SSAC2004.QE539.LV1.5
Earth’s Planetary Density – Constraining What
We Think of the Earth’s Interior
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).
Core Quantitative Issue
Weighted average
Supporting Quantitative Issues
Unit conversions
Solid geometry: Volume of spherical shell
Forward modeling: Inverse problem by trial and error
Integral: Concept
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
1
Version 10/04/06
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.
Crust
Mantle
Outer core
Inner core
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 2
Earth, which we knew before the first travel time was ever measured.
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
3
Lehmann in 1936 -- at 5150 km. (End note 4)
Problem and Overview
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.
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.
4
Slide 14 gives the end-of-module assignment.
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.
B
2
3
4
5
6
7
8
9
10
Shell
crust
mantle
outer core
inner core
C
D
3
Thickness (km) Density (g/cm )
1592.5
2.8
1592.5
5
1592.5
7
1592.5
9
SUM (km)
6370
3
SUMPRODUCT (km-g/cm )
WEIGHTED AVERAGE (g/cm 3)
37901.5
5.95
Recreate this spreadsheet.
= cell with a number in it
= cell with a formula in it
Note the units
5
But … Our spreadsheet won’t work
Our spreadsheet uses thickness as the weighting factor.
We need for volume to be the weighting factor.
Why:
Density is mass over volume,
 Earth 
M Earth
VEarth
(1)
The mass of the Earth is the sum of the masses of all of the shells,
M Earth 
Vi  i
(2)
shells
The volume of the Earth is the sum of the volumes of all of the shells,
VEarth 
Vi
(3)
shells
Combining the three equations produces the weighted average
 Earth 
Vi i
shells
Vi
shells
(4)
6
The weighting factor
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?
r1
r2
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)
An alternative
derivation
Caution: Not the same as = (4/3)*PI()*(r2 – r1)^3
7
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
Thickness (km)
D
Depth to base
E
R top
F
Volume
G
Density
crust
mantle
outer core
inner core
(km)
1592.5
1592.5
1592.5
1592.5
(km)
1592.5
3185
4777.5
6370
(km)
6370
4777.5
3185
1592.5
(km3)
6.26E+11
3.21E+11
1.18E+11
1.69E+10
(g/cm3)
2.8
5
7
9
SUM (km)
6370
2 Shell
3
4
5
6
7
8
9
10
11
12
SUM (km3)
1.08E+12
3
SUMPRODUCT (km -g/cm3)
WEIGHTED AVERAGE (g/cm 3)
4.34094E+12
4.01
8
Looking at your results
Look at the distribution of thickness and volumes in this model Earth
B
C
Thickness (km)
D
Depth to base
E
R top
F
Volume
G
Density
crust
mantle
outer core
inner core
(km)
1592.5
1592.5
1592.5
1592.5
(km)
1592.5
3185
4777.5
6370
(km)
6370
4777.5
3185
1592.5
(km3)
6.26E+11
3.21E+11
1.18E+11
1.69E+10
(g/cm3)
2.8
5
7
9
SUM (km)
6370
2 Shell
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
SUM (km3)
1.08E+12
3
SUMPRODUCT (km -g/cm3)
WEIGHTED AVERAGE (g/cm 3)
Volumes
Thicknesses
outer core
11%
inner core
25%
inner core
2%
crust
25%
mantle
30%
outer core
25%
4.34094E+12
4.01
mantle
25%
Notice how the
outer shells have
larger volumes
than the inner
shells in this
Earth in which all
the shells have
the same
thickness.
crust
57%
Add these pie
graphs to your
spreadsheets.
For help with
pie charts
9
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.)
B
C
Thickness (km)
D
