# Gravity and isostasy Theory of gravity

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EPS 122: Lecture 14 Gravity and isostasy

Gravity and isostasy

Reading: Fowler p193-206

EPS 122: Lecture 14 Gravity and isostasy

Theory of gravity

Use two of Newtons laws:

1) Universal law of gravitation: F F

m1 m2

r

We can combine them to obtain the gravitational acceleration of m2 toward m1:

2) Second law of motion:

The gravitational attraction due to sphere mass M is the same as placing all the mass at the center of the sphere as long as the mass being attracted is outside the sphere.

Universal gravitational constant G=6.67 x 10-11 Nm2/kg2

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EPS 122: Lecture 14 Gravity and isostasy

Theory of gravity

Gravitational potential: Definition: The gravitational potential, V, due to a point mass m1, at a distance r from m1, is the work done

by the gravitational force in moving a unit mass from infinity to to a position r from m1.

F F

m1 m2

r Gravitational acceleration:

Relationship:

The gravitational acceleration is equal to the rate of change in the potential field

EPS 122: Lecture 14 Gravity and isostasy

Gravitational acceleration on Earth

Is the Earths gravitational acceleration a constant?

If the Earth was a sphere the gravitational acceleration at the surface of the Earth would be:

Typical value: 9.81 m/s2

Note that g is not dependent on the mass of the object being accelerated toward the center of the Earth

Gravity unit: gal 1 gal = 0.01 m/s2 Typical value: g = 981 gal

Named after Galileo who did much of the pioneering studies into gravity

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EPS 122: Lecture 14 Gravity and isostasy

Gravitational acceleration on Earth

Earth is not a stationary sphere

The Earth is an oblate spheroid

Fatter at the equator and thinner at the poles

The Earth is rotating

Centrifugal acceleration reduces gravitational attraction. The further you are from the rotation axis the greater the centrifugal acceleration

Reference gravity formula

The mathematically determined gravitational acceleration on a rotating oblate spheroid

Where is the latitude, ge the gravitational acceleration at the equator ge = 9.7803185 m/s

2, and constants = 0.005278895 = 0.000023462

EPS 122: Lecture 14 Gravity and isostasy

Orbits of satellites

When a mass gets caught in the gravitational field of a planet it starts to orbit the planet.

The gravitational attraction of the planet balances the outward centrifugal force:

The distance of the satellite from the Earth is dependent on the gravitational attraction

By monitoring the orbits of man made satellites we can determine variations in g around the globe

Also, using satellite altimetry we can measure the distance of the Earth surface below the satellite and determine the Earths shape

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EPS 122: Lecture 14 Gravity and isostasy

The geoid

Mean sea level is an equipotential surface

it is the geoid

These figures show the differences between the geoid and the reference

ellipsoid/spheroid

EPS 122: Lecture 14 Gravity and isostasy

Geoid

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EPS 122: Lecture 14 Gravity and isostasy

Geoid anomalies Differences between the geoid and reference ellipsoid are due to lateral density anomalies

Due to topography

Due to higher and lower density material in the crust or mantle

Measured gravity anomalies are small compared to mean gravity

mean g = 9.81 m/s2

measure gravity anomalies of 10-8 of the mean surface value

Instrument sensitivity:

John Milne (1906): When a squad of 76 men marched to within 16 or 20 feet of the Oxford University Observatory it was found that a horizontal pendulum inside the building measured a deflection in the direction of the advancing load

EPS 122: Lecture 14 Gravity and isostasy

Isostasy

They used plumb bobs and expected them to be attracted toward adjacent mountain chains

eg the Andes and Himalayas

But the plumb bob was not attracted as much as expected

18th and 19th century surveys set out to measure the shape of the Earth.

