Black-Body Radiation

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\section*{Blackbody Radiation}

A blackbody is the simplest source: it absorbs and re-emits radiation with 100\% efficiency. The frequency content of blackbody radiation is given by the {\it Planck Function}: \begin{equation} B_\nu={2h\nu^3\over c^2(e^{\frac{h\nu}{kT}}-1)} \end{equation}

Since radio photons typically have very low energies compared with the kinetic energy of most emitters, so that $h\nu\ll kT$, it is usually safe in radio astronomy to take the {\it Raleigh-Jeans} approximation to the blackbody spectrum: \begin{equation} B_\nu\approx\frac{2kT}{\lambda^2} \end{equation} Even though we've written $\lambda$ in place of $c/\nu$ above, remember that we are talking about $B_\nu$, which is a per-bandwidth (i.e. per Hz) quantity.

The Raleigh-Jeans approximation is so common in radio astronomy that it is used to define a {\it brightness temperature}, $T_b$, given by: \begin{equation} \frac{2kT_b}{\lambda^2}\equiv I_\nu. \end{equation} The brightness temperature is the temperature a blackbody would have to be at to produce match the intensity $I_\nu$ at the frequency in question. Note that this is not saying that $I_\nu$ is a blackbody spectrum, or is at all thermal in nature. It's just matching the emission at a particular frequency to an equivalent temperature. Brightness temperatures are generally a function of frequency for non-thermal sources.

\subsection*{ Kirchoff's Law}

The CMB is the perfect blackbody because in the past, the surface of the black body (the edge of the universe) was not allowed to leak out energy. This begs the question: does it matter what the boundary of the black body is made of? The answer is: No. In the short term, the boundary may imperfectly reflect a photon, but then the resulting change in the temperature of boundary dictates that it will tend to reemit the energy into the photon gas. Overall: $$I_\nu I_{emitted}=I_\nu I_{absorbed}=B_\nu f_{abs}$$ However, $f_{emission}B_\nu\equiv I_\nu I_{emitted}$, so: $$\boxed{f_{emission}=f_{abs}}$$ \centerline{(Kirchoff's Law)}\par Note that Kirchoff's Law is true regardless of equilibrium.

\section*{Blackbody Sources}

There are not many examples of true blackbody sources of radiation, but what few there are tend to be quite important.

\subsection*{Cosmic Microwave Background}

The mother of all blackbodies, this is the relic 2.7K radiation left over from the hot plasma that was generated by the Big Bang. Photons were produced and absorbed via the thermal radiation of charged particles (mostly electrons) scattering off one another, converging quickly on a blackbody spectrum. Once the universe cooled to the point that protons and electrons could bind to form neutral hydrogen without being immediately ionized, these photons stopped scattering, and have been free-streaming ever since. As the universe expands, the wavelength of these photons stretch with it, causing photon energies (and hence, the characteristic temperature) to gradually decline with time. Note that, at the current temperature of 2.7K, the CMB peaks at 160.2 GHz.

\subsection*{Sun}

Stars are generally blackbody-like, although most of them are so faint in the radio band as to be nearly unobservable. Just by sheer proximity, our Sun is the brightest radio source in the sky, but, particularly at lower frequencies, not by a huge margin. Emission from the quiet Sun is dominated at radio frequencies by the photosphere (6000K) around 100 GHz, the chromosphere (10,000K) at 1 GHz, and the corona (1,000,000K) at 100 MHz. As a result, the spectrum of the quiet Sun within a relatively narrow band is blackbody-like, but broadly departs from the characteristic blackbody spectrum.

It should also be noted that during sunspot activity, which varies on an 11-year solar cycle, solar emission departs dramatically from that of a blackbody, and becomes dominated by intense emission from electrons trapped in the magnetic fields around sunspots. The perturbed sun can be orders of magnitude brighter than the quiet Sun, and with emission dominated by particular ``hot spots that rotate with the Sun.


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