Spectral radiance
= Spectral radiance
{title2=$L_\nu,\ L_\lambda$}
{wiki}
= Specific intensity
{synonym}
<Spectral radiance> is <radiance> per unit <frequency> or <wavelength> interval. Its astronomical synonym <specific intensity> is usually written $I_\nu=L_\nu$ or $I_\lambda=L_\lambda$; this $I_\nu$ is a spectral <radiance>, distinct from the <radiant intensity> $I$ in <radiometry>. For positive small intervals,
$$
d^3\Phi=L_\nu\cos\theta\,dA\,d\Omega\,d\nu
=L_\lambda\cos\theta\,dA\,d\Omega\,d\lambda.
$$
The two expressions refer to the same spectral slice, with the positive widths related through <frequency> $\nu$ and <wavelength> $\lambda$. For <frequency> measured in hertz, $\mathrm{Hz}=\mathrm{s^{-1}}$, the units are $[L_\nu]=\mathrm{W\,m^{-2}\,sr^{-1}\,Hz^{-1}}$. For <wavelength> measured in metres, $[L_\lambda]=\mathrm{W\,m^{-2}\,sr^{-1}\,m^{-1}}$. Integrating over a spectral band gives its <radiance>, and then multiplying by the collecting <projected area> and accepted <solid angle> gives a small uniform beam's <radiant flux>. A spectral measurement therefore retains information that the band-integrated <power> alone loses.
For an ideal <blackbody>, <Planck's law> gives $L_\nu=B_\nu(T)$ and $L_\lambda=B_\lambda(T)$ at <temperature> $T$, independent of outgoing direction. Its existing <frequency and wavelength peaks of Planck's law> illustrate why the coordinate matters. The simpler flat-band calculation below isolates that same <change of variables> without the thermal physics.