HY Vir: Eclipsing Binary Overshoot Benchmark

• HY Vir is a detached eclipsing binary whose precisely measured masses and radii serve as a benchmark for testing convective-core overshoot models.
• The system’s common-age and composition allow researchers to calibrate both diffusion-based and classical step overshoot prescriptions with high accuracy.
• Updated overshoot models reveal that partial mixing creates smooth hydrogen gradients, effectively extending main-sequence lifetimes and refining evolutionary tracks.

HY Vir is a detached eclipsing binary whose components, with masses $1.838 \pm 0.009\,M_\odot$ and $1.404 \pm 0.006\,M_\odot$, both possess convective cores and therefore provide a stringent empirical testbed for convective-core overshoot prescriptions. In the cited studies, the system is treated as two single stars sharing a common age and initial composition, because the period is long enough and there is no evidence of significant interaction. Within that framework, HY Vir has been used to calibrate diffusion-based overshoot mixing models, to compare them with classical fully mixed step overshoot, and to examine the resulting internal composition gradients, temperature gradients, and evolutionary tracks (Meng et al., 2014, Zhang, 2012).

1. System characterization and measured parameters

HY Vir is described as a well-observed detached eclipsing binary with component masses above $1.2$ solar masses, which makes it an effective calibrator of convective core overshoot. Both components are classified, in the terminology of the 2014 study, as low-mass stars with convective cores. Spectral classifications are not explicitly given in these papers, and the analysis is based on fundamental parameters rather than spectral type. The observational basis is primarily the work of Lacy & Fekel (2011), which provided new light curves and radial velocity curves and enabled accurate masses, radii, effective temperatures, and luminosities (Meng et al., 2014, Zhang, 2012).

Quantity	Primary	Secondary
Mass	$1.838 \pm 0.009\,M_\odot$	$1.404 \pm 0.006\,M_\odot$
Radius	$2.806 \pm 0.008\,R_\odot$	$1.519 \pm 0.008\,R_\odot$
$\lg T_{\rm eff}$	$3.836 \pm 0.008$	$3.816 \pm 0.008$
$1.404 \pm 0.006\,M_\odot$0	$1.404 \pm 0.006\,M_\odot$1	$1.404 \pm 0.006\,M_\odot$2

The 2012 study emphasizes the precision of these data: masses to $1.404 \pm 0.006\,M_\odot$3 for the primary and $1.404 \pm 0.006\,M_\odot$4 for the secondary, radii to $1.404 \pm 0.006\,M_\odot$5–$1.404 \pm 0.006\,M_\odot$6, $1.404 \pm 0.006\,M_\odot$7 to $1.404 \pm 0.006\,M_\odot$8 dex, and $1.404 \pm 0.006\,M_\odot$9 to $1.2$0 dex. The 2014 study lists a reference metallicity $1.2$1 and a reference age $1.2$2 Gyr from the literature, while also deriving updated calibrated values within its own modeling framework. Surface gravities, orbital period, eccentricity, and parallax or distance are not reported or used explicitly in the 2014 calibration (Meng et al., 2014, Zhang, 2012).

2. HY Vir as a calibration benchmark for overshoot mixing

The astrophysical utility of HY Vir arises from the simultaneous availability of precise parameters for two stars that must share the same initial composition and age. That common-origin constraint sharply limits acceptable stellar models and makes the system especially sensitive to the treatment of convective-core overshoot. In both studies, the masses are fixed to their observed values and the remaining free parameters are adjusted so that both stars are reproduced at a single age (Meng et al., 2014, Zhang, 2012).

The 2012 analysis calibrates three parameters—the overshooting mixing parameter $1.2$3, $1.2$4, and $1.2$5—by imposing a single-age solution in which the primary’s radius reaches its observed value, thereby fixing the age with the highest sensitivity, while the primary and secondary temperatures and the secondary’s radius simultaneously match observations. The procedure uses Newton iterations and requires agreement in $1.2$6, $1.2$7, and $1.2$8 within $1.2$9 dex for temperatures and within the observational errors for the remaining quantities. The 2014 analysis adopts the same binary logic but solves for $1.838 \pm 0.009\,M_\odot$0, $1.838 \pm 0.009\,M_\odot$1, $1.838 \pm 0.009\,M_\odot$2, and $1.838 \pm 0.009\,M_\odot$3 using the two stars’ radii and effective temperatures under the assumption of common age and composition, with error propagation detailed in its appendix (Zhang, 2012, Meng et al., 2014).

