White dwarf -- red giant star binaries as Type Ia supernova progenitors: with and without magnetic confinement

Various white-dwarf (WD) binary scenarios have been proposed trying to understand the nature and the diversity of Type Ia supernovae (SNe Ia). In this work, we study the evolution of carbon-oxygen WD -- red giant (RG) binaries (including the role of magnetic confinement) as possible SN Ia progenitors (the so-called symbiotic progenitor channel). Using the \textsc{mesa} stellar evolution code, we calculate the time dependence of the structure of the RG star, the wind mass loss, the Roche-lobe-overflow (RLOF) mass-transfer rate, the polar mass-accretion rate (in the case of magnetic confinement), and the orbital and angular-momentum evolution. We consider cases where the WD is non-magnetic and cases where the magnetic field is strong enough to force accretion onto the two small polar caps of the WD. Confined accretion onto a small area allows for more efficient hydrogen burning, potentially suppressing nova outbursts. This makes it easier for the WD to grow in mass towards the Chandrasekhar mass limit and explode as a SN Ia. With magnetic confinement, the initial parameter space of the symbiotic channel for SNe Ia is shifted towards shorter orbital periods and lower donor masses compared to the case without magnetic confinement. Searches for low-mass He WDs or relatively low-mass giants with partially stripped envelopes that survived the supernova explosion and are found in SN remnants will provide crucial insights for our understanding of the contribution of this symbiotic channel.


INTRODUCTION
Type Ia supernovae (SNe Ia) are known as one of the very energetic astronomical phenomena which are a key for understanding the evolution of the Universe; they have generally been considered the results of thermonuclear explosions of carbon-oxygen white dwarfs (CO WDs;Hoyle & Fowler 1960) in close binary systems. In both observational and theoretical studies, it is still hard to arrive at a robust conclusion on the binary stellar evolution pathways that lead to SNe Ia (for recent reviews see e.g., Maoz, Mannucci & Nelemans 2014;Maeda & Terada 2016;Livio & Mazzali 2018;Soker 2019;Jha et al. 2019), and indeed many different possible routes could contribute to the SN Ia population (see, e.g., Soker [2018] for a comparison of the different evolutionary scenarios). The SN Ia progenitor scenarios can be summarized as follows: (a) The single-degenerate (SD) scenario in which the CO WD accretes matter from a nondegenerate companion star and grows in mass towards the Chandrasekhar mass (e.g. Whelan & Iben 1973;Nomoto 1982;Li & van den Heuvel 1997;Langer et al. 2000;Han & Podsiadlowski 2004;Podsiadlowski et al. 2008;Lü et al. 2009;Ruiter et al. 2009;Di Stefano & Kilic 2012;Wang & Han 2012;Nelemans et al. 2013;Claeys et al. 2014; (CSM) interaction in SN 2020eyj, and they suggest the SD scenario with helium star donors (e.g., Ablimit 2022) might be the progenitor of SN 2020eyj. Interestingly, the expected CSM from CMD scenario (especially with helium donors, see Ablimit [2011] for more details) may also produce the properties of SN 2020eyj. Observations of SN 2002ic, SN 2005gj, SN 2006X, SN 2008J, PTF 11kx, SN 2015cp and SN 2018fhw (e.g. Hamuy et al. 2003Aldering et al. 2006;Patat et al. 2007;Taddia et al. 2012;Dilday et al. 2012;Graham et al. 2019;Vallely et al. 2019;Kollmeier et al. 2019) demonstrated the presence of a CSM, and the WD + RG channel (a variant of the SD scenario) has been suggested as a possible origin for at least some of these (i.e. SN 2005gj, SN 2006X, SN 2008J, SN2015cp and SN 2018fhw). Moreover, a number of observed symbiotic recurrent nova systems (i.e. RS Oph, T CrB and V407 Cyg) have been suggested to be observational counterparts of SN Ia progenitors in the WD + RG channel (e.g. Hachisu et al. 1999;Sokoloski et al. 2006).
