Residual Carbon in Oxygen-Neon White Dwarfs and its Implications for Accretion-Induced Collapse

# Residual Carbon in Oxygen-Neon White Dwarfs and its Implications for Accretion-Induced Collapse

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###### Abstract

We explore the effects of the residual present in oxygen-neon white dwarfs (ONe WDs) on their evolution towards accretion-induced collapse (AIC). We produce a set of ONe WD models using MESA and illustrate how the amount and location of the residual carbon depends on the initial mass of the star and assumptions about rotation and convective overshooting. We find a wide range of possible mass fractions roughly ranging from 0.1 to 10 per cent. Convection and thermohaline mixing that occurs as the ONe WDs cool leads to nearly homogeneous interior compositions by the time that AIC would occur. We evolve these ONe WD models and some toy WD models towards AIC and find that regardless of the carbon fraction, the occurrence of Urca-process cooling due to implies that the models are unlikely to reach carbon ignition before electron captures on occur. Difficulties associated with modeling electron-capture-driven convective regions in these ONe WDs prevent us from evolving our MESA models all the way to thermonuclear oxygen ignition and the onset of collapse. Thus, firm conclusions about the effect of carbon on the final fates of these objects await improved modeling. However, it is clear that the inclusion of residual carbon can shift the evolution from that previously described in the literature and should be included in future models.

stars: evolution — white dwarfs

0000-0002-4870-8855]Josiah Schwab \@AF@joinHubble Fellow \move@AU\move@AF\@affiliationDepartment of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA Corresponding author: Josiah Schwab

0000-0003-4474-6528]Kyle Akira Rocha \move@AU\move@AF\@affiliationDepartment of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA \move@AU\move@AF\@affiliationDepartment of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA

## 1 Introduction

Stellar evolution provides a variety of pathways that produce degenerate cores with oxygen-neon (ONe) compositions and masses near the Chandrasekhar mass. When the central density of the core reaches a critical value , electron-capture reactions on isotopes such as and set in motion a chain of events that lead to the destruction of the star (Miyaji1980; Miyaji1987). This has generally been thought to involve the collapse of the core and the formation of a neutron star (NS).

Single stars with masses in the range develop degenerate ONe cores while avoiding oxygen ignition (e.g., Nomoto1984a; GarciaBerro1994) and thus have the potential to produce an electron-capture supernova (ECSN; Nomoto1987b; Hashimoto1993). A related process can occur in binary systems that produce an ONe WD with a close companion, such that material is added to the ONe WD (Nomoto1979a): this could be a system where a non-degenerate companion donates material onto the WD or in a double WD system that merges. These two binary scenarios, which are in close analogy to the single and double degenerate scenarios for Type Ia supernovae (e.g., Maoz2014), are referred to as accretion-induced collapse (AIC) and merger-induced collapse (MIC) respectively. AIC and MIC, by virtue of operating in old stellar populations, have long been invoked as a method of producing young NSs in globular clusters (Lyne1996; Boyles2011). Stellar population synthesis calculations indicate AIC and MIC may occur at 10% of the thermonuclear supernova rate (Ruiter2018).

The final phase of ECSN/AIC/MIC occurs when the density-driven initiation of exothermic electron-capture reactions causes the thermonuclear ignition of oxygen fusion at or near the center of the ONe core. This leads to the development of an oxygen-burning deflagration wave that then propagates outward through the star. Further electron-capture reactions on the deflagration ashes sap the pressure support. The final fate of the star is set by a competition between these two processes. The timescales over which these processes occur are sensitive to density, and so the central density of the ONe core when oxygen ignition occurs has been understood as the primary determinant of the final fate.

One-dimensional calculations (e.g., Nomoto1991; Gutierrez1996) suggested that this process led to collapse to a NS. However, it was understood that this conclusion was sensitive to the presence and efficiency of mixing processes in the star. Current models of electron-capture-initiated collapse, which have only recently begun to harness the power of multidimensional hydrodynamics codes, are extremely close to the threshold between implosion and explosion. Jones2016 find that in some cases these objects do not implode, but instead explode leaving behind a low mass bound remnant. Leung2017 vary a variety of model parameters and find that the outcome switches between explosion and implosion within existing uncertainties.

