A New Approach to Convective Core Overshooting: Probabilistic Constraints from Color–Magnitude Diagrams of LMC Clusters

The treatment of convection in the stellar interior affects the effective temperature, luminosity, and age of the MSTO in low-mass stars and the hot extension of the blue loop in intermediate-mass He-burning stars. Core overshooting affects different parts of a CMD differently, depending on the age of the stellar population. Constraining core overshooting is important in astrophysics beyond the goal of precision stellar evolution models because uncertainties in core overshooting strength can be of the order of 5% of the MS lifetime (${\tau }_{\mathrm{MS}}$) for low-mass stars. At MC metallicities, 5% of an MS lifetime is a significant portion of subsequent evolutionary phases like the He-burning phase ($\sim 20 \% \ {\tau }_{\mathrm{MS}}$) and thermally pulsating AGB phase (TP-AGB; $\lesssim 1 \% \ {\tau }_{\mathrm{MS}}$). It is therefore a critical goal for those who study or use HB and TP-AGB models to push uncertainties in MS lifetimes smaller than the duration of the short-lived evolutionary phases after the MSTO.

Convection in stars is a complex, three-dimensional time dependent process, and while efforts are underway to apply 3D and 2D models of convection to 1D stellar models (e.g., Arnett et al. 2015), these techniques are still too computationally expensive to be applied across all stellar ages and masses needed to synthesize stellar populations. Instead, convective energy transport in 1D stellar models typically follows the mixing length formalism (MLT, Böhm-Vitense 1958), which defines the mixing length parameter ${\alpha }_{\mathrm{MLT}}$, as the mean distance a convective element travels before being reabsorbed into its surrounding medium. The Sun is the main target for calibrating ${\alpha }_{\mathrm{MLT}}$ (e.g., Basu et al. 2009).

The formalism to describe convective overshooting differs between stellar modeling groups. The main two varieties continue the formalism of MLT and parameterize the strength of convective overshooting in units of pressure scale height (${H}_{p}$). The PARSEC models (Bressan et al. 2012, 2013) define the parameter ${{\rm{\Lambda }}}_{{\rm{c}}}$ across the Schwarzschild boundary (Bressan et al. 1981), while other groups adopt the parameter ${\alpha }_{\mathrm{ov}}$ measured from above the Schwarzschild boundary. Whichever the preference, they can be compared following the relation, ${\alpha }_{\mathrm{ov}}\sim \tfrac{{{\rm{\Lambda }}}_{{\rm{c}}}}{2}{H}_{p}$.

Observational constraints on core overshooting for masses $M\lesssim 3{M}_{\odot }$ historically come from by-eye fitting of isochrones to MSTO morphology of open clusters, after determining or adopting values for distance, reddening, and membership. The range of overshooting parameters fill in the range $0\lesssim {{\rm{\Lambda }}}_{{\rm{c}}}\lesssim 0.5$, but are most commonly found to be 0.4 (e.g., Demarque et al. 1994; Kozhurina-Platais et al. 1997; Sarajedini et al. 1999; Woo & Demarque 2001). Bressan et al. (1993) suggested that ${{\rm{\Lambda }}}_{{\rm{c}}}$ was not one value for all masses, and used ${{\rm{\Lambda }}}_{{\rm{c}}}=0.25$ for $1-1.5{M}_{\odot }$ and ${{\rm{\Lambda }}}_{{\rm{c}}}=0.5$ for masses, ${\rm{M}}\geqslant 1.5{M}_{\odot }$. A more gradual increase of the overshooting efficiency with mass was introduced by Demarque et al. (2004).

In the early 2000's, "by-eye" isochrone fitting was gradually supplemented with more robust analyses based on the comparison with synthetic CMDs and luminosity functions (e.g., Bertelli et al. 2003; Woo et al. 2003). This change in methods was enabled by the better photometric quality in the MCs with large telescopes and later with HST. These improvements were particularly important for studying MC clusters, which are generally more populated than their Galactic counterparts, and are often projected over sparely populated Galactic fields, hence reducing uncertainties related to low stellar counts and unknown membership probabilities.

Woo et al. (2003), using Yale isochrones (Yi et al. 2001), found the overshooting strength of ${{\rm{\Lambda }}}_{{\rm{c}}}\sim 0.4\ {H}_{p}$ to best-fit the CMDs of the intermediate-age LMC clusters NGC 2173, SL 556, and NGC 2155 (cluster ages $\sim 1.5\mbox{--}3\,\mathrm{Gyr};$ which would correspond to MSTO stars $\sim 1.4\mbox{--}1.8$ ${M}_{\odot }$ according to PARSEC models). Similar results were also obtained for NGC 2173 (Mucciarelli et al. 2007). Girardi et al. (2009) was able to simultaneously fit the dual red clump (RC) and MSTO in the center of the SMC cluster NGC 419 by adopting ${{\rm{\Lambda }}}_{{\rm{c}}}={0.47}_{-0.04}^{+0.14}$ and $\mathrm{logAge}={1.35}_{-0.04}^{+0.11}$ Gyr (corresponding to an MSTO mass ∼1.8 ${M}_{\odot }$) and assuming uncertainties dominated by random errors. At higher masses, the young LMC cluster NGC 1866 has produced independent evidence of moderate core overshooting (Barmina et al. 2002), though the findings have been challenged (see, e.g., Brocato et al. 2003).

