Recent determinations of the total rate of the SUP12SUPCSUP12SUPC nuclear reaction show nonnegligible differences with the reference reaction rate commonly used in previous stellar simulations. In addition the current uncertainties in determining each exit channel constitute one of the main uncertainties in shaping the inner structure of super asymptotic giant branch stars that could have a measurable impact on the properties of pulsating ultramassive white dwarfs WDs. We explore how new determinations of the nuclear reaction rate and its branching ratios affect the evolution of WD progenitors. We show that the current uncertainties in the branching ratios constitute the main uncertainty factor in determining the inner composition of ultramassive WDs and their progenitors. We found that the use of extreme branching ratios leads to differences in the central abundances of SUP20SUPNe of at most 17 which are translated into differences of at most 1.3 and 0.8 in the cooling times and size of the crystallized core respectively. However the impact on the pulsation properties is small less than 1 s for the asymptotic period spacing. We found that the carbon burns partially in the interior of ultramassive WD progenitors within a particular range of masses leaving a hybrid CONecore composition in their cores. The evolution of these new kinds of predicted objects differs substantially from the evolution of objects with pure CO cores. Differences in the size of the crystallized core and cooling times of up to 15 and 6 respectively lead to distinct patterns in the period spacing distribution.
['DA stars', 'Stellar evolution', 'Nuclear abundances', '348', '1599', '1128', 'Astrophysics - Solar and Stellar Astrophysics']
Impact of Current Uncertainties in the SUP12SUPCSUP12SUPC Nuclear Reaction Rate on Intermediatemass Stars and Massive White Dwarfs
{'Impact of current uncertainties in the 12 C + 12 C nuclear reaction rate on intermediate-mass stars and massive white dwarfs': "F rancisco C. D e G er ' onimo , 1 M arcelo M. M iller B ertolami , 1 T iara B attich , 2 X iaodong T ang , 3, 4 M' arcio C atelan , 5, 6 A lejandro H. C' orsico , 1 Y unjun L i , 7 X iao F ang , 8 and L eandro G. A lthaus 1  \n1 Instituto de Astrof´ısica de La Plata, CONICET-UNLP, La Plata, Argentina  \n2 Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild Strasse 1, 85748 Garching, Germany  \n3 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  \n4 School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China  \n5  \nInstituto de Astrof´ısica, Pontificia Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, 7820436 Macul, Santiago, Chile  \n6 Millennium Institute of Astrophysics, Nuncio Monse˜nor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile  \n7 China Institute of Atomic Energy, Beijing 102413, China  \n8 Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China", 'ABSTRACT': 'Recent determinations of the total rate of the 12 C + 12 C nuclear reaction show non-negligible di ff erences with the reference reaction rate commonly used in previous stellar simulations. In addition, the current uncertainties in determining each exit channel constitute one of the main uncertainties in shaping the inner structure of super asymptotic giant branch stars that could have a measurable impact on the properties of pulsating ultra-massive white dwarfs (WDs). We explore how new determinations of the nuclear reaction rate and its branching ratios a ff ect the evolution of WD progenitors. We show that the current uncertainties in the branching ratios constitute the main uncertainty factor in determining the inner composition of ultra-massive WDs and their progenitors. We found that the use of extreme branching ratios leads to di ff erences in the central abundances of 20 Ne of at most 17%, which are translated into di ff erences of at most 1.3 and 0.8% in the cooling times and size of the crystallized core. However, the impact on the pulsation properties is small, less than 1 s for the asymptotic period spacing. We found that the carbon burns partially in the interior of ultra-massive WD progenitors within a particular range of masses, leaving a hybrid CONe-core composition in their cores. The evolution of these new kinds of predicted objects di ff ers substantially from the evolution of objects with pure CO cores. Di ff erences in the size of the crystallized core and cooling times of up to 15 and 6%, respectively leading to distinct patterns in the period spacing distribution.  \nKeywords: stars:evolution-stars:interiors-white dwarfs, nuclear reactions, abundances', '1. INTRODUCTION': "Stars more massive than ∼ 7 M ⊙ undergo carbon fusion in the core (Kippenhahn et al. 2013). 