The evolutionary pathway of WD + RG binaries has been studied for decades (e.g., van den Heuvel et al. 1992;O'Brien et al. 2006;Lü et al. 2009;Chomiuk et al. 2012;Lundqvist et al. 2020). It has been pointed out that the relatively higher and unstable mass-transfer rate may easily lead to CE evolution, which may reduce the contribution of the WD + RG channel to SNe Ia (e.g. Yungelson & Livio 1998;Han & Podsiadlowski 2004). There are many important unsolved physical processes, and the wind mass-loss process from RG stars is one of the particularly uncertain processes in the symbiotic channel. Some previous studies have suggested that it is a spherical stellar wind (e.g. Chomiuk et al. 2012;Lundqvist et al. 2020), while other studies consider it an aspherical wind lost from the RG star in the WD binary (O'Brien et al. 2006;Lü et al. 2009). Besides, the details of Roche-lobe (RL) mass transfer from a RG donor is still poorly understood (e.g. Pastetter & Ritter 1989;Chen et al. 2010), and these main physical processes need further investigation.
Magnetism may play a crucial role in the accretion and nuclear burning processes on the WD (see Mukhopadhyay & Bhattacharya [2022] for a recent review on magnetized compact stars). The magnetic field strength of the WD, the mass-transfer rate, the masses of the donor and the WD are the most relevant parameters for studying the effects of magnetism in WD binary evolution (e.g., Livio 1983;Ablimit & Maeda 2019a,b;Gupta et al. 2020;Hogg et al. 2021;Walters et al. 2021;Ablimit 2022). Magnetized WDs have been detected in symbiotic binaries and supersoft X-ray sources (Kahabka 1995;Sokoloski & Bildsten 1999;Osborne et al. 2001). Ablimit & Maeda (2019a) find that highly magnetized WDs, accreting H-rich material, can lead to different outcomes in detailed WD + MS binary evolution calculations for SNe Ia (see also Ablimit & Maeda 2019b), while Ablimit (2022) demonstrated that the contribution of the WD + He star channel to SNe Ia is moderately influenced by the magnetic field of the WD. This suggests that the role of magnetism in WD binary evolution also needs to be investigated for the WD + RG channel.
In this work, we investigate the evolution of WD + RG binaries as a potential channel for SN Ia progenitors by considering stable mass transfer via RL overflow (RLOF; i.e. avoid CE evolution), stellar-wind mass loss, non-magnetic and magnetic WDs with the binary version of the 1D stellar-evolution code mesa (Modules for Experiments in Stellar Astrophysics). The WD + RG channel for SNe Ia is also commonly referred to as the symbiotic channel. In §2, we provide the main description of the binary physical processes and parameters in the detailed mesa simulations in the symbiotic channel. Results and discussions are presented in §3, followed by the main conclusions in §4.

MESA BINARY STELLAR EVOLUTION SIMULATIONS
We simulate the detailed evolution of WD + RG binaries using the star and binary packages of version 15140 of the mesa code (Paxton et al. 2011(Paxton et al. , 2015(Paxton et al. , 2019. In the first step of our calculations, we use the star package to make a RG star module based on a typical Pop I composition with hydrogen mass fraction X = 0.70, He mass fraction Y = 0.28 and metallicity Z = 0.02. We set initial zf racs = 6 and kappa f ile pref ix = 'a09' to call the opacity tables which are built using the more recent available solar composition, and the Henyey theory of convection with mixing length alpha = 1.8 is used in the code. To generate a RG star model, we start the evolution with a pre-main sequence model and continue the evolution until the central helium mass fraction is ≥ 0.98 (at this stage it has a pure helium core and a RG radius). We construct RG star models with masses (M RG ) in the range from 0.5 to 2.0 M with mass intervals of 0.1 M (see also Ablimit 2023).
