Spectral luminosity ($L_{\nu }$)

Spectral luminosity is energy radiated per unit time per unit frequency. Its frequency integral is the total luminosity. For an isotropic source at small distance $r$, neglecting attenuation and cosmological effects, the spectral flux density is $L_{\nu }/(4\pi r^2)$.

For photon energy $\epsilon =h\nu$, equating radiated energy in corresponding spectral intervals gives $L_{\nu }d\nu =\epsilon L_{\epsilon }^{N}d\epsilon$. Thus $L_{\epsilon }^N=L_{\nu }/(h\epsilon )$. A quoted rate per joule must be multiplied by the number of joules per electronvolt or per $\mathrm{keV}$ to express it in that spectral unit. For homogeneous optically thin emission, $L_{\nu }=4\pi V{j}_{\nu }$ if $j_{\nu }$ is defined per solid angle.