Introduction to gravity
Saad M. Al-Humidan
Abstract
gravity is one the most used geophysical method to find objects and in this paper we will
tray to explain the physical and historical theory behind this method and we will discuss
some of the anomalies and how to correct them, also we will try to comprehend couple
of the theories that deal with the principal models of isostasy.
Introduction
Gravity is the name given to the force of attraction between all objects. Earth’s gravity
keeps us from floating away into space. Gravity also holds the oceans, atmosphere, trees,
plants, cars, building, animals, and everything else with mass, and keeps them from
drifting off into space. Rain, rocks, and rivers are all pulled downward by gravity.
Because they must grow upward to overcome it, gravity determines the growth of plants
and trees. Our bones and muscles have developed to withstand the constant force of
gravity. This is why our bones and muscles become very weak if we travel in space for
long periods of time with low levels of gravity. The fluid in our inner ear that gives us
our balance is controlled by gravity. It’s very hard to imagine what our lives would be
like without the influence of gravity.
Newton Low of gravitation
The young English physicist and mathematician Sir Isaac Newton (1643 AD-1727
AD) first became interested in the force of gravity one day as he sat in contemplation
under an apple tree and was suddenly struck on the head by a falling piece of fruit. The
incident left Newton with a sore skull and a remarkable idea: what if gravity, the force
that pulled apples out of trees, actually reached as high as the moon? From here, it was a
short leap to the concept that the Earth’s gravity was responsible for keeping the moon in
orbit and that the Sun’s gravity was responsible for keeping the planets in orbit.
But if such a force did exist, Newton realized, it must be governed by certain rules to
define how it should effect matter. Making use of Kepler’s laws of planetary motion,
Newton set out to discover these rules. He worked on the problem for a time and came up
with a partial answer. Yet his full solution was not made public until 1687 when he
published his three book volume Philosophiae Naturalis Principia Mathematica ,or
simply Principia for short. In this publication, which many consider to be the greatest
single piece of scientific literature ever written and which also contained his laws of
motion, Newton finally disclosed to the public his law of universal gravitation.
Newton’s law, which said that all matter in the universe attracts all other matter in the
universe, was a major step forward for scientific thought. For the first time in history,
people finally had an understanding of gravitation that went well beyond just the force
that keeps our feet on the ground. We now realized that gravity was a universal force.
Every single point mass attracts every other point mass by a force pointing along the
line combining the two. The force is proportional to the product of the two masses and
inversely proportional to the square of the distance between the point masses:
F is the magnitude of the gravitational force between the two point masses
G is the gravitational constant
m1 is the mass of the first point mass
m2 is the mass of the second point mass
r is the distance between the two point masses
Assuming SI units, F is measured in newtons (N), m1 and m2 in kilograms (kg), r in
metres (m), and the constant G is approximately equal to 6.67 × 10−11 N m2 kg−2.
Latitude Correction
Because the Earth is not a perfect sphere, the acceleration due to gravity depends on
latitude. The radius of the earth is greater at the equator than at the poles thusCorrection
subtracted from gobs that accounts for Earth's elliptical shape and rotation. The gravity
value that would be observed if Earth were a perfect (no geologic or topographic
complexities), rotating ellipsoid is referred to as the normal gravity.
gn = 978031.85 (1.0 + 0.005278895 sin2(lat) + 0.000023462 sin4(lat)) (mGal)
where lat is latitude
Free Air Correction
In geophysics, a free air anomaly is the difference between observed gravity and
theoretical gravity that has been computed for latitude and corrected for elevation of the
station above or below the geoid, by application of the normal rate of change of gravity
for change of elevation, as in free air.
According to the Encyclopedic Dictionary of Applied Geophysics (Robert E. Sheriff,
2002, Tulsa USA), the free to air correction is 0.3086 mGal/Meter
That mean the free-air correction accounts for gravity variations caused by elevation
differences in the observation locations. The form of the Free-Air gravity anomaly, gfa ,
is given by:
gfa = gobs - gn+ 0.3086h (mGal)
where h is the elevation (in meters) at which the gravity station is above the datum
(typically sea level).
