Modelling the density and mass of the Milky Way's proto-galaxy components with $APOGEE$-$Gaia$

Unravelling galaxy formation theory requires understanding galaxies both at high and low redshifts. A possible way to connect both realms is by studying the oldest stars in the Milky Way (i.e., the proto-Galaxy). We use the $APOGEE$-$Gaia$ surveys to perform a purely chemical dissection of Milky Way (MW) stellar populations, and identify samples of stars likely belonging to proto-Galactic fragments. The metallicity dependence of the distribution of old MW stars in the [Mg/Mn]-[Al/Fe] enables the distinction of at least two populations in terms of their star formation histories: a rapidly evolved population likely associated with the main progenitor system of the proto-MW; and populations characterised by less efficient, slower, star formation. In the Solar neighbourhood less efficient star forming populations are dominated by the $Gaia$-$Enceladus$/$Sausage$ accretion debris. In the inner Galaxy, they are largely associated with the $Heracles$ structure. We model the density of chemically defined proto-Galaxy populations, finding that they are well represented by a Plummer model with a scale radius of $a\sim3.5$ kpc, and an oblate ellipsoid with flattening parameters [$p\sim0.8$; $q\sim0.6$]; this finding indicates that the MW plausibly hosts a low-mass, metal-poor, bulge component. We integrate this density for $chemically$ $unevolved$ stars between $-2 < \mathrm{[Fe/H]} < -0.5$ to obtain a minimum stellar mass for the proto-Galaxy of $M_{*} (r<10~\mathrm{kpc}) = 9.1\pm0.2\times10^{8}$ M$_{\odot}$. Our results suggest the proto-Milky Way is at least comprised of two significant fragments: the main $in$ $situ$ progenitor and the $Heracles$ structure.