Impact of current uncertainties in the ¹²C+¹²C nuclear reaction rate on intermediate-mass stars and massive white dwarfs

Stars more massive than $\sim 7\,M_{\odot}$ undergo carbon fusion in the core (Kippenhahn et al., 2013).¹¹1The exact mass threshold depends on metallicity and treatment of convective boundary mixing. The burning of ¹²C occurs at temperatures above $6\times 10^{8}$ K through the formation of compound nuclear states of ²⁴Mg, denoted as ²⁴Mg^∗, with excitation energies of 14 to 17 MeV above the ground level. The unstable ²⁴Mg^∗ states then decay through at least five channels (Li et al., 2020), namely:

At the typical energies of carbon fusion in stars, the first two channels have similar (high) probabilities, making them many orders of magnitude more likely than the other channels. Consequently, the reactions ¹²C$(^{12}$C,$\alpha)^{20}$Ne and ¹²C$(^{12}$C$,p)^{23}$Na dominate the total ¹²C+¹²C cross section. The cross-section of these reactions must be known with high accuracy down to the Gamow peak energy $E_{G}=1.5\pm 0.3$ MeV (for 5$\times 10^{8}$ K; Rolfs & Rodney, 1988) since it not only affects the production of ²⁰Ne and ²³Na, but also the subsequent evolutionary stages. The branching ratios of the $\alpha$ and $p$ exit channels determine the total amount of ²⁰Ne and ²³Na produced inside stars. Previous works adopt 56% (Caughlan & Fowler, 1988, hereinafter CF88) and 65% (Monpribat et al., 2022) as the branching ratio for the $\alpha$ channel. However, the probability of each exit channel becomes very uncertain at the typical temperatures that characterize C-burning (Pignatari et al., 2013).

Despite considerable experimental efforts, the total ¹²C+¹²C reaction rate remains uncertain at stellar temperatures. On the one hand, the heavy ion fusion studies performed by Jiang et al. (2007a) suggested that the fusion cross-section may be hindered at low energies, resulting in rates lower than the standard ones from CF88. On the other hand, low-energy experiments by Spillane et al. (2007) hint at the presence of resonant structure effects at lower energies that are not considered in many works and would lead to an important enhancement of the ¹²C+¹²C fusion rate at stellar temperatures. In the last decade, a significant effort has been made by the nuclear physics community, both experimentally and theoretically, to understand the challenging regime of astrophysical low energies of the ¹² C+¹²C fusion reaction (Jiang et al., 2018; Chien et al., 2018; Li et al., 2020; Mukhamedzanov, 2022; Tang & Ru, 2022; Morales-Gallegos et al., 2023). As recently reported by Monpribat et al. (2022), the present uncertainty of the ¹²C+¹²C rate still covers orders of magnitude at the range of temperatures of interest for astrophysical applications.

Stars with initial masses in the range $7\,M_{\odot}\lesssim M_{\rm ini}\lesssim 10\,M_{\odot}$ might eventually become massive white dwarfs (WD). While most WDs comprise He or CO cores, these massive WDs have O and Ne as their main ingredients. These objects are the result of the evolution of progenitor stars that reach temperatures high enough to ignite their CO cores under degenerate conditions (Garcia-Berro & Iben, 1994), as they evolve into the so-called super asymptotic giant branch (SAGB) phase. Classic works by Garcia-Berro & Iben (1994), Ritossa et al. (1996), García-Berro et al. (1997), and Iben et al. (1997) showed that the C-flash and subsequent C-burning leads to an oxygen-neon (ONe) core and, consequently, to an ONe WD (see Siess, 2006, 2007, 2010; Camisassa et al., 2019, and references therein) or an electron-capture supernova (Tominaga et al., 2013; Doherty et al., 2017), depending on the intensity of winds.

The chemical structure of SAGB progenitors at the end of the C-burning phase, and thus at the WD stage, depends on how the C-burning proceeds. In this sense, the current uncertainties in the total reaction rate for C-burning and its branching ratios could have a non-negligible impact on the predicted structure of ultra-massive WD. The impact of the uncertainty of the ¹²C+¹²C fusion rate on the properties of massive stars has been studied in several works (Gasques et al., 2007; Bennett et al., 2012; Pignatari et al., 2013; Chieffi et al., 2021; Monpribat et al., 2022; Dumont et al., 2024). However, none of them explored the consequences of such uncertainties on the evolution of intermediate-mass stars and the final composition of ultra-massive WDs. Additionally, the properties of the nonradial $g$-modes of pulsating WDs depend also on the inner distribution of elements (see, for example, De Gerónimo et al., 2017, 2022; Althaus et al., 2021). In this regard, the internal chemical profile left at the end of the C-burning phase plays an important role in the pulsation properties of ultra-massive pulsating WDs (see Córsico et al., 2019a; De Gerónimo et al., 2019, and references therein).

