Gravity studies - folk.uio. History of gravity studies Gravity theory Measurement techniques Earth material

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  • Gravity studies

    As part of GEO-DEEP9300

    Maaike Weerdesteijn

    11-11-2019

    Courtesy: NASA Courtesy: red-leaf Courtesy: Airbus/GFZ Courtesy: macrovector Courtesy: EHT

  • Table of content

    • History of gravity studies

    • Gravity theory

    • Measurement techniques

    • Earth material characteristics

  • History of gravity studies

  • The first theories: Newton

    • Gravity field reflects mass distribution and shape of the Earth

    • Newton: shape of the Earth is an oblate body which had swollen in the direction of the equator

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    1642-1727

  • The first theories: J. Cassini

    • J. Cassini: shape of the Earth is longer along the north-south axis based on triangulation survey in France

    • Curvature of the Earth from the distance and latitude difference between the end points of a meridian arc

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    Pallikaris et al. (2009)

    1677-1756

  • Newton vs. J. Cassini

    • The French Academy of Sciences sent out a mission to find the truth • Bouguer to the equator in Ecuador

    • Maupertuis to the pole in Lapland

    • Meridian arc measurements close to the equator and close to the pole

    At pole

    - Meridian arc longer for fixed latitude difference

    - Smaller curvature: Earth flattened at poles

    • The Earth is flattened at the poles: Newton was right

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

  • The first gravity measurements

    • Huygens: Dutch geophysicist

    • Invention of precise clock pendulum for gravity measurements • Pendulum has same period when

    hung from its center of oscillation as when hung from its pivot

    • Distance between the two points was equal to the length of a simple gravity pendulum of the same period

    • Acceleration of gravity function of pendulum’s period, length, and amplitude

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    1629-1695

  • The first seaborne gravity measurements

    • Previous pendulum required stable platform

    • Prior to 1920: only continental measurements

    • 73% of Earth’s gravity field unknown

    • Vening Meinesz: Dutch geophysicist / geodesist • Invention of gravimeter with multiple pendulums

    • Mean periods of two pendulums

    • The mean not affected by horizontal disturbances

    • Seaborne gravity measurements

    • Increased Earth coverage

    Courtesy: Utrecht University archive

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    1887-1966

  • Gravity theory

  • Gravitational attraction

    • Newton’s law of gravitation • 𝐺 = 6.673 · 1011 Nm2kg-2

    • 𝐅𝟏 = −𝐺 𝑚1𝑚2

    𝑟21 2 𝐞𝟐𝟏

    • Newton’s second law of motion • 𝐅𝟏 = 𝑚1𝐚𝟏

    • Acceleration of 𝑚1 due to its attraction by 𝑚2 • 𝐚𝟏 = −𝐺

    𝑚2

    𝑟21 2 𝐞𝟐𝟏

    • Acceleration of attracted point mass is independent of its mass

    • Gravitational field 𝐠 𝐫

    • Gauss’s law: 𝛷 = −4𝜋𝐺𝑀, 𝑀 = σ𝑖𝑚𝑖

    • Gravitational field of a spherically symmetric body

    • 𝐠 𝐫 = −𝐺𝑀 𝐫

    𝐫 3

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

  • Gravitational potential

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    • Gravitation is a vector field: 𝐠 𝐫 = −𝐺𝑀 𝐫

    𝐫 3

    • Gravitational potential: 𝛻V 𝐫 = 𝐠 𝐫  V 𝐫 = 𝐺𝑀

    𝐫

    • The gravitational potential at point P VP is the work done to bring a unit mass from infinity to P

    • On a gravitational equipotential surface the gravitational potential VP is constant

    Courtesy: physbot

  • A rotating Earth: centrifugal potential

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    Gravitation ≠ gravity!

