1:1.2.1 Extraterrestrial Radiation

The radiant energy from the sun is practically the only source of energy that impacts climatic processes on earth. The solar constant, ISC, is the rate of total solar energy at all wavelengths incident on a unit area exposed normally to rays of the sun at a distance of 1 AU from the sun. Quantifying this value has been the object of numerous studies through the years. The value officially adopted by the Commission for Instruments and Methods of Observation in October 1981 is

$I_{SC} = 1367 W m^{-2} = 4.921 MJm^{-2} h^{-1}$

On any given day, the extraterrestrial irradiance (rate of energy) on a surface normal to the rays of the sun, $I_{0n}$, is:

$I_{0n} = I_{SC}E_0$ 1:1.2.1

where $E_0$ is the eccentricity correction factor of the earth's orbit, and $I_{0n}$ has the same units as the solar constant, $I_{SC}$. To calculate the irradiance on a horizontal surface, $I_{SC}$,

To calculate the irradiance on a horizontal surface,$I_0$,

$I_0 = I_{0n} \cos\theta_z = I_{SC}E_0\cos\theta_z$ 1:1.2.2

where $cos\theta_z$, is defined in equation 1:1.1.3.

The amount of energy falling on a horizontal surface during a day is given by

$H_0 = \int_{SR}^{SS} I_0dt = 2 \int_0^{SS} I_0dt$ 1:1.2.3

where $H_0$ is the extraterrestrial daily irradiation$(MJ m^{-2} d^{-1})$, SR is sunrise, and SS is sunset. Assuming that $E_0$remains constant during the one day time step and converting the time $dt$ to the hour angle, the equation can be written

$H_0 = \frac{24}{\pi} I_{SC}E_0\int_0^{\omega.T_{SR} }(\sin\delta \sin\phi+\cos\delta\cos\phi\cos\omega.t)d\omega.t$ 1:1.2.4

or

$H_0 = \frac{24}{\pi} I_{SC}E_0[{\omega.T_{SR} }(\sin\delta \sin\phi+\cos\delta\cos\phi\sin(\omega.T_{SR}))]$ 1:1.2.5

where $I_{SC}$ is the solar constant (4.921 $MJ m^{-2} h^{-1}$), E$_0$is the eccentricity correction factor of the earth's orbit, is the angular velocity of the earth's rotation ($0.2618 rad h^{-1}$), the hour of sunrise, $T_{SR}$, is defined by equation 1:1.1.4, δ is the solar declination in radians, and is the geographic latitude in radians. Multiplying all the constants together gives

$H_0 = 37.59E_0[{\omega.T_{SR} }\sin\delta \sin\phi+\cos\delta\cos\phi\sin(\omega.T_{SR})]$ 1:1.2.6