Bouguer Slab Correction
The Bouguer correction is a first-order correction to account for the excess mass
underlying observation points located at elevations higher than the elevation datum (sea
level or the geoid). Conversely, it accounts for a mass deficiency at observation points
located below the elevation datum. The form of the Bouguer gravity anomaly, gb, is
given by:
gb = gobs - gn + 0.3086h - 0.04193r h (mGal)
where r is the average density of the rocks underlying the survey area.
The Bouguer correction is made, as a rule, to sea level and by using an average density
value of the rocks, determined by the profile method. However, if the density of the rocks
forming the topography varies from place to place, then the proper density value should
be used for every topographical feature. The Bouguer correction with varying density
should be made only to a surface drawn through the low points of the topography. Below
this surface, and to sea level, a constant density value should be used all over in the area;
otherwise, nonexistent gravity anomalies may be introduced into the results of the survey,
or existing gravity anomalies may be distorted. A few examples are given for computing
Bouguer corrections with varying surface density, and to illustrate the errors introduced
by using improper density values.
Strictly interpreted, the Bouguer correction is added to the known value of gravity at
the reference station to predict the value of gravity at the measurement level. The
difference between the actual value and the predicted value is the gravity anomaly, which
results from differences in density between the actual Earth and reference model
anywhere below the measurement station.
Terrain Correction
Like Bouguer Slab Corrections, when computing Terrain Corrections we need to
assume an average density for the rocks exposed by the surrounding topography. Usually,
the same density is used for the Bouguer and the Terrain Corrections. Thus far, it appears
as though applying Terrain Corrections may be no more difficult than applying the
Bouguer Slab Corrections. Unfortunately, this is not the case.
To compute the gravitational attraction produced by the topography, we need to
estimate the mass of the surrounding terrain and the distance of this mass from the
observation point (recall, gravitational acceleration is proportional to mass over the
distance between the observation point and the mass in question squared). The specifics
of this computation will vary for each observation point in the survey because the
distances to the various topographic features varies as the location of the gravity station
moves. As you are probably beginning to realize, in addition to an estimate of the average
density of the rocks within the survey area, to perform this correction we will need a
knowledge of the locations of the gravity stations and the shape of the topography
surrounding the survey area.
Estimating the distribution of topography surrounding each gravity station is not a
trivial task. One could imagine plotting the location of each gravity station on a
topographic map, estimating the variation in topographic relief about the station location
at various distances, computing the gravitational acceleration due to the topography at
these various distances, and applying the resulting correction to the observed
gravitational acceleration. A systematic methodology for performing this task was
formalized by Hammer* in 1939. Using Hammer's methodology by hand is tedious and
time consuming. If the elevations surrounding the survey area are available in computer
readable format, computer implementations of Hammer's method are available and can
greatly reduce the time required to compute and implement these corrections.
Although digital topography databases are widely available, they may not be sampled
finely enough for computing what are referred to as the near-zone Terrain Corrections in
areas of extreme topographic relief or where high-resolution (less than 0.5 mGals) gravity
observations are required. Near-zone corrections are terrain corrections generated by
topography located very close (closer than 558 ft) to the station. If the topography close
to the station is irregular in nature, an accurate terrain correction may require expensive
and time-consuming topographic surveying. For example, elevation variations of as little
as two feet located less than 55 ft from the observing station can produce Terrain
Corrections as large as 0.04 mGals. Where possible, stations should be located to avoid
extreme terrain effects (for instance, away from cliffs or quarries). Fieldwork may
include estimation of the topography in the inner Hammer zones.
Gravimeter
A gravimeter is a device designed to measure the local gravitational field. Also
sometimes called a gravitometer, a gravimeter is a type of accelerometer.
According to the equivalence principle of Albert Einstein, the effects of a gravitational
field are rigorously equivalent to acceleration. Therefore, gravimeters are susceptible to
vibration which causes small, oscillatory accelerations. This is counteracted by integral
vibration isolation and signal processing.
