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A&A
Volume 699, June 2025
Article Number A35
Number of page(s) 11
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202554449
Published online 26 June 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Type Ia supernovae (SNe Ia) are excellent cosmological distance indicators that enable the measurement of cosmological parameters and thus can be used to determine the accelerating expansion of the Universe and whether dark energy exists (e.g., Riess et al. 1998; Perlmutter et al. 1999; Howell 2011; Meng et al. 2015). SNe Ia are crucial to understanding the evolution of galactic chemical compositions because they are the primary source of iron production in the Universe (e.g., Greggio & Renzini 1983; Matteucci & Greggio 1986). It is widely accepted that a SN Ia originates from the thermonuclear explosion of a carbon-oxygen white dwarf (CO WD) in binary systems. Both the Chandrasekhar mass and the sub-Chandrasekhar mass models are theoretically capable of producing explosions (e.g., Hoyle & Fowler 1960; Noebauer et al. 2017; Ruiter & Seitenzahl 2025). Based on the different properties of the companion star, progenitor models for SNe Ia are primarily divided into two categories: the double-degenerate (DD) model and the single-degenerate (SD) model (e.g., Podsiadlowski et al. 2008; Wang & Han 2012; Maoz et al. 2014; Ruiz-Lapuente 2014; Soker 2019; Liu et al. 2023). In the DD model, two WDs can produce a SN Ia when they merge (e.g., Iben & Tutukov 1984; Webbink 1984; Han 1998; Nelemans et al. 2001; Toonen et al. 2012). In the SD model, a CO WD accretes hydrogen-rich or helium-rich material from a non-degenerate companion, which could be a main-sequence (MS) star, a red giant (RG) star, or a helium (He) star (e.g., Whelan & Iben 1973; Nomoto et al. 1984; Li & van den Heuvel 1997; Langer et al. 2000; Han & Podsiadlowski 2004; Chen & Li 2007; Wang et al. 2009; Chen et al. 2011; Li et al. 2024). Both models have some supporting evidence, but both also face significant issues in terms of observations and theory.

In our study we focus on the SD model. The detection of circumstellar material around SNe Ia provides strong evidence for the SD scenario (e.g., Maguire et al. 2013; Vallely et al. 2019; Uno et al. 2023), as does the early optical and ultraviolet emission resulting from ejecta–companion interactions observed in some SNe Ia (e.g., Ganeshalingam et al. 2011; Cao et al. 2015; Liu et al. 2015; Piro & Morozova 2016).

However, the SD model faces some challenges, the primary being that the CO WD accretion rate must lie within a narrow range for a stable burning of accreted material, which makes it difficult to account for the observed SN Ia birth rate in the Galaxy (e.g., Wang & Han 2012; Maoz et al. 2014; Ruiz-Lapuente 2014; Wang 2018; Soker 2019; Liu et al. 2023). Many studies have attempted to address this issue by expanding the parameter space, defined by the initial donor star mass and orbital period, that can lead to SNe Ia (e.g., Hachisu et al. 1996, 1999a,b; Li & van den Heuvel 1997; Han & Podsiadlowski 2004; Chen & Li 2007; Hachisu et al. 2008; Meng et al. 2009; Meng & Yang 2010; Chen et al. 2011; Wheeler 2012; Hachisu et al. 2012; Meng & Podsiadlowski 2017, 2018; Ablimit & Maeda 2019), as well as by improving binary population synthesis (BPS; e.g., Mennekens et al. 2010; Ruiter et al. 2014; Piersanti et al. 2014; Toonen et al. 2014). While these efforts have led to significant progress, many uncertainties in stellar physics persist, which influences the outcomes of such studies.

Convective overshooting is one of the most significant sources of uncertainty in stellar physics. It can significantly affect the internal structure of a star and subsequently its evolution within a binary system. The extent of convective overshooting is often parameterized as δovHp, where δov is a free parameter and Hp represents the local pressure scale height. From the first principles, it is not possible to determine the distance of the convective overshooting region. Instead, δov is only constrained through observations (e.g., Prather & Demarque 1974; Maeder & Mermilliod 1981; Demarque et al. 1994; Ribas et al. 2000; Zhang 2012; Montalbán et al. 2013; Claret & Torres 2016; Viani & Basu 2020), such as measurements from asteroseismology. However, asteroseismic studies have revealed that stars with similar masses can have significantly different convective overshooting parameters. For instance, the value of δov is 0.2 for KIC 9812850 with M ≃ 1.48 M⊙ (Yang 2016), ∼1.4 for KIC 2837475 with M ≃ 1.50 M⊙ (Yang et al. 2015), 0.9 − 1.5 for Procyon with M ≃ 1.50 M⊙ (Guenther et al. 2014; Bond et al. 2015), ∼0.6 for HD 49933 with M ≃ 1.28 M⊙ (Liu et al. 2014), and 1.7 − 1.8 for KIC 11081729 with M ≃ 1.27 M⊙ (Yang 2015). These findings emphasize the significant uncertainties faced when determining convective overshooting.

