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Post-AGB stars are exquisite probes of AGB nucleosynthesis. However, theprevious lack of accurate distances jeopardised comparison with theoretical AGBmodels. The $Gaia$ Early Data Release 3 ($Gaia$ EDR3) has now allowed for abreakthrough in this research landscape. In this study, we focus on a sample ofsingle Galactic post-AGBs for which chemical abundance studies were completed.We combined photometry with geometric distances to carry out a spectral energydistribution (SED) analysis and derive accurate luminosities. We subsequentlydetermined their positions on the HR-diagram and compared this with theoreticalpost-AGB evolutionary tracks. While most objects are in the post-AGB phase ofevolution, we found a subset of low-luminosity objects that are likely to be inthe post-horizontal branch phase of evolution, similar to AGB-manqu\'e objectsfound in globular clusters. Additionally, we also investigated the observedbi-modality in the $s$-process enrichment of Galactic post-AGB single stars ofsimilar Teff, and metallicities. This bi-modality was expected to be a directconsequence of luminosity with the $s$-process rich objects having evolvedfurther on the AGB. However, we find that the two populations: the $s$-processenriched and non-enriched, have similar luminosities (and hence initialmasses), revealing an intriguing chemical diversity. For a given initial massand metallicity, AGB nucleosynthesis appears inhomogeneous and sensitive toother factors which could be mass-loss, along with convective andnon-convective mixing mechanisms. Modelling individual objects in detail willbe needed to investigate which parameters and processes dominate thephotospheric chemical enrichment in these stars.

the astrophysical journal. letters

Luminosities and masses of single Galactic Post-Asymptotic Giant Branch  (Post-AGB) stars with distances from $Gaia$ EDR3: The revelation of an  $s$-process diversity