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Solar turbidity factors – Linke turbidity coefficient

Atmospheric turbidity is an important parameter for evaluating not only air pollution in an environment, but also the main parameter that determines the weakened solar radiation that reaches the solar panel. The Linke turbidity factor model common is used to evaluate atmosferic turbidity. It It is based on a one-year measurement of horizontal irradiation in a certain area. The obtained results are further compared with the recommended values for various types of calculations. The SPAC application uses turbidity calculation methods for the most realistic results. And formulas bellow are implemnts in application.

Understanding of the turbidity coefficient as an important factor

The weakening of the solar energy that reaches the solar panel therefore depends on: the state of the sky, the density and shape of the clouds and the composition of the gases in the atmosphere and the humidity of the air. Solar irradiation is also weakened by the influence of permanent atmospheric factors. This attenuation has nothing to do with local atmospheric conditions and includes: ozone and moisture concentration. But that’s not all. The attenuation of sunlight reaching the solar panel is affected by two other important factors: water vapor and aerosols, and these are known as atmospheric haze.

Should we aproximate turbidity coefficient ?

This further means that some solar panel installers consider standard testing conditions *STC* as a realistic situation.

More about solar swindlers you can read here

In order to assess the atmospheric situation, annual data for the location are required:
Climate and climatic factors (severe winters with or without snow, length of spring and autumn with estimation of precipitation) Distribution and speed of winds during the year (wind is a good purifier of the atmosphere)

  • Amount of precipitation from rain and snow, rate of evaporation, retention of snow cover.
  • Mean monthly air temperature
  • Atmosphere pressure
  • Mean relative humidity

Hence, all these data are impractical to observe, so solar panel installers often avoid unnecessary complications.

Linke turbidity coefficient

There are several turbidity factors, but the most commonly used is Linke’s, which is marked with TL. Linke in 1922 defined this factor as the number of clean and dry atmospheres necessary to achieve the same effect on direct radiation as that produced by a clean atmosphere. It depends on the optical depth of the clean and dry atmosphere, which depends on the air mass, and the air mass depends on the altitude angle. This factor describes the optical depth of the atmosphere due to the scattering process of the sun’s rays. Ozone, water vapor, and carbon dioxide are supporting this scattering. Linke’s coefficient ranges from 1 to 10. For example, for a clean atmosphere, it is 1 for a clean, cold winter air, 2 for a clean, warm summer day, and 3. For humid, warm air, this coefficient is in the range of 4 to 6. In a polluted city atmosphere, it is 8.

Fog and haze in cities grow the Linke cofficient
Ideal suny day

The formula for calculating the Linke coefficient

TL = ln ( ( A*sin(beta) /IBH ) / ( sigma*m) ) and we have:

  • A – flux of extraterrestrial radiation entering the atmosphere  A = 1160 + 75 * sin ( 360/365 * (n – 275)) where n is the regular day of the year
  • IBH – direct irradiation in a horizontal surface
  • beta – altitude angle
  • sigma – Integral Rayleigh optical thickness due to scattering in the atmosphere
  • m – air mass coefficient : m = 1/sin(beta)
The formula for optical thickness can be calculated in several ways, the Kasten formula is most often used sigma = 1/ (6.6296 + 1.7543 * m -0.1202 *m2 + 0.0065 m3 – 0.00013 * m4) .. for m  <= 20 in other cases :  sigma = 1/ (10.4 +0.718m)

Deep dive into math

The calculation of the direct component IBH requires knowledge of the direct component of horizontal radiation ILiu Jordan’s formula for the decomposition of total horizontal radiation is used for this to the direct and diffuse component using the brightness index Kt. IDH:IH = 1.39 – 4.027 *Kt + 5.531 * Kt2 -3.108Kt3 IDH – diffuse component of horizontal irradiation ( IH>/sub> = IBH+IDH) The clearness index is defined as the mean ratio of horizontal insolation to mean horizontal extraterrestrial insolation on the earth’s surface. Kt = IH:Io

Mean horizontal extraterrestrial irradiation

Io = (24/pi)*SC*(1+0.034*cos(360*n/365)) * (cos L * cos(sigma) * sin(Hsr) + Hsr* sinL * sin (sigma))
and we have:
 SC: Solar Constant = 1377 W/m2
 Hsr: hour angle Hsr = arccos(-tan(L) * tan(sigma))

and in this formula we have :  
L: latitude
sigma: solar declination = 23.45 * sin(360/365 * (n-81))

All these formulas are applied in the SPAC application algorithm, so it was important to mention them.


If the Linke coefficient is low, it means the sky is clear and the air is clean. When looking at the Linke turbidity coefficient for a place with a moderately continental climate,
numbers between 1.8 and 4.5 will be found. Small amounts of straight horizontal irradiation that show up during the day are high-value effects.

Summer precipitation can cause humidity and increase the coefficient. Precipitation (snow and rain) as well as wind are good air purifiers. On the other hand, evaporation after rain can cause humidity, which is not favorable.

More about Linke coefficient you can find here


Mechanikal engineer and php programer, work in electric power industry more then25 years

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