# Earth's Planetary Density: Constraining What We Think about the Earth's Interior

This material was originally developed by Spreadsheets Across the Curriculum as part of its collaboration with the SERC Pedagogic Service.

#### Summary

In this Spreadsheets Across the Curriculum module, students explore combinations of densities and shell thicknesses that produce an aggregate planetary density of the Earth of 5.5 g/cm3. Students build a spreadsheet that calculates the average density of a planet consisting of four shells. They start constructing a model of the Earth with guessed densities of the shells and depths to the boundaries between the shells. One of the early scenarios is a planet in which the shells all have the same thickness. Next is an Earth with the discontinuities at their correct depths. The last scenario has all the depths correct, but densities correct only for the first and fourth shells (crust and inner core). For this last model, students need to find a combination of densities for the second and third shells (mantle, outer core) that produces Earth's planetary density.

Students use the spreadsheets to make pie charts showing how the Earth's volume is partitioned into the four shells. One of the end-of-module assignments asks them to add a column for mass and to construct a pie chart showing how the Earth's mass is distributed amongst the four shells.

The module includes links to material about seismology and the Earth's shells, and Andrija Mohorovicic, Beno Gutenberg, and Inge Lehmann.

## Learning Goals

Students will:
• Gain experience in solving problems calling for the weighted average.
• Apply their knowledge of the volume of a sphere to determine the volume of a spherical shell.
• Develop a spreadsheet to forward-calculate an answer and use the spreadsheet to find, by trial and error, values of input variables that produce a desired answer.
• Construct pie charts of the thickness and volume breakdown of the Earth.
In the process, the students will:
• Begin to recognize the power of using a spreadsheet to solve the inverse problem by trial and error.
• Gain insight to the concept of a constraint.
• Gain insight to the statement, "The solution is not unique."
• Gain insight to how modeling can involve trial-and-error fitting of parameters.
• Be impressed with how often a weighted average comes up in problem solving.
• Be startled with how much of the Earth is in the mantle.

## Context for Use

I use this module in my Computational Geology course, GLY 4866 (Acrobat (PDF) 39kB Sep25 06). The course is aimed at geology majors who have completed or are about to complete the mathematics courses required for the geology major. The course is a succession of interactive how-to-solve-it sessions.

This module comes in the fourth week of the semester. It is timed to coincide with the due date of their reading of Chapter 1 of Understanding our Quantitative World (Andersen and Swanson, 2005). This chapter, on functions, stresses four ways of representing a function: symbolically, verbally, by a graph, in a table.

The in-class session went roughly as follows:

• I started the session with the question, "Is the density of the Earth as a function of depth to the center of the Earth a function in the mathematical sense of the word?" After deciding that we will stipulate a radially symmetric Earth, the question then became "How would you represent rho(D) (four different ways)?
• We proceeded to try to draw a graph. The students were comfortable that it would look like a series of steps.
• The students were certainly aware of the four shells. Although they had seen illustrations of the four-layer model in numerous textbooks and/or Websites, it turns out that they hadn't looked closely at the depths of the discontinuities. Also, they did not have a feel for the magnitude of the shell densities, except for the crust.
• The first guess for the bottom three shells was timid: ca. 0.5 g/cm3 jumps at the discontinuities. I dutifully graphed the stairstep function with 0.5 g/cm3 risers. Then a student announced it was all nonsense – everything was less than the density of the Earth as a whole. That brought us nicely to the subject of the module – using the planetary density to constrain the model of thicknesses and densities.
• I told the students the depths to the discontinuities, a little about Andrija Mohorovicic, Beno Gutenberg and Inge Lehmann, and agreed-upon densities of the crust and inner core (as anchors of the graph). They divided into groups to work out the densities of the mantle and outer core.
Recognizing that this was a weighted-average problem and that they needed to use the volumes rather than the thicknesses of the shells posed no problem for the students. Neither did the formula for the volume of a shell. What turned out to be the biggest problem for some was that, in "Looking Back" (in Polya's terminology), the result that more than 80% of the Earth is in the mantle seems unreasonable.

The module "came live" on Blackboard during the in-class session. The students adjourned to work through it and hand in the end-of-module assignments for grading within a week.

## Description and Teaching Materials

PowerPoint SSAC2004.QE539.LV1.5-student version (PowerPoint 183kB Feb15 07)

The module is a PowerPoint presentation with embedded spreadsheets. If the embedded spreadsheets are not visible, save the the PowerPoint file to disk and open it from there.

This PowerPoint file is the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

This module is the revised edition of The Earth's Shells, A. Thicknesses and Densities (PowerPoint 159kB Jun11 12) and The Earth's Shells, B. Density vs. Depth (PowerPoint 317kB Jun11 12) from an earlier project Modules for Geological-Mathematical Problem Solving.

## Teaching Notes and Tips

The module is constructed to be a stand-alone resource. It can be used as a homework assignment or lab activity. It can also be used as the basis of an interactive classroom activity with just-in-time teaching of the relevant mathematics.

The module's final scenario asks the students to find a combination of two densities that produce the correct weighted average. One way of solving this problem with Excel is to guess one of the densities and then use the built-in solver to find the second density. I prefer that the students experience using Excel to find an answer by trial and error many times before I tell them about "Solver" near the end of the semester. However, with "Solver," one can easily find a wide range of combinations and plot one density against the other to examine their mutual dependence given the average as a constraint.

The module develops the formula for the volume of a spherical shell as the difference between the volume of two concentric spheres. The module contains a link to an appendix page that derives the same formula by integrating the surface area of a sphere with respect to the radius. I use this page to foreshadow a later module that uses numerical integration to solve volume problems.

## Assessment

The end-of-module questions can be used for assessment.

The instructor version contains a pre-test

## References and Resources

- Andersen, Janet and Swanson, Todd, 2005, Understanding our Quantitative World. The Mathematical Association of America, Washington D.C.