Transcript B. Density vs. Depth

Module 1-3B
The Earth’s Shells
B. Density vs. Depth
In Module 1-3A, we worked out a
model for the density structure of the
Earth – the density and thicknesses of
the four shells. How can we represent
this information graphically?
Quantitative Concepts and Skills
Weighted average
Bar and pie charts
Manipulation of XY graphs
Concept that an integral is a sum
1
Retrieve your spreadsheet from Slide 12 of Module 1-3A
B
2
3
4
5
6
7
8
9
10
11
C
D
r1
depth to
base (km)
surface
0
crust
50
mantle
2891
outer core
5150
inner core
6371
(km)
6371
6321
3480
1221
0
Sum
3
3
Sumproduct (km -g/cm )
3
12 weighted average (g/cm )
E
F
G
thickness
volume
density
(km)
(km )
(g/cm )
50
2841
2259
1221
2.53E+10
8.81E+11
1.69E+11
7.62E+09
2.8
4.6
10.5
13
6371
1.08E+12
3
3
This table is one representation
of the model for the variation of
density with depth, in that it
contains the information (i.e.,
Columns C and G tell the story).
But most people get more out of
a figure than a table.
6.00E+12
5.54
Problem: Develop a graph
that portrays this model for the
density structure of the Earth
2
PREVIEW
Slides 4 and 5 try to tell the story with bar graphs. Slide 4 shows density, but it
gives the impression that all densities are equally important. We remember that
volume is the weighting parameter, and so Slides 5 and 6 include a bar graph of
the volumes of the shells. The perhaps surprising result prompts us to make a
pie graph of volumes in Slide 7, and we explore the relation between thickness
and volume a little further in Slides 8 and 9
Because we wish to show both density and thickness (or depth, which is
cumulated thickness) on a single graph, we abandon bar and pie graphs for x-y
graphs, beginning in Slide 10. Our first choice of blindly plotting density vs.
depth produces a gross misrepresentation in Slide 10, because it ignores the
presence of the shells – that is, it assumes that the variation of density vs. depth
is a continuous function. Slide 11 gets us on track with a step function. Slides
11 and 12 apply the step function to the model of equally thick shells, so that we
can see what we are doing on the graph. Slide 13 completes the task by
plugging in the values that we found for the thickness and density of the shells.
The graph in Slide 13, then, portrays our finding from Module 1-3A.
3
One possibility is a bar graph showing the density of
the four shells (Column G).
2
3
4
5
6
7
8
9
10
11
C
D
r1
depth to
base (km)
surface
0
crust
50
mantle
2891
outer core
5150
inner core
6371
(km)
6371
6321
3480
1221
0
Sum
3
3
Sumproduct (km -g/cm )
3
12 weighted average (g/cm )
E
F
G
thickness
volume
density
(km)
(km )
3
50
2841
2259
1221
2.53E+10
8.81E+11
1.69E+11
7.62E+09
6371
1.08E+12
3
(g/cm )
2.8
4.6
10.5
13
15
Density (g/cm 3)
B
10
5
0
1
6.00E+12
2
3
4
Shells
5.54
This representation shows the density but it leaves out
very important information: the depths.
The depths are critically important here, especially in the context of
our constraint: average density = 5.5 g/cm3. Looking at the figure
(and not the table) one might get the impression that the average is
closer to 8 than to 5. Why?
4
The volumes are unequal, and so all the bars are not
the same importance in the weighted average.
B
C
2
3
4
5
6
7
8
9
10
11
D
r1
depth to
base (km)
surf ace
0
crust
50
mantle
2891
outer core
5150
inner core
6371
(km)
6371
6321
3480
1221
0
Sum
3
3
Sumproduct (km -g/cm )
3
12 weighted average (g/cm )
E
F
G
thickness
v olume
3
density
3
(km)
(km )
(g/cm )
50
2841
2259
1221
2.53E+10
8.81E+11
1.69E+11
7.62E+09
2.8
4.6
10.5
13
6371
1.08E+12
6.00E+12
5.54
Create a bar graph
showing the volumes of
the shells (Column F).
Density (g/cm 3)
15
10
Before that, look at the
table. What do you think
the relative sizes of the
bars will be?
5
0
1
2
3
4
Shells
Bar graph of Column G
5
The volumes are vastly unequal.
