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V. Appendix . Solar UV Irradiance.

The radiation intensity incident on a given surface has two terms:

(A1)

representing the direct component of UV photons arriving from the Sun without any deviation and

(A2)

the diffuse component of photons arriving from all directions. The variable z includes factors characterizing the geographical place (coordinates and height of the place and ground reflectivity), the solar zenithal angle q _z and the atmospheric components affecting the considered radiation (aerosols, ozone and other gases). The first one is given by the law of exponential attenuation and the second (transforming the variable wavelength in frequency) through the solution of the equation of radiative transfer:

(A3)

where s is the trajectory of the incident radiation beam and W the solid angle associated with this radiation and the considered point. is the spectral extinction coefficient and the spectral source function, given respectively by:

(A4)

being p the scattering phase function, y spectral absorption and dispersion coefficients respectively and the spectral albedo (reflectivity). The formula (A7) expressed the very well-known Planck’s law of the blackbody radiation.

The spectral irradiances in UV range (280 - 400 nm) is obtained with the following expression:

(A8)

With the integration of this expression the UV solar total irradiance is:

(A9)

The erythemal irradiance is obtained using a factor of weight, called erythemal action spectrum (skin alarm), B(l ) to the UV irradiance given for (A8).

(A10)

And expressed in SI units and multiplied by the factor 40, gives the Index of Solar Risk (UV Index) at any time during the day.

Figures 3 and 4 and table 1 shows the results obtained using formulas (A8 - A10) for different days and ozone profiles.