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The i-process and CEMP-r/s stars

We investigate whether the anomalous elemental abundance patterns in some of the C-enhanced metal-poor-s+r (CEMP-r/s) stars are consistent with predictions of nucleosynthesis yields from the i-process, a neutron-capture regime at neutron densities intermediate between those typical for the slow (s) and rapid (r) processes. Conditions necessary for the i-process are expected to be met at multiple stellar sites, such as the He-core and He-shell flashes in low-metallicity low-mass stars, super-AGB and post-AGB stars, as well as low-metallicity massive stars. We have found that single-exposure one-zone simulations of the i-process reproduce the abundance patterns in some of the CEMP-r/s stars much better than the model that assumes a superposition of yields from s- and r-process sources. Our previous study of nuclear data uncertainties relevant to the i-process revealed that they could have a significant impact on the i-process yields obtained in our idealized one-zone calculations, leading, for example, to ~0.7dex uncertainty in our predicted [Ba/La] ratio. Recent 3D hydrodynamic simulations of convection driven by a He-shell flash in post-AGB Sakurai's object have discovered a new mode of non-radial instabilities: the Global Oscillation of Shell H-ingestion. This has demonstrated that spherically symmetric stellar evolution simulations cannot be used to accurately model physical conditions for the i-process.


Introduction
The fraction of C-Enhanced Metal Poor (CEMP) stars (objects with [C/Fe]>1) increases with a decrease in metallicity, reaching 20% at [Fe/H]=-2 [1,3]. CEMP-r/s stars form a subclass characterized by simultaneous enhancements of both s-(e.g., Ba) and r-process (e.g., Eu) elements. [17,21,22]. The prevailing model of a CEMP-r/s star assumes that it is a component of a binary system initially pre-enriched with r-process material that has additionally accreted s-process material from its heavier AGB-star companion, which leads to a superposition of both processes [17]. However, this model fails to explain CEMP-r/s stars with large [hs/ls] 1 and low [La/Eu] ratios. An example of such a star is CS 31062-050 shown in Fig. 26 in [5]. The intermediate neutron-capture process (hereafter, the i-process, see [7]) operates at neutron densities between those characteristic of the s-and r-process, when H is convectively entrained and advected into a He-burning zone. When the advected fluid parcel has reached a depth with T ≈ 1.5×10 8 K, 13 N is produced in the reaction 12 C(p,γ) 13 N. The beta-decay of 13 N to 13 C followed by the reaction 13 C(α,n) 16 O result in a production of neutrons with a density N n ≈ 10 15 cm −3 . Possible astrophysical sites for the i-process have been identified in stellar evolution computations. These are the He-shell thermal pulses in the low-and zero-metallicity AGB stars [9], the very late thermal pulse (VLTP) in post-AGB stars (e.g., Sakurai's object, [13]) and the He-core flash in lowmetallicity low-mass stars ( [6]). The i-process in VLTP models with mixing assumptions motivated by 3D hydrodynamic simulations was found to reproduce the observed abundance distribution in Sakurai's object [13]. It may also explain the excess of 32 S and anomalous Ti isotopic ratios in presolar A+B grains [10,15]. In this paper, we investigate if the i-process nucleosynthesis can explain the observed abundance patterns in peculiar CEMP-r/s stars, such as CS 31062-050. We also briefly discuss the impact of nuclear physics uncertainties on the calculated i-process yields, as well as the results of 3D hydrodynamic simulations of nuclear burning and mixing under conditions relevant to the i-process in Sakurai's object recently reported in [14,24].

