Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Equations of motion wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Centripetal force wikipedia , lookup
Center of mass wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Mass versus weight wikipedia , lookup
Moment of inertia wikipedia , lookup
Earth's rotation wikipedia , lookup
0 Gravity, Rotation, Shape of the Earth 0.1 Inertia. Conservation laws. Gravity Inertia. Energy. Angular momentum Moment of inertia of Earth Gravity field and potential Inertia. Pseudo-gravity potential 0.2 Figure of equilibrium of the Earth. Condition for equilibrium in a fluid. Gravity field and potential Inertia. Pseudo-gravity potential Reference ellipsoid. Geoid. 0.3 Earth’s gravity field Variations in sea level gravity. Reduction of gravity measurements. Free air and Bouguer corrections. Anomalies. Isostasy. Post glacial rebound. Mantle viscosity. Post glacial adjustments and earth rotation. 0.4 Tides. Ocean and solid Earth Secular slowing down of Earth rotation. Periodic variations in Earth rotation. Length of day (LOD) variations . Chandler Wobble (T ≈ 435 j ) 1 0.5 Milankovicic cycles Earth’s orbit. Variations of orbital parameters: eccentricity (T ≈100,000 yr), inclination of rotation axis (T ≈ 41, 000 yr), perihelion and precession of equinoxes (T ≈ 25, 000 yr). Cartwright, D.E. (1999). Tides: A scientific history. Cambridge. pp. 292. Hays, J.D., Imbrie, J., & Shackelton, N.J. (1976). Variations in Earth orbit: pacemaker of the ice ages. Science, 194. 1121-1132. Lambeck, K. (1988). Geophysical geodesy. Oxford Science Publication. pp.718. Smylie, D.E., & Mansinha, L. (1971). The rotation of the Earth. Sci. Am., 225. 80-88. Watts, A.B. (2001). Isostasy and flexure of the lithosphere. Cambridge University Press. pp. 458. 2 1 Gravity potential. Gravity field. 1.1 Inertia Newton’s first law of motion: A body at rest or in uniform motion will not change its velocity unless forces are applied. (Galileo) Newton’s second law of motion: The relationship between an object’s mass m, its acceleration ~a, and the applied force f~ is m~a = f~ Newton’s third law: action = reaction 1.2 Gravity The gravity force F~ between two point masses m and m0 is obtained from Newton’s gravity law: ~ Gmm0~u Gmm0 R F~ = = R2 R3 where R is the distance between the two masses, G = 6.67×10−11 N m2 kg−2 is the gravity constant. The force is attractive and it is parallel to the line joining ~ is the vector joining the two masses and ~u = R/R ~ the two masses (R is the ~ unit vector in the direction of R). This an example of action and reaction: the attractive force of m to m0 is equal to that of m0 on m (examples Earth-Sun, Earth-Moon systems.) Although there is no obvious reason that it should be so, the property ”mass” that enters in the gravity law, and that in the inertia law have always been found to be proportional to each other: the proportionality constant was chosen to be 1. (General relativity is based on the equivalence principle that states that inertial and gravitational masses are identical). A particle of mass m under the gravity forces of other particles will experience an acceleration ~g that does not depend on its mass. This acceleration which characterizes any point in space is the gravity field, or the acceleration of gravity. Z ~ P 0 P dV (P 0 ) ~k X mk R ρ(P 0 )R = G ~g (P ) = G Rk3 RP3 0 P k 3 ~ k is the vector from P to mass mk (R ~ P P 0 = {(x0 −x), (y 0 −y), (z 0 −z)}) where R and ρ is a mass density distribution. The gravity field is measured in m s−2 . It is a conservative field, i.e. because the work done by gravity to go from one point to another is independent of the path, it can be derived from a potential function. The gravity potential U in point P represents work done by gravity when a unit mass is brought from P to ∞: W (P ) = It gives: Z ∞ P Z ˙r ~g (~r)d~ ρ(P 0 )dV (P 0 ) RP 0 P U (P ) = G with ~g = ∇P U The acceleration of gravity is thus normal to the equipotential surfaces. The gravity field of a spherically symmetric body with mass M is identical to the field of a point mass at the center of the sphere. Outside the sphere, the gravity