Circumpolar Equation Derivation Edit: Geometric answer from Math.stackexchange

I am trying to understand where the circumpolar equation comes from.

$\delta \geq \frac{\pi}{2} - l$

I can understand that the sort of "rotation" of the reference frame that one experiences in horizon coordinates. I'm not sure where to go from the fact that a star's lowest altitude must be greater than 0.



It is pretty simple to show that $a = l$ and $c = \frac{\pi}{2} - l$. $a$ in this case is the altitude of the north celestial pole. What I need to do is show that the angle of the cone made by a circumpolar star with $\delta + l \geq \frac{\pi}{2}$.

You can deduce that $a_{min} \geq 0$ for a star to be considered circumpolar. I just need to do this geometrically in terms of its declination from the celestial equator.

Edit: I think my latest attempt has found a solution.



For a star to be circumpolar, the star must, at the very least, come above the horizon. 

$0 \leq a \leq \frac{\pi}{2}$

In the celestial sphere coordinate system, $c$ can be found to be

$c = \frac{\pi}{2} - \delta$

In the horizon coordinate system, $c$ can be found to be

$c = l - a_{min}$

Therefore

$l - a_{min} = \frac{\pi}{2} - \delta$
$\\$
$l + \delta = \frac{\pi}{2} + a_{min}$

$a_{min}$ can be set to 0 because it is the lower bound for circumpolarity.

$l + \delta = \frac{\pi}{2}$

$\delta$ must be in the cone made by the star's rotation about its declination angle. Therefore

$\frac{\pi}{2} - l \leq \delta$
There is nothing wrong with your explanation.  You could add some explanatory notes like $a_{min} = l-c$ and $a_{max}=l+c$. Also, I suggest that you put the equation $l+\delta = \frac{\pi}2-a_{min}$ on a separate line.   

It is very good to figure out these equations on your own. 

$ $

Here is an alternative explanation with less algebra:


If you walk outside at night, you will notice that the north star has an elevation of $l$ degrees if your latitude is $l$.  
The north star does not move.  
The angular distance between stars does not change  (in the short term!).
In order for the a star to be circumpolar, it needs to be within $l$ degrees of the north star (otherwise it will dip below the horizon when its hour angle is 12 hours.)
So, the angle $\theta$ between the north star to a circumpolar star needs to be less than $l$.  That angle is $\theta=90^\circ-\delta = \pi/2 -\delta$ where $\delta$ is the declination of the star.  
Finally, 
$$
\begin{align} \theta &amp;\leq l\\
\pi/2 -\delta &amp;\leq l \\
\pi/2  &amp;\leq  \delta + l \\
\pi/2  - l &amp;\leq \delta. 
\end{align}
$$