Radiative Processes in Astrophysics

The Black Body

A blackbody is the simplest source: it absorbs and reemits radiation with 100% efficiency. The frequency content of blackbody radiation is given by the Planck Function:

$B_{\nu }={\frac {h\nu }{\lambda ^{2}}}{2 \over (e^{\frac {h\nu }{kT}}-1)}$

$B_{\nu }={\frac {2h\nu ^{3}}{c^{2}(e^{\frac {h\nu }{kT}}-1)}}\neq B_{\lambda }$

Derivation

The # density of photons having frequency between $\nu$ and $\nu +d\nu$ has to equal the # density of phase-space cells in that region, multiplied by the occupation # per cell. Thus:

$n_{\nu }d\nu ={\frac {4\pi \nu ^{2}d\nu }{c^{3}}}{\frac {2}{e^{\frac {h\nu }{kT}}-1}}$

However,

$h\nu {\frac {n_{\nu }c}{4\pi }}=I_{\nu }=B_{\nu }$

so we have it. In the limit that $h\nu \gg kT$:

$B_{\nu }\approx {\frac {2h\nu ^{3}}{c^{2}}}e^{-{\frac {h\nu }{kT}}}$

If $h\nu \ll kT$:

$B_{\nu }\approx {\frac {2kT}{\lambda ^{2}}}$

Note that this tail peaks at $\sim {\tfrac {3kT}{h}}$. Also, $\nu B_{\nu }=\lambda B_{\lambda }$