Convective regions are defined by the Schwarzschild criterion, i.e., they are regions where the fluid elements experience acceleration. However, at the convective boundary, although the acceleration of fluid elements is zero, their velocity remains nonzero. As a result, fluid elements overshoot the convective boundary, forming a region known as the convective overshooting zone. By enhancing the internal mixing processes, convective overshooting allows for the transport of hydrogen-rich material into the hydrogen-burning core, which in turn allows hydrogen to burn more efficiently and increases the mass of the core. Consequently, stars with convective overshooting tend to have larger radii at the same evolutionary stage compared to those without convective overshooting.

In this work we investigate the impact of convective overshooting on the progenitors of SNe Ia in WD + MS systems. We describe the methods for detailed binary evolution calculation and present the parameter space that leads to SNe Ia for three different convective overshooting parameters in Sect. 2. We calculate the SN Ia birth rate for three different convective overshooting parameters in Sect. 3. A discussion of our findings is provided in Sect. 4, followed by conclusions in Sect. 5.

2. Binary calculation and results

2.1. Methods and physics input

In this work we calculated the evolution of binary systems using the most updated and flexible stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA; version r24.08.1; Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023).

In our calculations, we adopted a typical Population I composition with a hydrogen mass fraction X = 0.70, helium mass fraction Y = 0.28, and metallicity Z = 0.02 for initial models. The convective regions are defined by the Schwarzschild criterion, with the mixing length parameter set to α = l/Hp, representing the ratio of the typical mixing length to the local pressure scale height, at a value of 2 (Pols et al. 1997; Schroder et al. 1997). We applied the “step” scheme for convective overshooting. We assumed that the mixing efficiency in the overshooting zones is equal to that at the boundary of the convective zones. To explore the impact of convective overshooting on SN Ia progenitors, we set the convective overshooting parameter, δov, to 0.00, 0.25, and 0.50. We focused on WD + MS systems. Roche lobe overflow (RLOF) is treated using the “Kolb” scheme (Kolb & Ritter 1990). In this scheme, stars have an extended atmosphere, and the mass transfer process is classified as optically thin or optically thick, depending on whether the Roche lobe radius is larger or smaller than the stellar radius. We treated WD as a point mass without calculating its internal structure. When the mass of WD reaches 1.378 M⊙, it is assumed that a SN Ia explosion occurs.

We employed the common-envelope wind (CEW) model (Meng & Podsiadlowski 2017), which is a new version of the SD model, as our progenitor model. The CEW model presents several advantages, notably its ability to operate effectively across any metallicity. Here, we provide a brief introduction to the CEW model. For detailed information, please refer to Meng & Podsiadlowski (2017). In the CEW model, when the mass transfer rate, | M ˙ 2 | $ |\dot{M}_{\mathrm{2}}| $, exceeds the critical rate, M ˙ cr $ \dot{M}_{\mathrm{cr}} $, a common envelope (CE) forms. The CE is divided into two regions: the inner region, which corotates with the binary system, and the outer region, which exhibits differential rotation. The differential rotation in the outer regions continuously extracts orbital angular momentum from the inner binary. Depending on the presence or absence of a CE, the accretion rate of the WD, M ˙ WD $ \dot{M}_{\mathrm{WD}} $, can be expressed as follows:

M ˙ WD = { M ˙ cr , if CE exist , η He η H | M ˙ 2 | , if CE does not exist , $$ \begin{aligned} \left.\dot{M}_{\mathrm{WD} }=\left\{ \begin{array}{ll}\dot{M}_\mathrm{cr} ,&\mathrm {if \ CE \ exist}, \\ \eta _\mathrm{He} \eta _\mathrm{H} |\dot{M}_2|,&\mathrm {if \ CE \ does \ not \ exist}, \end{array}\right.\right. \end{aligned} $$(1)

where ηH and ηHe are the mass accumulation efficiencies for hydrogen burning and helium flashes, respectively, and | M ˙ 2 | $ |\dot{M}_2| $ is the mass transfer rate. The critical accretion rate, M ˙ cr $ \dot{M}_{\mathrm{cr}} $, is expressed as (Hachisu et al. 1999b)

M ˙ cr = 5.3 × 10 − 7 ( 1.7 − X ) X ( M WD − 0.4 ) M ⊙ yr − 1 , $$ \begin{aligned} \dot{M}_{{\mathrm{cr} }} = 5.3\times 10^{-7}\frac{(1.7-X)}{X}(M_{{\mathrm{WD} }}-0.4) M_{\odot } \mathrm{yr} ^{-1}, \end{aligned} $$(2)

where X is the hydrogen mass fraction and MWD is the mass of the accreting WD (in M⊙).

