Wien’s Law Spectrum

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High-frequency Limit of a Blackbody: Wien’s Law Spectrum

In the high-energy, high-frequency, short wavelength limit, ($hv\gg kT$), we get the Wien’s law spectrum (or the Wien tail) from the formula for the blackbody spectrum. It can be expressed in several ways:

\[
\begin{align*}
B_{\nu} & \sim \frac{2h\nu^{3} kT}{c^{2}} e^{-h\nu/kT} \ \ \ {\rm erg \ cm^{-2} \ s^{-1} \ Hz^{-1} \ steradian^{-1}} \\
B_{\lambda} & \sim \frac{2c kT}{\lambda^{4}} e^{-hc/\lambda kT} \ \ \ {\rm erg \ cm^{-2} \ s^{-1} \ cm^{-1} \ steradian^{-1}} \\
B_{E} & \sim \frac{2E^{2} kT}{c^{2} h^{3}} e^{-E/kT} \ \ \ {\rm erg \ cm^{-2} \ s^{-1} \ erg^{-1} \ steradian^{-1}}
\end{align*}
\]

where $\nu$, $\lambda$, and $E$ are frequency, wavelength, and energy respectively. $T$ is the temperature in Kelvin. Example units are given to make clear the physical quantity that $B$ respresents.

In order to get formulas for the energy density (energy per unit volume per unit steradian), multiply any of the above formulas by $1/c$. Then multiply by $4\pi
$ if you need the total energy density of isotropic radiation, integrated over solid angle.