Gravity and the figureof the EarthEric CalaisPurdue UniversityDepartment of Earth and Atmospheric SciencesWest Lafayette, IN [email protected]://www.eas.purdue.edu/~calais/Objectives• What is gravity?• How and why does it vary (in space andtime)?• What do gravity variations tell us about theEarth?• How can we measure gravity?• How is gravity related to geologicalstructures and processes?Definitions• Gravimetry = measuring and analyzing the Earth’s gravityfield and its space and time variations.• Closely related to Geodesy = measuring and analyzing theshape and dimensions of the Earth.• Applications of gravimetry:– Internal structure of the Earth (from the surface to the core)– Exploration for ore, oil, water– Isostasy and mechanical properties of the lithosphere– Earth tides– Transfer of geophysical fluids between reservoirs: water, magma,ice => temporal variations of gravity– Artificial satellites: orbitography• Planetary gravimetryExample• Positive/negative anomalies• Correlation with topography• Interpretation?Right: map of gravityanomalies (France)Below: correspondingtopographic mapGravitation (1/2)• Newton’s second law:– Force (in Newtons) acting on mass m(in kg), responsible for itsacceleration a (in m.s-2)• Newton’s law of gravitation:– Two masses m and M attract eachother– This attraction results in a force:– Where r is the distance between the2 masses and G the constant ofuniversal gravitation, r the unitvector in the direction of r– G = 6.673 x 10-11 m3.kg-1.s-2amFrr=rrmMGFrr2=FrmMrrGravitation (2/2)•  force(s) acting on m•  force exerted by M on m• In the absence of any other force besides the onegenerated by M, one can write:• a = gravitational acceleration of mass m due tothe attraction of mass MamFrr=rrmMGFrr2=rrMGarrmMGamrrrr22=!=FrmMrrThe Earth’s gravitationalacceleration• a is usually called ‘g’• g should be expressed in m.s-2, but variations on theorder of 10-8-10-3 m.s-2• g is usually expressed in Gals:– 1 Gal (for Galileo) = 1 cm.s-2 = 10 -2 m.s-2– 1 mGal = 10-5 m.s-2– 1 µGal = 10-8 m.s-2• g on the surface of the Earth ~9.8 m.s-2 = 982 000 mGalOn other planets…• Table 2.1 L&VBesides the Earth’s gravitation• Previous formulas valid if only force = attractionof the mass of the Earth• But the Earth’s rotates ⇒ 2 effects:– Centrifugal acceleration that opposes gravity– Deformation of the Earth: polar flattening• Effects of other celestial bodies, in particularMoon and Sun:– Accelerations of the Earth on its orbit– TidesEffect of the Earth’srotation (1/2)• Recall that for a spherical, fixed,homogeneous Earth:(R = mean Earth radius = 6371 km)• Angular rotation = ω– Let us consider a plane parallel tothe equator:• Centrifugal acceleration:• Equator: ac max (r=R)• Pole: ac=0 (r=0)– Let us consider the spherical Earth:• Radial component of ac:• Then:2RMGg =rac2!=!=!= coscos2raacr"!="!=22coscos RaRrr#!rac2!=rR!!R2!!= cos2Rac"!=22cosRar"!= cosRr!slice viewEffect of the Earth’s rotation (2/2)• Because of its rotation, the Earth is not a spherebut is flattened at the poles.• The effect of the flattening on the gravity is:– (see Turcotte and Schubert p. 199)– J2 = dimensionless coefficient that quantifies theEarth’s flattening (J2 = 1.0827 x 10-3)– a = equatorial radius = 6,378 km (polar radius = 6,357km)( )1sin3232242!"JRGMaThe Earth’s gravity• Earth’s gravitational acceleration = sum of:– Earth’s gravitation (= “Newtonian” attraction)– Centrifugal acceleration• Earth’s gravity = sum of:– Gravitational acceleration + centrifugal acceleration– Flattening correction– g depends on latitude only• Is that all? No…– A point at the surface of the Earth is also submitted to the Newtonianattraction of celestial bodies (in particular Moon and Sun)– These formulas assume an homogeneous Earth (or spherical symmetry)! g =GMR2"3GMa22R4J23sin2# "1( )"$2Rcos2#Gravity variations• Consider a spherical volume nearthe Earth’s surface:– Radius r = 100 m– Depth to center d = 200 m– Density ρ1 = 2,000 kg/m3– Contribution of m1 to the gravityfield on the Earth’s surface?• Substituting a different material inthe same spherical volume:– Density ρ2 = 3,000 kg/m3 (basaltreplacing sandstones)– Contribution of m2 to the gravityfield on the Earth’s surface?PP..m1,,,ρ1m2,,ρ2EarthdrrEgrEgrGravity variationsContributions of m1 and m2 :• m1 ⇒ gravity below average Earth’sgravity• m2 ⇒ gravity below average Earth’sgravitySmall contributions:• Average gravity on Earth ~ 106 mGals• Contributions of m1 and m2 ~ 10-6• Precision needed to detect m1 and m2mGalsmsmggdrGgrVdVGdmGgmmmspherem4.1.104.1.101410431014.31021074343425264631123132211111=!=!"#!!!!!!!!=#=#===$$$$$%&%&Density of m2 = 1/3 greater than m1:⇒ Contribution to the gravity field 1/3 greater⇒ gm2 = 2.1 mGalPP..EarthdrrEgrEgr1mgdr2mgdrm1,,,ρ1m2,,ρ2Gravity variations• Increasing height abovesea level:– Increasing distance fromEarth center– Gravity decreases (1/r2)• What is the magnitude ofthis elevation effect ongravity?(hint: differentiate g=GM/r2)Mount Everest: 8,830 m (~ 29,000 feet)Gravity variationsAt 8,800 m (top of Mt. Everest), gravity variation (decrease):[0.3x10-5] x 8,800 = 2640x10 -5 m.s-2 = 2640 mGalsmpersmdrdgrgdrdgorrMGdrdgrMGg25632.103.0104.61022:2!!"!="!#$!=!=$=Density• Physical property of materials = mass/volume,unit = kg/m3 (S.I., but g/cm3 often used instead)• Density variations of the Earth materials creategravity variations:– e.g. denser materials ⇒ higher gravity– Higher gravity ⇒ denser materials ???• Density of some Earth’s materials: see Table• Density depends on porosity, water content,temperature, and pressure.• 3 easy to remember values:– Earth = 5,500 kg/m3 = 5.500 g/cm3– Continental crust = 2,670 kg/m3 = 2,67 g/cm3– Mantle = 3,300 kg/m3 = 3,300 g/cm3DensityMaterial0.88 – 0.92Ice2.65 – 2.75Gneiss2.7 – 3.1Basalts2.7 – 3.3Gabbros3.1 – 3.4Peridotite1.2 – 1.8Coal0.6 – 0.9Oil1.01 – 1.05Sea water7.3 – 7.8Iron8.8 – 8.9Copper10.1 – 11.1Silver15.6 – 19.4Gold2.5 – 2.7Dolerite2.5 – 2.7Granit2.4 – 2.8Limestone2.1 – 2.6Shales2.1 – 2.4Salt2.0 – 2.5Sandstone1.9 - 2.05Wet sand1.4 - 1.65Dry sandIn g/cm3What have we learned?• Gravity is an acceleration• Earth's gravity is caused by its gravitationalattraction and