2.2. Single star evolution tracks

We conducted a test to investigate the effect of the different convective overshooting parameters (δov = 0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.70) on the evolution of single star with a mass of 2.8 M⊙ (Fig. 1). The evolutionary tracks are obviously affected by the convective overshooting. The increase in δov enhances mixing, consistently transporting hydrogen-rich material from the outer layers to the core, and the lifetime of a star on the MS increases. The greater the amount of fuel in the core, the more massive the core becomes at the same evolutionary stage, which leads to a higher core temperature. To maintain equilibrium, the star expands, which results in an increase in radius and luminosity, while the effective temperature decreases.

thumbnail Fig. 1.

Evolution of a star with 2.8 M⊙ in the Hertzsprung–Russell diagram with different δov. The δov is set to 0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.70. A star symbol indicates the point at which the star radius equals the Roche radius, RL, where a WD mass of MWD = 1.0 M⊙ and orbital periods of log(Pi/days) = 0.8 are assumed. The purple line connecting the star symbols represents the equal-radius line in the Hertzsprung–Russell diagram.

For the same initial binary parameters (MWD = 1.0 M⊙ and log(Pi/days) = 0.8), for the above reasons, the star with a large δov fills its Roche lobe at an earlier evolutionary stage (see Fig. 1). If the star fills its Roche lobe during the red giant branch (RGB) phase, the system will enter dynamical instability. This suggests that convective overshooting significantly influences the parameter space that leads to SNe Ia.

2.3. Binary evolution results

Figures 2 and 3 present two representative examples from our binary evolution calculations. These figures show the evolutionary track of the donor star in the Hertzsprung–Russell diagram, the orbital period evolution, the mass of the WD and donor star, MWD and M2, the mass transfer rate, | M ˙ 2 | $ |\dot{M}_{\mathrm{2}}| $, and the accretion rate of the WD, M ˙ WD $ \dot{M}_{\mathrm{WD}} $. Figure 2 shows the evolution of a binary system with initial parameters: initial donor star mass M2i = 2.8 M⊙, initial WD mass MWD = 1.0 M⊙, and initial orbital period log(Pi/days) = 0.0. The donor star fills its Roche lobe on the MS, and the system undergoes Case A RLOF. The mass transfer rate quickly exceeds the critical accretion rate, which leads to the formation of a common envelope (CE) and the system entering the CE phase. The WD gradually increases its mass until it reaches MWDSN = 1.378 M⊙; at this moment it is assumed to explode as a SN Ia. At this point, the donor stars mass are M2SN = 1.1786 M⊙, 1.2121 M⊙, 1.2331 M⊙, and the orbital period is log(PSN/days) =  − 0.3703, −0.3802, and−0.3892 for δov = 0.00, 0.25, and 0.50, respectively. The figure shows that the evolution of the orbital period, the variations in the mass of the WD and the donor star, and the mass transfer rate are similar for different values of δov; the influence of the different δov on the binary evolution is not significant if mass transfer occurs on MS.

thumbnail Fig. 2.

Example of binary evolution. In panel (1), the evolutionary tracks of the donor stars are represented by solid lines and the orbital period evolution with dashed curves. In panel (2), the solid and dashed lines indicate the donor star mass (M2) and the WD mass (MWD), respectively. In panel (3), the solid, dashed, and dotted lines correspond to the mass transfer rate ( M ˙ 2 $ \dot{M}_2 $), the mass growth rate of the CO WD ( M ˙ WD $ \dot{M}_{\mathrm{WD}} $), and the mass of the CE (MCE), respectively. The black, red, and blue lines represent δov = 0.00, 0.25, and 0.50, respectively. The initial binary parameters and the parameters at the time of the SN Ia explosion are also provided in panel (1).

thumbnail Fig. 3.

Same as Fig. 2 except the initial orbital period is different and the donor initiates RLOF in the HG.