Depth to base
E
R top
F
Volume
G
Density
crust
mantle
outer core
inner core
(km)
50
2840
2260
1220
(km)
50
2890
5150
6370
(km)
6370
6320
3480
1220
(km3)
2.53E+10
8.81E+11
1.69E+11
7.61E+09
(g/cm3)
2.8
5
7
9
SUM (km)
6370
2 Shell
3
4
5
6
7
8
9
10
11
12
SUM (km3)
1.08E+12
3
SUMPRODUCT (km -g/cm3)
WEIGHTED AVERAGE (g/cm 3)
Close, but less than the known
density of the Earth. We need to
increase some of the shell densities.
5.72611E+12
5.29
10
A more realistic Earth
Now we can change the densities until Cell G12 agrees with the known
density of the Earth.
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.
B
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
C
Thickness (km)
D
Depth to base
E
R top
F
Volume
G
Density
crust
mantle
outer core
inner core
(km)
50
2840
2260
1220
(km)
50
2890
5150
6370
(km)
6370
6320
3480
1220
(km3)
2.53E+10
8.81E+11
1.69E+11
7.61E+09
(g/cm3)
2.8
5
7
9
SUM (km)
6370
2 Shell
3
4
5
6
7
8
9
10
11
12
Shell
SUM (km3)
1.08E+12
3
SUMPRODUCT (km -g/cm3)
WEIGHTED AVERAGE (g/cm 3)
5.72611E+12
5.29
Goal: Change
Cells G5, G6, and
G7 until G12 is
5.54.
G7 is easy: make
that one 13
11
A better model for the layered Earth
Here is one possibility
B
C
Thickness (km)
D
Depth to base
E
R top
F
Volume
G
Density
crust
mantle
outer core
inner core
(km)
50
2840
2260
1220
(km)
50
2890
5150
6370
(km)
6370
6320
3480
1220
(km3)
2.53E+10
8.81E+11
1.69E+11
7.61E+09
(g/cm3)
2.8
4.6
10.5
13
SUM (km)
6370
2 Shell
3
4
5
6
7
8
9
10
11
12
SUM (km3)
1.08E+12
3
SUMPRODUCT (km -g/cm3)
WEIGHTED AVERAGE (g/cm 3)
5.99544E+12
5.54
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.
12
Looking at our model for the Earth
Thicknesses
Inner Core
19%
The mantle’s thickness is
about half the radius of the
Earth, but the mantle makes
up more than 4/5 of the Earth.
Crust
1%
Mantle
45%
Outer Core
35%
Volumes
Inner Core
1%
Outer Core
16%
The crust is by far the
thinnest of the Earth’s
shells, but the inner core
has a larger volume.
Crust
2%
Mantle
81%
13 5
End note
End of Module Assignments
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?
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End Notes
1. For an overview of the basics: http://scign.jpl.nasa.gov/learn/plate1.htm.
Return to Slide 2
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
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
15
Volume of a spherical shell
Find the volume of a spherical shell with inner radius r1
and outer radius r2.
ri
r1
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.
m icroshell
thickness = dr
r2
An integral is a sum.
Let ri be the internal radius of any given
microshell.
Recall the area of a sphere with radius r:
Then, write down equation for volume of each microshell:
A  4r 2
Vi  4ri2 dr
Thickness, dr, is infinitesimal.
And, add them up: V 
r2

4
2
2
4

r
dr

4

r
dr

 r23  r13
 i

3
microshells
r
1

Return to Slide 7
16
To make a simple pie chart
B
C
Thickness (km)
D
Depth to base
E
R top
F
Volume
G
Density
crust
mantle
outer core
inner core
(km)
1592.5
1592.5
1592.5
1592.5
(km)
1592.5
3185
4777.5
6370
(km)
6370
4777.5
3185
1592.5
(km3)
6.26E+11
3.21E+11
1.18E+11
1.69E+10
(g/cm3)
2.8
5
7
9
SUM (km)
6370
2 Shell
3
4
5
6
7
8
9
10
11
12
SUM (km3)
1.08E+12
SUMPRODUCT (km3-g/cm3)
WEIGHTED AVERAGE (g/cm 3)
4.34094E+12
4.01
• 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
17