1 Introduction

Most of the elements heavier than iron are produced in stars by slow (s) and rapid (r) neutron-capture nucleosynthesis processes. In the slow-neutron-capture process (s-process) the timescale for neutron capture is usually much longer than that for the \(\beta \)-decay of unstable nuclei along the s-process path, whereas in the rapid-neutron-capture process (r-process) the timescale for neutron capture is much shorter than the \(\beta \)-decay process. While the s-process operates with a neutron density of 10\(^{6-10}\) cm\(^{-3}\) (Busso et al. 1999) in the inter-pulse phases of low-mass and intermediate-mass AGB stars (Gallino et al. 1998), the r-process requires very high temperatures and neutron fluxes (n > 10\(^{20}\) cm\(^{-3}\)) and is expected to occur during supernova explosions and neutron star mergers (Thielemann et al. 2011; Wehmeyer et al. 2015). These two processes are characterized by distinct elemental abundance patterns. A large fraction of carbon-enhanced metal-poor (CEMP) stars are known to exhibit enhancement of s-process elements (CEMP-s stars) in surface chemical composition, a few exhibit enhancement in r-process elements (CEMP-r stars), a sizeable fraction show enhancement of both s-process and r-process elements (CEMP-r/s stars), and a few stars do not show enhancement of heavy elements (CEMP-no stars) (Beers & Christlieb 2005; Aoki et al. 2007). In general, CEMP stars are characterized by [Fe/H]Footnote 1 < −1.0 and [C/Fe] > 1.0, and barium and europium are considered as the representative elements of s-process and r-process respectively (Beers & Christlieb 2005). Chemical composition studies on CEMP-r/s stars have revealed that the observed heavy-element abundances cannot be explained by either s-process or r-process nucleosynthesis alone (Aoki et al. 2015, 2017). To explain the abundance pattern of the CEMP-r/s stars different formation scenarios have been proposed, involving different production sites for the s-process and r-process elements (Jonsell et al. 2006; Lugaro et al. 2009; Abate et al. 2016). All these scenarios are, however, not free from certain uncertainties, either in explaining the observed frequency of these stars, or the observed abundance patterns. An alternative process called ‘i-process (intermediate process) nucleosynthesis’ has recently been suggested as a possible production mechanism for CEMP-r/s stars (Dardelet et al. 2014; Hampel et al. 2016, 2019). Although it has been known for long (Cowan & Rose 1977) that proton ingestion from the convective envelope to the intershell region of AGB stars can produce high neutron flux which can initiate i-process nucleosynthesis, it is only very recently that it has been explored to understand the observed abundance patterns of CEMP-r/s stars on the basis of i-process (Dardelet et al. 2014; Hampel et al. 2016, 2019). Different sites have been proposed for the proton-ingestion episodes (PIE), such as core-helium flash (Fujimoto et al. 1990; Lugaro et al. 2009, the most massive AGB stars (Jones et al. 2016), and the super AGB stars (Doherty et al. 2015). However, the physical conditions under which proton ingestion can take place still remain a matter of debate. Many details regarding the site of the i-process nucleosynthesis that operates with neutron densities n\(\sim \)10\(^{15}\) cm\(^{-3}\) also remain poorly understood.

In this work, to understand the origin of the abundance patterns of heavy elements in CEMP-r/s stars, we have chosen a sample of eight stars reported to be CEMP-r/s stars by various researchers (Goswami et al. 2006; Goswami & Aoki 2010; Allen et al. 2012; Hansen et al. 2019; Purandardas et al. 2019) based on their estimates of abundances and abundance ratios of heavy elements for these stars. We have performed a parametric-model-based study to delineate the contributions of s-process and r-process to the observed heavy-element abundances. With reference to the sample stars, we have discussed and examined different formation scenarios to understand the processes responsible for the double enhancement seen in these stars. We find that using i-process model yields we could reproduce the observed abundance patterns of heavy elements of the sample of CEMP-r/s stars chosen for this study.

