The Critical Density and the Effective Excitation Density of Commonly
  Observed Molecular Dense Gas Tracers
  The optically thin critical densities and the effective excitation densities
to produce a 1 K km/s (or 0.818 Jy km/s $(\frac{\nu_{jk}}{100 \rm{GHz}})^2 \,
(\frac{\theta_{beam}}{10^{\prime\prime}})^2$) spectral line are tabulated for
12 commonly observed dense gas molecular tracers. The dependence of the
critical density and effective excitation density on physical assumptions (i.e.
gas kinetic temperature and molecular column density) is analyzed. Critical
densities for commonly observed dense gas transitions in molecular clouds (i.e.
HCN $1-0$, HCO$^+$ $1-0$, N$_2$H$^+$ $1-0$) are typically $1 - 2$ orders of
magnitude larger than effective excitation densities because the standard
definitions of critical density do not account for radiative trapping and 1 K
km/s lines are typically produced when radiative rates out of the upper energy
level of the transition are faster than collisional depopulation. The use of
effective excitation density has a distinct advantage over the use of critical
density in characterizing the differences in density traced by species such as
NH$_3$, HCO$^+$, N$_2$H$^+$, and HCN as well as their isotpologues; but, the
effective excitation density has the disadvantage that it is undefined for
transitions when $E_u/k \gg T_k$, for low molecular column densities, and for
heavy molecules with complex spectra (i.e. CH$_3$CHO).
