Originally posted by zoobyshoe
A light is visible to the unaided human eye at a distance of 70 miles. How many lumens is required for this?
Stipulating the light is equally bright in all directions, can we calculate from the above lumens a reliable estimate of the joules needed to produce a light of that strength?
-zoob
If you turned the question around, you can get an estimate of the minimum amount of energy required (actually power, since you didn't say how long the light was visible for).
A sample calculation:
- assume the light is 'red', say 650 nm
- assume the human eye has a quantum efficiency of 5% (only 1 in 20 photons incident on the eyeball result in the firing of a neuron)
- assume the extinction of a column of 70 miles of air is x% (of the red photons which are emitted, only 1-x reach the eyeball)
- assume the human threshold for detection is [see below]
- assume the source can turn input energy into red photons with 100% efficiency
- assume the light is tightly collimated, and spreads to a circle of radius 7cm at distance 70 miles (in other words, hits both eyeballs, and surrounding part of the face, but not much else)
... just plug in the numbers, with the appropriate formulae, and the answer will fall out. You can then modify the assumptions to suit whatever case you're interested in investigating (isotropic emission, solar-spectrum light source, eyes which aren't dark adapted, ...).
I've only a dimly remembered 'rule of thumb' that can get you to the human eye threshhold: at the top of the atmosphere, the number of photons from a zero magnitude star with an A spectrum is 10,000 per square cm per angstrom per second (probably V or B band). More reliably, the visual threshold of the fully dark-adapted human eye (under truly dark skies) is 6.5 mag. If my 'A star' number is more or less correct, it's quite straightforward to calculate the number of photons, across the visible spectrum, that an eyeball needs to receive before the brain perceives a light.
Oh, if your source isn't a 'point', there are several effects which will need to be added to the above calculations.
Does this help?