GRAVITY AND THE
EARTH
Gravity is one of the four fundamental forces (the others are the
electromagnetic, the weak force and the strong force) and probably the least
well understood. The basic concepts were formulated by
However, gravitational attraction is an everyday experience and its
variations provide insights into Earth structure. Early studies used either the
periods of oscillation of pendulums or 'deflections of the vertical' measured
by observations of fixed stars. In 1775 the Astronomer Royal, Nevil Maskelyne, reported to the
Royal Society an estimate of the mass of the Earth obtained by observations of
the deflections on either side of Schiehallion, a
mountain of almost perfectly triangular cross-section in the Scottish
highlands. A few years earlier the French scientist Bouguer noted that the gravitational attraction of the
Basic Relations
Described by
F = G m1 m2 / r2
F = force acting between two point masses
m1, m2 = the masses
r = separation of the two masses
G = Universal gravitational constant = 6.67 x 10-11 Nm2kg-2
The gravitational field at any point is the force experienced by a unit mass,
and it follows that the field due to a point mass m at a distance r from the
observation point is:-
g = G.m / r2
The units are Newton.kg-1 (force per unit mass) or (more commonly),
m.s-2 (acceleration). Numerically, these are identical.
The field equation holds also for bodies made up of spherical shells of uniform density, the separation in these cases being the distance of the point of measurement from the common centre of the spheres. It also holds, approximately, for the Earth (mass M, radius R), i.e.
g = G.M / R2
Geologically, the density of earth is very important. If r is the average density of the Earth, then
r = mass/volume = M / [(4/3)pR3] = 3M / 4pR3
and we can substitute for M using the
relationship between it and g, i.e. M = R2g / G
Therefore: r = 3g / 4pRG
Thus if we know g, R and G, we can
calculate r. With current values:-
r = 5.52 x 103 kgm-3
Since most surface rocks have densities in the range 2-3 x 103kg.m-3, density must increase with depth in Earth. This has also been confirmed by seismology, since seismic velocities, which are strongly correlated with density, increase with depth. However, we would also like to monitor lateral density variations. We cannot easily measure density at depth, but it is quite easy to measure g at different places on the surface of the Earth.
Absolute values of g were originally obtained by measuring the periods of oscillation of pendulums, but observations of falling weights now provide much greater accuracy. Neither pendulums nor weight drop chambers are suitable for routine field use and instead we use spring-balance gravity meters to estimate changes in g. Linking such measurements to places where absolute values are known allows us to determine absolute values with gravimeters.
The variation of g over the surface of the globe is important because it provides information on variations in the shape and internal structure of the Earth.
If we rearrange the previous equation, we obtain:
g = 4p r RG / 3
If the Earth were a perfect sphere of uniform density, g would be constant over
its entire surface. But if the Earth deviates from spherical (i.e. if R varies)
or if there is a local density anomaly, g will vary.
Gravity and the Shape of the Earth
The Earth is not spherical, but an ELLIPSOID OF REVOLUTION ? i.e. it is flattened at the Poles ? this is a rotational effect (see detailed notes).
Satellite studies have provided a very accurate measure of ellipticity:-
equatorial radius = 6378 km.
polar radius = 6356.6 km.
Flattening = (6378 - 6356.6 )/ 6378 = 1 /
298.26
Now since g = GM / R2, g
will be larger where R is smaller. Therefore g at the poles is larger than g
at the Equator. g is also affected by the fact
that the Earth rotates and an observer on its surface therefore experiences a centrifugal
force. We can summarize by saying:
(1) If the Earth were a non-rotating perfect sphere, the acceleration due to gravity would be constant.
(2) Because of rotation, the Earth is flattened at poles. This affects g in
two ways:-
(a) g at the poles is greater than g at the equator because R at the poles is less than R at the equator.
(b) rotational force at the Earth surface is at right angles to the axis of rotation and proportional to the distance from that axis. It is therefore zero at the poles and a maximum at the Equator. It acts outwards, reducing g.
The net gravitational force at the surface of Earth is equal to the resultant
of the forces due to internal mass and the centrifugal action. If
the gravitational force due to M is g’ = GM / R2
and the centrifugal force is c, then the total effective gravitational force is
the vector sum of g" and c. However, g" does not act towards the centre
of Earth, but at right angles to the surface of the elliptical Earth,
since a perfectly homogeneous plastic body will deform until the combination of
g’ and c meets this criterion. In mathematical jargon, the surface of the
ellipsoid is an equipotential surface.
