  The standard visibility model in light pollution studies is the formula of
Hecht (1947), as used e.g. by Schaefer (1990). However it is applicable only to
point sources and is shown to be of limited accuracy. A new visibility model is
presented for uniform achromatic targets of any size against background
luminances ranging from zero to full daylight, produced by a systematic
procedure applicable to any appropriate data set (e.g Blackwell (1946)), and
based on a simple but previously unrecognized empirical relation between
contrast threshold and adaptation luminance. The scotopic luminance correction
for variable spectral radiance (colour index) is calculated. For point sources
the model is more accurate than Hecht's formula and is verified using
telescopic data collected at Mount Wilson by Bowen (1947), enabling the sky
brightness at that time to be determined. The result is darker than the
calculation by Garstang (2004), implying that light pollution grew more rapidly
in subsequent decades than has been supposed. The model is applied to the
nebular observations of William Herschel, enabling his visual performance to be
quantified. Proposals are made regarding sky quality indicators for public use.
