Every number here is computed from real atomic data (Trident line list, plus Morton 2003 and Verner et al. 1996 where Trident has no entry) and the same model as the paper figures: Doppler profiles, a Gaussian line-spread function, and matched-filter statistics with S/N quoted per resolution element.
A cloud imprints an optical depth τ(v) on the background light. The spectrograph blurs the absorbed flux with its line-spread function (LSF) of width c/R, then samples it with noise. Play the sequence, or step through it.
Two components separated by Δv blend when Δv is comparable to the effective width, FWHMeff = [(c/R)² + (1.665 b)²]1/2. The dip between them vanishes below the Sparrow separation ΔvS = 0.85 FWHMeff. A fit can still tell two from one if the noise is low enough. The orange dashed curve is the best single-component impostor, refit live; the lower strip shows what it cannot reproduce, against the noise per pixel.
Left: the weakest single absorber detectable at 3σ, for the current b. Right: the smallest separation at which two equal components (current log N, fiducial b) are resolved at 3σ. Click a map to move the spectrograph there; the ring marks the current setting and the dotted lines mark existing instruments.
For optically thin components the problem collapses to a single curve. Measure the separation in units of FWHMeff and the noise through Sdet, the significance with which the whole blend is detected. Equal pairs follow Δvmin ≈ 1.98 FWHMeff Sdet−1/4, so halving the separation costs 16× in S/N. The dot is your current setting.
Each cell gives the minimum detectable log N for one component at the fiducial b, then (smaller) the minimum resolvable Δv in km/s for two equal components each at τ0 = 1. Click a row to switch ion.
Model: Doppler cores (damping wings neglected), Gaussian LSF, constant continuum noise, 3 pixels per resolution element for the noise illustration. Fiducial b combines the thermal width at a representative temperature with 5 km/s of turbulence. Significances are expected (median) values. Source and paper-quality figures: github.com/tanmay-starslab/cgm-absorber-requirements. Companion page: mock quasar sightlines through a TNG50 galaxy (O VI, H I Lyβ, C III), degraded live to any resolution and S/N.