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2.1 Gravity and the Gravity Field of the Earth

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2.1 Gravity and the Gravity Field of the Earth 2.1 Gravity and the gravity field of the Earth Fundamentals The mass and shape of the Earth The Geoid Satellite measurements of gravity GRACE - Gravity Recovery and Climate Experiment Fundamentals The force that attracts all matter to the Earth is the gravity field; an example of the general phenomenon of attraction between any masses. Newton's Law describes the force of attraction between two point masses, M1 and M2 separated by r: MM12 FG 2 r (2.1.1) The force per unit mass, F/M2 defines the vector gravity field or the gravitational acceleration, g when M1 is the Earth (Me) and r is the radius of the Earth, Re. So F M = g = G e r M R2 1 e (2.1.2) [Note that vectors are denoted either in boldface text or with an overlying arrow, e.g. r .] The gravitational constant G has been determined experimentally and is: -1- 6.67259 x 10-11 m3 kg-1 s-2 ( SI units ) or 6.67259 x 10-8 cm3 g-1 s-2 with an uncertainty of 100 parts per million (ppm). The gravitational acceleration is also measured experimentally and at the equator is about 9.8 meters/sec/sec (m s-2) or 980 cm s-2. (In almost all gravity surveys results are presented in c.g.s units rather than SI units. In c.g.s the unit acceleration, 1.0 cm s-2, is called the gal (short for Gallileo). A convenient subunit for surveys is the milligal, mgal, 10-3cm s-2. Another unit that has been used is the gravity unit, gu, which is defined as 10-6 m s-2 or 0.1 mgal.) Some properties of the gravitational force field are required for explaining the variations in g that are observed on and above the Earth's surface and for interpreting these variations in terms of the desired distributions of densities in the subsurface. A test mass M gains potential energy U in being moved a distance S in the gravitational force field F according to: BB U F dl M g dl (2.1.3) AA Figure 2.1.1 -2- For example, moving a test mass vertically along the radius for uniform spherical earth the difference in potential energy is: RRR2F 2 2 1 GM GM dr g dr GM dr ee RRR e 2 1MRR 1 1 r 21 GM the function U e is the potential of that point in space. All points in space R that have the same potential define an equipotential surface. By definition the force field is everywhere perpendicular to the equipotential surface. Potentials cannot be measured directly, only their force fields. Potential fields are conservative meaning that the work done in moving from A to B in Figure 2.1.1 is independent of the path taken. It is obvious from the simple example of the force between two masses that the force Mg is simply the derivative of the potential in the direction between the masses. In general, for any conservative field; gU (2.1.4) Potentials are additive so that for any point P external to a volume V of matter: Gdm r d3 r U r' G (2.1.5) P VVr'' r r r -3- Figure 2.1.2 This is the basic formula used to calculate the potential of any subsurface density distribution. [Note that this integral has the form of a convolution of the density (the source function) and the mathematical form of the fall off with distance of the field from a source point (the Green’s function). This concept will be used later to derive a fast and efficient method to calculate the gravity anomalies from subsurface bodies.] Another property of this conservative force field is that the integral of the normal component of the gravitational acceleration over the surface S of the volume containing mass M satisfies Gauss's law: g n dS 44 G dm G dV (2.1.6) VV -4- This relationship is easy to verify for a sphere of uniform density, but Equation 2.1.6 is more general, and applies to any volume with arbitrary density. We will use this integral later in the section on interpretation to determine the excess mass of any finite body in the subsurface. The divergence theorem states that: g ×n ds = Ñ× g dV (2.1.7) òS òV and gU so, U 4 G dV V or 2U4 G dV vV In any region where no matter is present, for example in the space above the earth, this leads to: 2U 0 Laplace’s equation (2.1.8) The mass and shape of the Earth The mass of the earth can be estimated through Gauss’ law: -5- g n dS 44 G dV GM (2.1.9) e V The radius of the earth Re was determined remarkably accurately by Eratosthenes in the third century B.C. using the observations shown in the sketch of Figure 2.1.3. This was arguably the first geophysical