The Dependence of Convective Core Overshooting on Stellar Mass: A Semi-empirical Determination Using the Diffusive Approach with Two Different Element Mixtures

Theoretical research on the extent of convective cores in stars dates back to the beginning of the 1960s. Convective elements that travel past the boundary of the core as defined by the classical Schwarzschild criterion, and extend its size, give rise to the phenomenon typically referred to as convective core overshooting. The extra mixing has a strong influence on the evolution of stars. It extends their main sequence lifetimes, and increases the degree of mass concentration toward the center. The latter property can be measured directly by determining the rate of apsidal motion in close, eccentric binary systems (see, e.g., Claret & Giménez 2010). The bibliography on the subject of convective core overshooting is quite extensive, and for the most relevant sources of interest for the present work, we refer the reader to our earlier paper on the topic (Claret & Torres 2016, hereafter Paper I).

While it is now widely accepted that core overshooting is needed in order to explain the observations of stars in a variety of stellar populations (see the discussion in the aforementioned paper), the degree to which overshooting might depend on stellar mass has been much less clear. Many (though not all) current grids of stellar evolution models assume some form of mass-dependence for the extra mixing, but those prescriptions are largely ad hoc. Empirical support has been scant, and sometimes contradictory. The issue has commonly been addressed most directly using measurements of double-lined eclipsing binaries (DLEBs), and in recent years also using diagnostics from asteroseismology. In the framework of the classical prescription for overshooting, the extra distance traveled by the convective elements beyond the limits of the core is represented by ${d}_{\mathrm{ov}}={\alpha }_{\mathrm{ov}}\ {H}_{p}$, where H_p is the pressure scale height and ${\alpha }_{\mathrm{ov}}$ is the classical overshooting parameter. Studies using DLEBs have found results ranging from a strong dependence of the ${\alpha }_{\mathrm{ov}}$ on mass (Ribas et al. 2000) to a more uncertain and much less pronounced dependence (Claret 2007), and even no dependence at all (Meng & Zhang 2014; Stancliffe et al. 2015). The DLEB samples in these investigations have typically been small, not exceeding a dozen or so systems (often far fewer), and have generally been restricted to the main sequence. Asteroseismic studies of single stars have so far been performed over limited ranges in mass and evolutionary state, but do suggest changes in overshooting that are not inconsistent with results described below (see,e.g., Aerts 2013; Deheuvels et al. 2016).

The most comprehensive observational study of the dependence of core overshooting on stellar mass was presented in Paper I, where we made use of a much larger set of 33 DLEBs in the range of $1.2\mbox{--}4.4\,{M}_{\odot }$ with accurate determinations of their physical properties (masses, radii, effective temperatures, and in many cases metallicities). These systems were carefully selected to be evolved enough so that we could discern the effects of core overshooting and infer a possible dependence on the mass of the star. A clear relation emerged between ${\alpha }_{\mathrm{ov}}$ and stellar mass, indicating a steep rise to about $2\,{M}_{\odot }$ with little or no change beyond this mass. No measurable dependence on metallicity was seen. This semi-empirically determined pattern of overshooting for stars of different mass has found an important practical application for stellar evolution models and has now been incorporated into the new grid of Yale–Potsdam Stellar Isochrones (YaPSI; Spada et al. 2017).

In recent years, an alternate prescription for core overshooting has become increasingly popular, in which the region of convective overshooting is not fully mixed, as in the classical method. Rather, as they travel outward, the convective elements disintegrate following a diffusive process and the decay of the turbulent velocity field is exponential. Following Freytag et al. (1996) and Herwig et al. (1997), the diffusion coefficient in the overshooting region is given by

$$\begin{eqnarray}&&{D}_{\mathrm{ov}}={D}_{0}\,\exp \left(\displaystyle \frac{-2z}{{H}_{\nu }}\right),\end{eqnarray} \tag{ 1 }$$

where D₀ is the diffusion coefficient at the convective boundary, z is the geometric distance from the edge of the convective zone, ${H}_{\nu }$ is the velocity scale-height at the convective boundary expressed as ${H}_{\nu }={f}_{\mathrm{ov}}\,{H}_{p}$, and the coefficient ${f}_{\mathrm{ov}}$ is a free parameter governing the width of the overshooting layer.

Comparisons between models using both approximations to overshooting over limited ranges in mass indicate that the resulting internal structures are rather similar (e.g., Herwig et al. 1997; Magic et al. 2010; Noels et al. 2010), perhaps suggesting a rough scaling between ${\alpha }_{\mathrm{ov}}$ and ${f}_{\mathrm{ov}}$. However, some asteroseismic studies claim to be able to distinguish the internal structures from both types of models, implying they are in fact not quite the same. For example, Moravveji et al. (2015, 2016) find evidence that the diffusive formulation of overshooting seems better able to explain the observed pattern of period spacings of slowly pulsating B stars than the classical step-function formulation.

The "equivalence" between ${\alpha }_{\mathrm{ov}}$ and ${f}_{\mathrm{ov}}$ has yet to be explored more fully over a wide range of masses and evolutionary states. While a calibration of ${\alpha }_{\mathrm{ov}}$ with stellar mass now exists (Paper I), the precise quantitative dependence of ${f}_{\mathrm{ov}}$ with mass is not known at the present time, as has been pointed out by Magic et al. (2010), although one might expect it to be at least qualitatively similar. One of the recent series of publicly available evolutionary models that uses the diffusive approximation for overshooting is the large grid of MESA Isochrones and Stellar Tracks (MIST) by Choi et al. (2016), which is based on the Modules for Experiments in Stellar Astrophysics package (MESA; Paxton et al. 2011, 2013, 2015). These models assume ${f}_{\mathrm{ov}}$ is independent of stellar mass, and adopt a fixed value of ${f}_{\mathrm{ov}}=0.016$ for all stars with convective cores, tuned to reproduce the shape of the main sequence turnoff in the color–magnitude diagram (CMD) of the open cluster M67. If ${f}_{\mathrm{ov}}$ actually varies with mass in some fashion, as we expect, the impact of using a constant ${f}_{\mathrm{ov}}$ rather than a mass-dependent one may not be immediately obvious in fitting isochrones to CMDs because of the various degeneracies and uncertainties involved.³ On the other hand, DLEBs with well-measured parameters may provide more sensitive tests.

Without access to an empirical calibration of the mass-dependence of ${f}_{\mathrm{ov}}$, modelers wishing to use the now more common diffusive approximation are left with little choice but to adopt a fixed value of this parameter constrained in some way by observations, as in the case just described, or to fall back on artificial trends informed perhaps by theoretical expectations (see also Magic et al. 2010). This unsatisfactory situation provides the main motivation for our work, which is to explore the dependence of overshooting on stellar mass using the diffusive formulation, by means of DLEBs, much in the same way as we did in Paper I with the simpler step-function prescription. Our hope is that the results may provide an empirical basis for a more realistic implementation of diffusive convective core overshooting in future models of stellar evolution. An additional goal is to extend our earlier work and examine the impact on the results of the element mixture and the primordial helium abundance, which lead to differences in the opacities.

