Question

materials: ruler, string (at least 25cm), push pins (2), pencil, calculator, styrofoam board, paper (4), tape

objectives: upon completion of this laboratory exercise, students will be able to:

1. define the words eccentricity, ellipse, and foci
2. calculate the eccentricity of an ellipse.
3. observe how eccentricity changes as the distance between foci changes.
4. explain kepler’s first law of planetary motion.
5. describe how earth’s orbit compares to any given eccentric orbit
6. describe the shape of an ellipse with an eccentricity of 0.00 or 1.00
7. give an example of a celestial object with a highly eccentric orbit.
8. describe earth’s orbital shape.

before the following vocabulary terms in your own words:

1. eccentricity __________
2. ellipse __________
3. foci __________

procedure part a:
in this part of the laboratory exercise you will construct several ellipses.

1. tie the piece of string into a loop of approximately 10 cm.
2. place a piece of white paper on the styrofoam board and fix it into place with a small piece of tape.
3. label the piece of paper in the top right hand corner “#1” (for other ellipses, number them #2, #3, #4).
4. in the center of the piece of paper, place the two pushpins approximately 2.0 cm apart.

Explanation:

1. Eccentricity: In the context of conic sections (especially ellipses, which are relevant here for orbital motion), eccentricity (\(e\)) is a measure that describes how "stretched out" or non - circular an ellipse is. For an ellipse, it is calculated as the ratio of the distance between the center and a focus (\(c\)) to the length of the semi - major axis (\(a\)), so \(e=\frac{c}{a}\). The value of eccentricity ranges from \(0\) (for a perfect circle, where the two foci coincide at the center) to just less than \(1\) (for a very elongated ellipse). In the case of planetary orbits, which are ellipses with the Sun at one of the foci, the eccentricity tells us how much the orbit deviates from being a circle.
2. Ellipse: An ellipse is a closed - curve conic section. It can be defined as the set of all points in a plane such that the sum of the distances from two fixed points (called the foci) to any point on the curve is constant. Mathematically, if we have two foci \(F_1\) and \(F_2\), and a point \(P\) on the ellipse, then \(PF_1 + PF_2=2a\), where \(2a\) is the length of the major axis. In the context of astronomy, planetary orbits around the Sun are ellipses, with the Sun located at one of the foci. The shape of the ellipse (how "flat" or "round" it is) is related to its eccentricity.
3. Focus (Foci): In the context of an ellipse, a focus (plural: foci) is one of the two fixed points inside the ellipse. As mentioned in the definition of an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant. For a planetary orbit (which is an ellipse), the Sun is located at one of the foci. The position of the foci relative to the center of the ellipse determines the eccentricity of the ellipse: the farther the foci are from the center, the more eccentric (elongated) the ellipse is.

Answer:

1. Eccentricity: A measure of how non - circular an ellipse is, calculated as \(e = \frac{c}{a}\) (ratio of distance from center to focus to semi - major axis length), ranging from 0 (circle) to near 1 (elongated ellipse), relevant for orbital shapes.
2. Ellipse: A closed conic section; set of points where sum of distances from two foci is constant (\(PF_1+PF_2 = 2a\)), used to model planetary orbits.
3. Focus: One of two fixed points inside an ellipse; for planetary orbits, the Sun is at a focus, and focus position affects ellipse eccentricity.