Therefore, an important step in understanding the final fate of these objects is improving the modeling of the phase in which the WD approaches the Chandrasekhar mass. Such calculations inform the initial conditions for the collapse calculations. Current models of ECSN progenitors generally use relatively large nuclear networks (e.g., Jones2013; Takahashi2013), and so the compositions at the time of collapse are self-consistently set by the preceding stellar evolution. In contrast, AIC models have primarily used homogeneous degenerate cores where the focus is restricted to the most abundant isotopes , , and (e.g., Canal1992; Gutierrez1996; Schwab2015). Gutierrez2005 extend this to include the effects of and which can be present with mass fractions at the percent level. Gutierrez2005 find that can lead to low density explosions, but that has little effect. Using a more accurate treatment of the relevant weak reaction rates (Paxton2015; see also Fuller1985; Toki2013; MartinezPinedo2014; Suzuki2016), Schwab2017a demonstrate that Urca-process cooling due to has a significant effect on the thermal state of the WD.

In this paper, we revisit the role of in the evolution of massive ONe WDs towards AIC. In Section 2, we generate a set of massive ONe WD models and characterize their chemical compositions. In Section 3, we describe how the composition profile of the WD changes during its evolution towards collapse. Guided by these results, in Section 4 we use a set of simple WD models to demonstrate the effects of . In Section 5, we evolve our realistic WD models towards collapse. In Section 6, we conclude and indicate areas of important remaining uncertainty in their evolution towards collapse.

## 2 Composition of Oxygen-Neon WDs from Stellar Models

Super asymptotic giant branch (SAGB) stars that produce ONe WDs first produce partially degenerate CO cores. Off-center carbon ignition then occurs, leading to the formation of a convectively-bounded “flame” that propagates inward (e.g., Timmes1994). The ashes of carbon burning are dominated by and with and also being produced at mass fractions of . Some of the can remain unburned, with the amount and its distribution within the WD varying with the mass of the star and assumptions related to the propagation of the flame (Siess2006).

In order to produce a set of ONe WD models with self-consistent abundance profiles, we evolve a set of SAGB star models using Modules for Experiments in Stellar Astrophysics (MESA; Paxton2011; Paxton2013; Paxton2015; Paxton2018). In Sections 2.1 and 2.2 we describe these models and their compositions. In Section 2.3 we compare our results to the SAGB models of Siess2007A&A.

### 2.1 Setup of SAGB Mesa models

Farmer2015ApJ use MESA to investigate the properties of carbon burning in SAGB stars as a function of the initial stellar mass and rotation rate as well as mixing parameters including convective overshooting. They use the MESA nuclear network sagb_NeNa_MgAl.net which uses 22 isotopes to cover the H, He, and C burning phases (including the pp chains and the CNO, NeNa, and MgAl cycles). They use fine spatial and temporal resolution in order to carefully resolve the carbon flame. We therefore elect to use the MESA input files from Farmer2015ApJ as the framework for producing ONe WDs with realistic composition profiles. We updated the Farmer2015ApJ input files to be compatible with MESA version 9793. We then evolve models from the pre-main sequence to the SAGB and through the carbon flame phase.

Following SAGB stars through their final phases is computationally demanding. The challenges come from needing to track a large number of thermal pulses or contending with high-temperature hydrogen-burning during “dredge-out” events. For a recent review of some of the challenges of modeling stars in this mass range see Doherty2017PASA. If we let our MESA models continue indefinitely, they encounter numerical trouble before they eject their envelope. Since we run into numerical trouble only after the completion of the carbon flame phase (and thus after the core composition is done evolving), we decide to manually halt the evolution and eject the envelope.

We stop the evolution and eject the envelope once two conditions are met simultaneously. First, we check if the highest temperature in the region of the star where the energy generation rate is dominated by CNO-cycle hydrogen burning is greater than roughly . This condition is necessarily triggered before high-temperature proton ingestion can lead to numerical difficulties. Second, we check that the carbon core mass and the oxygen core mass are within 10% of each other. This condition ensures that we do not stop the evolution before the ONe core has finished forming. At this point, we remove the envelope using an artificially-enhanced stellar wind with . We turn off nuclear burning during this phase for numerical simplicity. After the envelope has been removed we are left with a hot ONe WD model with a thin hydrogen and helium envelope.