The paper is structured as follows. In Section 2 we present our observational sample consisting of 29 well-studied DLEBs. Section 3 describes the new prescription for overshooting that we use here, the stellar evolution code, and our methodology to derive for each star the semi-empirical values of ${f}_{\mathrm{ov}}$ and the mixing-length parameter ${\alpha }_{\mathrm{MLT}}$, as well as the system metallicity. Our results on the mass-dependence of ${f}_{\mathrm{ov}}$ are reported in Section 4 for two different element mixtures, and a general discussion of the results is given in Section 5. We summarize our findings in Section 6. Finally, in the Appendix we use a simple analytical model based on homology transformations and the differential equations of stellar structure to investigate the role of opacities, the equation of state, and nuclear energy generation rates on the enlargement of the convective core due to overshooting. This provides important insight into the behavior of ${f}_{\mathrm{ov}}$ as a function of mass, which is reported in Section 4.

In Paper I we assembled a set of 33 detached eclipsing binary systems with well determined masses and radii (typically good to 3% or better), selected so that they were sufficiently evolved for the effects of overshooting to be noticeable. Many of the more evolved systems belong to the Large or Small Magellanic Clouds (LMC, SMC), and are preferentially more massive and more metal-poor than systems from the solar neighborhood, which tend to have compositions near solar. Four of those field binaries, ${\chi }^{2}$ Hya, YZ Cas, V885 Cyg, and VV Crv, did not quite meet the requirement adopted there that the individual ages inferred for the components from a comparison with stellar evolution models be within 5% of each other. They were nevertheless retained in that study because their mass ratios are very different from unity and therefore provide stronger leverage for the model fits.

The sample selected for the present work draws heavily on the same set of binaries, though with a few differences. One of the problematic systems, V885 Cyg, now provides satisfactory fits to the models used here within the 5% cap on the age difference, and has been included. This is likely the result of the use of a different stellar evolution code than in Paper I, as described in the next section. On the other hand, for HY Vir (another system with dissimilar masses) we now find poor fits to the models employed for this work, and we have therefore preferred to exclude it. This binary and the three others dropped from the sample in Paper I are interesting in their own right, and a full investigation of the reasons for the poor fits, which may possibly provide further insights into the models, will be the subject of a future paper.

The final sample for this work contains 29 DLEBs. Additionally, in a few cases with primary and secondary masses that are indistinguishable within the uncertainties, the use of different models here than in Paper I has led to a different conclusion regarding which component of the system is more evolved. For the benefit of the reader, Table 1 lists the properties of the 29 systems we use for this analysis, with the binaries arranged in order of decreasing primary mass and the more evolved star given on the first line.

Table 1.  Binaries Systems in Our Sample

Name	Mass (${M}_{\odot }$)	Radius (${R}_{\odot }$)	${T}_{\mathrm{eff}}$ (K)	[Fe/H]	Source
SMC-108.1-14904	4.429 ± 0.037	64.05 ± 0.50	4955 ± 90	−0.80 ± 0.15	1
	4.416 ± 0.041	46.95 ± 0.53	5675 ± 105
OGLE-LMC-ECL-CEP-0227	4.165 ± 0.032	34.92 ± 0.34	6050 ± 160		2
	4.134 ± 0.037	44.85 ± 0.29	5120 ± 130
OGLE-LMC-ECL-06575	4.152 ± 0.030	39.79 ± 1.35	4903 ± 72	−0.45 ± 0.10	3
	3.966 ± 0.032	49.35 ± 1.45	4681 ± 77
OGLE-LMC-ECL-CEP-2532	3.90 ± 0.10	28.95 ± 1.4	6345 ± 150		4
	3.83 ± 0.10	37.7 ± 1.7	4800 ± 220
LMC-562.05-9009	3.70 ± 0.03	28.6 ± 0.2	6030 ± 150:		5
	3.60 ± 0.03	26.6 ± 0.2	6030 ± 150:
OGLE-LMC-ECL-26122	3.593 ± 0.055	32.71 ± 0.51	4989 ± 80	−0.15 ± 0.10	3
	3.411 ± 0.047	22.99 ± 0.48	4995 ± 81
OGLE-LMC-ECL-01866	3.574 ± 0.038	46.96 ± 0.61	4541 ± 85	−0.70 ± 0.10	3
	3.575 ± 0.028	28.20 ± 1.06	5327 ± 72
OGLE-SMC-113.3-4007	3.561 ± 0.025	48.4 ± 0.7	4813 ± 100		6
	3.504 ± 0.028	45.8 ± 0.7	4800 ± 100
OGLE-LMC-ECL-10567	3.345 ± 0.040	25.6 ± 1.6	5067 ± 73	−0.81 ± 0.20	3
	3.183 ± 0.038	36.0 ± 2.0	4704 ± 80
OGLE-LMC-ECL-09144	3.303 ± 0.028	26.18 ± 0.31	5288 ± 81	−0.23 ± 0.10	3
	3.208 ± 0.026	18.64 ± 0.30	5470 ± 96
OGLE-051019.64-685812.3	3.278 ± 0.032	26.05 ± 0.29	5300 ± 100		7
	3.179 ± 0.029	19.76 ± 0.34	5450 ± 100
OGLE-LMC-ECL-09660	2.988 ± 0.018	43.87 ± 1.14	4677 ± 75	−0.44 ± 0.10	3
	2.969 ± 0.020	23.75 ± 0.66	5352 ± 70
SMC-101.8-14077	2.835 ± 0.055	23.86 ± 0.31	5170 ± 90	−1.01 ± 0.15	1
	2.725 ± 0.034	17.90 ± 0.50	5580 ± 95
α Aur	2.5687 ± 0.0074	11.98 ± 0.57	4970 ± 50	−0.04 ± 0.06	8
	2.4828 ± 0.0067	8.83 ± 0.33	5730 ± 60
WX Cep	2.533 ± 0.050	3.996 ± 0.030	8150 ± 250		7
	2.324 ± 0.045	2.712 ± 0.023	8900 ± 250
V1031 Ori	2.468 ± 0.018	4.323 ± 0.034	7850 ± 500		7
	2.281 ± 0.016	2.978 ± 0.064	8400 ± 500
V364 Lac	2.333 ± 0.014	3.309 ± 0.021	8250 ± 150		7
	2.295 ± 0.024	2.986 ± 0.020	8500 ± 150
SZ Cen	2.311 ± 0.026	4.556 ± 0.032	8100 ± 300		7
	2.272 ± 0.021	3.626 ± 0.026	8380 ± 300
OGLE-LMC-ECL-25658	2.230 ± 0.019	27.57 ± 0.24	4721 ± 75	−0.63 ± 0.10	9
	2.229 ± 0.019	21.41 ± 0.15	4860 ± 70
V885 Cyg	2.228 ± 0.026	3.387 ± 0.026	8150 ± 150		7
	2.000 ± 0.029	2.346 ± 0.017	8375 ± 150
AI Hya	2.140 ± 0.038	3.916 ± 0.031	6700 ± 60		7
	1.973 ± 0.036	2.767 ± 0.019	7100 ± 65
AY Cam	1.905 ± 0.040	2.772 ± 0.020	7250 ± 100		7
	1.709 ± 0.036	2.026 ± 0.017	7395 ± 100
SMC-130.5-04296	1.854 ± 0.025	25.44 ± 0.25	4912 ± 80	−0.88 ± 0.15	1
	1.805 ± 0.027	46.00 ± 0.35	4515 ± 75
OGLE-LMC-ECL-03160	1.799 ± 0.028	37.42 ± 0.52	4490 ± 82	−0.48 ± 0.20	3
	1.792 ± 0.027	16.36 ± 1.06	4954 ± 83
EI Cep	1.7716 ± 0.0066	2.897 ± 0.048	6750 ± 100		7
	1.6801 ± 0.0062	2.330 ± 0.044	6950 ± 100
SMC-126.1-00210	1.674 ± 0.037	43.52 ± 1.02	4480 ± 70	−0.86 ± 0.15	1
	1.669 ± 0.039	39.00 ± 0.98	4510 ± 70
HD 187669	1.505 ± 0.004	22.62 ± 0.50	4330 ± 70	−0.25 ± 0.10	10
	1.504 ± 0.004	11.33 ± 0.28	4650 ± 80
OGLE-LMC-ECL-15260	1.426 ± 0.022	42.17 ± 0.33	4320 ± 81	−0.47 ± 0.15	3
	1.440 ± 0.024	23.51 ± 0.69	4706 ± 87
AI Phe	1.2336 ± 0.0045	2.932 ± 0.048	5010 ± 120	−0.14 ± 0.10	7
	1.1934 ± 0.0041	1.818 ± 0.024	6310 ± 150

