Gavity and the figue of the Eath Eic Calais Pudue Univesity Depatment of Eath and Atmospheic Sciences West Lafayette, IN 47907-1397 ecalais@pudue.edu http://www.eas.pudue.edu/~calais/
Objectives What is gavity? How and why does it vay (in space and time)? What do gavity vaiations tell us about the Eath? How can we measue gavity? How is gavity elated to geological stuctues and pocesses?
Definitions Gavimety = measuing and analyzing the Eath s gavity field and its space and time vaiations. Closely elated to Geodesy = measuing and analyzing the shape and dimensions of the Eath. Applications of gavimety: Intenal stuctue of the Eath (fom the suface to the coe) Exploation fo oe, oil, wate Isostasy and mechanical popeties of the lithosphee Eath tides Tansfe of geophysical fluids between esevois: wate, magma, ice => tempoal vaiations of gavity Atificial satellites: obitogaphy Planetay gavimety
Example Right: map of gavity anomalies (Fance) Below: coesponding topogaphic map Positive/negative anomalies Coelation with topogaphy Intepetation?
Gavitation (1/) F = ma Newton s second law: Foce (in Newtons) acting on mass m (in kg), esponsible fo its acceleation a (in m.s - ) Newton s law of gavitation: Two masses m and M attact each othe This attaction esults in a foce: F = G mm Whee is the distance between the masses and G the constant of univesal gavitation, the unit vecto in the diection of G = 6.673 x 10-11 m 3.kg -1.s - F m M
Gavitation (/) F = ma mm F = G foce(s) acting on m foce exeted by M on m In the absence of any othe foce besides the one geneated by M, one can wite: mm ma = G M! a = G a = gavitational acceleation of mass m due to the attaction of mass M F m M
The Eath s gavitational acceleation a is usually called g g should be expessed in m.s -, but vaiations on the ode of 10-8 -10-3 m.s - g is usually expessed in Gals: 1 Gal (fo Galileo) = 1 cm.s - = 10 - m.s - 1 mgal = 10-5 m.s - 1 µgal = 10-8 m.s - g on the suface of the Eath ~9.8 m.s - = 98 000 mgal
Table.1 L&V On othe planets
Besides the Eath s gavitation Pevious fomulas valid if only foce = attaction of the mass of the Eath But the Eath s otates effects: Centifugal acceleation that opposes gavity Defomation of the Eath: pola flattening Effects of othe celestial bodies, in paticula Moon and Sun: Acceleations of the Eath on its obit Tides
Effect of the Eath s otation (1/) Recall that fo a spheical, fixed, homogeneous Eath: M g = G R (R = mean Eath adius = 6371 km) Angula otation = ω Let us conside a plane paallel to the equato: Centifugal acceleation: a c =! Equato: a c max (=R) Pole: a c =0 (=0) Let us conside the spheical Eath: Radial component of a c : Then: a = a cos! = cos! c " = R cos! " a = # R cos! slice view = R cos!!! R! a c =! a = " R cos! a c = " R cos!! R!
Effect of the Eath s otation (/) Because of its otation, the Eath is not a sphee but is flattened at the poles. The effect of the flattening on the gavity is: 3 J GMa 4 R ( 3sin "! 1) (see Tucotte and Schubet p. 199) J = dimensionless coefficient that quantifies the Eath s flattening (J = 1.087 x 10-3 ) a = equatoial adius = 6,378 km (pola adius = 6,357 km)
The Eath s gavity Eath s gavitational acceleation = sum of: Eath s gavitation (= Newtonian attaction) Centifugal acceleation Eath s gavity = sum of: Gavitational acceleation + centifugal acceleation Flattening coection g = GM R " 3GMa J ( R 4 3sin # "1) "$ Rcos # g depends on latitude only Is that all? No A point at the suface of the Eath is also submitted to the Newtonian attaction of celestial bodies (in paticula Moon and Sun) These fomulas assume an homogeneous Eath (o spheical symmety)
Gavity vaiations Conside a spheical volume nea the Eath s suface: Radius = 100 m Depth to cente d = 00 m Density ρ 1 =,000 kg/m 3 Contibution of m 1 to the gavity field on the Eath s suface? Substituting a diffeent mateial in the same spheical volume: Density ρ = 3,000 kg/m 3 (basalt eplacing sandstones) Contibution of m to the gavity field on the Eath s suface? d Eath P g E P.. g E m 1,,, ρ 1 m,, ρ
Gavity vaiations P dg m1 P m1 & V gm = G = G 1 d d 4 3 Vsphee = % 3 3 4G& 1% # gm = 1 3d $ 4! 7! 10 # g = # g m m 1 1 " 14! 10 $ 6 11 m. s 3!! 10! 3.14! 10 4 3! 4! 10 $ = 1.4! 10 $ 5 m. s Density of m = 1/3 geate than m 1 : 6 $ = 1.4 Contibution to the gavity field 1/3 geate g m =.1 mgal mgal d Eath g E.. m 1,,, ρ 1 m,, ρ g E dg m Contibutions of m 1 and m : m 1 gavity below aveage Eath s gavity m gavity below aveage Eath s gavity Small contibutions: Aveage gavity on Eath ~ 10 6 mgals Contibutions of m 1 and m ~ 10-6 Pecision needed to detect m 1 and m
Gavity vaiations Inceasing height above sea level: Inceasing distance fom Eath cente Gavity deceases (1/ ) What is the magnitude of this elevation effect on gavity? (hint: diffeentiate g=gm/ ) Mount Eveest: 8,830 m (~ 9,000 feet)
Gavity vaiations g = G $ o dg d : M =! G M 3 dg d $ =! dg d g #! 10 6.4" 10 6 =! 0.3" 10! 5 m. s! pe m At 8,800 m (top of Mt. Eveest), gavity vaiation (decease): [0.3x10-5 ] x 8,800 = 640x10-5 m.s- = 640 mgals
Density Physical popety of mateials = mass/volume, unit = kg/m 3 (S.I., but g/cm 3 often used instead) Density vaiations of the Eath mateials ceate gavity vaiations: e.g. dense mateials highe gavity Highe gavity dense mateials??? Density of some Eath s mateials: see Table Density depends on poosity, wate content, tempeatue, and pessue. 3 easy to emembe values: Eath = 5,500 kg/m 3 = 5.500 g/cm 3 Continental cust =,670 kg/m 3 =,67 g/cm 3 Mantle = 3,300 kg/m 3 = 3,300 g/cm 3 Mateial Dy sand Wet sand Sandstone Salt Shales Limestone Ganit Doleite Gneiss Basalts Gabbos Peidotite Coal Oil Sea wate Ice Ion Coppe Silve Gold In g/cm 3 Density 1.4-1.65 1.9 -.05.0.5.1.4.1.6.4.8.5.7.5.7.65.75.7 3.1.7 3.3 3.1 3.4 1. 1.8 0.6 0.9 1.01 1.05 0.88 0.9 7.3 7.8 8.8 8.9 10.1 11.1 15.6 19.4
What have we leaned? Gavity is an acceleation Eath's gavity is caused by its gavitational attaction and its otation Eath's gavity changes fom place to place Changes eflect, fo instance, ock densities, altitude, latitude Gavity is expessed in Gals (mgals, 1 mgal = 10-5 m.s - )