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

• 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.

• 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.

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.

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.

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.

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

The instructor version contains a pre-test