I'm trying to estimate the distance and power I'd need for a green LED to appear visually roughy as bright as Venus. Here is what I have so far.

Be warned I am just ballparking it here.

The sun is visual magnitude -27, and five visual astronomical magnitudes are a factor of 100, so a zero magnitude star should appear to be a factor of $100^{-27/5} \approx 1.6 \times10^{-11}$ as bright as the sun.

The FWHM of the sensitivity of human vision [is about 100nm][2], however the center changes between about 550nm and 500nm depending on photopic (black) or scotopic conditions.

At sea level, direct sunlight is about $ 1.3 \ W/m^2/nm$, so for a 100nm wide bandpass that's  $130 \ W/m^2 $.  A zero visual magnitude object should then produce $ 2.1 \times10^{-9} \ W/m^2$.

If I have a say 555nm green LED with 30% external quantum efficiency, then $0.1 \ A$ of current should produce $0.22 \ W \times 0.3 \approx 0.067 \ W$ of light. If it is roughly uniform over a cone with a half-width of 10°, then the LED produces $ 0.7 \ W/Sr$, or  $ 0.7/r^2 \ W/m^2$ at a distance of $r$ meters.

That means I would have to move my 100 mA, 30% eQE 555nm LED with a 10° half-angle 18 kilometers away for it to look roughly as bright a 0 visual magnitude star!

Have I made some fundamental mistake here? Or - baring atmospheric absorption - could I actually see a green LED ~20km away (or on a balloon 20km straight up) on a dark night?


[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/RIvLe.png
  [2]: https://en.wikipedia.org/wiki/File:Luminosity.png