Eccentric Orbits

Setup and Procedure

• Knot the ends of the string together to make a large loop.
• Fold the paper in half lengthwise (hotdog style) and place it on the poster board.
• Place one push-pin on the fold near the center of the paper.
• Place a second push-pin on the fold between 2cm – 5cm from the first push-pin.
• Loop the string around the two push-pins.
• Put the pencil point inside of the loop created by the string.
• Use the out-stretched string as a guide and trace around the push pins.

Ask: Does your drawing look circular or elliptical to your eye?

Ask: Can can we measure how much the drawing deviates from a perfect circle?

Add the following labels to your drawing so it can make these measurements:

• Remove the push-pins and label the holes F1 and F2.

Additional Information: The “F” stands for “Focus.” Label one focus as “Sun.” It does not matter which focus is designated as the sun, but it is important to note that the sun is always at one of the foci. The second focus in our drawing is an empty point, but it’s location is measurable.

• Measure and place a mark (+) at the center point between F1 and F2 on the fold. Label this mark as “center.”
• Label the semi-minor and semi-major axis.

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Eccentricity can be measured with the following formula:

E = C/A

• E is Eccentricity
• C is the measurement from the center to the focus (F1 or F2)
• A is the measurement of the semi-major axis.

The value for E will vary depending on the length of string, the size of the loop created with the string and the distance between push-pins. However, with the materials outlined above, E should be between 0.1 and 0.3.

Flip over the sheet of paper and repeat the process above, except this time use one push-pin and trace your drawing using that one point. Follow all the steps, including labeling the drawing and placing measurements in the formula: E = C / A

Ask: By using one push-pin, E = 0. What questions do you have when E = 0?

• What happened to the second focus?
• If E = 0, did I make a mistake?
• What type of drawing did I make? A circle or an ellipse? How do I know?
• Based on the two drawings and the respective values for E, what can we conclude about measuring eccentricity?

Consider this: all orbits have two foci. When an orbit is a perfect circle, the foci overlap, giving the appearance of one focus. In our model, we represent this with one push-pin. By using one push-pin and a fixed length of string, the drawing is a perfect circle. The center-point between the foci and the distance between the center point and one of the focus is zero, so C = 0. If C = 0, then E = 0. When E = 0, there is no eccentricity. Any value over 0 indicates eccentricity, up to 0.9999.

How does your model’s eccentricity match up with Solar System planet eccentricity? Compare your values for E with the chart to the right.

Assessment

Return to the initial question:

• In 2018, during an event called Mars Opposition, Mars will make its closest approach to Earth since 2003. This close approach will not happen again until 2035.

• Due to the orbital periods of Mars and Earth, the planets make close approaches roughly once every two years. Why were the planets closer together in 2018 than in previous years?

Based on your understanding of orbits, eccentricity and the planet data table above, draw a model that represents why in 2018 Mars made a close approach to Earth.

Models may look similar to to the figure below:

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