Density - GEOL 114 The Earth's Dynamic Interior Lecture...

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GEOL 114 The Earth's Dynamic Interior Lecture Notes (Copyright © 2005 by Jeffrey S. Barker) 4. Density of the Earth Measuring and Weighing the Earth So far we have a model of the structure of the Earth based on measurements and inferences of P and S wave velocities. These velocities are more sensitive to variations in elastic properties (incompressibility and rigidity) than to variations in density. And yet density variations are more directly related to chemical composition and crystal structures of the materials in the Earth. (Elastic moduli give better information of the pressure/temperature regime). Therefore to understand the chemical composition of the Earth, we need to learn something about it's density variations. Recall that density is mass/volume. As a start, we'll determine the Earth's average density by weighing and measuring the Earth to obtain it's total mass and volume. Volume of the Earth First, lets "measure" the Earth's radius, R (the distance from the center to the surface). In about the year 200 BC, a Greek mathematician named Eratosthenes measured the radius of the Earth quite accurately. Of course, western Europeans conveniently forgot about this during the Middle Ages, so we make Columbus out as a hero for "proving" that the Earth is round. At any rate, let's look at Eratosthenes' method: At noon on the summer solstice, sunlight illuminated all the way to the bottom of a vertical well in Syene (modern Aswan) in Egypt. Therefore the sun must have been directly overhead. (What does this tell you about Syene's location?). The distance from Syene to Alexandria, Egypt, was measured to be 5,000 stadia (about 787 km) due north. By the way, Eratosthenes was the chief librarian in Alexandria at the time. At noon on the same day of the year, Eratosthenes set up a vertical rod in Alexandria (using a plumb line to define vertical) and measured the angle (7.2°) cast by GEOL 114 1 4. Density of the Earth
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the shadow. (So, the sun was not directly overhead in Alexandria. What does that tell you about its location?). Now, assume that the Earth is a sphere, so that "vertical" refers to the direction radially outward. Also, assume the sun is so far away that the sun's rays arrive essentially parallel at Earth. Then, with a bit of geometry, we can see that the angle of the shadow is the same as the angle between the radii from the Earth's center to Syene and Alexandria (7.2°). Since there are 360° in a circle, and the angle Eratosthenes measured was some fraction of a circle, the circumference of the Earth, C, may be determined from this fraction: 7.2 0 360 0 = 787 km C which Eratosthenes solved to get a circumference of 250,000 stadia (or 39,350 km). Since the circumference is simply C = 2 R , Eratosthenes' estimate of the Earth's radius was about 40,000 stadia or about 6,250 km. A modern estimate of the Earth's average radius is 6,371 km. Still assuming that the Earth is a sphere, the volume,
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This note was uploaded on 11/02/2010 for the course GEOLOGY 11 taught by Professor Howardr.naslund during the Spring '10 term at Binghamton University.

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Density - GEOL 114 The Earth's Dynamic Interior Lecture...

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