Eclipsing binaries have also been used to measure core overshooting, through core overshooting's effect on stellar radius (e.g., Schroder et al. 1997; Ribas et al. 2000). The range in overshooting strength was found to be $0.48\lesssim {{\rm{\Lambda }}}_{{\rm{c}}}\lesssim 0.64$ and increasing with increasing mass between 2.5 and 6.5 ${M}_{\odot }$. Claret & Torres (2016) found contradictory results from reanalyzing a well measured set of 33 double-lined eclipsing binaries in the Milky Way, LMC, and SMC. They found that ${{\rm{\Lambda }}}_{{\rm{c}}}$ is independent of metallicity but depends on mass, such that ${{\rm{\Lambda }}}_{{\rm{c}}}$ rises approximately linearly from 0 to 0.4 over the interval $1.2{M}_{\odot }\leqslant M\leqslant 2.0{M}_{\odot }$ and remains roughly constant for higher masses with a dispersion of ∼0.06 (their sample reaches 4.4 ${M}_{\odot }$).

With the burst of asteroseismology observations, new avenues for observational constraints have found some of the most extreme non-zero values of core overshooting, (Deheuvels et al. 2010, ${{\rm{\Lambda }}}_{{\rm{c}}}=0.34\pm 0.06{H}_{p}$ for a solar metallicity star with mass $M=0.95{M}_{\odot }$) and the highest (Guenther et al. 2014, ${{\rm{\Lambda }}}_{{\rm{c}}}=2\mbox{--}2.5$ for Procyon, $M\sim 1.5{M}_{\odot }$).

In summary, studies that are focused on individual stars are converging on convective core overshooting increasing with increasing mass at least up to masses of 6.5 ${M}_{\odot }$, while constraints from stellar populations find ${{\rm{\Lambda }}}_{{\rm{c}}}$ values between 0.4 and 0.5 ${H}_{p}$. Reported uncertainties or dispersions of core overshooting strength are typically around the 15%–20% level or 0.06 ${H}_{p}$.

ACS and WFC3 archival data were re-reduced using the University of Washington data reduction pipeline, which was developed to reduce large HST programs, e.g., ANGST and PHAT (Dalcanton et al. 2009, 2012). Its current capabilities are described in detail in Williams et al. (2014). Briefly, flt and flc images were downloaded from the Mikulski Archive for Space Telescopes (MAST; flt for WFC3/IR; flc for ACS and WFC3/UVIS), astrometrically aligned using the cross-camera alignment software developed for PHAT as part of astrometry.net, and cleaned of cosmic rays using the astrodrizzle package (Gonzaga et al. 2012). We then used DOLPHOT (Dolphin 2000) for PSF photometry and created three photometric catalogs, phot, st, and gst, and we use the gst catalogs for our analysis. These catalogs provide three different levels of measurement quality. The phot catalogs are the closest to the full photometric output table from DOLPHOT. The st catalogs are culled from the phot catalogs and are limited to S/N ≥ 4 and sharpness² values in at least one filter to be below 0.2, 0.15 for ACS and UVIS respectively. The gst catalogs are a subset of the st catalogs that have crowding values below 1.3. CMDs of the gst catalogs are shown in black in Figure 3 and the archival data are summarized in Table 2. Full details of our data reduction pipeline are postponed to our instrument paper (M. Fouesneau et al. 2017, in preparation).

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In order to characterize the photometric errors and completeness of the HST data, we placed ∼100k artificial stars for each cluster. Artificial stars are distributed rather uniformly in CMD space covering the full magnitude and color range of the data, and weighted such that fainter mags have relatively larger numbers of tests. Artificial stars are distributed spatially according to a King profile, literature values for center and half-light radius, fixed concentration, covering a range in radius out to four half-light radii, and bounded by the ACS or UVIS field of view.

We fed our gst catalogs into ASteCA (Perren et al. 2015), an automated stellar cluster analysis package, to estimate the cluster parameters and cluster membership. Table 3 lists the derived cluster centers, radii, and great circle distances between derived cluster centers and values from Bica et al. (2008).

Table 3.  ASteCA-Derived Cluster Parameters

Cluster	α Center	δ Center	${r}_{\mathrm{cluster}}$	${{\rm{\Delta }}}_{\mathrm{center}}$ a
	(J2000)	(J2000)	(arcsec)	(arcsec)
HODGE 2	5 17 48.816 ± 0.048	−69 38 41.640 ± 0.720	33.84 ± 1.80	1.24
NGC 1718	4 52 25.704 ± 0.072	−67 03 05.040 ± 1.080	38.88 ± 2.52	4.06
NGC 2203	6 04 43.392 ± 0.072	−75 26 19.320 ± 1.080	42.12 ± 2.16	6.17
NGC 2213	6 10 42.240 ± 0.072	−71 31 45.840 ± 1.080	29.52 ± 2.16	2.13
NGC 1644	4 37 39.792 ± 0.096	−66 11 55.680 ± 1.440	26.28 ± 2.52	5.25
NGC 1795	4 59 47.280 ± 0.096	−69 48 03.960 ± 1.440	57.60 ± 2.88	6.58

Note.

