  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. 
Modeling CO Line Profiles in Shocks of W28 and IC44