| Parameter | Value | Unit |
|---|
Pin multiple temperatures in the Lab tab to overlay their SPDs on the same canvas. Each pin is colour-coded. Use the CSV/JSON export to get tabulated data for all pinned temperatures simultaneously.
Planck’s Radiation Law (1900)
Max Planck derived the spectral radiance of a blackbody in thermal equilibrium, resolving the ultraviolet catastrophe predicted by classical Rayleigh-Jeans theory. The law introduces the quantum of action h = 6.626 070 15 × 10−34 J·s (exact, SI 2019).
Bλ(λ, T) = (2hc²) / (λ&sup5; · (ehc/λkBT − 1))
The first and second radiation constants: c1 = 2πhc² = 3.741 771 852 × 10−16 W·m², c2 = hc/kB = 1.438 776 877 5 × 10−2 m·K.
Reference: M. Planck, Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237 (1900).
Wien Displacement Law (1893)
The wavelength of peak spectral radiance is inversely proportional to the absolute temperature: λmax = b / T, where b = 2.897 771 955 × 10−3 m·K.
Frequency-domain Wien peak: νmax = 2.821… · kBT / h ≈ 5.879 × 1010 Hz·K−1 · T.
Reference: W. Wien, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1893).
Stefan-Boltzmann Law (1879/1884)
Total radiant exitance from a blackbody: M = σT4, where σ = 5.670 374 419 × 10−8 W·m−2·K−4 (exact SI).
Josef Stefan (1879) established the relationship empirically; Ludwig Boltzmann (1884) derived it theoretically from thermodynamics.
Reference: J. Stefan, Sitzungsberichte der kaiserlichen Akademie der Wissenschaften, 79, 391–428 (1879).
CIE 15:2004 — Colorimetry
Defines the CIE 1931 2° Standard Observer colour-matching functions x̄(λ), ȳ(λ), z̄(λ) used to integrate spectral data into XYZ tristimulus values.
This tool uses the Wyman et al. (2013) piecewise Gaussian analytical approximation for the CIE 1931 CMFs, valid from 380 to 780 nm with < 0.06% RMS error.
Reference: CIE, Colorimetry, 3rd Edition, CIE 15:2004; C. Wyman et al., JCGT 2(2), 2013.
IEC 61966-2-1 — sRGB Colour Space
XYZ tristimulus values are converted to sRGB using the 3×3 matrix from IEC 61966-2-1:1999. The perceived colour chip is normalised for maximum brightness to represent the chromaticity rather than absolute luminance.
[R] [ 3.2406 -1.5372 -0.4986] [X]
[G] = [-0.9689 1.8758 0.0415] [Y]
[B] [ 0.0557 -0.2040 1.0570] [Z]
Reference: IEC 61966-2-1:1999, Multimedia systems and equipment — Colour measurement and management — Part 2-1: Default RGB colour space — sRGB.
ISO 11664-2 — CIE Standard Illuminants
Temperature presets in this tool correspond to standard CIE illuminants: Illuminant A (2856 K, Planckian tungsten), D50 (5003 K), D55 (5503 K), D65 (6504 K — sRGB reference white). The Sun photosphere temperature is 5778 K.
Reference: ISO 11664-2:2007 / CIE S 014-2:2006.
SI 2019 Physical Constants
All constants use the 2019 SI redefinition exact values:
| Constant | Symbol | Value | Unit |
|---|---|---|---|
| Planck | h | 6.626 070 15 × 10−34 | J·s |
| Speed of light | c | 2.997 924 58 × 108 | m/s |
| Boltzmann | kB | 1.380 649 × 10−23 | J/K |
| Stefan-Boltzmann | σ | 5.670 374 419 × 10−8 | W·m−2·K−4 |
| Wien displacement | b | 2.897 771 955 × 10−3 | m·K |
| Photopic efficacy | Km | 683 | lm/W |
Reference: BIPM, The International System of Units (SI), 9th Edition (2019).
Planck’s Law Bλ(λ, T)
Spectral radiance (W·m−2·sr−1·m−1):
where c1 = 2πhc² = 3.741 771 852 × 10−16 W·m², c2 = hc/kB = 1.438 776 877 5 × 10−2 m·K.
Implementation: spdBlackbody(wl, T) — input in nm,
internally converted to metres.
Wien Displacement Law
νmax = 2.821 439 372 1 · kBT / h
For wavelength-domain: b = 2.897 771 955 × 10−3 m·K.
Example: Sun (5778 K) → λmax ≈ 501 nm (green).
Stefan-Boltzmann Total Radiant Exitance
σ = 5.670 374 419 × 10−8 W·m−2·K−4.
Example: Sun (5778 K) → M ≈ 6.32 × 107 W/m².
