The GALAH Survey: Chemical tagging and chrono-chemodynamics of accreted halo stars with GALAH+ DR3 and $Gaia$ eDR3

Since the advent of $Gaia$ astrometry, it is possible to identify massive accreted systems within the Galaxy through their unique dynamical signatures. One such system, $Gaia$-Sausage-Enceladus (GSE), appears to be an early "building block" given its virial mass $> 10^{10}\,\mathrm{M_\odot}$ at infall ($z\sim1-3$). In order to separate the progenitor population from the background stars, we investigate its chemical properties with up to 30 element abundances from the GALAH+ Survey Data Release 3 (DR3). To inform our choice of elements for purely chemically selecting accreted stars, we analyse 4164 stars with low-$\alpha$ abundances and halo kinematics. These are most different to the Milky Way stars for abundances of Mg, Si, Na, Al, Mn, Fe, Ni, and Cu. Based on the significance of abundance differences and detection rates, we apply Gaussian mixture models to various element abundance combinations. We find the most populated and least contaminated component, which we confirm to represent GSE, contains 1049 stars selected via [Na/Fe] vs. [Mg/Mn] in GALAH+ DR3. We provide tables of our selections and report the chrono-chemodynamical properties (age, chemistry, and dynamics). Through a previously reported clean dynamical selection of GSE stars, including $30 < \sqrt{J_R~/~\mathrm{kpc\,km\,s^{-1}}} < 55$, we can characterise an unprecedented 24 abundances of this structure with GALAH+ DR3. Our chemical selection allows us to prevent circular reasoning and characterise the dynamical properties of the GSE, for example mean $\sqrt{J_R~/~\mathrm{kpc\,km\,s^{-1}}} = 26_{-14}^{+9}$. We find only $(29\pm1)\%$ of the GSE stars within the clean dynamical selection region. We thus discuss chemodynamic selections (such as eccentricity and upper limits on [Na/Fe]).