The ideal (ellipsoidal) mean sea level surface is called the EARTH ELLIPSOID or (EARTH) SPHEROID. The gravitational force over the spheroid varies, with a maximum at the poles (where c = 0) and a minimum at the equator (where c is a maximum).
The gravitational acceleration on the surface of the spheroid is given by
the International Gravity Formula (IGF).
g = 9.780318 (1 + a sin2(l) - b sin2 (2l))
where g = sea-level gravitational acceleration on the
spheroid and l = latitude, and
a = 0.0053024 )
constants
b = 0.00000587 )
g at Equator (lat = 0) = 9.780318 m.s-2
g at Pole (lat = 90) = 9.832177 m.s-2
The difference amounts to approximately a half of one percent. The value of
the theoretical or normal gravity varies smoothly between the two extremes, but
inhomogeneities in the Earth produce shorter
wavelength perturbations in the smooth curve.
Effect of Inhomogeneities: the Geoid
Large-scale inhomogeneities produce departures of the measured values of g at sea-level from those predicted by the I.G.F. The fact that g does not vary smoothly from equator to pole provides evidence that there are lateral inhomogeneities within the Earth.
Values of g can be determined by surface measurements and by satellite studies.
The real sea level equipotential surface is known
as the GEOID and has "highs" and "lows"
relative to the spheroid. Contours of the geoid
give the height, above or below the spheroid, by which sea level actually
varies over the Earth's surface. Sea-level is +54 metres
higher in the
e.g.
Mid-Atlantic Ridge
The
The
Negative features are centred over old, inactive ocean basins and continents:-
e.g.
Major physical undulations (e.g. mountains) are NOT associated with geoid anomalies, and so they must be balanced by deeper
seated mass excesses or deficiencies (Isostasy
– see detailed notes for
more).
It is believed that long wavelength undulations in the geoid
reflect the convective system in the mantle, or some other deep phenomena (e.g.
undulations on the outer surface of the core). The problem is complex because
of the effects of flow dynamics. Thus, an upwelling should be characterised by low density, which would produce a
negative geoid anomaly, but the convective motion
deflects the surface and so produces a +ve anomaly. (see Fowler, "The Solid Earth" p. 178).
Crustal Gravity Anomalies
In addition to the global anomalies due to convection, there are smaller scale effects because of crustal inhomogeneities (sedimentary basins, intrusions, etc.). Their analysis is important in exploration for natural resources, but they also need to be taken into account in global scale investigations and surveys (see also detailed notes).
In a gravity survey we measure the difference in gravity between survey points (S) and a reference station (P), using a gravity meter. Ideally P is either an international gravity reference station or has been linked to such a station by gravity measurements. Inevitably, the differences will be small and the m.s-2 is far too large a unit. Gravity anomalies are therefore measured in GRAVITY UNITS.
1 g.u. = 10-6 m.s-2
(An older unit, the milligal, abbreviated as mGal, is still in common use. 1 mGal = 10 g.u.)
Since g is approximately 10 m.s-2, 1 g.u. is about one ten millionth of the absolute value of gravity at Earthís surface. Most gravity meters can detect changes in gravity of as little as 0.1 g.u., and need to, as mineable deposits of metals such as copper, lead, zinc, nickel and iron have been discovered on the basis of anomalies of less than 5 g.u. Underground cavities which could represent hazards to e.g. motorways or airstrips may give rise to effects of only a few tenths of a g.u.
The measured value of gravity at a field station might vary from the value at the base station for a variety of reasons, even if there were no crustal or geoid anomalies. Once the value has been obtained it must be CORRECTED to account for effects such as:-
(1) Latitude differences
(2) Elevation effects
(3) Topographic effects
Any differences that remain after these corrections have been made must be due
to real lateral variations in density.
Latitude Correction
We have seen that gravity on the surface of a homogeneous Earth varies from pole to equator because of effect of centrifugal force and polar flattening.Thus if stations are at different latitudes, we would expect gravity to be different. We use the IGF to describe the latitude effect.