measurement. Figure 2.1.3 It is now known even more accurately through geodetic measurements. Assuming a spherical earth with a mean value of observed g, the experimental measurements of -6- 2 G, and the radius, the mass of the Earth is found from g4 Ree GM to be 5.976 × 1024 kg. If there were a perfect sphere of uniform density the gravity field measured on its surface would be the simple Newtonian field of a point mass at the center of the earth plus the radial acceleration term caused by the rotation of the earth as shown in Figure 2.1.4. Figure 2.1.4 The Earth is not a perfect sphere and it is very close to the shape that a spinning fluid would have. Because of the rotation the Earth is flattened as shown in Figure 2.1.5 and to first order the shape is that of an oblate spheroid or ellipsoid. -7- Figure 2.1.5 Any section through the axis of rotation is an ellipse in which the minor axis is the polar diameter and the major axis is the equatorial diameter. The flattening is described by the flattening parameter: RRep f (2.1.10) Re Early measurements of f were made geodetically, that is by measuring the varying length on the surface of a degree of latitude. More accurate measurements are now made using satellites. The orbit of an earth satellite over a spherical earth of uniform radial density distribution is a stationary ellipse. Because of the bulge associated with the flattening the extra ring of mass exerts a pull or torque on any orbit that is inclined to the equatorial plane. Accurate measurement of the resulting rotation of -8- the plane of the satellite orbit, known as the precession, leads directly to an estimate of the flattening [See the classic text by Kaula (1960)]. The Geoid Because the Earth is rotating the observed gravity field has an added component due to the centrifugal acceleration. At point A on the surface at latitude and radius r, Figure 2.1.4, the centrifugal acceleration, grot, in the direction normal to the surface is given by: 2 2 2 grot Rcos cos R cos where is the angular frequency of rotation. This centrifugal grot is in the opposite direction to the gravitational g so the observed g is given by: 22 gobs g z r cos The value of gz on the oblate spheroid is not a simple calculation and the credit for the solution, including the rotation term, is due to Clairaut. Clairaut solved the problem for the gravitational potential on the surface of the oblate spheroid. This surface is in fact an equipotential surface; the gravitational acceleration vector is everywhere perpendicular to this surface. The actual surface of this equipotential is defined as the geoid. The ideal surface derived from the symmetric ellipsoid is called the reference ellipsoid and it does not coincide exactly with the actual geoid because the Earth is not homogeneous in density and there are deformations in the actual geoid caused by subsurface inhomogeneities that invalidate the fluid earth hypothesis. -9- The mathematical description of the gravitational acceleration on the reference ellipsoid incorporates the flattening information from satellite orbits, the angular frequency, the equatorial value of g, and the equatorial radius in a closed form expression for the value of g as a function of latitude on the reference spheroid. A reference field, World Geodetic System 1984 (WGS84), is given by: 1 0.00193185138639sin2 g 9.7803267714 0 2 1 0.00669437999013sin (2.1.11) For this reference spheroid: The equatorial radius, Re, is 6378.137 km The flattening parameter, f, is 1/298.257 -2 Acceleration at the equator, ge, is 9.7803267714 ms This reference field is used to remove the first order variations due to latitude that are observed in gravity surveys. While not corresponding to the actual shape of the earth the equipotential ellipsoid is in a way a mathematical construct that is used by geodesists as a reference shape for surface elevation mapping and satellite positioning. The gravity field of this reference ellipsoid fulfills the same role for geophysicists interested in perturbations of the gravity field caused by subsurface density variations. The actual equipotential surface, the geoid, is usually described in terms of its departure from the reference ellipsoid. Alternatively the values of g on the geoid can be described with respect to the values of g for the reference ellipsoid. The actual geoid is found to be a very complex shape with undulations in its surface with a wide range of length scales.
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