The Gravity Field of the Earth  Part 1 (Copyright 2002, David T. Sandwell)


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1 1 The Gavity Field of the Eath  Pat 1 (Copyight 00, David T. Sandwell) This chapte coves physical geodesy  the shape of the Eath and its gavity field. This is just electostatic theoy applied to the Eath. Unlike electostatics, geodesy is a nightmae of unusual equations, unusual notation, and confusing conventions. Thee is no clea and concise book on the topic although Chapte 5 of Tucotte and Schubet is OK. The things that make physical geodesy messy include: eath otation latitude is measued fom the equato instead of the pole; latitude is not the angle fom the equato but is efeed to the ellipsoid; elevation is measued fom a theoetical suface called the geoid; spheical hamonics ae defined diffeently fom standad usage; anomalies ae defined with espect to an ellipsoid having paametes that ae constantly being updated; thee ae many types of anomalies elated to vaious deivatives of the potential; and mks units ae not commonly used in the liteatue. In the next couple of lectues, I ll ty to pesent this mateial with as much simplification as possible. Pat of the eason fo the mess is that pio to the launch of atificial satellites, measuements of elevation and gavitational acceleation wee all done on the suface of the Eath (land o sea). Since the shape of the Eath is linked to vaiations in gavitational potential, measuements of acceleation wee linked to position measuements both physically and in the mathematics. Satellite measuements ae made in space well above the complications of the suface of the eath, so most of these poblems disappea. Hee ae the two most impotant issues elated to oldstyle geodesy. Elevation land othometic geometic geoid sea suface spheoid Pio to satellites and the global positioning system (GPS), elevation was measued with espect to sea level  othometic height. Indeed, elevation is still defined in this way howeve, most measuements ae made with GPS. The pesatellite appoach to measuing elevation is called leveling.
2 1. Stat at sea level and call this zeo elevation. (If thee wee no winds, cuents and tides then the ocean suface would be an equipotential suface and all shoelines would be at exactly the same potential.). Sight a line inland pependicula to a plumb line. Note that this plumb line will be pependicula to the equipotential suface and thus is not pointed towad the geocente. 3. Measue the height diffeence and then move the setup inland and to epeat the measuements until you each the next shoeline. If all measuements ae coect you will be back to zeo elevation assuming the ocean suface is an equipotential suface. Afte atificial satellites measuing geometic height is easie, especially if one is fa fom a coastline. 1. Calculate the x,y,z position of each GPS satellite in the constellation using a global tacking netwok.. Measue the tavel time to 4 o moe satellites, 3 fo position and one fo clock eo. 3. Establish you x,y,z position and convet this to height above the spheoid which we ll define below. 4. Go to a table of geoid height and subtact the local geoid height to get the othometic height used by all suveyos and mappes. Othometic heights ae useful because wate flows downhill in this system while it does not always flow downhill in the geometic height system. Of couse the poblem with othometic heights is that they ae vey difficult to measue o one must have a pecise measuement of geoid height. Let geodesists woy about these issues. Gavity The second complication in the pesatellite geodesy is the measuement of gavity. Intepetation of suface gavity measuement is eithe difficult o tivial depending whethe you ae on land o at sea, espectively. Conside the land case illustated below. would like to know gavity on this suface land obsevations geoid Small vaiations in the acceleation of gavity (< 106 g) can be measued on the land suface. The majo poblem is that when the measuement is made in a valley, thee ae masses above the obsevation plane. Thus, binging the gavity measuement to a common level equies
