S-Process Component Coefficients in Low-Metallicity AGB Stars

• S-Process Component Coefficients are quantitative metrics that capture neutron exposure, 13C-pocket efficiency, and mixing in low-metallicity AGB stars.
• They utilize diagnostic ratios like [hs/ls] and [Pb/hs] to trace the production of heavy elements, from light-s peaks to lead, under varying stellar conditions.
• These coefficients guide the interpretation of abundance patterns in CEMP-s stars and help refine stellar evolution models and Galactic chemical evolution studies.

S-process component coefficients quantify the relative contributions of the slow neutron-capture process (s-process) to the production of heavy elements in stars, particularly low-mass asymptotic giant branch (AGB) stars at low metallicity. These coefficients are fundamental parameters in both theoretical stellar models and spectroscopic analyses, encapsulating the degree of neutron exposure, the efficiency and extent of mixing phenomena (such as the 13C-pocket), and the resulting abundance patterns of s-process elements from the light-s (ls, N = 50) and heavy-s (hs, N = 82) peaks up to lead (Pb, N = 126). This framework enables interpretation of observed chemical signatures in metal-poor stars—especially in carbon-enhanced metal-poor (CEMP-s) stars—and supports broader modeling of Galactic chemical evolution.

1. Fundamental Principles and Diagnostic Ratios

In low-mass AGB stars (1.3 – 2.0 M$_\odot$) at low metallicity, the s-process is primarily driven by two neutron source reactions:

• $^{13}$C($\alpha$, n)$^{16}$O in the partial-mixing–induced “13C-pocket,” releasing neutrons in radiative conditions during interpulse phases,
• $^{22}$Ne($\alpha$, n)$^{25}$Mg during convective thermal pulses.

The relative abundances of s-process elements cluster around three “peaks” at magic neutron numbers: ls (e.g., Sr, Y, Zr at N = 50), hs (e.g., Ba, La, Ce, Nd, Sm at N = 82), and Pb (N = 126). To characterize the s-process enrichment independent of dilution or the absolute extent of dredge-up, two logarithmic indices are defined: $$\begin{align*} {[hs/ls]} & = {[hs/Fe]} - {[ls/Fe]}, \ {[Pb/hs]} & = {[Pb/Fe]} - {[hs/Fe]}. \end{align*}$$ Here, [X/Fe] denotes the standard spectroscopic logarithmic abundance ratio relative to solar values. High [hs/ls] indicates significant neutron exposure—neutron captures move much of the abundance past the ls bottleneck—while high [Pb/hs] signals that much of the s-process path terminated at lead due to high neutron-to-seed ratios, as is typical in metal-poor environments with efficient 13C pockets (Bisterzo et al., 2010, Bisterzo et al., 2011, Bisterzo et al., 2012).

2. Theoretical Modeling: Formation and Variation of Component Coefficients

The detailed nucleosynthesis calculations employ stellar evolution codes such as FRANEC, where key variables are the initial mass, metallicity (often with [Fe/H] as low as –3.6), and the parametrized efficiency of the 13C-pocket (designated as “ST”, “ST/12”, etc.).

• The 13C-pocket forms during the third dredge-up, with proton ingestion into the 12C-rich He intershell;
• The pocket’s efficiency controls the neutron exposure: more 13C yields more neutrons per Fe “seed,” pushing the s-process toward the hs and Pb peaks.

AGB models demonstrate that, at a fixed [Fe/H], varying the 13C-pocket strength produces a wide spread in predicted [ls/Fe], [hs/Fe], and [Pb/Fe], and thus in [hs/ls] and [Pb/hs]. The two indicators cleanly separate different nucleosynthetic histories, regardless of subsequent dilution or mixing in a binary companion (Bisterzo et al., 2010, Bisterzo et al., 2011, Bisterzo et al., 2012). These models predict that low-metallicity (e.g., [Fe/H] ≤ –2.5) systems show large neutron/seed ratios, enhancing hs and Pb.

