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Gravity 1

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Topics Gravity and gravitational potential Gravity and shape of the Earth Isostasy Gravity anomalies/corrections Gravity modeling Gravity and tectonics

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Gravity – basic principle mass of Earth (M E ) x mass (m) (distance between masses (R)) 2 Force = gravitational constant (G) x (Newton’s Universal Law of Gravitation) M m R F F (Newton’s Second Law of Motion) 1666

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Gravity – basic principle Mm R F F G – gravitational (Newtonian) const G = 6.67 x 10 -11 m 3 kg -1 s -2

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Fundamental Properties of Gravity 1.The acceleration due to gravity (g) does not depend on the mass m 1, attracted to the Earth. 2.The further from Earth’s centre of mass (R >>), the smaller is the gravitational acceleration (gravity obeys inverse square law). Galileo Galilei Galileo supposedly reached out from an upper balcony of the Leaning Tower of Pisa and let fall two stones of different weights. What happened? http://www.daviddarling.info/encyclopedia/A/accgrav.html (Gallileo is the name of a skyscraper in Frankfurt.)

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Why g is different? Three factors for the difference of g: 1. The spin (rotation) is greatest at the equator of the Earth but reduces to zero at the poles. 2. There is bulging (R > at the centre). 3. The added mass of the bulge creates more acceleration. 40% of g variations - due to the difference/variations in the spherical shape 60% of g variations – due to the rotation g is measured in m/s 2 or gal [ 1 gal = 10 -2 ms -2 ] centrifugal acceleration

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Gravitational Potential and Energy Gravitational Potential: Potential of the Earth (general form) - quantity of energy that is associated with the position of unit mass in the gravitational field Definition 1: The gravitational potential at a point is equal to “negative” of the work by the gravitational force as a particle of unit mass is brought from infinity to its position in the gravitational field. Definition 2: The gravitational potential at a point is equal to the work by the external force as a particle of unit mass is brought from infinity to its position in the gravitational field. [ m 3 kg -1 s -2 kg m -1 = m 2 s -2 ] or

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Gravitational Potential and Energy Gravitational Potential (defined for unit mass): [ m 3 kg -1 s -2 kg m -1 = m 2 s -2 ] Gravitational Potential Energy of a mass m (or m 2 ): Released potential energy if mass m 2 moves from p.1(r 1 ) to p.2 (r 2 ): Gravitational acceleration: or

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Gravitational Potential and Energy Gravitational acceleration : Generalization - from 1-D to 3-D: For a distribution of masses m i at positions r i : or - For infinitesimal masses dm at positions r

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Potential of a Spherical Shell The potential at a point P at distance r from the centre O of a shell is: t – thickness of the shell b – radius; - density; r - distance When P is outside, the potential is the same, as the whole mass is at point O. When P is inside, the potential is a const, which is independent of position of P: r a = 0 inside the shell (a – acceleration)

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What is oblate spheroid? An oblate spheroid is a rotationally symmetric ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane bisects it. Elipticity: Eccentricity: Not the same as … ReRe RpRp R e > R p f = 1/298.247

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Sphere vs Oblate spheroid? f = 1/298.247 “Radius” of the oblate spheroid: – latitude For a perfect sphere – the gravitational acceleration is the same in every point on the surface of the sphere. For the Earth surface this is not true.

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Centrifugal Acceleration (Acceleration due to the rotation of the Earth) angular frequency

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Reference Gravity Formula (Accepted 1967 by the International Association of Geodesy) Gravity observations are expresses as deviations from (1) (1)

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Reference Spheroid A mathematical figure whose surface is equipotential of the theoretical gravity field of a symmetrical spheroidal Earth Model with realistic radial variations in density, plus centrifugal potential (Formula 1)

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The shape of the Earth Geoid – the Earth’s reference: - The mean sea level over the ocean - The level at which the water would lie if there were imaginary canal cut through the continents Plumb bob Reference spheroid (reference ellipsoid)

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Geoid http://cddisa.gsfc.nasa.gov/926/egm96/egm96.html A hole ~100m

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Geoid

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Kepler’s Laws LAW 1: The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus. LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time. LAW 3: The squares of the periods of the planets are proportional to the cubes of their semi major axes: T a 2 / T b 2 = R a 3 / R b 3 (describe the motions of the planets around the Sun)

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Orbits of satellites T a 2 / T b 2 = R a 3 / R b 3 – Kepler’s Third Law As the gravitational force of the Earth acting on the satellite is balanced by the outward centrifugal force: The equation solved for angular frequency is: Alternatively, solving for the period of the satellite (T):

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Satellite over the Earth – Goce gravity satellite http://news.bbc.co.uk/1/hi/sci/tech/6919016.stm 1. Goce senses tiny variations in the pull of gravity over Earth 2. The data is used to construct an idealised surface, or geoid 3. It traces gravity of equal 'potential'; balls won't roll on its 'slopes' 4. It is the shape the oceans would take without winds and currents 5. So, comparing sea level and geoid data reveals ocean behaviour 6. Gravity changes can betray magma movements under volcanoes 7. A precise geoid underpins a universal height system for the world 8. Gravity data can also reveal how much mass is lost by ice sheets http://www.bbc.co.uk/mediaselector/check/player/nol/newsid_6920000/newsid_6920000?redirect=6920034.stm&news=1&bbwm=1&nbwm=1&bbram=1&nbram=1&asb=1

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Satellite over the Earth – Goce gravity satellite http://news.bbc.co.uk/1/hi/sci/tech/6919016.stm 1.The 1,100kg Goce 2.Solar cells produce 1,300W 3.The 5m-by-1m frame to stabilize the spacecraft 4.Goce's accelerometers ~ 1 part in 10,000,000,000,000 of the gravity experienced on Earth 5.Goce's mission will end when the 40kg fuel tank empties 6.S Band antenna: Data downloads to the Kiruna (Sweden) 7.GPS antennas: Precise positioning gradiometer

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Gravity on the Earth - units miliGals (mGal) 1 Gal = 1 cm/s 2 =0.01 m/s 2 =10 -2 m/s 2 1 mGal = 10 -3 Gal = 10 -3 cm/s 2 = 10 -5 m/s 2 g = 9.78 m/s 2 (at the equator) to 9.83 m/s 2 (at the poles) g = 0.05 m/s 2 = 5,000 mGal

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