1. 1 SSAC2004.QE539.LV1.5 Earth’s Planetary Density – Constraining What We Think of the Earth’s Interior Prepared for SSAC by Len Vacher – University of South Florida, Tampa © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2006 Supporting Quantitative Issues Unit conversions Solid geometry: Volume of spherical shell Forward modeling: Inverse problem by trial and error Integral: Concept Core Quantitative Issue Weighted average Any model for the thickness and density of Earth’s constituent shells must be consistent with the planetary density (5.5 g/cm 3 ), which is known from the value of g (9.81 m/sec 2 ). Version 10/04/06

2. 2 Crust Mantle Outer core Inner core The Earth’s Shells No doubt you know that the Earth consists of concentric shells: an outermost crust, a thick shell called the mantle, and an interior core (end note 1). You probably also know that the core can be subdivided into an outer core that is liquid and an inner core that is solid.end note 1 In your first geology course you learned that knowledge of these shells has come from the interpretation of travel times of seismic waves. Earthquakes occur near the surface of the Earth (up to depths of ~700 km), and so seismic waves (specifically P and S waves; 2) travel from one side of the Earth to the other passing through the Earth’s interior. Their arrival at the Earth’s surface is recorded by seismographs (3). The travel times of the P and S waves give an indication of the density of the material that the waves traversed.23 This module explores the magnitude of these densities – specifically, how our interpretation of their values is constrained by the overall average density of the Earth, which we knew before the first travel time was ever measured.

3. 3 Density as a function of depth Before seismology it was known –The Earth is a sphere with circumference 40,000 km, and therefore a radius of 6370 km. –The average density of the planet is 5.5 g/cm 3. –Nearly all rocks we see at or near the surface are less dense than the planet as a whole. In fact, except for unusual rocks such as ores, rocks that we experience first hand are about half as dense as the Earth as a whole. –Therefore, the Earth must be denser in the interior than it is near the surface. With early seismology it was known that the density of the interior changes abruptly at certain depths, that the interior of the Earth is structured into layers. The boundaries between the layers are named discontinuities, because they register as discontinuities in the graph of P and S velocity – and hence density – as a function of depth. The three boundaries are: –The Mohorovicic Discontinuity (1909), at 5-70 km depth. –The Gutenberg Discontinuity (1914), at 2890 km. –The boundary between liquid and solid discovered by Inge Lehmann in 1936 -- at 5150 km. (End note 4)End note 4

4. 4 The goal of this module is to make a first cut at describing how the density of the Earth varies as a function of depth. We know the depths of the discontinuities. The abrupt increases in seismic velocities at the discontinuities demonstrate that the densities increase from shell to shell as we go deeper within the Earth. And, we know the overall average density of the Earth. What combination of increasing shell densities produces the overall average? Clearly, the core mathematics of this module is the weighted average. Problem and Overview Slide 5 starts with a spreadsheet to find the average density of a rectangular stack of layers using a weighted average with thickness as the weighting factor. Slides 6-8 elaborate on why volume needs to be the weighting factor, considers the volume of a spherical shell, and adapts the previous spreadsheet accordingly for the case of shells of equal thickness. Slide 9 examines the difference between the distribution of thickness and volume in a layered sphere. Slides 10-13 consider a more realistic Earth. Slide 14 gives the end-of-module assignment.

5. 5 Getting started In order to start designing our spreadsheet, we will begin with an easier problem – a stack of layers, rather than concentric cells. We will also assume that the layers are all the same thickness. And, we will make some guesses for the densities. = cell with a number in it = cell with a formula in it Note the units Recreate this spreadsheet.

6. 6 Why: Density is mass over volume, (1) The mass of the Earth is the sum of the masses of all of the shells, (2) The volume of the Earth is the sum of the volumes of all of the shells, (3) Combining the three equations produces the weighted average (4) But … Our spreadsheet won’t work Our spreadsheet uses thickness as the weighting factor. We need for volume to be the weighting factor.

7. 7 The weighting factor r1r1 r2r2 We need the volume of the three spherical shells (crust, mantle and outer core) and the volume of the inner sphere (inner core) The volume of a sphere is no problem V sphere = (4/3)*PI()*r^3, in the language of Excel What about the volume of a spherical shell? Let r 1 = inside diameter r 2 = outside diameter Then think of subtracting the inside sphere from the outside sphere V shell = (4/3)*PI()*(r 2 ^3 – r 1 ^3) Caution: Not the same as = (4/3)*PI()*(r 2 – r 1 )^3 An alternative derivation

8. 8 Building your spreadsheet Now that we have the formula for the volume of a spherical shell, we can lay out a spreadsheet that calculates the average total density using the same values for the thickness and density of the shells. Insert three new columns between the one with thicknesses and the one with densities. In Column D, list the depth to the base of each shell by cumulating the thicknesses. In Column E, list the distance (radius) from the center of the earth to the top of each shell. Start with the radius of the Earth in E4 (=C9) and subtract the successive thicknesses. In Column F, calculate the volumes. Then complete Rows 10-12.

