1. List Kepler's Laws of Planetary Motion, and use them to describe the motions of bodies in space. In doing so, show how Newton's Laws of Motion and Gravity are responsible for the motions that we observe. For each planet's orbit, give the semi-major axis and eccentricity, and discuss how these quantities affect the planets' surface temperatures and, where appropriate, their weather.

Preparing an Outline/Discussion of the Answer
      To prepare your answer, you should construct an outline, or some kind of summary of the topics to be covered, and how you want to cover them. This should address every point brought up by the question, in more-or-less the same order, so that you can see how each part of the answer relates to the question.
      As you list the major topics, think about what laws or relationships might be related to each topic, and make notes in the appropriate places in the outline. Use parentheses or question marks to indicate areas where you may want to add or remove material. Think about whether tables, diagrams, or numerical examples might help better explain the topics. If so, make notes to remind yourself to use them. Remember that diagrams need labels or descriptions to point out their most important characteristics, and that word descriptions often need a quick sketch or numerical example to make their meaning clear.
      If you have constructed a good outline, you should find it fairly easy to write an actual essay, expanding on the outline. You should try doing this, so that you can see how long it will take to do the exam, and how difficult it is to write it out. If you find that you have too much (or too little) material for the time available (about 25 to 35 minutes for each essay), you can remove (or add) material until you can feel confident of covering everything in the time available. if you find that some parts of the essay are hard for you to remember or to flesh out, then you can spend extra time organizing and studying the material covered by the outline.

Additional Points to Consider for This Class
      You are going to have a very limited time to answer the questions, so you should not waste time on unimportant 'niceties'. Do NOT write down the question, or otherwise summarize what you are going to do. Just plunge right into your answer. Similarly, do NOT summarize your answer at the end.
      Do not waste time on historical discussions. Although I use them in class, they are only intended to put things in context, and to make it easier to follow the discussion. Only worry about the astronomy and physics.
      Use diagrams and tables wherever you can explain things faster by using them, than by using words. It will not only save you a lot of time writing things down, but also save me a lot of time hunting through your verbiage to figure out what you are trying to say. Don't forget to use words to explain what the diagrams or tables are showing, as otherwise I won't know if you actually understand what you put down, but do use every means at your command to save yourself time, so that you have more time to put down what you know.
      Don't worry about 'proper' grammar. Don't use poor grammar which makes it hard for me to understand what you are trying to say, but don't worry about explanatory phrases which add no information, and only make the answer 'read' better. As long as I can see what you mean, I will give you credit for it.
      Keep in mind that long, detailed, thorough, well-organized answers which show a considerable effort to learn and understand the material are expected for every single essay question on the exams. All of the essay questions are long, involved questions, and answers of less than a page cannot possibly cover them in adequate detail. Even a 'C' answer will usually run 2 to 3 pages, and an 'A' answer is usually 4 to 6 pages long.

Sample Outline/Discussion of Topics for an A-Plus Answer
A. Summarize Kepler's Laws, so that you don't forget to cover one of them.
     List each Law, in a reasonably complete form, so that you remember all the topics covered by that Law:
     (1) The planets move in elliptical orbits, with the Sun at one focus.
     (2) (called the Law of Areas) As a planet moves around its orbit, the radius vector from the Sun to the planet sweeps out equal areas in equal times.
     (3) (called the Harmonic Law) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit (P-squared = a-cubed).

B. Explain the meaning of each Law in some detail, including a brief discussion of how the laws of physics make things work in the way that they do.
     1. For Kepler's First Law, describe what an ellipse is like. Describe various types of ellipses, to make it clear that you understand how they differ from each other. Discuss how the semi-major axis and eccentricity describe the size and shape of the orbit, and the position of the Sun along the major axis. Discuss how the semi-major axis is related to the perihelion and aphelion distances and how the eccentricity describes the percent variation in distance. Explain how the physics of the situation causes the orbits to work this way. Point out that at any given time, the planet is moving in some direction, and 'wants' to keep moving in that direction (Newton's First Law, the Law of Inertia), unless some force acts on it. Explain that there is a force acting on it, the force of gravity, pointing toward the Sun (Newton's Law of Universal Gravitation). Show how, as a result of that force, the planet must gradually change its motion (Newton's Second Law, the Force Law), so that instead of following a straight line path out into interstellar space, it gradually curves around the Sun. Be sure to point out that because of the geometry of space, and the nature of the Law of Gravity, the curve happens to be a closed, plane curve, which repeats itself over and over again, specifically, an ellipse with the Sun at one focus of the ellipse, as stated by Kepler.
     2. For Kepler's Second Law, describe how the speed of the planet changes as it moves along the orbit, both in terms of the Law of Areas, and in terms of the forces involved. Either description should show that it moves faster near perihelion, and slower near aphelion. Where possible, use numeric examples (if 1% closer, moves 1% faster, if 1% further away, moves 1% slower, etc), and diagrams (show how, if it moved at a constant speed, the areas swept out in a given time would be larger when it is further away, so that to obey the Law of Areas, it must move slower when further away). Explain how the laws of physics cause these speed changes (when moving away from the Sun, the force of gravity is pulling the planet backwards, which, according to the Force Law, should slow it down, and when moving towards the Sun, the force of gravity is pulling the planet forwards, which should speed it up). Point out that the slowest speed is at aphelion, where it is furthest from the Sun, and the fastest at perihelion, where it is closest to the Sun.
      3. For Kepler's Third Law, describe how the average speeds depend on the size of the orbit (numeric examples, dividing size of orbit by orbit period?). Discuss how planets in big orbits, such as Pluto, take longer to go around partly because of greater distance, and partly because of slower speed (use a comparison with Mercury? 100 times bigger orbit for Pluto requires 100 times more time, but the actual time is 1000 times larger, so it must move slower, as well, by a factor of 10, which is exactly the same as the square root of 100). Point out that if a planet in a large orbit moved as fast as a planet in a small orbit, the weak pull of gravity at large distances wouldn't be adequate to hold onto the planet, and it would go off into interstellar space, along a hyperbolic orbit.

C. Discuss the differences between Kepler's original Laws and Newton's reformulation?
     Kepler didn't realize that the orbits could be any conic section, because open conic sections (parabolas and hyperbolas) wouldn't allow the planets to keep going around the Sun. Also, because he didn't know that the force involved was the force of gravity, he had no way of realizing, as Newton did, that the masses of the objects would affect the results, leading to a version of the 3rd Law which includes the masses, and allows measurements of planetary motions, satellite motions, stellar motions, or such, to be used to estimate the masses of the Sun, the planets, stars, and galaxies.

D. List the semi-major axes of the planetary orbits, and the eccentricities (use approximate values, since the instructor said that if a number is accurate to within 10%, it is accurate enough for the purposes of this class). If the significance of the numbers was not explained earlier, be sure to do so here.
      Point out that the surface temperatures of the planets should depend upon their distances from the Sun, and the fact that, other than for Venus, every single planet is colder than the next planet in towards the Sun. Also explain that if the distance changes substantially (that is, if the eccentricity is not close to zero), then the planet's temperature should change during an orbit. Explain that, in the case of the Earth, the eccentricity is so small (less than 2%) that there is no observable effect on our weather or temperature, but that in the cases of Mercury, Mars, and Pluto, there are observable temperature changes, and for Mars and Pluto, some minor changes in the weather. Briefly discuss these changes.