# Daylight hours

# Definition

The daylight hours are maximum possible duration of sunshine for a given day of the year (Allen et al. 1998).

# Formula

## Formulation according to FAO:

The daylight hours \(N\) are obtained as follows (Allen et al. 1998):

\[N=\dfrac{24}{\pi}\cdot\omega_s\]

where \(\omega_s\) is the sunset hour angle [rad], which is calculated as follows:

\[\omega_s=\arccos [-\tan\varphi\cdot\tan\delta]\]

where \(\delta\) is the solar declination [rad], and \(\varphi\) the latitude [rad].

The solar declination \(\delta\) is calculated as follows:

\[\delta=0.409\cdot\sin\bigg(\dfrac{2\pi}{365}\cdot J-1.39\bigg)\]

where \(J\) is the number of the day in the year between 1 (1 January) and 365 or 366 (31 December).

The conversion from decimal degrees to radians is obtained as follows:

\[\mbox{[Radians]}=\dfrac{\pi}{180} \mbox{ [decimal degrees]}\]

NB: for calculating potential evapotranspiration according to Thornthwaite, a daylight coefficient \(C\) is required, which corresponds to the duration of sunlight in units of 12 hours. This daylight coefficient \(C\) is thus given by:

\[C=\dfrac{N}{12}\]

## Formulation according to NFDRS

The procedure for determining daylight hours \(N_{nfdrs}\) proposed by (Cohen & Deeming (1985) in order to calculate some components of the NFDRS is slightly different from that proposed by (Allen et al. (1998):

\[N_{nfdrs}=24\cdot\bigg(1-\dfrac{\arccos(\tan\varphi\cdot\tan\delta)}{\pi}\bigg)\]

where \(\varphi\) is latitude [rad] and \(\delta\) the solar declination [rad].

The latitude \(\varphi\) [rad] is given by:

\[\varphi=\varphi_{deg}\cdot{0.01745}\]

where \(\varphi_{deg}\) is the latitude in decimal degrees.

The solar declination \(\delta\) is calculated as follows:

\[\delta=0.41008\cdot\sin\Big((J-82)\cdot{0.01745}\Big)\]

where \(J\) is the Julian date.

# Reference

Cohen & Deeming (1985)

Allen et al. (1998)