In the book by Goldstein (Third Edition, Page 95) on classical mechanics, Goldstein derives equation of the trajectory of particle moving under center force. The equation he derives is given by:

$$r= \frac{a(1-e^{2})}{1+e\cos(\theta-\theta')}.$$

My question is how $\theta$ is measured in the given equation i.e w.r.t which position $\theta$ is measured?


It should be clear from inspection that $r$ has it's smallest value when $$ \theta = \theta' + 2 n \pi \;,$$ for $n \in \mathbb{Z}$ (that is $n$ is any integer) and likewise has it's largest value for $$ \theta = \theta' + (2 n + 1) \pi \;.$$

Which tells you that in coordinate system $\theta'$ is the angular position of periapsis and $\theta' + \pi$ is the angular position of apoapsis.


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