Lyman-alpha absorption ($1s+\gamma \to 2p$)

A neutral hydrogen atom absorbs a photon near the Lyman-alpha resonance by excitation from $1s$ to $2p$. Absorption lines in a quasar spectrum can therefore locate hydrogen along the sightline by their redshift, subject to peculiar-velocity and line-profile effects.

For a population with comoving number density $n_c$ and proper interception area ${\sigma }_p$, the absorption incidence is the proper density multiplied by cross-section and proper light-path length $cdz/[H(z)(1+z)]$. The displayed relation assumes one counted absorption system per interception. Completeness, covering fraction, evolution and the distinction between observed and rest-frame wavelength separation must be stated in an inference from a spectrum.

A circular, opaque, geometrically thin disk has projected area $\pi R^2|\cos i|$. For isotropically oriented normals, $|\cos i|$ is uniform on $[0,1]$, so the mean interception area is half the face-on area. A nonunit covering fraction multiplies this cross-section. Finite thickness, orientation-dependent absorption thresholds and incomplete coverage can change it; this is an average over randomly placed disks, not the inclination distribution conditional on an observed interception.