In Section 2, we have discussed in brief the sample of CEMP-r/s stars taken from the literature for this study. Section 3 discusses the procedure and results of the parametric-model-based study. In Section 4.1, different formation scenarios have been discussed in the context of the abundance pattern observed in the sample stars. Section 4.2 discusses the i-process models and results of the comparison of the model predictions with the observed heavy-element abundances of our sample stars. We report our conclusions in Section 5.

2 Sample of CEMP-r/s stars: CD−28 1082, CS 29503−010, CS 29528−028, HD 209621, HE 0002−1037, HE 0059−6540, HE 0151−6007, HE 1305+0007

The objects were selected following CEMP stars criteria (i.e. [Fe/H] < \(-1\), and [C/Fe] > 0.7). In our sample, the CEMP-r/s stars have metallicity in the range −1.70 < [Fe/H] < −2.70. In Purandardas et al. (2019) we have derived the atmospheric parameters and, for the first time, estimated the elemental abundances of CD−28 1082. The object was found to be a CEMP-r/s star with [Ba/Fe] = 2.09, [Eu/Fe] = 2.07 and [Ba/Eu] = 0.02, with \(^{12}\)C/\(^{13}\)C ratio \({\sim }\) 16. The objects HE 0002−1037, HE 0059−6540 and HE 0151−6007 have been reported to be CEMP-r/s by Hansen et al. (2019). These three objects exhibit abundances of Ba and Eu in the ranges 1.7 < [Ba/Fe] < 2.3 and 1.5 < [Eu/Fe] < 2.3, with [Ba/Eu] < 0.5 in each case. Hansen et al. (2019) estimated the \(^{12}\)C/\(^{13}\)C ratio to be 24 and 1 for the objects HE 0002−1037 and HE 0059−6540 respectively. The kinematic analysis shows that all these four sample stars belong to the inner halo population in the Galaxy. Allen et al. (2012) classified the objects CS 29503−010 and CS 29528−028 to be CEMP-r/s stars with [Ba/Fe] = 1.81 and 2.49 and [Eu/Fe] = 1.69 and 2.16 respectively on the basis of their analysis. Elemental abundances of two CEMP-r/s stars HE 1305 \(+\) 0007 and HD 209621 have been taken from Goswami et al. (2006) and Goswami and Aoki (2010) respectively. Tsuji et al. (1991) and Goswami et al. (2006) estimated \(^{12}\)C/\(^{13}\)C ratio to be \(\sim \) 10 for both HD 209621 and HE 1305 \(+\) 0007.

The atmospheric parameters of the sample stars are presented in Table 1. The abundance ratios of neutron-capture elements and carbon with respect to Fe are presented in Table 2.

Table 1 Atmospheric parameters of the sample stars.
Full size table
Table 2 Abundance ratios of neutron-capture elements of the sample stars.
Full size table

3 Parametric-model-based study

To understand the origin of the observed abundances in the sample stars, it is important to identify the contribution of the dominant neutron-capture process. Following the procedure described in Goswami et al. (2010) and references therein, we have performed a parametric-model-based study to trace the contributions of s-process and r-process nucleosynthesis to the observed abundances of the heavy elements. We have normalized the Solar s-process and r-process isotopic abundances of the stellar models of Arlandini et al. (1999) to the barium abundances of the corresponding CEMP-r/s stars. The observed elemental abundances of the sample stars are then fitted with the parametric model function log \(\epsilon _{i}\) = C\(_{s}\)N\(_{is}\) + C\(_{r}\)N\(_{ir}\), where N\(_{is}\) indicates the normalized abundance from s-process, N\(_{ir}\) indicates the normalized abundance from r-process, C\(_{s}\) indicates the component coefficient that corresponds to contributions from the s-process, and C\(_{r}\) indicates the component coefficient that corresponds to contributions from the r-process.

The best fitting coefficients and \(\chi ^{2}\) values are presented in Table 3. Figure 1 shows the best model fits with the observed abundances of the sample stars. Goswami and Aoki (2010) performed a parametric-model-based analysis of the heavy-element abundances observed in HD 209621 and HE 1305\(+\)0007 and concluded that similar contributions of both s-process and r-process are required to explain the abundance pattern of the stars (see Figure 5 of Goswami & Aoki (2010)). We have confirmed these results based on our analysis.