Our goal is to produce a set of WD models that have a range of qualitatively different chemical profiles. Therefore, like Farmer2015ApJ, we vary a handful of stellar parameters including initial mass, rotation at the zero-age main sequence (ZAMS), and the location and strength of convective overshooting. Rotation in MESA uses the shellular approximation (Meynet1997) and is described in Paxton2013. Stellar models are given an initial solid body rotation rate at ZAMS. When we include rotation in our models, we use an initial rotation rate of 0.25 the critical rotation rate . Convective overshooting is implemented in MESA via an exponential decay of the convective mixing diffusion coefficient beyond the fully convective boundary (Herwig2000). The lengthscale of this decay, and hence the extent of the overshooting region, is controlled by the dimensionless parameter which multiplies the local pressure scale height. When we include overshooting, we use an overshooting efficiency of .

We run a set of non-rotating models without any convective overshoot (series NRNO). We run a set of models (series RO) with the fiducial rotation and overshooting parameters adopted by Farmer2015ApJ, which includes overshoot at all convective boundaries. The propagation of the carbon flame, and therefore the subsequent chemical profile, is sensitive to the treatment of mixing at the convective boundary near the flame (Siess2009; Denissenkov2013b). To explore this effect, we also run a set of models (series RMO) with an overshooting treatment where mixing below convective regions associated with carbon burning does not occur. This corresponds to the MESA options

Ω
Ψovershoot_f_below_burn_z_shell  = 0.000
Ψovershoot_f0_below_burn_z_shell = 0.000


This choice is motivated by the findings of Lecoanet2016ApJ who performed idealized hydrodynamic simulations of a carbon flame and found that little mixing occurs at this interface as convective plumes were not able to penetrate into the carbon flame.

### 2.2 Results of SAGB Mesa models

\twocolumngrid

We produced a total of 12 WD models. These models and their identifiers are listed in Table 2.2 together with the key parameters including initial mass, rotation at ZAMS, and convective overshooting. All but one model produce massive ONe WDs. Model 5 produces a hybrid CO-ONe WD.

Convective overshooting and rotation can have large effects on the chemical profile of the WD and especially that of C. Figure 2.2 shows the carbon profiles in our models; each panel compares different initial masses within the same model series. Broadly, models from more massive stars have less carbon than those from lower mass stars.333This is why is generally not important in stellar models for ECSNe which are necessarily more massive than those that produce ONe WDs. For example, after the end of core carbon burning, the ECSN progenitor model of Takahashi2013 has a mass fraction . This trend is particularly pronounced in the models that include overshooting (left and center panels). Models 1-8 from series RMO and RO have carbon depletion especially at larger radii while models 9-12 from series NRNO have comparatively homogeneous carbon profiles.

Stellar models that exhibit efficient convective overshooting below the carbon flame tend to extinguish the carbon flames earlier resulting in an increased abundance of carbon in the core. This effect is clearly visible in the left and center panels of Figure 2.2. The profiles of series RO and series RMO, which differ only by their assumption about mixing below the carbon flame, have similar carbon profiles in the outer half of the cores. However, series RO has systematically higher carbon abundances in the inner regions than series RMO. We find that all 12 WD models exhibit qualitative differences in their carbon profiles depending on their initial masses and modeling assumptions.

### 2.3 Comparison to Siess Models

Siess2007A&A constructed SAGB models using the stellar evolution code STAREVOL. These models do not take into account the effects of rotation. Like our MESA models, they use a decaying exponential treatment for convective overshooting with the same value of , though this is only applied at the edge of the convective core. We compare our models with their solar metallicity () models that include convective overshooting.

Figure 2.2 shows the central C abundance at the end of carbon burning as a function of initial mass. The models from Siess2007A&A are shown as gray squares. Models with higher initial masses have lower central carbon fractions. Our MESA models exhibit similar trends. Our models are not directly comparable to the Siess models at fixed initial mass because the different assumptions about rotation and overshooting imply a different mapping between initial mass and core mass. However, these independent sets of SAGB models display the same feature that stars in this transition mass range show a broad range of central carbon mass fractions ranging from tenths of a percent to tens of percent.

Figure 2.2 also illustrates the sensitivity of the central carbon fraction to convective overshooting below regions associated with carbon burning. The series RO models have central carbon mass fractions up to a factor of 10 greater than those in series RMO at fixed initial mass; again, the only difference between these models is the overshooting parameter at the lower boundary of the convection zone immediately above the carbon flame.

