Blackbody Spectrum Solar Vs Planetary Calculator

A blackbody is an idealized surface that absorbs every photon incident on it and re-emits a spectrum fixed entirely by its temperature. The Planck function B(lambda,T) gives the spectral radiance, and its extremum and its integral supply the two working rules used here: the Wien displacement law places the peak at lambda_max = b / T with b = 2.897772e-3 m K, and the Stefan-Boltzmann law gives the wavelength-integrated flux as sigma times T to the fourth, with sigma = 5.670374e-8 W per square meter per K to the fourth. For a sphere the total output is L = 4 pi R squared sigma T to the fourth, which returns about 3.83e26 W for the Sun at an effective 5772 K and roughly 1.2e17 W for Earth at 255 K. The practical payoff is the near-complete separation of the two curves. A 5772 K photosphere peaks near 0.50 micrometers in the green, while a 255 K planet peaks near 11.4 micrometers, so below about 4 micrometers a satellite sees reflected sunlight and above it sees thermal emission. That split defines the shortwave and longwave halves of every planetary energy budget and the 8 to 13 micrometer atmospheric window. Real bodies deviate from the ideal. The solar spectrum carries Fraunhofer absorption lines, limb darkening, and a chromospheric ultraviolet excess, so no single temperature fits it at all wavelengths; planetary spectra are cut by the 15 micrometer carbon dioxide band and the 9.6 micrometer ozone band, and surface emissivities of 0.90 to 0.99 depress emission below the Planck curve. A brightness temperature retrieved at one wavelength therefore rarely equals the physical temperature.

Gustav Kirchhoff introduced the concept at Heidelberg in 1859 and 1860, proving that at a given temperature and wavelength the ratio of emissive to absorptive power is the same for all bodies, and coining the term blackbody for the perfect absorber. Josef Stefan inferred the fourth-power law empirically in 1879, working in part from John Tyndall measurements of glowing platinum wire, and his student Ludwig Boltzmann derived it thermodynamically in 1884 by treating radiation pressure in a Carnot cycle. Wilhelm Wien published the displacement law in 1893 and received the 1911 Nobel Prize in Physics for his work on the laws governing heat radiation. Fitting the whole curve proved much harder. The Wien distribution of 1896 failed at long wavelengths while the Rayleigh-Jeans limit diverged at short ones, a failure later nicknamed the ultraviolet catastrophe. Max Planck presented the correct interpolation to the German Physical Society on 14 December 1900, and could justify it only by assuming that oscillator energy comes in discrete units of h times frequency; he received the 1918 Nobel Prize, and that date is conventionally taken as the birth of quantum theory. The constants have since been frozen by definition: the 2019 revision of the SI fixed the Planck constant at exactly 6.62607015e-34 J s and the Boltzmann constant at exactly 1.380649e-23 J per K, which makes the Stefan-Boltzmann and Wien constants exact derived quantities rather than measured ones.