This calibration strategy also defines what HY Vir does not constrain in these works. Rotation, mass loss, tidal effects, and orbital parameters are neglected, and the inference focuses on interior mixing physics. A plausible implication is that the system is used primarily as a controlled laboratory for core-boundary transport rather than as a general dynamical binary model (Meng et al., 2014).

3. Diffusive overshooting in the 2012 turbulent convection model study

The 2012 study formulates core overshooting through a non-local Turbulent Convection Model (TCM) solved simultaneously with the stellar structure equations. The TCM provides the turbulent kinetic energy $1.838 \pm 0.009\,M_\odot$4, heat flux $1.838 \pm 0.009\,M_\odot$5, and temperature variance $1.838 \pm 0.009\,M_\odot$6 throughout convection and overshooting zones. The local temperature gradient is modified through the turbulent heat flux according to

$1.838 \pm 0.009\,M_\odot$7

Composition evolution is treated as diffusion. In mass coordinate, the equation used in the code is

$1.838 \pm 0.009\,M_\odot$8

with the equivalent radial form

$1.838 \pm 0.009\,M_\odot$9

In convection zones, near-instantaneous mixing is enforced with

$1.404 \pm 0.006\,M_\odot$0

whereas in the overshoot region the diffusion coefficient is

$1.404 \pm 0.006\,M_\odot$1

Because $1.404 \pm 0.006\,M_\odot$2 decays roughly exponentially in the overshoot region, $1.404 \pm 0.006\,M_\odot$3 inherits an exponential decline. In the high-Péclet-number asymptotic regime, with $1.404 \pm 0.006\,M_\odot$4,

$1.404 \pm 0.006\,M_\odot$5

For HY Vir, this framework yields $1.404 \pm 0.006\,M_\odot$6, $1.404 \pm 0.006\,M_\odot$7, $1.404 \pm 0.006\,M_\odot$8, and $1.404 \pm 0.006\,M_\odot$9 Gyr. The adopted solar-calibrated TCM constants are $2.806 \pm 0.008\,R_\odot$0, $2.806 \pm 0.008\,R_\odot$1, $2.806 \pm 0.008\,R_\odot$2, $2.806 \pm 0.008\,R_\odot$3, $2.806 \pm 0.008\,R_\odot$4, $2.806 \pm 0.008\,R_\odot$5, and $2.806 \pm 0.008\,R_\odot$6. The study further reports that the same TCM constants and a solar $2.806 \pm 0.008\,R_\odot$7 reproduce HY Vir’s observed $2.806 \pm 0.008\,R_\odot$8 and $2.806 \pm 0.008\,R_\odot$9 within $1.519 \pm 0.008\,R_\odot$0, and therefore argues that the solar-calibrated TCM and diffusion parameter are applicable to this binary (Zhang, 2012).

4. The 2014 updated overshoot mixing model and its calibration on HY Vir

The 2014 study replaces classical fully mixed step overshoot with an updated overshoot mixing model grounded in fluid dynamics and implemented in the stellar evolution code YNEV. Its input physics includes OPAL opacities at high temperatures, Ferguson et al. (2005) at low temperatures, bicubic interpolation for smooth derivatives, the OPAL EOS, nuclear reaction rates from Angulo et al. (1999) with weak screening, an Eddington gray $1.519 \pm 0.008\,R_\odot$1–$1.519 \pm 0.008\,R_\odot$2 relation, and the non-local TCM of Li & Yang (2007). The TCM parameter is $1.519 \pm 0.008\,R_\odot$3, based on a solar calibration using AGSS09 composition. For comparison tracks, the paper uses MLT with $1.519 \pm 0.008\,R_\odot$4. Models evolve from the pre-main sequence starting at central temperature $1.519 \pm 0.008\,R_\odot$5 K, with a typical mesh of $1.519 \pm 0.008\,R_\odot$6 points (Meng et al., 2014).

