Formulas for Luminance Distribution on the Clear Sky and for Total Horizontal Illuminance from the Clear Sky

A formula for luminance distribution on the clear cloudless sky and a formula for total horizontal illuminance from that sky are proposed based on a scattering theory. The clear sky light consists of the scattered light of the direct sunlight and that of the reflected light from the ground surface. And the main scattered light in the clear sky is the 1st-order scattered light of the direct sunlight, but its higher order scattered light than or equal to the 2nd-order and the scattered light of the reflected light from the ground surface are considerably included in the clear sky light and they have effect on the variation of clear sky light, respectively. In the proposed formulas, these effects are estimated from the result of a theoretical analysis. In the conventional formulas for the clear sky luminance distribution which are derived based on observation, however, these effects are not done because of difficulties of the estimation by means of measurements. Therefore the luminance distributions from the proposed formula approximate more closely to measured data under the clear sky than the conventional formulas. And the proposed formula for total horizontal illuminance from the clear sky is applicable not only to the clear sky but to the overcast sky because the scattering characteristics of the overcast sky is equivalent to those of the clear sky where aerosole is included quite densely.