## 3 Homogenization during WD Cooling

In order for the ONe WD to evolve towards collapse, it must eventually accrete from a companion. This interaction is unlikely to begin immediately after the formation of the WD, but must instead wait for the companion star to evolve and fill its Roche lobe. Therefore, our modeling assumes that after formation there is a phase of evolution in which the WD does not interact with its companion and simply cools as an effectively single object.

As a WD cools and approaches an isothermal configuration, mixing can smooth out compositional gradients in the WD interior. Often in the literature this process is referred to as “rehomogenization”—framed as Rayleigh-Taylor instabilities smoothing regions with negative molecular weight gradients—and is often assumed to occur quasi-instantaneously (e.g., Althaus2007; Camisassa2018). However, more formally, the presence of a destabilizing composition gradient leads to mixing via thermocompostional convection. Depending on the magnitude of the stabilizing temperature gradient, this can take the form of overturning convection or double-diffusive fingering (thermohaline) convection. This distinction is important as these different processes occur over different timescales.

In CO WDs, the evolutionary effects of this kind of mixing are generally minor. However in hybrid CO-ONe WDs, where an ONe mantle overlays a CO core, such mixing is of critical importance as it will destroy the stratified core-mantle structure on a timescale significantly shorter than the likely timescale for the WD to grow to the Chandrasekhar mass (Brooks2017ApJ). The complex carbon profiles at formation in our ONe WD models (see Section 2.2) will similarly be affected by these mixing processes and this can substantially alter the central carbon fraction.

MESA uses the standard mixing length theory of convection (MLT; Cox1968). We evaluate convective stability using the Ledoux criterion. When we include the effects of thermohaline mixing in our MESA models, we use the prescription of Kippenhahn1980a with a dimensionless efficiency factor of 1. For a complete description of the mixing processes implemented in MESA see Paxton2013.

In order to illustrate the effects of the mixing processes, we show the evolution of our WD model 2 (see Table 2.2) during cooling. Figure 3 shows the model evolving assuming no thermohaline mixing occurs while Figure 3 shows the model evolving including thermohaline mixing. The lower panels of these plots show the electron fraction (): as the temperature becomes isothermal, convection alone will leave behind the neutrally buoyant  profile while the addition of thermohaline mixing leads to a flat  profile. The upper panels show the effects that mixing has on the C abundance. Figure 3 shows that without thermohaline mixing, a significant gradient in the C profile and  persists after 80 Myr. In contrast, the WD model with thermohaline mixing shown in Figure 3 has already reached a similar level of homogenization after 1 Myr and has almost completely homogeneous C and  profiles after only 11 Myr. This behavior is similar to what was found in Brooks2017ApJ.

Moving forward, all of our WD models are evolved using thermohaline mixing, as this is an expected physical effect. We do not include the effects of phase separation at crystallization which may further affect the central abundances (Segretain1993); such capabilities are not presently included in MESA. Since has the lowest charge of the isotopes present, the effect of phase separation would likely be to reduce the central carbon abundance.

After Myrs of cooling, all our WD models exhibit effectively homogeneous interior chemical profiles. Figure 3 shows the central mass fractions at this time; this value now reflects the abundance throughout the interior. The pale symbols show the mass fractions before cooling, illustrating that the central abundance of isotopes like C can change by a factor of a few. (The values pre- and post-cooling are also reported in Table 2.2.) In Section 2, we showed that varying initial mass, convective overshoot, and rotation produced ONe WDs having chemical profiles with qualitatively different shapes, but here we have shown that those differences are subsequently erased. However, Figure 3 shows that the overall range of central carbon abundance across models persists.

## 4 Effects of Residual Carbon

In Sections 2 and 3, we showed that ONe WD models from SAGB star evolution using MESA had residual mass fractions . In this section, we will outline the role of in the evolution of the WD towards collapse. Similarly, Gutierrez2005 evolve a set of ONe WD models with parameterized mass fractions in the range , motivated by the unburnt carbon in SAGB models (Dominguez1993; Ritossa1996; GilPons2001; GilPons2002). They find that that models with relatively small abundances can allow for ignition at densities . Ignition at these low densities seems likely lead to disruption of the star. Indeed, if carbon burning occurs before the exothermic electron captures, the evolution will be more analogous to the simmering phase experienced by near-Chandrasekhar mass CO WDs in which carbon burning leads to the development and growth of a large central convection zone (e.g., Lesaffre2006; Piro2008b). However, as we will demonstrate shortly, the presence of Urca-process cooling due to makes it unlikely that carbon ignition occurs in advance of the exothermic electron captures on and .

