Presentation Transcript

1. Gravity: mostly for exploration on a small scale

2. Inertial and Gravitational Mass

   There are two kinds of mass define by their equations: inertial (F=ma) and gravitational ( g=Gm/r2 ). But we now know that the two mass kinds are the same. Inertial mass is define as m = F/a. This mass can be measured by applying a force to an object and measuring its acceleration. This mass is thus a measure of the inertia of an object. And, inertia is the FACT that masses remain at rest or in straight-line constant velocity motion UNLESS acted upon by an unbalanced force. Gravitational mass is defined as m = g(r)*r2 / G. This mass can be measured by measuring the gravity field at a distance r. This mass is thus a measure of the gravitational acceleration field made by ALL objects (note: everything has mass).
3. Gravitational force law and Inertial force Law F is the magnitude of the force applied by mass 1 on mass 2 AND the force of mass 2 on mass 1. The direction of the two force is equal and opposite in direction. This unbalanced force will cause each mass to accelerate in inverse proportion to its mass: a = F/m.
4. Inverse square laws result from spherical area scaling The area of a spherically symmetric field increase as the radius squared; therefore, for the field energy to be conserved (as it must), then the field energy must decrease as the radius squared: e.g., the gravity law.
5. Calculate Assume G=1 m1=1 m2=2 Calculate all five term values for r=1 and r=4. Does the force of mass-1 ON mass-2 equal the force of mass-2 ON mass-1 ? Which mass make a greater gravitational acceleration field? Using the inertial force law F=ma, which mass accelerate more and why ? How can the gravity fields for the two masses be different, BUT the forces on each mass has the same magnitude (albeit opposite direction) ?
6. Gravitational force field around spheres
7. Calculating gravity for general shapes Problem: how to calculate gravity field between a point (zero dimension) mass (m2) and a general mass distribution (i.e., earth, asteroid)? Easy, divide the general shape into little squares (2-d) or cubes (3-d) and add up the vector forces applied the mass m2.
8. Calculating gravity of perfect spheres One caveat: the ability to treat a sphere’s mass distribution as a point is true ONLY IF the ms mass is beyond (outside) the radius of the sphere. This theorem found by Newton greatly simplifies the mathematics by treating the Me spherical mass as a point! Easy to apply to solar system as all the masses are near spherical. On the other hand, it is the Earth’s non-spherical ellipsoidal shape that makes the moon recede from the earth about 3.8 cm/yr and the earths rotation to slow by 0.002 s a day. When an object’s geometry can be approximated as a sphere, we integrate (add-up) the sphere’s gravity using spherical shells that extend over a small radial distance. In doing this integration, we find that the symmetry of the sphere makes the sphere’s gravity field to be the same as if ALL the mass was concentrated at the center of the sphere!!
9. Calculate gravity directly above a spherical mass anomaly The gravity signal directly above a sphere whose center is at 100 m depth and has a density contrast of 0.3 Mg/m3 is +1.048e-6 m/s2 . But, let us convert that answer to Mgal units. +1.048e-6 m/s2 * 1 Mgal/(10-5 m/s2 ) = 0.1 Mgal That is an large enough signal to be measured with a decent gravimeter.
10. Densities of Rocks and liquids Notice we are not using the MKS mass units of kilo-grams (kg). Mass is being reckoned in mega-grams with is a thousand (103 ) grams. Note that in general the substances density increase as follows: liquids, unconsolidated sediments, sediments, igneous/metamorphic rocks, minerals/ores. Remember that density is defined by the Greek letter rho as:
11. Calculating gravity at different points on surface To calculate the gravity effect of the irregular body above at point P1, the body is divided into small squares (parcels) and the many gravity vectors from all the parcels are added up to get the total gravity. Do the same analysis to get the gravity at the other points. An important detail. The gravity field is a vector quantity (has magnitude and direction). When measuring the gravity of the above situation, both the ‘pull of rest of Earth’ and the ‘total pull of excess mass of body’ are measured. Note that with respect to point P, the Earth and the ‘body’ pull at different directions. When can ignore this detail, because the Earth’s pull is so much greater, and just assume we are measuring the ‘vertical component of Fb‘.