In this updated formulation, the diffusion coefficient is different in convective and overshoot regions: $1.519 \pm 0.008\,R_\odot$7 where $1.519 \pm 0.008\,R_\odot$8 is the turbulent kinetic energy, $1.519 \pm 0.008\,R_\odot$9 is the turbulent dissipation rate, and $\lg T_{\rm eff}$0 is the Brunt–Väisälä frequency. The convective boundary is defined by the standard criterion $\lg T_{\rm eff}$1. Inside the convection zone, $\lg T_{\rm eff}$2 is very large and effectively enforces complete mixing; just outside the boundary, $\lg T_{\rm eff}$3 drops rapidly and then declines approximately exponentially through the overshoot region. Numerically, $\lg T_{\rm eff}$4 is capped at $\lg T_{\rm eff}$5 to avoid instability while still guaranteeing complete mixing in convective zones (Meng et al., 2014).

Applied to HY Vir, the updated model yields

$\lg T_{\rm eff}$6

together with $\lg T_{\rm eff}$7, $\lg T_{\rm eff}$8, and $\lg T_{\rm eff}$9 Gyr. The paper states that this $3.836 \pm 0.008$0 is fully consistent with the global recommendation $3.836 \pm 0.008$1 based on independent constraints, including solar modeling and limits on the equivalent fully mixed overshoot extent. In the broader four-binary sample of HY Vir, YZ Cas, $3.836 \pm 0.008$2 Hya, and VV Crv, no clear mass dependence of $3.836 \pm 0.008$3 is found over $3.836 \pm 0.008$4–$3.836 \pm 0.008$5 (Meng et al., 2014).

5. Internal-structure consequences: partial mixing, continuous profiles, and semi-convection

A central result of both studies is that overshoot mixing in HY Vir is partial rather than instantaneous. In the 2012 TCM-based treatment, incomplete mixing produces a continuous hydrogen abundance profile across the convective-core boundary. The e-folding location of the hydrogen profile is defined by

$3.836 \pm 0.008$6

with $3.836 \pm 0.008$7 and $3.836 \pm 0.008$8. On the main sequence, the e-folding length increases with stellar age. The study relates this to a diffusion timescale estimate,

$3.836 \pm 0.008$9

and, using the asymptotic TCM form,

$3.816 \pm 0.008$0

The paper notes that the scaling is valid in a quasi-static main-sequence background but not during rapid post-main-sequence core contraction (Zhang, 2012).

The same study reports that HY Vir’s core overshoot lies in the high-Péclet-number regime. In that regime, $3.816 \pm 0.008$1, $3.816 \pm 0.008$2, and $3.816 \pm 0.008$3 decrease exponentially beyond the boundary, while the anisotropy

$3.816 \pm 0.008$4

remains nearly constant at $3.816 \pm 0.008$5. Measured values include $3.816 \pm 0.008$6 to $3.816 \pm 0.008$7, $3.816 \pm 0.008$8 to $3.816 \pm 0.008$9, $1.404 \pm 0.006\,M_\odot$00 to $1.404 \pm 0.006\,M_\odot$01, and $1.404 \pm 0.006\,M_\odot$02–$1.404 \pm 0.006\,M_\odot$03 for primary models and $1.404 \pm 0.006\,M_\odot$04 for the secondary. For HY Vir’s core boundary, $1.404 \pm 0.006\,M_\odot$05, and with $1.404 \pm 0.006\,M_\odot$06 the model predicts a very thin adiabatic overshoot layer of $1.404 \pm 0.006\,M_\odot$07; beyond it, $1.404 \pm 0.006\,M_\odot$08 is close to $1.404 \pm 0.006\,M_\odot$09 (Zhang, 2012).

The 2014 updated model reaches a parallel qualitative conclusion through a different diffusion prescription. Inside the convective core, computed values reach $1.404 \pm 0.006\,M_\odot$10, though the code caps them at $1.404 \pm 0.006\,M_\odot$11. In the overshoot region, typical geometric mean diffusion coefficients are of order $1.404 \pm 0.006\,M_\odot$12 at $1.404 \pm 0.006\,M_\odot$13 beyond the boundary and $1.404 \pm 0.006\,M_\odot$14 at $1.404 \pm 0.006\,M_\odot$15, implying mixing time scales of $1.404 \pm 0.006\,M_\odot$16 s and $1.404 \pm 0.006\,M_\odot$17 s, respectively. This produces a smooth hydrogen abundance profile with near-zero gradient at the boundary, transitioning smoothly to small gradients further out. The same continuous partial mixing also removes the semi-convection that appears in standard local convection models of low-mass stars with convective cores, because composition gradients and radiative temperature gradients remain continuous and the region evolves naturally to convective neutrality without an artificial discontinuity (Meng et al., 2014).

