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 is D in our diagram on the previous page 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; (the volume of cosΘ 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 then...
Therefore, expressed in terms of the lumen output of an ideal point source of light with r (or D in this diagram) 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
Yes, 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 from the fixture into the sky. Summarizing...
Total Up-Light ∝ Direct Up-Light + (x)Reflected Up-light, Total Up-Light ≈ Direct Up-Light + (π/2 -1)Reflected Up-Light,
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 estimate the total up-light.
Things are just a little more complicated than this "ideal" light source or "Lambertian" surface. More details can be found in "The Sky-Glow Story".
