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\noindent The temperature of the oxygen core of SN~1987A is determined by analyzing the processes that heat and cool the gas and the processes that form and destroy carbon monoxide. The heating and ionization rates are calculated by examining the processes by which the energetic electrons created following the radioactive decay of $^{56}$Co lose energy in a mixed gas of atomic oxygen and carbon and carbon monoxide. The effects of non-uniform energy distribution in the supernova ejecta at early times due to the large optical depths of the $\gamma$-rays of $^{56}$Co are accounted for by reducing the energy input rate in the oxygen core according to the observed emission of CO. There is a large inhomogeneity in the thermal structure of the oxygen core because different cooling mechanisms are operative in different regions with different chemical compositions. The oxygen core contains a cold CO-emitting region which is effectively cooled by the vibrational emission of CO but heated by the O and C atoms as the result of nonthermal excitations by the energetic electrons, and a hot O-emitting region which contains neither CO nor SiO and is cooled by the metastable transitions of O. The temperature of the CO-emitting region is roughly a constant of $1800$~K in the first year and drops to $700$~K at $800$ days, consistent with the observed spectral shapes of CO. The temperature of the O-emitting region is relatively high and ranges from $4800$~K at $100$ days to $2200$~K at $800$ days, in agreement with that inferred from the observations of the [OI]$\lambda \lambda 6300,6364$ emission doublet. The masses of CO predicted by the thermal-chemical model and derived from the observations agree well and permit no microscopic mixing of helium into the oxygen core.
This work was supported by NSF grant AST-89-21939 and by NASA grant NAGW-1561.
