The mean distance between the earth and the sun is $1.496 X 10^8$ km and is called one astronomical unit (AU). The earth revolves around the sun in an elliptical orbit and the distance from the earth to the sun on a given day will vary from a maximum of 1.017 AU to a minimum of 0.983 AU. An accurate value of the earth-sun distance is important because the solar radiation reaching the earth is inversely proportional to the square of its distance from the sun. The distance is traditionally expressed in mathematical form as a Fourier series type of expansion with a number of coefficients. For most engineering applications a simple expression used by Duffie and Beckman (1980) is adequate for calculating the reciprocal of the square of the radius vector of the earth, also called the eccentricity correction factor, $E_0$, of the earth's orbit:

$E_0 = (r_0/r)^2 = 1+ 0.033 cos [(2\pi d_n /365)]$ 1:1.1.1

where $r_0$is the mean earth-sun distance (1 AU), r is the earth-sun distance for any given day of the year (AU), and $d_n$ is the day number of the year, ranging from 1 on January 1 to 365 on December 31. February is always assumed to have 28 days, making the accuracy of the equation vary due to the leap year cycle.

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.

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.

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

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

Total daylength, $T_{DL}$ is calculated:

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

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.

A number of basic concepts related to the earth's orbit around the sun are required by the model to make solar radiation calculations. This section summarizes these concepts. Iqbal (1983) provides a detailed discussion of these and other topics related to solar radiation for users who require more information.

The solar declination is the earth's latitude at which incoming solar rays are normal to the earth's surface. The solar declination is zero at the spring and fall equinoxes, approximately +23½° at the summer solstice, and approximately -23½° at the winter solstice. A simple formula to calculate solar declination from Perrin de Brichambaut (1975) is:

$\delta = sin^{-1} \{0.4sin [ 2 \pi /365] (d_n - 82)\}$ 1:1.1.2

where $\delta$ is the solar declination reported in radians and $d_n$ is the day number of the year.