8.5. The elliptical path of a comet is described using the equation Р 1 + ε cos 0 where r is the radial distance to the Sun, is the angular position, e is the eccentricity of the orbit, and p is a rectum parameter. In this exercise you are to use data for the comet 27P/Crommelin, which is given in Table 8.8. (a) Writing the model function as r = g(0), and using least squares, then the error function is 72 E(p, e) = (g(0) - r.1². i=1 What two equations need to be solved to find the value(s) of p and & that minimize this function? (b) By writing the model function as 1 + ε cos 0 Р explain how the nonlinear regression problem can be transformed into one with the model function R = V₁ + V₂ cos(0). Also, what happens to the data values? (c) Writing the model function in part (b) as R = G(0), and using the least squares error function 71 E(V₁, V₂)=G(0) - R.]², i=1 compute V₁ and V₂. Using these values, determine p and e.
(d) Redo part (e) but use the relative least squares error function ER (V₁, V₂): = -2 (G(0) G(0₂) - R₁ R₁ -R₁) ². im1 (e) Does part (c) or does part (d) produce the better answer? You need to provide an explanation for your conclusion, using a quantified comparison, graphs, and/or other information.
0₁ 0.00 1.57 3.14 4.71 6.28 Ti 0.62 1.23 16.14 1.35 0.62 Table 8.8 Data for the comet 27P/Crommelin, used in Exercise 8.5.
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