Note. The first line for each system corresponds to the more evolved star. Temperatures for LMC-562.05-9009 are listed as uncertain in the original source. The [Fe/H] value adopted here for OGLE-LMC-ECL-25658 is the average of the individual estimates reported. Sources are as follows: (1) Graczyk et al. (2014), (2) Pilecki et al. (2013), (3) Pietrzyński et al. (2013), (4) Pilecki et al. (2015), (5) Gieren et al. (2015), (6) Graczyk et al. (2012), (7) Torres et al. (2010), (8) Torres et al. (2015), (9) Elgueta et al. (2016), (10) Hełminiak et al. (2015).

Download table as:  ASCIITypeset image

In the classical formalism for convective core overshooting, sometimes referred to as the "step-function" approximation, the region beyond the core boundary as specified by the Schwarzschild criterion is assumed to be fully mixed. This simple prescription has found wide application in many of the current series of stellar evolution models, and a calibration of ${\alpha }_{\mathrm{ov}}$ as a function of stellar mass was presented in Paper I.

For the present work we have explored an alternate, exponentially decaying formulation for overshooting that is becoming more common, and is characterized by the free parameter ${f}_{\mathrm{ov}}$ described earlier that determines the width of the overshooting layer. The temperature gradient in this region is assumed to be radiative, and we adopt equal ${f}_{\mathrm{ov}}$ values above the hydrogen- and helium-burning regions. As mentioned before, recent evidence from an asteroseismic study of the slowly rotating pulsating B stars KIC 10526294 and KIC 7760680 by Moravveji et al. (2015, 2016) indicated that this prescription appears to be favored over the classical one, at least in these two cases.⁴

While our previous work in Paper I made use of the Granada stellar evolution code (Claret 2004), the diffusive formulation for overshooting is not yet implemented in that code, so for the present study evolutionary tracks were computed instead with the MESA code (Paxton et al. 2011, 2013, 2015), version 7385. A general description of the input physics of the MESA code is given in the above sources. For stars with convective envelopes we employed the standard mixing-length formalism (Böhm-Vitense 1958), where ${\alpha }_{\mathrm{MLT}}$ is a free parameter.⁵ The third-degree equation relating the temperature gradients was solved using the Henyey option in MESA. Microscopic diffusion was included, and the condition to determine the edge of the convection zone is the Schwarzschild criterion. As we did in our earlier work, our evolutionary calculations begin at the pre-main sequence phase. We have not considered stellar rotation in these models, and mass-loss was taken into account with the prescription by Reimers (1977) with the efficiency coefficient set to $\eta =0.2$. We note, though, that the effects of mass-loss in our sample are hardly significant, amounting to no more than 1% of the initial mass for our most massive giants (i.e., at the level of the observational errors in the masses). The high-temperature opacities were taken from the tables by Iglesias & Rogers (1996), and for lower temperatures we used the tables of Ferguson et al. (2005).

In our earlier study we had adopted the element mixture of Grevesse & Sauval (1998), which results in a present-day solar metallicity of ${Z}_{\odot }=0.0189$. The primordial helium content was set to Y_p = 0.24, and the adopted slope for the Galactic enrichment law was ${\rm{\Delta }}Y/{\rm{\Delta }}Z=2.0$. For the present work we have chosen to use the same element mixture initially to facilitate the comparison, but we also used the more recent mixture of Asplund et al. (2009), for which ${Z}_{\odot }=0.0134$, to explore the effect of a change in opacities. This new mixture was paired with an enrichment law specified by Y_p = 0.249 (Ade et al. 2016) and a slope ${\rm{\Delta }}Y/{\rm{\Delta }}Z=1.67$. Thus, we performed two sets of independent calculations:

• 1.  
  GS98 set: Grevesse & Sauval (1998) mixture, Y_p = 0.24, ${\rm{\Delta }}Y/{\rm{\Delta }}Z=2.0$ (same as Paper I);
• 2.  
  A09 set: Asplund et al. (2009) mixture, Y_p = 0.249, ${\rm{\Delta }}Y/{\rm{\Delta }}Z=1.67$.

These choices allow for two comparisons of interest: (1) to study the results from the two different overshooting formulations by comparing the GS98 set with the results of Paper I, which used the classical step-function approximation with the same mixture and helium enrichment law; and (2) to explore the influence of the element mixture and the helium content on overshooting using the same (diffusive) prescription for the phenomenon.

The fitting procedure applied here to infer the amount of overshooting for each star and the optimal abundance for each system is similar to the one used in Paper I, as described below. The observational constraints are the measured masses (M), radii (R), and effective temperatures (${T}_{\mathrm{eff}}$). The best match was sought between these observations and evolutionary tracks calculated for the measured masses of each component, allowing ${f}_{\mathrm{ov}}$ and ${\alpha }_{\mathrm{MLT}}$ to vary freely for each star, and letting Z also be variable but assumed to be the same for the two stars. As only about half of our systems have a measured chemical composition, we have preferred to use that information as a consistency check rather than a strict constraint.