Figure 1
figure 1

Solid curve represents the best fit for the parametric-model function \(log{\epsilon _i}\) = \(C_sN_{is}\) + \( C_r N_{ir}\), where \(N_{is}\) and \(N_{ir}\) represent the abundances due to s-process and r-process respectively (isotopic abundances from the stellar model of Arlandini et al. (1999) are normalized to the Ba abundances of the corresponding stars). The points with error bars indicate the observed abundances.

Full size image
Table 3 Best-fit coefficients and \(\chi ^{2}\) for the parametric model function \(log{\epsilon _i}\) = \(C_sN_{is}\) + \( C_r N_{ir}\).
Full size table

4 Discussion

4.1 Different scenarios for formation of CEMP-r/s stars

To explain the peculiar abundance pattern of CEMP-r/s stars, researchers have proposed several formation scenarios (Jonsell et al. 2006; Lugaro et al. 2009; Abate et al. 2016). Here, we have discussed briefly some of the scenarios that are considered relevant for the sample of CEMP-r/s stars under this study.

(i) Radiative levitation, where the partially ionized heavy elements having large photon absorption cross-sections are pushed outwards by radiative pressure, could be a possible scenario for the observed overabundance of heavy elements in CEMP-r/s stars. However, the simulations of Richard et al. (2002) and Matrozis and Stancliffe (2016) have shown that the process of radiative levitation can occur in the hot stars in their main-sequence and main-sequence turn-off phases of evolution due to their thin convective envelopes. The sample stars in this study are low-temperature objects with temperature in the range 4350 K − 7100 K, and the radiative levitation scenario may not be applicable to explain the observed abundance pattern of these stars. This scenario has also been discussed at length by Cohen et al. (2003), Jonsell et al. (2006) and Abate et al. (2016) and rejected as a possible mechanism for formation of CEMP-r/s stars.

(ii) There are two scenarios where the enhancement of r-process elements in the CEMP-r/s stars is attributed to the r-process-material-enriched ISM from which the star is formed. In these scenarios, it is proposed that the enrichment of s-process elements is either due to self-contamination in its AGB phase (Hill et al. 2000; Cohen et al. 2003; Jonsell et al. 2006) or AGB pollution in a binary system (Hill et al. 2000; Cohen et al. 2003; Ivans et al. 2005; Jonsell et al. 2006; Bisterzo et al. 2011).

In the self-contamination scenario, the star needs to pass through the AGB phase of evolution to undergo s-process nucleosynthesis. However, the evolutionary stage of CEMP-r/s stars studied so far reveals that they have not yet passed through the red-giant-branch phase to produce the s-process elements. For this reason, this scenario had been rejected by many researchers (Jonsell et al. 2006; Abate et al. 2016). Two objects in our sample, CS 29503−010 and CS 29528−028, exhibit log g values (\(\sim \) 4.0), similar to that of dwarfs or sub-giants. The rest of the stars show low values (\(\sim \) 1 − 24) of \(^{12}\)C/\(^{13}\)C implying the extrinsic nature of heavy elements and carbon in these stars. Thus, this scenario is not applicable to our sample stars.

In another scenario, the primary, being more massive than the other in the binary system formed from the r-process-material-enriched ISM, evolves faster and proceeds through the AGB phase, producing s-process elements along with carbon. The mass-transfer episodes in the AGB phase then make the secondary star enriched in s-process elements. Considering this scenario of pre-enrichment of the binary system with r-process elements from the r-rich molecular cloud, Bisterzo et al. (2011, 2012), tried to reproduce the observed [hs/ls] in the CEMP-r/s stars. Although they claimed to be successful in doing so (compatible within error bars), there are still some arguments against this scenario. It is found that the abundances of Ba and Eu are correlated in CEMP-r/s stars and the AGB models cannot explain this correlation in case of independent enrichment of s-process and r-process elements (Abate et al. 2016). Also, the large fraction of CEMP-r/s stars among the CEMP-s stars cannot be explained by this scenario (Jonsell et al. 2006; Lugaro et al. 2009).

