1. Gravity Methods Gravity is not a “constant” 9.78 m/s 2 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)

2. Gravitational theory Newton’s law: In terms of vectors: Universal gravitational constant:

3. 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

4. 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)

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 …)

6. 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!)

7. 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 )

8. 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:

9. Geoid

12. Geoid, spheroid

15. Units used in gravity prospecting Acceleration [m/s 2 ] Geophysical prospecting 1 gal = 1 cm/s 2 = 10 −2 m/s 2 cgs units: 1 mgal = 10 −3 gals = 10 −5 m/s 2 SI units: 1 gu = 10 −6 m/s 2 (a “gravity unit”) Conversion: 1 mgal = 10 gu. Earth gravity field: ( approximately) ≈ 10 m / s 2 = 1000 gals = 10 6 mgals = 10 7 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 mgals Sensitivity/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

16. 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 change in 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