^aSeparation between Bica et al. (2008) center coordinates and ASteCA-derived cluster center coordinates.

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Full details of the fitting algorithms are in the main ASteCA paper, in short, the cluster centers are determined by the maximum spatial density using a two-dimensional Gaussian kernel density estimator. The cluster radius is set to where the radial density profile becomes indistinguishable from the background stellar density. Contamination from non-cluster stars within the cluster radius are discussed in Section 4.2. Stars within the radius of the cluster and have at least 70% membership probability are used as input photometry and shown in red in Figure 3.

The Padova–Trieste Stellar Evolution Code (PARSEC Bressan et al. 2012, 2013) is a major update to the Padua models (Girardi et al. 2000). PARSEC adopts the solar metallicity value of Z_⊙ = 0.01524 and the scaled solar distribution of elements heavier than ⁴He are taken from Grevesse & Sauval (1998) except for Li, C, N, O, P, S, K, Fe, Eu, Hf, Os, and Th, which are taken from Caffau et al. (2011) and references therein. The initial He abundance (Y_i) for each metallicity set is calculated based on the primordial He abundance, Y_p = 0.2485, (Komatsu et al. 2011) and the helium-to-metals enrichment ratio, ${\rm{\Delta }}Y/{\rm{\Delta }}Z$ = 1.78, which was obtained in Bressan et al. (2012) using solar values. That is, ${Y}_{i}={Y}_{p}+({\rm{\Delta }}Y/{\rm{\Delta }}Z)\ {Z}_{i}$ (see Section 2 and Table 1 of Bressan et al. 2012).

PARSEC adopts an overshooting prescription that linearly increases in strength from no overshooting to the a maximum value (${{\rm{\Lambda }}}_{\max })$ between two mass steps (${M}_{O1},{M}_{O2}$). The Padua models (Girardi et al. 2000; Bertelli et al. 2008) set ${{\rm{\Lambda }}}_{\max }=0.5$, M_O1 = 1.0 ${M}_{\odot }$, and ${M}_{O2}=1.5{M}_{\odot }$ at all metallicities. In PARSEC V1.2S, ${{\rm{\Lambda }}}_{\max }$ is the same; however, the mass steps are derived separately for each metallicity and helium content.

4.1.1. The PARSEC Core Overshooting Model Grid

We relax the PARSEC setting of ${{\rm{\Lambda }}}_{\max }=0.5$ and calculate a grid of 3,560 stellar evolution tracks using PARSEC V1.2S (updates from Chen et al. 2014; Tang et al. 2014) beginning at the pre-main sequence and ending at the termination of He-burning. Core overshooting in our model grid is not only calculated for hydrogen-burning (MS) cores, but helium-burning cores (HB or HeB) are also calculated with the labeled core overshooting strength. Table 4 lists the details of the stellar model grid.

Table 4.  PARSEC Convective Core Overshooting Model Grid

Parameter	Values
Mixture (${Z}_{i},{Y}_{i}$)	0.0005, 0.249
	0.0010, 0.250
	0.0020, 0.252
	0.0040, 0.256
	0.0060, 0.259
	0.0080, 0.263
	0.0100, 0.267
${{\rm{\Lambda }}}_{c}$ (${H}_{p}$)	0.3–0.6: ${\rm{\Delta }}{{\rm{\Lambda }}}_{{\rm{c}}}=0.1$
Mass (${M}_{\odot }$)	$0.1\leqslant M\leqslant 2.4:{\rm{\Delta }}M\leqslant 0.05\ {M}_{\odot }$
	$2.6\leqslant M\leqslant 6.4:{\rm{\Delta }}M=0.20\ {M}_{\odot }$
	$7.0\leqslant M\lt 12.0:{\rm{\Delta }}M=1.0\ {M}_{\odot }$
	$12.0\lt M\leqslant 20.0:{\rm{\Delta }}M=2.0\ {M}_{\odot }$

Note. We interpolated the overshooting grid to obtain ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.45 and 0.55 models and extended the grid to ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.80 for select clusters.

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4.1.2. The Effect of Core Overshooting

Increasing core overshooting strength allows more fresh nuclear material to fuse, making for a larger and hotter core that leads to a longer main sequence or HB lifetime and a brighter and cooler MSTO. We now describe how such a physical change in the interior of a star is expected to affect observations of a nearly single age stellar population. First, we illustrate how the core fusion lifetimes change, followed by the Hertzsprung–Russell diagram (HRD) and CMD appearance of sample stellar evolution tracks chosen at masses relevant to the ages of the sample clusters, and finally three simple synthetic stellar populations as snapshots of different epochs in star formation. As we will show, longer core fusing lifetimes translate to higher stellar densities on the CMD than otherwise expected. A brighter and cooler MSTO changes the expected position of a star on a CMD, leading to possible misidentification of a star's mass, age, or distance.