(iii) This scenario explains how a star can acquire s-process and r-process elements in a triple-star system. The most massive one among the three stars evolves the fastest, and supernova explosion of this star makes the other two stars r-rich. Then, the more massive one among the other two stars evolves through the AGB phase and produces s-process elements along with carbon. AGB mass transfer from this star makes the tertiary a CEMP-r/s star (Cohen et al. 2003; Jonsell et al. 2006). However, it seems very unlikely that after the supernova explosion the triple system survives for further mass transfer. Abate et al. (2016) could not reproduce the observed frequency of CEMP-r/s stars among CEMP-s stars and hence dismissed this scenario.

(iv) There are two proposed scenarios that considered binary systems where the s-process elements in the CEMP-r/s star are assumed to come from the primary through its AGB phase, and r-process elements are attributed to either a type 1.5 supernova (Zijlstra 2004; Jonsell et al. 2006) or an accretion-induced collapse (AIC) (Qian & Wasserburg 2003; Cohen et al. 2003).

In the first scenario, the primary star, being more massive than the other in the binary system, evolves through the AGB phase. During the AGB phase, the star produces and transfers s-process-rich material to the companion. Then, the AGB star may explode as a type 1.5 supernova and pollute the companion with r-process elements, making the secondary a CEMP-r/s star. Iben and Renzini (1983) gave the name ‘type 1.5 supernovae’ to the process when the degenerate cores of high-mass AGB stars, due to low mass-loss efficiency at low metallicity, remain as massive as to reach the Chandrasekhar mass limit and explode (Zijlstra 2004). However, Nomoto et al. (1976), Iben and Renzini (1983) and Lau et al. (2008) stated that type 1.5 supernova can destroy the primary star and hence disrupt the binary system. As most of the CEMP-r/s stars have been reported to be found in binary systems (Lucatello et al. 2005; Abate et al. 2016) rejected this scenario.

In the other scenario, after transferring the AGB-processed material (s-process elements) to the companion star, the primary star becomes a white dwarf. Then, as time progresses, the secondary star evolves to the giant branch and transfers material back to the white dwarf. This mass transfer may trigger an accretion-induced collapse (AIC) and hence pollute the secondary star with r-process material (Qian & Wasserburg 2003; Cohen et al. 2003). This scenario demands that the secondary star remain in the giant branch to fill the Roche lobe for the second phase of mass transfer. This scenario may be rejected as the observed CEMP-r/s stars are not always giants (Lugaro et al. 2009). In many cases, these stars are seen in the main sequence turn-off, making the accretion process difficult. (Abate et al. 2016) stated that the three phases of mass transfer work properly only if the orbital separation of the binary system is narrow, but the observed frequency of CEMP-r/s stars could not be reproduced even by considering a narrow separation. Also, there are uncertainties regarding the efficiency of AIC to produce enough r-process elements to match the observed abundances of heavy elements in CEMP-r/s stars (Qian & Woosley 1996; Qian & Wasserburg 2003).

(v) A formation scenario named ‘intermediate neutron-capture process or i-process’, similar to the formation scenarios for CH, Ba and CEMP-s stars, has been proposed recently to explain the abundances of CEMP-r/s stars. This scenario considers a binary system where one star is slightly more massive than the other. The more massive star evolves faster and passes through the AGB phase, polluting the companion with AGB-processed material. The difference of this scenario from the s-process-enrichment scenarios is that in this scenario a neutron density (n \(\sim \) 10\(^{15}\) cm\(^{-3}\)), which is intermediate to the neutron densities of both s-process and r-process, can produce the double enhancement seen in CEMP-r/s stars (Cowan and Rose 1977; Dardelet et al. 2014; Hampel et al. 2016). Cowan and Rose (1977), for the first-time, suggested the possibility of occurrence of i-process in AGB stars. They found that significantly high neutron density can be achieved by mixing hydrogen-rich material into the intershell region of AGB stars. Using i-process models (Dardelet et al. 2014) and Hampel et al. (2016) have successfully reproduced the abundance distribution of a number of well-known CEMP-r/s stars.