9. 9 Looking at your results Look at the distribution of thickness and volumes in this model Earth Notice how the outer shells have larger volumes than the inner shells in this Earth in which all the shells have the same thickness. Add these pie graphs to your spreadsheets. For help with pie charts

10. 10 A more realistic Earth Now change the thicknesses to ones that are consistent with the known depths of the crust/mantle, mantle/core and outer/inner core boundaries. (Use 50 km to represent an average for the crust.) Close, but less than the known density of the Earth. We need to increase some of the shell densities.

11. 11 A more realistic Earth Now we can change the densities until Cell G12 agrees with the known density of the Earth. ShellDensity (g/cm 3 )Depth to base (km) Crust2.6-2.950 Mantle3.38-5.562891 Outer Core9.90-12.165150 Inner Core12.76-13.086371 Here are values from a standard textbook: Fowler, C.M.R., 1990, The Solid Earth: An Introduction to Global Geophysics, Cambridge University Press, p. 112. Goal: Change Cells G5, G6, and G7 until G12 is 5.54. G7 is easy: make that one 13

12. 12 A better model for the layered Earth Here is one possibility What we have done – Forward modeling and the inverse problem. We can calculate the aggregate density from the thicknesses and densities of the constituent shells. We know the thicknesses and some of the densities. We “guess” the other densities until the calculation produces the known aggregate density. That known aggregate density constrains the guesses. Thus by trial and error we find the unknown shell densities, hence solving the inverse problem. We have fit the forward model to the constraint. Our solution of the inverse problem ( for the densities) is not unique.

13. 13 Looking at our model for the Earth The mantle’s thickness is about half the radius of the Earth, but the mantle makes up more than 4/5 of the Earth. The crust is by far the thinnest of the Earth’s shells, but the inner core has a larger volume. End note 5

14. 14 1.According to Dante’s Inferno, Hell is at the center of the Earth. Assume that Hell is hollow with a radius of 1000 km, and that the rest of the Earth is the density of normal rocks (say 2.8 g/cm 3 ). What would the overall density of the Earth be? Submit a spreadsheet for such a two-layer Earth. 2.Submit a spreadsheet producing the correct overall average density of the Earth (Slide 12) with a different combination of densities for the mantle and outer core. 3.Add a column to the spreadsheet in Slide 12 that calculates the mass (kg) of each of the shells and the overall mass of the Earth. Make a pie graph for the distribution of masses. Submit the spreadsheet and pie graph. Note how does the distribution of masses differ from the distribution of volumes. 4.What is g at the surface of the Earth using Newton’s Law of Gravitation and the mass of the Earth from Question 3? For such a problem, it can be shown that we can consider all of the Earth’s mass to be at a point at the center of the Earth. Recall that g is the force of gravity per unit mass at the surface of the earth. Then, from Newton’s Law of gravitation, g is GM Earth /R 2, where R is the radius of the Earth. Google for the value of G. 5.What is g at the surface of the Earth described in Question 1? End of Module Assignments

15. 15 2. P and S waves are the two types of seismic waves that go through the Earth. They are body seismic waves, as opposed to surface seismic waves, which move along the Earth’s surface and do the great damage associated with earthquakes. For more: http://scign.jpl.nasa.gov/learn/eq6.htmhttp://scign.jpl.nasa.gov/learn/eq6.htm Return to Slide 2 1.For an overview of the basics: http://scign.jpl.nasa.gov/learn/plate1.htm.http://scign.jpl.nasa.gov/learn/plate1.htm Return to Slide 2 3. For a few basics about earthquakes, including an animation showing how passage of earthquake waves make a seismogram at a seismograph: http://www.discoverourearth.org/student/earthquakes/index.html. http://www.discoverourearth.org/student/earthquakes/index.html Return to Slide 2 4. Andrija Mohorovicic, 1857-1936: http://istrianet.org/istria/illustri/mohorovicic/http://istrianet.org/istria/illustri/mohorovicic/ Beno Gutenberg, 1889-1960: http://www.agu.org/inside/awards/gutenberg.htmlhttp://www.agu.org/inside/awards/gutenberg.html Inge Lehmann, 1888-1993: http://www.agu.org/inside/awards/lehmann2.htmlhttp://www.agu.org/inside/awards/lehmann2.html Return to Slide 3 5. For a summary account of the interior of the Earth, including a table of density as a function of depth: http://pubs.usgs.gov/gip/interior/http://pubs.usgs.gov/gip/interior/ Return to Slide 13 End Notes

16. 16 Then, write down equation for volume of each microshell: And, add them up: Recall the area of a sphere with radius r: Volume of a spherical shell Find the volume of a spherical shell with inner radius r 1 and outer radius r 2. Strategy: Divide the shell into a bazillion microshells each with incredibly small thickness dr. Find the volume of each of the microshells and add them up. Let r i be the internal radius of any given microshell. An integral is a sum. Thickness, dr, is infinitesimal. Return to Slide 7 riri

17. 17 To make a simple pie chart Block out C4 to C7 (or F4 to F7 for the volume). Click on the chart wizard. Select “Pie.” Click Next Click Next again. Type in your title. Click finish. Right-click on one of the pie slices. Select “Format Data Series.” Select “Data Labels” tab. Check “Percentages” and “Category Name”. Hit “OK”. Left-click twice on one of the labels. Change the numeral representing the category to the label (e.g., “mantle”) that you want. Right-click on the legend. Select clear. Return to Slide 9