Figure 4 shows as a function of mass, the difference in core fusing lifetimes of core overshooting strength compared to the canonical PARSEC value (${{\rm{\Lambda }}}_{{\rm{c}}}=0.5\ {H}_{p}$). Solid and dashed lines mark the extrema of the stellar evolution model grid metallicities and are plotted for each calculated overshooting value (a similar comparison to other stellar modeling groups is discussed in the Appendix). The effect of core overshooting on MSTO age quickly increases from low masses to a peak, which is set by the linear ramp-up of core overshooting from the lowest masses in PARSEC. Soon after ${{\rm{\Lambda }}}_{\max }$ is reached, the effect of the convective core on MSTO and He-burning age decreases, as expected, with increasing mass as core convection becomes less important. As an example, for a 1.5 ${M}_{\odot }$ star, an increase in core overshooting of ${\rm{\Delta }}{{\rm{\Lambda }}}_{{\rm{c}}}=0.1\ {H}_{p}$ leads to a $\sim 100\mbox{--}150\,\mathrm{Myr}$ ($\sim 5 \%$) longer MS lifetime depending on metallicity. A $\sim 100\mbox{--}\,150\,\mathrm{Myr}$ increase in MS lifetime is a small effect for the MS; however, for a 1.5 ${M}_{\odot }$ star, it is longer than the entire core He-burning lifetime, and longer than the thermally pulsating AGB lifetime (e.g., Rosenfield et al. 2016).

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Figure 5 shows an HRD (left panel) and a CMD (right panel) of evolutionary tracks from the core overshooting grid selected at initial masses of 1.5 ${M}_{\odot }$ and 2 ${M}_{\odot }$. The transformation from the HRD to magnitudes and colors are based on the tables of bolometric corrections from Girardi et al. (2008; revised for the latest ACS/WFC3 filter transmission curves) and rely on the ATLAS9 atmospheric models from Castelli & Kurucz (2004). The same transformations are also implemented in the MATCH routines (see Section 4).

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The MSTO is shifted to brighter and to cooler effective temperatures with increasing core overshooting. However, the amplitude of the brighter and cooler excursions decreases with increasing mass. There are also clear morphological differences around the MSTO, subgiant branch (SGB), and helium-burning phases. The extension between the minimum effective temperature on the MS and the MSTO increases with increasing core overshooting, the luminosity dip after the MSTO is more pronounced with increasing core overshooting, and the extent to hotter temperatures of the helium-burning tracks decreases with increasing core overshooting.

The age differences and the morphological changes in the stellar evolution tracks due to core overshooting strength culminate in the shape and number density of a stellar population on a CMD. Figure 6 shows synthetic stellar populations from models of each core overshooting strength produced using the fake routine in the MATCH package (see Section 4.2). The synthetic stellar populations are made of one burst of constant star formation lasting 60 Myr starting at 14 Myr (center left panel), 180 Myr (center right panel), and 1.4 Gyr (right panel), a constant initial metallicity of Z = 0.006 (corresponding to $[M/{\rm{H}}]=-0.40$ dex), a Salpeter (1955) IMF, a distance ${\mu }_{0}=18.50$, typical photometric uncertainties, and neglecting binaries and extinction.

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Figure 6 shows that observational signatures of core overshooting differ at different epochs of star formation. In the younger populations (center two panels) the core helium-burning stars show the most significant differences. Increasing overshooting strength decreases the extent of the blue loop, which manifests in the youngest CMD as brighter red and blue helium-burning stars (RHeB, BHeB; these stars have masses $\gt 12{M}_{\odot }$ and $\gt 3{M}_{\odot }$ in the center left and center right panels, respectively) with increasing overshooting. The numbers of RHeB and BHeB also decreases with increasing overshooting strength, which is more significant at the youngest aged population than the intermediate-aged population.

The oldest aged synthetic stellar population (right panel) exhibits different MSTO, SGB, and RC morphologies. Decreasing core overshooting strength increases the MSTO color, increases the number of SGB stars, lowers the total number of RC stars, and leads to a brighter, more compact RC.

CMDs are powerful tools for understanding the history of star formation in stellar populations. A CMD can be well approximated by a linear combination of bursts of star formation over cosmic time (Dolphin 2002). Exploiting the tenet, the MATCH software package (Dolphin 2016, and refs. therein), specifically, the calcsfh module was designed to derive the most likely SFH from a binned CMD (Hess diagram) of the photometry of a mixed-age stellar population.

To compare an observed Hess diagram to a model Hess diagram, MATCH first constructs the model Hess diagram given an input set of stellar models and user-specified prior on the IMF slope, binary fraction, metallicity, metallicity dispersion, and color and magnitude bin sizes. calcsfh will then iterate over distance, extinction, and epoch of SF burst, until either the most probable linear combination of ages is found (for mixed-age stellar populations), or until the likelihood is calculated for each epoch of SF (for near-single age stellar populations). We iterate calls to calcsfh to such that the single value priors become a distribution. We now describe how we set these prior distributions.

4.2.1. Prior Distributions

Table 5 lists our model priors and CMD fitting grid resolution. We chose our priors to be uninformative and flat distributions over a range set beyond derived literature values but limited for computational efficiency. We constrained values (IMF slope, binary fraction) that only effect the lower MS, where the photometric uncertainties are highest.