Figure 3 shows another example of an initial system, where the initial parameters are [M2i, MWD, log(Pi/days)] = [2.8 M⊙, 1.0 M⊙, 0.6]. When the mass of the WD reaches MWDSN = 1.378 M⊙, the binary parameters are as follows: M2SN = 0.4302 M⊙, 0.5339 M⊙, and 0.6507 M⊙, log(PSN/days) = 0.1004, −0.0660, and −0.2325, for δov = 0.00, 0.25, and 0.50, respectively. The main difference from the previous example is that the RLOF occurs in the Hertzsprung gap (HG), and the system undergoes the early Case B RLOF. From the figure, the evolution of the orbital period, the WD, the donor star, and the mass transfer rate are significantly different for different values of δov. Specifically, the mass transfer rate increases as δov increases, which results in earlier WD mass growth (see Fig. 3, middle panel).

To compare the effects of different δov on progenitor systems, we summarize the final outcomes of binary evolution calculation in the initial orbit-initial secondary mass (log Pi, M2i) plane (Figs. 4, 5 and 6). From the three figures for different δov, CO WDs can reach a mass of 1.378 M⊙ during the CE phase (filled squares), the supersoft X-ray source (SSS) phase (filled circles), or the recurrent nova (RN) phase (filled triangles), as shown in Meng & Podsiadlowski (2017). However, due to strong flashes (open circles) or unstable dynamical mass transfer (crosses), many CO WDs fail to reach the mass of 1.378 M⊙. Additionally, more systems with large δov explode during the CE phase than those with small δov.

thumbnail Fig. 4.

Final binary evolution calculations for δov = 0.00, shown in the initial orbital period-secondary mass (log Pi, M2i) plane, where Pi represents the initial orbital period and M2i is the initial mass of the donor star (for various initial WD masses, as indicated in each panel). Filled squares indicate SN Ia explosions during the CE phase (the red indicates MCE > 0.1 M⊙, the blue 0.1 M⊙ > MCE > 0). Filled circles denote SN Ia explosions in the SSS phase ( M ˙ cr ≥ | M ˙ 2 | ≥ 1 2 M ˙ cr $ \dot{M}_{\mathrm{cr}} \geq |\dot{M}_2| \geq \frac{1}{2} \dot{M}_{\mathrm{cr}} $ and MCE = 0), and filled triangles in the RN phase ( 1 2 M ˙ cr > | M ˙ 2 | ≥ 1 8 M ˙ cr $ \frac{1}{2} \dot{M}_{\mathrm{cr}} > |\dot{M}_2| \geq \frac{1}{8} \dot{M}_{\mathrm{cr}} $ and MCE = 0). Open circles represent systems that experience nova explosions, preventing the CO WD from reaching 1.378 M⊙ ( | M ˙ 2 | < 1 8 M ˙ cr $ |\dot{M}_2| < \frac{1}{8} \dot{M}_{\mathrm{cr}} $ and MCE = 0), while crosses show systems that are unstable due to dynamical mass transfer.

thumbnail Fig. 5.

Same as Fig. 4 but for δov = 0.25.

thumbnail Fig. 6.

Same as Fig. 4 but for δov = 0.50.

The left boundaries of the contours are decided by the condition at which the RLOF starts when the donor is still unevolved (i.e., on the zero-age MS). Beyond the right boundary, the donor evolves onto the RGB, where the system undergoes a dynamically unstable mass transfer. The upper boundaries represent regions with very large mass ratios q ( M 2 M 1 $ \frac{M_{\mathrm{2}}}{M_{\mathrm{1}}} $), which are prone to delayed dynamical instability. The lower boundaries arise from the requirement that the mass transfer rate must be sufficiently high for the WD to be able to grow, and the donor mass must be large enough to supply enough material for the WD to reach the MCh.

We show contours in the (log Pi, M2i) plane for four initial WD masses (0.7 M⊙, 0.75 M⊙, 0.90 M⊙, 1.20 M⊙) with three different δov in Fig. 7, which allows us to observe the impact of convective overshooting. The figure shows that the contours expand as δov increases when the WD mass exceeds 0.7 M⊙. The right boundaries are extended by convective overshooting, which allows systems with relatively long initial orbital periods to contribute to the progenitors of SNe Ia (see also Fig. 1). The star with a larger δov has a larger stellar radius at the same evolutionary stage than those with a smaller δov. Consequently, companions with large δov can initiate RLOF relatively early in their evolution for the same initial orbital period. For example, a companion with a large δov can start RLOF in the HG, whereas a companion with a small δov would initiate RLOF in the RGB (see also Fig. 1).

thumbnail Fig. 7.

Contours of initial parameters in the (log Pi, M2i) plane for different WD masses and varying δov values, representing the regions where SNe Ia are expected. The initial masses of the WDs are indicated in the lower-right corner of each panel.

The upper boundaries move upward with the increase in δov, which suggests that convective overshooting raises the critical mass ratio, qcrit. Ratios above this maximum will lead to the delayed dynamical instability.

