Question #1 (25%)
For each of the nine planets of 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 P, and then compute the ratio of the semimajor axis of the orbit cubed to the period squared:

R³/P²

Name of the planet	R (in AU)	P (in years)	R3/P2
Mercury	0.387	0.241	0.9979
Venus	0.723	0.615	0.9992
Earth	1	1	1.0000
Mars	1.524	1.881	1.0004
Jupiter	5.203	11.86	1.0014
Saturn	9.539	29.46	1.0001
Uranus	19.182	84.01	1.0000
Neptune	30.058	164.79	1.0000
Pluto	39.44	247.7	0.9999

The reference we used: W. J. Kaufmann, III, "Universe", 4th edition, Freeman & Company, New York, 1994.

Question #2 (15%)
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	0.387	0.387
Venus	0.723	0.723
Earth	1	1
Mars	1.524	1.524
Jupiter	5.203	5.203
Saturn	9.539	9.539
Uranus	19.182	19.182
Neptune	30.058	30.058
Pluto	39.44	39.44

We noticed that: they are equal. (If you used different references for those two numbers you may find them slightly different but still quite close. That's because the data get better with time as measurements improve, and so an older reference might have slightly less accurate data - but if you noticed that they are very close, that is as good as saying that they are exactly equal.)

Question #3 (20%)
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.

There are actually two different explanations, each is sufficient for the full credit.

• In this model the planets stay at the same distance from the Earth, but since the Sun moves in space, they will be at different distances from the Sun at different times, and therefore the amount of sunlight falling on them will change with time. Since planets do not emit their own light, only reflect the sunlight, their brightness will vary with time as the amount of sunlight they can reflect varies with time.
• The planets in this model will also have different phases at different times (similar to the phases of the Moon): the sun only shines on one half of a planet, and from the Earth we can only see one half of the planet (like one side of the Moon). But those two halves do not have be the same. Sometimes we will see the fully lit half of a planet (like a full Moon), sometimes we will see a half which is only partially lit (like a crescent Moon), and occasionally we will see the dark side of a planet (like a new Moon). Thus, the planet will appear to change its brightness with time just as Moon's brightness changes with time.

Question #4 (20%)
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.

The Sun goes around the sky in 1 day (24 hours), so in that model the Sun will go around its orbit in 1 day. But the Sun's height above the horizon at a given time (say, at noon) changes during the year - during winter the Sun does not rise as high in the sky as during summer, which is the reason for the seasons. In Z.'s model the Sun makes only one motion, and thus in his model there are no seasons on the Earth. Obviously, such a model disagrees with the observations.

Question #5 (20%)
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 and the semiminor axis of 20 AU. Sketch the the orbits of both asteroids.

Make your sketch in the space below.

An orbit of an asteroid is an ellipse. Asteroid A's ellipse is "fat", and B's ellipse is "skinny". Skinny ellipse has its foci farther apart (and thus closer to the edge) than a fat one. The Sun is at the focus of each orbit, thus two ellipses have one common focus. Thus, B comes closer to the Sun in its closest approach than A, but moves farther from the Sun in its farthest point.

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

• Period of asteroid A: 125 years
• Period of asteroid B: 125 years