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Table of Contents
Orbital Mechanics
Student Activity
Drawing Closed Orbits
Drawing Circles
Large circles are easily drawn with string and a pencil. The simplicity of circle geometry is one of the reasons that early theories of the solar system were based on circles.
Unfortunately simple circles were insufficient and models of the solar system included circles within circles within even more circles.
The strict adherence to circles lead to some amazingly complex, although surprisingly accurate mechanical models of the solar system which predicted the motion of the planets, the Sun and the Moon.
Drawing Ellipses
Ellipses are not as easy to draw as circles but with some simple equipment very elegant ellipses can be drawn.
The illustration to the left shows how this can be done.
The distance from the centre to the widest part of the ellipse (a) is called the semi-major axis, and the distance from the centre to the narrowest part of the ellipse (b) is called the semi-minor axis.
It is the elliptical shape that is important in orbital mechanics. Even a perfectly circular orbit is really nothing more than a special case of an elliptical orbit.
Activity
To begin you will need to acquire the following items, as shown in the photo:
Several sheets of graph paper.
A pencil or fine-point marker.
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String or heavy thread or fishing line.
Two "push-pins".
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An empty pizza box or sheet of corrugated cardboard.
Exploring Ellipses
Set Up
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Begin by drawing the x-axis and the y-axis on a sheet of graph paper. The origin should be near the centre of the page.
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Cut a length of string and tie two small loops in each end so that they are about 20cm apart when the string is stretched.
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Place the graph paper on top of the cardboard.
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Insert one push pin into each end of the string (through the loops) and then push each of the pins into the x-axis equidistant from the origin (about 3cm from the origin to start).
Challenge 1
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Tension the string with the point of a pencil and, keeping the string taut at all times, trace out an ellipse.
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Change the position of the pins along the x-axis while maintaining symmetry with respect to the origin, and determine the effect of this change on the shape of the ellipse.
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How is the length of the string related to quantity 2a, the length of the major axis of the ellipse?
Drawing Big Ellipses
Quantifying Ellipses
In working with spacecraft and other celestial objects we often wish to describe in a simple way how elliptical an orbit is. Is it long and narrow, or almost circular?
A very useful quantity which is used in describing an elliptical orbit is its eccentricity.
eccentricity e = c/a
Where a is the semi-major axis and c is the distance from the origin to either focus of the ellipse.
Challenge 2
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Draw several ellipses and calculate their eccentricity.
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Can two (or more) ellipses of different size have the same eccentricity? Explain your reasoning.
Challenge 3
The eccentricity of the orbit of the planet Pluto is 0.25 (the largest of all the planets). Using a sheet of graph paper, and using the method above for drawing orbits, construct an ellipse that has the same eccentricity as the orbit of Pluto.
HINT: Begin with the piece of string having loops at the end to accommodate the push-pins. The length of the string defines the major axis (2a) of your ellipse. Use squares on the graph paper as your units of measurement.
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Remove the pins from your graph. Mark the location of the Sun on your graph of Pluto's orbit.
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When in its orbit does Pluto move the fastest? The slowest? Mark these locations. Which of Kepler's Laws predicts this?