They calculated that the observed deflection could be explained if the excess mountain mass was matched by an equal mass

deficiency beneath

The mountains were in isostatic equilibrium

The Earths lithosphere is floating on the denser asthenosphere

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EPS 122: Lecture 14 Gravity and isostasy

Airys hypothesis to account for the mass deficiency beneath mountains

Two densities, that of the rigid upper layer, u, and that of the substratum , s

Mountains therefore have deep roots

The compensation depth is the depth below which all pressures are hydrostatic

Equating the masses in vertical columns above the compensation depth:

A mountain height h1 is underlain by a root of thickness:

Ocean basin depth, d, is underlain by a anti-root of thickness:

1

1

12

2 3

3

EPS 122: Lecture 14 Gravity and isostasy

Pratts hypothesis to account for the mass deficiency beneath mountains

The depth of the base of the upper layer is constant

Mountains therefore have low density roots

The compensation depth is the depth below which all pressures are hydrostatic

Equating the masses in vertical columns above the compensation depth:

A mountain height h1 is underlain by low density material, density 1:

Ocean basin depth, d, is underlain by a high density material, density d:

3

2

1

1 12 3

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EPS 122: Lecture 14 Gravity and isostasy

Which hypothesis? is operating on the Earth?

For a given location we must ask ourselves

Is there isostatic equilibrium?

Which process is operating?

Is it a combination of the two? Pratt and Airy are end members

We can start to address some of these questions with gravity measurements

But first, we must correct our measurements

EPS 122: Lecture 14 Gravity and isostasy

Gravity corrections

Observations still subject to extraneous effects unrelated to subsurface geology

Must make corrections

1. Latitude correction

2. Free-air correction

3. Bouguer correction

4. Terrain correction

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EPS 122: Lecture 14 Gravity and isostasy

Latitude correction correct for the spheroid

m/s2

This formula provides the variation in g due to the spheroid

It is a function of latitude only

Calculate the correction for latitude of observation point

Where is the latitude, ge the gravitational acceleration at the equator ge = 9.7803185 m/s

2, and constants = 0.005278895 = 0.000023462

Reference formula:

Do we add or subtract the correction?

EPS 122: Lecture 14 Gravity and isostasy

Free-air correction Accounts for the 1/r2 decrease in gravity with distance from the center of the Earth. A given gravity measurement was made at an elevation h, not at sea level, recall:

The gravity at elevation h above sea level is approximated by:

The free-air correction is therefore:

g0 is the gravity at sea level, ie

Do we add or subtract the correction?

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EPS 122: Lecture 14 Gravity and isostasy

Free-air anomaly

A gravity anomaly suggests the difference between a theoretical and observed value

The free-air anomaly is calculated by correcting an observation for expected variations due to (1) the spheroid and (2) elevation above sea level

Then the free-air anomaly is:

EPS 122: Lecture 14 Gravity and isostasy

Australia

Topography and bathymetry

Free-air anomaly

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EPS 122: Lecture 14 Gravity and isostasy

Bouguer correction

h

Accounts for rock thickness between observation and sea level

Treat the rock as an infinite horizontal slab, the Bouguer correction is:

Sea level

This additional slab of rock between the observation point and sea level causes an additional attraction

Do we add or subtract the correction?

EPS 122: Lecture 14 Gravity and isostasy

Terrain correction

When Bouguer correction is inadequate, also use terrain correction

hill

valley

Approaches to correction

Rectangular grid

Hammer segments both use elevation differences between station and surround

For crustal scale studies the terrain corrections are usually insignificant so ignore

Do we add or subtract the correction?

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EPS 122: Lecture 14 Gravity and isostasy

Hammer correction Terrain correction

EPS 122: Lecture 14 Gravity and isostasy

Bouguer anomaly

Apply all the corrections:

watch the signs!

With the Bouguer anomaly

We have subtracted theoretical values for the latitude and elevation

We have removed the rock above sea level so the anomaly represents the density structure of material below sea level

This is comparable to the free-air anomaly over the oceans and both

have been corrected to sea level

Bouguer anomaly for offshore gravity:

Replace the water with rock

Apply terrain correction for seabed topography

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EPS 122: Lecture 14 Gravity and isostasy

Australia Topography and

bathymetry

Free-air anomaly Bouguer anomaly

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