gravity of the earth gravitational acceleration a distance r from a sphere of density ρ is this...
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Gravity of the Earth
• Gravitational acceleration a distance r from a sphere of density ρ is
• This result is independent of radial density variations
3
2 2
43
G GMa
r r
b
Gravity of the Earth
• The earth is not a perfect sphere but is approximately an oblate spheroid – an ellipse rotated about the short axis
• The ellipticity is given by
1
298e p
e
R Rf
R
Gravity of the Earth
• The radius of an oblate spheroid is
• Centrifugal acceleration due to rotation of the earth decreases gravitational acceleration felt by an observer on the surface
2(1 sin )er R f
Gravity of the Earth
• These two effects are accounted for in the reference gravity formula adopted by the International Association of Geodesy which gives the gravitational acceleration as a function of latitude
2 4
2
3
5
) sin sin )
9.7803185 /
5.278895
( (1
10
2.3462 10
e
e
g g
g m s
x
x
Gravity of the Earth
• The earth is not a perfect oblate spheriod• We define a reference surface called the geoid
which is the gravitational equipotential– Over the oceans, this is simply the mean sea level– A plumb bob points perpendicular to the local
equipotential surface – in the direction of the potential gradient
• The reference spheroid is the theoretical equipotential that is the closest fit to Earth’s field
http://www.ncgia.ucsb.edu/education/curricula/giscc/units/u015/figures/figure04.gif
Gravity of the Earth
• Shape of the geoid can be determined from gravity measurements– We can measure the geoid very accurately from
space– Altimetry satellites can measure the geoid over
the oceans • Gravity anomalies are very small compared to
the mean surface gravity– g = 981 gal, but we usually work in mgal or μgal
http://www.csr.utexas.edu/grace/
http://www.csr.utexas.edu/grace/gravity/
Gravity anomaly (mgal)
• The large ice sheets are not in equilibrium– More mass is being lost
through ice bergs calving into the ocean than is being gained from winter snows inland
– Contributing to sea level rise
Ice loss from the Greenland and Antarctic Ice Sheets
Gravity Anomalies
• Isostasy– 1735-1745 French surveying expedition under Pierre
Bouguer found that the Andes mountains deflected their plumb bob line much less than expected
– 1830-1843 the Great Trigonometric Survey of India, completed under Sir George Everest, found the same result due to the Himalayas
– 1855 J.H. Pratt and Sir George Airy proposed two separate hypothesis to explain this observation which depend on Archimedes principal – termed isostasy
Gravity Anomalies
• Calculate a gravity anomaly by subtracting the reference gravity value– Compute the free-air correction (correct for
elevation change or Δr2
0
0
0 0
( )
2( ) 1
2( )F
Rg h g
R h
hg h g
R
hg g h gg
R
Gravity Anomalies
• Now the free-air gravity anomaly is given by
• The Bouguer correction compensates for density of rocks between h and sea level
2( ) ( ) 1F obs F obsg
hg gg g
Rg
2Bg G h
Assignments due next tuesday (10/20)
• Reading– Larsen, C.F., Motyka, R.J., Freymueller, J.T.,
Echelmeyer, K.A., Ivins, E.R., 2005, Rapid viscoelastic uplift in southeast Alaska caused by post-Little Ice Age glacial retreat: Earth and Planetary Science Letters, v. 237, 548-560.
• Homework– Chpt 5. – 14, 19, 21, 22, 24
Gravity Anomalies
• Terrain correction compensates for devations from a horizontal plane – very small except for mountainous regions and often ignored
• The Bouguer anomaly combines all of these corrections
• The Bouguer anomaly is then the difference between the observed value and the theoretical value at particular latitude, referenced to sea level
( )B obs BF Tgg gg g g
Gravity Anomalies and Isostasy
• The free air anomaly is primarily used to determine if a region is in isostatic equilibrium– If a region is in equilibrium there should be no
excess or lack of mass above the compensation depth and hence no gravity anomaly
• The Bouguer anomaly indicates mass variations that are not due to topography
Gravity Anomalies
UK continental marginMid Atlantic Ridge
Gravity Anomalies
Oahu
Gravity Anomalies and Isostasy
• If the structure is about 10 times the compensation depth the free air anomaly will be very small away from the edges
• Uncompensated structures have zero Bouguer anomaly
• Compensated structures have negative Bouguer anomaly – why?
Gravity anomalies and Isostasy
http://www.csr.utexas.edu/grace/gravity/
Density anomalies under the mid atlantic ridgeAll four density models (c, e, f, g) describe the observed gravity anomaly. There are actually an infinity number of density models that will produce the same gravity anomaly. This in another example of non-uniqueness.
What are the differences between each of the density models?
Model c is most consistent with what is understood about spreading centers.
Geoid height anomalies
• Lateral variations in density distribution produce variations in the geoid height. This anomaly is given by:
• A trough in the geoid height anomaly is associated with a negative gravity anomaly (low density) and a peak is associated with a positive gravity anomaly (high density)
g h V
Geoid height anomalies
• Overall geoid height anomalies are small indicating that isostatic equilibrium is typical for large features – This implies– The mantle is weak (has finite viscosity) on a
geologic time scale– The crust is strong and can support small scale
features
http://www.ncgia.ucsb.edu/education/curricula/giscc/units/u015/figures/figure04.gif
Geoid height anomalies
• The geoid height anomaly for an isostatic density distribution is not zero
• Where g is the reference gravity value, Δρ(z) is the anomalous density distribution at depth z beneath point P, and D is the compensation depth
0( )
2 DGz
gdzh z
Geoid height anomaly
• Hawaiian island ridge bends the crust causing flexure
• Long wavelength topography caused by mantle upwelling
• Increase in topography more than compensates for decreased mantle density and produces long wavelength positive gravity anomaly