Due: Friday February 9

**Scale Model**

- 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?

- Similarly determine the locations for Mercury, Venus & Mars.
- 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 - 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? - 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. -
For a planet with eccentricity e, and semimajor axis a, write expressions for
the planet's orbital distance from the Sun at perihelion (r
_{p}) and aphelion (r_{a}).

- 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).
**Reading: Croswell, Ch 2&3** - In 1897, Kapteyn discovered a remarkable star. What was special about it and why does it have this property?
- What mistake did Kapteyn make in his attempt to map
the Milky Way? Describe the observations that provided the first
hints at what he had overlooked.
- Compute the velocity of the Sun's orbit around the Milky Way. Assume a circular orbit. Use the values given in Ch. 1 of Croswell for the Sun's distance from the Galactic center and its orbital period around the Galaxy. Hint: conversion factors, e.g. from lightyears to meters, etc, may be found on the inside front cover of Carroll and Ostlie (Appendix A).