4.2 Comparison of the abundance pattern of the sample stars with i-process model

Dardelet et al. (2014) and Hampel et al. (2016) calculated i-process-model yields with slightly different approaches, but both groups could successfully reproduce the observed abundance pattern of CEMP-r/s stars. Single-zone nuclear network calculations were used in both the studies. Assuming proton ingestion from the H-rich envelope by the He-pulse-driven convective zone (PDCZ) to be responsible for the generation of higher neutron densities (\(n \sim 10^{15} {\text{cm}}^{-3}\)), Dardelet et al. (2014) used a constant combined mass fraction of C+H (= 0.7) in their simulations. They considered the termination time of the i-process as a free parameter for their calculations. On the other hand, Hampel et al. (2016) calculated the yields of the neutron-capture nucleosynthesis, assuming the nucleosynthesis to operate in the intershell region of an AGB star, at different constant neutron densities starting from n\(\sim \)\(10^{7}\) cm\(^{-3}\) to \(10^{15}\) cm\(^{-3}\). Dardelet et al. (2014) assumed the temperature and density for the He PDCZ to be \(2.0 \times 10^{8}\) K and 10\(^{4}\) g cm\(^{-3}\) respectively. These physical input parameters are chosen so as to prevent proton capture by \(^{13}\)N and allow the \(^{13}\)C(\(\alpha \), n)\(^{16}\)O reaction for neutron release. As a test, Hampel et al. (2016) tried to calculate the yields with a range of temperatures (\(1 \times 10^{8}\) K to \(2.2 \times 10^{8}\) K) and densities (800 g cm\(^{-3}\) to 3200 g cm\(^{-3}\)), but did not see significant changes in the results. However, for the final simulations they adopted the parameters (T = \(1.5 \times 10^{8}\) K and \(\rho \) = 1600 g cm\(^{-3}\)) of the intershell region of a low-metallicity (z = 10\(^{-4}\)), low-mass (M = 1 M\(_{\odot }\)) AGB star (Stancliffe et al. 2011). As initial abundances of the He PDCZ, Dardelet et al. (2014) considered solar abundances (except C and O), scaled down to z = 10\(^{-3}\). The abundances of C and O are taken to be X(\(^{12}\)C)=0.5 and X(\(^{16}\)O)=0.05, which are typical abundances for the He PDCZ. Hampel et al. (2016) adopted the constituents of the intershell region from Abate et al. (2015). A high neutron exposure of \(\tau \) \(\sim \) 495 mb\(^{-1}\) is ensured by adjusting the run times of the models. Due to such high neutron exposures, the abundance pattern of heavy elements and the seed nuclei comes to an equilibrium, which makes the element-to-element ratio a function of constant neutron density.

In the model of Dardelet et al. (2014), almost all of the \(^{12}\)C isotopes get transformed into \(^{13}\)N during the first second of run-time. Then, in 9.97 minutes, \(^{13}\)N decays to form \(^{13}\)C, which captures \(\alpha \) to release neutrons with high neutron densities through the reaction \(^{13}\)C(\(\alpha \), n)\(^{16}\)O. The neutron exposure (\(\tau \)) increases with time, reaching up to 10 − 50 mb\(^{-1}\), and subsequently the heavier elements are produced. This model could successfully reproduce the observed abundance pattern of three CEMP-r/s stars. On the other hand, Hampel et al. (2016) noticed that, when the neutron exposure was kept switched on, for lower neutron densities (n \(\sim \) \(10^{7}\) cm\(^{-3}\)) typical s-process abundance pattern is produced, with stable peaks of ls (Sr, Y, Zr) and hs (Ba, La, Ce) elements. But with higher neutron densities (n = \(10^{12}-10^{15}\) cm\(^{-3}\)), both ls and hs element peaks shift to lighter elements. In particular, a peak at \(^{135}\)I is formed due to the i-process neutron densities. Then, the neutron exposure is turned off for t = 10 Myr. During this time, it is noticed that unstable isotopes decay to produce stable isotopes at ls and hs peaks. The decay of \(^{135}\)I produces \(^{135}\)Ba. With increasing neutron densities, abundances of Ba and Eu are found to increase. This is how the abundance pattern gets modified due to i-process. Using this i-process model, Hampel et al. (2016) could successfully reproduce the observed abundance pattern of twenty CEMP-r/s stars, including the three previously reproduced by the i-process model of Dardelet et al. (2014).