For this work we have again opted for the mixed approach applied in our earlier study that combines a relatively coarse grid search of parameter space with manual fine-tuning to arrive at the final best-fit values. The manual adjustment has the advantage over a far more computationally expensive fine grid search that the solutions are easily inspected at each step of the process to avoid inconsistencies in the location of the components in the temperature-radius diagram. This is particularly valuable in rapid phases of evolution that could lead to unphysical situations (e.g., the more massive star being less evolved). For Paper I we had computed a grid of evolutionary tracks spanning a range of ${\alpha }_{\mathrm{ov}}$ values from 0.00 to 0.40 in steps of 0.05, and ${\alpha }_{\mathrm{MLT}}$ values between 1.0 and 2.0 (extended in some cases up to 2.7) with a step of 0.1. We have reused this grid here, not for actual fits but only in a differential sense, to gauge the direction in which predicted stellar parameters (temperature, radius, age) change as the overshooting or mixing lengths are modified, and aid our manual search for better fits. As the relevant variable for this work is ${f}_{\mathrm{ov}}$ rather than ${\alpha }_{\mathrm{ov}}$, we assumed a rough scaling between the two (see also Section 5.1) such that ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}\approx 10$ (e.g., Herwig et al. 1997; Noels et al. 2010), and used it to re-label our coarse grid in terms of ${f}_{\mathrm{ov}}$, which we then used as indicated above to guide our solutions. Based on this information and initial estimates of ${f}_{\mathrm{ov}}$, ${\alpha }_{\mathrm{MLT}}$, and Z we computed finer grids of MESA models with the diffusive overshooting approximation over shorter ranges tailored to each system. Small manual adjustments to the parameters were made in the final approach to the optimal fits until the predicted effective temperatures and radii agreed with the measured values approximately within their uncertainties, requiring also that the binary components have a similar age. To account for imperfections in the models and for observational errors, we allowed the derived ages to differ by up to 5%. Satisfactory fits with this condition were found for all 29 of our DLEBs. Initial estimates for Z were based on the measured metallicities for the 16 systems that have them, and for those that do not we used photometric estimates in some cases, or even values from the literature derived from fits to stellar evolution models, if available. We found that the final best-fit Z values occasionally deviated significantly from the initial values, as we discuss later.

Average uncertainties in ${f}_{\mathrm{ov}}$, for which we adopt 0.004 for unevolved stars (dwarfs) and 0.003 for giants, were estimated by varying this parameter keeping ${\alpha }_{\mathrm{MLT}}$ and the chemical composition fixed, with the requirement that the predicted radii and temperatures of the binary components be within the observational uncertainties and that the age difference be no larger than 5%. While these errors are considerably more conservative than those sometimes seen in the literature, we believe them to be realistic. The larger estimate for dwarfs reflects the lower sensitivity to overshooting on the main sequence. A similar procedure was followed for ${\alpha }_{\mathrm{MLT}}$, examining the effect of changes at constant values of ${f}_{\mathrm{ov}}$ and the chemical composition. The uncertainties in ${\alpha }_{\mathrm{MLT}}$ are estimated to be 0.20.

As a sanity check on our methods we chose the well-studied case of Capella (α Aur) to generate a grid of evolutionary tracks varying ${f}_{\mathrm{ov}}$ over the range 0.000–0.025 in increments of 0.005, and stepping ${\alpha }_{\mathrm{MLT}}$ over the range from 1.7 and 1.9 every 0.1, bracketing previous determinations for the components that used the same MESA code (see Torres et al. 2015). The observationally well-constrained Z value from the same study was held fixed. In addition to verifying that a grid search in this unambiguous case yielded essentially the same answer as our manual method, the exercise also supports our adopted uncertainties in ${f}_{\mathrm{ov}}$ and ${\alpha }_{\mathrm{MLT}}$.

The results of our calculations for the GS98 mixture are presented in Table 2, with the systems listed in the same order as in Table 1, and including the inferred values of the overshooting and mixing-length parameters (${f}_{\mathrm{ov}}$, ${\alpha }_{\mathrm{MLT}}$) for each component, as well as the optimal abundance Z and mean age for each system. Table 3 contains the corresponding results for the A09 mixture. Representative fits for four of our systems are shown in Figure 1, with the top row corresponding to the GS98 set and the bottom row to A09. The morphology of the evolutionary tracks, as well as the inferred evolutionary state of the components, are quite similar for the two mixtures.

Zoom In Zoom Out Reset image size

Table 2.  Fitted Parameters for Our Sample of DLEBs Using the GS98 Mixture

	Primary		Secondary
Name	${f}_{\mathrm{ov}}$	${\alpha }_{\mathrm{MLT}}$	${f}_{\mathrm{ov}}$	${\alpha }_{\mathrm{MLT}}$	Z	Mean Age (Myr)
SMC-108.1-14904	0.0210	1.80	0.0200	1.85	0.0025	130
OGLE-LMC-ECL-CEP-0227	0.0180	2.22	0.0180	2.26	0.0018	139
OGLE-LMC-ECL-06575	0.0190	2.05	0.0180	2.15	0.0080	160
OGLE-LMC-ECL-CEP-2532	0.0170	2.10	0.0200	1.90	0.0022	163
LMC-562.05-9009	0.0150	2.30	0.0150	2.30	0.0025	198
OGLE-LMC-ECL-26122	0.0190	1.80	0.0170	2.13	0.0080	235
OGLE-LMC-ECL-01866	0.0150	2.10	0.0150	2.10	0.0070	229
OGLE-SMC-113.3-4007	0.0205	2.30	0.0205	2.34	0.0035	224
OGLE-LMC-ECL-10567	0.0200	2.12	0.0180	2.00	0.0045	254
OGLE-LMC-ECL-09144	0.0150	2.10	0.0180	1.80	0.0035	252
OGLE-051019.64-685812.3	0.0190	2.40	0.0140	2.40	0.0045	275
OGLE-LMC-ECL-09660	0.0180	2.06	0.0180	2.06	0.0035	341
SMC-101.8-14077	0.0160	2.25	0.0160	2.28	0.0020	366
α Aur	0.0200	1.85	0.0200	1.80	0.0150	608
WX Cep	0.0160	1.85	0.0190	1.85	0.0220	523
V1031 Ori	0.0208	1.85	0.0190	1.85	0.0185	608
V364 Lac	0.0200	1.85	0.0200	1.85	0.0185	611
SZ Cen	0.0195	1.85	0.0190	1.85	0.0090	653
OGLE-LMC-ECL-25658	0.0181	2.06	0.0181	2.05	0.0045	805
V885 Cyg	0.0190	1.85	0.0180	1.85	0.0130	712
AI Hya	0.0160	1.80	0.0180	1.80	0.0370	897
AY Cam	0.0165	1.80	0.0165	1.80	0.0175	970
SMC-130.5-04296	0.0120	2.32	0.0150	2.04	0.0015	983
OGLE-LMC-ECL-03160	0.0800	1.94	0.0080	2.15	0.0025	1023
EI Cep	0.0130	1.90	0.0130	1.90	0.0150	1309
SMC-126.1-00210	0.0120	2.00	0.0120	2.05	0.0020	1240
HD 187669	0.0090	1.80	0.0090	1.82	0.0100	2330
OGLE-LMC-ECL-15260	0.0050	2.00	0.0050	2.11	0.0030	2143
AI Phe	0.0000	1.78	0.0000	2.05	0.0120	4383

