String design, ellipses, and orbital mechanics It is a good time of year to create a little art and maybe even understand why two gravitational forces will create an elliptical orbit. With string, a pencil, and graph paper, I can create an ellipse. I created a loop of string on my graph paper and drew an ellipse by dragging my pencil as far as I could reach within the loop. Figure 1 My ellipse looks pretty good. The two off-center points that I used to anchor my string are called the foci of the ellipse (focus is the singular term). The distance between the foci and the length of my string determine the dimensions and the shape (fatness and skinniness) of the resulting ellipse. Before you actually draw your ellipse, see if you can imagine what the result would look like if you made the following adjustments:
Transcript
String design, ellipses, and orbital mechanics
It is a good time of year to create a little art and maybe even understand why two gravitational forces will create an elliptical orbit.
With string, a pencil, and graph paper, I can create an ellipse.
I created a loop of string on my graph paper and drew an ellipse by dragging my pencil as far as I could reach within the loop.
Figure 1
My ellipse looks pretty good. The two off-center points that I used to anchor my string are called the foci of the ellipse (focus is the singular term). The distance between the foci and the length of my string determine the dimensions and the shape (fatness and skinniness) of the resulting ellipse. Before you actually draw your ellipse, see if you can imagine what the result would look like if you made the following adjustments:
1. If you shortened the string length so that it was just a little longer than the distance between the foci how would you expect your final ellipse's shape to change?
2. If you moved your foci closer together and left the string at the same length, how would you expect your ellipse to change?
3. If you made your string really long, how might that affect the shape of your resulting ellipse? Begin creating your string creation.
Decide what you would like your elliptical string design to look like. • thin and long? • nearly circular?
Glue your graph paper onto a piece of card stock (to give the paper more rigidity) and mark where you would like to place your foci.
Punch a hole at each foci ... a compass point works well. (If you are working on a table, it is a good idea to put a thick magazine beneath your card stock to allow you to sufficiently pierce the stock and not mar the table.)
Using a piece of string, decide an appropriate length for your string that will allow you to create the ellipse of your desired shape.
Thread your string ends through the graph paper/card stock and tie a knot on each end to hold your string in place.
Backside of string design
Draw your ellipse by keeping your pencil point taut along your looped string on the front of your card stock/graph paper as in Figure 1 above.
To be able to sew your string design through the card stock, you will need to punch regularly spaced holes through the graph paper/card stock for threading your design.
With embroidery floss, sewing thread, or crochet thread and a large-eyed needle, begin to sew your design.
Skipping a consistent number of holes will create a very symmetrical design.
Gradually changing the number of skipped holes will create a delightfully distorted design.
Back to the mathematics of Ellipses A mathematical way of knowing whether an ellipse will be nearly circular or long and skinny is to measure its eccentricity.
In the figure on the next page, notice that there are two lines that span the ellipse and divide it into halves. Those are the major and minor axis of the ellipse.
The longest axis is called the major axis. The distance from the center of the figure (where the major and minor axis cross) to the drawn ellipse is called the semi-major axis. This distance is !
" of that major axis length.
The minor axis is the shortest length across the shape. The semi-minor axis is again !"
of that length.
4. Judging from the 3 drawing pictures that I've shown below, how long do you suppose my string is in terms of the finished ellipse?
5. Use any math that you know (including the Pythagorean Theorem) to calculate the dimensions of the semi-major, semi-minor, and distance between the foci for an ellipse drawn with a 10-inch string and a 4-inch semi-minor axis.
6. What is the eccentricity of our example ellipse?
7. Below are two extreme ellipses. Decide what you think that their eccentricities will be close to. Please explain your reasoning.
The eccentricity of an ellipse = the following ratio;
8. What do you expect that the range of eccentricity values could be?
9. If you knew what the eccentricity value of an ellipse was without seeing the ellipse, what might you be able to conclude?
Real World Ellipses
Our Earth is hurtling through space and orbiting our Sun. Because of the incredible speed of our Earth's journey (nearly 70,000 miles per hour or 108,000 kilometers per hour), the Earth is barely contained by our Sun's gravity into an orbit. The Sun and its gravity are one of the elliptical orbit's foci and the various gravities of other heavenly bodies (including the Earth's gravity) are the other elliptical focus. The eccentricity of our orbit around the Sun is currently measured as about 0.0167.
10. What does that eccentricity tell you about the Earth's orbit around the Sun?
Pluto is the farthest planet from the Sun in our solar system. Its elliptical orbit has an eccentricity of 0.24880766.
11. How does the shape of Pluto's orbit compare to the Earth's?
As Earth travels around the Sun, there are times when it is closer or farther from the Sun. Earth is closest the Sun in early January.
• Perihelion is the place in an orbit that is closest to its orbiting body.
• Aphelion is the place in an orbit that is furthest from its orbiting body.
12. How does an object's perihelion and aphelion relate to our ellipse study?
13. Lastly, Halley's Comet orbits the Sun in an orbit with eccentricity of 0.967. What does that tell you?