### 4.1 Carbon Ignition and Electron Captures

The models in Gutierrez2005 start with central conditions and . From their Figure 3, it appears that these models almost immediately begin carbon burning. For lower carbon mass fractions, the carbon-burning runaway takes slightly longer, allowing additional compression to occur during the temperature increase. The lowest carbon mass fraction model () appears to run out of fuel before reaching temperatures and then subsequently rejoins the compression-driven evolution towards higher density and the eventual electron captures that is characteristic of their carbon-free models.

The fact that the models of Gutierrez2005 appear to be burning carbon at or near their initial central conditions is puzzling. Figure 4 of Gasques2005 shows ignition lines (defined as where the energy generation rate is equal to the thermal neutrino losses) and burning timescales for pure carbon that cover the relevant range of densities and temperatures. The initial central conditions from Gutierrez2005 are well below even a burning timescale of order a Hubble time.

Figure 4.1 shows the carbon ignition lines for a range of carbon mass fractions. These are evaluated with the same input physics as our subsequent MESA models. We use the carbon burning rate from Caughlan1988 and the “extended” screening treatment in MESA as described in Paxton2011. This combines the results of Graboske1973 in the weak regime and Alastuey1978 with plasma parameters from Itoh1979 in the strong regime. The neutrino loss rates are calculated via the fitting formulae of Itoh1996a. For comparison, the solid grey line shows the MESA implementation of the rate given by Gasques2005 for pure carbon444The extension of these results to multicomponent plasmas and hence smaller carbon mass fractions is given by Yakovlev2006, but this is not presently implemented in MESA. (). At low temperature (roughly in the regime where is below the ion plasma temperature), these ignition curves disagree, reflecting differences in the screening treatment. Figure 4 of Gasques2005 shows the theoretical uncertainty associated with the ignition curves in this regime; the pure carbon MESA ignition curve in Figure 4.1 lies within that range.

Figure 4.1 also indicates where the key weak reactions will become important. The dashed lines show where the electron capture timescale for each of , , , and is equal to a characteristic compression timescale of (Schwab2015), with more rapid electron captures occurring to the right of the line. These curves were evaluated using the rate tabulations of Suzuki2016 assuming .

### 4.2 Simple MESA Models

In order to illustrate the potential role of carbon, we follow the approach of Schwab2015; Schwab2017a by constructing quasi-homogeneous WD models with parameterized abundances. This provides a simple way to isolate the effects of . In Section 5 we will apply this understanding to the more realistic WD models described in previous sections. The calculations shown in this section use MESA version 10108. All input files will be made publicly available at http://mesastar.org.

We first create a pre-main sequence star of . We relax its composition to one of the compositions indicated in Table 4.2. These are the fiducial compositions from Schwab2015; Schwab2017a plus a variable amount of carbon. We then have the models accrete a 50/50 oxygen-neon mixture555We accrete material that is free of hydrogen, helium and carbon in order to avoid the complications associated with following the nuclear burning on the surface of the WD. at a rate until the WD has reached . This value is selected to be below the density at which electron captures on the isotopes present in the initial composition will first occur. This procedure creates near-Chandrasekhar mass WD models with homogeneous inner regions of the desired compositions.

We then take these models and continue letting them accrete at . This choice represents an accretion rate that is roughly characteristic of any near-Chandrasekhar WD accretor stably burning hydrogen or helium (e.g., Wolf2013; Brooks2016). We evolve until the model experiences carbon ignition (defined as the energy release from carbon burning locally exceeding thermal neutrino losses).

Schwab2017a demonstrated that models that experience electron captures after significant Urca-process cooling has occurred can develop convectively-unstable regions near the electron capture front. Proper modeling of this convection likely requires both theoretical and numerical progress and remains an important open question. We discuss the difficulties of evolving MESA models in this phase more specifically in Section 5. To circumvent these issues, we artificially suppress the action of convection in the MESA models using the control mlt_option = ’none’. This is an important caveat that applies to the evolutionary tracks shown in Figure 4.2 after electron captures . Nonetheless, these models still provide insight into when carbon burning first begins to affect the evolution.