Download table as:  ASCIITypeset image

Table 3.  Fitted Parameters for Our Sample of DLEBs Using the A09 Mixture

	Primary		Secondary
Name	${f}_{\mathrm{ov}}$	${\alpha }_{\mathrm{MLT}}$	${f}_{\mathrm{ov}}$	${\alpha }_{\mathrm{MLT}}$	Z	Mean Age (Myr)
SMC-108.1-14904	0.0190	1.80	0.0190	1.80	0.0018	123
OGLE-LMC-ECL-CEP-0227	0.0150	1.95	0.0150	2.08	0.0022	138
OGLE-LMC-ECL-06575	0.0170	2.02	0.0170	2.20	0.0070	155
OGLE-LMC-ECL-CEP-2532	0.0140	2.00	0.0170	1.97	0.0021	155
LMC-562.05-9009	0.0132	2.35	0.0128	2.35	0.0025	188
OGLE-LMC-ECL-26122	0.0190	1.80	0.0170	2.13	0.0070	234
OGLE-LMC-ECL-01866	0.0150	2.10	0.0150	2.10	0.0070	229
OGLE-SMC-113.3-4007	0.0161	2.22	0.0161	2.22	0.0020	207
OGLE-LMC-ECL-10567	0.0140	2.25	0.0140	2.00	0.0035	233
OGLE-LMC-ECL-09144	0.0145	2.00	0.0175	1.75	0.0040	247
OGLE-051019.64-685812.3	0.0160	2.08	0.0140	2.33	0.0048	271
OGLE-LMC-ECL-09660	0.0170	2.15	0.0170	2.15	0.0035	333
SMC-101.8-14077	0.0150	2.31	0.0150	2.27	0.0020	353
α Aur	0.0200	1.83	0.0200	1.80	0.0120	594
WX Cep	0.0170	1.85	0.0170	1.85	0.0180	537
V1031 Ori	0.0180	1.85	0.0180	1.85	0.0190	634
V364 Lac	0.0170	1.85	0.0170	1.85	0.0150	611
SZ Cen	0.0170	1.90	0.0160	1.85	0.0090	655
OGLE-LMC-ECL-25658	0.0170	1.94	0.0170	1.94	0.0025	728
V885 Cyg	0.0175	1.95	0.0170	1.85	0.0130	715
AI Hya	0.0160	1.87	0.0140	1.87	0.0220	956
AY Cam	0.0150	1.83	0.0140	1.83	0.0150	1023
SMC-130.5-04296	0.0100	2.33	0.0140	2.04	0.0012	929
OGLE-LMC-ECL-03160	0.0070	2.13	0.0070	2.13	0.0033	1034
EI Cep	0.0110	1.85	0.0110	1.80	0.0130	1299
SMC-126.1-00210	0.0100	1.95	0.0100	1.95	0.0013	1156
HD 187669	0.0090	1.80	0.0090	1.80	0.0075	2204
OGLE-LMC-ECL-15260	0.0040	2.00	0.0040	2.25	0.0040	2197
AI Phe	0.0000	1.70	0.0000	1.95	0.0090	4108

Download table as:  ASCIITypeset image

The behavior of ${f}_{\mathrm{ov}}$ for each element mixture as a function of stellar mass is displayed in Figure 2, in which the size of the symbols is proportional to the surface gravity $\mathrm{log}g$. All systems more massive than about $2.5\,{M}_{\odot }$ are giants. To avoid clutter we show the mean error bars for evolved and unevolved systems on the lower right, rather than on each point.

Zoom In Zoom Out Reset image size

We find a clear dependence of the overshooting parameter on mass for both mixtures, with a strong rise in ${f}_{\mathrm{ov}}$ up to about $2\,{M}_{\odot }$ that flattens out thereafter. This represents the first semi-empirical mass calibration of ${f}_{\mathrm{ov}}$. We defer a discussion of the nature of this relation until Section 5.2. The trend appears qualitatively similar for GS98 and A09, though there may be slight differences. For example, if we consider the regime of stars more massive than $2\,{M}_{\odot }$, where ${f}_{\mathrm{ov}}$ is effectively constant, we find an average overshooting parameter for the sample of ${f}_{\mathrm{ov}}=0.0181\pm 0.0003$ for the GS98 set (mean of estimates for 40 stars) and ${f}_{\mathrm{ov}}=0.0164\pm 0.0003$ for the A09 set.⁶ While these appear statistically distinct, the formal uncertainties do not include possible systematic errors, which are difficult to quantify and may be much larger than the statistical errors, so we hesitate to conclude there is a real difference.

Concerning our fitted ${\alpha }_{\mathrm{MLT}}$ values, which rely on the 1D MESA code, we find that they are in reasonable agreement with the predictions from 3D radiative hydrodynamic simulations by Magic et al. (2015), restricted to the same range of metallicities, effective temperatures, and surface gravities as the binaries in our sample. This is noteworthy considering the significant differences between the input physics of the MESA code and the 3D models of Magic et al., and the presence of observational errors in our own estimates.

We note also that the model comparisons with the GS98 mixture lead to best-fit Z values that are on average slightly but systematically smaller than the measured abundances, for the 16 systems for which empirical metallicity estimates are available. A similar but more pronounced effect was seen in Paper I, and is most obvious for some of the metal-poor systems in the LMC and SMC. The average disagreement is at about the 2σ level, corresponding to a [Fe/H] difference of 0.11 ± 0.05 dex. We see no such discrepancy on average with the A09 mixture (${\rm{\Delta }}[\mathrm{Fe}/{\rm{H}}]=0.02\pm 0.05$ dex). Possible explanations for the difference in behavior may be related to the opacities themselves, an incorrect slope ${\rm{\Delta }}Y/{\rm{\Delta }}Z$, or an inaccurate value for the adopted primordial helium content for the GS98 mixture. We return to this in the next section.

5.1. The Diffusive and Step-function Implementations of Overshooting Compared

A first interesting conclusion we draw from the results is that the qualitative way in which the strength of the overshooting depends on stellar mass (Figure 2) seems essentially independent of the opacities and helium enrichment laws involved. Furthermore, the run of ${f}_{\mathrm{ov}}$ with mass from both the GS98 and A09 sets is in turn qualitatively very similar to that obtained in Paper I for ${\alpha }_{\mathrm{ov}}$, which used the step-function approximation. Thus, the details of how the phenomenon is parameterized also seem unimportant, at least when it comes to the mass-dependence. While this last conclusion might have been anticipated from the rough similarity between the internal stellar structures inferred theoretically using the two overshooting formulations (e.g., Magic et al. 2010), empirical verification such as what we report here remains essential can lead to further insights, and now provides a practical recipe for incorporating the ${f}_{\mathrm{ov}}$ mass-dependence into models, in the same way as our earlier work in Paper I supplied one for the classical ${\alpha }_{\mathrm{ov}}$ implementation. We note, however, that with our current understanding of the phenomenon, one recipe cannot necessarily be inferred from the other.

A number of authors (e.g., Herwig et al. 1997; Noels et al. 2010) have in fact suggested an approximate scaling between the two overshooting parameters such that ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}\approx 10$, though very few semi-empirical estimates of this relation seem to be available in the literature. One estimate of ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}\approx 13$ was reported in an asteroseismic study of the (single) slowly pulsating B star KIC 7760680 by Moravveji et al. (2016). A study of the eclipsing binary TZ For by Valle et al. (2017) gave ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}\approx 12$. Magic et al. (2010) made a direct comparison between models for masses ranging from 2 M_☉ to 6 M_☉ using the step-function approximation (with ${\alpha }_{\mathrm{ov}}$ held fixed at 0.20) and calculations based on the diffusive implementation, and obtained a best-fit value with the latter of ${f}_{\mathrm{ov}}=0.018$, corresponding to a ratio ${\alpha }_{\mathrm{ov}}$/${f}_{\mathrm{ov}}$ ≈ 11. However, their study did not address the more critical regime below 2 M_☉, or stars that have evolved off the main sequence.

