Temperature of the Sun using Wien's law

Problem

Estimate the temperature of the Sun’s surface knowing that its black-body emission peaks at $\lambda_\text{max} \approx 500\;\text{nm}$. (Use Wien’s law: $\lambda_\text{max}\,T \approx 2{,}9\cdot 10^{-3}\;\text{m·K}$.)

Solution

Wien’s displacement law. The emission peak of a black body shifts towards shorter wavelengths as the temperature increases, with $\lambda_\text{max}\,T = b = 2{,}9\cdot 10^{-3}\;\text{m·K}$. Solving for $T$:

$T = \frac{b}{\lambda_\text{max}} = \frac{2{,}9\cdot 10^{-3}\;\text{m·K}}{500\cdot 10^{-9}\;\text{m}}$

$\ev{T \approx 5{,}8\cdot 10^{3}\;\text{K} \approx 5800\;\text{K}}$

Consistent with the Sun’s surface temperature ($\approx 5800\;\text{K}$). It’s no coincidence that the peak falls in the visible range: the human eye evolved precisely where the Sun emits the most.

Links

Topics: Quantum physics Concepts: Black body and Planck’s hypothesis Skills: Fermi estimates Objects: Black body