Modeling CO Line Profiles in Shocks of W28 and IC44
  Molecular emission arising from the interactions of supernova remnants (SNR)
shock waves and molecular clouds provide a tool for studying the dispersion and
compression that might kick-starts 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 of $\sim 10^4$ cm$^{-3}$ and
$35 \leq$ V$_{\rm Shock}$ $\leq 50$ 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 robust
constraints of shock velocity and preshock density while requiring fewer CO
lines compared to diagnostics using a rotational energy level diagram.
