Black bodies, Wien, Boltzmann – very briefly

Screen Shot 9.pngA black body is an idealised body which absorbs and emits all radiation incident upon it. For example, this piece of iron which glows orange-red when heated is an approximate black body. The colour tells us the temperature of the metal. If we continue heating the metal, eventually it will glow orange, then yellow then white. By then, the metal would have melted and boiled off. Theoretically, if we kept on heating it, it’d glow blue-white, eventually emitting UV and even X-rays. Similarly, a star is an approximate black body radiator.  For hotter stars, the maximum wavelength emitted shifts to shorter wavelengths, as shown by the graph.

The black body radiation curves for different temperatures peak at a wavelengths inversely proportional to the temperature.

The plot is valid for determining the temperature of any object which is considerably hotter than its surroundings. Wien’s Displacement Law relating wavelength to temperature is shown on the diagram and the displacement constant shown.

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Our Sun has an external T of about 5300K – the intensity peak is at a wavelength of 550nm.

The thermal energy radiated by a hot body per unit area per second  (power/area) is proportional to the fourth power of temperature – Stefan-Boltzmann’s Law

Screen Shot 5.pngSo, for our Sun, at a temperature of 5300K, every square metre of the surface radiates almost 45MW of power. An object which absorbs all of the energy which falls on it is an ideal absorber or blackbody. For such a body e = 1, where e is the emissivity. Most emitters aren’t black bodies, however, so we can amend Stefan-Boltzmann thus:Screen Shot 8.pngwhere e is a number between 0 and 1. e is zero for a shiny mirror (absorption =0) and 1 for a black body.

By contrast, Albedo (in Latin ‘whiteness’) is the fraction or percentage of incident solar infrared energy  reflected from the Earth back into space and is a measure of how reflective the earth’s surface is. Ice, especially with snow on top of it, has a high albedo – up to 90%: most sunlight hitting the surface bounces back into space, whereas the albedo of a summer forest is only about 0.1 or 10%. Recently, a chunk of ice the size of Scotland fell off the Larsen C ice shelf in Antarctica, reducing its size by 12%. Ultimately, it will either melt or break up. Given that sea water has an albedo of only 6%, the reader might like to speculate about the effect this might have on global warming.

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