1.E+12
2
3
4
5
6
7
8
9
10
11
C
D
r1
depth to
base (km)
surf ace
0
crust
50
mantle
2891
outer core
5150
inner core
6371
(km)
6371
6321
3480
1221
0
Sum
3
3
Sumproduct (km -g/cm )
3
12 weighted average (g/cm )
E
F
G
thickness
v olume
density
(km)
(km )
(g/cm )
8.E+11
50
2841
2259
1221
2.53E+10
8.81E+11
1.69E+11
7.62E+09
2.8
4.6
10.5
13
7.E+11
6371
1.08E+12
3
9.E+11
3
6.00E+12
Volume (km 3)
B
6.E+11
5.E+11
4.E+11
3.E+11
5.54
2.E+11
Density (g/cm 3)
15
1.E+11
10
Average weighted by
volume = 5.5 g/cm3
5
0
1
2
3
Shells
4
Visually, the bar graph
of volumes explains
why the Earth’s
average density is so
close to the density of
the mantle.
0.E+00
1
2
3
4
shells
Bar graph of Column F
What percentage of the Earth’s
volume is in the Earth’s mantle?
Draw a pie graph of Column F
6
The volumes are vastly unequal.
C
2
3
4
5
6
7
8
9
10
11
D
r1
depth to
base (km)
surf ace
0
crust
50
mantle
2891
outer core
5150
inner core
6371
(km)
6371
6321
3480
1221
0
Sum
3
3
Sumproduct (km -g/cm )
3
12 weighted average (g/cm )
E
F
G
thickness
v olume
density
3
3
(km)
(km )
(g/cm )
50
2841
2259
1221
2.53E+10
8.81E+11
1.69E+11
7.62E+09
2.8
4.6
10.5
13
6371
1.08E+12
6.00E+12
5.54
Volume (km3)
Density (g/cm 3)
15
1%
16%
1.E+12
9.E+11
8.E+11
Volume (km 3)
B
7.E+11
6.E+11
5.E+11
4.E+11
3.E+11
2%
10
2.E+11
5
1.E+11
0.E+00
0
81%
1
2
3
Shells
Average weighted by
volume = 5.5 g/cm3
4
Pie graph of Column F. When you
looked at the table, did you notice that
the mantle occupies >75% of the Earth,
and the crust has more volume than the
inner core?
1
2
3
4
shells
Bar graph of Column F
7
Use the same type of representations to illustrate our earlier model of the
Earth where all the shells are the same thickness (Module 1-3A, Slide 10).
B
2
3
4
5
6
7
8
C
depth to
base (km)
surface
0
crust
1592.5
mantle
3185
outer core
4777.5
inner core
6370
D
r1
(km)
6370
4777.5
3185
1592.5
0
E
F
thickness
volume
(km)
1592.5
1592.5
1592.5
1592.5
G
3
(km )
density
3
(g/cm )
6.26E+11
3.21E+11
1.18E+11
1.69E+10
Bar graph
of thickness
2.8
5
7
9
Bar graph
of volume
Pie graph
of volume
Average weighted by volume (4.01 g/cm3)
Average weighted by thickness (5.95 g/cm3)
8
Thickness
2
3
4
5
6
7
8
C
depth to
base (km)
surface
0
crust
1592.5
mantle
3185
outer core
4777.5
inner core
6370
D
r1
(km)
6370
4777.5
3185
1592.5
0
E
F
G
thickness
volume
2000
thickness (km )
B
density
3
3
(km)
(km )
(g/cm )
1592.5
1592.5
1592.5
1592.5
6.26E+11
3.21E+11
1.18E+11
1.69E+10
2.8
5
7
9
1500
1000
500
0
1
2
3
4
shells
3
Volume (km )
11% 2%
volume (km 3)
8.E+11
6.E+11
30%
4.E+11
57%
2.E+11
0.E+00
1
2
3
4
Volume
shells
Average weighted by volume (4.01 g/cm3)
Average weighted by thickness (5.95 g/cm3)
Notice that for the shells of equal thickness, the volumes of outer
shells are larger than the volumes of inner shells.
9
So how can we show both depth and density for our
model of the density structure of the Earth?
