The impact of wind accretion in Evolving Symbiotic Systems

Symbiotic stars are binary systems in which a white dwarf (WD) accretes material from its late-type companion, producing a variety of electromagnetic signatures (Luna et al., 2013). Studying accretion in symbiotic systems is crucial, because they are very likely the progenitors of unique objects, such as Barium stars (Bidelman & Keenan, 1951) and carbon- and $s$-element-enhanced metal-poor stars (Beers & Christlieb, 2005). Besides cataclysmic variables, symbiotic systems also produce nova-like events and are potential progenitors of type Ia supernovae with cosmological implications (see Mukai, 2017; Chomiuk et al., 2021).

The accretion process in symbiotic systems can be studied indirectly through various types of observations, such as the rotation of accretion discs (e.g., Leedjarv et al., 1994; Robinson et al., 1994; Zamanov et al., 2024), associated X-ray emission (e.g., Toalá et al., 2023; Vasquez-Torres et al., 2024; Zhekov & Tomov, 2019), or the flickering effect observed mainly in optical wavelengths (e.g., Sokoloski et al., 2001; Gromadzki et al., 2006; Merc et al., 2024). Despite these insights, the specific mechanisms by which material from the late-type companion is accreted onto the WD are still under investigation.

For decades, it has been widely accepted that accretion through wind capture in a Bondi-Hoyle-Lyttleton (BHL)-like scenario (Hoyle & Lyttleton, 1939; Bondi & Hoyle, 1944; Bondi, 1952) is not applicable to symbiotic systems, particularly when the velocity of the stellar wind ($\varv_{\mathrm{w}}$) of the mass-losing, late-type star is lower than the orbital velocity ($\varv_{\mathrm{o}}$) of the accreting WD (e.g., Boffin, 2015; Hansen et al., 2016). For $\varv_{\mathrm{w}}<\varv_{\mathrm{o}}$, the standard implementation of the BHL formalism in binary stars overestimates the resulting mass accretion rate, predicting in some cases a non-physical value exceeding unity. This challenge has been extensively discussed in several numerical simulations (see, e.g., Huarte-Espinosa et al., 2013; Malfait et al., 2024; Saladino et al., 2019; Theuns et al., 1996, and references therein).

However, the problem has persisted since theoretical works still turn to BHL-like prescriptions to study accreting WDs given its simplicity (see, e.g., Chen et al., 2018; Vathachira et al., 2024). Several groups tried to alleviate the mass accretion efficiency problem by including arbitrary efficiency factors to the BHL formalism (e.g., Nagae et al., 2004; Malfait et al., 2024; Richards et al., 2024; Saladino & Pols, 2019; Li et al., 2023) or cut-off values (e.g., Saladino et al., 2019), and those recipes are included in population synthesis studies (see, e.g., Hurley et al., 2002; Izzard et al., 2009; Osborn et al., 2024; Andrews et al., 2024).

To address this issue, Tejeda & Toalá (2024) (hereinafter Paper I) have recently presented analytical modifications to the standard BHL formalism applied to accreting binaries that resulted in more consistent mass accretion efficiencies when compared with numerical simulations in the $\varv_{\mathrm{w}}<\varv_{\mathrm{o}}$ regime. Paper I showed that for circular orbits, the mass accretion efficiency $\eta$ can be expressed simply by two dimensionless quantities. Moreover, their approach allowed them to peer into the properties of accreting objects in eccentric orbits, resulting in intricate and more complex evolution of such systems.

In this paper, we build on the model proposed in Paper I to investigate the impact of accretion processes in evolving symbiotic systems. Specifically, we examine the detailed evolution of an accreting WD in circular orbit around mass-losing, Solar-like stars with initial masses of 1, 2, and 3 M_⊙. A direct comparison with known symbiotic systems is also presented. We give some predictions for the final destination of symbiotic systems as possible type Ia supernova. Note that an extension of this work but for the case of eccentric orbits will be presented in a subsequent paper.

This paper is organized as follows. In Section 2 we present our methodology, which includes the usage of stellar evolution models in combination with dynamical simulations. In Section 3 we present our results and a discussion is presented in Section 4. A summary is finally presented in Section 5.

