Length of day Vs latitude $\varphi$  at solstices

We'll do this problem by computing the intersection of a tilted circle with a plane that is normal to the line between the earth and the sun and which includes earth center as a point with a tilted circle of radius $a=R\cos (\varphi )$ centered at

$$(x_0,z_0)=(R\sin \varphi \tan \theta ,R\sin \varphi )$$

(1.1)

where $\theta$ is the earth tilt angle, 23.44 degrees, and where $\varphi$ is the latitude.

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The description of the tilted circle is then

$$\begin{array}{l}x=a\cos \varphi \cos \psi -x_0\\ y=a\sin \varphi \end{array}$$

(1.2)

The vertical plane is described by

$$x=0$$

(1.3)

We can solve the first equation for $\psi$ :

$$\begin{array}{l}R\cos \varphi \cos \psi -R\sin \varphi \tan \theta =0\\ \psi ={\cos }^{-1}\left[\frac{\sin \varphi \tan \theta }{\cos \varphi }\right]\end{array}$$

(1.4)

Parameter $\psi$ is called the hour angle and is simply related to the hour at sunset and sunrise.