Filament Gas Density Profiles: Central Densities, Electron Columns and Line Masses for a Stated Intercluster Filament

Intercluster filaments are the densest part of the intergalactic medium outside halos, yet their gas is diffuse enough that every observable derived from them — dispersion, absorption, scattering of background radiation — is a line-of-sight integral over a density profile that is rarely written down explicitly. This paper adopts one stated profile, a cylindrical form falling as one over one plus the squared radius in units of a 0.4 Mpc core radius, with central gas overdensity 30 against the cosmic mean, and integrates it exactly. The central hydrogen density is 5.73e-06 per cubic centimetre and the central electron density 6.64e-06 per cubic centimetre. A sightline through the centre accumulates an electron column of 2.25e+19 per square centimetre, equivalent to a dispersion measure of 7.3 pc per cubic centimetre from a single filament. The gas mass per unit length is 3.05e+11 solar masses per megaparsec, and the mean overdensity inside a one-megaparsec radius is 9.51, a factor of three below the central value, which quantifies how strongly aperture-averaged measurements dilute the peak conditions. All numbers are computed from the stated profile and a standard cosmic baryon density; none is fitted.