In this section we report the evolution of symbiotic systems across a range of initial conditions and accretion models. We considered primary star masses initially set to $m_{1,0}=1$, 2, and 3 M_⊙, accreting WD masses initially set to $m_{2,0}=0.7$ and 1.0 M_⊙, and initial binary separations of $a_{0}=4$, 8, 16, 20, 40, 80, 160, and 200 AU. This resulted in 48 unique initial configurations. As described in the previous section, for each configuration we conducted three types of dynamical simulations: i) no accretion, ii) accretion using the modified model described in Paper I, and iii) accretion using the standard BHL model, leading to a total of 144 simulations.

Each simulation spanned the evolution of the binary system from the onset of the RGB phase past the end of the TPAGB phase, specifically until the effective temperature of the donor star reached $\log_{10}(T/\mathrm{K})=3.7$ after the TPAGB phase (see Fig. 1). Simulations were terminated earlier if the WD accreted enough mass to reach the Chandrasekhar limit (1.4 M_⊙), at which point it would theoretically explode as a Type Ia supernova (a phenomenon not modelled in this study).

Fig. 4 provides a representative example of the time evolution of a binary with initial masses $m_{1,0}=1$ M_⊙, $m_{2,0}=0.7$ M_⊙, showcasing the results for all eight initial separations and the three types of accretion prescriptions. This figure illustrates the time evolution of key parameters: binary separation ($a$), orbital velocity ($\varv_{\mathrm{o}}$), mass-accretion efficiency ($\eta$), mass accretion rate onto the WD ($\dot{M}_{\mathrm{acc}}$), WD mass ($m_{2}$), and the dimensionless mass ratio ($q$). The time origin ($t=0$) corresponds to the onset of the RGB phase $t_{\mathrm{iRGB}}$, as defined in the stellar evolution model summarized in Table 1.

Simulations without accretion serve as the baseline for comparison. Due to continuous mass loss from the primary star, these systems experience orbital expansion. This is evident in the upper-left and upper-middle panels of Figure 4, which show an increasing orbital separation ($a$) and a decreasing orbital velocity ($\varv_{\mathrm{o}}$).

In contrast to the no-accretion scenario, from this figure we can also appreciate the significant role accretion plays in shaping the evolution of symbiotic systems, particularly during the transition from the RGB to the TPAGB phases. The initial orbital separation also emerges as a key factor influencing the final outcomes and accretion efficiencies. Notably, models incorporating accretion (solid and dashed lines) exhibit shorter final orbital separations than those without accretion, an effect amplified by smaller initial orbital separations and more prominent during the TPAGB phase.

Furthermore, the choice of accretion model impacts the system’s evolution. The standard BHL formalism (dashed lines) generally predicts larger mass accretion efficiencies ($\eta$) than the modified model from Paper I (solid lines), as shown in the top-right panel of Fig. 4. Consequently, the orbits of systems modelled with the standard BHL model are less affected by accretion, as the total mass of the system is better conserved due to the higher accretion efficiency. Fig. 5 provides a closer look at the final TPAGB phase for the $m_{1,0}=1$ M_⊙ model, highlighting these differences. Models with more massive donor stars exhibit similar evolutionary trends, although less pronounced, as shown in Figures 6 and 7 for $m_{1,0}=2$ and 3 M_⊙, respectively. Similarly, simulations with a white dwarf companion of initial mass $m_{2,0}=1$ M_⊙ exhibit comparable evolutionary trends as those presented in Figs. 4–7.

Across all models, the simulations including the modified wind accretion from Paper I predict mass accretion rates $\dot{M}_{\mathrm{acc}}$ between $10^{-10}$ and $\lesssim 10^{-5}$ M_⊙ yr^-1 during the TPAGB phase, as shown in the bottom-left panels of Figs. 5–7. These accretion rates evidently reflect the pulsating behaviour of the corresponding evolving stars, with $\dot{M}_{\mathrm{acc}}$ increasing in tandem with the stellar mass-loss rate $\dot{M}_{\mathrm{w}}$ (see Fig. 3). Notably, the more massive models exhibit higher mass-loss rates (see Fig. 2), but their dynamical response is tempered by lower mass accretion efficiencies $\eta$ (top-right panels of Figs. 5–7).