A star-by-star comparison between the ${f}_{\mathrm{ov}}$ values from the present work and the ${\alpha }_{\mathrm{ov}}$ values from Paper I (both using the GS98 mixture and the same enrichment law) enables us to revisit this issue. We find that on average ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}=11.36\pm 0.22$ (mean of 56 ratios, excluding the two stars in AI Phe with ${f}_{\mathrm{ov}}=0$). This is consistent with previous estimates, but is based on a much larger sample of semi-empirical determinations. The two parameters plotted against each other are shown graphically in Figure 3. Closer inspection suggests, however, that the connection may be more complex, and there may be a slight dependence of the ${\alpha }_{\mathrm{ov}}$/${f}_{\mathrm{ov}}$ ratio on the surface gravity $\mathrm{log}g$, or possibly Z, mass, or ${T}_{\mathrm{eff}}$. Unfortunately, these quantities are strongly correlated with each other in our sample. This is because the majority of the cool giants are in the LMC and SMC and are both more massive and significantly more metal-poor than the field dwarfs, so it is not possible to ascertain which variable is the most relevant one. Nonetheless, as an illustration, if we split the sample at ${T}_{\mathrm{eff}}=6500$ K we obtain averages of ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}=10.50\pm 0.25$ for the hotter stars and ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}=11.71\pm 0.27$ for the cooler ones. Identical numbers result for the dwarfs and giants, respectively, if we split the sample at $\mathrm{log}g=3.0$. The difference is formally at the 3σ level. An even larger sample with a range of uncorrelated stellar properties would be needed to investigate this issue further.

Zoom In Zoom Out Reset image size

Overshooting has a significant impact on stellar ages. Between the present work and that of Paper I we now have semi-empirical estimates of the overshooting parameter, as well as age estimates derived in the framework of two separate approaches (${f}_{\mathrm{ov}}$ and ${\alpha }_{\mathrm{ov}}$) for 29 binary systems, using the same element mixture (GS98) and the same helium enrichment law. We find that the mean ages for these 29 systems in common are rather similar for the two overshooting implementations, with an average ratio of $\mathrm{Age}({f}_{\mathrm{ov}})/\mathrm{Age}({\alpha }_{\mathrm{ov}})=0.975\pm 0.002$ (statistical error) and a range of values between 0.88 for AI Phe and 1.10 for OGLE-LMC-ECL-26122. Although the mean ratio is formally smaller than unity, we suspect the difference may not be significant given that unquantified systematics probably dominate the error budget, and especially since there are also differences between the Granada and MESA evolutionary codes used in each case, both in terms of their physical ingredients and their numerical details.

5.2. The Size of the Convective Core and the Nature of the Mass-dependence of ${f}_{\mathrm{ov}}$

In this section we comment on the underlying reasons for the initial increase in ${f}_{\mathrm{ov}}$ with mass shown in Figure 2, starting around $M\approx 1.2\,{M}_{\odot }$, and the subsequent flattening of the relation after about 2 M_☉ and up to the mass limit of our sample. For this we focus on the behavior of the size of the convective core, as indicated by the models, which is a quantity more closely linked to the interior physics. Using our best-fit values for ${f}_{\mathrm{ov}}$, ${\alpha }_{\mathrm{MLT}}$, and the chemical composition for each star, we have extracted from the corresponding evolutionary tracks the mass of the convective core Q_c that the star has at the zero-age main sequence (ZAMS),⁷ which is directly determined by the strength of the overshooting as parameterized by ${f}_{\mathrm{ov}}$. We restrict ourselves to examining the physics at the ZAMS in order to avoid effects from the evolution that occurs later, and because of the simpler, nearly homogeneous chemical structure of ZAMS stars. The top panel of Figure 4 shows how this quantity Q_c, which is normalized to the total mass of the star, varies as a function of mass for the A09 element mixture in the diffusive approach to overshooting used in this paper (filled symbols). We find a very similar trend for the GS98 mixture (not shown), indicating the opacities have little influence.

Zoom In Zoom Out Reset image size

The open symbols in the top panel of Figure 4 represent the normalized mass of the convective core at the ZAMS in the absence of overshooting (i.e., from "standard" models that use the Schwarzschild criterion to set the core boundary). As before, the results are similar for GS98 and A09. The separation between the filled and open symbols at a given mass represents the enlargement of the convective core due to overshooting. This net growth is seen to increase rapidly with mass up to about $2\,{M}_{\odot }$, but not much thereafter, reflecting the same trend observed in Figure 2. An equivalent representation is displayed in the lower panel of Figure 4, which shows the fractional change in Q_c. For both element mixtures we see an obvious trend such that the fractional increase in Q_c is larger for the less massive stars, reaching ∼60% at about 1.5 M_☉ and then decreasing to 20%–30% for the higher-mass stars in our sample.

The physical reasons for this behavior of the fractional size of the convective core as a function of stellar mass, with a change in character around $2\,{M}_{\odot }$, have to do with concomitant changes in the opacities, the equation of state, and the nuclear reaction rates. We explore this in further detail in the Appendix using a simple analytical toy model involving homology transformations and the basic differential equations of stellar structure. We show there that it is indeed possible to understand the trends in Figures 2 and 4 as due at least in part to the role of those ingredients at different masses.

Finally, we note that the way in which Q_c varies with stellar mass is essentially the same as seen in Figure 5 of Paper I for the step-function approximation to overshooting, and similarly with the fractional change in Q_c (Figure 5 of Paper I, lower panel). In the latter case, though, the effect was somewhat magnified because the models without overshooting were computed for a single representative solar metallicity, rather than the Z value of each system that we use in the present work. Thus, once again the formulation of overshooting appears not to matter much in terms of the general trend, at least to first order. This is not entirely surprising given the approximate scaling reported earlier between the two free parameters, ${\alpha }_{\mathrm{ov}}\approx 11.4{f}_{\mathrm{ov}}$, which to some extent implies a correspondence in the net effect of overshooting, even though the details of how it operates in the two implementations are not the same.

5.3. Relation between the Best-fit Metallicities and the Primordial Helium Abundance

The key result of this paper is the determination of the explicit dependence of the diffusive convective core overshooting parameter ${f}_{\mathrm{ov}}$ on stellar mass in a semi-empirical way, relying on a set of nearly 30 eclipsing binary systems with well-measured properties. This study extends our previous work (Claret & Torres 2016) that used the classical step-function approximation for overshooting with its free parameter ${\alpha }_{\mathrm{ov}}$ and a similar set of binaries, and completes our investigation of the phenomenon with the two most commonly used prescriptions. Additionally, in the present work we have tested two different element mixtures for the opacities, adding to the Grevesse & Sauval (1998) mixture used in Paper I a more recent mixture by Asplund et al. (2009). We find a clear dependence of ${f}_{\mathrm{ov}}$ on stellar mass rising to about 2 M_☉ and then leveling off to the upper mass limit of our sample (∼4.4 M_☉), regardless of the opacities. This qualitative behavior is similar to the one found before for ${\alpha }_{\mathrm{ov}}$. While many recent grids of stellar evolution models build in variations in the strength of overshooting with mass in somewhat arbitrary ways, this study and that in Paper I now provide practical recipes for doing this that are grounded on observations, rather than expectations from theory. Our ${f}_{\mathrm{ov}}$ calibration therefore represents an important step forward, although there is still ample room for improvement before it can be considered definitive. For example, our binary sample has few stars near the low end of the mass range where overshooting begins to ramp up, and about half of the systems are lacking empirical metallicity estimates that could be used to further constrain the fits. Observational errors and shortcomings in the models themselves must also be kept in mind.

