# Solar Noon, Sunrise, Sunset, and Daylength

The angle between the line from an observer on the earth to the sun and a vertical line extending upward from the observer is called the zenith angle, $$\theta_z$$ (Figure 1:1-1). Solar noon occurs when this angle is at its minimum value for the day.

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![Figure 1:1-1 Diagram illustrating zenith angle](https://1348478613-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MTFpIHUB1K63DLhLJaf%2F-MjGBev3ej9wmQ0MKvRQ%2F-MjGFrcHvsW8m3ytvBrt%2FFigure_1.png?alt=media\&token=f577f06c-1f1f-4960-aaad-8fe59dcd5072)

For a given geographical position, the relationship between the sun and a horizontal surface on the earth's surface is:

$$\cos\theta_z = \sin\delta\sin\phi + \cos\delta \cos\phi\cos\omega t$$                                       1:1.1.3

where $$\delta$$ is the solar declination in radians, $$\phi$$ is the geographic latitude in radians, $$\omega$$ is the angular velocity of the earth's rotation (0.2618 rad $$h^{-1}$$ or 15˚ $$h^{-1}$$ ), and t is the solar hour. t equals zero at solar noon, is a positive value in the morning, and is a negative value in the evening. The combined term $$\omega t$$ is referred to as the hour angle.&#x20;

Sunrise, $$T_{SR}$$, and sunset, $$T_{SS}$$, occur at equal times before and after solar noon. These times can be determined by rearranging the above equation as:

$$T_{SR} = +(\cos^{-1}[-\tan\delta \tan\phi]/\omega)$$                                                1:1.1.4

and&#x20;

$$T_{SS} = - (\cos^{-1}[-tan \delta  tan\phi]/\omega)$$                                                    1:1.1.5

Total daylength, $$T_{DL}$$ is calculated:&#x20;

$$T_{DL} = (2 \cos^ {-1}[-1\tan \delta \tan \phi]/\omega)$$                                                1:1.1.6

At latitudes above $$66.5\degree$$or below $$-66.5\degree$$, the absolute value of \[ $$\tan\delta \tan\phi$$ ] can exceed 1 and the above equation cannot be used. When this happens, there is either no sunrise (winter) or no sunset (summer) and $$T_{DL}$$ must be assigned a value of 0 or 24 hours, respectively.&#x20;

To determine the minimum daylength that will occur during the year, equation 1:1.1.6 is solved with the solar declination set to $$-23.5\degree$$ (-0.4102 radians) for the northern hemisphere or $$+23.5\degree$$ (0.4102 radians) for the southern hemisphere.

The only SWAT+ input variable used in the calculations reviewed in Section 1:1.1 is given in Table 1:1-1.

Table 1:1-1: SWAT+ input variables that are used in earth-sun relationship calculations.

<table data-header-hidden><thead><tr><th width="176">Variable Name</th><th width="191.28689431071913">Definition</th><th>File Name</th></tr></thead><tbody><tr><td>Variable Name</td><td>Definition</td><td>File Name</td></tr><tr><td>SUB_LAT</td><td>Latitude of the subbasin (degrees).</td><td>.sub</td></tr></tbody></table>

