A comet is observed to have a speed $\displaystyle v$ when it is at a distance $\displaystyle r$ from the Sun, and its direction of motion makes an angle $\displaystyle \phi$ with the radius vector from the Sun. Show that the major axis of the elliptical orbit of the comet makes an angle $\displaystyle \theta=\cot^{-1}(\tan\phi-\frac{2}{V^2R}\csc(2\phi))$ with the initial radius vector of the comet, where $\displaystyle V=\frac{v}{v_{E}}$ and $\displaystyle R=\frac{r}{a_{E}}$ are the dimensionless ratios. Use the numerical values that we used for the eccentricity calculation to now calculate a value for the angle $\displaystyle \theta$.

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