Sub-Unit 2.2 – Earth’s Figure and Gravitation
 The figure of the
Earth
 Gravity and its
potential
 Normal gravity
 Satellite geodesy
Objectives:
•
•
•
•
•
•
Define the figure of the earth
Identify the cause of polar flattening
Define Normal gravity
Know what a Geoid is
Define Satellite Geodesy
Know how Satellite Laser Ranging is use to
identify plate movement
• Know the Method of Global Position System
THE EARTH’S FIGURE AND GRAVITY
What is that true physical surface our Earth?
Assume surface of the Earth
-smooth ellipsoid
True physical surface of the Earth is:
• uneven and irregular,
• partly land and partly water.
The figure of the Earth
• assumed smooth shape is an
oblate ellipsoid or Spheroid.
Spheroid shape is:
• the equipotential surface of gravity,
• correspond to mean sea level
• called International Reference
ellipsoid
Reference ellipsoid has two radius
values:
i. equatorial radius (a) = 6378.137
km and
ii. polar radius (c) = 6356.752 km
Comparison with an equivalent sphere
radius
• radius of the equivalent sphere (R) =
6371.000 km
Overlap of Equivalent Sphere and Ellipsoid
• spheroid is flattened by about 14.2 km at
each poles (𝑅 − 𝑐 = 14.2𝑘𝑚)
• equator bulges by about 7.1 km. (𝑎 −
𝑅 = 7.1𝑘𝑚)
Polar Flattening Effect
𝒇=
𝒂−𝒄
𝒂
(a) = 6378.137 km and
(c) = 6356.752 km
𝒇=
(𝟔𝟑𝟕𝟖. 𝟏𝟑𝟕𝒌𝒎) − (𝟔𝟑𝟓𝟔. 𝟕𝟓𝟐 𝒌𝒎)
= 𝟎. 𝟎𝟎𝟑𝟑𝟓𝟐𝟖 (𝟎. 𝟑%)
(𝟔𝟑𝟕𝟖. 𝟏𝟑𝟕𝒌𝒎)
What causes flattening ?
flattening's are zero for
a circle (a = c)
The cause is:
• centrifugal acceleration – due to the rotation of the earth
Flattening caused by centrifugal acceleration
Increase rotational speed
Centripetal acceleration
reduces towards equator
Centrifugal force
increases towards equator
Flattening occur
𝑀
𝑎𝐺 = −𝐺 2 𝑟
𝑟
𝑣2
𝑎𝑐 =
𝑟
Effect of Centrifugal acceleration
Centrifugal acceleration is:
•
maximum at the equator where the gravitational and
centripetal acceleration is the smallest
• Minimum at poles where 𝑎𝑐 and g is max.
Normal Gravity
Normal gravity
Normal Gravity (g)
Define as sum of mean gravity
acceleration (𝑎𝐺 ) and centrifugal
acceleration (𝑎𝑐 )
• direction at a point is perpendicular to
the equipotential surface through the point
• g is not radial except equator and poles
Latitude of the Spheroid
observed point
2 types of latitude:
1. Geocentric latitude (𝝀′)
• angle of the line connecting an
observed point to the geocentre,
relative to equatorial plane.
2. Geographic Latitude (𝝀) =
• angle between the ellipsoidal normal
relative to the plane of equator
• angle 𝜆 slightly greater than
geocentric latitude 𝜆′.
