CALIBRATING THE UPDATED OVERSHOOT MIXING MODEL ON ECLIPSING BINARY STARS: HY Vir, YZ Cas, χ² Hya, and VV Crv

The stellar evolutionary code YNEV (Zhang 2014) is adopted to calculate the stellar evolutionary models. The opacities are interpolated from OPAL opacity tables (Iglesias & Rogers 1996) and the F05 low temperature opacity tables (Ferguson et al. 2005). The equation of state (EOS) functions are interpolated from the OPAL-EOS tables (Rogers & Nayfonov 2002). A bicubic polynomial is used in the interpolations of opacity and in the EOS tables in order to obtain continuous derivatives. The rates of nuclear reactions are based on Angulo et al. (1999) and are enhanced by the weak screening model (Salpeter 1954). The T-τ relation of the Eddington gray model of stellar atmospheres is adopted in the atmosphere integral. In the YNEV code, two theories of stellar convection are optional: the mixing length theory (MLT) and the turbulent convection model (TCM) developed by Li & Yang (2007). The latter is a non-local turbulent convection theory which is based on hydrodynamic equations and some modeling assumptions. In this paper, we focus on the effects of overshoot mixing, and thus the non-local TCM is adopted to deal with turbulent convection in the stellar interior. The implements of the TCM in the YNEV code are described by Zhang (2012c).

The traditional overshoot mixing model extends the convective boundary by a distance of l_OV = α_OVH_P to become the boundary of the fully mixed region, where α_OV is a parameter and H_P is the pressure scale height. The temperature gradient in this extending region, i.e., the overshoot region, is adiabatic or radiative (radiative means that it ignores the convective flux in the overshoot region). The illustration is based on ballistic overshoot models (e.g., Shaviv & Salpeter 1973; Maeder 1975; Bressan et al. 1981), which trace the average fluid element overshooting from the convection zone into the radiative region. However, these ballistic overshoot models are physically unreasonable (Renzini 1987; Zhang 2013) and inconsistent with helioseismic investigations. Helioseismic investigations (Christensen-Dalsgaard et al. 2011) have shown that the temperature gradient smoothly changes from adiabatic to radiative near the convective boundary, so the ballistic overshoot models are excluded because they show an adiabatic overshoot region and a jump of the temperature gradient at the boundary of the overshoot region. It has been suggested that only the TCMs (e.g., Xiong 1981; Xiong et al. 1997; Canuto 1997, 2011; Canuto & Dubovikov 1998; Deng et al. 2006; Li & Yang 2007; Li 2012) can fit the restriction.

Another popular overshoot mixing model is the diffusion model with the diffusion coefficient D based on the characteristic turbulent velocity v and the characteristic length l, i.e., D∝vl. However, in most diffusion overshoot mixing models (e.g., Freytag et al. 1996; Ventura et al. 1998; Lai & Li 2011; Zhang & Li 2012; Ding & Li 2014), the characteristic length l is assumed to be comparable with H_P. This is an analogy used to assume that the characteristic length in the overshoot region is similar with the characteristic length in the convection zone. Deng et al. (1996) have found that setting l ∼ H_P in D ∼ vl/3 leads to almost full mixing and have suggested a small characteristic length of l ∼ 10⁻⁵l₀ ∼ 10⁻⁵H_P for the overshoot mixing in order to obtain a mixing timescale comparable with the evolutionary timescale. Zhang (2012a, 2012b, 2012c) have shown that when the characteristic length is assume to be comparable with H_P, the dimensionless parameter in D = C_XvH_P should be small C_X ∼ 10⁻¹⁰ in order to fit some observations. This excessively small dimensionless parameter makes the assumption l ∼ H_P doubtful.

Recently, Zhang (2013) has developed an updated overshoot mixing model based on hydrodynamic equations and some modeling assumptions. This model focuses on the turbulent flux of the chemical component and calculates the diffusion coefficient for convective/overshoot mixing. It is found in the model that the diffusion coefficient in the overshoot region is different from that in the convection zone. In the convection zone,

$$\begin{eqnarray} &&D_{{\rm CZ}} = C_{{\rm CZ}} \frac{k^2}{\varepsilon }, \end{eqnarray} \tag{ 1 }$$

and in the overshoot region,

$$\begin{eqnarray} &&D_{\rm OV} = C_{\rm OV} \frac{\varepsilon }{N^2}, \end{eqnarray} \tag{ 2 }$$

