Classical barium stars are binary systems which consist of a late-type giant enriched in carbon and slow neutron capture (s-process) elements and an evolved white dwarf (WD) that is invisible at optical wavelengths. The youngest observed barium stars are surrounded by planetary nebulae (PNe), ejected soon after the wind accretion of polluted material when the WD was in its preceeding asymptotic giant branch (AGB) phase. Such systems are rare but powerful laboratories for studying AGB nucleosynthesis as we can measure the chemical abundances of both the polluted star and the nebula ejected by the polluter. Here we present evidence for a barium star in the PN Hen 2-39. The polluted giant is very similar to that found in WeBo 1. It is a cool (Teff=4250 +/- 150 K) giant enhanced in carbon ([C/H]=0.42 +/- 0.02 dex) and barium ([Ba/Fe]=1.50 +/- 0.25 dex). A spectral type of C-R3 C_24 nominally places Hen 2-39 amongst the peculiar early R-type carbon stars, however the barium enhancement and likely binary status mean that it is more likely to be a barium star with similar properties, rather than a true member of this class. An AGB star model of initial mass 1.8 Msun and a relatively large carbon pocket size can reproduce the observed abundances well, provided mass is transferred in a highly conservative way from the AGB star to the polluted star (e.g. wind Roche-lobe overflow). The nebula exhibits an apparent ring morphology in keeping with the other PNe around barium stars (WeBo 1 and A 70) and shows a high degree of ionization implying the presence of an invisible hot pre-WD companion that will require confirmation with UV observations. In contrast to A 70, the nebular chemical abundance pattern is consistent with non-Type I PNe, in keeping with the trend found from nebular s-process studies that non-Type I PNe are more likely to be s-process enhanced. (abridged)
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