• g (gravitation potential) is the vertical
at observation point
Difference of the latitude
• Zero at equator and poles
• Maximum at 45°
Equatorial
plane
Geocentre
Formula for gravity normal to the ellipsoid:
𝒈𝒏 = 𝒈𝒆 (𝟏 + 𝜷𝟏 𝒔𝒊𝒏𝟐 𝝀 + 𝜷𝟐 𝒔𝒊𝒏𝟐 𝟐𝝀)
𝒈𝒆 = 𝟗. 𝟕𝟖𝟎 𝟑𝟐𝟕𝒎𝒔−𝟐
Gravity value at equator
𝜷𝟏 = 𝟓. 𝟑𝟎𝟐𝟒𝟒 × 𝟏𝟎−𝟑
𝜷𝟐 = −𝟓. 𝟖 × 𝟏𝟎−𝟔
Normal gravity increase as geographic
latitude (𝝀) increases
Normal gravity formula is important in:
• analysis of gravity measurements on the Earth, because it gives the theoretical
variation of normal gravity (𝒈𝒏 ) with latitude on the surface of the reference ellipsoid
The geoid
Before looking at Geoid, lets look at the shape of Reference ellipsoid
International Reference Ellipsoid
Homogenous material
International Reference
Ellipsoid - Theoretical
Mathematical model of
smooth equipotential surface
• From Assuming that the
distribution of mass beneath
the ellipsoid is
Homogeneous (same
material, same mass)
Smooth ellipsoid
Effect of different mass material inside parts of Reference ellipsoid causing changes to
equipotential surface:
• Greater mass (access mass) regions – ellipsoid warp above normal line (g increase )
• Lesser mass regions – ellipsoid warp below normal line (g decrease)
(reference ellipsoid = equipotential surface)
𝐹𝑔 = 𝑚𝑔 − 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑚𝑎𝑠𝑠
access mass
Equal mass ellipsoid
9.8
9.8
lesser mass
9.8 shifted above
10
Heterogeneous material
Rough ellipsoid
Is called Geoid
The geoid
Geoid is define as physical equipotential surface of gravity on
Earth’s surface or surface of earth with equal gravity.
Geoid:
• shows true distribution of mass inside the Earth
• differs from the theoretical ellipsoid by small amounts
• is at same level as mean sea level far from land
• Is affected by mass of continent and variation of mass below
ellipsoid
Geoid undulation – displacement above or below reference
ellipsoid
i) Geoid line over the sea or on land without variation of mas present
Geoid line
ellipsoid
Mean sea level
ii) Positive Undulation - with access mas present above (like mountains) and
below, ellipsoid creates positive undulation
pull
Centre of mass
outside ellipsoid
pulls geoid line up
push
push
iii) negative undulation – bending downward of geoid due to the presence of less mass
body
geoid
ellipsoid
Less mass body
As a result of the uneven topography and heterogeneous internal
mass distribution of the Earth, the geoid is a bumpy equipotential
surface
A
B
C
• Orthometric Height or Geodetic Height - vertical distance from a location
on the Earth's Surface distance to the geoid (Red surface).
Because the earth geoid is set at the level of the average sea level it is often
called the elevation at Mean Sea Level (MSL).
• Ellipsoidal Height of that same point of the Earth Surface is the vertical
distance from that point to the ellipsoid
The largest negative undulation (-105 m) is in
Causes are yet to understand
the Indian Ocean south of India, and the largest positive
undulation (+73 m) is in the equatorial Pacific Ocean
north of Australia (Papua New Guinea).
Satellite Geodesy
Satellite Geodesy
- Study of geodesy by using artificial satellites
The purpose of using artificial satellite is to give more accurate
information on:
• the form and dimensions of Earth,
• locating objects on earth’s surface
• Enhance Geoid model
• give accurate figure of the Earth's gravity field
What is an artificial satellite?
Satellite - a craft capable of traveling
in outer space
• Have ability to collect data, more
quickly, than instruments on the
ground
Artificial Satellite motion
Satellites:
• move around the Earth forming a loop path called its orbit
• plane of the orbit intersects the equatorial plane in the line of nodes
Plane of orbit
line of nodes
Equatorial plane
Artificial satellite motion in Earth
orbits are influenced by:
• Earth’s mass distribution
- balance between centrifugal force
and gravitational attraction of the
Earth’s mass, which increase and
decrease the radius of the satellite’s
orbit
• attraction of the equatorial bulge
on the satellite causes the orbit of
the satellite to precess around
the rotation axis.
Illustration of precession:
• satellites first orbit = line 𝐶𝑁1
• satellites second orbit = line 𝐶𝑁2
• precession of the orbit has moved the nodal line to a new
position line 𝐶𝑁2
The orbital precession in this
case is retrograde; the nodal line
regresses
Shifting of Nodal line direction
i. satellite orbiting in the same sense as the Earth’s rotation the
longitude of the nodal line shifts gradually westward;
ii. orbital sense is opposite to the Earth’s rotation the longitude of
the nodal line shifts gradually eastward.
Because of the precession of its orbit
the path of a satellite eventually
covers the entire Earth between the
north and south circles of latitude
defined by
the inclination of the orbit
Observations of satellite orbits are so precise that small disturb of the
orbit can be related to the gravitational field and to the geoid.
Orbital inclination of satellites
Define as the angle between the plane of an orbit and the
equator.