In this paper, we explore how the new measurements of the ¹²C+¹²C nuclear reaction rate and the branching ratio adopted during C-burning affect the final chemical composition and pulsations of ultra-massive WDs. The paper is organized as follows: in Section 2, we discuss the current status of the uncertainties in both the nuclear reaction rate and its branching ratios. In Section 3, we introduce the most important features of the computation of our numerical models, while in Sections 4 and 5 we explore the impact of those uncertainties on the properties of SAGB progenitors and pulsating ultra-massive WDs. Finally, in Section 6, we provide some concluding remarks.

CF88 set a milestone in developing analytical formulae for several nuclear reaction rates, later used as standards in the computation of astrophysical numerical models. The ¹²C+¹²C nuclear reaction stands out as very important for stellar evolution and yet remains subject to large uncertainties. This reaction has been studied intensively in recent decades (see Patterson et al., 1969; Becker et al., 1981; Spillane et al., 2007; Bucher et al., 2015, and references therein). Despite the experimental efforts made and the consistent results obtained for energies near the Coulomb barrier, the total ¹²C+¹²C fusion reaction rate remains uncertain at temperatures of astrophysical interest, with different experiments differing substantially (see Pignatari et al., 2013; Li et al., 2020; Monpribat et al., 2022, and references therein for a detailed discussion). This is due both to the fact that the experimental background noise is high at low energies, as well as to the fact that extrapolation of the experimental data to energies below the Coulomb barrier is affected by the strong resonant behavior of the ¹²C+¹²C cross-section. Recent studies suggest that the latter may suffer from hindrance phenomena at low energies (Jiang et al., 2007b), resulting in a lower rate than the widely used one from CF88. Additionally, resonant structures that would increase the reaction rate have been found at low energies (Spillane et al., 2007).

Recently, Monpribat et al. (2022) derived the most up-to-date reaction rates for carbon fusion, based on the measurement published by the STaged ELectron Laser Acceleration (STELLA) experiment (Fruet et al., 2020). In that work, the authors provided two updated formulae that account for the fusion hindrance phenomenon,²²2The fusion hindrance phenomenon corresponds to a sudden fall-off of the cross-section. labeled HIN and HINRES in their study, based on different assumptions for the resonant behavior at low energies. Specifically, the HINRES rate differs from the HIN one in that the former includes the effect of a possible low-energy resonance (Spillane et al., 2007), better fitting the measured cross-section. These formulae can be incorporated into the computation of stellar models. In Fig. 1 we show the ratio between the total HIN (green line) and HINRES (red line) reaction rates and the standard value from CF88 as a function of the temperature $T_{9}$ (in units of $10^{9}$K). For the typical temperatures characterizing C-burning regions inside stars (shaded region), the HINRES rate behaves very similarly to the one provided by CF88, whereas the difference with respect to the HIN rate can reach more than a factor of ten.

In addition to the uncertainties in the total ¹²C+¹²C$\,\longrightarrow\,^{24}$Mg^∗ cross-section, there are large uncertainties in the ²⁴Mg^∗ decay channels. The uncertainty in the relative strengths of the ²⁴Mg^∗ $\alpha$- and $p$-decay channels has a relevant impact on stellar nucleosynthesis calculations. The original branching ratio provided by CF88 was [56/44] for the [$\alpha/p$] channels, respectively. Pignatari et al. (2013) adopted branching ratios of [65/35] instead, and explored the consequences of extreme values, [5/95] and [95/5] using a simple single-zone post-processing calculation. Recently, a Chinese-lead international collaboration has been reanalyzing the ¹²C+¹²C reaction (Zhang et al., 2020a, b), and, in particular, its branching ratios (Li et al., 2020). Preliminary results indicate that, in the range of relevant stellar temperatures ($0.6<T_{9}<0.9$), the relative strength of the $\alpha$-channel to the $p$-channel decreases by about 30% with temperature. More importantly, the uncertainty in the relative strengths of both decay channels encompasses one order of magnitude (see next sections).

