Radiative transfer equation and optical depth

Specific intensity $I_\nu$ is power per projected area, frequency interval, and solid angle, with SI units $\mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$. A ray crossing an area at angle theta to its normal contributes an additional cosine projection. The spectral flux through that oriented surface is the angular integral of intensity times cosine, with units watts per square metre per hertz. It can be zero for an isotropic radiation field even when photons and energy density are present. Intensity itself is not a signed flux.

Define the mean intensity as the full angular average and the spectral energy density as its angular integral divided by the speed of light. For a uniform outward hemisphere, the flux equals pi times its constant intensity, whereas a uniform full sphere gives zero net flux. These factors differ from four pi multiplying an isotropic volume emissivity. Distance dilution of a source's received flux also does not imply dilution of its resolved specific intensity along an empty ray in flat stationary space. Its apparent angular area changes instead.

Let s increase along a ray toward its endpoint, often the observer. Define $j_\nu$ as energy emitted per volume, time, frequency, and solid angle, with units $\mathrm{W\,m^{-3}\,Hz^{-1}\,sr^{-1}}$. The extinction coefficient alpha nu has inverse-length units; a mass opacity kappa nu instead has square metres per kilogram and satisfies $\alpha_\nu=\rho\kappa_\nu$. Emission adds j times the path increment, while extinction removes alpha times the current intensity times that increment. Extinction may include scattering out of the beam as well as genuine absorption.

Optical depth measured from the incident boundary increases along this same s direction. In pure attenuation the surviving intensity is its boundary value times the exponential of minus optical depth. An optical depth of one therefore leaves e to the minus one of the unscattered original beam, not zero. The inverse extinction coefficient is a local mean free path only when the coefficient can be treated as uniform over that scale. A source function can later add radiation back into the ray, so attenuation alone is not a complete solution for an emitting cloud.

For a homogeneous plane-parallel slab, a ray with direction cosine magnitude mu relative to the slab normal travels a distance equal to the normal thickness divided by mu. Its optical depth is correspondingly the normal optical depth divided by mu. A grazing ray can therefore be thick even when the normal path is comparatively thin. This geometric relation assumes a laterally extended slab; a finite cloud can instead be exited through a side boundary, so arbitrarily long grazing paths should not be inferred without checking its shape.

Stellar-atmosphere notation often defines optical depth inward from the visible surface while the outgoing coordinate points outward. That depth decreases along an escaping ray, reversing the apparent sign in the differential transfer equation. For outward-normal direction cosine mu and source function $S_\nu=j_\nu/\alpha_\nu$, the usual inward-depth form is $\mu\,dI_\nu/d\tau_{\nu,z}=I_\nu-S_\nu$. It is consistent with our propagation-depth form after applying the chain rule. Always state the depth origin, coordinate direction, and incoming boundary condition before comparing formulas from different geometries.