The surface area and solid angle integration results an expression of luminosity per wavelength Lλ =4π^2R^2Bλ. Integrate Lλ over all wavelength to obtain an expression of bolometric luminosity. Keep the final expression symbolic – do not use any numerical values. Feel free to do the integral using any tools you prefer (octave, matlab etc.), or online integral calculators. E.g. https://www.wolframalpha.com/calculators/integral- calculator/ b) Compare the expression you have in part a) with the Stefan-Boltzmann law, and show how σSB is related to fundamental physics constants such as k, h, c etc.
In class we discussed stars can be approximated as blackbodies, and the blackbody radiation is described by the Planck function Bλ = 2hc^2/λ^5 (e^(hc/λkT) − 1)^−1. We also discussed that the Stefan-Boltzmann law says the (bolometric) luminosity of an object is L = 4π*R^2*σ_SB*T^4, and this comes from integrating the Planck function over surface area, solid angle, and wavelength. In this problem, we will get a better understanding of the connection by carrying out some of the calculations.
a) The surface area and solid angle integration results an expression of luminosity per wavelength Lλ =4π^2R^2Bλ. Integrate Lλ over all wavelength to obtain an expression of bolometric luminosity. Keep the final expression symbolic – do not use any numerical values. Feel free to do the integral using any tools you prefer (octave, matlab etc.), or online integral calculators. E.g. https://www.wolframalpha.com/calculators/integral-
calculator/
b) Compare the expression you have in part a) with the Stefan-Boltzmann law, and show how σSB is related to fundamental physics constants such as k, h, c etc.
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