David R. Brooks, March 2006

The simple model of solar irradiance (insolation) presented on the Web page from which this
document is linked provides a tool for predicting insolation and
understanding, at least roughly, the
effects of the atmosphere on insolation. It is important to understand that this is only an
*approximate* model that provides some insight into the basic physical processes that
control the amount of sunlight reaching Earth's surface under "clear sky" (no cloud) conditions.
It is *not* intended as a replacement for actual measurements of
insolation. Comparisons between the model and measured values should be viewed with caution, if for
no other reason than the fact that the biggest deficiency of this model (and other models, too) is
that it cannot take into account the primary effect of clouds on insolation. However, it is
very useful as a learning tool that shows how location, time of year, and time of day affect
insolation.

Insolation is defined as solar irradiance on a horizontal surface at Earth's surface. It is controlled primarily by the seasons and the weather. Maximum daily total insolation is greatest during the summer when the sun is highest in the sky. Even under clear sky conditions, the atmosphere reduces the amount of sunlight that reaches Earth's surface. Gas molecules (including water vapor) and aerosols (small particles suspended in the atmosphere) scatter sunlight, some of which is returned to space. Gases (including water vapor) and aerosols also absorb sunlight and some of this energy is re-radiated back to space. Of course, clouds have a major effect on insolation because they reflect a great deal of sunlight back to space.

The conditions that determine insolation are difficult to model with high accuracy even under clear skies. However, here's a simple model that accounts at least conceptually for the factors that reduce clear-sky insolation.

S = S_{o}cos(z)T_{r}T_{g}T_{w}T_{a}/R^{2}

where S_{o} is the solar constant, z is the solar zenith angle ( z = 90° - elevation angle), and
R is the Earth-distance in Astronomical Units (average distance is 1 AU).

Transmission factors are dimensionless values between 0 and 1 that account for reductions in transmission
of sunlight through the atmosphere to Earth's surface. One paper (see references below) has given these
factors for molecular scattering (also called "Rayleigh
scattering," hence the "r" subscript), gas absorption and scattering,
water vapor
absorption, and aerosol absorption and scattering as:

T_{r}T_{g} = 1.021 - 0.084 [m (949 px10^{-6} + 0.051)]^{1/2}

T_{w} = 1.0 - 0.077 (PW m)^{0.3}

T_{a} = A^{m}

where p is barometric pressure in millibars, PW is total column precipitable water vapor in units of
cm of H_{2}O, A is
an aerosol transmission factor, and m is the relative air mass. (The relative air mass is 1
when the sun is directly overhead and varies approximately as 1/cos(z).) The values given here are typical
values. For example, 1.42 cm of H_{2}O is the value assigned to a so-called "standard
atmosphere" that scientists use for modeling the behavior of the atmosphere. In very dry or high-elevation
locations, PW can be as low as a few millimeters. In very wet locations, it can be as high as
6 or 7 cm. 1013 millibars is the standard
atmospheric pressure at sea level.
At any elevation above sea level, you need to use "station pressure" -- the actual barometric pressure. With only
a few exceptions for research sites, weather
reports *always* give pressure converted to sea level pressure. If you are at a higher
elevation, you need to convert this value to your elevation. An approximate conversion is:

station pressure = (sea level pressure) - (elevation in meters)/9.2

That is, pressure decreases very roughly 1 millibar for each 10 meters increase in elevation.

You can change the typical values given here if you like, but you may get very strange results if you change them arbitrarily!

The solar constant -- the energy available at the average Earth-sun distance of 1 Astronomical Unit -- is
about 1375 W/m^{2}. (S_{o} is not really a "constant," and varies a little due to
fluctuations in solar activity.)
On a clear summer day in temperate latitudes, the maximum insolation near sea level
is around 1100 W/m^{2} -- a little less in northern hemisphere summer and a little more
in southern hemisphere summer. (Why? Because the sun is *farther* from Earth during
northern hemisphere summer than it is during southern hemisphere summer.) If all this energy could be
converted to electrical
energy, it would power about 10 100-W lightbulbs.
Unfortunately, we are a long way from being able to do that!

