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Suppose I am given that a planet's position with respect to some star is of the form $\textbf{r} = r\textbf{e}_{r}$. Then of course $\textbf{v} = \dot{r}\textbf{e}_{r} + r\dot{\theta}\textbf{e}_{\theta}$.

Is it correct to say that, when the planet is at its closest and at its furthest from the star, the radial component of the velocity vector will always be zero, i.e. $\dot{r}=0$? In which case, the planet will still have some velocity purely in the $\textbf{e}_{\theta}$ direction?

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Yes, the radial component of the velocity vector $\dot{r}$ is 0, since the distance between the planet and the origin is simply $r$. Assuming the planet is not on a purely radial trajectory, i.e. $\dot{\theta}$ is not zero at any point, the planet will also have velocity purely in the $e_\theta$ direction at this point.

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