Optical depth (${\tau }_{\nu }$)

Optical depth satisfies $d{\tau }_{\nu }={\alpha }_{\nu }ds$ along a path and measures attenuation through $I_{\nu }\propto e^{-{\tau }_{\nu }}$ in the absence of emission.

An optically thick medium has optical depth $\tau \gg 1$ along the relevant path. Photons undergo many interactions before escaping. Radiative diffusion applies under suitable near-isotropy conditions, but a large scattering depth alone does not guarantee thermalization when true absorption is weak.

An optically thin medium has $\tau \ll 1$, so attenuation and emission may be expanded to first order in optical depth.

For isotropic emission without significant absorption, integrating the spontaneous photon-production rate along the sightline gives the displayed energy intensity per solid angle. Photon intensity omits the factor $h\nu$. Summing unresolved transitions is necessary before comparing with an observed blended spectral feature.

The absorption coefficient is the fractional reduction of beam intensity per unit path length: $d{I}_{\nu }/ds=-{\alpha }_{\nu }{I}_{\nu }$ when emission and scattering are absent.

A collision or transient molecular pair can possess a dipole even when the isolated molecules do not. The resulting opacity can be important in a dense molecular hydrogen atmosphere. Pair absorption per unit length scales approximately as the product of the two partner number densities, so its mass absorption coefficient is generally density-dependent. It supplies continuum absorption between ordinary spectral lines.

For dust-grain number density $n_d$ and absorption cross-section ${\sigma }_d$, the dust optical depth along a path is ${\tau }_d=\int n_d{\sigma }_{d}ds$.

The mass absorption coefficient is absorbing cross-section per unit mass. For identical spherical dust grains of radius $a$, mass $m_d$, and absorption efficiency $Q_{\nu }$, it is ${\kappa }_{\nu }=\pi a^2{Q}_{\nu }/m_d$.

For isothermal optically thin dust at distance $D$, its measured spectral flux density determines its mass through $M_d=F_{\nu }{D}^2/[{\kappa }_{\nu }{B}_{\nu }(T)]$. Temperature and opacity uncertainty commonly dominate the inferred mass uncertainty.