• Zero orbital inclination is directly above the equator,
• 90° crosses right above the pole
TYPES OF ORBITS
TYPES OF ORBITS
High Earth Orbit / Geosynchronous Orbit
• When a satellite reaches exactly 42,164 kilometers from the center of the Earth
(about 36,000 kilometers from Earth’s surface),
• it enters a “sweet spot” - its orbit matches Earth’s rotation.
• satellite seems to stay in place over a single longitude
• high Earth orbit is called geosynchronous.
Function:
• Weather monitoring
• Communication
Medium Earth Orbit/ Intermediate circular orbit (ICO)
• Orbit at the height of 26,560 kilometers from the center of
the Earth (about 20,200 kilometers above the surface)
• takes 12 hours to complete an orbit
• used by the Global Positioning System (GPS) satellites
Function:
• navigation, communication, and geodetic/space
environment science
The Telikom PNG deal - SES’ NSS-9 satellite
located at the 183 degrees East and on NSS6 at 95 degrees East
• 35,801 kilometres
Low Earth Orbit (LEO)
• Orbit at an altitude of less than 1000 km but could be as low as
160 km above Earth.
• Speed is 7.8 km per second (satellite takes approximately 90
minutes to circle Earth)
Function:
• Satellite Imaging
• International Space Station (ISS)
2000 star link satellites
have been launched by
Elon Musk.
• circular orbit at an
altitude of about 330
kilometers.
Calculating the Orbital height and velocity of satellite
An object having Right
velocity at a height above the
earth will make a continuous
loop above the earth
10𝑣1
ℎ
𝑣1
50𝑣1
1000𝑣1
100000𝑣1
Calculating the Orbital height and velocity of satellite
Find the orbital height and velocity of a satellite have a period of 144.8 𝑚𝑖𝑛
𝐹𝑐 = 𝐹𝐺
𝑚2
𝑣2
𝑀1 𝑚2
𝑚2
=𝐺
𝑟
𝑟2
𝑣2
2𝜋𝑟
𝑇
𝑀1
=𝐺
𝑟
2
=𝐺
𝑟
Mass feel
two force at
the same
time
𝑀1
𝑀1
𝑟
4𝜋 2 𝑟 2
𝑀1
=𝐺
𝑇2
𝑟
𝑟3
earth
𝑣2
𝐹𝑐 = 𝑚2
𝑟
𝑀1 𝑚2
𝐹𝐺 = 𝐺
𝑟2
𝑀1 𝑇 2
=𝐺
4𝜋 2
𝐺 = 6.673 × 10−11 𝑁𝑚2 /𝑘𝑔2
𝑟 = 𝑟𝐸 + ℎ
𝑟𝐸 = 6371𝑘𝑚
𝑑 2𝜋𝑟
=
𝑡
𝑇
𝑀1 = 5.98 × 1024 𝑘𝑔
𝑣=
𝑇 = 144.8 𝑚𝑖𝑛
𝑚2
𝑀1 = 5.98 × 1024 𝑘𝑔
𝑑 2𝜋𝑟
𝑣= =
𝑡
𝑇
earth
𝑟
𝑀1
𝑟𝐸 = 6371𝑘𝑚
𝑣2
𝐹𝑐 = 𝑚2
𝑟
𝑀1 𝑚2
𝐹𝐺 = 𝐺
𝑟2
Mass feel
two force at
the same
time
𝑇 = 144.8 𝑚𝑖𝑛
𝐺 = 6.673 × 10−11 𝑁𝑚2 /𝑘𝑔2
𝑟3
𝑟3
𝑟 = 𝑟𝐸 + ℎ
𝑀1 𝑇 2
=𝐺
4𝜋 2
= (6.673
× 10−11 𝑁𝑚2 /𝑘𝑔2 )
3
𝑟=
(6.673 × 10−11 𝑁𝑚2 /𝑘𝑔2 )
𝑟 = 7.83 × 106 𝑚
𝑟 = 𝑟𝐸 + ℎ
ℎ = 𝑟 − 𝑟𝐸
× 1024 𝑘𝑔
60𝑠
144.8 𝑚𝑖𝑛 × 1𝑚𝑖𝑛
4𝜋 2
2
5.98 × 1024 𝑘𝑔
60𝑠
144.8 𝑚𝑖𝑛 ×
1𝑚𝑖𝑛
4𝜋 2
2
5.98
Or 7830𝑘𝑚
ℎ = 7830𝑘𝑚 − 6371𝑘𝑚 = 1455𝑘𝑚
Velocity of the satellite
𝑑 2𝜋𝑟
𝑣= =
𝑡
𝑇
𝑣=
7830𝑘𝑚
60𝑠
144.8𝑚𝑖𝑛 × 1𝑚𝑖𝑛
𝑇 = 144.8 𝑚𝑖𝑛
𝑟 = 𝑟𝐸 + ℎ = 7830𝑘𝑚
= 7.15𝑘𝑚/𝑠
Satellite laser-ranging
Satellite Laser-ranging:
A techniques that uses the laser light to locate a satellite in space.