This behavior can be understood from the entropy profile of the donor star with different δov. A donor star with a larger δov has a more massive and hotter core, which leads to a radiation envelope with a larger radius but a lower mass, consequently leading to a steeper entropy gradient within the donor star. When RLOF begins, a steeper entropy gradient enables the donor star to adjust its radius more efficiently in response to the changing Roche lobe radius. Thus, an increase in δov delays the onset of dynamical instability, and in some cases, it may not occur even at the end of the mass transfer phase. Figure 8 shows the entropy profiles of the donor star after losing different amounts of mass. The figure shows that the entropy gradient increases with the increase in δov. Dynamical instability has already occurred when △M = 1.1 M⊙ for the case of δov = 0.00.

thumbnail Fig. 8.

Entropy profiles of the donor star after it loses different amounts of mass for varying δov. The solid line represents δov = 0.50, the dashed line δov = 0.25, and the dotted line δov = 0.00. Black, blue, red, and purple represent the donor losing masses △M = 0.0 M⊙, △M = 0.8 M⊙, △M = 1.1 M⊙, and △M = 1.7 M⊙, respectively. The initial parameters are MWDi = 1.1 M⊙, M2i = 3.4 M⊙, and log Pi(days) = 0.5.

We have only qualitatively discussed the impact of convective overshooting on delayed dynamical instability. However, the exact relationship between the qcrit and convective overshooting parameters is uncertain.

Furthermore, as the mass of the WD decreases, the influence of convective overshooting becomes progressively weaker. This is because a low WD mass corresponds to a narrower mass range for the companion star, which also has a low mass. Low-mass stars typically have relatively small convective cores, which reduces the impact of convective overshooting.

However, when the mass of WD is less than 0.75 M⊙, the effect of convective overshooting is not monotonic (as shown in Fig. 7). The minimum WD mass (MWDmin) also follows a non-monotonic trend, with values of 0.65 M⊙, 0.63 M⊙, and 0.675 M⊙ for δov = 0.00, δov = 0.25, and δov = 0.50, respectively, as illustrated in Figs. 4, 5, and 6. This is because a low-mass WD requires more accreted material to reach MCh. For a given evolutionary stage of the donor star, the mass transfer rate increases with δov. For large δov, the system has a higher mass transfer rate. However, WD can only accrete material at a rate up to the M ˙ cr $ \dot{M}_{\mathrm{cr}} $, which leads to substantial material loss. Although the lost mass can accumulate in the CE, the strong wind from the CE surface causes the envelope to dissipate quickly, which prevents the WD from reaching MCh. Conversely, for small δov, the lower mass transfer rate fails to supply sufficient material for the WD. By contrast, at δov = 0.25, the mass transfer rate is moderate – neither excessively high nor too low – which enables the WD to accumulate enough mass to reach MCh.

3. Birth rate estimate

We adopted Eq. (1) from Iben & Tutukov (1984) and followed a similar procedure as Chen et al. (2011) to enable a direct comparison of the effects of convective overshooting on the SN Ia birth rate in the Galaxy:

ν = 0.2 yr − 1 Δ q Δ log 10 a ∫ M A M B M − 2.5 d M , $$ \begin{aligned} \nu = 0.2\mathrm{~yr} ^{-1}\Delta q\Delta \log _{10}a\int _{M_\mathrm{A} }^{M_\mathrm{B} }M^{-2.5}dM, \end{aligned} $$(3)

where MA and MB represent the typical progenitor mass of WDs at each end of the mass range. The integrand is the initial mass function (IMF). For the quantity Δlog10a, we set Δ log 10 a ≈ 2 3 Δ log 10 P i $ \Delta\log_{10}a \approx \frac{2}{3} \Delta\log_{10}P^{\mathrm{i}} $ (Hachisu et al. 1999a). The initial range of mass ratios Δq for a given WD mass is

Δ q = M u M A − M l M B , $$ \begin{aligned} \Delta q=\frac{M_{\mathrm{u} }}{M_{\mathrm{A} }}-\frac{M_{\mathrm{l} }}{M_{\mathrm{B} }}, \end{aligned} $$(4)

where Mu and Ml represent the upper and lower limits of the SN Ia progenitor regions, which are directly identified from Figs. 4, 5, and 6.

