  Molecules with hyperfine splitting of their rotational line spectra are
useful probes of optical depth, via the relative line strengths of their
hyperfine components.The hyperfine splitting is particularly advantageous in
interpreting the physical conditions of the emitting gas because with a second
rotational transition, both gas density and temperature can be derived. For HCN
however, the relative strengths of the hyperfine lines are anomalous. They
appear in ratios which can vary significantly from source to source, and are
inconsistent with local thermodynamic equilibrium. This is the HCN hyperfine
anomaly, and it prevents the use of simple LTE models of HCN emission to derive
reliable optical depths. In this paper we demonstrate how to model HCN
hyperfine line emission, and derive accurate line ratios, spectral line shapes
and optical depths. We show that by carrying out radiative transfer
calculations over each hyperfine level individually, as opposed to summing them
over each rotational level, the anomalous hyperfine emission emerges naturally.
To do this requires not only accurate radiative rates between hyperfine states,
but also accurate collisional rates. We investigate the effects of different
sets of hyperfine collisional rates, derived via the 'proportional method' and
through direct recoupling calculations. Through an extensive parameter sweep
over typical low mass star forming conditions, we show the HCN line ratios to
be highly variable to optical depth. We also reproduce an observed effect
whereby the red-blue asymmetry of the hyperfine lines (an infall signature)
switches sense within a single rotational transition.
