# solar zenith angle derivations

1. solar zenith angle from solar elevation angle

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\alpha_{0}\) | solar elevation angle | \(deg\) | solar_azimuth_angle {:}
   \(\theta_{0}\) | solar zenith angle | \(deg\) | solar_zenith_angle {:}

   The pattern : for the dimensions can represent {time}, or no dimensions at all.

   \[\theta_{0} = 90 - \alpha_{0}\]

2. solar zenith angle from latitude and solar declination/hour angles

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\theta_{0}\) | solar zenith angle | \(deg\) | solar_zenith_angle {time}
   \(\delta\) | solar declination angle | \(deg\) | solar_declination_angle {time}
   \(\phi\) | latitude | \(degN\) | latitude {time}
   \(\omega\) | solar hour angle | \(deg\) | solar_hour_angle {time}

   \[\theta_{0} = \frac{180}{\pi}\arccos(\sin(\frac{\pi}{180}\delta)\sin(\frac{\pi}{180}\phi) - \cos(\frac{\pi}{180}\omega)\cos(\frac{\pi}{180}\delta)\cos(\frac{\pi}{180}\phi))\]