Gravity and isostasy

Size: px

Start display at page:

Download "Gravity and isostasy"

Gloria Osborne
6 years ago
Views:

1 Gavity and isostasy Reading: owle p60 74 Theoy of gavity Use two of Newton s laws: ) Univesal law of gavitation: Gmm = m m Univesal gavitational constant G=6.67 x 0 - Nm /kg ) Second law of motion: = ma We can combine them to obtain the gavitational acceleation of m towad m : Gm a = The gavitational attaction de to sphee mass M is the same as placing all the mass at the cente of the sphee as long as the mass being attacted is otside the sphee.

2 Theoy of gavity Gavitational acceleation: Gm a = m m Gavitational potential: Relationship: Gm = V V a = Definition: The gavitational potential, V, de to a point mass m, at a distance fom m, is the wok done by the gavitational foce in moving a nit mass fom infinity to to a position fom m. The gavitational acceleation is eqal to the ate of change in the potential field Gavitational acceleation on ath If the ath was a sphee the gavitational acceleation at the sface of the ath wold be: g = GM R Typical vale: 9.8 m/s Note that g is not dependent of the mass of the object being acceleated towad the cente of the ath Gavity nit: gal gal = 0.0 m/s Typical vale: g = 98 gal Named afte Galileo who did mch of the pioneeing stdies into gavity Is the ath s gavitational acceleation a constant?

3 Gavitational acceleation on ath ath is not a stationay sphee! The ath is an oblate spheoid atte at the eqato and thinne at the poles The ath is otating Centifgal acceleation edces gavitational attaction. The fthe yo ae fom the otation axis the geate the centifgal acceleation Refeence gavity fomla The mathematically detemined gavitational acceleation on a 4 g( λ ) = g ( + α sin λ + β sin otating oblate spheoid e Whee λ is the latitde, g e the gavitational acceleation at the eqato g e = m/s, and constants α = β = Obits of satellites When a mass gets caght in the gavitational field of a planet it stats to obit the planet. The gavitational attaction of the planet balances the otwad centifgal foce: GM m = mω The distance of the satellite fom the ath is dependent on the gavitational attaction By monitoing the obits of man made satellites we can detemine vaiations in g aond the globe Also, sing satellite altimety we can mease the distance of the ath sface below the satellite and detemine the ath s shape

4 The geoid Mean sea level is an eqipotential sface it is the geoid These figes show the diffeences between the geoid and the efeence ellipsoid/spheoid Geoid anomalies Diffeences between the geoid and efeence ellipsoid ae de to lateal density anomalies De to topogaphy De to highe and lowe density mateial in the cst o mantle Meased gavity anomalies ae small compaed to mean gavity mean g = 9.8 m/s mease gavity anomalies of 0-8 of the mean sface vale Instment sensitivity: John Milne (906): When a sqad of 76 men mached to within 6 o 0 feet of the Oxfod Univesity Obsevatoy it was fond that a hoizontal pendlm inside the bilding meased a deflection in the diection of the advancing load

5 Isostasy 8 th and 9 th centy sveys set ot to mease the shape of the ath. They sed plmb bobs and expected them to be attacted towad adjacent montain chains eg the Andes and Himalayas Bt the plmb bob was not attacted as mch as expected They calclated that the obseved deflection cold be explained if the excess montain mass was matched by an eqal mass deficiency beneath The montains wee in isostatic eqilibim The ath s lithosphee is floating on the dense asthenosphee Aiy s hypothesis 3 to accont fo the mass deficiency beneath montains Two densities, that of the igid ppe laye, ρ, and that of the sbstatm, ρ s Montains theefoe have deep oots The compensation depth is the depth below which all pesses ae hydostatic qating the masses in vetical colmns above the compensation depth: 3 tρ ρ = h + t ) ρ tρ + ρ = dρ + t d ) ρ + ( ) ρ + s ( + s w ( s A montain height h is ndelain by a oot of thickness: h ρ ρ ρ = s Ocean basin depth, d, is ndelain by a anti-oot of thickness: d( ρ ρw) 3 = ρ ρ s

Patt s hypothesis 3 to accont fo the mass deficiency beneath montains The depth of the base of the ppe laye is constant Montains theefoe have low density oots The compensation depth is the depth

density ρ : D ρ = ρ h + D Ocean basin depth, d, is ndelain by a high density mateial, density ρ d : ρ D ρ d w ρd = D d Which hypothesis? is opeating on the ath?