3 3 knowledge of the mass distibution above the obsevation point. This equies knowledge of both the geometic topogaphy and the 3D density. We could assume a constant density and use leveling to get the othometic height but we need to convet to geometic height to do the gavity coection. To calculate the geometic height we need to know the geoid but the geoid height measuement comes fom the gavity measuement so thee is no exact solution. Of couse one can make some appoximations to get aound this dilemma but it is still a poblem and this is the fundamental eason why many geodesy books ae so complicated. Of couse if one could make measuements of both the gavity and topogaphy on a plane (o sphee) above all of the topogaphy, ou toubles would be ove. Ocean gavity measuements ae much less of a poblem because the ocean suface is nealy equal to the geoid so we can simply define the ocean gavity measuement as feeai gavity. We ll get back to all of this again late when we discuss flateath appoximations fo gavity analysis. GLOBAL GRAVITY (Refeences: Tucotte and Schubet Geodynamics, Ch. 5. p ; Stacey, Physics of the Eath, Ch. +3; Jackson, Electodynamics, Ch. 3; Fowle, The Solid Eath, Ch. 5.) The gavity field of the Eath can be decomposed as follows: the main field due to the total mass of the eath; the second hamonic due to the flattening of the Eath by otation; and anomalies which can be expanded in spheical hamonics o fouie seies. The combined main field and the second hamonic make up the efeence eath model (i.e., spheoid, the efeence potential, and the efeence gavity). Deviations fom this efeence model ae called elevation, geoid height, deflections of the vetical, and gavity anomalies. Spheical Eath Model The spheical eath model is a good point to define some of the unusual geodetic tems. Thee ae both fundamental constants and deived quantities. paamete desciption fomula value/unit R e mean adius of eath m M e mass of eath x 10 4 kg G gavitational constant x10 11 m 3 kg 1 s ρ mean density of eath M e (4/3π R 3 e ) kg m 3 U mean potential enegy GM e R 1 e 6. 6x107 m s needed to take a unit mass fom the suface of the eath and place it at infinite distance g mean acceleation on the suface of the eath δu/δ = GM e R  e 9.8 m s δg/δ gavity gadient o feeai coection δg/δ = GM e R e 3 = g/r e x106 s
4 4 We should say a little moe about units. Deviations in acceleation fom the efeence model, descibed next, ae measued in units of milligal (1 mgal = 103 cm s  =105 m s = 10 gavity units (gu)). As notes above the vetical gavity gadient is also called the feeai coection since it is the fist tem in the Taylo seies expansion fo gavity about the adius of the eath. g g ( ) = gr ( R e) + ( e)+... (1) Example: How does one measue the mass of the Eath? The best method is to time the obital peiod of an atificial satellite. Indeed measuements of all longwavelength gavitational deviations fom the efeence model ae best done be satellites. GMe m v GMem = mv () GM e = ω GM = ω 3 s e s The mass is in obit about the cente of the Eath so the outwad centifugal foce is balanced by the inwad gavitational foce; this is Keple s thid law. If we measue the adius of the satellite obit and the obital fequency ω s, we can estimate GM e. Fo example the satellite Geosat has a obital adius of 7168 km and a peiod of sec so GM e is x10 14 m 3 s. Note that the poduct GM e is tightly constained by the obsevations but that the accuacy of the mass of the eath M e is elated to the accuacy of the measuement of G. Ellipsoidal Eath Model The centifugal effect of the eath's otation causes an equatoial bulge that is the pincipal depatue of the eath fom a spheical shape. If the eath behaved like a fluid and thee wee no convective fluid motions, then it would be in hydostatic equilibium and the eath would assume the shape of an ellipsoid of evolution also called the spheoid.