With the binary package of mesa, we then simulate the evolution of CO WD + RG binaries with initial orbital periods (P orb,i ) in the range of 0.5 − 2000 d, using the RG donor models discussed above. For the accreting WDs, treated as point masses in the code, two initial masses, 1.2 and 1.0 M , are adopted. All relevant mechanisms for angular-momentum evolution from the system (including magnetic braking, gravitational-wave radiation, and mass loss) during the binary evolution are taken into account (see Paxton et al. 2015). If matter is lost from the system, we assume that it carries the angular momentum of either the RG or the WD, whichever is appropiate. The most important physical parameters that determine the evolution of the binaries are the initial orbital periods, initial masses of the donor and the accretor, and the mass-transfer process. Because giant stars have low surface gravities and extended atmospheres, it is not so straightforward to model the RLOF mass-transfer process (see the discussions in Pastetter & Ritter 1989;Chen et al. 2010). Here, we do not consider the complications of extended atmospheres and use the general technique developed by Kolb and Ritter (1990) to compute mass transfer, which should yield acceptable results for stars at different evolutionary stages, whereṀ RL is the RLOF mass-transfer rate, Γ 1 is the first adiabatic exponent, P ph and P RL are the pressures at the photosphere and at the radius when the radius of the donor is equal to its RL radius, respectively. T is the temperature of the donor, κ B is the Boltzmann constant, and µ ph is the mean molecular weight. The effective RL radius (Eggleton 1983) of the RG donor star (R RL ) can be calculated as, where q = M RG /M WD and a is the orbital separation.Ṁ 0 iṡ where m p is the proton mass, and T eff is the effective temperature of the donor. µ ph and ρ ph are the mean molecular weight and density at its photosphere. The fitting function F (q 2 ) (q 2 = M accretor /M donor ) is F (q 2 ) = 1.23 + 0.5 log(q 2 ), for 0.5 q 2 10, the same as in the mesa code. Other physical assumptions are the same as in the instrumental mesa papers (e.g., Paxton et al. 2015). There are different options for configuring mass loss for RG and AGB stars in mesa; we used the following wind options for the RG branch (RGB) stellar-wind mass loss, cool wind RGB scheme = Reimers , cool wind AGB scheme = Blocker , RGB to AGB wind switch = 1d − 4, Blocker scaling factor = 0.0003d0.
It is worth noting that the wind mass-loss rate before RLOF is less than 3×10 −8 M yr −1 most of the time (furthermore, only a small fraction of the mass lost from the spherically (isotropic) stellar wind moves toward the WD); thus the RLOF mass-transfer rate dominates for the mass accretion during the WD binary evolution. The RLOF mass-transfer rate (Ṁ RL ) plays the decisive role in determining the nature of hydrogen and helium burning on the WD and the stability of burning affects its mass retention efficiency and how the WD grows in mass. The mass growth rate of the WD is usually written asṀ where we adopt the prescription of Hillman et al. (2015Hillman et al. ( , 2016 for the efficiency of hydrogen burning (η H ) and the methods of Kato & Hachisu (2004) for the mass accumulation efficiency of helium (η He ). The different episodes of these burning efficiencies strongly affect the results of the mass-accretion phase. The adopted prescriptions here are widely used, but different models for carbon burning would somewhat affect the growth of the WD (Brooks et al. 2017). The stream/confined accretion and related emission on a magnetized WD has to be treated differently compared to spherically symmetric accretion onto a non-magnetic WD (e.g., Fabian et al. 1977;Livio 1983;King & Shaviv 1984;Hameury et al. 1986;King 1993;Wickramasinghe & Ferrario 2000;Wickramasinghe 2014;Ferrario et al. 2015;Mukhopadhyay et al. 2017;Ablimit 2019Ablimit , 2022. For a sufficiently magnetized WD, the mass flow onto the WD, once RLOF has started, will be magnetically channeled and not through an accretion disk connected to the WD (e.g., Cropper 1990;Frank et al. 2002; see the schematic figure of Ablimit & Maeda (2019a) and Ablimit (2019)). The strong magnetic field of the WD controls the motion of the accreting matter near the WD; as the WD's magnetic pressure increases more rapidly than the ram pressure of the accreting material as it approaches the WD's surface, there will be a radius, the magnetospheric radius, at which the magnetic pressure is equal to the ram pressure (Frank et al. 2002). Below this radius, matter will flow along magnetic field lines and fall onto the magnetic poles of the WD through an accretion column. The minimum physical condition for magnetically confined accretion is that the magnetic field strength (B) satisfies (Livio 1983), whereṀ is the RLOF mass transfer rate (Ṁ RL in this work ). The pressure at the base of the accreted matter (P b ) is related to the properties of the WD, the size of the polar cap regions, and the accreted mass, and is taken as P b = 5 × 10 19 dyne cm −2 (following Livio (1983)). We also adopt the mass (M WD ) -radius (R WD ) relation of Nauenberg (1972) for the WD. For WDs with no magnetic field or an intermediate-strength magnetic field, the mass-transfer rate has to be 5 × 10 −8 M yr −1 to avoid nova