Table 5.  Priors and the calcsfh Grid Search Space

Parameter	Range	Step Size
IMF (Γ)	1.35	(Salpeter 1955 fixed)
Binary fraction	0.30	(See Milone et al. 2009, 2016, fixed)
Distance (${\mu }_{0};$ mag)	18.3–18.7a	0.05
Extinction (AV; mag)	0.0–0.5a	0.05
Age (Gyr)	1.0–2.5	0.06
Metallicity ([Fe/H]; dex)	−0.85 to 0.15	0.10
Core overshooting strength (${{\rm{\Lambda }}}_{{\rm{c}}};$ ${H}_{p}$)	0.3–0.6b	0.1; 0.05 between ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.4–0.6
Color (mag)	∼0.0–2.0 (varies by cluster)	0.05
Magnitude (mag)	∼16–24 (varies by cluster)	0.10

Notes.

^aNGC 1718 distance modulus prior was 18.5–18.9, and its extinction prior was 0.0–1 with the same listed step sizes. ^bWe extended the core overshooting grid to ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.80 for NGC 1718 and NGC 2203.

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Binary Fraction—Observations of pre-MS stellar systems (e.g., Kroupa & Petr-Gotzens 2011), N-body simulations (e.g., Marks et al. 2011), and theoretical arguments (e.g., Goodwin & Kroupa 2005) suggest most stars are likely formed in binaries. The cluster radius, stellar type, cluster density, and age, among other factors contribute to the binary fraction. Sollima et al. (2007) found that the binary fraction varies from 0.1 to 0.5 for low-density Galactic globular clusters. Galactic field populations have measured binary fractions from $\sim 0.2\,\mathrm{to}\,0.8$ (e.g., Marks & Kroupa 2011, and references therein), with the fraction decreasing with decreasing stellar mass. In the MCs, Milone et al. (2009, 2016) determined the binary fraction ranges from $\sim 0.19\,\mathrm{to}\,0.46$ for several MC clusters. We set the binary fraction to the approximate median found in the Milone et al. (2009) papers, 0.3, with a uniform mass ratio distribution from $0.1\,\mathrm{to}\,1.0{M}_{\odot }$. We will explore variations of binary fractions in subsequent work that includes LMC and SMC clusters of differing age.

IMF Slope—We do not attempt to constrain the low-mass MS stars in this study, and adopt the Salpeter (1955) IMF slope of ${\rm{\Gamma }}=1.35$. The lowest mass stars to be included in our analysis have M = 0.8 ${M}_{\odot }$.

Distance—We adopt a true distance modulus range of ${\mu }_{0}=18.30\mbox{--}18.70$ mag and step size 0.05 mag, which encompasses common literature values of ${\mu }_{0}=18.36\mbox{--}18.54$ mag with the exception of NGC 1718, which has a derived literature distance of 18.73 ± 0.07 (Kerber et al. 2007). Therefore, we extended the distance modulus range to 18.9 mag for NGC 1718 to ensure the best-fitting distance was not at the edge of the grid.

Extinction—Following the method to set our distance priors, values of A_V from the literature range from 0.03 to 0.58 mag. With A_V step size of 0.05 mag, we set our prior limits from 0 to 0.6 mag, again extending the grid edge for NGC 1718 to 1.0 mag.

Age—Clusters were selected because their literature ages were around 1.5 Gyr, we limited the age prior to 1.5 ± 0.75 Gyr for computational efficiency.

Metallicity—Most isochrone fitting of the clusters in our sample set the metallicity to either Z = 0.008 ([Fe/H] =−0.28 dex) or Z = 0.006 ([Fe/H] = −0.4 dex). We set our prior limits to Z = 0.002–0.01 ([Fe/H] = −0.85−0.15 dex) with a step size of 0.1 dex.

CMD Range and Binning—Using the simulated stellar populations described in Section 4.3, we ran calcsfh setting the color and magnitude bin sizes at all combinations of values 0.01, 0.05, 0.1, and 0.15 mag. We confirmed the heuristic tenet from Dolphin (2002): CMD bin sizes should be set smaller than the important observed CMD features, in our case, the MSTO and the HB. We adopt the color bin size of 0.05 mag and magnitude bin size of 0.10 mag.

Cluster Contamination—In the ASteCA package, the user may calculate the star-by-star probability of cluster membership by invoking a non-parametric Bayesian decontamination algorithm based on the method of Cabrera-Cano & Alfaro (1990), which was originally applied to open clusters. We limit the input photometry to stars within the cluster radius with at least 70% membership probability (shown in red in Figure 3).

Age and Metallicity Resolution—For stellar clusters, it is useful to measure the goodness of data-model fit of a simple stellar population (SSP) as a function of age. The minimum possible SSP age resolution in MATCH is set by an internal pre-compiled grid of partial CMDs, for our core overshooting grid this resolution is d[Fe/H] = 0.05 dex and dlog Age = 0.01 (log year). This high-resolution grid allows us to test SF in age bins $\gtrsim 20\,\mathrm{Myr}$ at ages of 1 Gyr (2%). However, we found there was no added improvement in the fitting between resolutions of 2% and 6%, so we adopted an SSP age resolution of 60 Myr because it provided an optimal balance between computational time and sensitivity to cluster age spreads.

