  Molecular emission arising from the interactions of supernova remnant (SNR)
shock waves and molecular clouds provide a tool for studying the dispersion and
compression that might kick-start star formation as well as understanding
cosmic ray enhancement in SNRs. Purely-rotational CO emission created by
magneto-hydrodynamic shock in the SNR - molecular cloud interaction is an
effective shock tracer, particularly for slow-moving, continuous shocks into
cold inner clumps of the molecular cloud. In this work, we present a new
theoretical radiative transfer framework for predicting the line profile of CO
with the Paris-Durham 1D shock model. We generated line profile predictions for
CO emission produced by slow, magnetized C-shocks into gas of density ~ 100,000
cm-3 with shock speeds of 35 and 50 km s-1. The numerical framework to
reproduce the CO line profile utilizes the Large Velocity Gradient (LVG)
approximation and the omission of optically-thick plane-parallel slabs. With
this framework, we generated predictions for various CO spectroscopic
observations up to J=16 in SNRs W28 and IC443, obtained with SOFIA, IRAM-30m,
APEX, and KPNO. We found that CO line profile prediction offers constraints on
the shock velocity and preshock density independent of the absolute line
brightness and requiring fewer CO lines than diagnostics using an rotational
excitation diagram.
