The Earths Shells

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Module 1-3B
The Earths Shells
B. Density vs. Depth
In Module 1-3A, we worked out a model for the
density structure of the Earth the density and
thicknesses of the four shells. How can we
represent this information graphically?
Quantitative Concepts and Skills Weighted
average Bar and pie charts Manipulation of XY
graphs Concept that an integral is a sum
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Retrieve your spreadsheet from Slide 12 of Module
1-3A
This table is one representation of the model for
the variation of density with depth, in that it
contains the information (i.e., Columns C and G
tell the story). But most people get more out of
a figure than a table.
Problem Develop a graph that portrays this
model for the density structure of the Earth
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PREVIEW
Slides 4 and 5 try to tell the story with bar
graphs. Slide 4 shows density, but it gives the
impression that all densities are equally
important. We remember that volume is the
weighting parameter, and so Slides 5 and 6
include a bar graph of the volumes of the shells.
The perhaps surprising result prompts us to make
a pie graph of volumes in Slide 7, and we explore
the relation between thickness and volume a
little further in Slides 8 and 9 Because we wish
to show both density and thickness (or depth,
which is cumulated thickness) on a single graph,
we abandon bar and pie graphs for x-y graphs,
beginning in Slide 10. Our first choice of
blindly plotting density vs. depth produces a
gross misrepresentation in Slide 10, because it
ignores the presence of the shells that is, it
assumes that the variation of density vs. depth
is a continuous function. Slide 11 gets us on
track with a step function. Slides 11 and 12
apply the step function to the model of equally
thick shells, so that we can see what we are
doing on the graph. Slide 13 completes the task
by plugging in the values that we found for the
thickness and density of the shells. The graph
in Slide 13, then, portrays our finding from
Module 1-3A.
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One possibility is a bar graph showing the
density of the four shells (Column G).
This representation shows the density but it
leaves out very important information the depths.
The depths are critically important here,
especially in the context of our constraint
average density 5.5 g/cm3. Looking at the
figure (and not the table) one might get the
impression that the average is closer to 8 than
to 5. Why?
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The volumes are unequal, and so all the bars are
not the same importance in the weighted average.
Create a bar graph showing the volumes of the
shells (Column F). Before that, look at the
table. What do you think the relative sizes of
the bars will be?
Bar graph of Column G
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The volumes are vastly unequal.
Average weighted by volume 5.5 g/cm3
Bar graph of Column F
Visually, the bar graph of volumes explains why
the Earths average density is so close to the
density of the mantle.
What percentage of the Earths volume is in the
Earths mantle? Draw a pie graph of Column F
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The volumes are vastly unequal.
Pie graph of Column F. When you looked at the
table, did you notice that the mantle occupies
75 of the Earth, and the crust has more volume
than the inner core?
Bar graph of Column F
Average weighted by volume 5.5 g/cm3
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Use the same type of representations to
illustrate our earlier model of the Earth where
all the shells are the same thickness (Module
1-3A, Slide 10).
Bar graph of thickness
Bar graph of volume
Pie graph of volume
Average weighted by volume (4.01 g/cm3)
Average weighted by thickness (5.95 g/cm3)
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Thickness
Volume
Average weighted by volume (4.01 g/cm3)
Average weighted by thickness (5.95 g/cm3)
Notice that for the shells of equal thickness,
the volumes of outer shells are larger than the
volumes of inner shells.
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So how can we show both depth and density for our
model of the density structure of the Earth?
One might think to plot an x-y graph of density
vs. depth (Column G vs. Column C)
This would be a horrible choice. Why?
Specifically, why would this grossly misrepresent
this model for the density structure of the Earth?
For example, what does the graph say is the
density at a depth of 5000 km? Is this what the
table says?
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The boundaries between the shells are
discontinuities, because the materials change.
The density jumps up in value at the boundaries.
The graph of density vs. depth is discontinuous.
It can be plotted like a series of steps.
Modify the spreadsheet of Slide 9 (equally thick
shells) to produce a step function when density
is plotted against depth.
This graph is not satisfying to geologists. We
like to see depths increase downward on the
vertical axis.
So reconfigure the graph so that density
increases to the right, and depth increases
downward.
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Now that you have the layout for your density vs.
depth plot, revise the depths and densities in
Columns D and E to conform with our model in
Slide 4.
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Here is the graph we want. It shows the
discontinuous variation of density vs. depth
within the Earth four shells and three
discontinuities.
Major seismic discontinuities in the Earth
Mohorovicic Discontinuity Crust/mantle boundary
(Andres Mohorovicic, 1909)
Gutenburg Discontinuity Mantle/core boundary
(Beno Gutenburg, 1912)
Lehman Discontinuity Inner/outer core boundary
(Ingrid Lehmann, 1936)
Of course, the density isnt constant within the
shells, so the picture is more complicated. How
would you solve that one?
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End of Module Assignments

• Add Column H, for mass of the shell, to the
  spreadsheet in Slides 4-7. Draw bar and pie
  charts for the distribution of mass in the
  Earths four shells.
• Modify the spreadsheets and graphs of Slide 9, so
  that, for the given densities, the thicknesses
  vary in such a way that the volumes are all
  equal.
• Suppose a planet has no discontinuities, but
  rather a continuously increasing density with
  depth. Suppose that the density at the surface
  of the planet is 2.8 g/cm3, that the density at
  the center of the planet is 13 g/cm3, and that
  between the surface and center, there is a
  straight-line variation between the extremes.
  Finally, suppose that the planets radius is 6000
  km. What is the density of such a planet?
  Remember that an integral is simply a weighted
  sum, so divide the radius into a large number
  (20-100) of equally thick shells, and do the same
  type of calculation that you did in Module 3A.
  Also plot volume of shell vs. depth.
• Actually, the density in the shells vary from
  small values at shallow depths to large values at
  greater depths. Thus the graph of density vs.
  depth has sloping lines within the shells
  instead of vertical lines in representations
  such as that in the previous slide. The range of
  densities for the shells given in Slide 11of
  Module 1-3A reflect that top-to-bottom gradation.
  So, create a spreadsheet that calculates the
  average density and graphs density vs. depth for
  a many-layer Earth with the density variation
  given in Slide 11 of Module 1-3A. Again,
  remember that an integral is a weighted sum.