TOPoS VI. The metal-weak tail of the metallicity distribution functions
  of the Milky Way and of the Gaia-Sausage-Enceladus structure
  Context. The TOPoS project has the goal to find and analyse Turn-Off (TO)
stars of extremely low metallicity. To select the targets for spectroscopic
follow-up at high spectral resolution, we have relied on low-resolution spectra
from the Sloan Digital Sky Survey. Aims. In this paper we use the metallicity
estimates we have obtained from our analysis of the SDSS spectra to construct
the metallicity distribution function (MDF) of the Milky Way, with special
emphasis on its metal-weak tail. The goal is to provide the underlying
distribution out of which the TOPoS sample was extracted. Methods. We make use
of SDSS photometry, Gaia photometry and distance estimates derived from the
Gaia parallaxes to derive a metallicity estimate for a large sample of over 24
million TO stars. This sample is used to derive the metallicity bias of the
sample for which SDSS spectra are available. Results. We determined that the
spectroscopic sample is strongly biased in favour of metal-poor stars, as
intended. A comparison with the unbiased photometric sample allows to correct
for the selection bias. We select a sub-sample of stars with reliable
parallaxes for which we combine the SDSS radial velocities with Gaia proper
motions and parallaxes to compute actions and orbital parameters in the
Galactic potential. This allows us to characterize the stars dynamically, and
in particular to select a sub-sample that belongs to the Gaia-Sausage-Enceladus
(GSE) accretion event. We are thus able to provide also the MDF of GSE.
Conclusions. The metal-weak tail derived in our study is very similar to that
derived in the H3 survey and in the Hamburg/ESO Survey. This allows us to
average the three MDFs and provide an error bar for each metallicity bin.
Inasmuch the GSE structure is representative of the progenitor galaxy that
collided with the Milky Way, that galaxy appears to be strongly deficient in
metal-poor stars compared to the Milky Way, suggesting that the metal-weak tail
of the latter has been largely formed by accretion of low mass galaxies rather
than massive galaxies, such as the GSE progenitor.