5 5 c a θ θ g paamete desciption fomula value/unit (WGS84) GM e x m 3 s a equatoial adius m c pola adius m ω otation ate x 105 ad s 1 f flattening f = (a  c)/a 1/ θ g geogaphic latitude   θ geocentic latitude   The fomula fo an ellipse in Catesian coodinates is x a y z + + = 1 (3) a c whee the xaxis is in the equatoial plane at zeo longitude (Geenwich), the yaxis is in the equatoial plane and at 90 E longitude and the zaxis points along the spinaxis. The fomula elating x, y, and z to geocentic latitude and longitude is x = cosθcosφ y = cosθsinφ z = sinθ (4) Now we can ewite the fomula fo the ellipse in pola coodinates and solve fo the adius of the ellipse as a function of geocentic latitude.  1 / = cos θ sin θ + a 1 f θ (5) ( sin ) a c
6 6 Befoe satellites wee available fo geodetic wok, one would establish latitude by measuing the angle between a local plumb line and an extenal efeence point such as Polais. Since the local plumb line is pependicula to the spheoid (i.e., local flattened suface of the eath), it points to one of the foci of the ellipse. The convesion between geocentic and geogaphic latitude is staightfowad and its deivation is left as an execise. The fomulas ae c tanθ = tanθg o tanθ = ( 1 f ) tanθ g (6) a (Example: What is the geocentic latitude at a geogaphic latitude of 45. The answe is θ = 44.8 which amounts to a km diffeence in location!) Flattening of the Eath by Rotation Suppose the eath is a otating, self gavitating ball of fluid in hydostatic equilibium. Then density will incease with inceasing depth and sufaces of constant pessue and density will coincide. The suface of the eath will be one of these equipotential sufaces and it has a potential U o. U = V ( o, θ) 1 ω cos θ (7) whee the second tem on the ight side of equation (7) is the change potential due to the otation of the eath at a fequency ω. The potential due to an ellipsoidal eath in a nonotating fame can be expessed as GM V J a e P J a = 1 1 1( θ) P ( θ )... (9) The cente of the coodinate system is selected to coincide with the cente of mass so, by definition, J 1 is zeo. Fo this model, we keep only J (dynamic fom facto o "jay two" = 1.08 x 103 ) so the final efeence model is GMe GMeJa V = + ( 3sin θ 1 3 ) (10) This paamete J is elated to the pincipal moments of inetia of the eath by MacCullagh's fomula. Fo a complete deivation see Stacey, p Let C and A be the moments of intetia about the spin axis and equatoial axis espectively. Fo example
7 7 C = ( x + y ) dm V Afte a lot of algeba one can deive a elationship between J and the moments of inetia. (11) J C A = Ma (1) In addition, if we know J, we can detemine the flattening. This is done by inseting equation (10) into equation (7) and noting that the value of U o is the same at the equato and the pole. Solving fo the pola and equatoial adaii that meet this constaint, one finds a elationship between J and the flattening. f a c a = ω = J + (13) a GM Thus if we could somehow measue J, we would know quite a bit about ou planet. Measuement of J Just as in the case of measuing the total mass of the eath, the best way to measue, is to monito the obit of an atificial satellite. In this case we measue the pecessional peiod of the inclined obit plane. To second degee, extenal potential is GMe GMeJa V = + ( 3sin θ 1) (14) 3 The foce acting on the satellite is  V. V g = θ θ 1 V V ˆ ˆ 1 ˆ cosθ φ φ (15) If we wee out in space, the best way to measue J would be to measue the θcomponent of the gavity foce. g θ V GMea J = 1 3 =  sinθcos θ (16) 4 θ This component of foce will apply a toque to the obital angula momentum and it should be aveaged ove the obit. Conside the diagam below.
8 8 L L m g θ i Fo a pogade obit, the pecession ω p is etogade; that it is opposite to the eath's spin diection. The complete deivation is found in Stacey [1977, p. 76] and the esult is ω ω p s a = 3 J i cos (17) whee i is the inclination of the satellite obit with espect to the equatoial plane, ω s is the obit fequency of the satellite and ω p is the pecession fequency of the obit plane in inetial space. Example  LAGEOS As an example, the LAGEOS satellite obits the eath evey seconds at an aveage adius of 1,65 km, and an inclination of Given the paametes in the table above and J = 1.08 x 103, the pedicted pecession ate is /day. This can be compaed with the obseved ate of /day. The figue on the next page is an illustation of the LAGEOS satellite. Hydostatic Flattening Given the adial density stuctue, the eath otation ate and the assumption of hydostatic equilibium, one can calculate the theoetical flattening of the eath (see Galand, The Eath's Shape and Gavity, Pemagon, 1977, Appendix ). This is called the hydostatic flattening f h = 1/99.5. Fom the table above we have the obseved flattening f = 1/98.57 so the actual eath is flatte than the theoetical eath. Thee ae two easons fo this. Fist, the eath is still ecoveing fom the last ice age when the poles wee loaded by heavy ice sheets. When the ice melted pola dimples emained and the glacial ebound of the viscous mantle is still incomplete. Second, the mantle is not in hydostatic equilibium because of mantle convection. Finally it should be noted that J is changing with time due to the continual postglacial ebound. This is called "jay two dot" and it can be obseved in satellite obits as a time vaiation in the pecession ate.
9
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