where k is the turbulent kinetic energy, ε is the turbulent dissipation rate, N² describes the Brunt-Väisälä frequency, C_CZ is a parameter of the magnitude of the order of unity, and C_OV is another parameter which is recommended to be on the magnitude of the order of C_OV ∼ 10⁻³ based on the adopted TCM (and its parameters) and some observational restrictions (Zhang 2013). The physical meanings of Equations (1) and (2) have also been pointed out. Equation (1) is equivalent to the model D∝vH_P. However, Equation (2) is physically different from D∝vH_P. The diffusion coefficient in the overshoot region being Equation (2) is for the reason that fluid elements moving around their equilibrium location so the characteristic length is v/N (Zhang 2013).

In this paper, we use Zhang's (2013) overshoot mixing model (i.e., Equation (2)). The turbulent dissipation rate ε is calculated using TCM (Li & Yang 2007). Since the value of the dimensionless parameter C_OV for overshoot mixing is not determined in the theoretical model, the main aim of this paper is to use the observations of eclipsing binary stars to restrict the value of C_OV.

The masses and radii of the two components of an eclipsing binary star (by this we mean detached eclipsing binary star in this paper) can be observed via analyses of light curves. The masses, radii, and effective temperatures can be used to restrict the stellar evolution theory: the evolutionary track of a star with a given mass should pass the observed radius and effective temperature. For a main-sequence star with a convective core, the evolutionary track is sensitive to convective overshoot. Therefore, an eclipsing binary is a good tool to investigate convective overshoot. This has been performed on some eclipsing binary stars, e.g., CO And by Lacy et al. (2010) and AQ Ser by Torres et al. (2014). Ribas et al. (2000) and Claret (2007) have studied the dependence of the size of the fully mixed overshoot region on the stellar mass, and suggested that a classical overshoot region with a size of 0.2 ⩽ α_OV ⩽ 0.25 is the best overall.

In the investigations mentioned above, the overshoot region is assumed to be fully mixed. It is necessary to study the updated overshoot formula via eclipsing binary stars.

In the cases of standard stellar models with convective cores, the structure of a star is fixed when the mass, initial hydrogen abundance X, initial metallicity Z, age t, an overshoot parameter (i.e., α_OV for the classical overshoot or C_OV for the updated overshoot model), and a convection parameter α (i.e., α_MLT for the MLT theory or α_TCM for the TCM theory) are all fixed. Observations of a binary star give four restrictions for two stars with given masses, i.e., the radii and effective temperatures of the two components. We assume that the age and the chemical composition are the same for the two components. In this case, the initial hydrogen abundance X, initial metallicity Z, age t, and overshoot parameter of the binary star can be mathematically fixed when we adopt a fixed convection parameter because the number of variables is equal to the number of equations (Equations (A1) for two components). The standard errors can also be obtained based on the method described in the Appendix. Based on those properties, Zhang (2012c) tested the previous diffusion formula of the overshoot on the binary star HY Vir.

However, this method of calibration does not work in some cases. When the mass ratio q ≈ 1, the masses, radii, and effective temperatures of the two components are very close to each other. This leads to the problem that the Equations (A1) for the primary are almost identical to the Equations (A1) for the secondary, so there are only two independent equations. Therefore, we can only perform the calibration on eclipsing binary stars where the masses of the two components are obviously different. The effects of overshoot mixing on the stellar radius and effective temperature accumulate as the stellar age increases. Therefore, we cannot calibrate the overshoot parameter by using the stars near the zero-age main sequence (ZAMS) stage (e.g., UZ Dra Lacy et al. 1989 and V335 Ser Lacy et al. 2012).

Observation data for intermediate- and high-mass eclipsing binaries is scarce and not accurate enough. For high-mass stars, mass loss (Chiosi et al. 1978; Brunish & Truran 1982; Chiosi & Maeder 1986; Maeder& Meynet 1987; Meynet et al. 1994) and rotation (Meynet & Maeder 2000; Brott et al. 2011; Maeder& Meynet 2012) can also significantly affect stellar structure and evolution. Since rotation and mass loss are not well studied at the moment, we do not attempt to calibrate overshoot in high-mass stars. We focus on low-mass eclipsing binaries. In this paper, we use the methods above to find a suitable value of the overshoot parameter C_OV for the updated overshoot mixing model for four eclipsing binaries: HY Vir, YZ Cas, χ² Hya, and VV Crv.

