Overall Directions:
Draw an ellipse of the right orbital shape (a) for Earth, Moon or Venus , (b) for either Mars or Mercury, and (c) for one other world. Detailed instructions follow. Compare your drawings with results of your partner (each of you should have a different choice from (a) and from (b or c)).
Turn in: the placement of pins (measurements), the drawings, and your description of the resulting shapes.
You need:
Calculate Rperihelion and Raphelion (the closest approach to Sun and the farthest distance, respectively):
Rp = a*(1-e)
Ra = a*(1+e)
a = semi-major axis = average distance (basically, the longest
"radius")
e = eccentricity (a number between zero and one)
Multiply by some appropriate number to scale it down to a reasonable model size (10-20 cm is fairly reasonable). For classroom use, choose 10 cm to fit on the paper. Note: You don't need the actual orbital distance because we're just looking at shapes, and are scaling to 10 cm as the semi-major axis.
EXAMPLE: for an eccentricity of 0.5 (half squashed), use
Ra = 10cm * (1 + 0.5) = 15 cm and
Rp = 10 cm * (1.0 - 0.5) = 5 cm. Any
multiple of these would also work.
Pin your paper to the cardboard using the flat-head pins. Draw a horizontal line approximately centered on your paper (no need to be too picky). Mark a spot on this line and label it "Sun". Measure the aphelion distance from Sun to one side, and mark it (15 cm to the right for the example). Measure perihelion distance from Sun to the other side, and mark that (5 cm to the left for the example).
Place one push-pin at the point marked "Sun"; you'll leave this pin in place from now on. (Using a yellow or white pin can be handy.)
Place a pin at the aphelion mark. Tie a string snugly around the two. (That makes the string twice 15 cm for the example.)
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.
Remove the aphelion pin.
Pull the string toward the perihelion mark, and insert a pin within the loop, at the perihelion mark. The string should go around the Sun pin and the perihelion pin, and will generally NOT be completely snug (it may, in fact, be very loose). Leave this pin and the Sun pin in place for the moment.
Pull the string toward the aphelion (right) side using another tack. Place the tack along the line so that the string is snug. This new pin becomes the second focus. (For the example, this second pin ends up at 15 cm - 5 cm = 10 cm to the right of the "Sun" pin.)
Remove the perihelion pin (the one that, for the example, was 5 cm to the left of the "Sun" pin), replace it with a pencil, and draw the ellipse. (Run the pencil around the two pins by holding it so that it keeps the string snug. The string will guide the pencil. Try to keep the pencil vertical and the string near the pencil tip. If you've done it well, the ellipse should go through both perihelion and aphelion.)
Pull the pin at the second focus and remove the string. Take a look at the results. How round or squashed is the ellipse? Where is the Sun relative to the center?
Use the same position of "Sun" for the second ellipse. Try to make the orbits roughly the same size, so that you can compare their shapes.
Last update: 21 Feb 2001, EER
URL: http://www.nthelp.com/eer/HOAorbit.htm