A simple i-proces model for the CEMP-r/s stars
We compare the abundance pattern in CS 31062-050 with results of our nucleosynthesis simulations of the i-process in one zone with constant temperature and density corresponding to the physical conditions in a He pulse-driven convective zone (PDCZ) from 1D stellar evolution models of AGB stars. Our assumed values of T = 2 × 10 8 K and ρ = 10 4 g cm −3 prevent destruction of 13 N via the (p,γ) channel, but they allow the neutron release via 13 C(α, n). In the He PDCZ of a real star, with H entrained by convection from the surrounding H-rich envelope, the reactions 12 C(p,γ) 13 N and 13 C(α,n) 16 O are spatially separated, the first occurring close to the upper and the second close to the lower convective boundary. 13 N produced in the first reaction decays to 13 C while being transported down by convection. The one-zone model reproduces the characteristic i-process neutron density (N n ∼ 10 15 cm −3 ) and neutron exposures of 10 . . . 50 mbarn −1 . The initial abundances are taken from the solar abundance distribution [11], with isotopic ratios from [20], Figure 1: The evolution of the neutron density N n with time for the initial H mass fractions 0.2, 0.1 and 0.05 (upper panel). The best fit of the abundance pattern in the CEMP-r/s star CS 31062-050 with the i-process model (lower panel, with observational data from [16]). Shown in the bottom-right corner are the fitted values of the neutron density N n and exposure (tau). scaled down to the metallicity Z = 10 −3 . The abundances of C and O are additionally modified, so that they are close to those found in the He PDCZ, i.e. X( 12 C) = 0.5 and X( 16 O) = 0.05. A specified fraction of H, assumed to be ingested from the H-rich envelope, is added to the mixture, with the combined mass fraction of C+H being always set to 0.7. The i-process nucleosynthesis is followed with the NuGrid single-zone PPN code [12].
During the first second, 12 C is almost all transformed into 13 N which then decays into 13 C on the timescale of 9.97 min. After the decay of 13 N, the reaction 13 C(α,n) 16 O releases neutrons with a high neutron density. The upper panel of Fig. 1 shows the evolution of the neutron density N n with time for different initial mass fractions of H. As the simulation proceeds, the neutron exposure τ = t 0 N n v T dt (v T is the neutron thermal velocity) increases with time, and the abundance distribution is shifted to heavier elements. The model results shown in the lower panel of Fig. 1 give the best representation of the abundance pattern in CS 31062-050. Its corresponding neutron density and exposure are indicated in the panel's bottom-right corner. Hydrodynamic simulations suggest that the i-process nucleosynthesis in the stellar environment is terminated when the energy released through the combustion of ingested H leads to a significant perturbation of the spherically symmetric convective shell structure. This self-quenching will depend on the exact configuration of the convective shells at the time of H-ingestion and can be expected to vary. Therefore, a range of the termination time is expected for the i-process, which motivates us to consider it as a free parameter in this simple model. However, in spite of its simplicity, our one-zone i-process model reproduces surprisingly well the entire heavy-element abundance distribution from Y to Ir within the observational errors in CS 31062-050 (Fig. 1). Specifically, this includes the large ratio of [hs/ls] ≈ 1, the [La/Eu] ratios between 0.0 and 0.5, which are between the ratios predicted for the s-and r-processes, as well as high abundances in the Er-W region compared to those in the Os-Ir region, which are characteristic signatures of the i-process nucleosynthesis at high neutron density, and which are all present in the abundance pattern of CS 31062-050. On the other hand, the one-zone model is not suited to correctly describe the evolution of the Fe group elements and Pb, because of its unrealistic depletion of the n-capture seed elements. The observed abundance distributions in the CEMP-r/s stars HE 0338-3945 [17] and CS 22898-027 [2] are also well reproduced by the one-zone i-process model (Fig. 2).

Nuclear physics uncertainties relevant to the i-process
When comparing predicted nucleosynthesis yields from the i-process with observations, it is important to understand what and how strongly nuclear physics uncertainties can affect the calculated abundances of isotopes along the i-process nucleosynthesis path, that goes four to five species off the valley of stability, as shown in Fig. 3. The first i-process uncertainty analysis was done in [4], where propagating systematic uncertainties of nuclear reaction cross sections from different theoretical models were investigated for elements of the second s-process peak. The analysis considered Ba, La and Eu. The same i-process one-zone model described in Section 2 was used. Fig.  6 in [4] shows changes of the [La/Eu] versus [Ba/La] ratio for cases when nuclear reaction rates are estimated using three different theoretical Hauser-Feshbach codes: NON-SMOKER [23] (with rates from JINA REACLIB [8]), CoH 3.3 [18] and TALYS 1.4 [19]. The changes were found to be strongly model-dependent. The resulting [Ba/La] and [La/Eu] ratios varied, at least, by factors of ∼ 0.7 dex and ∼ 0.3 dex, respectively, between the models. The conclusions are that the nuclear physics uncertainties strongly limit, at present, the predictive power of i-process simulations [4] and that it would be very important to further study the impact of nuclear physics uncertainties on the i-process element production at various sites.

The i-process conditions in Sakurai's object
3D hydrodynamic simulations of mixing and H entrainment in the He PDCZ of the post-AGB star Sakurai's object are now possible at fine enough grids, so that the entrainment process is numerically converged [24]. Combustion of ingested protons in the reaction 12 C(p,γ) 13 N and its energy feedback on mixing was taken into account [14]. Two runs with 768 3 and 1536 3 grid resolutions were performed and showed quantitatively quite similar behaviour. The continued H ingestion led to its accumulation in the upper part of the convective zone. At some point, a large amount of H fuel ignited there and a resulting burning pocket rose and collided with the stiff convective boundary. The mass-conservation law then made the pocket to spread and move along the boundary to the antipode, where the two fronts collided causing a strong downdraft of the material, and the process repeated several times. That new mode of non-radial instabilities, coined by the authors "The Global Oscillation of Shell H-ingestion (GOSH)", first appeared after nearly 850 min in the 1536 3 run. It is shown in Fig. 4 at 1149 min [14,24]. Such a GOSH event becomes more and more violent as it is repeated for about a dozen times in a row, leading to entrainment of entropy from the stable layer. In the run with 768 3 grid resolution, a new upper convective boundary forms, while the run with 1536 3 cells has not been followed to this point yet. It is not clear yet how the GOSH can be taken into account in 1D stellar evolution calculations.

Conclusion
Simple one-zone simulations of i-process nucleosynthesis give yields that can fit almost all of the observed heavy-element abundances from Y to Ir, within their observational errors, in some of the CEMP-r/s stars. The observed high enrichments with s-and r-process material in these stars cannot be explained by the currently prevailing model that assumes a superposition of yields from different s-and r-process sources. These CEMP-r/s stars could therefore represent an i-process site. Nuclear physics uncertainties are a main obstacle to obtaining reliable yields from the iprocess nucleosynthesis calculations, as found by [4]. Mixing and H entrainment relevant to the i-process in the post-AGB star Sakurai's object have recently been investigated, which has revealed a new mode of non-radial instabilities, the GOSH [14].