We solve Equation (A1) for two components to calibrate the overshoot parameter C_OV, composition (X and Z), and age t for four eclipsing binaries: HY Vir, YZ Cas, χ² Hya, and VV Crv. The mass range in the samples is about 1.3 < M/M_☉ < 3.6, which is comprised of seven low-mass stars and an intermediate-mass star (i.e., the primary of χ² Hya). The results of the calibrations are shown in Table 1. The radii and the effective temperatures of the calibrated stellar models match the observations in the accuracies of ΔR/R_☉ < 10⁻³ and ΔlgT_eff < 10⁻³.

The calibration results in Table 1 show that the best value of C_OV is about 1 × 10⁻³. This is consistent with the suggested value via the test of the solar model and the classical restriction on convective core overshoot (Zhang 2013). Although the masses of the two components are different and there may possibly be a relationship between C_OV and the stellar mass, we fix C_OV in the calibration of each eclipsing binary. However, the results do not support an obvious dependency of C_OV on stellar mass. The calibration results show that the standard errors of C_OV are significant. It seems that the observational errors of M/M_☉, R/R_☉, and especially lgT_eff should be less than 1}, otherwise the corresponding error of the overshoot parameter is too large. The metallicities and ages of those eclipsing binaries have been suggested in the references as shown in Table 1 (e.g., Ref. Z and t) by comparing the stellar model based on the fully mixed classical overshoot region with the observations. It has been found that our results for metallicities and ages based on the updated overshoot mixing model are similar to the results of the classical overshoot model. Only the suggested metallicity of χ² Hya is significantly higher than our calibration. This may result from the fact that Clausen & Nordström (1978) have used old opacity tables in modeling the stars. The calibrations of the chemical composition on HY Vir and YZ Cas support a helium enrichment law ΔY/ΔZ ≈ 2. The results of the chemical composition of χ² Hya and VV Crv are not accurate enough to validate the law.

The evolutionary tracks for effective temperature versus radius for the stellar models of the binaries are shown in Figures 1–4. The thick solid lines, solid lines, dashed lines, and the dotted lines represent the evolutionary tracks with C_OV = 1 × 10⁻³, the calibrated stellar models, the stellar models with a 0.3 H_P fully mixed classical overshoot region, and the standard stellar models without mixing outside the convection zones, respectively. The stellar models with a 0.3 H_P fully mixed overshoot region and the standard stellar models without mixing are calculated for comparison, and the MLT is adopted in those models to deal with the convective flux. The parameter α_MLT = 1.75 in the MLT is based on solar calibration. The dashed lines are almost identical to the thick solid lines, indicating that the updated overshoot model with C_OV = 1 × 10⁻³ leads to a similar mixing efficiency as for a 0.3 H_P fully mixed overshoot region in the stars with masses in the studied range. It is shown that the stellar effective temperature of the models in the PMS stage and near ZAMS is slightly affected by the adopted convection theory (MLT is used in the dashed lines and dotted lines, and the non-local TCM is used in the solid lines and thick solid lines). However, the differences are small because both of the turbulent dissipation parameters in the MLT and the TCM are based on solar calibrations and both convection theories show adiabatic convection in the convective core. It is interesting to note that for the stars with M/M_☉ > 2 (i.e., the two components of χ² Hya and the primary of YZ Cas), there are significant differences in the PMS stage, i.e., where R/R_☉ ≈ 2.5 for the primary of χ² Hya, R/R_☉ ≈ 2.1 for the secondary of χ² Hya, and R/R_☉ ≈ 1.9 for the primary of YZ Cas. This can be explained as follows. At those locations, ¹²C is burned to be ¹⁴N in the center. Overshoot mixing could affect this process because of the refueling of ¹²C in the convective core. In the stellar models with a 0.3 H_P fully mixed overshoot region, overshoot mixing is assumed to be instantaneous. In the updated overshoot model, overshoot is a diffusion process, and the diffusion coefficient is not high enough to result in significant mixing in the short timescale of the PMS stage. Therefore, in Figure 3 and the left panel in Figure 2, the solid lines and the thick solid lines are located between the dotted lines and the dashed lines, and are very close to the dotted lines.