We calculate the Galactic SN Ia birth rates to be 2.1 × 10−3yr−1, 3.7 × 10−3yr−1, and 3.0 × 10−3yr−1 for δov = 0.00, 0.25, and 0.50, respectively (see Tables 1, 2, and 3). While the parameter space of massive WDs (≥0.75 M⊙) increases with δov, the IMF in Eq. (3), ∝M−2.5, indicates that higher-mass systems constitute a smaller fraction of the population. Therefore, although the parameter space for massive WDs is larger, they contribute only a small fraction of SNe Ia. In contrast, SNe Ia are primarily contributed by WDs with a mass around 0.75  M⊙ (e.g., Meng et al. 2009). Meanwhile, the minimum WD mass and the parameter space for low-mass WDs do not vary monotonically with δov. Ultimately, this results in a non-monotonic relation between the computed SN Ia birth rate and δov.

Table 1.

SN Ia birth rate for δov = 0.00.

Table 2.

SN Ia birth rate for δov = 0.25.

Table 3.

SN Ia birth rate for δov = 0.50.

4. Discussion

4.1. Uncertainties in our model

For our model we adopted the “step” scheme and assumed that the mixing efficiency in the overshooting zones is equal to that at the boundary of the convective zones. However, the treatment of convective overshooting remains highly uncertain. Various approaches have been proposed, such as exponential diffusive overshooting (Herwig 2000) and turbulent entrainment (Meakin & Arnett 2007). These different methods can yield varying results in certain aspects.

Additionally, we simplified our model by assuming that all stars have the same convective overshooting parameter. However, previous studies suggest that there could be a certain correlation between stellar mass and convective overshooting, which the δov increases with star mass. (Schroder et al. 1997; Claret & Torres 2016;Higl et al. 2021; Jermyn et al. 2022; Ahlborn et al. 2022). If δov increases with stellar mass, it would extend both the right and upper boundaries of the parameter space for massive WD. It may also reduce the minimum WD mass required for SNe Ia, which could potentially increase the SN Ia birth rate of the SD model. However, the relationship remains poorly quantified.

4.2. Mass transfer rate

As previously mentioned, systems with a large δov have high mass transfer rates when the donor star is at a given evolutionary stage. For simplicity, we used the thermal timescale mass transfer rate to reflect the actual mass transfer rate qualitatively:

M ˙ th ≃ ( M 2 i − M 1 i ) τ KH , $$ \begin{aligned} \dot{M}_{\mathrm{th} }\simeq \frac{\left(M_2^\mathrm{i} -M_1^\mathrm{i} \right)}{\tau _\mathrm{KH} }, \end{aligned} $$(5)

where M1i and M2i are the initial masses of the accretor and donor, respectively, and τKH is the Kelvin-Helmholtz timescale, given by

τ KH ≃ G M 2 2 2 R 2 L 2 , $$ \begin{aligned} \tau _{\mathrm{KH} }\simeq \frac{GM_2^2}{2R_2L_2}, \end{aligned} $$(6)

where G is the gravitational constant, and M2, R2, and L2 represent the mass, radius, and luminosity of the donor star, respectively.

Figure 9 shows the evolution of the Kelvin-Helmholtz timescale for stars with different convective overshooting parameters(δov = 0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70). The figure shows that during the MS and HG phases, the τKH decreases as the star evolves. Moreover, the τKH reduces with an increase in δov. As a result, M ˙ th $ \dot{M}_{\mathrm{th}} $ increases with δov, which leads systems with large δov to exhibit a high mass transfer rate. It should be noted that for systems with the same initial binary parameters, a large δov does not necessarily result in a high mass transfer rate. This depends on the evolutionary stage of the donor star when it begins RLOF. For instance, in a system with δov = 0.00, the donor star may start RLOF in the HG, whereas in a system with δov = 0.50, the donor star begins RLOF during the MS. At this point, the thermal timescale for δov = 0.00 could be shorter than that for δov = 0.50, which results in a higher mass transfer rate for the δov = 0.00 system.

thumbnail Fig. 9.

Evolution of the Kelvin-Helmholtz timescale for stars with 2.8 M⊙ and different δov. The δov is set to 0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.70.

Additionally, excessive mass transfer does not necessarily increase the birth rate of SNe Ia. This is because the WD can only accrete material up to the critical accretion rate M ˙ cr $ \dot{M}_{\mathrm{cr}} $, any additional material is lost from the system. Conversely, if the mass transfer rate is too low, the donor cannot provide sufficient material for the WD to reach the Chandrasekhar mass (Mch). Only an optimal mass transfer rate allows the birth rate of SNe Ia to reach its maximum. Therefore, to address the issue of the birth rate in the SD, it is crucial to investigate the accretion process of WDs. Currently, the accretion rate of WDs remains a significant uncertainty (e.g., Wang & Han 2012; Wang et al. 2015; Wang 2018).