We have used the model predictions ([X/Fe]) for neutron densities ranging from n \(\sim \) \(10^{9}\) to n \(\sim \) \(10^{15}\) cm\(^{-3}\) ( Hampel et al. 2016), and compared them with the elemental abundance pattern of our sample of CEMP-r/s stars. To examine whether the i-process models could reproduce the observed abundances of the sample stars we have followed the procedure discussed in Hampel et al. (2016) and used the equation

$$\begin{aligned} X = X_{i} \times (1-d) + X_{\odot }\times d, \end{aligned}$$
(1)

where \(X_{i}\) is the model yield, \(X_{\odot }\) is solar-scaled abundance, and d is a dilution factor.

Table 4 presents the values of the fit parameters neutron density, ‘d’ and ‘\(\chi ^{2}\)’ for each star. The best-fit neutron density for each of the sample stars is chosen for which we got the minimum value of ‘\(\chi ^{2}\)’. The best-fit models with appropriate neutron densities and corresponding dilution factors are shown in Figure 2. We have found that i-process models with neutron densities of n \(\sim \) \(10^{12}\) cm\(^{-3}\) to \(10^{14}\) cm\(^{-3}\) closely fit the observed abundances of the sample stars. We have obtained a neutron density of n \(\sim \) 10\(^{14}\) cm\(^{-3}\) for the star HE 1305\(+\)0007, which is same as that reported by Hampel et al. (2016) for the object. Hampel et al. (2019) tried to understand i-process nucleosynthesis through the abundances of Pb in CEMP-r/s stars. In their i-process models (Hampel et al. 2019) considered neutron exposure (\(\tau \)) as a free parameter along with dilution factor (d). Varying ‘d’ and ‘\(\tau \)’ at different constant neutron densities, they could fit the observed abundances of HE 1305+0007 and HD 209621 with models with neutron densities n \(\sim \) 10\(^{14}\) cm\(^{-3}\) and 10\(^{13}\) cm\(^{-3}\) respectively. We too got the best fit at n \(\sim \) 10\(^{13}\) cm\(^{-3}\) for the object HD 209621 taking only ‘d’ as a free parameter with constant ‘\(\tau \)’.

Figure 2
figure 2

Best-fitting i-process model (solid blue curve) for the sample stars. The points with error bars indicate the observed abundances.

Full size image
Table 4 Fit parameters of i-process model for the sample stars.
Full size table

5 Conclusions

In this study, we have examined if the currently available theoretical i-process stellar yields could adequately explain the enhancement in both s-process and r-process elements observed in a selected sample of CEMP-r/s stars. We have considered eight stars, of which the abundances of the heavy elements for three objects are taken from our previous studies (Goswami et al. 2006; Goswami & Aoki 2010; Purandardas et al. 2019) and those for the rest from other sources (Allen et al. 2012; Hansen et al. 2019).

With the help of a parametric-model-based study, we have estimated the contributions from s-process and r-process nucleosynthesis to the observed elemental abundances. It is found that similar contributions from both the processes are required to explain the observed abundance pattern of heavy elements in these stars.

With reference to the double enhancement seen in the sample stars, we have discussed different scenarios for formation of CEMP-r/s stars. The scenarios involving two stellar sites for the production of s-process and r-process elements are found to be not applicable to the sample of CEMP-r/s stars under this study. However, we have seen that the i-process models of Hampel et al. (2016) can satisfactorily reproduce the observed overabundance of heavy elements in these stars. The i-process stellar yields required to fit the observed abundance patterns are found to correspond to neutron densities as high as \(10^{12}\) cm\(^{-3}\) to \(10^{14}\) cm\(^{-3}\). The estimated low values of \(^{12}\)C/\(^{13}\)C ratio observed in the stars agree well with i-process predictions, indicating the extrinsic nature of carbon and the heavy elements.