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:
- $\sin(\gamma) = \sin(\theta) \sin(i)$
- $\theta \sim U(0,2\pi)$
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?