Modeling the the phenomenon of the artificial sky glow

Already the first measurements of the artificial sky glow have been combined with attempts to model this phenomenon. One of the first models was created in the 70s of the 20^th century by M.F. Walker [88] [89] [91] . Walker sought the relationship between the population of the city and the distance from it, where it causes the sky brightening of 0.2 mag at a height of 45° above the horizon. This value was accepted as the limit of the the city impact. On the based of the field measurements he set the relationship between the population P and the distance limit D: P ~ D ^2.5 for a range of population from 4,3⋅10³ to 1,2⋅10⁶. This allowed him to create a map of light polluted areas of California and Arizona.

A more sophisticated model of the impact of light pollution sources (cities) on the environment has been developed by P.J. Treanor SJ of the Vatican Observatory [83] . This model assumed only the forward scattering (ie only aerosols and dust), in a small cone angle, in a homogeneous atmosphere, with a negligible heigh of the dispersion medium relative to the from the city center. R.L. Berry [3] , adapting the Treanor's model to their measurements, rejected the assumption of this negligible heigh of the dispersin medium. He assumed also, on the base of measuring the brightness of the sky in the city centers, that the sky brightness is proportional to the square root of the city population. Again, the effect of this model was creation of the light pollution maps for southern Ontario.

Another development of a Treanor's and Berry's models was a model of R.H. Garstang[29][30][31]. Garstang resigned from the assumption that light scattering takes place only at one height. The consequence of this was the resignation of simplifying the homogeneous atmosphere - he adopted the simple model of density changing exponentially with height , however, introduced a different density distributions of molecules and aerosols. He examines not only the forward scattering (on the dust and aerosols), but also the Rayleigh scattering (on the molecules of the air components). As for the geometry of the problem, the model does not already assumed, as before, a point source of light pollution, but assumes that the source of this pollution is the circle area with a uniform surface brightness. A detailed description of the model can be found in [31] . This model is still under development and is verified by the confrontation with the measured data.

Treanor and Garstang models served as the basis for calculating the brightness of the night sky, which results as the light pollution atlas created by Falchi and Cinzano. Originally, these authors used a modified Treanor's model [28] , which resulted in a new light pollution atlas for Italy, and then adapted to their needs the Garstang's model, creating light pollution atlas for the entire world, but without taking into account the terrain [15] [16] [18] . Currently, the team is developing the light pollution model taking into account the terrain [17] [19] .