Computing color and brightness of a hot material

Question

Every blackbody color calculator I've managed to find only calculates hue and saturation; they completely ignore brightness, which severely limits their usefulness if you're trying to model the actual appearance of a hot object.

I've tried looking for how to calculate it myself, but Google is choked by discussions about color temperature for monitors; every discussion I come across is focused on particular applications besides finding the color for hot objects.

I understand the range of perception for the eye changes depending on ambient light. We're looking at hot objects in space, from 500 to ~7000 K, so let's assume it's equivalent to a moonlit night (lower end of mesopic vision).

I'm guessing what I need is some form of Planck's Law or the Stefan-Boltzmann Law, for visible light only. Then I need to figure out how to convert the power given by those laws into a luminance value for graphics software (ranges between 0% for a non-emitting object and 100% for a glowing white object). The sources are surfaces, not points.

I think I can use the SB Law, I'm just not sure how to get from there to the final brightness value.

Thanks for your help.

Answer 1

You've got your work cut out for you, so this answer can only point you in a direction. This calculation can be difficult because there is a profusion of pieces of terminology with subtly different meanings. Find a book/other source (but for the straight dope I suggest a book) which discusses radiometry and the differences between radiant flux, radiance, spectral radiance, etc., etc.

The overall starting point you are looking for is Planck's law which says that for a blackbody the spectral radiance (energy emitted per meter squared of surface per unit solid angle per unit frequency) is

$B(\nu, T) = \frac{2h\nu^3}{c^2}\frac{1}{\exp{(\frac{h\nu}{k_B T})}-1}$

where $c$ is speed of light, $k_B$ is Boltzmann's constant, $T$ is temperature in Kelvin, $\nu$ is frequency and $h$ is Planck's constant. This is the starting point that you need. This will give you both the hue (because it tells you the relative amounts emitted in each frequency) and the total brightness. But getting the brightness involves knowing your way around radiometric quantities (what to integrate by what to get what you are looking for...) which would take many pages of explanation. So grab a book that covers radiometry. The Stefan-Boltzman law can get you straight to the total brightness, but tells you nothing about hue. S-B law is what you get when you integrate Planck's law over all frequencies and angles.

An additional complication is that for any real object the spectrum deviates from that of a black body. This gets you into emissivity and lots of other fun.