12. Gravity field of a sphere on the 2-d surface If we measured the gravity at every point in the 45 km square plane, this is what the gravity field would look like for a buried spherical mass. What is the sign of the mass difference between the spherical mass anomaly and the background material ?
13. Gravity anomalies of a sphere a cylinder The y-axis is the gravity anomaly and the x-axis is distance from the center of the sphere/cylinder. This is just a cross-section through these 3-d objects. The cylinder extends to +/- infinite in and out of the page. This is why, for the same depth object and mass anomaly, the cylinder and sphere anomalies are different. Important: Note that the peak amplitude reduces and the anomalies ‘half-width’ widens as the anomaly is placed deeper. This is just a consequence of gravity being an ‘inverse square law’.
14. Gravity anomalies of dipping narrow mass sheets Note three gravity effects: As the depth to the top of the anomalies increases the peak amplitude of the anomaly decreases. As the sheet anomaly dips to one side, the anomaly’s peak value moves to that side. As the sheet anomaly dips to one side, the anomaly has a long-tail on that side, and a short tail on the other side. From these effects, we can determine the dip of the sheet anomaly.
15. Gravity of Horizontal sheet The gravity anomaly of an infinite horizontal sheet at depth d and width t provides an interesting rendezvous with infinity. First, note that the sheet depth d is NOT in the equation. Second, even though the mass sheet anomaly extends to infinity, the gravity is finite because of the ‘inverse square law’. Third, the gravity effect is the same everywhere as demonstrated above where it can be seen that the gravity at points P1 and P2 are the same. Thus, an infinite sheet anomaly makes the gravity everywhere change by the same values as given by Eqn. 8.6. Therefore, one cannot detect an infinite mass sheet. But, we will use this concept to calculate the Bouguer gravity correction.
16. Gravity effects of half-sheets Figure (a) shows a stratigraphic section with different layer densities that has be offset by a fault. Figure (b) shows the layer densities changed into density contrasts so that we can easily calculate the total gravity anomaly associated with the variable vertical distribution of density. From this the motion of the fault can be determined. Note two effects: The deeper sheet has a smaller peak anomaly. The deeper sheet has a wider anomaly half-width. What happens if the mass anomaly sign is reversed ?
17. Measuring gravity with a gravimeter In a simple sense, a gravimeter measures gravitational variations by using Hooke’s Law of elasticity that states that the force required to extend a spring is F = k * dx where k is the spring stiffness and dx is a small displacement of the spring. So, as the force of gravity applied to the mass (m) varies, the mass position changes (dx) proportional to the spring stiffness (k). Note: we are only measuring relative variations, not the absolute value of gravity.
18. Measuring gravity redux In practice, measuring gravitational variations requires a very precise instrument that costs >40,000 dollars. The temperature must be keep stable to <1° C using thermistors and heaters. The beam must be very accurately engineered so that the change in force associated with the change in the spring-beam angle is compensated. A modern instrument can measure changes in gravity to one part in a million (i.e., 1 inch height variation).
19. Gravity Corrections As is true of most all measurement of physical properties, there are always effects that change the measured values that we are NOT interested in and that we desire to remove as accurately as possible. In the case of gravity , there are five gravity effects to correct for: Latitude Free-air: Distance above mean sea-level (or another datum). Bouguer: mass changes associated with movement from sea-level. Topography: the irregular pull of the mountains and valleys. Eotvos: moving E-W with a non-zero velocity (airplane or boat) adds or subtracts to the earth’s Centrifugal force. An important point is that we measure gravity at whatever value our gravimeter reads, and THEN we correct that data for these different effects that we are not interested in. This is called the free-air or Bouguer or isostatic corrected data.