CIE 1931 2° Observer — Piecewise Gaussian
Wyman et al. (2013) analytical approximation using sums of piecewise Gaussians:
where d = λ − μ, σk = σ1 if d < 0 else σ2
x̄(λ) = 1.056 · g(599.8, 37.9, 31.0) + 0.362 · g(442.0, 16.0, 26.7) − 0.065 · g(501.1, 20.4, 26.2)
ȳ(λ) = 0.821 · g(568.8, 46.9, 40.5) + 0.286 · g(530.9, 16.3, 31.1)
z̄(λ) = 1.217 · g(437.0, 11.8, 36.0) + 0.681 · g(459.0, 26.0, 13.8)
Source: C. Wyman, P.-P. Sloan, P. Shirley, JCGT 2(2), 2013.
Spectrum → XYZ Integration
Y = ∑ S(λ) · ȳ(λ) · Δλ
Z = ∑ S(λ) · z̄(λ) · Δλ
Normalised: X /= k, Y /= k, Z /= k
where k = ∑ ȳ(λ) · Δλ
Integration from 380 to 780 nm in 5 nm steps (81 points).
XYZ → sRGB Conversion
[Glin] = [-0.9689 1.8758 0.0415] [Y]
[Blin] [ 0.0557 -0.2040 1.0570] [Z]
Gamma: CsRGB = Clin1/2.2 (simplified)
Brightness normalised: divide by max(R,G,B) before gamma
Chromaticity display: colours are normalised for maximum brightness to show hue/saturation independent of absolute luminance.
Correlated Colour Temperature (CCT)
For a Planckian radiator, the CCT equals the physical temperature. This tool verifies by computing the CIE 1931 (x, y) chromaticity coordinates and confirming they lie on the Planckian locus.
For non-Planckian SPDs, Ohno (2014) or Robertson (1968) methods would be used to find the nearest Planckian temperature. Since our source is inherently Planckian, CCT = T by definition.
Reference: Y. Ohno, Opt. Express 22(16):19457 (2014); A.R. Robertson, JOSA 58:1528 (1968).
Luminous Efficacy of Radiation
Km = 683 lm/W (maximum photopic luminous efficacy at 555 nm). V(λ) is the CIE photopic luminosity function, approximated here by ȳ(λ) normalised to unity at peak.
Example: Sun (5778 K) → K ≈ 93 lm/W; Tungsten (3200 K) → K ≈ 15 lm/W.
- M. Planck, “Über das Gesetz der Energieverteilung im Normalspectrum,” Annalen der Physik, 309(3):553–563, 1901. Original derivation of the Planck radiation law.
- W. Wien, “Über die Energievertheilung im Emissionsspectrum eines schwarzen Körpers,” Annalen der Physik, 294(8):662–669, 1896. Wien displacement law and Wien approximation.
- J. Stefan, “Über die Beziehung zwischen der Wärmestrahlung und der Temperatur,” Sitzungsberichte der kaiserlichen Akademie der Wissenschaften, 79:391–428, 1879. Empirical T&sup4; law.
- L. Boltzmann, “Ableitung des Stefan’schen Gesetzes,” Annalen der Physik, 258(6):291–294, 1884. Thermodynamic derivation of the Stefan-Boltzmann law.
- CIE, Colorimetry, 3rd Edition, CIE 15:2004. CIE 1931 2° Standard Observer colour-matching functions.
- C. Wyman, P.-P. Sloan, P. Shirley, “Simple Analytic Approximations to the CIE XYZ Color Matching Functions,” Journal of Computer Graphics Techniques, 2(2):1–11, 2013. Piecewise Gaussian approximation used in this tool.
- IEC, Multimedia systems and equipment — Colour measurement and management — Part 2-1: Default RGB colour space — sRGB, IEC 61966-2-1:1999. XYZ-to-sRGB conversion matrix and transfer function.
- Y. Ohno, “Practical Use and Calculation of CCT and Duv,” Optics Express, 22(16):19457–19472, 2014. Modern CCT estimation algorithm.
- A. R. Robertson, “Computation of Correlated Color Temperature and Distribution Temperature,” Journal of the Optical Society of America, 58(11):1528–1535, 1968. Classic CCT computation by isothermal line method.
- BIPM, The International System of Units (SI), 9th Edition, 2019. Exact values of h, c, kB used in all computations.
- ISO 11664-2:2007 / CIE S 014-2:2006, CIE Standard Illuminants for Colorimetry. Definitions of standard illuminants A, D50, D55, D65, D75.
- R. W. G. Hunt, M. R. Pointer, Measuring Colour, 4th Edition, Wiley, 2011. Comprehensive reference on colorimetric measurement including blackbody radiator chromaticity.
Automated survey from 1000 K to 40 000 K in 500 K steps. Each row computes: Wien peak λmax, Stefan-Boltzmann M, CIE 1931 (x, y) chromaticity, sRGB hex, luminous efficacy K (lm/W).
| T (K) | λmax (nm) | M (W/m²) | x | y | Hex | K (lm/W) | Colour |
|---|