Class I Methanol (CH$_{3}$OH) Maser Conditions near Supernova Remnants

We present results from calculations of the physical conditions necessary for the occurrence of 36.169 ($4_{-1}-3_{0}\, E$), 44.070 ($7_{0}-6_{1}\,A^+$), 84.521 ($5_{-1}-4_{0}\,E$), and 95.169 ($8_{0}-7_{1}\,A^+$) GHz methanol (CH$_3$OH) maser emission lines near supernova remnants (SNRs), using the MOLPOP-CEP program. The calculations show that given a sufficient methanol abundance, methanol maser emission arises over a wide range of densities and temperatures, with optimal conditions at $n\sim 10^4-10^6$ cm$^{-3}$ and $T>60$ K. The 36~GHz and 44~GHz transitions display more significant maser optical depths compared to the 84~GHz and 95~GHz transitions over the majority of physical conditions. It is also shown that line ratios are an important and applicable probe of the gas conditions. The line ratio changes are largely a result of the $E$-type transitions becoming quenched faster at increasing densities. The modeling results are discussed using recent observations of CH$_3$OH and hydroxyl (OH) masers near the supernova remnants G1.4$-$0.1, W28, and Sgr A East.