Gravity 2

Geoid, spheroid

Measurement of gravity

• Absolute measurements

• Relative measurements

True gravitational Acceleration

Difference in gravitational acceleration

Measurement of absolute gravity

2o

2o

t/)tvz(2g

gt2

1tvz

)T/4(Lg

g/L2T22

z – distance the object fallst – time to fall the distance zvo – initial velocityg – absolute gravity

T – period of the pendulumL – length of the pendulum

Expensive and time consuming measurements

Measurement of relative gravity

• F = mg

g1 < g2 F1 < F2

• g=F/m• L ~ g

gravimeter

Worden gravimeter

Measurement of relative gravity

• F = mg

g1 < g2 F1 < F2

• g=F/m• L ~ g

Base station – absolute gravity – calibration of L (L g) Other stations – relative gravity – change in L

Problems with surveying in the ocean

Eötvös correction(required for seaborne, airborne measurements)

• Apparent gravity is affected by motion of moving platforms

Eastward ship (aircraft) travel: adds to the earth's rotation, increases centrifugal forces and decreases the gravity readings.

Westward travel: increases the gravity reading.

North-south travel: is independent of rotation, and decreases the gravity reading in either case.

Correction required:

Isostasy

Deflection of a plumb bob• Expected deflection

due to the attraction of the mass of the mountain

• Actual deflection for Andes and Himalayas (less than expected due to a deficiency of mass beneath the mountains)

Measurements:

- in 1735-1745 – Bouguer - Andes - mid -1800’s – Sir George Everest - Himalayas

Local Isostasy (1885 -1889)

• Block of different density

• The same pressure from all blocks at the depth of compensation (crust/mantle boundary)

• Blocks of the same density but different thickness

• The base of the crust is exaggerated, mirror image of the topography

Hydrostatic (lithostatic) pressure

P = gh

P – pressure – density

z – thickness

Archimedes – a floatingbody displaces its ownweight of water

“Floating”rigid surfacelayer

Denser substratum

The isostasy restores the equilibrium

Hydrostatic (lithostatic) pressure

• Pratt modelP= 2gh2= 3gh3= 4gh4

= 5gh5

P/g= 2h2= 3h3= 4h4= 5h5

• Airy modelP/g= 2h5= (2h4+ 1h4’)

= (2h3+ 1h3’)= (2h3+ 1h3’)= (2h3+ 1h3’)

1 – mantle density2 – crustal density (constant)1 > 2

P = gh

Airy Isostatic Model

1) P/g= aha+ whw+ chc+ mhm= Constant Total pressure

2) T = ha + hw + hc + hm = Constant Total thickness

a –airc – crustw – waterm – mantle

5-8 times

Regional Isostasy

Compensation directly below the load, no rigidity

(like water)

Take lithospheric strength into account, there is flexural rigidity

Regional Isostasy – Elastic Plate

Elastic thickness Flexural Rigidity (Bending)

Regional Isostasy – Elastic Plate(Turcotte and Schubert, 1982)

Elastic thickness Flexural Rigidity (Bending)

D(d4w/d4x)+(b - a)gw=q(x) – differential

equationD – flexural rigidityw – vertical deflection at x

g – gravityx – horizontal distance from the loadq(x) – load applied to the top of the plate at xIndexes: “a” – above, “b” - below

Regional Isostasy – Elastic Plate

Elastic thickness Flexural Rigidity (Bending)

D(d4w/d4x)+(b - a)gw=q(x)

Small amplitude of w, long wavelength

Large w, smaller wavelengthPeripheral bulge + depression

Collapse into local equilibrium

Regional Isostasy – Examples

Elastic thickness Flexural Rigidity (Bending)

~ bending of diving plateLoad – topography of accretionary plate wedge + volcanic arc

~ bending of elastic plateLoad – mass of high mountains

Isostatic Rebound

Historical levels of Lake Superior

due to post-glacial rebound

Isostatic Rebound

Present-day uplift rate, horizontal velocity, free air gravity anomaly, and rate of change in gravity (Wu, 2001)