6. Evolutionary interpretation, comparison with classical overshoot, and limitations

In evolutionary terms, both studies conclude that overshoot mixing refreshes hydrogen in the burning core, effectively enlarges the mixed core, and extends the main-sequence lifetime. The 2012 paper states that larger $1.404 \pm 0.006\,M_\odot$18 produces hotter and more luminous terminal main-sequence tracks, and that no-overshoot models are cooler and less luminous at terminal main-sequence conditions. Both the calibrated $1.404 \pm 0.006\,M_\odot$19 model and the solar $1.404 \pm 0.006\,M_\odot$20 model match the observed $1.404 \pm 0.006\,M_\odot$21 and $1.404 \pm 0.006\,M_\odot$22 of both components within $1.404 \pm 0.006\,M_\odot$23 (Zhang, 2012).

The 2014 paper compares its updated diffusion-based overshoot model directly to classical step overshoot. For HY Vir, the evolutionary tracks computed with $1.404 \pm 0.006\,M_\odot$24 are nearly identical to dashed tracks computed with a classical fully mixed overshoot extent of $1.404 \pm 0.006\,M_\odot$25, whereas no-overshoot tracks deviate significantly. The paper therefore concludes that, for HY Vir’s mass range, the updated model reproduces the effective refueling and evolutionary impact of a classical $1.404 \pm 0.006\,M_\odot$26, but does so with physically motivated partial mixing and smooth gradients rather than an instantaneous fully mixed extension. In this context, the classical overshoot model is described there as physically unreasonable and inconsistent with helioseismic investigation (Meng et al., 2014).

The calibrated age and composition are mutually consistent across the two studies but not numerically identical. The 2012 model gives $1.404 \pm 0.006\,M_\odot$27 Gyr, $1.404 \pm 0.006\,M_\odot$28, and $1.404 \pm 0.006\,M_\odot$29, implying $1.404 \pm 0.006\,M_\odot$30. The 2014 model gives $1.404 \pm 0.006\,M_\odot$31 Gyr, $1.404 \pm 0.006\,M_\odot$32, and $1.404 \pm 0.006\,M_\odot$33, implying $1.404 \pm 0.006\,M_\odot$34. Both are consistent with the literature age $1.404 \pm 0.006\,M_\odot$35 Gyr cited from Lacy & Fekel (2011). The 2014 study further notes that, together with YZ Cas, the results suggest a helium enrichment law $1.404 \pm 0.006\,M_\odot$36 (Meng et al., 2014, Zhang, 2012).

The principal limitations are also explicit. The 2014 calibration notes that the observables, especially $1.404 \pm 0.006\,M_\odot$37, must be known to better than $1.404 \pm 0.006\,M_\odot$38 to tightly constrain overshoot, and it omits rotation, mass loss, tidal effects, and orbital parameters. The 2012 error analysis finds that $1.404 \pm 0.006\,M_\odot$39 is the dominant contributor to parameter uncertainties, accounting for $1.404 \pm 0.006\,M_\odot$40 of $1.404 \pm 0.006\,M_\odot$41, $1.404 \pm 0.006\,M_\odot$42 of $1.404 \pm 0.006\,M_\odot$43, and $1.404 \pm 0.006\,M_\odot$44 of $1.404 \pm 0.006\,M_\odot$45. It also sets the diffusions of $1.404 \pm 0.006\,M_\odot$46 and $1.404 \pm 0.006\,M_\odot$47 to zero for numerical convergence, which produces the thin adiabatic overshoot layer. These caveats do not alter the central role of HY Vir in the literature: it is a precisely measured detached eclipsing binary whose shared-age, shared-composition structure provides a stringent observational calibration of convective-core overshoot and a clear demonstration of the difference between fully mixed step prescriptions and diffusion-based partial mixing (Meng et al., 2014, Zhang, 2012).