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Summation of Surface Reflectivity:

Not so well known is the ratio of reflected up-light to direct up-light from the same luminaire. There is a difference. You can always contact me if you find a problem with the following.

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 a perfect 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 reflected light from an ideal source (and ideal Lambertian flat surface), and a 1x correction factor for any direct light (shining directly from the fixture into the sky). Summarizing...

Total UpLight ∝ DirectUplight + (π/2 -1)ReflectedUplight, For a full-cutoff fixture only a reflected component contributes to up-light.

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

Putting in some values for the light-balance of a classic cobra-head having some up-light by direct and reflected light, we can confidently estimate the total up-light. Things are just a little more complicated than this "ideal" light source or "Lambertian" surface, details of which can be found in "The Sky-Glow Story".