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The hydrogen abundances in the stellar interior models are shown in Figure 5. The models for 1.5 M_☉ in the three cases, i.e., the standard model, the model with classical overshoot with 0.3H_P, and the updated overshoot model with C_OV = 10⁻³, and with different center hydrogen abundance (X_C ≈ 0.57, 0.40, 0.23) are shown. It is found that the updated overshoot model results in a smooth profile of the hydrogen abundance. Near the convective boundary, the gradient is close to zero, indicating a large diffusion coefficient. Compared with the standard stellar models and the classical overshoot stellar models, the stellar models with the updated overshoot model show effects similar to the classical overshoot model with 0.3 H_P for refueling the core. This can be validated by estimating the difference of the areas below the solid lines and corresponding dotted lines. The distinction is that the hydrogen abundances of the updated overshoot model are always smooth, unlike the fact that there are no derivatives at the fully mixing boundaries in the classical overshoot model. The updated overshoot model showing effects similar to the classical overshoot model with 0.3 H_P explains that, as mentioned above, the evolutionary tracks of the two cases are identical and the calibrated metallicity and age are consistent with the suggested values based on stellar models with classical overshoot.

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The formula for the diffusion coefficient of the updated overshoot mixing model (e.g., Equation (26) in Zhang 2013) has shown that the diffusion coefficient is very large in the convection zone and is low in the overshoot region, which intrinsically ensures a fully mixed convection zone and a partially mixed overshoot region (Zhang 2013). The approximation of the formula for the diffusion coefficient in the overshoot region is Equation (2), which is adopted in the numerical calculations. Figure 6 shows the profile of the diffusion coefficient for overshoot mixing. The stellar model is for the 1.5 M_☉ star with X_C = 0.4 and C_OV = 10⁻³. According to Equation (1), the diffusion coefficient in the convection zone is as large as D ∼ 10¹⁶. We set the upper limit of D to be 10¹⁰ in the calculations because an excessively large diffusion coefficient may lead to numerical instability and D > 10¹⁰ ensures complete mixing. It is shown that D quickly decreases in a thin layer near the convective boundary and then exponentially decreases in the most of the overshoot region. After the quick decreasing, the geometric mean diffusion coefficient in the overshoot region is typically 10² and 10⁰ for 0.5 H_P and 1 H_P, respectively. Accordingly, the timescale for a length L showing obvious mixing is τ ∼ L²/D, i.e., about 10¹⁶s for 0.5 H_P and about 10²⁰s for 1 H_P (H_P ≈ 10¹⁰cm here). The former is of the same order of magnitude as the evolutionary timescale and the latter is much larger than the evolutionary timescale.

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In the standard stellar model of a low-mass main sequence star with a convective core, a phenomenon called semi-convection occurs. There is a region near the convective boundary where ∇_R > ∇_ad when this region is not mixed into the convective core, or ∇_R < ∇_ad when this region is fully mixed into the convective core. This leads to the contradiction that the fully mixed region is not the convection boundary. In the framework of the classical ideal of local convection, this region should be partially mixed to reach the convective neutral condition ∇_R = ∇_ad, since the mixing process cannot continue when the neutral condition is satisfied (Schwarzschild & Härm 1958). The intrinsic reason for this phenomenon is that the mixing timescale is much less than the evolutionary timescale (the mixing timescale is zero in classical ideal local convection, e.g., instantaneous mixing), and thus we do not have sufficient time resolution to trace the variation of ∇_R during the mixing process. However, semi-convection does not appear in our stellar models with the updated overshoot mixing model. Figure 7 shows the profiles of ∇_R and ∇_ad in stellar models with M = 1.5 M_☉ and X_C ≈ 0.4. The left panel shows the standard model and the right panel shows the model with updated overshoot mixing. In the region denoted as "SC", it is clearly shown that ∇_R > ∇_ad when the region is not mixed and ∇_R < ∇_ad when the region is fully mixed into the convective core. The radiative temperature gradient ∇_R is discontinuous at the fully mixing boundary due to the jump of the chemical abundance. In the model with updated overshoot, there is no such phenomenon. The radiative temperature gradient is continuous because the updated overshoot model describes a weak mixing process and the profile of chemical abundance is continuous. Xiong (1981, 1986) has found that semi-convection in massive stars results from the local convection theory and can be removed by using the non-local turbulent convection theory. Our calculations show the similar result that the non-local effect of turbulent convection, i.e., overshoot mixing, can remove semi-convection in low-mass stars.

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