Page 336 - Physical Principles of Sedimentary Basin Analysis

318 Gravity and gravity anomalies

uncompensated compensated

crust

mantle

Figure 10.2. The load to the left, with short wavelength, is supported by the elastic strength of the
lithosphere. The load to the right, with long wavelength, is supported by buoyancy from the ductile
mantle, and it is therefore in isostatic equilibrium. The mass of the load is compensated by a mass
deficiency from low-density crustal roots.

and the acceleration is the vector
M
g =− G n r . (10.4)
r 2
There is also a force of equal size acting in the opposite direction from mass m to
mass M.
The masses in Newton’s law of gravity are point masses, which is an idealization where
all the mass is concentrated in one point. The distance r in Newton’s law is therefore
between two points with masses m and M, respectively. The Earth is almost a point mass
at the length scale of the solar system and Newton’s law of gravity applies. The situation
might be different on the surface of the Earth, because the Earth is then a mass with a
large lateral extent. The question is now how we can apply Newton’s law when the mass
occupies a very large volume. The acceleration of gravity from a body like the Earth can
be obtained by dividing it into a large (infinite) number of small (infinitesimal) cells and
adding the contribution from each cell. The contribution to the gravity from one single
(small) cell a distance r away is

dV
dg = G n r (10.5)
r 2
where the mass of the cell is dm = dV . The acceleration of gravity from the entire body
of volume V is then the sum of the contributions from all cells in the volume, which is the
integral over the entire volume V :

g = G n r dV. (10.6)
V r 2
An interesting example of this integral is for a spherical object like the Earth, when the
density is only dependent on the distance r from the center of the sphere, as shown