outbursts (e.g., Hillman et al. 2015Hillman et al. , 2016. Once the magnetic field strength of the WD meets the condition for magnetic confinement, nova outburst can be suppressed at a much lower mass-transfer rate. In this work, we adopt the approach of Ablimit & Maeda (2019a) and Ablimit (2019Ablimit ( , 2022 for simulating the accretion phase of the WD, and consider WDs with no magnetic fields and WDs with intermediate and high magnetic field strengths (B) and the effects of the magnetic fields on the binary evolution. We take a sufficiently magnetized WD with a fixed initial magnetic field strength of 2.5 × 10 7 G to realize magnetic confinement (cf. Livio 1983). For the purposes of deciding whether nova outbursts occur and to calculate the accretion efficiencies in Eq. 6, we define an equivalent, isotropic polar mass-transfer rate (Ṁ p ) in the case of magnetic confinement as (see also Ablimit & Maeda 2019a) where the ratio of the surface area of the WD and the two polar regions of the WD (on which material is accreted) is S/∆S = 2R m /(R WD cos 2 θ), where we take θ = 0 (the angle between the rotation axis and the magnetic field axis; see Ablimit (2022) for more information) for the simulations in this work. R m is the magnetospheric radius which is related with Alfven radius R A (Lamb et al. 1973;Norton & Watson 1989;Frank et al. 2002), where φ is a parameter (≤ 1) that takes into account the departure from the spherically symmetric case, µ = BR 3 WD is the magnetic moment of the WD in units of 10 33 G cm 3 ,Ṁ acc is the mass-accretion rate in units of 10 16 g s −1 and M WD is the WD mass in solar units.

RESULTS AND DISCUSSION
The Hertzsprung-Russell (HR) diagram in Figure 1 shows that the RG stars modeled with the mesa code in this work fit with our current understanding of stellar evolution (note that the figure only shows theoretical evolution tracks without observational constraints). Figure 2 shows the outcomes of the binary evolution sequences in the initial RG donor mass -initial orbital period plane, where the blue solid lines enclose the regions that produce SNe Ia (i.e. regions where the WDs grow in mass and reach the Chandrasekhar mass limit, taken as M Ch = 1.38M )), for different magnetic field strengths. The upper panels of Figure  , the ranges from both models (those with and without magnetic confinement) are different for the following reasons: 1. The treatment of mass transfer varies for the different stellar evolution codes. Here, we use the mesa code with the Kolb scheme for the RLOF mass transfer. 2. Both wind mass loss and RLOF mass transfer are considered at the same time in our simulations. This is important as RG stars can experience substantial wind mass loss, which tends to widen the orbits prior to the beginning of RLOF. Some RG stars (near the upper mass range) can lose up to ∼ 32% of their mass through their stellar winds, which can increase the orbital period by ∼ 21% prior to RLOF. This may help to stabilize the RLOF mass transfer in some binaries. We take both wind mass loss and RLOF mass transfer into account while previous studies only considered one of them. Compared to the case without magnetic confinement, the initial donor mass can be lower for the high magnetic-field case, because magnetic confinement leads to a higherṀ p , which in turn allows matter to burn more stably and increasesṀ WD even in lower-mass donor stars. The ranges of initial orbital periods that produce SNe Ia shrink because the mass loss from the system is generally higher for the higher values oḟ M p (the WD eject some of the transferred mass whenṀ p > 10 −6 M yr −1 ). Figure 3 shows the detailed binary evolution of one WD + RG system without and with magnetic confinement. Without magnetic confinement (left panels of Figure 3), the RG with an initial mass of 0.8 M cannot let the WD (M WD,i = 1.2 M ) grow in mass to the Chandrasekhar mass limit as the low mass-transfer rate leads to nova outbursts with no significant accretion. With the effect of the WD's strong magnetic field (right panels of Figure 3), the transferred matter can be confined to the polar cap regions, and the higher polar mass-accretion rate (red line) alters the mass-accretion phase of the binary. The main difference of these two cases with the same RLOF mass transfer from the donor is that the hydrogen burning efficiency (mass retention efficieny) on the WD due to the magnetic confinement is higher than that in the spherical accretion case. As the RG star loses its mass, the WD gains mass smoothly due to the magnetic confinement, and this mass accretion leads to a contraction of the orbit. In contrast, the WD's mass remains constant (i.e., experiences no significant accretion) without magnetic confinement, and the transferred mass will be lost from the binary, and this mass loss, including the RG stellar wind mass loss, will take angular momentum with it. Thus, the orbital period evolves from 10 to 75 days in the non-magnetic case. In the magnetic confinement case, the orbital period evolves in a much slower way because the system loses only a small amount of mass through the RG stellar wind (the WD accretes the mass transferred through RLOF from the RG). Because of the different mass accretion, the radius of the RG star expands more than in the case of a RG without magnetic confinement.
