How much light-pollution any lamp will produce can be determined by:
1 - The choice of the fixture;
2 - The brightness level (the lumens) of the lamp inside this fixture;
3 - The manner in which the unit is installed outdoors.
The spectral makeup of the lamp is also crucial in terms of how light-rays behave as they leave the fixture, but it will
be ignored here. See the next paragraph.
Looking at the three determinants above, it should be evident to everyone that the application of the illuminating
structure & lamp system is of a central consequence regarding up-light. We're not interested in glare here unless it
causes up-light! For glare & trespass review this page. We're also not interested in extinction-dimming and reddening of any ray of light, because this characteristic is atmospheric related and not fixture related.
No matter what type of light-source is inside a housing, if the lamp rays are aimed into the night sky, either
directly from an inherently poor fixture (point 1), from excessive illumination on the ground (point 2), or from the tilt by an installer (point 3), the sky-bound rays will produce sky-glow according to the night sky conditions and scattering laws at
play. The latter, the atmospheric part, indeed plays a major role in how much sky-glow is created, but we will ignore
that here in order understand the relative polluting power of typical fixtures actually installed outdoors.
The Basic Lighting Laws:
The physical laws of lighting determine how much up-light is generated. The well known basic laws that apply to outdoor lighting are presented here for review.
Consider light coming from a fixture or any other light source from above:
Where:
H = Vertical Height (the mounting height) of the light-source above the ground.
D = Linear Distance from the source to the ground.
Θ = Angle between the vertical and the direction of D where the light strikes the ground.
x = Horizontal Distance between the point directly below the source to the point in question.
Inverse Square Law: Light reduction with increasing distance. If the surface is normal to the direction of the incident light, then the illumination E at a point on a surface varies directly with the luminous intensity I of the source and inversely with the square of the distance D from the source: E = I/D2.
Lambert's Law or the Cosine Law: Light reduction with increasing angle of incidence. If the surface at the point receiving the light is not normal to the source, the illumination will vary with the cosine of the angle of incidence. Combining this with the Inverse Square Law yields: E = I cosΘ/D2.
Cosine Cubed Rule: By substituting H/cosΘ for D we get: E = I cos3Θ/H2. Thus the illumination E at any point on the ground with angle of incidence Θ can be expressed simply as a function of the luminous intensity I of the source and its mounting height H.
Graphically: By plotting E versus Θ, we can see the drastic drop-off of the light into vertical angles above and beyond 75°.
For E = I/H2 cos3Θ, where I is the luminous intensity and H is the lamp height.
From the graph above, you can see the reason why the refractor was created for cobra-heads back in the 1950s.
This graph compares the illumination drop-off for typical and ideal full-cutoff heads (in black & blue) with that of a classic cobra-head with refractor (red curve). Note the shaded reddish area on the right. This is the portion of direct up-light from the fixture. No illumination falls on the ground with an angle of incidence of 90° or greater. The green distribution is for the cosine function as calculated above.
What do the light patterns actually look like on the ground?
These HPS heads were on a string of street-lights on an industrial road. Do you see the yellow 15 sticker? This is why I
put up this page.
The head on the left was an older model (a Cooper OVZ) with a dropped refractor. It sheds light as in the red curve in the
graph. The head on the right was a more recent, full-cutoff (FCO) model (a GE M-250R2), a terrible FCO head. This represents the black curve in the graph.
The stickers indicate 150 watt lamps for both; in fact the entire string were 150 watt HPS. City technicians often kept
the same watts for trouble free maintenance or available HPS lamp stocks. A good practice for them perhaps, not for
astronomers. The lux on the ground from the FCO head is 3-4 x the old cobrahead lux.
