solar hour angle derivations

1. solar hour angle from datetime and longitude

   symbol	description	unit	variable name
   \(EOT\)	equation of time	\(minutes\)
   \(t\)	datetime (UTC)	\(s\) since 2000-01-01	datetime {time}
   \(\eta\)	orbit angle of the earth around the sun	\(rad\)
   \(\lambda\)	longitude	\(degE\)	longitude {time}
   \(\omega\)	solar hour angle	\(deg\)	solar_hour_angle {time}

   \begin{eqnarray} A & = & 2\pi \left( \frac{t + 10 \cdot 86400}{365.2422 \cdot 86400} - \lfloor \frac{t + 10 \cdot 86400}{365.2422 \cdot 86400} \rfloor \right) \\ B & = & A + 2 \cdot 0.0167 \sin( 2\pi \left( \frac{t - 2 \cdot 86400}{365.2422 \cdot 86400} - \lfloor \frac{t - 2 \cdot 86400}{365.2422 \cdot 86400} \rfloor \right) ) \\ C & = & \frac{A - \arctan(\frac{\tan(B)}{cos(\frac{\pi}{180} 23.44)})}{\pi} \\ EOT & = & 720 \left( C - \lfloor C + 0.5 \rfloor \right) \\ \omega & = & \lambda + 360 \left( \frac{t}{86400} - \lfloor \frac{t}{86400} \rfloor + \frac{EOT}{24 \cdot 60} \right) - 180 \end{eqnarray}

   The solar hour angle will be mapped to [-180,180].