In Figure 4, we show another example for the evolution of a WD + RG binary, where the WD (M WD,i = 1.2 M ) can reach M Ch with and without magnetic confinement, and the mass transfer in both cases proceeds on a thermal timescale. The main difference between the evolution of the two cases is that the accretion rate per unit area on the WD is higher with magnetic confinement. In this binary, the wind mass-loss rate of the RG star with an initial mass of 1.1 M is higher than 10 −10 M yr −1 and can be as high as 10 −8 M yr −1 ; it takes more angular momentum away from the binary; thus, the wind mass loss widens the orbit significantly compared to the previous example. With the higher-mass donor (higher RLOF mass-transfer rate), the polar mass-transfer rate will be higher, and the outcomes will accordingly be different. In the magnetic confinement case (right panels), some of the transferred mass via RLOF would be lost from the binary and take away angular momentum; the mass loss rate will be higher ifṀ p (red line) is higher (when it exceeds 10 −6 M yr −1 ), and some of the transferred matter can not burn stably on the WD and escape from the WD. Thus, the RG donor loses more mass in order to allow the WD to grow in mass to M Ch , and this mass loss also widens the binary orbit more compared to the case of no magnetic confinement (left panels).
There are some differences in other properties (i.e., radius, see Figure 4; luminosity and effective temperature, see Figure 5) of the giant donors in the two cases due to the different mass accretion/mass loss (or/and different initial conditions). The giant donor in the magnetic confinement case loses more mass and evolves further towards lower mass than the final donor in the case without magnetic confinement. Thus, the total ranges of orbital period and donor properties (mass, orbital velocity, luminosity, effective temperature) at the time of the SN explosion are different in the two cases. For the case without magnetic confinement, combining the results for M WD,i = 1.2 and 1.0 M , the ranges of orbital periods, donor mass, orbital velocity, luminosity and effective temperature are 21.3 − 2521.5 d, 0.59 − 1.23 M , 13.5 − 57.7 km s −1 , 1.94 − 3.62 (in log(L/L )), and 2960 − 4458 K, respectively, while for the magnetic confinement case they are 5 − 652 d, 0.41 − 1.21 M , 22.3 − 100.5 km s −1 , 1.08 − 2.83 (in log(L/L )), and 3268 − 4700 K, respectively. In both examples, the ages of the RG stars are very long, implying that the WD + RG channel has a long delay time between the star-formation phase and the time of the SN explosion, and hence this channel would contribute to the old population of SNe Ia. The giant stars will survive after the SN explosion and finally evolve to become WDs. The existence of single WDs (especially low-mass He WDs with < 0.45 M ) could be the survivors from SNe Ia produced by this channel. Indeed, a population of low-mass, apparently single WD, presumably He WDs, has been found in the cluster NGC 6791 (Kilic et al. 2007) and in the field (Kawka et al. 2006;Bergeron & Leggett 2002) and have been proposed to be SN Ia survivors (Justham et al. 2009). Besides, some observed symbiotic novae appear to occur in binaries with very massive WDs and relatively low-mass giant companions; these systems are potential observational counterparts of SN Ia progenitors, e.g., RS Oph (e.g., Brandi et al. 2009;Mikolajewska & Shara 2017), T CrB (e.g., Belczynski & Mikolajewska 1998), V745 Sco (e.g., Drake et al. 2016;Orlando et al. 2017). The properties of these symbiotic nova systems can be well reproduced by the WD + RG binary evolutionary sequences with and without magnetic confinement. For future observations, it will be very interesting and challenging to find the surviving He WDs or stripped giant stars whose envelopes have been partially lost by the stellar wind or/and during the masstransfer phase of in the supernova explosion, which would be less massive and hotter than comparable single stars. In addition, detailed observations of SN Ia environments can provide further clues, and the CSM properties from SNe Ia observations may constrain this model as the wind mass loss may create a CSM-like environment around the system in this symbiotic channel (but see also Moriya et al. 2013). In addition, some observational clues such as continued observations of late-time light curves of nearby SNe Ia, new high-cadence survey capacities in the short term (Ivezić et al. 2019) and future direct observations via gravitational waves (Korol et al. 2018) will provide crucial information on the nature of the SN Ia progenitors. Additionally, the evolution of oxygen-neon-magnesium composition WD -RG binaries via accretion-induced collapse provides a promising channel to form peculiar neutron star X-ray binaries (Ablimit 2023).