4.3. SNe Ia-CSM

Type Ia supernovae that interact with the circumstellar medium (CSM; hereafter SNe Ia-CSM) are characterized by narrow H/He emission lines, and they maintain a high luminosity over a long period (> 1 yr). The presence of narrow H/He lines and high luminosity suggests that the progenitor systems underwent significant mass loss, producing a massive, H/He-rich CSM before the supernova explosion (e.g., Maguire et al. 2013; Vallely et al. 2019; Uno et al. 2023). SNe Ia-CSM are most frequently found in late-type and irregular galaxies, which indicates that they likely arise from relatively young stellar populations (Silverman et al. 2013).

In the CEW model, if a WD explodes during the CE phase, and the mass of CE exceeds 0.1 M⊙, hydrogen lines can be directly detected in the spectra (Meng & Podsiadlowski 2018). A massive CE is typically formed in systems with massive companions. These massive stars have short evolutionary timescales, which leads to shorter delay times. This aligns with the observation that SNe Ia-CSM are associated with relatively young stellar populations.

Figures 4, 5, and 6 show that a large δov significantly increases the number of systems capable of forming a CE with MCE > 0.1 M⊙. This markedly enhances the predicted proportion of SNe Ia-CSM. We calculated the birth rates and proportions of SNe Ia-CSM for different δov, δov = 0.00, 0.25, and 0.50: 1.1 × 10−4, 4.20%; 3.1 × 10−4, 8.40%; and 6.9 × 10−4, 22.89%, respectively. Therefore, convective overshooting significantly increases the theoretically predicted proportion of SNe Ia-CSM.

However, most SNe Ia lack H/He lines. Dilday et al. (2012) estimated that the fraction of SNe Ia-CSM that exhibit prominent circumstellar interaction near maximum light ranges from 0.1% to 1%. Although this fraction could be higher if cases with weaker H signatures have been overlooked or if some cases with stronger signatures have been misclassified as Type IIn SNe, SNe Ia-CSM remain rare. However, theoretical calculations predict a significantly higher proportion of SNe Ia-CSM. For instance, the proportion of SNe Ia-CSM is as high as 22.89% for δov = 0.50. This might require a time delay between the WD reaching 1.378 M⊙ and the subsequent supernova explosion. The delay would need to be long enough for the CE to dissipate. One possible mechanism is the spin-up/spin-down model (e.g., Justham 2011; Di Stefano et al. 2011; Di Stefano & Kilic 2012; Hachisu et al. 2012; Wang et al. 2014). During this spin-down phase, the CE can dissipate. Meng & Podsiadlowski (2017) discuss this problem in detail, suggesting that no more than 1.7 in 100 SNe Ia belong to the SN Ia-CSM class if the spin-down timescale exceeds 105 years. However, the timescale for the WD’s spin-down remains uncertain (e.g., Justham 2011; Di Stefano et al. 2011; Di Stefano & Kilic 2012; Hachisu et al. 2012; Meng & Podsiadlowski 2013; Wang et al. 2014).

However, the birth rate and the proportion of SNe Ia-CSM could be overestimated by Eq. (3) because the calculated region is a rectangular area that includes some points where SNe Ia cannot be produced. However, this does not affect the trends in either the SN Ia birth rate or the proportion of SNe Ia-CSM for different δov.

5. Conclusions

We have studied the impact of convective overshooting on the SD model of SNe Ia. We obtained the parameter space (log Pi, Mi) for SNe Ia by calculating binary evolution in detail. Subsequently, we calculated the SN Ia birth rate for different convective overshooting parameters. The main conclusions of our study are as follows:

  1. Convective overshooting expands the parameter space for systems with massive WDs (≥0.75 M⊙). However, the minimum WD mass and the parameter space for low-mass WDs do not vary monotonically with δov, which results in a non-monotonic trend when calculating the SN Ia birth rate.

  2. The upper boundaries of the initial parameter space are extended by convective overshooting, which enables systems with massive companions to contribute to the production of SNe Ia. Consequently, the qcrit increases with δov.

  3. The mass transfer rate in the system increases with δov when the donor is in the same evolutionary stage.

  4. The δov has a notable impact on the initial parameter space for SNe Ia-CSM, which increases with δov.

Acknowledgments

We are grateful to Zhengwei Liu, Zhenwei Li, Shunyi Lan, Boyang Guo, Lifu Zhang, Jinxiao Luo, Junda Zhou, and Tan Liu for their fruitful discussions. This work is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (grant Nos. XDB1160303, XDB1160000), the National Natural Science Foundation of China (Nos. 12288102 and 12333008), and the National Key R&D Program of China (No. 2021YFA1600403). X.M. acknowledges support from Yunnan Fundamental Research Projects (Nos. 202401BC070007 and 202201BC070003), International Centre of Supernovae, Yunnan Key Laboratory (No. 202302AN360001), the Yunnan Revitalization Talent Support Program Science & Technology Champion Project (No. 202305AB350003), and the science research grants from the China Manned Space Program with grant no. CMS-CSST-2025-A13.

