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
  Post-AGB stars are exquisite probes of AGB nucleosynthesis. However, the
previous lack of accurate distances jeopardised comparison with theoretical AGB
models. The $Gaia$ Early Data Release 3 ($Gaia$ EDR3) has now allowed for a
breakthrough in this research landscape. In this study, we focus on a sample of
single Galactic post-AGBs for which chemical abundance studies were completed.
We combined photometry with geometric distances to carry out a spectral energy
distribution (SED) analysis and derive accurate luminosities. We subsequently
determined their positions on the HR-diagram and compared this with theoretical
post-AGB evolutionary tracks. While most objects are in the post-AGB phase of
evolution, we found a subset of low-luminosity objects that are likely to be in
the post-horizontal branch phase of evolution, similar to AGB-manqu\'e objects
found in globular clusters. Additionally, we also investigated the observed
bi-modality in the $s$-process enrichment of Galactic post-AGB single stars of
similar Teff, and metallicities. This bi-modality was expected to be a direct
consequence of luminosity with the $s$-process rich objects having evolved
further on the AGB. However, we find that the two populations: the $s$-process
enriched and non-enriched, have similar luminosities (and hence initial
masses), revealing an intriguing chemical diversity. For a given initial mass
and metallicity, AGB nucleosynthesis appears inhomogeneous and sensitive to
other factors which could be mass-loss, along with convective and
non-convective mixing mechanisms. Modelling individual objects in detail will
be needed to investigate which parameters and processes dominate the
photospheric chemical enrichment in these stars.
