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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