Geoscience Reference
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down through the core can be calculated using the Adams-Williamson equation.
Because the total mass in the model must equal the mass of the Earth, successive
guesses at the density at the top of the core must be made until this constraint is
satisfied.
Although such a self-compression density model for the Earth satisfies the
seismic-velocity data from which it was derived, it does not satisfy data on the
rotation of the Earth. In particular, the Earth's moment of inertia, which is sen-
sitive to the distribution of mass in the Earth, is significantly greater than the
moment of inertia for the self-compression model. (To appreciate the importance
of mass distribution, test the difference between opening the refrigerator door
when all the heavy items in the door compartments are next to the hinge and
when they are all next to the handle). There must be more mass in the mantle
than the self-compression model allows. To determine the reasons for this dis-
crepancy, it is necessary to re-examine the assumptions made in determining the
Adams-Williamson equation. First, it was assumed that the temperature gradient
in the Earth is adiabatic (Section 7.7). However, since we know that convec-
tion is occurring both in the mantle (Section 8.2) and in the liquid outer core
(Section 8.3), the temperature gradients there must be superadiabatic. Equation
(8.15) can be modified to include a non-adiabatic temperature gradient:
d
d
r
=−
GM
r
ρ
(
r
)
r
2
+
αρ
(
r
)
τ
(8.18)
φ
where
the difference between the
actual temperature gradient and the adiabatic temperature gradient. This modi-
fication means that, in the case of a superadiabatic gradient (
α
is the coefficient of thermal expansion and
τ
0), the density
increases more slowly with depth. Conversely, in the case of a subadiabatic tem-
perature gradient, the density increases more rapidly with depth. This means that
corrections for the temperature gradient, which, in practice, are found to be fairly
small, act in the opposite direction to that required to explain the missing mantle
mass. Another explanation must be found.
The second assumption made in deriving the Adams-Williamson equation
was that there were no chemical or phase changes in the Earth (other than dif-
ferences amongst crust, mantle and core, which have already been included in
the model). This assumption provides the answer to the problem of the missing
mantle mass. In the mantle transition zone (400-1000 km), there are jumps in
seismic velocity that seem to be due to changes of state (phase changes). An
example of a change of state is the change from liquid to solid such as occurs
when water freezes. This is not the only type of change of state possible; there are
also solid-solid phase changes in which the atoms in a solid rearrange and change
the crystal structure. Examples of this are the change of carbon from graphite to
diamond under increasing pressure and the changes which take place in the trans-
formation from basalt to greenschist to amphibolite to pyroxene granulite (at
high temperatures) or to blueschist (at low temperatures) and finally to eclogite
τ>
