Chapter 34: Problem 18

Spacecraft instruments measure the radiation from an asteroid, and the data show that the power per unit wavelength peaks at \(40 \mu \mathrm{m} .\) Assuming the asteroid is a blackbody, find its surface temperature.

Short Answer

Expert verified

The surface temperature of the asteroid, assuming it is a blackbody, is approximately 72.425 K or -200.725 °C.

Step by step solution

Identify relevant information

From the exercise, the following information is established for analysis: the wavelength at which power is highest, \( \lambda_{max}\) is given as \(40 \mu m\) or \(40 \times 10^{-6} m\). Secondly, Wien's displacement constant \( b \) is a known physical constant equalling \(2.897 \times 10^{-3} m \cdot K\).

Use Wien's Law

Apply Wien's law which is stated mathematically as \( \lambda_{max} = \frac{b}{T}\). The exercise's objective is to find the temperature \( T \), so the equation must be rearranged to find T: \( T = \frac{b}{\lambda_{max}}\).

Substitute and compute

Finally, substitute the given values for \( \lambda_{max} \) and Wien's displacement constant \( b \) into the equation: \( T = \frac{2.897 \times 10^{-3}}{40 \times 10^{-6}} \). Solve for T to find the blackbody temperature of the asteroid.

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Spacecraft instruments measure the radiation from an asteroid, and the data show that the power per unit wavelength peaks at \(40 \mu \mathrm{m} .\) Assuming the asteroid is a blackbody, find its surface temperature.

Short Answer

Step by step solution

Identify relevant information

Use Wien's Law

Substitute and compute

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