Geoscience Reference
In-Depth Information
we can write
d
ρ
d
V
=−
m
V
2
=−
V
(8.11)
Substituting this into Eq. (8.9)gives
d
P
d
ρ
K
=
ρ
(8.12)
Equation (8.8) can therefore be written
d
d
r
=−
GM
r
ρ
(
r
)
r
2
ρ
(
r
)
K
(8.13)
Combining Eqs. (8.1) and (8.2)gives
K
ρ
4
3
β
=
α
2
−
2
(8.14)
so that
d
d
r
=−
GM
r
ρ
(
r
)
r
2
α
GM
r
ρ
(
r
)
r
2
2
=−
(8.15)
4
3
β
2
−
φ
2
,
Equation (8.15)isthe
Adams-Williamson equation
and is used to determine
density as a function of radius. To use the equation, it is necessary to start at the
Earth's surface and to work inwards, applying the equation successively to shells
of uniform composition. It is important to remember that
M
r
is the mass within
radius
r
:
2
4
where
φ
, the seismic parameter, is equal to
α
−
3
β
a
=
r
0
ρ
(
a
)
a
2
M
r
=
4
π
d
a
(8.16)
a
=
a
=
R
ρ
(
a
)
a
2
M
r
=
M
E
−
4
π
d
a
(8.17)
a
=
r
where
M
E
is the mass of the Earth (known from study of periods of rotation
of satellites and from direct measurements of gravity, see Chapter 5) and
R
is
the radius of the Earth. Thus, at each stage of the calculation, working from the
Earth's surface inwards, all the terms on the right-hand side of the equation refer
to material outside or at the radius
r
and so have already been determined.
A density structure for the whole Earth obtained by using the Adams-
Williamson equation is called a
self-compression model
because the density at
each point is assumed to be affected only by compression by the material above
it. In practice, it is pointless to assume that the whole Earth has uniform com-
position since we know this to be untrue, so the density determination is usually
begun at the top of the mantle, with a chosen density (the crustal thickness and
density vary widely). Then densities can be calculated all the way to the base of
the mantle using the Adams-Williamson equation. The core clearly has a differ-
ent composition from the mantle since the dramatic changes in seismic velocity
occurring at the core-mantle boundary could hardly be due to pressure alone. A
new starting density is therefore chosen for the top of the core, and the densities
