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, 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?

I'm trying to estimate the distance and power I'd need for a green LED to appear visually roughy as bright as a relatively bright star - say a visual magnitude of zero. 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 and peaks roughly in the green part of the spectrum, however the center changes between about 550nm and 500nm depending on photopic 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?