Gravity Methods

• Gravity is not a “constant” 9.78 m/s2

• Responds to local changes in rock density

• Widely used in oil and gas, mineral exploration, engineering site surveys

• “Natural source” – only measure an existing field

• Major difficulty is extreme weakness of field variations (one part in 100 million)

• Variations in elevation and latitude are much greater, requires correction, and detailed surveying and levelling

• Gravity from ships, aircraft are much more difficult (expensive)

Gravitational theory

Newton’s law:

In terms of vectors:

Universal gravitational constant:

Gravitational theory

Force:

Acceleration (=f/m):

For the earth:

This is non-uniform, because:

• the shape of the earth is ellipsoidal, oblate

• the earth rotates

• topography is irregular

• there are internal variations in density

Gravitational potential“Potential” – equivalent to “work done”

What is the work done (per unit mass) in moving an object in from an infinite distance?

Gravity fields are “conservative” – work done is path independent

Integrating force x distance:

(2.5)

Gravitational potential

“Equipotential surface” – a surface over which the potential is a constant (e.g., surface of a fluid, earth’s oceans)

If

then the potential starts at zero, and decreases to negative infinity at the centre of the earth.

Exercise: Sketch the gravitational potential predicted using equation (2.5) as a function of distance, r. What is wrong with the prediction? (Hint: so far we have not considered anything other than point masses). Sketch a more accurate potential function for a solid earth. (Hint: see the next slide …)

Gravitational potential

How do we get gravitational acceleration from potential?

Answer: Gravity is the spatial derivative of potential

• first spatial derivative in any direction gives the component of gravity in that direction

• in mathematical language, g is the gradient of U, or

Use the equation for the gradient (2.6) and the formula for the potential (2.5) to rederive the formula for gravitational acceleration. (Hint: make use of the symmetry of the system!)

Gravitational potential

Gravity is always perpendicular to equipotential surfaces

(this is mathematically true)

Local variations in density:

• outside a perfect sphere, gravity effect is exactly that of a “point mass”

• gravity varies, in part, due to variations in density

If there is an excess of mass (increase in density) under the ocean, will this cause the equipotential surface to move A: down toward the centre of the earth, or B: up, away from the centre of the earth? Sketch the equipotential surface. Work out a reasoned argument for your answer. Add arrows to your sketch showing the local gravity vectors. Do these point slightly toward the density anomaly or slightly away from the anomaly? (Hint: make use of )

Geoid, spheroid – equipotential surfaces

Geoid: the actual equipotential surface of the earth (mean sea level)

• on continents, think of imaginary canels connected to the oceans

• responds to rotation, shape of the earth, and responds to density variations

Reference spheroid: a mathematical surface for a perfect, rotating, spheroidal earth

• takes into account radial variations in density, flattening, centrifugal effects

• accepted international formula is:

Geoid

Geoid

Geoid

Geoid, spheroid

Geoid, spheroid

Geoid, spheroid

Units used in gravity prospecting

Acceleration [m/s2] Geophysical prospecting 1 gal = 1 cm/s2= 10−2 m/s2

cgs units: 1 mgal = 10−3gals = 10−5 m/s2

SI units: 1 gu = 10−6 m/s2 (a “gravity unit”)Conversion: 1 mgal = 10 gu.Earth gravity field: ( approximately) ≈ 10 m / s2 = 1000 gals

= 106 mgals = 107 gu

Variation: Over the whole surface of the earth the gravity field varies by about 7,000 mgals Local effects: Due to local density changes alone, the variation is of the order of 10 mgalsSensitivity/accuracy: In order to make useful measurements, typical gravimeters need to detect changes in gravity of the order of 0.01 mgals (.1 gu). The absolute accuracy of a typical gravimeter is only about 0.1 mgal.

Therefore

Next lecture: Instruments used in gravity prospecting

Fundamental design of almost all gravity instruments uses a mass on a spring:

A change in gravity should cause a changein length given by

The problem with systems of this nature is they are natural oscillators

• restoring force overcompensates, mass overshoots equilibrium point

• solution is a system with no effective restoring force, an “unstable gravimeter”

• such a system has inherent periodicity, and mechanical instability