The properties of the dust in the cold and hot gas phases of early-type galaxies (ETGs) are key to understand ETG evolution. We thus conducted a systematic study of the dust in a large sample of local ETGs, focusing on relations between the dust and the molecular, atomic, and X-ray gas of the galaxies, as well as their environment. We estimated the dust temperatures and masses of the 260 ETGs from the ATLAS3D survey, using fits to their spectral energy distributions primarily constructed from AKARI measurements. We also used literature measurements of the cold (CO and HI) and X-ray gas phases. Our ETGs show no correlation between their dust and stellar masses, suggesting inefficient dust production by stars and/or dust destruction in X-ray gas. The global dust-to-gas mass ratios of ETGs are generally lower than those of late-type galaxies, likely due to dust-poor HI envelopes in ETGs. They are also higher in Virgo Cluster ETGs than in group and field ETGs, but the same ratios measured in the central parts of the galaxies only are independent of galaxy environment. Slow-rotating ETGs have systematically lower dust masses than fast-rotating ETGs. The dust masses and X-ray luminosities are correlated in fast-rotating ETGs, whose star formation rates are also correlated with the X-ray luminosities. The correlation between dust and X-rays in fast-rotating ETGs appears to be caused by residual star formation, while slow-rotating ETGs are likely well evolved, and thus exhausting their dust. These results appear consistent with the postulated evolution of ETGs, whereby fast-rotating ETGs form by mergers of late-type galaxies and associated bulge growth, while slow-rotating ETGs form by (dry) mergers of fast-rotating ETGs. Central cold dense gas appears to be resilient against ram pressure stripping, suggesting that Virgo Cluster ETGs may not suffer strong related star formation suppression.
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