We have examined the nature of the mass-dependence of ${f}_{\mathrm{ov}}$ by focusing on the related fractional increase in the size of the convective core at the ZAMS (${\rm{\Delta }}{Q}_{c}/{Q}_{c}$) caused by diffusive overshooting. This quantity is more closely linked to the physics of stellar interiors, and also shows a mass-dependence such that stars with smaller masses have larger relative increases in the core mass Q_c up to 60%, which becomes smaller (20%–30%) for larger stars. The behavior is similar for both element mixtures used here, and also mirrors that seen in Paper I with the step-function implementation of overshooting. The use of a simple model based on homology transformations and the basic differential equations of stellar structure succeeds in explaining this trend in a qualitative way, revealing the role of opacities, the equation of state, and nuclear reaction rates. A deeper understanding of the behavior of ${f}_{\mathrm{ov}}$ as a function of stellar mass displayed in Figure 2 would benefit from an investigation into the physical conditions at the edge of the convective cores, and how they affect the convective plumes. While such a study is beyond the scope of the present work, we plan to undertake it in a future publication.

Our use of two different helium enrichment laws associated with the adopted GS98 and A09 mixtures has also helped to shed light on a nagging systematic difference between the fitted and measured metallicities noted originally in Paper I, and seen also here with the GS98 set to a lesser degree. We find that the discrepancy is largely removed by adopting the higher primordial helium abundance Y_p = 0.249 established in recent studies (Ade et al. 2016), instead of the value Y_p = 0.24 we used previously.

Comparing our ${f}_{\mathrm{ov}}$ determinations that use the GS98 mixture and the diffusive approximation with the ${\alpha }_{\mathrm{ov}}$ estimates from Paper I based on the step-function prescription and the same mixture (as well as the same enrichment law), we establish a semi-empirical scaling relationship between the two overshooting parameters such that ${\alpha }_{\mathrm{ov}}/{f}_{\mathrm{ov}}=11.36\pm 0.22$ (mean of 56 estimates), with a tentative dependence on either temperature or $\mathrm{log}g$, or possibly mass or Z (all strongly correlated in our sample). This connection between ${f}_{\mathrm{ov}}$ and ${\alpha }_{\mathrm{ov}}$ is consistent with, but now more firmly established than previous estimates.

We are grateful to A. Dotter for expert assistance and advice in using the MESA module, as well as C. Aerts for references and helpful comments on the subject of this paper. We also thank our two anonymous referees for constructive criticism. The Spanish MEC (AYA2015-71718-R and ESP2015-65712-C5-5-R) is gratefully acknowledged for its support during the development of this work. G.T. acknowledges partial support from the NSF through grant AST-1509375. This research has made use of the SIMBAD database, operated at the CDS, Strasbourg, France, and of NASA's Astrophysics Data System Abstract Service.

Here, we explore and attempt to understand the nature of the stellar mass-dependence of ${\rm{\Delta }}{Q}_{c}/{Q}_{c}$ (the relative change in the core mass shown in the lower panel of Figure 4), which is directly linked to the ${f}_{\mathrm{ov}}$ trend of Figure 2, as revealed by our semi-empirical estimates for our sample of 29 eclipsing binary systems. We will use simple ideas based on homology transformations, in which the two stellar models to be scaled are required to have the same relative mass distribution. One of the most useful tools to investigate how mass is distributed in stellar interiors is the Radau differential equation, whose solution provides the apsidal-motion constant k_j of order j that gives an accurate indication of the mass distribution. Standard models and those computed with core overshooting for the same mass present similar apsidal-motion constants at the ZAMS, indicating that they also have similar relative mass distributions (see, e.g., Claret & Giménez 1991, Figures 2, 4, and 6). The same holds when analyzing the properties of the respective cores. Therefore, we may apply homology transformations to both standard homogeneous models as well as those with extra mixing from overshooting. Due to the complexity of the various ingredients of stellar physics (opacities, energy generation, equation of state, etc.) homology transformations do not guarantee accurate solutions, but they are still very useful for exploring certain problems in a qualitative way, and even quantitatively to some degree, providing insights into the physics involved.

For the problem at hand—understanding the change in ${\rm{\Delta }}{Q}_{c}/{Q}_{c}$ as a function of stellar mass, which is closely related to the change in ${f}_{\mathrm{ov}}$ as a function of mass—we begin with a star in hydrostatic equilibrium with a convective core of mass Q₀ that satisfies the Schwarzschild criterion, and that is surrounded by a shell of negligible mass in radiative equilibrium. Consider now a second model to be scaled from the first, with a large convective core ${Q}_{0}+{\rm{\Delta }}Q$. Our reference shell is displaced upward to a position ${R}_{0}+{\rm{\Delta }}R$ simulating the effect of core overshooting. We apply simple homology relations and the basic differential equations of stellar structure and evolution to study in an approximate quantitative way the impact of the increase in radius (and mass) of the convective core.

The net rate of nuclear energy generation inside a sphere of radius r is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dL}}_{g}(r)}{{dr}}=4\pi {r}^{2}\rho \epsilon ,\end{eqnarray} \tag{ 2 }$$

where L_g denotes the rate of energy production by nuclear processes, is the thermonuclear energy generation rate per unit mass ($\epsilon =b\rho {T}^{\nu }$), ρ is the density, T is the temperature, b is a constant, and r is the radial distance. Here, we focus on homogeneous (ZAMS) models to compare with the data displayed in Figure 4. Let Q₀, R₀, and ${\mu }_{0}$ be the mass, radius, and the mean molecular weight of the standard convective core (without extra mixing) and Q, R, and μ the same variables for a configuration with extra mixing, i.e., with radius $R={R}_{0}\,+{\rm{\Delta }}R$, core mass $Q={Q}_{0}+{\rm{\Delta }}Q$, and mean molecular weight $\mu ={\mu }_{0}+{\rm{\Delta }}\mu$. The homology transformations will be used to scale one model to the other and evaluate the changes ${\rm{\Delta }}Q$ and ${\rm{\Delta }}R$ produced by the additional mixing. Inserting the usual homology transformations for the density and temperature, assuming an ideal non-degenerate gas with no radiation pressure, and integrating the above equation, we obtain

$$\begin{eqnarray}{L}_{g}(x) & = & \displaystyle \frac{Q}{{Q}_{0}}{\displaystyle \int }_{0}^{x}{\left(\displaystyle \frac{b}{{b}_{0}}\right)}_{x^{\prime} }\displaystyle \frac{{{dL}}_{\mathrm{go}}(x^{\prime} )}{{dx}^{\prime} }{dx}^{\prime} \\ & = & {L}_{\mathrm{go}}(x)\left(\displaystyle \frac{b}{{b}_{0}}\right){\left(\displaystyle \frac{\mu }{{\mu }_{0}}\right)}^{\nu }{\left(\displaystyle \frac{Q}{{Q}_{0}}\right)}^{2+\nu }{\left(\displaystyle \frac{R}{{R}_{0}}\right)}^{-3-\nu },\end{eqnarray} \tag{ 3 }$$

where x = r/R.

