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Applied MATH Honors Calculus III Applied Project: Keplers Laws of Planetary Motion Due: Wed. Mar. 9, 2005
Winter 2005
You can add up to ve extra points to your nal score in the course by completing this project. Preliminaries: Before you start working on the problems described below, you should familiarize yourself with Keplers laws of planetary motion. The information in the textbook (Section 14.4) should be enough to work out the problems listed/described below, but further reading is highly recommended (see references below). The book proves Keplers rst law (pg. 912). It would be very dicult to complete this project without rst reading and understanding that proof. Before attempting to solve the problems below, you should make sure that youve understood the coordinate system used to describe the orbits of planets, the notation (i.e., understand what every vector in Fig. 8, pg. 913 represent), and the notation of Newtons law of gravitation. To this end, I recommend you take some notes as you read the proof of the rst law. You may want, for example, to write a list describing the variables and constants that are used in the proof, and, possibly, replicate the proof as you read it, adding comments that help you understand it. Grading Policy: You need to show all your work to receive credit (full or partial). Follow and label the steps suggested in the problems and organize your work clearly. The maximum score you can get in this assignment is 30 points, 15 points per problem (I will divide it by 6 to gure out how many points to add to your nal grade). Partial credit will be given for right ideas that may have lead you to the proof even if you couldnt complete it, but arriving at the right conclusions based on incorrect assumptions/statements will not earn you any credit. Part 1: Problem 1 (prove Keplers Second Law) or Problem 2 (prove Keplers Third Law) on page 916 of Stewart. Your choice. Part 2: Solve the following problem presented by N. Grossman in his book The Sheer Joy of Celestial Mechanics (Exercise V.2, pg. 91): In the movie Journey the to Far Side of the Sun [1969], an astronaut blasts o from Earth to begin a journey to Venus. An explosion occurs soon after lifto, and the astronaut wakes up in a hospital. Soon he begins to notice disturbing dierences in chirality1 : most people are left-handed and wear wedding rings on the right hand, coats button on the wrong side, and so on. He learns that he is on Anti-Earth, which is the same mass and shape as the Earth and which revolves around the Sun in the same orbit but always on the other side of the Earths diameter. Anti-Earth cannot be seen through Earth-based telescopes. Anti-Earth does know about Earth and naturally is hostile to it. An annihilating invasion is about to be launched. The astronaut manages to escape with the help of a pretty nurse, and they get back to warn Earth. Use the following plan to show that the existence of an Anti-Earth could have (and would have) been deduced long ago. [a] Suppose that the Earth is represented as a point of mass m revolving in a
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see reference [3] for a denition of chirality
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circular orbit of radius a around a center of force (the Sun) of mass M . [b] Increase the usual Newtonian attraction of the Sun by the attraction of the hidden Anti-Earth and compute the chan...