The detectability of mm-wave molecular rotational transitions
  Elaborating on a formalism that was first expressed some 40 years ago, we
consider the brightness of low-lying mm-wave rotational lines of strongly polar
molecules at the threshold of detectability. We derive a simple expression
relating the brightness to the line of sight integral of the product of the
total gas and molecular number densities and a suitably-defined
temperature-dependent excitation rate into the upper level of the transition.
Detectability of a line is contingent only on the ability of a molecule to
channel enough of the ambient thermal energy into the line and the excitation
can be computed in bulk by summing over rates without solving the multi-level
rate equations or computing optical depths and excitation temperatures. Results
for \hcop, HNC and CS are compared with escape-probability solutions of the
rate equations using closed-form expressions for the expected range of validity
of our {\it ansatz}, with the result that gas number densities as high as $10^4
\pccc$ or optical depths as high as 100 can be accommodated in some cases. For
densities below a well-defined upper bound, the range of validity of the
discussion can be cast as an upper bound on the line brightness which is 0.3 K
for the J=1-0 lines and 0.8 - 1.7 K for the J=2-1 lines of these species. The
discussion casts new light on interpretation of line brightnesses under
conditions of weak excitation, simplifies derivation of physical parameters and
eliminates the need to construct grids of numerical solutions of the rate
equations.
