Solar Elevation Angle (Height of the Sun above the Horizon)

The solar elevation angle $\alpha$ is how high the Sun stands above the horizon, found from three angles: the latitude $\varphi$, the Sun's declination $\delta$ and the hour angle $h$. The declination is the Sun's angle north or south of the celestial equator, and it runs between minus 23.44 and plus 23.44 degrees through the year. The hour angle is 0 at solar noon and 15 degrees for each hour from it, positive in the afternoon. The elevation and the zenith angle, which is measured from straight up, add to 90 degrees. Solar designers use the elevation for shading, row spacing and the length of the day, and the airmass that sets the clear-sky irradiance follows from it.

The formula has two terms. $\sin\varphi\,\sin\delta$ does not change through the day, while $\cos\varphi\,\cos\delta\,\cos h$ is the daily swing, largest at solar noon when $h = 0$. At solar noon the elevation is 90 degrees less the difference between the latitude and the declination. The page's example is 49.28 degrees north on the June solstice, with a declination of 23.44 degrees: the zenith angle is 25.84 degrees and the Sun stands 64.16 degrees above the horizon.

Other cases show how the three angles interact. At the equator at an equinox noon the Sun is overhead. At an equinox the Sun is on the horizon at an hour angle of 90 degrees, 6 o'clock solar time, whatever the latitude, because $\cos h$ is zero there. At 45 degrees north at an equinox, 3 hours after noon ($h = 45^{\circ}$), $\sin\alpha = \cos45^{\circ}\times\cos45^{\circ} = 0.5$, so $\alpha = 30^{\circ}$. At 70 degrees north with a declination of 23.44 degrees, the Sun at solar midnight ($h = 180^{\circ}$) is 3.44 degrees above the horizon: the midnight Sun. A negative elevation means the Sun is below the horizon.

The formula is geometric: it gives the elevation without the small lift that atmospheric refraction adds near the horizon, and the hour angle is in solar time, not clock time. Solved backwards, the inverse trigonometry has branches. A noon elevation of 64.16 degrees with a declination of 23.44 degrees fits 49.28 degrees north, but also 2.40 degrees south; the page returns the larger latitude. Likewise a noon elevation of 64.16 degrees at 49.28 degrees north fits a declination of 23.44 degrees or 75.12 degrees, and the second is not a Sun. The relation is from Duffie and Beckman, Solar Engineering of Thermal Processes (4th ed., 2013).