We have seen that varying core overshooting strength propagates to CMD in ways that depend on age, which is the manifestation of the importance of convection in the stellar core (see Section 4.1.2). On an optical CMD, a stellar population that is brighter and cooler could mean that it is in fact closer and has less extinction than assumed. For example, it is tempting to point out that the F814W mag of the MSTO of the youngest synthetic population in Figure 6 shows ∼0.3 mag spread depending on the strength of core overshooting. These would correspond to bright stars with negligible photometric uncertainty. However, in these optical filter sets and at that population age, a large 0.3 spread in core overshooting strength at the MSTO is nearly indistinguishable from a ∼0.2 mag uncertainty in the distance modulus. This underscores the importance of using the entire CMD to test models of uncertain evolutionary phases. It also behooves us to test the sensitivity of CMD fitting and our core overshooting grid.

We ran the stellar population synthesis module fake within MATCH to simulate simple stellar populations. The fake module takes as input the same user-specified parameters as listed in Section 4.2, including artificial star tests to convolve with the model Hess diagrams and returns a synthetic CMD. We ran fake to simulate a constant burst of SF at 1.5 Gyr $\pm 30\,\mathrm{Myr}$, [Fe/H] = −0.40 dex, a metallicity dispersion of 0.10 dex, at each grid value of core overshooting strength, and convolved the CMD with a typical cluster artificial star test uncertainty profile.

Using the MATCH fake photometric catalogs as input to calcsfh, we derived the best-fitting CMD by searching over interstellar extinction, age, metallicity, and core overshooting strength (see Table 5). In all cases, that is, for each of the four mock data input catalogs, calcsfh clearly recovered the input parameters.

4.3.1. Likelihood

The best-fitting model is found by minimizing the Poisson-equivalent of ${\chi }^{2}$ (see Dolphin 2002):

$$\begin{eqnarray}&&{\chi }_{P}^{2}=2\sum {m}_{i}-{n}_{i}+{n}_{i}\mathrm{ln}\left(\displaystyle \frac{{n}_{i}}{{m}_{i}}\right),\end{eqnarray} \tag{ 1 }$$

where m_i is the number of model points and n_i is the number of data points in the Hess diagram bin i.

To visualize the likelihoods, we produce marginalized PDFs of each parameter, and joint-marginalized PDFs for each parameter pair. Joint-marginalized PDFs are comparable ${\chi }^{2}$ maps for Gaussian distributions; in our case, they are ${\chi }_{P}^{2}$ maps.

The posterior distributions provide the full story of the uncertainties and correlations between parameters, given the PARSEC models and our priors. However, it is useful to note the most probable value and estimate the uncertainties for each fitted parameter. To do so, we report the "best fit" as the maximum posterior probability and take the 16th and 84th quartiles of a polynomial fit to posterior distribution as uncertainties. For a Gaussian distribution, these values would correspond to the mean and $1\sigma$.

To explore the effect of core overshooting strength on the derived cluster parameters, we marginalized the mock data results over the true value of ${{\rm{\Lambda }}}_{{\rm{c}}}$ (i.e., ${{\rm{\Lambda }}}_{{\rm{c}}}$ of the input synthetic stellar population) and the assumed value of ${{\rm{\Lambda }}}_{{\rm{c}}}$ (i.e., ${{\rm{\Lambda }}}_{{\rm{c}}}$ used to derive the cluster parameters).

Figure 7 shows the marginalized PDFs derived with an assumed value of ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.50 and all calculated true values. Red vertical lines show the (true) input values used to create the synthetic stellar populations, including the ${{\rm{\Lambda }}}_{{\rm{c}}}$ values noted on the right vertical axes.

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The systematic offsets introduced as a function of increasing true ${{\rm{\Lambda }}}_{{\rm{c}}}$ follow from the discussion in Section 4.1.2 and Figures 4–6. For example, we have seen that increasing core overshooting strength increases core fusion lifetimes, therefore underestimating core overshooting strength will bias derived cluster ages older. This effect can be seen comparing the age panels in the top row and the third row of Figure 7 (the third row being where the assumed ${{\rm{\Lambda }}}_{{\rm{c}}}$ matches the true ${{\rm{\Lambda }}}_{{\rm{c}}}$). We can further estimate the systematic offsets as a function of core overshooting strength expected for a population aged ∼1.5 Gyr. The median offsets of the maximum posterior probabilities are ${A}_{V}\sim -0.04$ mag, ${\mu }_{0}\sim +0.002$ mag, Age $\sim -120\,\mathrm{Myr}$, and log $Z\sim +0.1$ dex, when increasing ${{\rm{\Lambda }}}_{{\rm{c}}}$ by $0.1{H}_{p}$. In other words, for intermediate-aged stellar clusters, distance, and extinction seem to be immune from uncertainties of core overshooting strength; however, age and metallicity may have disconcertingly large systematic offsets when ${{\rm{\Lambda }}}_{{\rm{c}}}$ is uncertain by more than $\pm 0.05{H}_{p}$.

Figure 8 shows the joint-marginalized PDFs when the assumed value matches the true value (the rest are in the figure set), and Figure 9 shows the marginalized PDFs. Effectively, this is a visualization of how well parameters can be recovered given a typical artificial star test uncertainty profile.

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View figure set (4 panels)

Tarball of figure set images

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Using our core overshooting grid as the back-end stellar evolution models to MATCH, the ${\mathtt{ASteCA}}$-derived cluster members as input photometry, and the artificial star tests to account for photometric uncertainty and completeness, we evaluate Equation (1) using MATCH or iterating calls to MATCH such that all combinations of parameters listed in Table 5 are searched.

