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Some of you may recognize this equation as the latitude of a sub satellite point for a circular orbit. I have seen papers that provide the solution, but not the steps.

Given:

  1. $\sin(\gamma) = \sin(\theta) \sin(i)$
  2. $\theta \sim U(0,2\pi)$

find the PDF as a function of $\gamma$ and $i$, given that $i$ is constant for a given PDF and $ -i < \gamma < i$.

The solution in several texts is

$$f(\gamma) = \frac{\cos\gamma}{\pi\sqrt{\sin^2 i - \sin^2\gamma}}; \qquad -i < \gamma < i$$

Can anyone provide guidance on the derivation?

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  • $\begingroup$ After looking around I have found some documentation about "change of variables" and starting with the know CDF of $\theta$ to calculate the CDF of $f(\gamma)$ $\endgroup$
    – S moran
    Commented Feb 12, 2020 at 17:16

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