Elliptical Orbits

Introduction

Johannes Kepler observed that the planets orbit the Sun in elliptical orbits—not circular orbits as originally thought. As a consequence, Kepler successfully determined the orbiting properties of any object. With the Elliptical Orbit Overhead Platform, show your students the elliptical shape of a planet’s orbit around the Sun and discuss Kepler’s laws of planetary motion.

Concepts

• Elliptical orbits
• Kepler’s laws of planetary motion

Materials

Safety Precautions

Please follow normal classroom safety guidelines.

Disposal

Materials may be saved for future use.

Prelab Preparation

1. Obtain the sheet of four rubber feet. Peel the bumper feet from the sheet and stick one at each corner of the transparent plate (see Figure 1).
   {13918_Preparation_Figure_1_Elliptical orbits demonstration setup}
2. Measure and cut two lengths of string—one 25 cm long and the other 40 cm long.
3. Form two loops of string by tying each string’s ends together with a knot as shown in Figure 2. Tie the knot close to the ends of the string.
   {13918_Procedure_Figure_2}
4. Obtain the washer. Press the washer on the tip of the overhead transparency pen firmly so that it becomes “stuck” to the pen body. The washer will act as a guide for the string to keep the string at the same height when drawing the ellipses.

Procedure

1. Obtain an overhead projector, Elliptical Orbit Overhead Platform, nuts and bolts, marking pen and a ruler.
2. Place the two bolts in the holes of the platform that are two spaces from the center hole (see Figure 3). Secure each bolt with two nuts. (The additional height of the two nuts prevents the string from slipping too low down the bolts.)
3. Loop the 25-cm string around both bolts and the transparency pen as shown in Figure 3. Make sure the string rests above the washer on the pen so that it does not slip down. Pull the pen away from the bolts so that the string becomes taut. Note: If the sting loop is too long and it extends over the edges of the plate, measure and cut shorter lengths of string that will fit with the corresponding positions of the bolts.
4. Draw the first ellipse as shown in Figure 3 by allowing the string to act as the guide for the pen. Keep the pen perpendicular to the plate when drawing the ellipse.
5. After drawing the ellipse, make two marks anywhere on the ellipse with the pen.
6. Use a ruler or straightedge to draw straight lines from the two foci (bolts) to each point marked on the edge of the ellipse.
7. Measure the total length from one focus to a point on the ellipse and back down to the second focus. Repeat for the second point on the ellipse. Are the total lengths the same for the two separate points? (See the Discussion section.)
8. Repeat steps 2–7 with the 40-cm string. Adjust the position of the bolts as necessary to the classroom discussion.
9. Discuss the mathematics and shape of an ellipse, as well as the properties of the planetary orbits with the class. Note: The information in the Discussion section may be used as a guide if necessary.

Teacher Tips

• Enough materials are provided in this kit to perform the demonstration indefinitely. An extra bolt, two nuts and a washer are provided for replacement purposes, if necessary.
• A colored protective film (e.g., blue, red) may be on the Elliptical Orbit Overhead Platform when it is received. Peel off the protective film before using.
• A damp paper towel can be used to wipe down the Elliptical Orbit Overhead Platform and remove the ellipse drawings.
• To draw the approximate elliptical shape of Mercury’s orbit, measure and cut the string to about 30 cm. Tie the ends together to form a loop with a circumference of approximately 27–28 cm. Position the bolts in the two holes adjacent to the center hole. Draw the elliptical “orbit.” The resulting ellipse shape will have an eccentricity of approximately 0.2, close to that of Mercury’s.
• To draw the approximate elliptical shape of Halley’s comet, use the same size string as above (28 cm circumference). Position the bolts in the farthest holes from the center hole. Draw the ellipse. This ellipse will have an eccentricity of approximately 0.95. Note: Since there will not be much space between the pen and the bolts with this elliptical shape, do not use the washer as a guide on the pen body.
• For additional instruction, provide students with a table similar to Table 1 with blanks. Then, have the students calculate the period of revolution or the distance from the Sun for different planets. Students will need one value (the period or the distance) in order to calculate the other.

