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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

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/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 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...

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 "

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