Page 347 - Physical Principles of Sedimentary Basin Analysis

10.7 Gravity from a buried sphere 329

(b) The case r = 0 is shown by integration over a volume that includes the origin. The
left-hand side then becomes

#
# #
2 1 1 n r · n
∇ dV = ∇ · n dS =− dS =− d =−4π (10.54)
V r S r S r 2
where the volume integral was converted to a surface integral using the divergence theorem.
We have used that ∇r = n r (the unit vector in the direction of r), and that is a solid angle,
see equation (10.35). Figure 10.4 shows the difference between r = 0insidethe volume
and r = 0 on the outside of the volume.

Exercise 10.6 Show that the potential

(r )

U(r) =−G dV (r ) (10.55)

V |r − r|
is a solution of Poisson’s equation. Hint: use equation (10.51).

10.7 Gravity from a buried sphere
A buried object with a different density than the surrounding rock leads to a weak
perturbation of the gravitational acceleration at the surface. This perturbation is now
calculated for some simple objects. It is important to keep in mind that the grav-
ity from different bodies can be superposed. We can therefore compute the gravity
for one body at a time and add the contribution from all bodies to obtain the total
effect. Another point is that perturbation of gravity from a buried body is caused by
the density difference between the body and the surrounding rock. The buried body
can be viewed as two overlapping bodies, one with the density of the surrounding
rock and another with the density difference. It is the latter object that perturbates the
gravity.
We have already shown that the gravitational acceleration from a spherical object with a
constant density is the same as for a point mass

GM
g = (10.56)
r 2
3
where M = (4/3)πa is the excess mass of a sphere with radius a. The difference
in density between the sphere and the surrounding rock is . The vertical compo-
nent of the acceleration is g z = g cos α where cos α = h/r (see Figure 10.8a). The
vertical component g z in position (x, y) in the xy-plane from a sphere at depth h is
therefore
GMh
g z (x, y) = (10.57)
2 3/2
2
2
(x + y + h )