Infrared continuum observations provide a means of investigating the physical composition of the dust in the ejecta and swept up medium of the Cas A supernova remnant. Using low resolution Spitzer IRS spectra (5-35 $\mu$m), and broad-band Herschel PACS imaging (70, 100, and 160 $\mu$m), we identify characteristic dust spectra, associated with ejecta layers that underwent distinct nuclear burning histories. The most luminous spectrum exhibits strong emission features at $\sim9$ and 21 $\mu$m and is closely associated with ejecta knots with strong Ar emission lines. The dust features can be reproduced by magnesium silicate grains with relatively low Mg to Si ratios. Another dust spectrum is associated with ejecta having strong Ne emission lines. It has no indication of any silicate features, and is best fit by Al$_2$O$_3$ dust. A third characteristic dust spectrum shows features that are best matched by magnesium silicates with a relatively high Mg to Si ratio. This dust is primarily associated with the X-ray emitting shocked ejecta, but it is also evident in regions where shocked interstellar or circumstellar material is expected. However, the identification of dust composition is not unique, and each spectrum includes an additional featureless dust component of unknown composition. Colder dust of indeterminate composition is associated with emission from the interior of the SNR, where the reverse shock has not yet swept up and heated the ejecta. Most of the dust mass in Cas A is associated with this unidentified cold component, which is $\lesssim0.1$ $M_{\odot}$. The mass of warmer dust is only $\sim 0.04$ $M_{\odot}$.
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