Learning Library

# Orbital Eccentricity

### Problem

What determines the shapes of the planets’ orbits?

### Materials

• Pencil
• Ruler
• 2 pushpins
• 12-inch (31-cm), square piece of poster board
• 12-inch (31-cm), square piece of thick cardboard
• 10-inch (26-cm) length of string

### Procedure

1. Draw a straight line through the center of the poster board.
2. Mark two dots on the line 4 inches apart at the center of the board.
3. Place the poster board on top of the cardboard and stick the pushpins into the dots.
4. Tie the loose ends of the string together to make a loop.
5. Drape the string loop around the pushpins. Place the pencil inside the loop as well and use it stretch the loop into a triangle.
6. Keeping the string taut, guide the pencil around the pushpins while tracing out a path on the poster board below. Continue around the pins until you’ve drawn a closed loop. Describe the shape of the curve you’ve drawn. Is the diameter across the loop always the same, or does it change? Where is it widest? Where is it narrowest?

1. Repeat the above steps on the same poster board with the pins at different distances from each other: 0 inches (just use one pin), 3 inches, and 5 inches. How does the shape of the loop change? How does the distance across the widest and narrowest parts change as the pins get closer together?

### Results

You have four ellipses on the poster board. Each pushpin marks a point in the ellipse called a focus (plural: foci). The widest diameter across the ellipse is called the major axis; the narrowest diameter is known as the minor axis. The foci sit along the major axis, equidistant from the center of the ellipse. As the foci get closer together, the ellipse looks more like a circle. An ellipse with only one focus is a circle (the major and minor axes are the same length).

### Why?

In 1609, Johannes Kepler figured out that the planets travel along elliptical paths with the sun sitting at one focus of the ellipse. He called this his First Law of Planetary Motion. As a planet moves along its orbit, the distance between it and the sun changes. The point on the ellipse where the planet is closest to the sun is called the perihelion; the point where it is farthest is the aphelion. The Earth passes its perihelion in early January and goes through aphelion in early July. On your ellipses, can you mark where the perihelion and aphelion might be?

The eccentricity of an orbit is a single number, between 0 and 1, which describes how stretched out the orbit is. Zero means the orbit is perfectly circular. An eccentricity close to 1 means the orbit is extremely elongated; only comets coming from the outer reaches of the solar system get close to this value.

You can calculate the eccentricity of your ellipses using the following equation:

• where e is the eccentricity,
• a is the aphelion distance, and
• p is the perihelion distance.

For each ellipse, pick a focus where the sun should sit (either one will do), then measure the aphelion and perihelion distances. Calculate the eccentricity, and record your results in Table 1.

 Distance between foci Aphelion Perihelion Eccentricity 0 inches 3 inches 4 inches 5 inches

Table 1. Calculating the eccentricity of your ellipses

How does the eccentricity change as the foci get farther apart?

Now you know enough to figure out the eccentricities of the planets in the solar system. Table 2 lists the aphelion and perihelion distance of all the major planets (plus one famous comet). The distances are in astronomical units (AU), where 1 AU is the average distance between the Earth and Sun (93 million miles).

 Name Aphelion (AU) Perihelion (AU) Eccentricity Mercury 0.47 0.31 Venus 0.73 0.72 Earth 1.02 0.98 Mars 1.67 1.38 Jupiter 5.46 4.95 Saturn 10.12 9.05 Uranus 20.08 18.38 Neptune 30.44 29.77 Comet Halley 35.1 0.59

Table 2. Orbits in the solar system

### Going Further

Draw scale models of the orbits of all the planets! The first thing you’ll need to do is decide what sort of scale to use. For the inner planets (Mercury, Venus, Earth, and Mars), using 1 AU = 10 cm should work pretty well. For the outer planets, you will need to use either a much larger sheet of poster board or switch to a different scale. Try and figure out what scale would be useful for drawing Jupiter, Saturn, Uranus, and Neptune on a single poster board. What challenges do you run in to if you try to draw all the orbits using one scale?

You also need to calculate how far apart the foci need to be based on the orbital data in Table 2. The distance between the foci is simply the difference between the aphelion and perihelion distances: f = a – p.

1. Convert the aphelion and perihelion distances of the planets to centimeters and record your results in Table 3. Example: Earth aphelion = 1.02 AU x 10 cm/AU = 10.2 cm.
2. Calculate the distance between the foci for each orbit. Record these distances in Table 3.
3. Follow the steps from the original drawings to place the pushpins the right distance apart and set up the string and pencil. To get the string to be the right length, you will need to cut it to be equal to twice the sum of the aphelion and perihelion (plus a few centimeters extra to make room for the knot). Record the length of the string you’ll need in Table 3.
4. Draw the orbits for each of the inner planets on a single poster board. Which is the most eccentric? Which is the least?
 Name Aphelion (cm) Perihelion (cm) Foci distance (cm) String length (cm) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

Table 3.

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