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All Tables

Table 1.

SN Ia birth rate for δov = 0.00.

In the text
Table 2.

SN Ia birth rate for δov = 0.25.

In the text
Table 3.

SN Ia birth rate for δov = 0.50.

In the text

All Figures

thumbnail Fig. 1.

Evolution of a star with 2.8 M⊙ in the Hertzsprung–Russell diagram with different δov. The δov is set to 0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.70. A star symbol indicates the point at which the star radius equals the Roche radius, RL, where a WD mass of MWD = 1.0 M⊙ and orbital periods of log(Pi/days) = 0.8 are assumed. The purple line connecting the star symbols represents the equal-radius line in the Hertzsprung–Russell diagram.
In the text
thumbnail Fig. 2.

Example of binary evolution. In panel (1), the evolutionary tracks of the donor stars are represented by solid lines and the orbital period evolution with dashed curves. In panel (2), the solid and dashed lines indicate the donor star mass (M2) and the WD mass (MWD), respectively. In panel (3), the solid, dashed, and dotted lines correspond to the mass transfer rate ( M ˙ 2 $ \dot{M}_2 $), the mass growth rate of the CO WD ( M ˙ WD $ \dot{M}_{\mathrm{WD}} $), and the mass of the CE (MCE), respectively. The black, red, and blue lines represent δov = 0.00, 0.25, and 0.50, respectively. The initial binary parameters and the parameters at the time of the SN Ia explosion are also provided in panel (1).
In the text
thumbnail Fig. 3.

Same as Fig. 2 except the initial orbital period is different and the donor initiates RLOF in the HG.
In the text
thumbnail Fig. 4.

Final binary evolution calculations for δov = 0.00, shown in the initial orbital period-secondary mass (log Pi, M2i) plane, where Pi represents the initial orbital period and M2i is the initial mass of the donor star (for various initial WD masses, as indicated in each panel). Filled squares indicate SN Ia explosions during the CE phase (the red indicates MCE > 0.1 M⊙, the blue 0.1 M⊙ > MCE > 0). Filled circles denote SN Ia explosions in the SSS phase ( M ˙ cr ≥ | M ˙ 2 | ≥ 1 2 M ˙ cr $ \dot{M}_{\mathrm{cr}} \geq |\dot{M}_2| \geq \frac{1}{2} \dot{M}_{\mathrm{cr}} $ and MCE = 0), and filled triangles in the RN phase ( 1 2 M ˙ cr > | M ˙ 2 | ≥ 1 8 M ˙ cr $ \frac{1}{2} \dot{M}_{\mathrm{cr}} > |\dot{M}_2| \geq \frac{1}{8} \dot{M}_{\mathrm{cr}} $ and MCE = 0). Open circles represent systems that experience nova explosions, preventing the CO WD from reaching 1.378 M⊙ ( | M ˙ 2 | < 1 8 M ˙ cr $ |\dot{M}_2| < \frac{1}{8} \dot{M}_{\mathrm{cr}} $ and MCE = 0), while crosses show systems that are unstable due to dynamical mass transfer.
In the text
thumbnail Fig. 5.

Same as Fig. 4 but for δov = 0.25.
In the text
thumbnail Fig. 6.

Same as Fig. 4 but for δov = 0.50.
In the text
thumbnail Fig. 7.

Contours of initial parameters in the (log Pi, M2i) plane for different WD masses and varying δov values, representing the regions where SNe Ia are expected. The initial masses of the WDs are indicated in the lower-right corner of each panel.
In the text
thumbnail Fig. 8.

Entropy profiles of the donor star after it loses different amounts of mass for varying δov. The solid line represents δov = 0.50, the dashed line δov = 0.25, and the dotted line δov = 0.00. Black, blue, red, and purple represent the donor losing masses △M = 0.0 M⊙, △M = 0.8 M⊙, △M = 1.1 M⊙, and △M = 1.7 M⊙, respectively. The initial parameters are MWDi = 1.1 M⊙, M2i = 3.4 M⊙, and log Pi(days) = 0.5.
In the text
thumbnail Fig. 9.

Evolution of the Kelvin-Helmholtz timescale for stars with 2.8 M⊙ and different δov. The δov is set to 0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, and 0.70.
In the text

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