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. Our
methodology will improve future studies of accreted structures and their
importance for the formation of the Milky Way.
