# Selected Topics in Astronomy and Space Studies

## CONTENTS OF CURRICULUM UNIT 96.06.11

## An Interdisciplinary Approach to Kepler’s Laws

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## BASIC CONCEPTS

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Distance
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The measurement of distance in space requires serious attention. Because of the vastness of space, the conventional unit of measuring distance on Earth becomes too cumbersome if applied to distances in space. Thus, the application of light year and Astronomical Units, become an essential part of the discussion. A basic interpretation of the light year could be contrasted with a real life situation. For example, a car traveling at a speed of 1 mile per minute will travel a distance of 60 miles in one hour. In two hours at the same speed, it will travel a distance of 120 miles. Students could be asked to predict how far it would travel in ten hours if it travels at the same average speed. Next, students could be expected to imagine that the car never stops and it maintains the average speed of 60 miles an hour for one day. How far would it have traveled in one day? The result could be named “one car day”. Students could then be required to calculate the distance traveled by the same car under the same conditions for one week, then one month, then one year. This would provide working definitions for a “car week, “car month “ and a “car year”.

The same situation is applied to the concept of a light year. Light travels at a speed of about 186,000 miles per second or about 300,000 kilometers per second. That distance could be immediately classified as one light second. The same distance multiplied by 60 would provide the distance light travels in one minute, which is about 11,160,000 miles or 18,000,000 kilometers. If students are required to calculate up to a day, they might have a greater appreciation of what is meant by one light year—the distance light travels in space (vacuum) for one year. The Sun is about 93,000,000 miles or about 150,000,000 kilometers from the Earth. Light from the sun takes about eight minutes to reach the Earth, a distance which could be rightly classified as eight light minutes. Bodies in the Universe are so far apart that the use of light year becomes a useful and practical unit for measuring distance.

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Astronomical Units
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The Earth’s distance of about 93,000,000 miles from the Sun is also used as a unit for measuring distances in space, especially when we deal with distances in our Solar System. One Astronomical Unit (AU) is equivalent to the distance of 93,000,000 million miles or 150,000,000 kilometers. Students should be required to convert large distances to Astronomical Units. A meaningful exercise would be to convert the planetary distances from miles to Astronomical Units, AU.

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Rotation
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It is of primary importance the students clearly understand the concept of rotation and revolution. To simply explain the concept of rotation, a globe which has a base can be placed in a stationary position and the spherical portion is spun. The idea is that a rotation is more or less an object completing circular motions (360 degrees) in some what of a stationary position.
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Revolution
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If the teacher picks up the globe and allow it to rotate but at the same time move around the desk, then the idea of a revolution could be introduced . The idea is that in a revolution, the object moves around another object. In this case, the globe is rotating and at the same time it is revolving around the desk: orbiting and revolving should be used interchangeably. The Earth makes one complete rotation approximately every 24 hours, which allows us to experience night and day. However, as it rotates, it orbits or revolves around the Sun. It takes the Earth about 365.25 days or one year to complete one orbit or one revolution around the Sun.

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Velocity
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Students should be made to understand that velocity is somewhat like speed. Speed tells us how far an object travels in a certain time—60 miles an hour , 60 miles in one hour. Velocity also tells us how far the object travels in a certain time but it has a direction. However, at the seventh grade level, speed and velocity can be used inter-changeably.

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Ellipse
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At the seventh grade level, it might be safe to represent an ellipse as an oval shape. Students could be asked to identify objects which they are familiar with, which have oval shapes (a football, an egg). The basic features of an ellipse are the major axis, minor axis, center and foci. The students could be given several drawn ellipses, and as the instruction proceeds, they could be required to put in the elements. It would be fair to elicit from the students definitions of the various elements of the ellipse. This information they should be able to extract from the diagrams they have been working with. These definitions should be refined to fit the formal concept.

The maximum diameter or the major axis would represent the line drawn from the two farthest points on the ellipse which passes through the center. The minimum diameter or the minor axis would represent the line drawn at right angle to the major axis passing through the center of the ellipse. The two points, one on either side of the center of the major axis, are called the foci. These points determine the shape of the ellipse. The further apart the foci are, the more elongated the ellipse will be. The opposite is true if the two points get closer. If both points coincide, then there is a perfect circle. The distance between the foci compared to the major axis is called the eccentricity of the ellipse. When it is zero, it is a circle. Between zero and one, it is an ellipse. Whenever it is less than zero or greater than one, it produces some shapes other than ellipses.

There is a constant term which is associated with ellipses. If both foci A and B are identified, and a point C is chosen on the rim of the ellipse, the sum of the distances, AC + BC is equal to the length of the major axis. This is true for any point on an ellipse.

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Kepler’s Laws
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This is a classic example of the application of a mathematical solution to a theory which had existed for centuries. The common belief that the Earth was the center of the Universe, was dispelled convincingly by Kepler, who was mathematician and an ardent astronomer.
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Kepler’s First Law
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Kepler’s first law stated that the planetary orbits around the Sun are elliptical, contrary to the long held belief that they were perfect circles. This law can be demonstrated by constructing an ellipse with the Sun at one focus. At a point on the ellipse, the orbiting planet could be drawn. Here, the students could from observation infer that at some points on the orbit, the planet is closer to the Sun, and at other times, it is farther away.

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Kepler’s Second Law
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Kepler’s second law states that when an orbiting body is closest to the Sun, its orbiting velocity will increase so as to sweep through an equal area within the same time interval as when it was farther away. This means that if it takes X days for the planet to cover an area of B square units on a portion of the orbit which is farther away from the Sun, then when the planet gets to the section of the orbit closest to the Sun, for it to cover the same area of sweep in the same time of X days, it will have to increase its velocity. This law further reinforces the principle that the orbital path of the planet is elliptical.

Students could have a clearer image of this law if it is sketched on grid paper. Here they will be able to develop and practice their estimating skills. Instead of trying to formally calculate the area of sweep on an ellipse, students could count the number of squares which would be needed to match the given time of orbit.

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Kepler’s Third Law
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The time that the planet takes to complete on orbit around the Sun is called the Sidereal Period. Kepler stated that the square of the Sidereal Period of a planet is proportional to the cube of the mean distance from the Sun, P
^{
2
}
= D
^{
3
}
. Where P is equal to the sidereal period and D is the average distance of the Sun from the Earth in Astronomical Units, AU. If the Sidereal Period of a planet is known, its distance from the Sun can be derived from the same law. Similarly, the Sidereal Period can be calculated from the known distance from the Sun.

Here students are provided with the interesting opportunity of learning and practicing the art of substituting in a formula. Firstly, there must be a clear understanding of what the variables represent. The cube of a number should be expressed in an expanded form to provide clarity. At times, some students think that it is a trivial mistake to express exponential functions as merely the exponent times the base. Therefore, the expression five to the third power is 5 x 5 x 5 = 125 and not 3 x 5 = 15.

Given the grade level, it might be more productive to give the formula where P is equal to the square root of the cube of D. And D is equal to the cube root of P squared.