1 The burning of 12 C occurs at temperatures above 6 × 10 8 Kthrough the formation of compound nuclear states of 24 Mg, denoted as 24 Mg ∗ , with excitation energies of 14 to 17 MeV above the ground level. The unstable 24 Mg ∗ states then decay through at least five  \nchannels (Li et al. 2020), namely:  \n12 C + 12 C → 24 Mg ∗ → 20 Ne + α ( Q = 4 . 62 MeV) , → 23 Na + p ( Q = 2 . 24 MeV) , → 23 Mg + n ( Q = -2 . 60 MeV) , → 16 O + 8 Be ( Q = -0 . 20 MeV) , → 24 Mg + γ ( Q = 14 . 93 MeV) . (1)  \nAt 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 12 C( 12 C, α ) 20 Ne and 12 C( 12 C , p ) 23 Na dominate the total 12 C + 12 C cross section. The cross-section of these reactions must be known with high accuracy down to the Gamow peak energy EG =  \n1 . 5 ± 0 . 3 MeV (for 5 × 10 8 K; Rolfs & Rodney 1988) since it not only a ff ects the production of 20 Ne and 23 Na, but also the subsequent evolutionary stages. The branching ratios of the α and p exit channels determine the total amount of 20 Ne and 23 Na produced inside stars. Previous works adopt 56% (Caughlan & Fowler 1988, hereinafter CF88) and 65% (Monpribat et al. 2022) as the branching ratio for the α channel. However, the probability of each exit channel becomes very uncertain at the typical temperatures that characterize C-burning (Pignatari et al. 2013).  \nDespite considerable experimental e ff orts, the total 12 C + 12 C 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 e ff ects at lower energies that are not considered in many works and would lead to an important enhancement of the 12 C + 12 C fusion rate at stellar temperatures. In the last decade, a significant e ff ort 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 + 12 C 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 12 C + 12 C rate still covers orders of magnitude at the range of temperatures of interest for astrophysical applications.  \nStars with initial masses in the range 7 M ⊙ ≲ M ini ≲ 10 M ⊙ 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), Garc'ıa-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.  \nThe 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  \nuncertainty of the 12 C + 12 C 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; Chie ffi 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 Ger'onimo 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 C'orsico et al. 2019a; De Ger'onimo et al. 2019, and references therein).  \nIn this paper, we explore how the new measurements of the 12 C + 12 C nuclear reaction rate and the branching ratio adopted during C-burning a ff ect 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.", '2. UNCERTAINTIES IN THE NUCLEAR REACTION RATE AND BRANCHING RATIOS': '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 12 C + 12 C 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 e ff orts made and the consistent results obtained for energies near the Coulomb barrier, the total 12 C + 12 C fusion reaction rate remains uncertain at temperatures of astrophysical interest, with di ff erent experiments di ff ering 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 a ff ected by the strong resonant behavior of the 12 C + 12 C cross-section. Recent studies suggest that the latter may su ff er from hindrance phenomena at low energies (Jiang et al. 2007b), resulting in a lower rate than the widely used one from CF88. Addition-  \nFigure 2. Run of the maximum temperature of the flame (upper panel), the size of the CO core (middle panel), and the total 20 Ne content left (lower panel) for all the reaction rates adopted for a M ZAMS = 8 M ⊙ model during the lapse of the carbon burning phase.  \n<!-- image -->  \nFigure 1. Ratio of the total carbon burning reaction rate between Monpribat et al. (2022) and CF88 as a function of the temperature T 9 = T / 10 9 . The green line represents the HIN rates, whereas the HINRES rates are shown in red. The dashed blue line represents a ratio of 1. The shaded region depicts the range of typical temperatures characterizing 12 C burning inside SAGB stars.  \n<!-- image -->  \nally, resonant structures that would increase the reaction rate have been found at low energies (Spillane et al. 2007).  \nRecently, Monpribat et al. (2022) derived the most up-todate 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, 2 labeled HIN and HINRES in their study, based on di ff erent assumptions for the resonant behavior at low energies. Specifically, the HINRES rate di ff ers 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 di ff erence with respect to the HIN rate can reach more than a factor of ten.  \nIn addition to the uncertainties in the total 12 C + 12 C -→ 24 Mg ∗ cross-section, there are large uncertainties in the 24 Mg ∗ decay channels. The uncertainty in the relative strengths of the 24 Mg ∗ α - and p -decay channels has a relevant impact on stellar nucleosynthesis calculations. The original branching ratio provided by CF88 was [56 / 44] for  \nthe [ α/ 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 12 C + 12 C 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 α -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).', '3. STELLAR MODELS': "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 , 2 H, 3 , 4 He, 7 Li, 7 Be, 8 B, 12 , 13 C, 13 -15 N, 16 -18 O, 19 F, 20 -22 Ne, 21 -23 Na, 24 -26 Mg, 25 -27 Al, 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 di ff usion coe ffi cient 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  \nFigure 3. Chemical profiles for the most abundant elements at the end of the carbon burning phase for a M ZAMS = 8 M ⊙ model for the three nuclear reaction rates adopted, namely CF88 (solid lines), HINRES (dashed lines), and HIN (dotted lines). The branching ratio was set to 65% and 35% for the α and p channels, respectively. In this plot, Xi is the element abundance per mass, whereas Mr is the Lagrangian mass coordinate.  \n<!-- image -->  \nC-ignition by about 2 M ⊙ . CBM at the C-burning convective zones and at the bottom of the convective envelope was not included.  \nOur evolutionary sequences were computed from the zeroage 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 ≤ M ZAMS / M ⊙ ≤ 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 di ff erent 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 Cburning phase, we took into account three di ff erent reaction  \nFigure 4. The S ∗ ratio of the alpha and proton channels. The measurements of Kettner et al. (1980), Aguilera et al. (2006), and Spillane et al. (2007) are shown as filled circles in the colors of dark green, red, and brown, respectively. The best fit, upper and lower limits are shown as red lines. These limits are set at factors of 2.5 and 0.5 of the best fit, respectively  \n<!-- image -->  \nrates for the 12 C + 12 C 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).  \nThe adoption of the Schwarzschild Criterion for convective instability in our pre-WD models implies that RayleighTaylor unstable regions are not identified as such in our models. In our models, the CO-core during the early AGB develops a o ff -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.", '4.1. From uncertainties in the total rate': "In this section, we computed the evolutionary sequences adopting di ff erent reaction rates with a fixed branching ratio of 65% for the α channel. In Fig. 2, we show the impact of the di ff erences in the reaction rates shown in Fig. 1 on the properties of carbon burning. Specifically, we show the evolution of the maximum temperature of the flame ( T max, upper panel), the size of the CO core (middle panel), and the total 20 Ne produced (lower panel) as a function of the age, for an 8 M ⊙ progenitor model. As it is known, in the case of burning shells on top of electron-degenerate cores, the temperatures of the burning shell and the core increase with an increment of the size of the core (Miller Bertolami 2022). Sequences computed with a higher 12 C + 12 C reaction rate re-  \nFigure 5. Ratio of the 12 C( 12 C, α ) 20 Ne to the 12 C( 12 C,p) 23 Na channel as a function of temperature. The color band indicates the region between the estimated upper and lower limits.  \n<!-- image -->  \nre slightly lower core temperatures to ignite carbon and undergo the carbon flash, which thus takes place at lower core masses and temperatures. Conversely, the sequences computed with the less e ffi cient reaction rates are characterized by a late onset of the C-flash, larger cores at the moment of the C-flash, and higher C-burning temperatures, as also seen in the study of more massive stars (Bennett et al. 2012; Pignatari et al. 2013). Once carbon starts to burn quiescently, more massive cores are forced, by hydrostatic equilibrium, to burn carbon faster. Consequently, those sequences with lower 12 C + 12 Creaction rates that ignited carbon with slightly more massive cores display shorter carbon-burning lifetimes. Di ff erences in the size of the cores at the moment of the Cflash, in their T max, and in the lifetime of the C-burning phase amount to 0.5, 13, and 40%, respectively.  \nDespite the di ff erences in the burning temperature and the e ffi ciency of the reaction rate, the total 20 Ne content for each sequence is practically identical, as seen in the lower panel of Fig. 2. This is also observed with the distribution of the most important elements (see Fig. 3). The most noticeable di ff erences in the chemical structure arise in the outermost part of the core, as shaped at the final stages of C-burning. However, as a general trend, di ff erences in the resulting distribution of elements are small.  \nIn the next sections, we will study the impact of such di ff erences on the age, crystallization, and pulsation properties of ultramassive DAV stars, which are H-rich WDs that present g -mode pulsations (C'orsico et al. 2019a; Catelan & Smith 2015). Before, we will analyze the uncertainties coming from the branching ratios of the 12 C + 12 C nuclear reaction.", '4.2. From uncertainties in the branching ratios': "As mentioned in Sect. 1, the 12 C( 12 C, α ) 20 Ne and 12 C( 12 C , p ) 23 Na reactions dominate carbon fusion inside  \nstars, leaving 20 Ne and 23 Na as the main end products. The probability of each exit channel becomes very uncertain for the typical temperatures at which C-burning takes place in stars (Pignatari et al. 2013; Tang & Ru 2022).  \nFrom the measurements made by Kettner et al. (1980), Aguilera et al. (2006), and Spillane et al. (2007), we assessed the estimation of the ratio of the α - to the p -channel, S ∗ α / S ∗ p , after correcting for the branching ratio of the missing channels with the procedure described in Li et al. (2020). The theoretical prediction is calculated using TALYS (Koning et al. 2005). The TALYS prediction is scaled by a factor 1.33 to match with the average of the experimental data. The results are shown in Fig. 4. We recommend upper and lower limits to be set by scaling the best fit with factors of 2.5 and 0.5, respectively. These limits covers all the measurements above E c . m . = 3.3 MeV. At lower energies, the measurement of Kettner et al. (1980) agrees with the limits while the measurement of Spillane et al. (2007) becomes significantly higher than that of Kettner et al. (1980) and exceeds the upper limit by a factor of 2.7 around E c . m . > 3.16 MeV. Further experimental and theoretical investigations are needed to resolve the discrepancy.  \nThe reaction rates of the α - and p -channels are calculated based on the extrapolated and experimental S ∗ factor or cross-sections. At Ec . m . > 2.7 MeV, S ∗ α and S ∗ p obtained from the experimental measurements are used after correcting the missing channels in their measurement. S ∗ n is estimated with the experimental data and theoretical extrapolation from Bucher et al. (2015) and Bucher (2014). At lower energies, the S ∗ factor of 12 C + 12 C is generated with a uniform distribution bound by the upper and lower limits recommended in Li et al. (2020). S ∗ α and S ∗ p are generated with the branching ratios of the α - to p -channels with a uniform distribution bound by the upper and lower limits shown in Fig. 4. The reaction rate ratio of the Ne and Na channels are shown in Fig. 5. Our results show a decrease with temperature by about 40% from T 9 = 0 . 6 to T 9 = 0 . 9, and are significantly higher than the value α/ p = 1.27 recommended by CF88, at the temperatures of interest.  \nTo quantify the impact of these uncertainties on the chemical composition of stars at the end of the SAGB phase, we performed calculations considering a wide range of branching ratios. To this end, we computed evolutionary sequences for stellar models with masses 6 . 80 ≤ M ZAMS / M ⊙ ≤ 8 . 50 considering the CF88 and HIN reaction rates and, from the results shown in Fig. 5 that indicates α/ p > 1 in the range of temperatures of interest, branching ratios of [ α/ p ] of [90 / 10], [80 / 20], [65 / 35], [60 / 40], [55 / 45], and [50 / 50]. Higher branching ratios for the α channel translate into smaller initial masses needed for the occurrence of the C-flash (by < 1 . 5%). This happens because the total energy released per burned 12 C nuclei by the α channel is higher. Consequently,  \na lower burning rate is required to reach the carbon-burning luminosity at which the runaway occurs. This means that the core ignites carbon at a slightly lower core temperature, and consequently a smaller core mass. These di ff erences are minor, and the onset of the C-burning flash occurs almost at the same age, and di ff erences in the sizes of the cores are at most of 0.1% Conversely, the lifetime of the C-burning phase increases by up to 20% when the α channel is more e ffi cient due to the larger energy released per burnt nuclei.  \nFor some of our sequences, the flame is quenched before reaching the star's center. This means that 12 C does not burn completely inside these objects and therefore the chemical structure left at the end of this stage is that of a core composed of CO surrounded by an ONe mantle. This depends on the reaction rate adopted and its branching ratio. Particularly, if the CF88 (HIN) reaction rate is adopted with the [65 / 35] branching ratio, this happens for models with initial masses 6 . 85 ≤ M ZAMS / M ⊙ < 7 . 10 (7 . 25 < M ZAMS / M ⊙ < 7 . 40). The resulting object will be a CO-core or hybrid 3 WD, depending on the total amount of 12 C burned. In Fig. 6, we show the chemical structure for a M ZAMS = 6 . 