Satellite tracking station
A retro-reflector :
• consists of three orthogonal mirrors that
form the corner of a cube;
• reflects an incident beam of light back along its path
LAGEOS 1 flies at 5858–5958 km altitude,
the inclination of its orbit is 𝟏𝟏𝟎°
LAGEOS – Laser Geodynamics Satellite 1
designed by NASA and launched in 1976
Starlink satellite
SLR can detect the motion of tectonic plates
Changes in the position
of two stations denote
the movement of
tectonic plates.
Rate of their movement
is deduced
∆𝑥
∆𝑥
Eurasia
Yaragadee tracking station in Australia
and the tracking station in Hawaii
crosses the converging plate boundary
between the Indo-Australian and Pacific
plates
𝑟𝑎𝑡𝑒 = 63 ∓ 3𝑚𝑚/𝑦𝑟
∆𝑥
Eurasia
∆𝑥
Satellite altimetry
•
•
•
measures the time taken for a radar pulse to travel from the satellite to the sea surface and back
to the satellite.
altitude of a spacecraft can be determined relative to the reference ellipsoid
difference between the satellite’s height above the ellipsoid and above the Earth’s surface gives
the height of the topography relative to the reference ellipsoid
.
𝑻𝒐𝒑𝒐𝒈𝒓𝒂𝒑𝒉𝒚
= 𝒔𝒂𝒕𝒆𝒍𝒍𝒊𝒕𝒆 𝒂𝒍𝒕𝒊𝒕𝒖𝒅𝒆 𝒂𝒃𝒐𝒗𝒆 𝒆𝒍𝒍𝒊𝒑𝒔𝒐𝒊𝒅 − 𝒔𝒂𝒕𝒆𝒍𝒍𝒊𝒕𝒆 𝒂𝒍𝒕𝒊𝒕𝒖𝒅𝒆 𝒂𝒃𝒐𝒗𝒆 𝒆𝒂𝒓𝒕𝒉 𝒔𝒖𝒓𝒇𝒂𝒄𝒆
Satellite Altimetry is used:
• to determine the topography of the
surface of the earth
Features interpreted from Altimetry
measurement
• ocean ridge systems and
seamount chains - (geoid) is raised.
• fracture zones – high geoid
• deep ocean trenches - Very dark
areas mark the locations of deep
• Seaward of the deep ocean
trenches the mean sea surface is
raised as a result of the upward
flexure of the lithosphere
before it plunges downward in a
subduction zone
Satellite-based global positioning systems (GPS)
Global Positioning System (GPS)
• is a satellite constellation supporting highly accurate positioning,
navigation and timing measurements worldwide
• GPS system consists of 24 satellites at altitude of around 20,200 km
• There are four satellites in each of six orbital planes, equally spaced at 60°
intervals around the equator and inclined to the equator at about 55°
GPS works by triangulation" or "trilateration“ method
• GPS receiver measures the distance between itself and each of the four
satellites
• atomic clocks on board each satellite send radio frequency to GPS receiver.
• signal transit time divide by the speed of light gives distance of GPS .
• Comparing position with reference station, location of GPS receiver is known.
Measurement of gravity and the geoid from
orbiting satellites
The GRACE (Gravity Recovery and Climate
Experiment )mission
• uses two nearly identical satellites in nearcircular polar orbits (inclination 85.5° to
the equator), initially about 500 km above
Earth’s surface.
As the satellite-pair orbits the Earth, it
traverses variations in the gravity field due to
the inhomogeneous mass
distribution in the Earth is recorded.
• If there is a mass excess, the
equipotential surface bulges upward, and
gravity is enhanced locally
Global Gravity Anomaly Recorded by GRACE satellites
Gravity Anomaly –
unusual concentration
of mass in an area
High gravity
anomaly – contains
more massive body
Low gravity anomaly
– contains less
massive body