Also given in the above calculations is the maximum clear-sky insolation for the specified date, assuming atmospheric conditions such as barometric pressure do not change between the original time and the time of maximum insolation. To do this precisely, you have to know when "solar noon" occurs. This is the time at which the sun crosses your meridian. (Your meridian is the imaginary line running from north pole to south pole through your longitude.) Solar noon is, in general, never exactly the same as clock noon. It varies with time of year and the observer's longitude and can occur several minutes before or after clock noon. The calculations require the astronomically derived value known as the "equation of time." This gives the time by which actual solar noon occurs at the Greenwich Meridian relative to clock noon. Clock time is based on the apparent motion of a fictitious "mean sun" around Earth, periodically adjusted during leap years to keep seasons in sync with the sun over long periods of time.

The equations for the equation of time and the time of solar noon at your observing longitude are complicated. If you are interested, you can see all the calculations by viewing the source code for this Web page. These calculations haven't been checked exhaustively, but they appear to give values for the equation of time that agree to within less than 0.1 minute with the values given by NOAA's Surface Radiation Research Branch at their online solar position calculator.

"Local clock time" is based on 15° time zones, with the "0" time zone centered around the Greenwich Meridian, and ignores Daylight Savings Time and local anomalies in selecting a time zone. For example, on the East coast of the United States, UT is 5 hours later than Eastern Standard Time, but only 4 hours later than Eastern Daylight Savings Time. One of the output form fields shows the number of hours these calculations assume you have to add to your local clock time to get to Universal Time. Be sure you understand the relationship of this information to your actual local time! (The endless possibilities for confusion about this point is why scientific and astronomical calculations always use Universal Time rather than local clock time!)

These calculations include an *approximate* day length, to assist in calculating daily
averaged insolation. This daylength calculation is based on the time at which the elevation
of the center of the sun, based on astronomical calculations of the sun's geometric position relative
to Earth, is 0°.
This is *not* the same as actual or perceived sunrise or sunset for two reasons. First, the sun
is not a "point," but a disc with a size of about 0.5°. Second, atmospheric refraction
bends light traveling from the sun and makes it visible even when it is geometrically below
the horizon. Thus, perceived sunrise, when the top of the sun just appears on the
horizon, occurs earlier than the present calculation and perceived sunset, when the top
of the sun just disappears below the horizon, occurs
later. Nonetheless, this geometric calculation is a reasonable figure to use for calculating
average daily insolation.

As a "zero order" approximation, clear-sky insolation varies as the cosine of the of the solar zenith angle, symmetrically around local solar noon, and reaches 0 at sunrise or sunset. This is not a very good approximation because insolation is due to both direct and scattered sunlight. Scattered sunlight does not follow a cosine curve at all and its relative contribution to total insolation varies during the day. Direct sunlight is attenuated as it passes through more atmosphere when it is near the horizon, after sunrise or before sunset. A considerably better approximation is to model insolation as a function of time as a somewhat "pinched" cosine curve:

insolation = (solar noon insolation)•cos(πt/D)^{x}

where D is the daylength, t varies between -D/2 and +D/2, and x is an exponent greater than 1.

**Some Comments and Precautions:**

For the output fields, I have rounded off displayed calculated values to what I consider to be
an appropriate number of digits to
the right of the decimal point.

These calculations have been tested extensively but not exhaustively. In any event, common sense dictates a healthy skepticism about any results from online applications, especially when results do not appear to make sense. Values for latitudes poleward of the Arctic or Antarctic Circles may cause arithmetic errors leading to "values" of "NaN" appearing in their respective form fields when there is no sunlight.

If you find errors in the model calculations, please e-mail me.

References:

Duchon and O'Malley. *Journal of Applied Meteorology*, **38**, pp 132-141, 1999.

Meeus, Jean, *Astronomical Algorithms*. Willmann-Bell, Inc., Richmond, VA, 1991.