The restriction mentioned above concerning the equation of state is necessary because, even if we consider an ideal gas, the addition of radiation pressure will not lead to permanent homology relationships. On the other hand, as we are dealing with homogeneous models, we may take $\mu ={\mu }_{0}$, and we may also assume that the constants b and b₀ are independent of x for small ${\rm{\Delta }}R$. In this case, the relative change in the nuclear energy generation will be given by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{L}_{g}}{{L}_{g}}=(2+\nu )\displaystyle \frac{{\rm{\Delta }}Q}{{Q}_{0}}-(3+\nu )\displaystyle \frac{{\rm{\Delta }}R}{{R}_{0}}.\end{eqnarray} \tag{ 4 }$$

Additionally, the equation of radiative transfer can be written as

$$\begin{eqnarray}&&\displaystyle \frac{\partial T}{\partial r}=-\displaystyle \frac{3}{16\pi {ac}}\displaystyle \frac{\kappa \rho {L}_{\mathrm{rad}}}{{r}^{2}{T}^{3}},\end{eqnarray} \tag{ 5 }$$

where ${L}_{\mathrm{rad}}$ is the net rate of energy carried out by radiation, a is the radiation pressure constant, and c is the speed of light. As before, we apply the homology relation for the density assuming a perfect ideal gas, no radiation pressure, and assuming a Kramers complete opacity law given by $\kappa =B{\rho }^{n}{T}^{-s}$, where B is a constant. The same remarks from above concerning b and b₀ hold also for the constants B and B₀. Finally, the resulting relative change of energy ${L}_{\mathrm{rad}}$ is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{L}_{\mathrm{rad}}}{{L}_{\mathrm{rad}}}=(3-n+s)\displaystyle \frac{{\rm{\Delta }}Q}{{Q}_{0}}-(3n-s)\displaystyle \frac{{\rm{\Delta }}R}{{R}_{0}}.\end{eqnarray} \tag{ 6 }$$

We note that the relative change in ${L}_{\mathrm{rad}}$ is insensitive to the rate of nuclear energy generation because it is independent of ν. Under conditions of thermal equilibrium we have ${\rm{\Delta }}{L}_{g}/{L}_{g}={\rm{\Delta }}{L}_{\mathrm{rad}}/{L}_{\mathrm{rad}}$, from which the fractional increase in the convective core mass due to the extra mixing is easily derived as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}Q}{{Q}_{0}}=\displaystyle \frac{(3+3n+\nu -s)}{(n+\nu -s-1)}\displaystyle \frac{{\rm{\Delta }}R}{{R}_{0}}.\end{eqnarray} \tag{ 7 }$$

Note also that ${L}_{\mathrm{rad}}$ depends only on how energy is transported (opacities, equation of state), whereas L_g depends on how it is generated (thermonuclear energy generation rate per unit mass). The functional form of the slope at each mass in Equation (7) is the result of the interaction between both processes. This interaction is a consequence of the imposition of the condition of thermal equilibrium. The above equation may also be derived in a different way using permanently homologous models by expanding $Q/{Q}_{0}$ in a series up to 2nd order.

The slope $(3+3n+\nu -s)/(n+\nu -s-1)$ in Equation (7) presents some interesting features. For pure Thomson scattering or for Kramers opacities it is larger for lower values of ν, that is, for stars for which the proton–proton chain contributes significantly to the energy generation. In other words, the slope is more pronounced for less massive stars than for more massive ones. For larger values of ν the slope has a near-asymptotic behavior, independently of the adopted opacity law.

The dependence on temperature of the two major nuclear sources of energy during the main sequence (the proton–proton chain and the CNO cycle) shows three main characteristics: (1) for $\mathrm{log}T\lt 7.25$ the dominant source is the proton–proton chain, (2) for $\mathrm{log}T\approx 7.25$ the two processes have a similar contribution, and (3) for $\mathrm{log}T\approx 7.3$ (corresponding to a stellar mass larger than about 2 M_☉) the CNO cycle contributes about 10 times more than the proton–proton chain, and at a slightly higher temperature of $\mathrm{log}T=7.4$ it becomes about 100 times more important. Due to the complicated nature of the nuclear energy rates, and the fact that in a given star both nuclear processes may be acting at the same time in different regions, it is difficult to determine an effective value of ν for each mass. It is common to assign a typical value of $\nu =4$ for stars of small mass and $\nu =17$ for more massive stars, though without explicitly stating a dependence on mass. To provide a continuous relation between these two regimes, for our purposes we adopt a simple linear expression motivated by the properties of the nuclear energy production summarized above, of the form ${\nu }_{\mathrm{eff}}=(M-1.25)+4.8$, to be inserted in Equation (7). In this expression M is the stellar mass in solar units. For simplicity, we also assume that the opacity is given by the classical Kramers law (n = 1, $s=7/2$).

With these ingredients, we may now compare the relative increase in core mass due to extra mixing, as given by Equation (7), with the semi-empirical data from our 29 DLEBs. As we cannot establish the values of ${\rm{\Delta }}R/{R}_{0}$ for the binary components a priori (from first principles), we consider a few fixed values and draw iso-${\rm{\Delta }}R/{R}_{0}$ curves in Figure 5 for comparison with the observations. The triangles in this figure represent the semi-empirical values of ${\rm{\Delta }}{Q}_{c}/{Q}_{c}$ for each star obtained using the A09 mixture. We explore three values of ${\rm{\Delta }}R/{R}_{0}$ as shown by the three curves in the figure (0.15, 0.10, and 0.05, from top to bottom). Detailed estimates of ${\rm{\Delta }}R/{R}_{0}$ for selected binaries in our sample using the best-fit models give values within this range, indicating they are realistic.

Zoom In Zoom Out Reset image size

Despite the simplicity of our analytical model, the theoretical iso-${\rm{\Delta }}R/{R}_{0}$ curves show a similar pattern of variation as the semi-empirical data, with the less massive stars displaying higher ${\rm{\Delta }}{Q}_{c}/{Q}_{c}$ values than more massive stars. Additionally, it can be seen that the previously mentioned near-asymptotic behavior of the slope for the more massive stars is consistent with the semi-empirical results. While these calculations were made specifically for the A09 mixture, we found the same features when using the GS98 set.

In summary, building on very simple approximations Equation (7) is able to successfully predict the pattern of changes in the relative increase ${\rm{\Delta }}{Q}_{c}/{Q}_{c}$ in the mass of the convective core due to overshooting, illustrating the connection between this slope and stellar mass, the type of nuclear process, the opacities, and the equation of state.