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Astronomy 300: Homework Assignment #2

Due: Friday February 9

Scale Model

1. Make a scale model of the Solar System on the First Floor of Thornton Hall, as follows.
• Borrow a basketball from Physics Equipment Technician Anthony Kelly or Pete Verdone in TH 104. This will represent the Sun.
• Measure the diameter of the ball, look up the diameter of the Sun, and compute the scale ratio of this model.
• Using data from Appendix C in C&O, calculate the number of solar diameters in one AU. Round off this number and try to remember it. This will determine the location of the Earth. How far away should Earth be placed?

2. Similarly determine the locations for Mercury, Venus & Mars.

3. A true scale model also requires proportionally sized objects to represent the planets. Compute the ratio of the Sun's diameter (or radius) to Earth's. Round off this number and try to remember it. What object should you use to represent Earth?

Reading: The Watershed , Ch. 6 and C&O, Ch 2

4. Kepler first modeled Mars' orbit as a circle, with the Sun offset from the Center, around which Mars moved non-uniformly. But, as viewed from one point, the Equant, (puntus equans) Mars' motion did appear uniform. After 900 pages of calculations, he finally determined the parameters of this model, eg. the radius of this circle. Why did Kepler then reject the model?

5. On p. 149 of The Watershed, Koestler states that Kepler "knew his inverse-ratio 'law' (between the planet's speed and distance) was incorrect". That is, a planet's velocity is not simply proportional to the reciprocal of its distance from the Sun. How, exactly, does a planet's velocity depend on its distance from the Sun? Consult C&O, and write down one equation which gives the relationship between the two.

6. For a planet with eccentricity e, and semimajor axis a, write expressions for the planet's orbital distance from the Sun at perihelion (rp) and aphelion (ra).

7. What is the maximum variation (Delta r) in a planet's distance from its host star as a function of a and e? Evaluate this quantity for the Earth (with e=0.017): how much closer to the Sun is Earth at perihelion (Jan. 4th) than aphelion (July 4th)? Express your answer in both km and AU (ie a fraction).