Depending on the stellar evolution model, the TPAGB phase is not the only stage characterised by significant mass-loss rate. In particular, the 1 M_⊙ stellar evolution model exhibits substantial mass ejection during the RGB phase, just before the He-flash (see Fig. 2). The bottom-left panel of Fig. 4 reveals that during the RGB phase, $\dot{M}_{\mathrm{acc}}$ is comparable to values predicted for the TPAGB phase. Particularly, the $m_{1,0}=1$ M_⊙ stellar evolution model produces the ejection of about 0.3 M_⊙ during the RGB phase (Fig. 4, bottom-middle panel). Such behaviour, however, is less pronounced in the more massive stellar evolution models.

The mass accretion efficiency history shown in Figs. 4–7 illustrates that the highest accretion rates are attained in systems with the smallest initial orbital separations. This is a natural consequence of the dependence of $\eta$ on the velocity ratio $\varw=\varv_{\mathrm{w}}/\varv_{\mathrm{o}}$ (see Eq. 2 and 3), where closer orbital separations result in higher orbital velocities and thus smaller values of $\varw$. In addition, models with larger $m_{1,0}$ have lower accretion efficiencies due to their smaller mass ratios $q$.

Fig. 8 shows the evolution of a representative system with initial masses $m_{2,0}=0.7$ M_⊙ and $m_{1,0}=3.0$ M_⊙, and an initial orbital separation of $a_{0}=8$ AU in the $\eta$ vs. $\varw$ parameter space. The evolutionary tracks are illustrated for both the modified accretion prescription from Paper I and the standard BHL model. The main trend for both prescriptions is an increasing wind-to-orbital velocity ratio $\varw$. However, this trend can temporarily reverse during the TPAGB phase in response to the thermal pulses of the primary star, as the system undergoes episodes of increased mass loss and enhanced wind velocities. The BHL model consistently predicts higher accretion efficiency than the Paper I model, with the difference exceeding an order of magnitude at the start of the evolution. While the BHL model generally shows a decreasing accretion efficiency, the Paper I model exhibits a more complex behaviour. It starts with a constant value $\eta\simeq q^{2}<1$, then increases during the TPAGB phase, eventually reaching a turnover point and declining slightly.

Fig. 9 summarizes the evolution of all accreting models in the $\eta$ vs. $\varw$ parameter space. The left panels display the results for the Paper I model, and the right panels show those for the standard BHL model. In the top row, colours distinguish models with different initial primary masses, while in the bottom row, colours differentiate models with different initial orbital separations. As a reference, dashed lines in all panels show the $\eta-\varw$ evolution for fixed $q$ corresponding to the initial configuration of the simulations ($q_{1}$, $q_{2}$, and $q_{3}$). The parameter $q_{0}$ corresponds to the final configuration of the $m_{1,0}=1$ M_⊙ simulations when the accretor reaches 0.8 $\mathrm{M_{\odot}}$.

The evolutionary tracks in the left panels of Fig. 9 generally resemble those described in Fig. 8 for initial distances $a_{0}\leq 16\,$AU. However, models with larger $a_{0}$ do not exhibit a marked turnover, but rather follow a decreasing mass accretion efficiency trend, asymptotically approaching $\eta\approx q^{2}/\varw^{4}$ (Eq. 2) in the large $\varw$ limit, where the Paper I model approximates the standard BHL theory. The early evolution of all models occurs in the region where $\eta\approx q^{2}$ for small $\varw$, dominated by the modified wind accretion formulation. These two limiting behaviours are illustrated in the bottom-left panel of Fig. 9 with dotted lines.

The right-column panels of Fig. 9 show the results for the standard BHL accretion approximation. As demonstrated in Paper I, the calculations for $\varw>1$ are consistent between the two accretion prescriptions, but for $\varw<1$, the BHL calculations yield non-physical $\eta_{\mathrm{BHL}}$ values larger than 1. To avoid these non-physical values, which have been extensively discussed in previous works (see references in Section 1), these cases were restricted to $\eta_{\mathrm{BHL}}=1$. Notably, all models using the standard BHL accretion approximation exhibit larger $\eta$ values than those with the same configuration but adopting the accretion regime from Paper I.