The SAGB models employed in this work were computed with the stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA) version r21.12.1 (Paxton et al., 2011, 2013, 2015, 2018, 2019). Most of the adopted input physics corresponds to the default options that are described in detail in those papers, and will thus not be repeated here. The nuclear network adopted (sagb_NeNa_MgAl.net) accounts for 28 isotopes n, ^1,2H, ^3,4He, ⁷Li, ⁷Be, ⁸B, ^12,13C, ^13-15N, ^16-18O, ¹⁹F, ^20-22Ne, ^21-23Na, ^24-26Mg, ^25-27Al, including all the dominant nuclear reactions. Most of these reactions are taken from CF88 and Angulo et al. (1999), with some exceptions. Convective boundary mixing (CBM) was only adopted during core H and He burning, with the suggestion of Freytag et al. (1996) of an exponentially decaying velocity field. Here, the diffusion coefficient is adopted according to Herwig et al. (1997), with the free parameter $f=0.017$. The main impact of CBM in these stages is to decrease the initial mass required for a progenitor star to reach C-ignition by about $2\,M_{\odot}$. CBM at the C-burning convective zones and at the bottom of the convective envelope was not included.

Our evolutionary sequences were computed from the zero-age main sequence (ZAMS) through central hydrogen and helium burning, up to the end of the carbon burning stage, before the thermally unstable phase, for models with initial masses $6.85\leq M_{\rm ZAMS}/M_{\odot}\leq 8.50$ and metallicity $Z=0.02$, assuming a solar-scaled composition. Each model was computed with a spatial resolution greater than 6000 spatial points from center to surface, while the temporal resolution achieved is of the order of 14 yrs, similar to those adopted in Farmer et al. (2015). The evolutionary behavior of SAGB progenitors before carbon ignition is very similar to that of intermediate-mass stars that end up as CO WDs, well documented in previous works (Garcia-Berro & Iben, 1994; Siess, 2006, 2010). The development of carbon burning, under the hypothesis of a strict Schwarzschild criterion (Schwarzschild, 1906) for the delimitation of the convective region, is characterized by two different stages (Garcia-Berro & Iben, 1994; Siess, 2006). The first corresponds to the ignition of C at the point of maximum temperature inside the partially degenerate CO core, inducing a thermal runaway called the carbon flash. The sudden energy injection by the C-flash leads to a convective zone, which extends outward from the point of maximum temperature. The second stage corresponds to the development of a flame that propagates to the center and transforms the CO core into an ONe core (Garcia-Berro & Iben, 1994; Siess, 2006). During the C-burning phase, we took into account three different reaction rates for the ¹²C+¹²C reaction, namely the aforementioned HIN and HINRES ones from Monpribat et al. (2022) and the value from CF88, which has been widely used in previous evolutionary computations. Mass loss during AGB was considered adopting the Bloecker’s wind prescription with a scaling factor of 0.1 (Bloecker, 1995).

The adoption of the Schwarzschild Criterion for convective instability in our pre-WD models implies that Rayleigh-Taylor unstable regions are not identified as such in our models. In our models, the CO-core during the early AGB develops a off-centered peak in the oxygen profile (Salaris et al., 1997). Such a profile would be unstable to Rayleigh-Taylor instabilities (Stevenson & Salpeter, 1977). A proper assessment of mixing processes driven by chemical gradients during the early- and TP-AGB phases would require the computation of these processes during the formation of the oxygen peak. For simplicity, in our computations, the chemical profile of the CO core is homogenized by an ad-hoc algorithm just before the WD cooling phase.

This section is devoted to analyzing the effect of the different assumptions studied previously on the chemical structure and composition of ultramassive WDs, their cooling timescales, crystallization, and pulsation properties.

Each WD evolutionary sequence was computed from the point of maximum temperature of the cooling track at high luminosities, down to the development of the Debye cooling at low surface luminosities. Initial WD structures were constructed by mapping the detailed chemical structure of the H-free core at the end of the central C-burning stage into an already existing WD thermal structure. Our DA WDs models have a H content of $M_{\rm H}\sim 1.5\times 10^{-6}M_{\rm WD}$. The evolution and structure of the WD models presented in this section were computed with the LPCODE evolutionary code (for details, see Althaus et al., 2003, 2005, 2015, 2021; Miller Bertolami, 2016). During crystallization, we took into account the release of latent heat and changes in the core chemical composition resulting from phase separation upon crystallization, using phase diagrams suitable for C/O or O/Ne plasmas (Camisassa et al., 2019, 2022). For the sake of clarity, we selected a fiducial model of initial mass $7.50\,M_{\odot}$, which corresponds to a $M_{\rm WD}\sim 1.15\,M_{\odot}$.