85 M ⊙ model. An unburnt CO core is surrounded by an ONe mantle similar to those found when overshooting is adopted during the C-burning phase (Denissenkov et al. 2013; De Ger'onimo et al. 2022). However, 3D hydrodynamic simulations revealed that this quenching of the carbon flame might be an artificial consequence of overshooting prescriptions adopted in 1D models (see Lecoanet et al. 2016). Conversely, our computations do not include any extra-mixing process during the C-burning stage, and consequently, the quenching of the C-flame is of a di ff erent nature. Depending on the size of the ONe mantle, i.e. the total amount of 20 Ne produced, the chemical structure at the WD phase, post RayleighTaylor homogenization, could be a WD with a core composed mostly of CO (see next sections). It should be noted that the initial mass ranges for the di ff erent regimes discussed in this paper are dependent on the assumptions on convective boundary mixing. More extended convective boundary mixing in the hydrogen burning core on the main sequence would imply lower initial masses for C-ignition (Wagsta ff et al. 2020). Convective boundary mixing at the carbon-burning flame is also expected to strongly a ff ect the development of the carbon flash (Denissenkov et al. 2013; De Ger'onimo et al. 2022).  \nIn Fig. 7, we show the chemical profiles for the 7 . 50 M ⊙ model at the end of the C-burning phase, assuming the branching ratios adopted in the literature, namely [55 / 45] (CF88) and [65 / 35] (Pignatari et al. 2013), together with the  \nFigure 6. Chemical structure for a 6 . 85 M ⊙ model computed with the CF88 reaction rate, at the end of the C-burning phase. The flame is quenched before reaching the star's center, leaving a core composed of CO surrounded by an ONe mantle.  \n<!-- image -->  \nFigure 7. Chemical structure for a 7 . 50 M ⊙ model at the end of the C-burning phase, for models adopting the CF88 reaction rate with [55 / 45], [65 / 35] and [90 / 10] branching ratios (dashed, dotted, and solid lines, respectively).  \n<!-- image -->  \nextreme case [90 / 10]. Depending on which of these branching ratios is selected, di ff erences in the central abundances of O and Ne can reach from 3% to 17%. The larger differences arise when comparing the calculations adopting the [55 / 45] and [90 / 10] branching ratios. For the [90 / 10] branching ratio, the 20 Ne production exceeds that of 16 O in the outermost part of the core ( Mr / M ⊙ > 0 . 8). This is because, as it moves inwards, the temperature the C-burning decreases. These di ff erences in the core abundances of 16 O and 20 Ne will be critical for the crystallization process during the WD cooling phase.", '5. CONSEQUENCES FOR THE PROPERTIES OF WDS': "This section is devoted to analyzing the e ff ect of the different assumptions studied previously on the chemical structure and composition of ultramassive WDs, their cooling timescales, crystallization, and pulsation properties.  \nEach 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 H ∼ 1 . 5 × 10 -6 M 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 / Oor O / Ne plasmas (Camisassa et al. 2019, 2022). For the sake of clarity, we selected a fiducial model of initial mass 7 . 50 M ⊙ , which corresponds to a M WD ∼ 1 . 15 M ⊙ .  \nAs discussed in the previous sections, the adoption of extreme reaction rates (CF88 and HIN) during the C-burning phase leads to small di ff erences 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 di ff erent 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 ∼ 12 000 K), the cooling times and the size of the crystallized core are of 1743 (1721) × 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%.  \nRegarding the adoption of di ff erent branching ratios, the di ff erences 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 log(1 -Mr / M ∗ ) 4 pre- and post-RayleighTaylor 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 23 Na content remains always below ∼ 8% and, as higher production rates are adopted, the 20 Ne central content increases from  \n<!-- image -->  \nFigure 8. Distribution of the most important chemical elements for the M ZAMS = 7 . 50 M ⊙ models computed with the CF88 rate and di ff erent branching ratios (from upper left, in clockwise order, 5050, 65-35, 90-10, and 80-20) at the beginning of the WD's cooling path ( ∼ 10 5 K), pre- and post-Rayleigh-Taylor rehomogenization (upper panel, dashed and filled lines respectively) and at the ZZ Ceti stage ( ∼ 12 000 K, lower panel). The grey region reflects the crystallized part of the star, with a crystallization level above 90%.  \n<!-- image -->  \n30% up to ∼ 50% for the [90 / 10] branching ratio. In the latter case, the 20 Ne content produced at the outer part of the core surpasses the 16 O 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 16 O and 20 Ne. The C / O mantle on top of the O / Ne core is identical for each case.  \nIn the lower panel of Fig. 8, we show the same chemical profiles but at the ZZ Ceti stage ( T ∼ 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 di ff erences in the percentage of the crystallized core, which amounts to up to 0.8%, come strictly from the di ff erences in their composition. The combined e ff ect of the di ff erent degrees of crystallization and inner composition leads to di ff erences in the WD cooling times of at most 1.3%.  \nFigure 9. Distribution of the most abundant elements for a hybrid CONe-core model at 10 5 K, pre- and post-Rayleigh Taylor rehomogenization (upper and lower panel, respectively).  \n<!-- image -->  \nAs 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 WD / M ⊙ = 1 . 05 ( M ZAMS / M ⊙ = 6 . 85) pre- and postRayleigh-Taylor re-homogenization (upper and lower panel, respectively). The total 20 Ne content produced during the short-lived C-burning phase is then redistributed (as are the other species) throughout the core. Consequently, after rehomogenization, the WD model's structure resembles that of a pure CO-core WD, though with a non-negligible 20 Ne content ( ∼ 10%) in the core. These new kinds of objects predicted by our study, which can have masses up to M WD / M ⊙ ∼ 1 . 11, di ff er substantially from objects with pure CO cores. The presence of 20 Ne modifies the structure of the WD (particularly its central density) so that di ff erences in the evolution are significant. This can be seen in Fig. 10, where we show the evolution of the e ff ective temperature for a pure CO and the hybrid model. 5 Both models start at the same evolutionary point in the cooling track but di ff er in their central compositions, which are 30%-67% ( 12 C16 O) for the pure COcore model and 23%-61%-10% ( 12 C16 O20 Ne) 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 × 10 6 yr for the hybrid composition, while for the pure CO-core model, it starts at 870 × 10 6 yr. At 12 000 K (grey line), the age di ff erences are of ∼ 6%.  \nThe 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  \nFigure 10. Evolution of the e ff ective temperature T e ff as a function of the age, for both the pure M ZAMS = 6 . 85 M ⊙ CO core and hybrid models. The black horizontal dotted line indicates a T e ff = 12 000 K.  \n<!-- image -->  \nFigure 11. Chemical profile of the M ZAMS = 6 . 85 M ⊙ hybrid and pure CO-core models (upper and lower panel, respectively) at 12 000 K. The shaded region depicts the crystallized portion of the core, which reach crystallization levels of 56.5 and 41.8%, respectively.  \n<!-- image -->  \nabsence of 20 Ne in the pure CO-core model and the crystallized size of the core are the most prominent di ff erences at this point, while the 4 He-bu ff er region and pure 1 H-envelope remain the same.", '5.1. Asteroseismology': "The period spectrum and mode-trapping properties of g -modes of DAV WDs depend sensitively on the precise shape of the Brunt-Vaisala frequency across the interior of the star, and, in particular, the location and shape of the bumps produced by the chemical composition interfaces (De Ger'onimo et al. 2019; Althaus et al. 2010). Thus, any change in the  \n<!-- image -->  \nFigure 12. Comparison of the period spacing ∆Π i as a function of the periods of the modes Π i for the CF88 and HIN models (upper and lower panel, respectively). The green dashed line shows the value of the asymptotic period spacing.  \n<!-- image -->  \nchemical profiles translate into changes in the WD's expected pulsation properties.  \nThe computation of the pulsation properties for all our ultra-massive pulsating DA WD models was done using the LP-PUL pulsation code described in C'orsico & Althaus (2006), previously employed in the study of the properties of ultra-massive WD models (De Ger'onimo et al. 2019, 2022) and used to perform asteroseismic studies of ultra-massive ZZ Ceti stars (C'orsico et al. 2019b; Kilic et al. 2023). Element di ff usion was included for all models from the beginning of the WD cooling track. The 'hard sphere' boundary conditions were adopted when accounting for the e ff ects of crystallization on the pulsation properties of the g -modes. These conditions assume that the amplitude of the eigenfunctions of g -modes is drastically reduced below the solid and liquid interface, as compared with the amplitude in the fluid region (Montgomery & Winget 1999).  \nFor the comparison of the pulsation properties, we computed the forward period spacing ( ∆Π k = Π k + 1 -Π k ), a quantity frequently used in asteroseismic analyses, that reflects the mode-trapping features of the models. In Fig. 12, we show the distribution of period spacing as a function of the periods for dipole modes, resulting from the assumption of di ff erent reaction rates. As all the models have the same mass, the similarity in ∆Π i reflects the fact that the crystallized portion of the star is similar in both cases, meaning that most of the different core-chemical features do not have a significant impact on the g -mode pulsations that might otherwise have allowed us to tell the di ff erent models apart. Consequently, the resulting asymptotic period spacing values are nearly identical. The period spacing distributions are also very similar when considering di ff erent branching ratios of the 12 C + 12 C reaction rate during the progenitor evolution, as shown by Fig. 13.  \nFigure 13. Same as Fig. 12, but for the CF88 model computed with di ff erent branching ratios, as indicated in the upper left corner of each panel.Figure 14. Same as Fig. 12, but for the CO-core and hybrid WD models of Fig. 11 (upper and lower panels, respectively).  \n<!-- image -->  \nIn contrast, the hybrid model shows perceptible di ff erences in the trapping characteristics of the pulsation modes compared to the pure CO-core models, as seen in Fig. 14. For Π i > 300 s, the CO-core model shows more frequent minima (a shorter trapping cycle) in their distribution of ∆Π i values than does the hybrid model. The reason for this is relatively straightforward. These trapped modes at long periods ( Π i > 500 s) are modes with larger amplitudes in the homogeneous region above the crystallized core. As the outer border of the homogeneous region stays basically fixed, the larger the crystallized core, the smaller the homogeneous cavity in which the modes resonate. As shown by De Ger'onimo et al. (2019), this results in longer trapping cycles in that regime. It is worth noting that this di ff erence in the trapping cycle of the two types of models would not be present if the mixed region were neutrally buoyant, as discussed by Montgomery & Dunlap (2024). The actual tem-  \nre profile above the crystallized core is a matter of ongoing discussion in the field (e.g. Castro-Tapia et al. 2024). Additionally, di ff erences in the asymptotic period amount to about ∼ 2 s, this being a consequence of di ff erences in the crystallization degree of the core.", '6. CONCLUSIONS': "In this work, we implemented the recently derived total nuclear reaction rate for carbon fusion, 12 C + 12 C, 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 12 C + 12 C rate from CF88 is used. We also explored how the current uncertainty in the branching ratios for the 12 C + 12 C reaction's α 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.  \nOur 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 ≤ M ZAMS / M ⊙ ≤ 8 . 50 and a metallicity Z = 0 . 02. In addition to exploring the 12 C + 12 C rates from Monpribat et al. (2022) and CF88, a wide range of [ α/ 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.  \nWe found that using less e ffi cient nuclear reaction rates results in a late onset of the C-burning phase, larger cores, higher burning temperatures, and, consequently, shorter Cburning phase lifetimes. Despite these di ff erences 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 e ffi -ciency of the α and p exit channels constitute the most important uncertainty in determining the final chemical structures of SAGB progenitors. Di ff erences in the central 20 Ne abundances can reach up to 17% within the range of [ α/ p ] ratios explored in our study, in the sense that a higher 20 Ne content is achieved when the α channel is more dominant. Moreover, we found that higher production of 20 Ne translates into smaller initial masses needed for C-ignition and longer Cburning phase lifetimes.  \nAn 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 ≤ M ZAMS / M ⊙ <  \n7 . 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 20 Ne produced during this stage, the progeny could be a CO-core WD or a hybrid CONe-core WD.  \nAs for the impact on the ONe-core WD evolution due to the use of di ff erent nuclear reactions and branching ratios, we found di ff erences 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.  \nRegarding the hybrid CONe-core WDs, we found that they di ff er substantially in their evolution, compared with those composed of pure CO cores. Our results show that even as little as 10% of 20 Ne 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.  \nIn conclusion, our study reveals that current uncertainties in both the total 12 C + 12 C 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 12 C + 12 C rate also impact the advanced evolutionary stages of highermass stars. Given its far-reaching astrophysical implications, further experimental work is thus urgently needed to properly constrain the 12 C + 12 Crate and branching ratios at astrophysically relevant energies.  \nSoftware: Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011, 2013, 2015, 2018, 2019), MESASDK 20.3.1 (Townsend 2020), LPCODE (Althaus et al. 2021), LP-PUL (C'orsico & Althaus 2006).", 'ACKNOWLEDGEMENTS': '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.'}