By incorporating the accretion formalism proposed in Paper I, we were able to investigate the evolution of symbiotic systems and their accretion history throughout the lifetimes of both stars in the system, in realistic accretion efficiency regimes. Our results indicate a general trend: symbiotic systems with small initial separations $a_{0}$ or systems early in their evolution exhibit the $\eta\sim q^{2}$ asymptotic behaviour. Over time, and best seen in systems with large $a_{0}$, the systems transition to the $\eta\sim q^{2}/\varw^{4}$ regime, characteristic of the standard BHL formulation for wind accretion in binary systems for large $\varw$ values.

A WD with initial mass of $m_{2,0}=0.7$ M_⊙ orbiting mass-losing stars with initial masses of $m_{1,0}=1$, 2, and 3 M_⊙ result in mass accretion rates ranging from $\dot{M}_{\mathrm{acc}}=[10^{-10}$–$10^{-5}$] M_⊙ yr^-1. These rates increase significantly toward the end of the TPAGB phase where the mass-loss rate of the donor star is the highest. While this range is broad, the predicted $\dot{M}_{\mathrm{acc}}$ values align well with estimates derived from optical and X-ray observations of symbiotic systems (e.g., Pujol et al., 2023; Vasquez-Torres et al., 2024; Zhekov & Tomov, 2019; Guerrero et al., 2024a; Marchev & Zamanov, 2024; Boneva & Yankova, 2021; Luna et al., 2018). However, determining the mass accretion efficiency $\eta$ is more challenging. Accurate measurements require precise estimates of $\dot{M}_{\mathrm{w}}$, which depend on detailed characterization of the late-type companion in the system and/or dedicated observational campaigns (e.g., Ramstedt et al., 2018).

The simulations predict that symbiotic systems have mass accretion efficiencies in the $\eta=[10^{-4}-10^{-1}]$ range as illustrated in the $q$–$\varw$ space of Fig. 10, where we also plot contours of $\eta$. The bottom panel in this figure display the expected locations of a sample of symbiotic systems listed in Appendix A, with data extracted from the New Online Database of Symbiotic Variables (Merc et al., 2019). For these systems, a standard wind velocity of $\varv_{\mathrm{w}}=12\pm 8$ km s^-1 is adopted, consistent with reported wind velocities for late-type stars (see Ramstedt et al., 2020). Fig. 10 shows a general agreement of the observed properties of symbiotic systems and the models presented here including the modified accretion scheme of Paper I. Notably, the system $o$ Ceti stands out, having one of the most extended orbits with a period of 498 yr. This results in a small orbital velocity $\varv_{\mathrm{o}}$ and, consequently, a large $\varw$, placing it as the rightmost data point in the figure.

The panels in Fig. 10 include a gray-shaded region representing the part of the $q$–$\varw$ space where the analytical approximation of Paper I is not applicable. That work assumes that the mass-losing star $m_{1}$ returns to its unperturbed state between consecutive passages of the accretor $m_{2}$. To evaluate this condition, Paper I introduced a refill time ($t_{\mathrm{fill}}$), which must be smaller than the system’s orbital period $T$. For circular orbits, this condition can be expressed as:

Fig. 10 shows that most models evolve while avoiding the non-valid region of the $q$–$\varw$ parameter space. All models evolve from within the non-valid region of the $q$–$\varw$ diagram due to the extremely low stellar wind velocities ($\varv_{\mathrm{w}}<2$ km s^-1) adopted prior to the onset of the peak of the RGB phase (see Fig. 3). A notable exception is seen in the models with $m_{1,0}=1$ M_⊙, which re-enter the non-valid region of the $q$–$\varw$ diagram during the evolutionary stage between the He-flash and the onset of the TPAGB phase. This occurs because the wind velocity decreases significantly during this interval (Fig. 3, top left panel). We predict that during these specific evolutionary phase, no significant mass is accreted, because $\dot{M}_{\mathrm{w}}$ is extremely low ($<10^{-11}$ M_⊙ yr^-1) making the impact of this intermediate phase negligible.

