Decoding the age-chemical structure of the Milky Way disk: An application of Copulas and Elicitable Maps

In the Milky Way, the distribution of stars in the $[\alpha \mathrm{/}\mathrm{F}\mathrm{e}]$ vs. $[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]$ and $[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]$ vs. age planes holds essential information about the history of star formation, accretion, and dynamical evolution of the Galactic disk. We investigate these planes by applying novel statistical methods called copulas and elicitable maps to the ages and abundances of red giants in the APOGEE survey. We find that the low- and high-$\alpha$ disk stars have a clean separation in copula space and use this to provide an automated separation of the $\alpha$ sequences using a purely statistical approach. This separation reveals that the high-$\alpha$ disk ends at the same [$\alpha$/Fe] and age at high $[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]$ as the low-$[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]$ start of the low-$\alpha$ disk, thus supporting a sequential formation scenario for the high- and low-$\alpha$ disks. We then combine copulas with elicitable maps to precisely obtain the correlation between stellar age $\tau$ and metallicity $[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]$ conditional on Galactocentric radius $R$ and height $z$ in the range $0<R<20$ kpc and $\mathrm{\mid }z\mathrm{\mid }<2$ kpc. The resulting trends in the age-metallicity correlation with radius, height, and [$\alpha$/Fe] demonstrate a $\approx 0$ correlation wherever kinematically-cold orbits dominate, while the naively-expected negative correlation is present where kinematically-hot orbits dominate. This is consistent with the effects of spiral-driven radial migration, which must be strong enough to completely flatten the age-metallicity structure of the low-$\alpha$ disk.