  The knowledge of the contribution $b_d(d)$ to the artificial sky luminance in
a given point of the sky of a site produced by the sources beyond a given
distance $d$ from it is important to understand the behaviour of light
pollution in diffusely urbanized areas and to estimate which fraction of the
artificial luminance would be regulated by norms or laws limiting the light
wasted upward within protection areas of given radii.
  I studied the behaviour of $b_d(d)$ constructing a model for the propagation
of the light pollution based on the modelling technique introduced by Garstang
which allows to calculate the contribution to the artificial luminance in a
given point of the sky of a site of given altitude above sea level, produced by
a source of given emission and geographic position. I obtained $b_d(d)$
integrating the contribution to the artificial luminance from every source
situated at a distance greater than $d$. I also presented an analitical
expression for $b_d(d)$ depending mainly from one parameter, a core radius,
well reproducing model's results.
  In this paper I present the results for $b_d(d)$ at some Italian Astronomical
Observatories. In a diffusely urbanised territory the artificial sky luminance
produced by sources located at large distances from the site is not negligible
due at the additive character of light pollution and its propagation at large
distances. Only when the core radius is small, e.g. for sites in the inner
outskirts of a city, the sky luminance from sources beyond few kilometers is
negligible. The radii of protection zones around Observatories needs to be
large in order that prescriptions limiting upward light be really effective.
