Objective: Calculate the eccentricity of an ellipse and draw models of planetary orbits.

Background: Around the turn of the 17th century, Johannes Kepler discovered that the orbits of planets are ellipses.  The shape of an ellipse varies.  Its shape is described by its eccentricity (a number that ranges from 0-1.  An ellipse with an eccentricity of _ is a perfect circle.  As eccentricity increases to 1, the ellipse becomes more stretched out.
 Ellipses are drawn between two points called focus points.  See Figure 1.  In the orbits of the planets, the sun is one focus.  The other focus is an empty point in space.  The distance between the two focus points is the focal distance (c).  The major axis (a) is the longest diameter of the ellipse.  Look again at Figure 1.

Using these values, you can find the eccentricity (e) of an ellipse.

  Eccentricity(e) =   focal distance (c)
                             Major axis (a)

You are to work on Part A by yourself.  Part B will be your group activity
Procedure Part A  Calculating Eccentricity
1. Look at the three ellipses in Figure 1
2. List the ellipses in order from the most circular to the most stretched out
3. Now calculate the eccentricity of each ellipse.  Use the formula given in the Background.  List the ellipses in order of increasing eccentricity
4. Compare your results from step 2 and 3.
 
  

Part A Questions
1. How do your ellipses’ shapes compare with your list of eccentricities?
 
2. Describe how an ellipse with an eccentricity of 0.5 might look?
 
 
 

Part B Models of Elliptical Orbits

    Materials:  Cardboard,  large sheet of paper, pencil, push pins, 30 cm length of string, metric ruler

    Procedure
        1. Fasten the large sheet of paper to the cardboard with the pushpins
        2. Using the ruler, draw a line through the middle of paper the entire length of the paper
        3. Draw a dot near the middle of the line.  Label the dot “SUN”
        4. Firmly place a push pin through the dot.
        5. Place a second dot 10 mm from the first, along the line.
        6. Tie a knot in the string as close to the ends as possible.
        7. Using your pencil stretch the loop of string around the tacks as shown (Figure 2)

        8. Draw an orbit around the sun.  This orbit is an ellipse.  It resembles the shape of Jupiter's ellipse. Label the orbit as     Jupiter's orbit
        9. Measure and record the major axis of your ellipse.
        10. Measure and record the focal distance
        11. Repeat the same process for Earth, Pluto and Halley’s comet.
        12. Look on the data table for the distance between the tacks.  Never remove the tack representing the sun
        13. Calculate all the eccentricities of your ellipse. (round to the nearest hundredth)

    Data Table: Eccentricity Measurements of Celestial Bodies
 

Celestial Body	Distance between tacks	Major Axis	Focal Distance	Eccentricity
Jupiter	.	.	.	.
Earth	.	.	.	.
Pluto	.	.	.	.
Halley’s Comet	.	.	.	.

        
COPY DATA TABLE and QUESTION/ANSWER ONTO YOUR LARGE PIECE OF PAPER.
MAKE SURE EVERYONE’s NAME (first and last) and PERIOD ARE ON THE SHEET.

Question

1.  Which orbit is most elliptical?  Support your answer by comparing eccentricities of all four orbits?