Converting radiometric units (W/m^2) to photometric units (Lux)

[cs4809]

1. I am doing a lab about solar radiation. We collected data using a lightmeter which measures illuminance in lux. Now to calculate the optical depth of the atmosphere I either need to find an expression for the solar constant in lux or be able to convert illuminance values in lux to W/m². The conversion is straightforward for monochrome radiation but for a range of frequencies it becomes much more complicated because each wavelength has a different conversion factor. I've been told that integrating a blackbody intensity curve is a good place to start but any approximate methods for this conversion would also be welcome.

Homework Equations

I₀ is the intensity of radiation at the source (Solar constant) and I is the observed intensity after a given path, then optical depth τ is defined by the following equation
[URL]http://upload.wikimedia.org/wikipedia/en/math/6/6/2/6622b2af6bba780c8a709106b0ec0f5b.png[/URL]
Solar constant: 1.361 kilowatts per square meter (kW/m²)
converting monochrome light:
1/683 watt of 555 nanometre green light provides one lumen: this is the peak of the weighting function and also the peak of the suns spectrum as a blackbody. All other wavelengths are "worth" progressively fewer lumens.

This website gives an idea of the problem but no solution for photometric to radiometric (and neither does the book he mentions in the last section)
http://www.optics.arizona.edu/palmer/rpfaq/rpfaq.htm

3. So far I have tried integrating Planck's law, by hand and using maple, between the visible light region limits to try to get the energy per second radiated by the sun but can't get the integral (integrating between 0 and ∞ gives the Stefan–Boltzmann law but other ranges end up with a reimann zeta function so cannot be done). I have assumed the sun is a blackbody at 5800K. I've been working on this for 2 full days but keep coming to dead ends no matter how i try to solve the problem.

 

[Nate810]

Homework EquationsPlanck's law:http://upload.wikimedia.org/math/e/d/f/edf8b2cba6ac45d6cfd5f7e9a0a61a3b.pngA possible solution is to use the spectral energy distribution of solar radiation (or other light source) and integrate it using a numerical integration method. This would involve using the spectral energy distribution in watts per square meter per nanometer to calculate the total energy in watts per square meter for a given wavelength range and then convert this to lux by multiplying it by the conversion factor for the desired wavelength range. For example, for the visible light region from 400nm to 800nm, you could use the conversion factor of 683 lm/W to convert the total energy in watts per square meter to lux.