Figure 4.2 shows the evolution of the central temperature and density in the models. The thin lines show the results of models with the SQB15C composition, which do not include Urca-process cooling since there is no or . These models are less physically realistic, but we show them primarily for comparison with the results of Gutierrez2005. Models with experience carbon ignition at densities below where electron captures on would occur. Energetically, it requires burning approximately of carbon to heat the WD to the conditions for dynamical burning (Piro2008a). Therefore, since these models have roughly that much carbon, if we continued their evolution they may be able to reach the conditions for thermonuclear explosions. Note that the model is similar to what one would expect from a fully-mixed hybrid CO-ONe WD. Models with very low carbon abundances evolve much like carbon-free models; the thin orange and blue lines are nearly identical and continue until oxygen ignition. Intermediate abundance models with experience carbon ignition as a result of the electron captures. This range encompasses the carbon fractions in many of the ONe WD models shown in Section 3.

However, the presence of Urca-process cooling changes this picture, causing models across a range of compositions and accretion rates to reach (see Equation 15 in Schwab2017a). The thick lines in Figure 4.2 show the results of models with composition SBQ17C; the presence of and leads to significant cooling. Models that experience this cooling will therefore almost certainly reach the electron captures before igniting carbon. Models with very low carbon abundances evolve much like carbon-free models (though again, we note that these models do develop convectively unstable regions that are artificially not allowed to mix). Models with experience carbon ignition as a direct result of the electron captures.

Stellar models indicate significant abundances of the odd mass number isotopes (in particular ) that are responsible for the Urca-process cooling. Thus the SBQ17C models are the ones that should guide our understanding of WDs with self-consistently set compositions. Urca-process cooling prevents carbon ignitions from occurring in advance of the exothermic electron captures on . The SBQ17C models do show carbon ignition triggered by the electron captures above a critical carbon fraction of around 1 per cent by mass. Since this is within the range seen in our self-consistent stellar models of ONe WDs, this is an indicator that carbon may play an important role in at least some systems.

## 5 Evolution to Collapse

In this section, we continue the schematic binary evolution of the WD models produced in Sections 2 and 3 and examine the role of carbon in the evolution towards their final fates.

### 5.1 Setup of accreting WD models

In the overall evolutionary scenario for AIC, the ONe WD will stop cooling once it begins accreting from its companion via Roche lobe overflow. This can be achieved through expansion of the secondary driven by stellar evolution processes or a shrinking of the orbit through angular momentum losses. Either process results in a massive WD accreting material.

We model the effects of accretion from a companion by adding a constant accretion rate of . We again select a rate of as roughly characteristic of any near-Chandrasekhar WD accretor stably burning hydrogen or helium. Accreting oxygen simplifies the models as it avoids having to follow the numerically challenging hydrogen, helium, or carbon flashes that could occur during the accretion process. We turn rotation off on all models to avoid numerical problems with angular momentum transport in the WD. This is consistent with past work that has primarily studied non-rotating WDs.

We use a version of the original nuclear network sagb_NeNa_MgAl.net modified to add the isotopes , , , , , , , , , , and and the weak reactions that link them to the isotopes originally present in our network. We use the reaction rate tables from Suzuki2016 which are specifically calculated for the conditions in high-density ONe cores. We adopt EOS options666 Schwab2017a include all isotopes in the nuclear network in the PC (Potekhin2000) EOS calculations by using the control mass_fraction_limit_for_PC = 0d0. In MESA r9793 doing so with networks including neutrons causes the program to halt unexpectedly. We backport the one-line fix from MESA commit r10361 to our version. and spatial/temporal resolution controls similar to those of Schwab2017a, as it is demonstrated that those choices lead to numerically converged models (see their Appendix B).

### 5.2 Model evolution

We show the evolution of the central density and temperature of a subset of our models in Figure 5.2. In particular, we select models 2 and 6 from Table 2.2. These models both descend from 7.6  ZAMS models, with the only difference between them being the efficiency of overshooting associated with the carbon flame. These produce similar WD models that differ mainly in their carbon abundance; model 2 and model 6 have central carbon mass fractions of 0.007 and 0.03 respectively.

Compared to the toy models in Section 4, these models begin somewhat cooler. This is because the toy models had been steadily accreting since they were much lower mass and had reached a balance between compressional heating and neutrino cooling (Paczynski1971d). In contrast, our realistic ONe WD models had already cooled below this temperature, and being initially more massive, have not accreted for long enough to reheat.

