Download Problem Set 2

Problem Set 2
1. Kepler’s Laws
a. In the northern hemisphere, during which season is the Earth moving fastest around the sun? Explain, using information from our first class, and one of Kepler’s laws.
b. How long will it take a 1 kg asteroid located at 4 AU (AU = astronomical unit) to orbit the sun?
c. How about a 2 kg asteroid?
2. Understanding Eccentricity
The closest approach of a planet to the sun is called the perihelion distance, (p, in the figure above)
and is equal to a × (1 − e), where a is the semi-major axis of the ellipse, and e is the eccentricity.
a. For the Earth, a =1 AU (1 astronomical unit, by definition) and e = 0.02. What is Earth’s perihelion distance?
b. For Mercury, a = 0.39 AU and e = 0.2. What is Mercury’s perihelion distance?
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c. Use the diagram above to find an expression for the farthest distance a planet gets from the sun,
and write it below. This is known as the aphelion distance
d. What is Earth’s aphelion distance?
e. What is Mercury’s aphelion distance?
f. Would you expect the eccentricity of Earth’s or Mercury’s orbits to significantly affect temperatures
on one/both/neither of these planets? Explain.
The distance between the sun and the center of the ellipse formed by a planetary orbit is given by
d=e×a
where d is the distance between the sun and the center of the ellipse, e is the eccentricity and a is the
semi-major axis of the orbit.
g. For Mercury, e = 0.2. We are going to create a scaled-down orbit for which the semi-major axis of
Mercury’s orbit is 1 inch. In this scaled-down orbit, what is the distance between the sun (the focus)
and the center of the ellipse, in inches?
h. What is the distance between the two foci, in inches?
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i. Get a string, two pins and a pencil. Tie the string so that the length of string between the pins is
2 inches. Then, mark two dots, separated by the distance you calculated in part (h.) and place your
pins on the dots. Draw an ellipse below, as we did in class.
j. Describe the ellipse you drew. Is it more or less ‘circular’ than you expected? (Remember, Mercury
is the most eccentric planet in the solar system)
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