20. Latitudinal gravity corrections Gravity varies from 9.78 m/s2 at the equator (lat=0°) to 9.83 m/s2 at the poles (lat: north = +90°; south = -90°). This is a huge change: a 0.052 m/s2 variation equals 5200 mgals! This is much larger than other gravitational effects. The gravity varies with latitude for two reasons: The Earth is not a sphere, but a flattened spheroid with an equatorial radius of 6,378 km and a polar radius of 6,356 km (21 km different). Thus, the gravity is LESS at the equator because it is FARTHER AWAY from the Earth’s center of mass. The Earth is a non-inertial reference frame because it is a rotating body that spins once per day. At the equator any object has a rotational velocity of 465 m/s, whereas at the poles the rotational velocity is zero! Physics requires that a rotational reference frame has non-inertial (fictitious) forces such as the outward directed centrifugal force. The centrifugal force is the force that any mass rotating with the planet ‘feels’ in response to the centripetal force that the planet’s gravity field provides to continually curve an object’s path into a circular path. Recall Newton’s first law says that all masses go in a straight line in a INTERTIAL reference frame unless acted on by an unbalanced force (it is gravity that provides the centripetal acceleration). The International gravity formula that describes latitudinal (Ѳ) gravity variations in m/s2 units is:
21. Topographic corrections The free-air correction using sea-level as a datum is 0.3086 Mgal/m. If gravity is measured above one’s datum the effect is subtracted and if gravity is measured below one’s datum the effect is added. The Bouguer correction uses the infinite sheet gravity equation to approximate the gravity of the material above or below sea-level. The relevant quantities are the thickness of the sheet (h), the sheets density, and the sign of h (positive if above base station and negative if below base station). Combining the free-air and Bouguer gravity effects gives Eqn. 8.11
22. All together: adding or subtracting the gravity corrections It is very important to keep physical track of the sign of the corrections; if you do not, you will get the wrong answer. Remember, we are correcting all the data to remove unwanted effects. The free-air effect is added if you are above sea-level and is subtracted if you are below sea-level. The Bouguer effect is subtracted if you are above sea-level (+h) and subtracted if you are below sea-level (-h). Total correction to Bouguer: Bouguer = observed – latitude + free-air – Bouguer Total correct to Free-air: Free-air = observed – latitude + free-air
23. Regional and residual anomalies Note that there are three different ‘structures’ - dyke, granite, dipping strata - associated with mass anomalies that create different gravitational effects. Often, we surveying at a small scale (e.g., for the dyke and granite bodies), we are NOT interested in the larger scale ‘regional’ gravity effects (e.g., the dipping strata). Thus, we ‘reduce’ the data by subtracting an ‘eyeball’ estimate of the regional gravity trend (dotted line). After subtracing the regional gravity trends, we can more easily see the short scale ‘residual’ features we are interested in studying.
24. Gravity model non-uniqueness (the inversion problem) Note that in (a) that three different gravity (mass) models make the same gravity! Note in (b) that two different gravity (mass) models make the same gravity!
25. Another gravity non-uniqueness example A positive mass excess on the left side make the same relative gravity profile as a negative mass excess on the right side!
26. Depth/half-width rules for different geometry mass anomalies Note systematic variations between the depth to bodies and the half-width of the gravity profiles. Sphere. Horizontal cylinder. Steeply dipping sheet. For the Irregular body, the peak anomaly and maximum slope value are required.
27. Modelling a basin To model irregular shapes like a basin, a set of rectangular mass anomalies can be used. Then, the total gravity effect of the basin is found by adding up all the gravity from the rectangles. Volume is the TOTAL volume under the 2-d gravity profile! If you know, the density of the body and its surroundings, the mass of the anomaly maybe calculated (e.g., weight of gold!).
28. Wyoming isostatic and bouguer gravity Isostatic Bouguer
29. Colorado Bouguer/Isostastic gravity Isostatic Bouguer
30. Chicxulub K-T Impact crater Bouguer gravity: Yucatan, Mexico
31. Free air gravity from Satellites
32. Temporal gravity variations
33. Mars Free Air Gravity