#### Blackbody Radiation [Eqn]

Like spherical cows, true blackbodies do not exist. Not because “black objects are dark, duh”, as I’ve heard many people mistakenly say — black here simply refers to the property of the object where no wavelength is preferentially absorbed or emitted, and all the energy input to it is converted into radiation. There are many famous astrophysical cases which are very good approximations to perfect blackbodies — the 2.73K microwave background radiation left over from the early Universe, for instance. Even the Sun is a good example. So it is often used to model the emission from various objects.

The blackbody spectrum is

$$B_{\nu}(T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{h \nu / k_B T} – 1} ~~ {\rm [erg~s^{-1}~cm^{-2}~Hz^{-1}~sr^{-1}]} \,,$$

where ν is the frequency in [Hz], *h* is Planck’s constant, *c* is the speed of light in vacuum, and *k _{B}* is Boltzmann’s constant. The spectrum is interesting in many ways. Its shape is characterized by only one parameter, the radiation temperature

*T*. A spectrum with a higher

*T*is greater in intensity at all frequencies compared to one with a lower

*T*, and the integral over all frequencies is

*σ T*, where $$\sigma \equiv \frac{2\pi^5k_B^4}{15 c^2 h^3}$$ is the Stefan-Boltzmann constant. Other than that, the normalization is detached, so to speak, from

^{4}*T*, and differences in source luminosities are entirely attributable to differences in emission surface area.

The general shape of a blackbody spectrum is like a rising parabola at low ν (which led to much hand-wringing in the late 19th century about the Ultraviolet Catastrophe) and an exponential drop at high ν, with a well-defined peak in between. The frequency at which the spectrum peaks is dependent on the temperature, with

$$\nu_{\rm max} = 2.82 \frac{k_B T}{h}$$,

or equivalently,

$$ \lambda_{\rm max} = \frac{2.9\cdot10^4}{T} ~~ {\rm[\AA]} \,,$$

where *T* is in [degK].

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