The solar zenith angle can be calculated in the following manner for any particular location (i.e., latitude and longitude), day of the year (dn), and time of day as described by Spencer (1971) and Madronich (1993). First, one needs to calculate what is known as the local hour angle (ih), which is defined as the angle (in radians) between the meridian of the observer and that of the sun:
+ (longitude/180)] + EQT, where GMT is Greenwich mean time converted from the local time, longitudes (in degrees) west of the Greenwich meridian are negative, and EQT is the "equation of time," given by
- 4.0849 X 10"2 sin 2N. where N is defined as
The day of the year, dn, is defined as the day number (0-364), with 0 corresponding to January f and 364 to December 31.
The second derived parameter that is needed for calculating the solar zenith angle at a particular time and place is the solar declination, 8, defined as the angle between the direction of the sun and the equatorial plane of the earth. The value of <5, which is 0° at the spring and fall equinoxes and falls between +23.45° (June 2f) and - 23.45° (December 2f), can be calculated from the following:
8 (in radians)
= 6.918 X 10"3 - 0.399912cos N + 0.070257sin N - 6.758 X 10~3 Xcos2N + 9.07 X 10"4 Xsin2Ar - 2.697 X 10"3 cos 3N + 1.480 X 10"3 sin 3N.
The solar zenith angle (0) for that particular time and place is then determined from:
cos 0 = sin 8 sin(latitude)
+ cos 8 cos(latitude)cos th, where 8 and th are calculated as already described and latitudes north of the equator (expressed in radians) are positive and south are negative. If all of the input parameters are in radians, 6 is also obtained in radians and can be converted to degrees using 1 rad = 57.296°. For example, at Los Angeles, California (latitude = 34.03°N, longitude = 118.14°W) on September 21 at noon PST, GMT = 20.0, N = 4.53, EQT = 0.0301, th = 0.0626, 8 = 0.0179 rad, and cos 6 = 0.837, giving a solar zenith angle of 0.579 rad, or 33°.
particles and is actually a sum of four terms,
where sg = light scattering by gases, ag = light absorption by gases, sp = light scattering by particles, and ap = light absorption by particles.
Gases scatter light by molecular, or Rayleigh, scattering. The intensity, /(A,©) of light of wavelength A scattered at an angle 0 to the direction of incident light is determined by a number of factors. These include the incident light intensity, the angle 0, the distance from the scattering molecule, and the index of refraction and size of the scattering molecule. In addition, and most importantly, Rayleigh scattering varies inversely with the fourth power of the wavelength.
Making the simplifying assumptions of a homogeneous atmosphere of fixed height of 7.996 X 105 cm and of uniform temperature and pressure throughout, Rayleigh scattering can be simplified for application to the atmosphere; as discussed in detail by Leighton (1961), the attenuation coefficient for scattering by gases, isg, becomes
where n()A is the index of refraction of air at wavelength A and the pressure and temperature of interest.
Zenith angle 8 (deg) |
m = sec 8 |
Air mass (m) |
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