As discussed in the previous sections, the adoption of extreme reaction rates (CF88 and HIN) during the C-burning phase leads to small differences in the chemical structure of the ultramassive WD progenitors at the end of the C-burning phase. The next evolutionary stage is the thermally unstable phase and consists of the build-up of the most external part of the core. After the WD starts to cool and the Rayleigh-Taylor re-homogenization has taken place, the chemical structures of the WDs computed with different nuclear reaction rates are almost indistinguishable. As the WD keeps cooling, crystallization gets underway and, by the time the star reaches the ZZ Ceti (DAV) instability strip (at a temperature of $\sim$ 12 000 K), the cooling times and the size of the crystallized core are of 1743 (1721) $\times 10^{6}$ yr and 92.9 (92.2) %, respectively, if the CF88 (HIN) rate is adopted, meaning small differences of 0.5 and 1.2%.

Regarding the adoption of different branching ratios, the differences found in the distribution of the most important chemical elements are more noticeable. In the top panel of Fig. 8, we show the chemical structure in terms of the outer mass fraction $\rm log(1-M_{r}/M_{*})$⁴⁴4 $\rm M_{r}$ and $\rm M_{*}$ stands for the mass coordinate and total mass of the star, respectively. pre- and post-Rayleigh-Taylor re-homogenization, at high temperatures, for the fiducial model that accounts for the following branching ratios: [50/50], [65/35], [80/20], and [90/10]. The central ²³Na content remains always below $\sim$ 8% and, as higher production rates are adopted, the ²⁰Ne central content increases from 30% up to $\sim$ 50% for the [90/10] branching ratio. In the latter case, the ²⁰Ne content produced at the outer part of the core surpasses the ¹⁶O content before rehomogenization. However, the final chemical structure of the WD is that of a typical ONe core WD, but composed of almost equal parts of ¹⁶O and ²⁰Ne. The C/O mantle on top of the O/Ne core is identical for each case.

In the lower panel of Fig. 8, we show the same chemical profiles but at the ZZ Ceti stage ($T\sim 12\,000$ K, when the crystallization of the core has reached more than 90%, indicated as a shaded region in the plot). As every model has the same mass, the differences in the percentage of the crystallized core, which amounts to up to 0.8%, come strictly from the differences in their composition. The combined effect of the different degrees of crystallization and inner composition leads to differences in the WD cooling times of at most 1.3%.

As noted in Section 4.2, under some of the conditions discussed in the present paper, the carbon flame is quenched prematurely, forming a hybrid core WD. In Fig. 9, we show the chemical structure for the particular case of a hybrid core, with $M_{\rm WD}/M_{\odot}=1.05$ ($M_{\rm ZAMS}/M_{\odot}=6.85$) pre- and post-Rayleigh-Taylor re-homogenization (upper and lower panel, respectively). The total ²⁰Ne content produced during the short-lived C-burning phase is then redistributed (as are the other species) throughout the core. Consequently, after re-homogenization, the WD model’s structure resembles that of a pure CO-core WD, though with a non-negligible ²⁰Ne content ($\sim$10%) in the core. These new kinds of objects predicted by our study, which can have masses up to $M_{\rm WD}/M_{\odot}\sim 1.11$, differ substantially from objects with pure CO cores. The presence of ²⁰Ne modifies the structure of the WD (particularly its central density) so that differences in the evolution are significant. This can be seen in Fig. 10, where we show the evolution of the effective temperature for a pure CO and the hybrid model.⁵⁵5The pure CO-core model was computed from the evolution of the hybrid CONe-core model progenitor but quenching C-ignition. Both models start at the same evolutionary point in the cooling track but differ in their central compositions, which are 30%-67% (¹²C-¹⁶O) for the pure CO-core model and 23%-61%-10% (¹²C-¹⁶O-²⁰Ne) for the hybrid model. As the models cool down, the hybrid one starts crystallizing earlier than the CO-core model. The crystallization onset occurs at 751$\times 10^{6}$ yr for the hybrid composition, while for the pure CO-core model, it starts at 870$\times 10^{6}$ yr. At 12 000 K (grey line), the age differences are of $\sim 6\%$.

The comparison of their chemical structures at this point can be seen in Fig. 11, where we show the distribution of the most important chemical species for both models. Both the absence of ²⁰Ne in the pure CO-core model and the crystallized size of the core are the most prominent differences at this point, while the ⁴He-buffer region and pure ¹H-envelope remain the same.

