DRAWING ORBITS (ELLIPSES)

Overall Directions:
Draw an ellipse of the right orbital shape for 3 worlds: Earth, your world, and a comet with orbital eccentricity e=0.5. Compare the overall shapes and the relative location of Sun.

Turn in:

• the placements of pins (measurements - for each one - in a table)
• your drawing (it should have all three orbits well-labeled, possibly color-coded)
• your description of the resulting shapes and relative location of Sun.

You need:

• a piece of pressed board for a surface and to push pins into. Warning: you may need to try several times to be satisfied with the result, so be prepared. This is particularly true for highly eccentric orbits.
• string.
• 3 push-pins (preferably round or tapered), and 2-4 flat thumbtacks to secure paper.
• pencil or other writing utensil. (Useful: 3 colors for highlighting and labeling.)
• ruler or other measuring device.
• the eccentricity of each orbit to be drawn.

Secure the paper to the board with the flat pins. Draw a faint horizontal line across the center of the page; you'll use this to mark the measurements.

Calculate relative distances: R_Aphelion and R_Perihelion

R_A = a*(1+e)R_A = 10 cm * (1 + e) for this activity

R_P = a*(1-e)R_P = 10 cm * (1 - e) for this activity

e = eccentricity (a number between zero and one)
a = semi-major axis = average distance (basically, the longest "radius").

Use a=10 cm for all the orbits - this way, you're only comparing shape.

EXAMPLE: for an eccentricity of 0.5 (half squashed), use
R_A = 10 cm * (1.0 + 0.5) = 15 cm and
R_P = 10 cm * (1.0 - 0.5) = 5 cm. (Any multiple of these would also work.)

1. On the horizontal line, mark a spot approximately in the center for the Sun; label it "Sun" or "S". Measure the aphelion and perihelion distances as shown, and mark the points ("A" and "P" would do). See diagram.

2a. Place a (yellow) push-pin at the position of the Sun ("S"). This pin will remain throughout.

2b. Place Pin #2 at the aphelion point "A" (to the right in the diagram, at a distance R_A from the Sun (15 cm for the example)).

3. Tie a string snugly around the two. (That makes the string twice 15 cm for the example.) Make sure the string is around just the shanks of the pins, not the thicker plastic parts.

To tie securely, it helps to have 3 hands. Tie an overhand knot (like starting your shoelaces) and pull until snug. Have your partner place a finger on this knot. Tie another knot, preferably reversed from the first knot (left over right instead of the usual right over left), and as you pull it snug, slip it under your partner's finger. When snug, your partner may lift the finger.

4. Release Pin #2 from the aphelion point "A".

5. Place Pin #2 at the perihelion point "P". Loop the string around Pin #2 and the "Sun" pin.

6. Put Pin #3 into the loop and pull the string toward aphelion (to the right).

7. When the string loop is snug along the drawn line, push Pin #3 into the horizontal line. The pin shank should not catch the fibers of the string. The point where the pin goes in will be the second focus of the orbit. Label it "2nd focus" or "F". (For the example, this second pin ends up at 15 cm - 5 cm = 10 cm to the right of the "Sun" pin.)

8. Remove Pin #2 from the perihelion point "P".

9. Place a pencil into the loop, and pull snugly to the perihelion point "P".

10. Run the pencil around in an ellipse by holding it so that it keeps the string snugly around the shanks of the two pins. The string will guide the pencil so that you draw the ellipse. Try to keep the pencil vertical and the string down near the pencil point. (Mechanical pencils are useful if you're having difficulty.) If you've done it correctly, the ellipse should pass through both perihelion and aphelion points.

Pull Pin #3 from "F". Remove the string. Leave the "Sun" pin in place for the other orbits. Label the measured points and the orbit line, perhaps in color, so you can distinguish them from the other orbits on the same page.

Take a look at the results. The ellipse should be 20 cm across - the major axis is twice the semi-major axis (10 cm) by definition. How big is the minor axis (the short distance across)? Where is the Sun compared to the center of the orbit?

Use the same position of "Sun" for the remaining ellipses. Use a=10 cm for all orbits, so that you can compare their shapes. (We're not worried about comparing sizes for this particular activity.) Measure along the same horizontal line in each case, in the same direction. Record the measurements and describe the results as above.