1. 1 The Earth’s Shells Quantitative Concepts and Skills Weighted average Bar and pie charts Manipulation of XY graphs Concept 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

2. 2 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

3. 3 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.

4. 4 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/cm 3. Looking at the figure (and not the table) one might get the impression that the average is closer to 8 than to 5. Why?

5. 5 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?

6. 6 The volumes are vastly unequal. Bar graph of Column F Average weighted by volume = 5.5 g/cm 3 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.

7. 7 The volumes are vastly unequal. Bar graph of Column F Average weighted by volume = 5.5 g/cm 3 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?

8. 8 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). Average weighted by volume (4.01 g/cm 3 ) Average weighted by thickness (5.95 g/cm 3 ) Bar graph of thickness Bar graph of volume Pie graph of volume

9. 9 Average weighted by volume (4.01 g/cm 3 ) Average weighted by thickness (5.95 g/cm 3 ) Thickness Volume Notice that for the shells of equal thickness, the volumes of outer shells are larger than the volumes of inner shells.

10. 10 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?

11. 11 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. 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.

12. 12 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.

13. 13 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 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 constant within the shells, so the picture is more complicated. How would you solve that one?

14. 14 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.