = sin 2 " g sin 2 2" g g e = GM # 1" 3 a 2 2 J " m & % 2 ( = m/s 2 $ ' This allows us to ompute the pola gavity: g p = m/s 2 Clint Conad 3-4 Univesity of Hawaii

5 GG612 Letue 3 2/1/11 5 The polewad inease in gavity is 5186 mgal, and thus only about 0.5% of the absolute value (gavity is typially measued in units of mgal = 10-5 m/s 2 ). Gavity deeases towad to pole beause the pole: (1) is lose to the ente of Eath than the equato (6600 mgal) (2) does not epeiene entifugal aeleation (3375 mgal) But the equato has moe mass (beause of the bulge), whih ineases the equatoial gavity. Togethe these thee affets yield the 5186 mgal diffeene. Eath s Geoid The geoid is the equipotential sufae that defines sea level, and is epessed elative to the efeene ellipsoid. Tempoal vaiations in the geoid ae aused by lateal vaiations in the intenal densities of the Eath, and by the distibution of masses (pimaily hydologial) upon the sufae of the Eath. Mass eess (eithe subsufae eess density o positive topogaphy) deflets the geoid upwads. Clint Conad 3-5 Univesity of Hawaii

6 GG612 Letue 3 2/1/11 6 Clint Conad 3-6 Univesity of Hawaii

7 GG612 Letue 3 2/1/11 7 Spheial Hamonis The geoid (and any funtion on a sphee) an be epessed in tems of spheial hamonis of degee n and ode m: Y m n = ( a m n osm" + b m n sinm" )P m n ( os# ) Top view Side view Top view Side view The powe spetum of the geoid is given by: P n = n a 2 2 "( nm + b nm ) m=0 The dominane of the low-hamoni degees in the geoid powe spetum indiate that the dominant shape of the geoid is ontolled by stutues deep within the mantle. Clint Conad 3-7 Univesity of Hawaii