For small N-S distances (up to a few km) the difference in gravity due to
latitude at latitude l is
approximately:-
8.1 sin 2l g.u./km
Free-air Correction
If stations are at different elevations, we would expect gravity to be different because of the different distances to the centre of the Earth. The effect for a positive height (h) above sea-level is approximately equal to 3.086 g.u./metre and is negative, i.e. an increase in height produces a decrease in gravity. The correction, known as the FREE-AIR CORRECTION (because, in the derivation, it is assumed that the only material between the station and the reference surface is air), must therefore be positive.
Note that if the gravity anomaly is to be measured to within 0.1 g.u., the station elevation (h) must be known to within 3
cm! Gravity corrections require accurate elevations, and getting these is often
the most expensive part of a gravity survey.
Bouguer Correction
The free-air correction assumes that only air exists between the station and the reference surface. In reality, a normal gravity station will be underlain by rock, of density r, which exerts a positive (downwards) gravitational pull. The Bouguer correction uses a simple approximation for the effect of this rock column. We assume that the gravity effect of the real topography can be approximated by the effect of a uniform flat plate, density r and thickness h, extending to infinity. This effect is given by:-
2p g r h = 41.91 x 10-5r h g.u.
Since the effect is positive (increases the gravity field), the correction must
be negative. For granite r is
approximately 2.67 Mg m-3, and this has been adopted as a ëstandardí density for the upper
crust, giving a correction of 1.118 g.u./metre.
Other densities may be used to suit the local geology, but use of the standard
density has the virtue of ensuring compatability
between maps of adjacent areas. Since the free-air correction is 3.086 g.u./metre, the net
elevation correction is about 1.968 g.u./meter,
implying that elevations of gravity stations should be known to the nearest 5
cm.
Topographic Correction
Although the Bouguer correction works surprisingly well, it is inadequate for high precision surveys or for surveys carried out in topographically rugged areas. If the station is next to a mountain or valley, the mass difference of the topographic feature from the Bouguer plate will affect the measured gravity field.
A mountain will attract upwards, and so reduce the value of gravity measured. A valley will not attract as much as it should if it were filled with rock and so will also give rise to a gravity value which is smaller than would be expected. Thus the topographic or TERRAIN correction must be added to give corrected gravity differences. Values can be obtained from standard tables for average elevations estimated using graticules overlaid on maps with topographic contours, or by computer programs operating on some form of Digital Terrain Model (DTM).
Once all the corrections have been made, the REDUCED
gravity records variations in gravity field due solely to subsurface density
variations. If only the latitude and free-air corrections have been
applied, the quantity calculated is known as the FREE-AIR GRAVITY (free-air
anomaly). If, IN ADDITION, the Bouguer correction has
been applied, the quantity is known as the (SIMPLE) BOUGUER GRAVITY (or
anomaly). If, IN ADDITION, terrain corrections have been made, the quantity is
known as the EXTENDED BOUGUER GRAVITY or COMPLETE BOUGUER GRAVITY (or anomaly).
Bouguer Gravity (Anomaly)
This is measured as follows:-
(1) Measure the gravity difference between the site and the base station.
(2) Add the absolute value of gravity at the base station
(3) Subtract the IGF value of gravity for a site at the same latitude but at sea-level.
(4) Correct for
- free-air
- Bouguer
- topography
as appropriate to give the required REDUCED GRAVITY.
Density variations below land areas are best studied via the Bouguer gravity, since this takes into account all relief effects and leaves data corrected down to sea-level (unless there are significant density variations within the topography above sea level). A gravity survey across a mountain range will show a negative Bouguer gravity, because mountains have low density roots. This ISOSTATIC balance is responsible for their elevation. The reality of isostasy is confirmed by the measurable uplift of Fennoscandia during the last two hundred years as a consequence of the unloading accompanying the melting of the ice sheets.
At sea, free-air gravity is generally used (measuring points in surface ships being generally at sea level), although sometimes a Bouguer correction is made by infilling the sea with imaginary rock. Bouguer gravity in the oceans is normally high, because the mantle surface (MOHO) is at shallow depths, but free-air gravity is low because the oceans are in isostatic equilibrium.
Useful websites
http://www.ngdc.noaa.gov/mgg/announcements/announce_predict.html
http://membres.lycos.fr/sawece/tecinfo/generaldef/gravity.html
http://www.jpl.nasa.gov/earth/features/watkins.html
http://www.csr.utexas.edu/grace/gallery/animations/ggm01/index.html