The star formation properties of early-type galaxies (ETGs) are currently the subject of considerable interest, particularly whether they differ from those of gas-rich spirals. We perform a systematic study of star formation in a large sample of local ETGs using polycyclic aromatic hydrocarbon (PAH) and dust emission, focusing on the galaxies star formation rates (SFRs) and star formation efficiencies (SFEs). Our sample is composed of the 260 ETGs from the ATLAS3D survey, from which we use the cold gas measurements (HI and CO). The SFRs are estimated from stellar, PAH and dust fits to spectral energy distributions created from new AKARI measurements and literature data from WISE and 2MASS. The mid-infrared luminosities of non-CO-detected galaxies are well correlated with their stellar luminosities, showing that they trace (circum)stellar dust emission. CO-detected galaxies show an excess above these correlations, uncorrelated with their stellar luminosities, indicating that they likely contain PAHs and dust of interstellar origin. PAH and dust luminosities of CO-detected galaxies show tight correlations with their molecular gas masses, and the derived current SFRs are typically 0.01-1 Msun/yr. These SFRs systematically decrease with stellar age at fixed stellar mass, while they correlate nearly linearly with stellar mass at fixed age. The majority of local ETGs follow the same star-formation law as local star-forming galaxies, and their current SFEs do not depend on either stellar mass or age. Our results clearly indicate that molecular gas is fueling current star formation in local ETGs, that appear to acquire this gas via mechanisms regulated primarily by stellar mass. The current SFEs of local ETGs are similar to those of local star-forming galaxies, indicating that their low SFRs are likely due to smaller cold gas fractions rather than a suppression of star formation.
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