In this work, we implemented the recently derived total nuclear reaction rate for carbon fusion, ¹²C+¹²C, from Monpribat et al. (2022), with the goal of evaluating its impact on computations of the structure and evolution of ultra-massive WDs and their progenitors, as compared to the case where the canonical ¹²C+¹²C rate from CF88 is used. We also explored how the current uncertainty in the branching ratios for the ¹²C+¹²C reaction’s $\alpha$ and $p$ exit channels, which are the dominant ones at temperatures of astrophysical interest, affect the internal composition and evolution of these stars and their progenitors.

Our extensive numerical experiments were carried out using MESA. Specifically, we computed evolutionary sequences, from the ZAMS up to the end of the C-burning phase, for models with initial masses $6.85\leq M_{\rm ZAMS}/M_{\odot}\leq 8.50$ and a metallicity $Z=0.02$. In addition to exploring the ¹²C+¹²C rates from Monpribat et al. (2022) and CF88, a wide range of $[\alpha/p]$ branching ratios was also considered, from [50/50] up to [90/10]. The resulting structures were subsequently evolved, using LPCODE, along the WD cooling track down to the ZZ Ceti (DAV) instability strip. When the models reached the latter phase, we computed their pulsation properties using the LP-PUL code.

We found that using less efficient nuclear reaction rates results in a late onset of the C-burning phase, larger cores, higher burning temperatures, and, consequently, shorter C-burning phase lifetimes. Despite these differences in the structure and evolution of SAGB progenitors, the impact on the distribution of the chemical elements is almost negligible. In contrast, the existing uncertainties in the relative efficiency of the $\alpha$ and $p$ exit channels constitute the most important uncertainty in determining the final chemical structures of SAGB progenitors. Differences in the central ²⁰Ne abundances can reach up to 17% within the range of $[\alpha/p]$ ratios explored in our study, in the sense that a higher ²⁰Ne content is achieved when the $\alpha$ channel is more dominant. Moreover, we found that higher production of ²⁰Ne translates into smaller initial masses needed for C-ignition and longer C-burning phase lifetimes.

An interesting result, derived from the exploration of the minimum mass needed for C-ignition, is that, for a specific range of masses that depend on the adopted total nuclear reaction rate and branching ratio (e.g., $6.85\leq M_{\rm ZAMS}/M_{\odot}<7.10$, for the CF88 rate and a branching ratio of [65/35]), carbon burns partially in the models’ interiors. In such cases, the final chemical structure at the end of the C-burning phase consists of a CO-core surrounded by an ONe mantle. Depending on the amount of ²⁰Ne produced during this stage, the progeny could be a CO-core WD or a hybrid CONe-core WD.

As for the impact on the ONe-core WD evolution due to the use of different nuclear reactions and branching ratios, we found differences in the cooling times and the size of the crystallized core of at most 1.3 and 0.8%, respectively. We found that the impact on the pulsation properties of these stars is also negligible.

Regarding the hybrid CONe-core WDs, we found that they differ substantially in their evolution, compared with those composed of pure CO cores. Our results show that even as little as 10% of ²⁰Ne in its interior can modify the structure of the WD in such a way that crystallization starts earlier and, by the time the star reaches the ZZ Ceti instability strip, differences in the crystallized portion of the star and its cooling time can reach up to 15% and 6%, respectively. This result is also reflected in the pulsation properties. By comparing its forward period spacing, we find that the hybrid WD has less frequent minima (larger trapping cycle) than its CO-core counterpart. This is because the hybrid model has a larger crystallized core and, consequently, a smaller resonating cavity.

In conclusion, our study reveals that current uncertainties in both the total ¹²C+¹²C reaction rate and its branching ratios can have a significant impact upon the late stages of evolution of intermediate-mass stars and their progeny. As demonstrated by other authors, uncertainties in the ¹²C+¹²C rate also impact the advanced evolutionary stages of higher-mass stars. Given its far-reaching astrophysical implications, further experimental work is thus urgently needed to properly constrain the ¹²C+¹²C rate and branching ratios at astrophysically relevant energies.

We wish to acknowledge the suggestions and comments of the anonymous referee who strongly improved the original version of this work. This work was supported by PIP 112-200801-00940 grant from CONICET, grant G149 from the University of La Plata, PIP-2971 from CONICET (Argentina) and by PICT 2020-03316 from Agencia I+D+i (Argentina). This research has made use of the NASA Astrophysics Data System.