3. Application to Binary and Carbon-Enhanced Metal-Poor Stars

Many observed CEMP-s stars are found in binaries with no intrinsic dredge-up: their s-process abundances are attributed to past mass transfer from a more massive AGB companion (Bisterzo et al., 2010, Bisterzo et al., 2011). The observed abundance pattern is modeled as the sum of two sources, with possible dilution after transfer: $$[\mathrm{El}^{\mathrm{obs}}/\mathrm{Fe}] = \log \left(10^{[\mathrm{El}_\mathrm{AGB}/\mathrm{Fe}] - dil} + 10^{[\mathrm{El}_\star/\mathrm{Fe}]}(1 - 10^{-dil})\right),$$ where the dilution factor $$dil = \log(M^\mathrm{obs}_\star / \Delta M^\mathrm{trans}_\mathrm{AGB})$$ describes mixing into the envelope of the secondary star.

However, the diagnostic ratios [hs/ls] and [Pb/hs] are largely insensitive to this dilution, precisely capturing the neutron exposure, 13C-pocket strength, and initial mass of the AGB star responsible for the enrichment (Bisterzo et al., 2010, Bisterzo et al., 2011, Bisterzo et al., 2012).

4. Observational Implications and Constraints

High-resolution spectra of metal-poor stars reveal a broad range in the s-process indicator ratios across the CEMP-s population, implying a distribution of 13C-pocket strengths and neutron exposures. Stars classified as CEMP-sII ([hs/Fe] > 1.5) tend to exhibit larger [hs/ls] and [Pb/hs] ratios than CEMP-sI ([hs/Fe] < 1.5), reflecting stronger neutron fluence and greater lead production, and supporting the model-based dichotomy in AGB donor efficiency (Bisterzo et al., 2011, Bisterzo et al., 2012).

The observed [hs/ls] and [Pb/hs] serve not only as fingerprints of the underlying s-process but also as constraints on the physics of proton mixing and nucleosynthesis in AGB envelopes. Outlier cases or broad observed spreads imply that the formation and efficiency of the 13C-pocket is not determined solely by metallicity or mass, but is affected by complex mixing processes.

5. Key Nuclear Reactions and Effects of Metallicity

The efficiency and pattern of the s-process depend on the neutron sources and metallicity:

• $^{13}$C(α,n)$^{16}$O dominates at low mass and low metallicity, enabling a primary (metal-insensitive) neutron exposure, favoring production of Pb;
• The $^{22}$Ne(α,n)$^{25}$Mg neutron source, partially activated during convective TPs, is secondary at low metallicity but can (i) allow branching at higher neutron densities (influencing, e.g., $^{96}$Zr), (ii) slightly raise the neutron density, and (iii) potentially act as a neutron poison or seed-regenerator via reaction chains involving primary 22Ne and Fe (Bisterzo et al., 2010).

At low [Fe/H], the high neutron-to-seed ratio (driven by primary 13C and a paucity of Fe seeds) shifts the s-process abundance pattern toward the heavy-s peak and lead—quantitatively captured by the elevated [hs/ls] and [Pb/hs] ratios.

6. Impact, Limitations, and Prospects

Use of s-process component coefficients enables stringent testing and validation of AGB nucleosynthesis models against observations, and vice versa. Limitations arise from uncertain physics of mixing and proton transport beneath the convective envelope, as well as nuclear reaction rate uncertainties for key neutron poisons, such as 22Ne(n,γ)23Ne, and for branch-point nuclei. Nevertheless, [hs/ls] and [Pb/hs] remain robust empirical tools for constraining the operating conditions, especially in low-metallicity binary systems and the broader context of Galactic chemical evolution.

Tables of predicted [ls/Fe], [hs/Fe], and [Pb/Fe] for different masses, metallicities, and 13C-pocket efficiencies, as generated by modern AGB models, are crucial for researchers aiming to map detailed stellar evolutionary scenarios to observed abundance features (Bisterzo et al., 2010, Bisterzo et al., 2011).

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In summary, s-process component coefficients—primarily [hs/ls] and [Pb/hs]—distill the complex nuclear and stellar physics of AGB nucleosynthesis in low-metallicity environments into quantitative diagnostics that probe neutron source operation, neutron/seed ratios, and mixing physics. Their empirical measurement and theoretical interpretation enable investigation of the s-process enrichment patterns in single stars, binaries, and the Galactic halo. The strict reliance on these indices in current model-observation comparisons underscores their centrality in advancing our understanding of the origin and evolution of the heavy elements.