2
C
D
r1
depth to
3
4
5
6
7
8
9
10
11
base (km)
surf ace
0
crust
50
mantle
2891
outer core
5150
inner core
6371
(km)
6371
6321
3480
1221
0
Sum
3
3
Sumproduct (km -g/cm )
3
12 weighted average (g/cm )
E
F
G
thickness
v olume
3
density
3
(km)
(km )
(g/cm )
50
2841
2259
1221
2.53E+10
8.81E+11
1.69E+11
7.62E+09
2.8
4.6
10.5
13
6371
1.08E+12
15
Density (g/cm 3)
B
10
5
0
0
6.00E+12
2000
4000
6000
8000
Depth (km)
5.54
One might think to plot an x-y graph of density
vs. depth (Column G vs. Column C)
This would be a horrible choice.
Why? Specifically, why would this grossly misrepresent this model for
the density structure of the Earth?
For example, what does the graph say is the density at a depth of
5000 km? Is this what the table says?
10
The boundaries between the shells are discontinuities, because the
materials change. The density jumps up in value at the boundaries.
The graph of density vs. depth is discontinuous. It can be plotted like
a series of steps.
B
Layer
2
3
4
Crust
5
6
Mantle
7
8 Outer core
9
10 Inner core
11
C
top
bottom
top
bottom
top
bottom
top
bottom
D
Depth
(km)
0
1592.5
1592.5
3185
3185
4777.5
4777.5
6370
E
Density
(g/cm3)
2.8
2.8
5
5
7
7
9
9
Density (g/cm 3)
Modify the spreadsheet of Slide 9 (equally thick shells) to produce a step
function when density is plotted against depth.
10
8
6
4
2
0
0
2000
4000
6000
8000
Depth (km)
This graph is not satisfying to
geologists. We like to see depths
increase downward on the vertical axis.
So reconfigure the graph so that density
increases to the right, and depth
increases downward.
11
density (g/cm3)
C
2
3
4
Crust
5
6
Mantle
7
8 Outer core
9
10 Inner core
11
top
bottom
top
bottom
top
bottom
top
bottom
D
Depth
(km)
0
1592.5
1592.5
3185
3185
4777.5
4777.5
6370
E
Density
(g/cm3)
2.8
2.8
5
5
7
7
9
9
0
5
10
0
1000
2000
depth (km)
B
Layer
3000
4000
5000
6000
7000
Now that you have the layout for your density vs. depth
plot, revise the depths and densities in Columns D and
E to conform with our model in Slide 4.
12
Here is the graph we want. It shows the
discontinuous variation of density vs.
depth within the Earth: four shells and
three discontinuities.
density (g/cm3)
0
5
10
15
0
1000
Major seismic discontinuities in the Earth
2000
Gutenburg Discontinuity: Mantle/core
boundary (Beno Gutenburg, 1912)
Lehman Discontinuity: Inner/outer core
boundary (Ingrid Lehmann, 1936)
Of course, the density isn’t constant
within the shells, so the picture is more
complicated. How would you solve that one?
3000
depth (km)
Mohorovičić Discontinuity: Crust/mantle
boundary (Andres Mohorovičić, 1909)
4000
5000
6000
7000
13
End of Module
Assignments
1.
Add Column H, for mass of the shell, to the spreadsheet in Slides 4-7. Draw bar and
pie charts for the distribution of mass in the Earth’s four shells.
2.
Modify the spreadsheets and graphs of Slide 9, so that, for the given densities, the
thicknesses vary in such a way that the volumes are all equal.
3.
Suppose a planet has no discontinuities, but rather a continuously increasing density
with depth. Suppose that the density at the surface of the planet is 2.8 g/cm3, that the
density at the center of the planet is 13 g/cm3, and that between the surface and center,
there is a straight-line variation between the extremes. Finally, suppose that the planet’s
radius is 6000 km. What is the density of such a planet? Remember that an integral is
simply a weighted sum, so divide the radius into a large number (20-100) of equally thick
shells, and do the same type of calculation that you did in Module 3A. Also plot volume
of shell vs. depth.
4.
Actually, the density in the shells vary from small values at shallow depths to large
values at greater depths. Thus the graph of density vs. depth has sloping lines within
the shells – instead of vertical lines – in representations such as that in the previous
slide. The range of densities for the shells given in Slide 11of Module 1-3A reflect that
top-to-bottom gradation. So, create a spreadsheet that calculates the average density
and graphs density vs. depth for a many-layer Earth with the density variation given in14
Slide 11 of Module 1-3A. Again, remember that an integral is a weighted sum.