The essential characteristics of gravity can be explained in terms of mass, density and the gravity equation. Mass (m) is the amount of the matter contained in an object. Density (ρ) is the mass contained in a unit volume of the matter, i.e. mass per unit volume, and is a measure of the concen- tration or compactness of a material’s mass. It is a funda- mental property of all matter and depends on the masses and spacing of the atoms comprising the material. Both quantities are scalars (seeSection 2.2.2). Mass is given by:
Mass¼ density volume ð3:1Þ
and from this
Density¼ mass
volume ð3:2Þ
The SI unit of mass is the kilogram (kg) and the unit for volume is cubic metres (m3), so fromEq. (3.2)density has the units kg/m3; however, it is common in the geosciences to use grams per cubic centimetre (g/cm3). The two units differ by a factor of 1000 and, based on 1000 kg being equal to 1 tonne (t), sometimes densities are specified as t/m3
, i.e. 2650 kg/m3¼ 2.65 t/m3¼ 2.65 g/cm3. The definition of a gram is the mass of 1 cm3of pure water at 4 °C. This means that a density is quantified relative to that of an equal volume of water, i.e. a substance with a density of 2.65 g/cm3is 2.65 times as dense as water.
The mass distribution and shape of an object are linked by the object’s centre of mass. It is the mass-weighted average position of the mass distribution and, therefore, the point through which gravity acts on the object. For symmetrical objects of uniform density, like a sphere, cube, sheet etc., the centre of mass coincides with their geometric centres (Fig. 3.2a).
3.2.1.1 The gravity equation
All objects attract one another with a force proportional to their masses and, for spherical masses whose sizes are much smaller than the distance between them, inversely proportionally to the square of the distance between their
centres of mass. This is known as the Universal Law of Gravitation and is the reason that objects are ‘pulled’ towards the Earth. The attractive force (F) between the two masses (m1 and m2) separated by a distance (r) is
given by the gravity equation (Fig. 3.2b): F ¼ Gm1m2
r2 ð3:3Þ
The constant of proportionality (G) is known as the universal gravitational constant. In the SI system of measurement it has an approximate value of 6.6726 10–11m3kg-1s-2.
If we suspend an object in a vacuum chamber (to avoid complications associated with air resistance), and then release it so that it falls freely, it will be attracted to the Earth in accordance with the Universal Law of Gravitation. The object’s velocity changes from zero, when it was suspended, and increases as it falls, i.e. it accelerates. This is acceleration due to gravity and it can be obtained fromEq. (3.3)as follows. Consider a small object, with mass m2, located on the Earth’s
surface. If the mass of the Earth is m1and its average radius is
r, entering the relevant values intoEq. (3.3)gives:
F¼ 9:81m2 ð3:4Þ
The attractive force acting on the object, due to the mass of the Earth, is the object’s weight, which is proportional to the object’s mass. For a body with unit mass (m2¼ 1 kg), the
average acceleration caused by the mass of the Earth, i.e. gravity, at sea level is approximately equal to 9.81 m/s2. Acceleration due to gravity is the same for all objects of any mass at the same place on the Earth. However, it does vary over the Earth owing to the Earth’s rotation, variations in its radius and variations in its subsurface density; and also varies with height above the Earth’s surface (see Section 3.4). Accordingly, a body’s weight changes from place to place. 3.2.1.2 Gravity measurement units
Changes in gravitational acceleration associated with dens- ity changes due to crustal geological features are minute in comparison with the average strength of the Earth’s gravity field. The SI unit of acceleration is metres/second/second (m/s2) and in the cgs system of measurement the unit is the gal (1 gal¼ 1 cm/s2), but they are so large as to be imprac- tical for gravity surveying. Instead, a specific unit of gravity has been defined in the cgs system of measurement and is known as the milligal (mgal, where 1 mgal¼ 10–3gal¼ 10–5m/s2). It is still in common use as there is no defined SI unit of gravity. An alternative unit known as the gravity unit (gu), which is 1μm/s2(10–6m/s2), is also used (1 mgal is
equal to 10 gu). We use gu throughout our description of the gravity method. Rates of spatial change in gravity, i.e. grav- ity gradients, are defined in terms of a unit known as the Eötvös (Eo), which is a gradient of 10–6mgal/cm, equal to 1 gu/km, 10–3gu/m, 10–9m s-2m-1or 1 ns-2.
