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

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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 IssuesUnit conversionsSolid geometry: Volume of spherical shellForward modeling: Inverse problem by trial and errorIntegral: 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/cm3), which is known from the value of g (9.81 m/sec2).

Version 10/04/06

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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.

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.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.

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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/cm3.

– 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)

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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.

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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.

B C D

2 Shell Thickness (km) Density (g/cm3)3 crust 1592.5 2.84 mantle 1592.5 55 outer core 1592.5 76 inner core 1592.5 978 SUM (km) 63709 SUMPRODUCT (km-g/cm3) 37901.5

10 WEIGHTED AVERAGE (g/cm3) 5.95

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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)

Earth

EarthEarth V

M

shells

iiEarth VM

shells

iEarth VV

shellsi

shellsii

Earth V

V

But … Our spreadsheet won’t work

Our spreadsheet uses thickness as the weighting factor. We need for volume to be the weighting factor.

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The weighting factor

r1r2

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

Vsphere = (4/3)*PI()*r^3, in the language of Excel

What about the volume of a spherical shell?

Let r1 = inside diameter r2 = outside diameter

Then think of subtracting the inside sphere from the outside sphere

Vshell = (4/3)*PI()*(r2^3 – r1^3)

Caution: Not the same as = (4/3)*PI()*(r2 – r1)^3

An alternative derivation

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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.

B C D E F G

2 Shell Thickness (km) Depth to base R top Volume Density

3 (km) (km) (km) (km3) (g/cm3)4 crust 1592.5 1592.5 6370 6.26E+11 2.85 mantle 1592.5 3185 4777.5 3.21E+11 56 outer core 1592.5 4777.5 3185 1.18E+11 77 inner core 1592.5 6370 1592.5 1.69E+10 989 SUM (km) 6370

10 SUM (km3) 1.08E+1211 SUMPRODUCT (km3-g/cm3) 4.34094E+1212 WEIGHTED AVERAGE (g/cm3) 4.01

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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

B C D E F G



10 SUM (km3) 1.08E+1211 SUMPRODUCT (km3-g/cm3) 4.34094E+1212 WEIGHTED AVERAGE (g/cm3) 4.0113 1415161718192021222324252627282930

Thicknesses

inner core25%

outer core25%

mantle25%

crust25%

Volumes

crust57%

mantle30%

inner core2%

outer core11%

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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.

B C D E F G


3 (km) (km) (km) (km3) (g/cm3)4 crust 50 50 6370 2.53E+10 2.85 mantle 2840 2890 6320 8.81E+11 56 outer core 2260 5150 3480 1.69E+11 77 inner core 1220 6370 1220 7.61E+09 989 SUM (km) 6370


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A more realistic Earth

Now we can change the densities until Cell G12 agrees with the known density of the Earth.

Shell Density (g/cm3) Depth to base (km)

Crust 2.6-2.9 50

Mantle 3.38-5.56 2891

Outer Core 9.90-12.16 5150

Inner Core 12.76-13.08 6371

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

B C D E F G


3 (km) (km) (km) (km3) (g/cm3)4 crust 50 50 6370 2.53E+10 2.85 mantle 2840 2890 6320 8.81E+11 56 outer core 2260 5150 3480 1.69E+11 77 inner core 1220 6370 1220 7.61E+09 989 SUM (km) 6370


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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.

B C D E F G


3 (km) (km) (km) (km3) (g/cm3)4 crust 50 50 6370 2.53E+10 2.85 mantle 2840 2890 6320 8.81E+11 4.66 outer core 2260 5150 3480 1.69E+11 10.57 inner core 1220 6370 1220 7.61E+09 1389 SUM (km) 6370


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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.

Thicknesses

Crust1%

Inner Core19%

Outer Core35%

Mantle45%

Volumes

Inner Core1% Crust

2%Outer Core16%

Mantle81%

End note 5

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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/cm3). 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 GMEarth/R2, 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

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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.htm

Return to Slide 2

1. For an overview of the basics: 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.

Return to Slide 2

4.Andrija Mohorovicic, 1857-1936: http://istrianet.org/istria/illustri/mohorovicic/ Beno Gutenberg, 1889-1960: http://www.agu.org/inside/awards/gutenberg.html Inge Lehmann, 1888-1993: http://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/

Return to Slide 13

End Notes

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m ic ro she llth ic kne ss = d r

r1r2

Then, write down equation for volume of each microshell: drrV ii24

And, add them up: 31

32

22

3

444

2

1

rrdrrdrrVr

rsmicroshelli

Recall the area of a sphere with radius r: 24 rA

Volume of a spherical shell

Find the volume of a spherical shell with inner radius r1 and outer radius r2.

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 ri be the internal radius of any given microshell.

An integral is a sum.

Thickness, dr, is infinitesimal.

Return to Slide 7

ri

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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

B C D E F G




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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