Question

Problem 6.9: A satellite viewing a surface location under cloud free conditions measures a 12 μm radiance of 6.2 W m-2μm-Isr (a) Compute the brightness temperature TB. (b) Compute the actual temperature, assuming that the atmosphere is completely transpar- ent, and that the surface in question is known to have an emissivity of 0.9 at this wavelength. (c) Is the ratio of the brightness tempera- ture to the actual temperature equal to the emissivity?

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Answer 1

Given value Wave length lambda = 12 mu m = 12 times 10^-6 m Intensity, 1 = 6.2 wm^-2 mu m^-1 sr^-1v = 6.2 times 10^-6 wm^3 sr^-1 (a) The Brightness temperature using planks formula is given h B (T) = 2 hc^2 /lambda^5 1/hc/e lambda kT - 1 e hc/lambda KT - 1 = 2hc^2 /lambda^5 1/B(T) e^hc/lambda KT = 2 hc^2 /lambda^5 1/B(T) + 1 take log on both side ln (e^hc/lambda KT) = ln (2hc^2 /lambda^5 1/B (T) + 1 hc/lambda KT = ln [2 hc^2 /lambda^5 1/B (T) + 1 T = h (c/lambda)/k ln [2 hc^2 /lambda^2 1/B (T) + 1] \r\n = h (c/lambda)/k ln [2 hc^2/lambda^5 1/B(T) + 1] = (6.63 times 10^34 m^2 kg/s) times 3 times 10^8 m/s /12 times 10^-6 m /(1.38 times 10^-23 m^2 kg 5^2 k^-1) ln (2 (6.63 times 10^34 m^2 kg/s) (3 times 10^8 m/s)^2 /(12 times 10^-6m)^5 middot 1/6.2 times 10^6 kg s^-2s^-2 sr^-1 + 1)