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).