Discussion

Long before Isaac Newton (1642–1727) “discovered” gravity and developed his laws of motion, Johannes Kepler (1571–1630) observed the planets and formulated his own laws of motion. In 1543, just before his death, Nicolaus Copernicus (1473–1543) proposed that the Sun was the center of the solar system and that the planets orbit the Sun in a circular path. In 1609, Kepler published a book entitled Law of Planetary Periods. In his manuscript, Kepler presented observations made by the well-respected astronomer Tycho Brahe (1546–1601), as well as his own observations and calculations, to show that the planets do not travel in perfect circular orbits. Instead, they travel in oblong circular paths known as ellipses. An ellipse is an elongated circle that has two separate points of rotation, or foci, instead of one central point (see Figure 4).

The “oblongness” of the ellipse is known as its eccentricity. The eccentricity (e) is calculated as the ratio of the distance between the foci to the length of the major axis of the ellipse (Equation 1).

f is the foci separation
a is the major axis length

An ellipse with an eccentricity of zero is more commonly known as a circle. The larger the eccentricity, the more flattened the ellipse is. An eccentricity of one (1) corresponds to the shape of a parabola—a non-enclosed curve. The eccentricities of most of the planets are very close to zero, so on a small, classroom-scale model, it is difficult to see the difference between a circular orbit and an elliptical orbit of a planet. Mercury, Pluto and Halley’s comet are exceptions (see Table 1). These orbiting bodies have larger eccentricities and therefore follow more elongated paths.

Table 1

*AU is an abbreviation for the Astronomical Unit which is equal to the Earth’s average distance to the Sun (1.50 x 1011 m).
†One Earth year is equal to 365.26 days.

One focal point for the planets in our solar system is located at the center of the Sun. Since the Sun is located at a focal point of the elliptical orbit of the planets, and not at the origin of the elliptical shape, a planet’s distance to the Sun is constantly changing. When a planet crosses the major axis of its elliptical orbit the planet will be either at its farthest distance from the Sun, or at its closest distance. When a planet is at its farthest distance from the Sun, it is at a position known as the aphelion. When the planet is at its closest distance to the Sun, it is at a position known as the perihelion.

As a consequence of the conservation of angular momentum, the orbital speed of a planet is inversely proportional to its distance from the Sun. Therefore, as a planet’s distance from the Sun increases, its orbital speed decreases. The closer a planet is to the Sun, the faster it orbits. A planet’s fastest and slowest orbiting speeds occur at the perihelion and aphelion, respectively. Kepler noticed this change in the orbiting speed of the planets (from Brahe’s observations) and discovered that the area of space swept by a planet in a given amount of time is always the same, no matter where the planet is in its orbit. Kepler called this the Law of Equal Areas (see Figure 6). This law follows from the basic property of an ellipse observed in step 7 and shown in Figure 7—for any point on the ellipse, the sum of the distance from the two foci is constant. The area of space produced by arc A in Figure 6 is equal to the area of space of arc B. The sum of distances a and b is equal to the sum of distances c and d.

Kepler’s observations also allowed him to formulate an equation for the period of revolution of the planets. Kepler calculated that the square of a planet’s period of revolution is proportional to the “radius” of the elliptical orbit cubed (Equation 2). An ellipse does not have a true “radius,” but it can be shown that one-half the length of the major axis (also known as the semi-major axis) is equal to the average separation of a planet from the Sun as it travels in its elliptical orbit. Therefore, the semi-major axis is used as the “radius” of the ellipse.

T is the planet’s period of revolution
k is the proportionality constant
R is the length of semi-major axis

Years later, Newton validated Kepler’s Law of Planetary Periods when he derived Kepler’s equation from his own equations of universal gravitation.

References

http://ssd.jpl.nasa.gov/ (accessed November 2003)

http://www.nd.edu/~hahn/orbital/ (accessed November 2003)