The annotations on the lower part of the figure indicate the location of key weak reactions. We see some difference from the toy models shown in Figure 4.2. Mildly exothermic captures on occur; this species is not included in our toy models. The Urca-process cooling due to is less than in the toy models, due both to the initially lower temperature and the lower abundance of this isotope. The cooling from is more dramatic, though the small glitch near the end of cooling is an artifact of the reaction rate tables and does correspond to slightly too little cooling (see Appendix D in Schwab2017a). The separation (in density) of the and electron captures is caused by different assumed strengths of the (currently unmeasured) non-unique second forbidden transitions (see Section 5 in Schwab2017a).

All model tracks are similar up until the electron captures on . The heating from these is sufficient to ignite carbon. We are unable to evolve the models including the effects of convection beyond this point. By artificially suppressing convection, we are able to continue their evolution. These tracks are marked “no convection” in Figure 5.2. Subsequently, the difference between the two model tracks is apparent, with the a bifurcation reflecting the difference in the carbon abundance. The low carbon fraction model 2 continues to high density while the low carbon fraction model 6 experiences a relatively low density carbon-assisted oxygen ignition.

### 5.3 Convective Instability and Carbon Burning

Both in Section 4 and in this section, we encountered numerical difficulties when convective regions developed in the regions of the star undergoing electron-capture reactions. We circumvented this problem by artificially turning off convection. The further evolution of the models is thus unphysical, though hopefully remains schematically useful.

To understand why it is difficult to evolve the models beyond this point, it is first important to recall why these models become convective. The onset of exothermic electron captures leads to a strongly destabilizing temperature gradient. However, this is locally accompanied by the development of a strongly stabilizing composition (electron fraction) gradient. When using the Ledoux criterion, this can lead to overall convective stability (e.g., Miyaji1987; Schwab2015). The crossed temperature and composition gradients mean that this region is unstable to double diffusive convection, though this process is thought to operate too slowly to have a significant effect given the short evolutionary timescale. When including the effects of Urca-process cooling, the models of Schwab2017a became convectively unstable because thermal conduction causes a more rapid spread of the thermal gradient and leads to a situation in which the thermal and compositional gradients can no longer balance locally because their spatial extents are no longer well-matched. We emphasize that the development of these convective regions is unrelated to the presence of and occurs even in carbon-free models.

However, once the exothermic electron captures raise the temperature enough for carbon burning to begin, its energy release can further contribute to convective instability.777The heating from carbon burning is accompanied by some stabilizing change in composition, as immediate electron captures on carbon-burning products can occur at these densities. Since these electron captures are super-threshold they also deposit additional thermal energy. In the standard CO WD simmering case, the reactions have been carefully worked out by Piro2008a, Chamulak2008, and Forster2010. However, in the ONe WD case the abundances are lower and the abundances are higher. This means that the protons and particles produced in carbon burning might go through somewhat different pathways than in the standard simmering case. The development of a convective core has important consequences in the overall evolution of the star. Most importantly, it distributes the entropy generated by continued carbon burning and exothermic electron captures over a larger mass. This has been shown to lead to oxygen ignition at higher densities (relative to convectively stable models), favoring collapse to an NS (Miyaji1980; Miyaji1987).

The numerical difficulties that occur once a convection zone develops are essentially the same as those encountered in previous work, thus we briefly recapitulate the discussion from Section 6 of Schwab2017a.

When using MLT in MESA, cells that are convective have roughly the adiabatic temperature gradient while non-convective cells have the radiative temperature gradient. Therefore, when cells change between convective and radiative, they change the overall temperature profile and in turn, this can alter the convective stability of neighboring cells. (This is further complicated by the fact that the sub-threshold electron-capture reaction rates are exponentially sensitive to temperature.) As MESA iterates to find the solution to its equations, convective boundaries change location iteration-to-iteration and the solver fails to converge.

It is clear that regions become convectively unstable, but an understanding of how and whether these convective regions grow remains an outstanding problem. If a long-lived central convection zone does form, matters are further complicated as this allows for the operation of the convective Urca process (Paczynski1973b). This continues to be difficult to model in stellar evolution codes (e.g., Lesaffre2005b) and current versions of MESA produce physically inconsistent results during such phases (Schwab2017b).

