The physical laws of lighting determine how much up-light is generated and the amount of sky-glow produced.

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.

For

Incidence

Angle cos

0° 1.0

10° 0.955

20° 0.830

30° 0.650

40° 0.450

50° 0.266

60° 0.125

70° 0.040

80° 0.005

85° 0.001

90° 0.0

From this graph, you can see the reason why the refractor was created for the classic cobra-heads back in the 1950s.

Not so well known is the ratio of reflected light to direct light from the same luminaire.

You can always contact me if you find a problem with the following math.

The luminous intensity of an ideal point source of light can be expressed in terms of its luminous flux (lumens) by

ALL of the light from a "simple" full-cutoff luminaire (with internal reflector but no direction of the light) is shed into just one hemisphere. Here ω = 2π, and I = F/2π.

The illumination on a flat surface is related to the intensity of the light source through the cosine law, E = I/D

The radius r = D in our diagram above and varies by H/cosΘ, so that r

Expressed by the luminous (lumen) output, the illumination

The total illumination the ground sees will be found by the area under cosΘ rotated about the E axis for our function; or the volume of cos(Θ) about the y axis.

Total illumination E is the

Using the shell method with incremental area dA taken parallel to the y axis:

The volume V = (Length)(Height)(Width), which are 2πx, f(x), and the increment is dx, respectively. Integrating...

Therefore, expressed in terms of the lumen output of an ideal point source of light with r (or D) being the linear distance from the source to the ground, the total illumination the ground receives (sees, not reflects) is

That's right, there's always a correction factor of 0.571x for the TOTAL reflected light from an ideal source (and ideal Lambertian flat surface),

and a 1x correction factor for any direct light (light shining directly from the fixture into the sky). Summarizing...

That "p" is actually the Greek Rho (ρ) and it's the coefficient of reflectivity. I measured the average reflectivity of dry asphalt to be at ~ 13%.

Putting in some values for the light-balance of a classic cobra-head, we can confidently estimate the total up-light.

Things are a little more complicated than this "ideal" light source or surface, details of which can be found in "

Back to the sky-glow story.