Before exploring the effects of uncertain core overshooting strength, it is useful to understand the cluster parameter uncertainties and their correlations while assuming the canonical PARSEC value of ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.50 ${H}_{p}$. In effect, this is a robust means to derive the cluster parameters if we were certain the most likely value of ${{\rm{\Lambda }}}_{{\rm{c}}}$ was indeed 0.50 ${H}_{p}$.

Figure 10 shows the joint marginalized PDFs for each cluster assuming ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.50 ${H}_{p}$ and the maximum posterior probabilities with the 16th and 84th quartiles of the distributions are listed in Table 6. The most likely parameters agree reasonably well with previous work (see Table 1) given the different stellar models, fixed parameters, and fitting methods. The best agreement between our derivation and that in the literature is Goudfrooij et al. (2014). Many of the derived values agree to within Goudfrooij et al. reported uncertainties and most derived values agree to within our more conservative $\sim 1\sigma$ constraints from the PDFs.

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Table 6.  Most Likely Cluster Parameters Given the PARSEC Model and Canonical Value of ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.50 ${H}_{p}$

Cluster	AV	${\mu }_{0}$	Age (Gyr)	Z
HODGE 2	${0.14}_{-0.16}^{+0.08}$	${18.40}_{-0.15}^{+0.06}$	${1.305}_{-0.179}^{+0.167}$	${0.007}_{-0.002}^{+0.001}$
NGC 1718	${0.55}_{-0.13}^{+0.15}$	${18.65}_{-0.08}^{+0.13}$	${1.833}_{-0.372}^{+0.274}$	${0.006}_{-0.003}^{+0.002}$
NGC 2203	${0.19}_{-0.19}^{+0.11}$	${18.50}_{-0.10}^{+0.11}$	${1.784}_{-0.417}^{+0.527}$	${0.006}_{-0.003}^{+0.002}$
NGC 2213	${0.15}_{-0.25}^{+0.07}$	${18.50}_{-0.09}^{+0.11}$	${1.591}_{-0.561}^{+0.381}$	${0.008}_{-0.004}^{+0.001}$
NGC 1644	${0.05}_{-0.27}^{+0.00}$	${18.51}_{-0.09}^{+0.10}$	${1.504}_{-0.340}^{+0.243}$	${0.008}_{-0.004}^{+0.000}$
NGC 1795	${0.26}_{-0.13}^{+0.10}$	${18.45}_{-0.12}^{+0.10}$	${1.541}_{-0.262}^{+0.221}$	${0.008}_{-0.003}^{+0.001}$

Note. Most likely cluster parameters listed are the maximum posterior probability, given our priors and assuming ${{\rm{\Lambda }}}_{{\rm{c}}}$ = 0.50. Conservative uncertainties listed are the 16 and 84 percentiles of a polynomial fit to the posterior distributions (see Figures 9 and 10).

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The main disagreements between Goudfrooij et al. (2014) and this work are in the fitting of NGC 1718, NGC 2203, and NGC 2213. Parameter differences for NGC 1718 and NGC 2203 are driven in part by the metallicity since Goudfrooij et al. select the best-fitting isochrone at either Z = 0.008 or Z = 0.006 and do not test intermediate values. For NGC 2213, the most likely metallicities agree, but we derive a distance of $\sim 0.7 \%$ farther (but closer to the mean LMC distance modulus of 18.49 ± 0.09 de Grijs et al. 2014) and a most probable age of ∼100 Myr younger. We find the distance to NGC 1718 $\sim 1 \%$ farther than Goudfrooij et al. (2014) and $\sim 0.7 \%$ closer than Kerber et al. (2007), though beyond the mean LMC distance modulus.

Perhaps the most important aspect underlying the disagreements in cluster measurements are that these three clusters have eMSTOs. Determining the exact MSTO may be method dependent, especially for isochrone fitting. In other words, the nearly equal probable age over a span of ages evident in the PDFs make recovering the exact parameters difficult with the standard methods of isochrone fitting. The age-eMSTO connection reported in Goudfrooij et al. (2014) and others' work is recovered in the relatively extended widths of the marginalized age PDFs. Since the morphology of the MSTO is the clearest signal on a CMD of the underlying population age, a spread MSTO age would certainly manifest as a spread in color and magnitude around the MSTO. However, we refrain from commenting on the cause of the eMSTO until fully rotational models are included within this framework.

Regardless of the level of agreement between derived parameters, the joint marginalized PDFs (Figure 10 and Figure Set) reveal obvious correlations beyond the well-known age–metallicity relationship. Metallicity appears correlated with extinction, distance, and age for each cluster. At the very least, these findings should give hesitation to heavily weighting parameters reported from isochrone fitting methods that fix values before attempting to constrain other parameters.

Relaxing the core overshooting strength prior has the effect of spreading all the PDFs, though no significant changes are seen in the maximum posterior probabilities (see Figure 11) with the exception of NGC 1718, which dropped 180 Myr in age and −0.2 dex in metallicity in order to land on its best-fit core overshooting value of ${{\rm{\Lambda }}}_{{\rm{c}}}=0.6\ {H}_{p}$. The PDFs are also more complex compared to those in Figure 10. Core overshooting strength has a complex effect on CMDs, and some values of ${{\rm{\Lambda }}}_{{\rm{c}}}$ seem to align well with different values of metallicity (e.g., NGC 2213). Asymmetric or lopsided PDFs are not signs of poor data quality or unreliable models, they are only signs that Gaussian and perhaps other functional approximations will likely inadequately describe the distribution.