For decades, numerical simulations found that the standard implementation of the BHL wind accretion model for binary systems did not produce reliable results for the case where the stellar wind velocity $\varv_{\mathrm{w}}$ was smaller than the orbital velocity of the accretor $\varv_{\mathrm{o}}$. A situation that is of outmost importance for symbiotic systems. The standard BHL scheme predicts non-physical mass accretion efficiencies larger than 1 for $\varv_{\mathrm{w}}<\varv_{\mathrm{o}}$. Recently, Tejeda & Toalá (2024) (Paper I) proposed a geometric correction to the standard implementation of the BHL model for accreting binaries and found a better agreement between their analytical predictions and numerical simulations.

In this paper, we studied the impact of the wind accretion model of Paper I for the case of evolving symbiotic systems. We modelled 48 different symbiotic systems, each consisting of a WD with an initial mass of $m_{2,0}=0.7$ or 1.0 M_⊙ and a Solar-like star with an initial mass of $m_{1,0}=1$, 2, and 3 M_⊙. The stars were placed in initially circular orbits with separations ranging from 4 to 200 AU. The evolution of the Solar-like stars was computed using the stellar evolution code mesa, and the orbital evolution of the systems was calculated making use of the N-body package rebound. For each configuration, we conducted three types of simulations: accretion using the modified model from Paper I, accretion using the standard BHL model, and no accretion.

Our main findings are summarized as follows:

• •

  The wide range of masses and orbital configurations modelled highlights distinct evolutionary paths. The adopted accretion prescription significantly affects the final destiny of the modelled symbiotic system. None of the simulations including the wind accretion from Paper I reached the Chandrasekhar limit (1.4 M_⊙). This means that for this model to produce recurrent nova (e.g., T CrB) and type Ia supernova events in symbiotic systems, the accreting WD should start with an initial mass in the range known for ultra-massive WDs. This means, that the WD should be the descendent of a massive ($>7$ M_⊙) progenitor. On the other hand, compact systems ($a_{0}<20$ AU) modelled with the standard BHL scheme evolve ultimately taking the accreting WDs to this mass limit for the two adopted masses of the accretor. This discrepancy suggests that binary simulations and population synthesis studies relying on the BHL approximation may require revision, even in cases where arbitrary efficiency factors are adopted.

• •

  Simulations including the wind accretion from Paper I generally agree with binary and orbital properties of a number of observed symbiotic systems. The accretion properties of well-characterised symbiotic binaries, such as R Aqr and Y Gem, are nicely reproduced by the wind accretion scheme of Paper I. The nova recurrence time of R Aqr is estimated to be a few times 10⁶ yr.

• •

  Simulations predict that accretors can transition back and forth through different recurrence nova times, and even between the three main accretion regimes (e.g., recurrence nova, steady burning, and giant envelope). Models that do not end in type Ia supernova events end up in the giant envelope accreting regime, given that the last thermal pulses from the donor star exhibit extreme events of mass loss.

• •

  The models produce bolometric luminosities consistent with optical and X-ray observations of symbiotic systems, supporting the reliability of the accretion scheme used in our simulation to match observed systems.

• •

  The total energy and angular momentum of modelled systems remain nearly constant during the RGB phase, with significant changes only occurring during the TPAGB phase. These changes are configuration-dependent and emphasize the influence of mass loss and accretion dynamics on system evolution.

• •

  Symbiotic systems with accretion maintain nearly circular orbits throughout their evolution, contrasting with systems without accretion, which develop increased eccentricities during the TPAGB phase. We confirm that wind accreting binaries exhibit $\dot{T}\geq 0$, but the orbital period variation increase reflecting the evolution of $\dot{M}_{\mathrm{w}}$ from each stellar evolution model.

This study demonstrates the critical role of the adopted accretion prescription in shaping the evolution of symbiotic systems and highlights the limitations of standard BHL-based models. The results also emphasize the importance of wind accretion dynamics in reproducing observed systems, offering new insights into the progenitors and long-term evolution of symbiotic binaries. The next step will be to study the accretion scheme defined in Paper I but for the more complex interactions unveiled by eccentric cases (R. F. Maldonado et al. in prep.).

R.F.M. thanks UNAM DGAPA (Mexico) and SECIHTI (Mexico) for postdoc fellowships. J.A.T. and J.B.R.-G. acknowledge support from the UNAM PAPIIT project IN102324. This work has made extensive use of NASA’s Astrophysics Data System.

The model results underlying this article will be shared on reasonable request to the corresponding author.