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

The Basic Lighting Laws:
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:

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.

    Angle      cos3Θ
      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.

Summation of Surface Reflectivity:

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 I = F/ω (lum/sr), where ω is the solid angle of the radiating sphere. ω = area/radius2 (in steradians) and area = 4πr2 for the sphere. The biggest solid angle possible is 4π.

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/D2cosΘ. The cosine cubed rule is E = I cos3Θ/H2.

The radius r = D in our diagram above and varies by H/cosΘ, so that r2 = H2/cos2Θ. The cosine cubed rule arises because of our desire to express the illumination as a function of the light-source height H. Putting I = F/2π into either, we get the same result.

Expressed by the luminous (lumen) output, the illumination E = (F/2πr2)cosΘ.

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 volume, not just the area under the cosine. The volume is the area under one half arch of the curve y = cos(x), from 0 to π/2, rotated about the y axis. We have y = cos(x), from 0 to π/2.

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 E(total) = F/r2 (π/2 -1), or E(total) = 0.571 F/r2

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),

Up-Light ∝ DirectUplight + (π/2 -1)ReflectedUplight,

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 "The Sky-Glow Story".

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