Geoscience Reference
In-Depth Information
Let us assume that the Earth is made up of a series of infinitesimally thin,
spherical shells, each with uniform physical properties. The increase in pressure
d
P
which results during the descent from radius
r
+
d
r
to radius
r
is due only to
the weight of the shell thickness d
r
:
d
P
=−
g
(
r
)
ρ
(
r
)d
r
(8.3)
(
r
)isthe density of that shell and
g
(
r
) the acceleration due to gravity
4
at
radius
r
.
On writing Eq. (8.3)inthe form of a differential equation, we have
d
P
d
r
=−
g
(
r
)
ρ
(
r
)
where
ρ
(8.4)
where d
P
d
r
is simply the gradient of the hydrostatic pressure. There is a minus
sign in Eqs. (8.3) and (8.4) because the pressure
P
decreases as the radius
r
increases. The gravitational acceleration at radius
r
can be written in terms of the
gravitational constant
G
and
M
r
, the mass of the Earth within radius
r
:
/
GM
r
r
2
g
(
r
)
=
(8.5)
Therefore, Eq. (8.4) becomes
d
P
d
r
=−
GM
r
ρ
(
r
)
r
2
(8.6)
To determine the variation of density with radius, it is necessary to determine
d
P
d
r
.
Using Eq. (8.6), we can write
d
/
d
r
=
d
P
d
r
d
d
P
(8.7)
GM
r
ρ
(
r
)
r
2
d
d
P
=−
(8.8)
The compressibility or bulk modulus for adiabatic compression
K
(Eq. (A2.31))
is used to obtain d
ρ/
d
P
, the variation of density with pressure, as follows:
increase in pressure
fractional change in volume
K
=
d
P
d
V
/
V
=−
(8.9)
There is a minus sign in Eq. (8.9) because volume decreases as pressure
increases. Since density
ρ
is the ratio of mass to volume,
m
V
ρ
=
(8.10)
4
Outside a spherical shell the gravitational attraction of that shell is the same as if all its mass were
concentrated at its centre. Within a spherical shell there is no gravitational attraction from that
shell. Together, the preceeding statements mean that, at a radius
r
within the Earth, the gravitational
attraction is the same as if all the mass inside
r
were concentrated at the centre of the Earth. All
of the mass outside radius
r
makes no contribution to the gravitational attraction and so can be
ignored. This is proved in Section 5.2.
