Kepler's Law In 1601 Johannes Kepler began analyzing the data that Tycho Brahe had painstakingly collected over 13 years on the orbit of Mars. We follow Kepler and look for a relationship between the orbit and period of planetary objects. The table below shows the semimajor axis (in Astronomical Units, where 1AU is the average distance from the Earth to the Sun) and orbital period for the planets, dwarf planets () and plutoids (**) in Earth years? This data is included in the attached Mathematica notebook. (a) We want to transform the model T=Cra so that we can fit a straight line to the transformed data. Use an appropriate transformation to build this transformed model, and plot a graph of the data with appropriate variables on each axis. Does the data fit the model? Why? Explain how this will help you determine the unknown constants C and a. (b) Use the method of least-squares, and Mathematica, to calculate the parameters C and a using the method you developed in part (a). You can use the formulas from the text/lectures, but you must show how you reached your parameter values; don't just use a built-in Mathematica command. You might want to begin with one of the Mathematica notebooks for section 3.3 and make appropriate changes. State the final model. (c) On a single graph, plot the transformed planetary data, together with your calculated line of best fit. See sample notebooks for help with this. (d) Now plot the planetary data for T against radius r, together with the curve of best fit obtained using the calculations above. Kepler's Law In 1601 Johannes Kepler began analyzing the data that Tycho Brahe had painstakingly collected over 13 years on the orbit of Mars. We follow Kepler and look for a relationship between the orbit and period of planetary objects. The table below shows the semimajor axis (in Astronomical Units, where 1AU is the average distance from the Earth to the Sun) and orbital period for the planets, dwarf planets () and plutoids (**) in Earth years? This data is included in the attached Mathematica notebook. (a) We want to transform the model T=Cra so that we can fit a straight line to the transformed data. Use an appropriate transformation to build this transformed model, and plot a graph of the data with appropriate variables on each axis. Does the data fit the model? Why? Explain how this will help you determine the unknown constants C and a. (b) Use the method of least-squares, and Mathematica, to calculate the parameters C and a using the method you developed in part (a). You can use the formulas from the text/lectures, but you must show how you reached your parameter values; don't just use a built-in Mathematica command. You might want to begin with one of the Mathematica notebooks for section 3.3 and make appropriate changes. State the final model. (c) On a single graph, plot the transformed planetary data, together with your calculated line of best fit. See sample notebooks for help with this. (d) Now plot the planetary data for T against radius r, together with the curve of best fit obtained using the calculations above