A large research solar furnace focuses sunlight with a mirror array whose focusing…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Geometrical optics and imaging

Optics and Photonics, Ch Radiometry and photometry

Problem

A large research solar furnace focuses sunlight with a mirror array whose focusing cone has effective f-number f/0.85f/0.85 (that is, the marginal ray in the image-side cone satisfies sin⁡θ′=1/(2×0.85)\sin\theta' = 1/(2\times 0.85)). The solar constant at the top of the atmosphere is 1361 W m−21361\ \text{W m}^{-2} and the Sun's angular half-radius as seen from Earth is 4.65×10−3 rad4.65\times10^{-3}\ \text{rad}. The Sun's surface temperature is a measured value, about 5800 K5800\ \text{K}; the connection between temperature and radiance is not derived in this course. ((a)) Find the Sun's radiance LsunL_{\rm sun}. ((b)) Find the irradiance delivered at the furnace's focal image, and explain, using conservation of radiance, why no target placed there can ever be raised above 5800 K5800\ \text{K}, however large the mirror array is made. ((c)) Given that sunlight has a luminous efficacy of about 93 lm W−193\ \text{lm W}^{-1} (so that a radiance in W m−2sr−1\text{W m}^{-2}\text{sr}^{-1} converts directly to a luminance in cd m−2\text{cd m}^{-2} by multiplying by this factor), find the Sun's luminance.

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