Gravity Anomaly Over a Buried Point Mass
Previously we defined the gravitational acceleration due to a
point mass as
where G is the gravitational constant, m is the
mass of the point mass, and r is the distance between
the point mass and our observation point. As shown in the figure below, lets compute how
the gravitational acceleration varies over a buried point mass as we change the location
of our observations point.
where z is the depth of burial of the point mass and x is the horizontal
distance between the point mass and our observation point.
Notice that the gravitational acceleration caused by the point mass is in the
direction of the point mass; that is it's along the vector r. Before taking a
reading, gravity meters are leveled so that they only measure the vertical component
of gravity; that is we only measure that portion of the gravitational accleration caused
by the point mass acting in a direction pointing down. The vertical component of the
gravitational acceleration caused by the point mass can be written in terms of the
angle &theta as
Now, it is inconvient to have to compute r and &theta for various values of
x before we can compute the gravitational acceleration. Lets now rewrite the above
expression in a form that makes it easy to compute the gravitational acceleration as
a function of horizontal distance x rather than the distance between the point
mass and the observation point r and the angle &theta.
&theta can be written in terms of z and r using the trigometric relationship
between the cosine of an angle and the lengths of the hypotenuse and the adjacent
side of the triangle formed by the angle.
Likewise, r can be written in terms of x and z using the relationship
between the length of the hypotenuse of a triangle and the lengths of the two other sides
known as Pythagorean Theorem.
Substituting these into our expression for the vertical component of the gravitational
acceleration caused by a point mass we obtain
Knowing the depth of burial, z, of the point mass, its mass, m, and the
gravitational constant, G, we can compute the gravitational acceleration we
would observe over a point mass at various distances by simply varying x in the
above expression. An example of the shape of the gravity anomaly we would observe over
a single point mass is shown above.
Therefore, if we thought our observed gravity anomaly
was generated by a mass distribution within the earth that approximated a point mass,
we could use the above expression to generate predicted gravity anomalies for given
point mass depths and masses and determine the point mass depth and mass by matching
the observations with those predicted from our model.
Although a point mass doesn't appear to be a geologically plausible dnesity distribution,
as we will show next, this simple expression for the gravitational acceleration forms
the basis by which gravity anomalies over any more complicated
density distribution
within the earth can be computed.