AGB stars in the Magellanic Clouds III. The rate of star formation across the Small Magellanic Cloud

Aims. This article compares the K-s magnitude distribution of Small Magellanic Cloud asymptotic giant branch stars obtained from the DENIS and 2MASS data with theoretical distributions. Methods. Theoretical K-s magnitude distributions have been constructed using up-to-date stellar evolution calculations for low- and intermediate-mass stars, and in particular for thermally pulsing asymptotic giant branch stars. Separate fits of the magnitude distributions of carbon- and oxygen-rich stars allowed us to constrain the metallicity distribution across the galaxy and its star formation rate. Results. The Small Magellanic Cloud stellar population is found to be on average 7-9 Gyr old but older stars are present at its periphery and younger stars are present in the direction of the companion galaxy the Large Magellanic Cloud. The metallicity distribution traces a ring-like structure that is more metal rich than the inner region of the galaxy. Conclusions. The C/M ratio discussed in Paper I is a tracer of the metallicity distribution only if the underlying stellar population is of intermediate-age.