# Thread: Central Force Motion on elliptical orbit

1. ## Central Force Motion on elliptical orbit

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 $\displaystyle \theta$.

Thanks a lot~.

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

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

These are the given values and konwn formulas