The choice to artificially suppress convection corresponds to the assumption that these convective regions do not grow. So long as the energy and composition changes from nuclear reactions remain approximately local, the evolution of our “no convection” models should be roughly accurate. In this case, the presence of carbon leads to ignition at a factor of lower density than in the carbon-free case. In the case that a large convective core does form, it seems clear that this leads to higher density oxygen ignition than in a case without a convective core; in the carbon-free models of Gutierrez1996, non-convective models ignite with ignition in convective cores occurring at a central density a factor of higher. However, the presence of carbon is likely to reduce the density relative to a carbon-free model, as the additional heating source means that fewer electron captures are required. Since we cannot follow the models through a convective Urca phase, we are unable to reliably estimate the size of this shift.

## 6 Conclusions

We have explored the role that residual present in ONe WDs can play during their evolution towards AIC. We used MESA to produce 12 ONe WD models with chemical abundance profiles self-consistently generated via stellar evolution models of SAGB stars. We explored the range of expected for different initial masses and showed its dependence on assumptions about rotation and convective overshooting (Figure 2.2). Our results were consistent with previous work preformed using an independent stellar evolution code (Siess2007A&A).

While the carbon profiles at the time of WD formation were complex and varied, we showed that after of cooling the individual interior chemical profiles become nearly spatially uniform due to the effects of convection and thermohaline mixing. We typically expect WDs on the way to AIC to cool post-formation, as time is required for their binary companion to evolve and donate sufficient material. Post-cooling, a significant spread in central carbon abundances remains with mass fractions ranging from 0.1 to 10 per cent (Figure 3).

We then construct simple homogeneous ONe WD models with varying carbon fractions and allow them to grow via accretion. This allows us to map out the effects of the in a controlled way. In contrast to Gutierrez2005, we did not find the presence of low-density carbon ignition. Instead, we find that the occurrence of Urca-process cooling implies that models are unlikely to reach carbon ignition before electron captures begin around . We find that models with very low carbon abundances (mass fraction ) evolve similarly to carbon-free models whereas models with higher carbon abundances (mass fraction ) ignite carbon burning during the electron captures. This change in behavior happens within the range of carbon fractions found in our more realistic ONe WD models.

Ultimately, the goal of studies such as this one is to advance our understanding of the final fates of accreting ONe WDs that approach the Chandrasekhar mass. A key indicator of the likely outcome is the central density at the time of thermonuclear oxygen ignition. The exact line between explosion and collapse has not been firmly delineated, but seems to lie around (e.g., Timmes1992b; Leung2017). In the case of explosion, the models of Jones2016 leave behind a low mass bound remnant. Intriguingly, several WDs with low masses and peculiar compositions have recently been discovered (Gansicke2010; Kepler2016b; Vennes2017a; Raddi2018). Understanding the connection between these objects and AIC is an important motivator in understanding the boundary between explosion and collapse.

Difficulties associated with modeling convective regions in these degenerate interiors prevent us from evolving our MESA models all the way to oxygen ignition. On one hand, the presence of carbon provides an additional energy source that, once ignited at the lower densities associated with the electron captures, can help heat the WD to oxygen ignition conditions. On the other hand, this energy release also seems likely to push the model towards convective instability and the development of a convective core can defer oxygen ignition to higher densities. Firm conclusions await improved modeling techniques.

Nonetheless, this work indicates that the inclusion of carbon burning alters the evolution from that previously described in the literature. Intriguingly, the range of carbon abundances seen in our WD models produce evolutionary variations of a magnitude such that one might expect them to lead to a diversity of outcomes. Including the effects of residual in ONe WDs, and more generally moving towards increasingly realistic progenitor modeling, is an important ingredient in modeling of AIC going forward.

We thank Jared Brooks, Lars Bildsten and Eliot Quataert for their involvement in early explorations in this vein. We thank them and Ken Shen for helpful discussions. We thank Enrico Ramirez-Ruiz for his co-supervision of KR. KR also thanks the Koret Scholars program and the Lamat REU program for financial support. Support for this work was provided by NASA through Hubble Fellowship grant # HST-HF2-51382.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. This research made extensive use of NASA’s Astrophysics Data System.

Software: MESA (Paxton2011; Paxton2013; Paxton2015; Paxton2018), Python (available from python.org), matplotlib (matplotlib), NumPy (numpy), py_mesa_reader (pmr)

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