1 The Earth’s Shells Quantitative Concepts and Skills Weighted average Bar and pie charts...

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The Earth’s Shells

Quantitative Concepts and SkillsWeighted averageBar and pie chartsManipulation of XY graphsConcept that an integral is a sum

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?

Module 1-3B

B. Density vs. Depth

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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 thedensity structure of the Earth

B C D E F G

2 depth to r1 thickness volume density

3 base (km) (km) (km) (km3) (g/cm3)4 surface 0 63715 crust 50 6321 50 2.53E+10 2.86 mantle 2891 3480 2841 8.81E+11 4.67 outer core 5150 1221 2259 1.69E+11 10.58 inner core 6371 0 1221 7.62E+09 139

10 Sum 6371 1.08E+1211 Sumproduct (km3-g/cm3) 6.00E+12

12 weighted average (g/cm3) 5.54

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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.

0

5

10

15

1 2 3 4

Shells

Den

sity

(g

/cm

3)

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?

B C D E F G





5

B C D E F G





The volumes are unequal, and so all the bars are not the same importance in the weighted average.

Bar graph of Column G

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?0

5

10

15

1 2 3 4

Shells

Den

sity

(g

/cm

3)

6

The volumes are vastly unequal.

0

5

10

15

1 2 3 4

Shells

Den

sity

(g

/cm

3)

0.E+00

1.E+11

2.E+11

3.E+11

4.E+11

5.E+11

6.E+11

7.E+11

8.E+11

9.E+11

1.E+12

1 2 3 4

shells

Vo

lum

e (k

m3)

Bar graph of Column F

Average weighted byvolume = 5.5 g/cm3

What percentage of the Earth’s volume is in the Earth’s mantle? Draw a pie graph of Column F

Visually, the bar graph of volumes explains why the Earth’s average density is so close to the density of the mantle.

B C D E F G





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The volumes are vastly unequal.

0

5

10

15

1 2 3 4

Shells

Den

sity

(g

/cm

3)

0.E+00

1.E+11

2.E+11

3.E+11

4.E+11

5.E+11

6.E+11

7.E+11

8.E+11

9.E+11

1.E+12

1 2 3 4

shellsV

olu

me

(km

3)

Bar graph of Column FAverage weighted byvolume = 5.5 g/cm3

Volume (km3)

81%

16%

1%

2%

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?

B C D E F G





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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).

012

34567

89

10

1 2 3 4

ShellsAverage weighted by volume (4.01 g/cm3)

Average weighted by thickness (5.95 g/cm3)

Bar graphof thickness

Bar graph of volume

Pie graph of volume

B C D E F G


3 base (km) (km) (km) (km3) (g/cm3)4 surface 0 63705 crust 1592.5 4777.5 1592.5 6.26E+11 2.86 mantle 3185 3185 1592.5 3.21E+11 57 outer core 4777.5 1592.5 1592.5 1.18E+11 78 inner core 6370 0 1592.5 1.69E+10 9

9

0

500

1000

1500

2000

1 2 3 4

shells

thic

kn

es

s (

km

)

012

34567

89

10

1 2 3 4

Shells

0.E+00

2.E+11

4.E+11

6.E+11

8.E+11

1 2 3 4

shellsvo

lum

e (k

m3)

Average weighted by volume (4.01 g/cm3)

Average weighted by thickness (5.95 g/cm3)

Thickness

Volume

Volume (km3)

57%30%

11% 2%

Notice that for the shells of equal thickness, the volumes of outer shells are larger than the volumes of inner shells.

B C D E F G


3 base (km) (km) (km) (km3) (g/cm3)4 surface 0 63705 crust 1592.5 4777.5 1592.5 6.26E+11 2.86 mantle 3185 3185 1592.5 3.21E+11 57 outer core 4777.5 1592.5 1592.5 1.18E+11 78 inner core 6370 0 1592.5 1.69E+10 9

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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?

0

5

10

15

0 2000 4000 6000 8000

Depth (km)

Den

sity

(g

/cm

3)

For example, what does the graph say is the density at a depth of 5000 km? Is this what the table says?

B C D E F G





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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.

0

2

4

6

8

10

0 2000 4000 6000 8000

Depth (km)

Den

sity

(g

/cm

3)

B C D E2 Layer Depth Density

3 (km) (g/cm3)4 Crust top 0 2.85 bottom 1592.5 2.86 Mantle top 1592.5 57 bottom 3185 58 Outer core top 3185 79 bottom 4777.5 7

10 Inner core top 4777.5 911 bottom 6370 9

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.

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.

0

1000

2000

3000

4000

5000

6000

7000

0 5 10

density (g/cm3)

dept

h (k

m)

B C D E2 Layer Depth Density3 (km) (g/cm3)4 Crust top 0 2.85 bottom 1592.5 2.86 Mantle top 1592.5 57 bottom 3185 58 Outer core top 3185 79 bottom 4777.5 710 Inner core top 4777.5 911 bottom 6370 9

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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.

0

1000

2000

3000

4000

5000

6000

7000

0 5 10 15

density (g/cm3)

dept

h (k

m)

Major seismic discontinuities in the Earth

Mohorovičić Discontinuity: Crust/mantle boundary (Andres Mohorovičić, 1909)

Gutenburg Discontinuity: Mantle/core boundary (Beno Gutenburg, 1912)

Lehman Discontinuity: Inner/outer core boundary (Ingrid Lehmann, 1936)

Of course, the density isn’t constantwithin the shells, so the picture is more

complicated. How would you solve that one?

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End of Module Assignments

1. 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 Earth’s four shells.

2. 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.

3. 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 planet’s 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.

4. 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.

1 The Earth’s Shells Quantitative Concepts and Skills Weighted average Bar and pie charts...

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