How it is calculated
The spectral radiance of a blackbody at temperature \(T\) is given by Planck's law. A gray body scales it by a wavelength-independent emissivity \(\varepsilon\):
The spectral exitance (power per area per wavelength into the hemisphere) of a Lambertian surface is \(\pi B_\lambda\). Integrating over all wavelengths gives the Stefan–Boltzmann law, \(M = \varepsilon\sigma T^4\), with \(\sigma = 5.670374\times10^{-8}\) W m⁻² K⁻⁴.
Where is the peak?
The answer depends on the variable. Per unit wavelength, \(B_\lambda\) peaks at Wien's displacement \(\lambda_\text{max} = b/T\), with \(b = 2.897772\times10^{-3}\) m·K. Per unit frequency (or wavenumber), \(B_\nu\) peaks at \(h\nu_\text{max} = 2.821439\,k_BT\). That corresponds to a wavelength about 1.76× longer than \(\lambda_\text{max}\). For the Sun's 5772 K effective temperature, the peaks are at 502 nm per unit wavelength and 339 THz (884 nm) per unit frequency. Neither one is "the" color of the source.
Band fraction
The fraction of total power emitted between 0 and \(\lambda\) depends only on the product \(\lambda T\). With \(x = hc/\lambda k_BT\), it can be summed exactly:
The in-band fraction is \(F_{0\to\lambda_2} - F_{0\to\lambda_1}\). The photon-number fraction uses the same idea with \(t^2\) in place of \(t^3\), normalized by \(2\zeta(3)\). The total photon exitance is \(M_p = 1.5205\times10^{15}\,\varepsilon T^3\) photons s⁻¹ m⁻² K⁻³.
| λT (µm·K) | F0→λ |
|---|---|
| 1000 | 0.000321 |
| 2000 | 0.06673 |
| 2898 (Wien peak) | 0.2501 |
| 5000 | 0.6337 |
| 10 000 | 0.9142 |
| 50 000 | 0.9989 |
Worked example
Take a 5772 K blackbody. \(M = \sigma T^4 = 6.294\times10^{7}\) W/m², or 6294 W/cm². The spectral radiance at 500 nm is \(2.624\times10^{4}\) W m⁻² sr⁻¹ nm⁻¹. The 200–400 nm band carries 12.0 % of the total power, and the 400–700 nm visible band carries 36.6 %. Raising the temperature to 15 000 K moves \(\lambda_\text{max}\) to 193 nm and puts most of the power in the ultraviolet.
Plasmas, arcs, and lamps
Hot plasmas and arc or discharge lamps are often described by an effective, brightness, or color temperature. In local thermodynamic equilibrium, Kirchhoff's law means no thermal source can exceed the blackbody spectral radiance at its own temperature. An optically thick region approaches that limit, while optically thin regions emit less. Real plasma spectra also contain atomic and ionic lines on top of a continuum, and their emissivity varies with wavelength. Treat the Planck curve as an upper bound and a scaling guide, not a prediction of a specific source.
Assumptions and limits
- Gray body. \(\varepsilon\) is constant with wavelength. Real materials (metals, oxides, glasses) have strongly wavelength-dependent emissivity.
- Lambertian emitter, hemispherical exitance. Radiance is the same in all directions. Exitance and power count emission into the full hemisphere from one side of the area.
- Emission only. Net radiative transfer to cooler surroundings is \(\varepsilon\sigma(T^4 - T_\text{env}^4)\) for a small body in a large enclosure.
- Vacuum wavelengths. In a medium of index \(n\), the radiance scales by \(n^2\) and wavelengths shorten by \(n\).
References
- J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer, 6th ed., CRC Press (2016) — blackbody functions and band fractions.
- E. Tiesinga, P. J. Mohr, D. B. Newell, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018,” Rev. Mod. Phys. 93, 025010 (2021).
- IAU 2015 Resolution B3 on nominal solar and planetary values (nominal solar effective temperature 5772 K).