CONCLUSION
As already discussed in the Introduction, both observational and theoretical studies to date suggest that a number of progenitor models may contribute to the SN Ia population. In this study, we employed the mesa stellar evolution code to simulate a large grid of WD + RG binaries as potential SN Ia progenitors, considering WDs without magnetic field, with intermediate, and high magnetic-field strengths in these binaries. In the simulations, the RLOF mass-transfer rate is always higher than the wind mass-loss rate, and the mass-accretion phase is dominated by RLOF mass transfer, while the stellar wind causes mass loss from the systems (which mainly affects the orbital period by taking away angular momentum).
Compared to systems with WDs with no or intermediate magnetic-field strength (where magnetic confinement does not occur), highly magnetized WDs can confine the transferred matter to their polar caps and can increase the burning efficiency even with a lower mass-transfer rate. With magnetic confinement, systems with a lower-mass RG star can drive the WD to grow in mass to experience a SN explosion, as shown in the initial parameter space of the RG mass and orbital period. In the case without magnetic confinement, the initial parameter space (initial donor mass and initial orbital period) derived in this study for producing SNe Ia (for M WD,i = 1.2 and 1.0 M ) are 1.1 − 1.8 M and 10 − 750 d, and 1.3 − 1.7 M and 35 − 400 d, respectively. For the highly magnetized WDs (for M WD,i = 1.2 and 1.0 M ), they are 0.7 − 1.8 M and 5 − 120 d, and 1.2 − 1.7 M and 20 − 28 d, respectively. Based on the two examples for which we derived the post-SN properties, we suggest that finding a He WD or a RG with an envelope that has at least partially been stripped, lower-mass giant (or/and hotter) stars after the SN or inside a SN remnant will provide a conclusive test for this symbiotic scenario. In the future we will also consider the possibility of super-Chandrasekhar WDs and the contribution of highly magnetized WDs in binaries with MS, RG and stripped helium star companions to the class of overluminous SNe Ia (Ablimit et al. in preparation).
Software: Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011Paxton et al. , 2015Paxton et al. , 2019 ACKNOWLEDGMENTS We thank Noam Soker for useful comments and discussions. This work is supported by NSFs.

DATA AVAILABILITY
The data underlying this article will be shared on reasonable request to the corresponding author.   . Detailed evolution of CO WD + RG binaries as a function of time using the mesa code. The left panels show the case of a non-magnetic WD and the right panels the case of a magnetized WD (with B = 2.5 × 10 7 G). All other initial parameters are the same: the initial masses of the WDs and donor stars are 1.2 M and 0.8 M , respectively, with an initial orbital period of 10 d. The evolution of mass transfer, WD/donor mass, orbital period, and the radius of the donors are shown. Wind mass loss is very low (< 5 × 10 −13 M /yr) and is not shown for clarity. RLOF M dot and Polar M dot in the figures arė MRL andṀp in the text, respectively. . Another example for the detailed evolution of CO WD + RG binaries using the mesa code: evolution of mass transfer, wind mass loss, WD/donor mass, orbital period, and radius of the donors are shown as a function of time. The right and left panels are for a non-magnetized and highly magnetized WD (B = 2.5 × 10 7 G) with magnetic confinement, respectively. The initial masses of the WD and RG donor are 1.2 M and 1.1 M , and the initial orbital period is 120 days for both cases. Figure 5. Example of the evolution of a RG star in the Hertzsprung-Russell diagram for the CO WD + RG binaries without and with magnetic confinement.