Though the essential principle of design is the same as accelerometers, gravimeters are
typically much more sensitive than accelerometers in order to measure very tiny changes
within the earth's gravity of 1 g). In contrast, accelerometers are often designed to
measure 1000 g or more, and many perform multi-axial measurements. The constraints
on temporal resolution are usually less for gravimeters, so that resolution can be
increased by processing the output with a longer "time constant".
In current use, there are two basic types of gravimeter: relative and absolute.
A relative gravimeter is basically a weight on a spring, and by measuring the amount
by which the weight stretches the spring, local gravity can be measured. However, the
strength of the spring must be calibrated by placing the instrument in a location with a
known gravitational acceleration.
An absolute gravimeter is used to calibrate relative gravimeters, and operates by
letting a mass free-fall in vacuum and measuring its rate of acceleration. The mass
includes a retroreflector and terminates one arm of a Michelson interferometer. By
counting and timing the interference fringes, the velocity of the mass can be measured.[1]
A more recent development is a "rise and fall" version that tosses the mass upward and
measures both upward and downward motion.[2] This allows cancellation of some
measurement errors, and twice the measurement time for a given instrument
The field of gravimetry deals with measuring gravitational fields. A high-grade,
calibrated gravimeter such as the portable LaCoste-Romberg gravimeter can measure the
earth's gravitational field to within 1 μgal, or 1 ng (10-9 g). Measurements of the surface
gravity of the earth are part of geophysical analysis.
There has also been some interest in the use of gravimeters to detect possible changes
in the gravitational field in various anti-gravity experiments.
Gal (unit)
The Gal or Galileo is the CGS unit of acceleration. It is defined as 1 centimeter per
second squared. The symbol for the Gal is Gal. It has nothing to do with the liquid
measure gallon, although both are sometimes abbreviated gal.
In SI units: 1 Gal = 0.01 m/s².
The acceleration due to Earth's gravity at its surface is 976 to 983 Gal; the variation is
mainly with latitude and elevation.
Variations in gravitational acceleration, due to mountains or masses of lesser density
within the Earth's crust, are typically tens to hundreds of milliGals (mGal). This is
especially important in the field of geophysics.
The Gal is named after Galileo Galilei, a physicist who made the first measurements of
the Earth's gravity.
The Theory Isostasy
Isostasy is a term used in Geology to refer to the state of gravitational equilibrium
between the Earth's lithosphere and asthenosphere such that the tectonic plates "float" at
an elevation which depends on their thickness and density. It is invoked to explain how
different topographic heights can exist at the Earth's surface. When a certain area of
lithosphere reaches the state of isostasy, it is said to be in isostatic equilibrium. Certain
areas (such as the Himalayas) are not in isostatic equilibrium, which has forced
researchers to identify other reasons to explain their topographic heights (in the case of
the Himalaya, by proposing that their elevation is being "propped-up" by the force of the
impacting Indian plate).
In the simplest example, isostasy is the principle observed by Archimedes in his bath,
where he saw that when an object was immersed, an amount of water equal in volume to
that of the object was displaced. On a geological scale, isostasy can be observed where
the Earth's strong lithosphere exerts stress on the weaker asthenosphere which, over
geological time flows laterally such that the load of the lithosphere is accommodated by
height adj Three principal models of isostasy are used:
The Airy-Heiskanen Model
where different topographic heights are accommodated by changes in crustal thickness.
The Pratt-Hayford Model
where different topographic heights are accommodated by lateral changes in rock density.
The Vening Meinesz Model
- where the crust acts as an elastic plate and its inherent rigidity spreads topographic
loads over a broader region. This hypothesis was put forward to explain how features of
grand magnitude like the himalayas could be explained using a regional isostatic
compensation rather than a localized one which is the case for the first two models.
References
http://library.thinkquest.org/27585/frameset_intro.html
http://geoinfo.nmt.edu/geoscience/projects/astronauts/gravity_method.html
http://www-geo.phys.ualberta.ca/~vkrav/Geoph110/L3erthdim.htm
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=GPYSA70000210
00004001004000001&idtype=cvips&gifs=yes
Hammer, Sigmund, 1939, Terrain corrections for gravimeter stations, Geophysics, 4,
184-194
Lisitzin E. (1974) "Sea level changes". Elsevier Oceanography Series, 8