Blackbody Radiance (Planck's Law)
Spectral radiance, photon flux, Wien peak, and total exitance of a greybody target — the front end of every EO-IR link budget.
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The engineering
Planck's law gives the spectral radiance a thermal source pours into a steradian at each wavelength — the starting point for any EO-IR detection-range estimate, NETD budget, or radiometric calibration against a cavity blackbody. The emissivity field turns the ideal blackbody into a greybody: paint and oxidized metal run 0.85–0.95, polished aluminum can be under 0.1, which is why bare-metal airframes look cold to an LWIR seeker. The photon radiance row is what photon-counting detector models (MCT, InSb) actually want.
Sanity checks worth memorizing: a 300 K scene peaks near 9.7 µm — squarely in the LWIR band — and puts out about 10 W/(m²·sr·µm) there; hot tailpipes around 500–700 K shift the peak into MWIR. Watch the units: EO-IR datasheets mix W/m² and W/cm² bases freely (a factor of 10⁴), and this card reports per-µm spectral quantities — integrate across your band, don't multiply by the band edges' radiance blindly if the curve is steep through the band.
The total-exitance row is Stefan-Boltzmann (ε·σ·T⁴), integrated over all wavelengths and the full hemisphere — a quick upper bound on what any in-band calculation can return.
Where this math comes from
The blackbody problem was industrial before it was quantum. In the 1890s the German Physikalisch-Technische Reichsanstalt needed an absolute standard to rate electric lamps against gas lighting, so Lummer, Pringsheim, Rubens, and Kurlbaum built precision cavity radiators and measured their spectra to unprecedented accuracy — and the data flatly refused to fit Wien's 1896 formula at long wavelengths.
Rubens showed Max Planck the discrepancy over Sunday tea in October 1900; Planck produced an interpolating formula that same evening and, weeks later, justified it by quantizing the cavity oscillators' energy in units of hν — a step he later called 'an act of desperation.' He presented it to the German Physical Society on 14 December 1900, the accepted birthday of quantum theory. The constants h and k in this card were fixed to exact values by the 2019 SI redefinition, so Planck's law is now exact by definition.
- 1859Gustav KirchhoffDefines the blackbody and proves its spectrum is a universal function of temperature alone.
- 1879Josef StefanFinds total emission scales as T⁴; Boltzmann derives it thermodynamically in 1884.
- 1893Wilhelm WienDisplacement law — λ_peak·T is constant, the 2898 µm·K rule.
- 1900Max PlanckPresents the quantized radiation law that fits Rubens and Kurlbaum's cavity data at all wavelengths.
- 2019BIPM / SI redefinitionFixes h and k to exact values, making Planck's law exact in SI units.
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