Consider the following data relating the semi-major axis r and orbital period t of the planets in the solar system: plan

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Consider The Following Data Relating The Semi Major Axis R And Orbital Period T Of The Planets In The Solar System Plan 1 (174.53 KiB) Viewed 60 times

Consider the following data relating the semi-major axis r and orbital period t of the planets in the solar system: planet r (AU) t (years) Mercury 0.38710 0.2408426 Venus 0.72333 0.6151865 Earth 1 1 Mars 1.52366 1.880815 Jupiter 5.20336 11.86241 Saturn 9.53707 29.50694 Uranus 19.1913 84.01537 Neptune 30.0690 164.7884 In 1618, using similar data from the six planets known at the time, Kepler conjectured a relationship of the form t = krb for some constants k and b. Taking logarithms, this becomes log t = a + blogr, where a = log k. (a) Use R to find the 8 x 2 matrix A such that a --- logt A (b) Use the method of least squares to find the best constants a and b that fit the data. (c) Compute the value of k corresponding to a. Why is it so close to 1? (d) What value of b do you think Kepler guessed?