We have derived abundances of 33 elements and upper limits for 6 additional elements for the metal-poor ([Fe/H] = -2.42) turn-off star HE 0338-3945 from high-quality VLT-UVES spectra. The star is heavily enriched, by about a factor of 100 relative to iron and the Sun, in the heavy s-elements (Ba, La, ..). It is also heavily enriched in Eu, which is generally considered an r-element, and in other similar elements. It is less enriched, by about a factor of 10, in the lighter s-elements (Sr, Y and Zr). C is also strongly enhanced and, to a somewhat lesser degree, N and O. These abundance estimates are subject to severe uncertainties due to NLTE and thermal inhomogeneities which are not taken into detailed consideration. However, an interesting result, which is most probably robust in spite of these uncertainties, emerges: the abundances derived for this star are very similar to those of other stars with an overall enhancement of all elements beyond the iron peak. We have defined criteria for this class of stars, r+s stars, and discuss nine different scenarios to explain their origin. None of these explanations is found to be entirely convincing. The most plausible hypotheses involve a binary system in which the primary component goes through its giant branch and asymptotic giant branch phases and produces CNO and s-elements which are dumped onto the observed star. Whether the r-element Eu is produced by supernovae before the star was formed (perhaps triggering the formation of a low-mass binary), by a companion as it explodes as a supernova (possibly triggered by mass transfer), or whether it is possibly produced in a high-neutron-density version of the s-process is still unclear. Several suggestions are made on how to clarify this situation.
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