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The marginalized PDFs of convective core overshooting vary dramatically from cluster to cluster. For example, Figure 12 shows the joint marginalized PDFs for NGC 1795. There are clear peaks in each PDF denoting the maximum posterior probabilities, which are listed in Table 7. The general trends in correlations between parameters in the top four rows are very similar to in Figure 10 when ${{\rm{\Lambda }}}_{{\rm{c}}}$ was fixed to the PARSEC canonical values. However, in the bottom row there are now correlations between ${{\rm{\Lambda }}}_{{\rm{c}}}$ and other cluster parameters. The most apparent is the correlation between core overshooting and age. Next, there are slight correlations with ${{\rm{\Lambda }}}_{{\rm{c}}}$ and distance and ${{\rm{\Lambda }}}_{{\rm{c}}}$ and metallicity (the ${{\rm{\Lambda }}}_{{\rm{c}}}$-metallicity correlation is built into the PARSEC models). These correlations apparent in the joint marginalized PDF of NGC 1795 are also seen in all other clusters (see the figure set).

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Table 7.  Most Likely Cluster Parameters Given the PARSEC Model

Cluster	AV	${\mu }_{0}$	Age (Gyr)	Z	${{\rm{\Lambda }}}_{{\rm{c}}}$
HODGE 2	${0.14}_{-0.08}^{+0.18}$	${18.40}_{-0.04}^{+0.17}$	${1.354}_{-0.222}^{+0.277}$	${0.008}_{-0.004}^{+0.001}$	${0.500}_{-0.127}^{+0.014}$
NGC 1718	${0.66}_{-0.27}^{+0.09}$	${18.61}_{-0.11}^{+0.17}$	${2.015}_{-0.712}^{+0.213}$	${0.004}_{-0.001}^{+0.004}$	${0.603}_{-0.177}^{+0.100}$
NGC 2203	${0.22}_{-0.14}^{+0.18}$	${18.47}_{-0.07}^{+0.10}$	${1.867}_{-0.699}^{+0.308}$	${0.006}_{-0.003}^{+0.002}$	${0.552}_{-0.100}^{+0.004}$
NGC 2213	${0.15}_{-0.06}^{+0.29}$	${18.49}_{-0.08}^{+0.10}$	${1.599}_{-0.480}^{+0.693}$	${0.008}_{-0.004}^{+0.001}$	${0.500}_{-0.028}^{+0.157}$
NGC 1644	${0.05}_{-0.01}^{+0.30}$	${18.50}_{-0.11}^{+0.09}$	${1.524}_{-0.294}^{+0.353}$	${0.008}_{-0.004}^{+0.000}$	${0.516}_{-0.067}^{+0.039}$
NGC 1795	${0.27}_{-0.12}^{+0.14}$	${18.45}_{-0.10}^{+0.14}$	${1.541}_{-0.344}^{+0.383}$	${0.008}_{-0.004}^{+0.001}$	${0.500}_{-0.029}^{+0.029}$

Note. Most likely cluster parameters listed are the maximum posterior probability, given our priors. Conservative uncertainties listed are the 16 and 84 percentiles of the PDFs (see Figures 11 and 12).

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One of many robust ways of discerning if the effect one measures is actually due to the parameter in question is by removing the parameter and re-running the analysis, and still understanding the results. By presenting our PDFs pedagogically, that is without varying core overshooting, and then by varying core overshooting, we have effectively done the necessary reliability test but in reverse. All the changes in the PDFs introduced by allowing core overshooting to vary are expected from the preceding discussion on the effects of core overshooting. For a couple clusters (e.g., NGC 2213), a higher value of overshooting with a lower value of metallicity fit nearly as well as the most probable values with canonical overshooting.

As discussed in Section 1.1, the emerging trend in recent studies focused on individual (or binary) stars is toward core overshooting strength increasing with increasing stellar mass (or age). It is interesting in this context that we find a strong correlation between age and ${{\rm{\Lambda }}}_{{\rm{c}}}$ in exactly the same direction over an age range of ∼1–2.5 Gyr (the exact age limits depend on the cluster). In other words, a younger age will be derived from a stellar population with true core overshooting strength that is lower than the model (see also the bottom two panels of Figure 7). Researchers constraining core overshooting (or perhaps any stellar evolutionary parameter) should be vigilant of possible degeneracies and their implications on their results.

We combined the marginalized PDFs of core overshooting strength (by summing the log likelihoods of all six clusters), and evaluated the resulting maximum posterior probability, the 16th, and the 84th quartiles. Given the PARSEC models, we find the most likely value of core overshooting for clusters with ages $\sim 1.3\mbox{--}2.0\,\mathrm{Gyr}$ is ${{\rm{\Lambda }}}_{{\rm{c}}}={0.500}_{-0.134}^{+0.016}\,{H}_{p}$. Our results fit within the previous work of Girardi et al. (2009), but are slightly more efficient values than expected in the relationships presented by Claret & Torres (2016).