Question #1 (25%) - required
For each of the nine planets of the solar system verify the third law of Kepler: fill in the table below with the values of the semimajor axis of the planet orbit in astronomical units R and the period of rotation in years P, and then compute the ratio of the semimajor axis of the orbit cubed to the period squared:

R³/P²     (press "Help me!" button after you put in the numbers to get help with math)

Name of the planet	R (in AU)	P (in years)	R3/P2
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto

The reference I used was:

  Question #2 (15%) - required
Find also the mean distance from the planet to the Sun and compare it to the semimajor axis of a planet's orbit. (A good reference should contain both.) Fill the table below and write your conclusion at the bottom.

Name of the planet	Semimajor axis (in AU)	Mean distance from the Sun (in AU)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto

I noticed that:

Question #3 (20%) - optional, non-math
Even in ancient times, astronomers knew that planets vary in brightness over the course of several months or even years. Explain why this observation cannot be used to rule out the geocentric model, in which all planets and the Sun orbit the Earth on circular orbits. Ignore the epicycles, i.e. assume that the distance from a planet to the Earth does not change with time.

Question #4 (20%) - optional, non-math
After the Kepler's laws were discovered, an astronomer Z., who still believed that the Earth was not moving, decided to improve the world model designed by Tycho Brahe. In his modified Brahe model, planets go around the Sun on ellipses, just like the Kepler's law require, but the Sun orbits the unmovable Earth, again on an elliptic orbit. There are no epicycles in this model. Make a (strong) argument against such a world model. Recall that the parallax was not measured by the time Kepler invented his laws, so counting on observing the parallax will not work.

Question #5 (20%) - optional, non-math
Asteroid A has a semimajor axis of its orbit of 25 AU and a semiminor axis of 5 AU. Asteroid B has the same semimajor axis as Asteroid A, but a semiminor axis of 20 AU. Sketch the the orbits of both asteroids around the Sun.

Make your sketch in the space below.

Question #6 (20%) - optional, math
compute the periods of rotation around the Sun for the two asteroids from Question #5:

Period of asteroid A:
Period of asteroid B:

Note: The total score from 6 questions adds up to 120%. This means that in order to get the full credit, you need to complete the two required questions (1 and 2) and three of the four optional ones (3 to 6) of your choosing. You may also try to do all 6 questions so that if you get only partial credit for some of the questions, you can still get 100% of the score. If you score is more than 100%, it will be truncated to 100% (you cannot get more than 100% even if you do all 6 questions correctly).