3.2.1.3 Excess mass
A buried body with higher density than the surrounding country rocks, i.e. a positive density contrast (Δρ), pro- duces an increase in mass above that which would be present if the body had the same density as the country rock. This extra mass is known as excess mass (Me) and
is given by:
Me¼ ΔρV ¼ ðρbody ρcountryÞV ð3:5Þ
where V is the volume of the body, and ρbody andρcountry
are the densities of the body and the country rock, respectively.
It is the excess mass that gives rise to a positive gravity anomaly (see Section 3.2.2). When the body has lower density than the country rock, i.e. a negative density contrast, it exhibits a mass deficiency and produces a nega- tive gravity anomaly, cf.Fig. 2.4. Note that it is the excess mass (or mass deficiency as the case may be) that gives rise to the gravity anomaly and not the body’s absolute mass.
Excess mass may be estimated from modelling (see
Section 3.10.3) and the absolute mass (Ma) of the source
determined as follows: Ma¼ ρbody Δρ Me¼ ρbody ρbody ρcountryMe ð3:6Þ
3.2.2
Gravity anomalies
Figure 3.3shows the effect on the gravityfield of a spherical source in the subsurface that is denser than its surrounds. For the simple model shown, gravity measurements are made on a horizontal surface above the source, and the gravitational attraction due to the Earth is taken as constant over the area of the target response. As shown inFig. 3.3d, the source’s field is radially directed towards its centre of mass. The Earth’s field is radially directed toward the centre of the Earth, but the Earth is so large that thefield lines are effectively parallel in the area depicted by thefigure. In the presence of an excess mass, thefield lines deflect towards the anomalous mass; but the effect is negligible because the strength of the Earth’s gravity field is substantially larger than that of any excess mass in the crust. The gravity fields of the Earth and the
sphere add to form the resultant total gravity field. The gravity instrument is aligned in the direction of the totalfield (defining the vertical, seeSection 3.2), so the measured grav- ity anomaly (change in gravity) is simply the magnitude of the variation in the totalfield, which is the vertical compon- ent of thefield due to the anomalous mass (Fig. 3.3c). Note that because a gravity source is a monopole, its gravity anom- aly has single polarity (sign) and has its peak centred directly above the source. This simple model illustrates how a high- density body in the subsurface, such as a mineral deposit, can
a) b) Source Depth X’ X D c) X’ X Earth’s field Gravity Gravity anomaly G E D C B A F G E D C B A F ’ X X Vertical component of source’s field at surface
Depth Source’s field
Earth’s field ’ X X d) Source
Figure 3.3 Gravitationalfield of a sphere. (a) Gravity measured on a horizontal surface above a dense spherical source (b). (c) Variation in the vertical component of gravity along the principal profile
(X–X0) due to the combined effects of the Earth’s field and that of the
sphere. (d) The radially directed gravityfield of the source and its
vertical component. Note that the strength of the Earth’s gravity field is many times greater than that of the source, but has been reduced here for clarity.
be remotely detected through its effect on the gravityfield. It should be noted that variations in the vertical component of gravity measured beside or below the source, e.g. downhole or with underground measurements, will be different; explanations of these are beyond our scope.
More information about the shape and location of the source of a gravity anomaly can be obtained if, instead of measuring just gravity, tensor measurements are made (see
Section 2.2.3). Three-component gradient or full-tensor
gradiometers measure the gravity gradient in three perpen- dicular directions. Data of this kind provide more infor- mation about the source than gravity alone. However, and as shown inFig. 3.4, the individual gradients are not easily correlated with the overall shape of the source, but collect- ively they are very useful for quantitative interpretation using inverse modelling methods (seeSection 2.11.2.1).