field does not depend on the radial density distribution. The potential and the field at distance R from the center of mass: M R M g = −G 2 R U =G the negative sign indicates an acceleration toward the center of the sphere. These equations could have been derived from Gauss theorem that states that the flux of the gravity field across a closed surface is proportional to the total mass enclosed by the surface I S Z ~g · ~ndS = −4πG V ρdV = −4πGM 1.3 Units Acceleration is measured in m s−2 . The practical unit in geophysics is the mGal (1 mGal = 10−5 m s−2 . For the potential, units m2 s−2 or J kg−1 are equivalent. Note that the potential represents an energy per unit mass. Geoid anomalies are measured in m. 4 2 Moment of inertia. Angular momentum and rotational energy The moment of inertia I of a body relative to an axis: I= X Z mk rk2 = ρr2 dV k where rk is the distance between the axis and mass mk . Moment of inertia is given in kg m2 . In general, for a non spherically symmetric body, the value of moment of inertia depends on the direction of the axis. One should thus use the tensor of inertia. In a coordinate system with origin at the center of mass, there are three perpendicular axes, the principal axes of d’inertia, respective to which the moment of inertia is maximum, minimum, and intermediary. Note that for a spherically symmetric body, any axis through the center is principal. For an ellipsoid of revolution, the axis of symmetry is principal axis. Any axis in the equatorial plane is also principal axis. There are only two distinct principal values for the tensor of inertia. For a uniform sphere, with radius a , I = 0.4M a2 . For Earth, the polar moment of inertia (i.e. relative to the rotation axis) C = 0.33M a2 . This value implies an increase in density toward the center of Earth, i.e the existence of a dense core. 2.1 Conservation laws The rotation of a body around an axis is described by the rotation vector ω ~ which is parallel to the rotation axis and such that the velocity of a point in the body: ~v = ω ~ × ~r where ~r is a vector from the axis of rotation to the point. The magnitude of the rotation vector is ω = 2π/T (where T is the period of rotation) The rotation is stable only if the rotation axis is minimum or maximum principal axis of inertia (This is indeed the case for a gyroscope, a spinning top, and the Earth). The kinetic energy of the body: Ecin = 1X 1X 1 mk vk2 = mk ω 2 rk2 = Iω 2 2 k 2 k 2 5 The net momentum is zero. The angular momentum is the sum of the moments of the momenta relative to a point on the axis of rotation. The angular momentum J~ X X J~ = mk~rk~vk = mk rk2 ω ~ = I~ω k k Without a torque exerted by external forces the angular momentum must be conserved. A mass attached to a rotating body experiences inertia. The effect of inertia is sometimes referred to as the centrifugal acceleration. ~a = −ω 2~r with ~r is the distance from the axis. And the sign indicates an acceleration away from the axis. This acceleration corresponds to a pseudo-potential V : V = −ω 2 r2 2 r is the distance to the axis of rotation. On Earth, it depends on latitude λ r = R cos(λ) 3 Figure of equilibrium of a fluid. The condition for a fluid to be in equilibrium is that all the forces acting on a parcel balance each other. If there is no motion, the only forces are differences in pressure and gravity. The condition is thus that ~g + ∇P = 0 This implies that all the isobaric surfaces are equipotentials, and in particular that the outer surface of the fluid is an equipotential. (For example, mean sea level corresponds to an equipotential of the gravity. For Earth, it can be shown that the equilibrium figure is an ellipsoid with the major inertia axis is the rotational axis. The gravity potential of an ellipsoid can be written as: U (R) = G 1 −GM 3 + 3 (C − A) cos2 θ − R R 2 2 with C moment of inertia relative to the rotation axis and A equatorial moment of inertia (in the direction perpendicular to the axis of rotation), θ 6 colatitude. Including the pseudo-potentiel of rotation, we get: −GM GM a2 J2 U (R) = + R R3 3 1 1 cos2 θ − − ω 2 R2 sin2 θ 2 2 2 with J2 = (C − A)/M a2 ) (it is a dimensionless number). If the equatorial et polar radii are a et c. The flattening is defined f = (a−c)/a. The flattening is determined by the condition that the potential is the same at the pole and the equator. 