    • Acceleration of gravity = gravitational acceleration + centrifugal acceleration

    • 𝐠 𝐫 = 𝐚𝐠𝐫𝐚𝐯 𝐫 + 𝐚𝐜𝐞𝐧𝐭 𝐫

    • 𝐚𝐜𝐞𝐧𝐭 𝐫 = ω 2𝐩 𝐫

    • Centrifugal potential

    • 𝛻Z 𝐫 = 𝐚𝐜𝐞𝐧𝐭 𝐫  Z 𝐫 = 𝜔2

    2 𝐩2

    Courtesy: P. Ditmar

  • Gravity potential

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    • Gravity potential = gravitational potential + centrifugal potential

    • W = V + Z

    • Total acceleration of a mass at the Earth

    • 𝐠 𝐫 = 𝛻W 𝐫

  • Equipotential surfaces and geoid

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    • Vertical direction of gravity at a point: plumb line, unit vector 𝐧

    • Constant W: equipotential surface

    • Surface of the oceans approximately coincides with an equipotential surface

    • Mean sea is an surface equipotential surface: geoid

    Courtesy: P. Ditmar

  • Finding the geoid on land

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    • Geoid coincides with mean sea surface, but how about on land?

    • Orthogonal trajectory to the equipotential surface: line of force

    • Gravity vector is tangential to line of force

    • Distance H along a line of force: from point P at Earth’s surface to the geoid

    • Orthometric height

    Courtesy: P. Ditmar

  • Reference ellipsoid

    • Geoid surface W 𝐫 can be approximated by an ellipsoid of revolution

    • Ellipsoid level surface: reference ellipsoid

    • Difference between geoid and ellipsoid surface: geoid height N

    • Approximate gravity potential such that ellipsoid is equipotential surface

    • Normal gravity potential U 𝐫

    • 𝛻U 𝐫 = 𝛄(𝐫): normal gravity vector

    EGM96 model

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

  • Geoid heights and deflections of the vertical

    • Point P above reference ellipsoid

    • Normal projection of point P on ellipsoid: point Q

    • Distance between point P and Q: ellipsoidal height h

    • Deviation between plumb line and

    ellipsoidal normal: deflection of the vertical • ξ: deflection in North-South direction

    • η: defection in East-West direction

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    Courtesy: P. Ditmar

  • Disturbing potential

    • Relation between the geoid height N, the orthometric height H and the ellipsoidal height h: 𝑁 = ℎ − 𝐻

    • Difference between gravity potential at geoid W and at ellipsoid U • Disturbing or anomalous potential T: T 𝐫 = W 𝐫 − U 𝐫

    • T can be related to geoid height N: 𝑁 = 𝑇

    𝛾 is Bruns formula

    • Decomposition of gravity field W into normal field U and anomalous field T practical • U is large but can be described by very

    limited number of parameters

    • T is irregular but small:

    linear approximation often sufficient

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    Courtesy: P. Ditmar

  • Gravity disturbance and gravity anomaly

    • Gravity disturbance vector: 𝛿𝐠 = 𝐠 − 𝛄

    • Gravity disturbance: 𝛿𝑔 = 𝐠 − 𝛄 = 𝑔 − 𝛾

    • 𝛻T 𝐫 = 𝛿𝐠 𝐫

    • 𝛿𝑔 ≈ − 𝜕𝑇

    𝜕𝑛

    • Obtaining gravity disturbance practically • 𝐠 : measured • 𝛄 : computed • Precise ellipsoidal height needs to be known

    • Nowadays: from GPS

    • Before GPS: computation of gravity anomalies

    History of gravity studies Gravity theory Measurement techniques Earth material characteristics

    Courtesy: P. Ditmar

  • Gravity disturbance and gravity anomaly

    • Gravity anomaly: Δ𝑔 = 𝑔𝑃 − 𝛾𝑄

    = 𝑔𝑃 − 𝛾𝑃 + 𝛾𝑃 − 𝛾𝑄

    = 𝛿𝑔𝑃 + 𝛾𝑃 − 𝛾𝑄

    • After derivations I won’t bore you with…

    • Spherical approximation of fundamental equation of physical geodesy:

    • 𝜕𝑇

    𝜕𝑟 +

    2

    𝑅 𝑇 + Δ𝑔 = 0: gravity anomalies Δ𝑔 disturbi