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Transcript
MATH 112 SPECIAL PLANETARY PROBLEM In this problem you will find the parametric equations of Neptune and Pluto. We will only consider the two dimensional orbits viewed from the north pole of the Sun way above the solar system. The Sun is at the origin. All the numbers in the equations will be in astronomical units (AU). An astronomical unit is the average distance from the Sun to the Earth. The orbits are ellipses or approximately circles. Your final equations will have the form: x(t ) = a cos( Bt + C ) + h , y (t ) = b sin( Bt + C ) + k where a is ½ the axis of the ellipse along the x-axis, b is ½ the axis of the ellipse along the y-axis, and h and k are horizontal or vertical shifts respectively. Note that C must be the same constant in the equation for x and y. The same is true for B. If the planet’s orbit is approximately a circle with radius r, and the Sun at the center, then a = b = r , and the shifts are zero, and the above equations become: x(t ) = r cos( Bt + C ) . y (t ) = r sin( Bt + C ) B is determined by the period, and C is determined by the initial position of the planets at t = 0. Neptune orbits the Sun every 163.7 years and its orbit is approximately a circle with a radius of 30.05 astronomical units (AU). Pluto orbits the Sun every 248.54 years and its orbit is an ellipse. The coordinate system has been chosen so that the x-axis lines up with the major axis of Pluto to make this problem easier. The major axis of this ellipse is 78.48 AU, while its minor axis is 76.1 AU. The center of this ellipse is shifted right by 10 AU from the origin where the sun is located. The following figure shows the orbits of Neptune and Pluto. We will let t = 0, be Jan. 1st, 2000. At t = 0 Neptune was at ( x, y ) = (−5.4,−29.6) AU, and Pluto was at ( x, y ) = (−28.0,−9.5) AU. The Problem: I) Find the complete parametric equations for Neptune and Pluto with all the constants a, b, B, C, h, and k determined. You must do all calculation and solve for all the constants algebraically and show all the steps. Pay particular attention to choosing the constants that will place the planets in the correct quadrants. II) Find the location of the two planets (both the (t, x, y) values in a table and the locations on the graphs) on Jan 1, 2000, 2050, 2100, 2150, 2200, and 2250. Label these clearly on your graph. Your report: Your report should include: I) A cover sheet. II) A complete statement of the problem, something like what I have written above, but in your own words. Do not attach this paper that you are now holding in your hand to your report. III) All your calculations. IV) All the final equations and graphs and tables. The location of the planets should be indicated on the graphs of the orbits and in a table of (t, x, y) values. You can present the graphs neatly by hand if you don’t have the means of transferring the graphs from your calculator to a computer and then to paper. I can also print very nice graphs for you with MAPLE if you bring your equations to me during my office hours. Do not use my graph above in your report. V) A short conclusion of what this project has contributed to your cosmic consciousness. Due Date: This problem is due Dec. 6th, 2004. At that time Neptune will be at (x, y)=(0.3, -30.0) AU, and Pluto will be at (-26.5, -14.0) AU. The morning sky just before sunrise will feature Venus and Mars only a few degrees apart in the eastern sky about 30 degrees from the horizon, Jupiter higher up in the southeastern sky, and Saturn in the southwestern sky. Saturn is the last planet visible without a telescope. Points: This problem will be worth 30 points, and will count toward your homework assignment. I always like to give extra points for especially good work and presentations for those who will like to make up low points on their first exam.