3 ω 2 a2 f = J2 + 2 2ge with ge gravity at the equator. As the Earth polar radius is c is about 20km shorter than the equatorial radius a, the flattening is about 1/300. It has been measured and calculated very precisely. The observed and predicted values of the flattening ≈ 1/300 are very close, but there is a small difference due to recent changes in Earth’s rotation rate. The theoretical equipotential surface defining mean sea level is the reference ellipsoid. The observed mean sea level equipotential is the geoid. Geoid anomalies are due to lateral variations in density within the Earth. They are measured in m. The fact that they are small is another indication that the Earth is close to hydrostatic equilibrium. 4 Gravity anomalies The gravity acceleration is the direction perpendicular to the equipotential. It is not constant on an equipotential. (The acceleration is inversely proportional to the distance between equipotentials.) As the polar radius is shorter than the equatorial, it is thus expected that g will be larger at the pole that at the equator. For a homogeneous (rotating) Earth, sea level gravity depends only on latitude φ and is given by: gref = ge (1 + α sin2 φ + β sin2 (2φ)) with ge = 9.780327m s−2 , α = 0.0053024, β = 0.0000058. This is the International Gravity Formula 1967. The difference between observed and calculated density defines the gravity anomalies. As gravity measurements are made at an elevation different from 7 sea level, the observed field must be reduced to sea level with different corrections. • The free air correction accounts for the changing distance to Earth center of mass. This is a positive correction (0.3086mgal m−1 ) when gravity is measured above sea level. • The Bouguer correction accounts for the masses between the observation surface and sea level. It is calculated for an infinite slab of density ρ and thickness H. It is equal to 2πGρH. It depends on the density of the rocks and is −0.118mgal m−1 for rock density = 2670kg m−3 . Note that the Bouguer correction is exactly half the correction that we would make for a spherical shell of thickness H. The correction is 4πρH. There are other corrections to account for variations in topography. Very precise measurements even require tidal loading corrections. Thus, free air (or Faye) anomaly is the difference: ∆gF = gmes + 0.3086H − gref with H is the elevation and g is measured in mGal. The Bouguer anomaly is ∆gB = gmes + 0.3086H − 2πGρH − gref Bouguer anomalies are negatively correlated with elevation, while free air are not. This observation is the basis for the concept of isostasy, i.e. topography of the Earth surface must be compensated at depth. 5 Tides Tides are caused by the difference between the gravitational pull of Moon and Sun, and the inertial forces due to Earth’s orbital rotation around the Sun and Moon (or more precisely the center of mass of the Earth-Moon system). At the center of mass of the Earth, gravity forces balance the inertia. But gravity varies with distance, so the gravity pull is stronger on the day side (toward the Sun) and weaker on the night side. So on both sides there will be a small force away from the center of Earth. Likewise for the Moon. The solid Earth, itself is deformed by these tidal forces and the the effect of this deformation can be observed in the time variations of the gravity field. 8 5.1 Lois de Kepler Les observations astronomiques de Kepler lui avaient permis de formuler certaines lois du mouvement des planétes autour du soleil. La troisième loi relie la période T de l’orbite d’une planète à sa distance moyenne au soleil R. Cette loi peut-être dérivée de façon très simpliste en exprimant que l’accélération due à la force gravitationnelle doit être égale à l’accélération inertielle. Pour une orbite circulaire de rayon R autour d’un corps de masse M, il faut donc: ω2R = GM R2 En remplacant ω = 2π/T , on obtient: T2 = 4π 2 R3 ∝ R3 GM 9