  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.