6 Patt s hypothesis 3 to accont fo the mass deficiency beneath montains The depth of the base of the ppe laye is constant Montains theefoe have low density oots The compensation depth is the depth below which all pesses ae hydostatic qating the masses in vetical colmns above the compensation depth: 3 Dρ = ( h + D ρ Dρ = dρ + ρ ( D d) ) w d A montain height h is ndelain by low density mateial, density ρ : D ρ = ρ h + D Ocean basin depth, d, is ndelain by a high density mateial, density ρ d : ρ D ρ d w ρd = D d Which hypothesis? is opeating on the ath? o a given location we mst ask oselves Is thee isostatic eqilibim? Which pocess is opeating? We can stat to addess Is it a combination of the two? some of these qestions Patt and Aiy ae end membes with gavity measements Bt fist, we mst coect o measements

Teain coection Latitde coection coect fo the spheoid Refeence fomla: = e 4 g( λ ) g ( + α sin λ + β sin m/s Whee λ is the latitde, g e

7 Gavity coections Obsevations still sbject to extaneos effects nelated to sbsface geology Mst make coections. Latitde coection. ee-ai coection 3. Boge coection 4. Teain coection Latitde coection coect fo the spheoid Refeence fomla: = e 4 g( λ ) g ( + α sin λ + β sin m/s Whee λ is the latitde, g e the gavitational acceleation at the eqato g e = m/s, and constants α = β = This fomla povides the vaiation in g de to the spheoid It is a fnction of latitde λ only Calclate the coection fo latitde of obsevation point

8 ee-ai coection Acconts fo the / decease in gavity with distance fom the cente of the ath. A given gavity measement was made at an elevation h, not at sea level, ecall: The gavity at elevation h above sea level is appoximated by: The fee-ai coection is theefoe: g δg h R ( h) = g0 h ( g R = g0 g h) = GM g = R 0 g 0 is the gavity at sea level, ie g 0 = g( Gavity deceases with distance fom the cente of the ath The fee-ai coection is theefoe added ee-ai anomaly A gavity anomaly sggests the diffeence between a theoetical and obseved vale The fee-ai anomaly is calclated by coecting an obsevation fo expected vaiations de to () the spheoid and () elevation above the spheoid (ie the geoid ie sea level) Then the fee-ai anomaly is: g = g = g = g obs obs obs g( + δg h g( + g( R h g( R

9 Boge coection Acconts fo ock thickness between obsevation and sea level h Sea level Teat the ock as an infinite hoizontal slab, the Boge coection is: g B δ = πgρh This additional slab of ock between the obsevation point and sea level cases an additional attaction The Boge coection is sbtacted fom the obsevation Teain coection When Boge coection is inadeqate, also se teain coection hill valley Appoaches to coection Rectangla gid Hamme segments both se elevation diffeences between station and sond o cstal scale stdies the teain coections ae sally insignificant so ignoe

10 Teain coection Hamme coection Boge anomaly Apply all the coections: g B = g = g obs δg B + δg T g( + δg With the Boge anomaly We have sbtacted theoetical vales fo the latitde and elevation We have emoved the ock above sea level so the anomaly epesents the density stcte of mateial below sea level This is compaable to the fee-ai anomaly ove the oceans and both have been coected to sea level δg B + δg T watch the signs! Boge anomaly fo offshoe gavity: Replace the wate with ock Apply teain coection fo seabed topogaphy

Gravity. Gravitational Acceleration.

Gravity. Gravitational Acceleration. Gavity Using the Eath s gavitational field to study its intenal stuctue has poven to be an extemely effective means of pobing stuctual vaiations at all levels depths and length scales within the Eath.