Chemical separation of stellar populations: analytic solutions for chemical evolution models with metallicity-dependent yields

TL;DR

The paper develops a cohesive, analytic framework for single-zone chemical evolution with metallicity-dependent yields, showing that metallicity dependence can be treated as a system-dependent delay approximately equal to the depletion time. By solving for the metallicity-independent baseline $Z^0_x(t)$ and the metallicity-dependent correction $Z^1_x(t)$ via a derivative with respect to the depletion-time inverse, the authors provide closed-form expressions for multiple star formation histories (exponential, constant, linear-exponential) and for yield saturation. They extend the formalism to abundance planes, notably the [Al/Fe]-[Mg/Fe] diagnostic used to distinguish in-situ from accreted populations, and demonstrate how parameter choices like the mass-loading factor $\eta$, star formation efficiency, and metallicity-gradient $g_{Al}$ shape the tracks. The work enables rapid exploration of chemical evolution scenarios and offers practical steps and code to apply the models to APOGEE-like data, with extensions to more channels and phases discussed for future work.

Abstract

Stellar abundances of elements with production channels that are metallicity-dependent (most notably aluminium) have provided an empirical route for separating different Galactic components. We present 'single-zone' analytic solutions for the chemical evolution of galaxies when the stellar yields are metallicity-dependent. Our solutions assume a constant star formation efficiency, a constant mass-loading factor and that the yields are linearly dependent on the interstellar medium abundance (with the option of a saturation of the yields at high metallicity). We demonstrate how the metallicity dependence of the yields can be mathematically considered as a system-dependent delay time (approximately equal to the system's depletion time) that, when combined with system-independent delay times arising from stellar evolutionary channels, produces the separation of different systems based on their star formation efficiency and mass-loading factor. The utility of the models is highlighted through comparisons with data from the APOGEE spectroscopic survey. We provide a comprehensive discussion of the chemical evolution models in the [Al/Fe]-[Mg/Fe] plane, a diagnostic plane for the separation of in-situ and accreted Galactic components. Extensions of the models are presented, allowing for the modelling of more complex behaviours largely through the linear combination of the presented simpler solutions.

Figures (6)

• Figure 1: The net yield from different Type II (left column), Type Ia (centre) and AGB (right) models normalised by the production at solar metallicity. Each panel shows a different element. The Type II models are from Nugrid1Nugrid2ChieffiLimongi2004Kobayashi2006WoosleyWeaver1995Prantzos2018, the Type Ia models from Bravo2019Gronow2021aGronow2021bKobayashi2020LeungNomoto2018Seitenzahl2013 and the AGB models from Nugrid1Nugrid2KarakasLugaroLugaro2012Karakas2018Fruity1Fruity2. The dots show the metallicities at which models are available and the lines linearly interpolate between these and extrapolate using a constant value outside the available model range (some smoothing has been applied). The thicker dashed grey line in the top left panel shows a linear function.
• Figure 2: Critical metallicity above which metallicity dependence of yields is 'significant' (computed as the inverse of the yield metallicity gradient from finite differencing models at $[\mathrm{M/H}]=-1.8$ and $[\mathrm{M/H}]=-0.1$). At the critical metallicity, the metal dependence of the yields for element X approximately gives rise to shifts of $\log_{10}[\mathrm{X}/\mathrm{O}]=0.43$ (see equation \ref{['eqn:delta_metal_shift']}). The Type II models (orange squares) are Nugrid Nugrid1Nugrid2, Kobayashi2006, ChieffiLimongi2004, LimongiChieffi2018, Prantzos2018 and WoosleyWeaver1995, the Type Ia models (green triangles) are Gronow2021b, Kobayashi2006, Seitenzahl2013, Bravo2019, Kobayashi2020, Kobayashi2020, LeungNomoto2018, LeungNomoto2020 and Shen2018, and the AGB models (blue circles) are Nugrid Nugrid1Nugrid2, KarakasLugaroLugaro2012KarakasLugaroKarakas2018 and FRUITY Fruity1Fruity2, all in order of increasing point size. Note in particular that the critical metallicity for Al falls exactly at the separation between the typical metallicity of in-situ Milky Way stars (i.e. solar) and those in dwarf galaxies $\mathrm{[M/H]}\approx-1$.
• Figure 3: Evolution of abundances for Mg, a representative prompt element with metallicity-independent yields and Al, a representative prompt element with metallicity-dependent yields. The top row shows evolution with time for [Al/H] in solid lines (and in faint dashed when metallicity dependence is switched off, $g_\mathrm{Al}=0$). The middle row shows the derivative of [Al/H] with respect to the logarithm of time. The bottom row shows the evolution of the abundance ratio [Al/Mg] with [Mg/H]. The dots correspond to times of $0.1$ and $1\,\mathrm{Gyr}$ and the model is generated for a total of $5\,\mathrm{Gyr}$. Each line shows a different model configuration: the default setup (blue) is an exponential star formation history with $\tau_\star=1\,\mathrm{Gyr}$, $\eta=1$ and $\tau_\mathrm{s}=2.5\,\mathrm{Gyr}$ (appropriate for the Milky Way thick disc) and is shown in both left and right panels; left: orange doubles the depletion time $\tau_\mathrm{d}=\tau_\star/(1+\eta)$ by altering $\eta$); dark pink introduces a non-zero delay time for Al of $\tau_\mathrm{p}=1.5\,\mathrm{Gyr}$; red lowers the metallicity gradient of the yields, $g$, by a factor $4$; right: green uses a linear-exponential star formation history; brown uses a constant star formation history and pink saturates the growth of the yields with metallicity at $t_\mathrm{c}=0.35\,\mathrm{Gyr}$. The greyscale shows the density of the APOGEE DR17 dataset.
• Figure 4: Example solutions for our analytic chemical evolution models plotted over APOGEE DR17 data (blue logarithmic density). The solid red line is an example "Milky Way" track and the dashed red line is an example Gaia-Sausage Enceladus (GSE) track. The dots (circles) are spaced by $1\,\mathrm{Gyr}$.
• Figure 5: Linear-exponential model variations when changing different parameters. The background blue distribution is APOGEE DR17 stars as shown in Fig. \ref{['fig::example_tracks']}. Each row corresponds to varying a different parameter as denoted in the panel: first, mass-loading factor, $\eta$, second, star formation history timescale, $\tau_\mathrm{s}$, third, star formation efficiency timescale, $\tau_\star$ and fourth, the gradient of Type II Al yields with metallicity, $g_\mathrm{Al}$. The colours of the lines correspond to the colour bar in the leftmost panels. The default parameters chosen to emulate the Milky Way thick disc are $\eta=1$, $\tau_\star=1\,\mathrm{Gyr}$ (so $\tau_\mathrm{d}=0.5\,\mathrm{Gyr}$), $\tau_\mathrm{s}=2.5\,\mathrm{Gyr}$, and $g_\mathrm{Al}m_\mathrm{O}=12$ (see Table \ref{['tab:yields']}). The tracks are all computed for $5\,\mathrm{Gyr}$ and the dots show $t=0.15\,\mathrm{Gyr}$ (when the first Type Ia supernovae start contributing) and $t=2\,\mathrm{Gyr}$.