Central Force Motion on elliptical orbit

**zorop** · July 26th 2009, 01:32 PM

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

Thanks a lot~.

**zorop** · July 27th 2009, 05:19 PM

Where $v_{E}$ is the speed of Earth, and $a_{E}$ is the orbital radius. Then the eccentricity calculation of the comet is $\varepsilon=\sqrt{1+(V^2-\frac{2}{R})(RVsin\phi)^2)}$

And the numerical values we used here is $\phi=30$ , R=4 and V=1/2 so that $\varepsilon$ =0.866

These are the given values and konwn formulas

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