THE OVERLOOKED ROLE OF STELLAR VARIABILITY IN THE EXTENDED MAIN SEQUENCE OF LMC INTERMEDIATE-AGE CLUSTERS

Delta Scuti (hereafter δ Sct) are variable stars that present both radial and non-radial pulsations produced by the well-known κ mechanism (Baker & Kippenhahn 1962; Cox 1980), with periods between 30 minutes up to 8 hr. They are main sequence stars with spectral types from late A to early F and masses between 1.5 and 2.5 ${M}_{\odot }$ (see, e.g., Breger 2000, for a review).

Figure 1 shows two isochrones from BaSTI models (Pietrinferni et al. 2004) of 1 and 3 Gyr with [Fe/H] = –0.5, representative of the intermediate-age cluster population in the LMC (e.g., Grocholski et al. 2006). Overlaid in dashed red lines are the limits of the empirical Galactic lower instability strip at the MSTO level from Rodríguez & Breger (2001). Since the Rodríguez & Breger (2001) sample is mostly confined to solar neighborhood variables, their metallicity is close to solar. To more properly address the instability strip at lower metallicities we use photometry of δ Sct in the body of the LMC from Poleski et al. (2010; black symbols) and the Carina dSph (Vivas & Mateo 2013; green symbols). We define new empirical limits of the instability strip (solid red lines) that encompass the bulk of the δ Sct in these two galaxies, with the caveat that these samples are likely contaminated with δ Sct of lower metallicities and probably metal-poor SX Phe variables, making their red limit uncertain.

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Stars in the upper MS, MSTO, and sub-giant branch for stellar populations with ∼1–3 Gyr will therefore experience pulsations. But unlike RR Lyrae stars, which lie at the intersection of the horizontal branch and the instability strip, not all MS stars inside the instability strip develop pulsations. The percentage of pulsating stars compared to the total number of stars inside the instability strip is known as the incidence. The incidence of δ Sct in the MW, from the study of the solar neighborhood and open clusters, is somewhere between a quarter and half of the stars within the instability strip (Breger 2000; Poretti et al. 2003; Balona & Dziembowski 2011), although the majority of these have very small amplitudes of a few millimagnitudes.

The incidence values for δ Sct in extragalactic systems are highly uncertain because it is expected that a large portion of them will not be detected given their low amplitudes, implying a lower incidence. In Carina, Vivas & Mateo (2013) report a lower limit of incidence of 8%. However, the number of high amplitude δ Sct in extragalactic systems like Carina and Fornax seems to be intrinsically higher than those observed in the field of the MW (Poretti et al. 2008; Vivas & Mateo 2013). In Carina, the peak of the amplitude distribution is A_V = 0.5 mag (Vivas & Mateo 2013), while in the LMC is ∼0.3 mag (Poleski et al. 2010).

In this paper we explore the influence of δ Sct on the morphology of the MSTO by using synthetic CMDs modified by the presence of these variable stars.

Since the great majority of extended MSTO have been detected using HST imaging (e.g., Milone et al. 2009), our models try to reproduce those CMDs. From DAOphot catalogs of HST/ACS photometry of NGC 1846 (GO: 10595, PI: Goudfrooij) obtained from the Hubble Legacy Archive,⁸ photometric errors at the MSTO level are of the order of 0.008 mag. Conservatively we assume an error of 0.01 mag per filter for the entire CMD. Our models use as a starting point BaSTI synthetic CMDs (Pietrinferni et al. 2004) with [Fe/H] = –0.5, scaled to solar mixture and convective overshooting. Stars follow a luminosity function implied by the Salpeter (1955) IMF, using a total number of stars resulting in a population of ${M}_{V}\sim \,-7.5$, typical for a fairly massive intermediate-age LMC cluster. Different ages between 0.75 and 2.5 Gyr were used (see below).

The instability strip defined in Section 1.1 was used to set the color boundaries from which stars would be selected to be modeled as variables. Stars within the instability strip were chosen randomly for values of the incidence between 10% to 50%. The selected stars were then modeled as variables using the cosine expansion of Simon & Lee (1981),

$$\begin{eqnarray}&&I={A}_{0}+\displaystyle \sum _{i=1}^{N}{A}_{i}\cos (i\omega t+{\phi }_{i}),\end{eqnarray} \tag{ 1 }$$

where A₀ is the "static" magnitude coming from the CMD modeling described above.

Amplitudes and phases are usually grouped in the form ${R}_{{ij}}\equiv {A}_{i}/{A}_{j}$ and ${\phi }_{{ij}}\equiv {\phi }_{i}-i{\phi }_{j}$, and we took these values from the study of δ Sct in the first OGLE bulge campaign of Morgan et al. (1998). The Morgan et al. (1998) values were preferred over the Poleski et al. (2010) values from OGLE-III since the former studied the Fourier parameters up to the 4th term, while the latter gives values for R₂₁ and ${\phi }_{21}$ only. In practice, for each star, values for the Fourier parameters were drawn from uniform distributions, ${R}_{21}=U(0.05,0.45)$, ${R}_{31}=U(0,0.2)$, ${R}_{41}=U(0,0.15)$, and ${\phi }_{1}=U(0,0.15)$, ${\phi }_{21}=U(3.5,4.5)$, ${\phi }_{31}=U(1,3)$, ${\phi }_{41}=U(3.5,8)$, following the results of Morgan et al. (1998), and we used the notation $U(x,y)$ to describe a uniform distribution with limits x and y. The most relevant quantity is A₁, which dominates the total amplitude of the light variation. Its impact was studied by drawing its distribution from several uniform distributions with maximum values for A₁ between 0.1 and 0.2. This is a conservative approach considering the largest amplitudes for δ Sct in the LMC reach ∼0.8 mag in I (Poleski et al. 2010).

Since Fourier parameters exist only for the observations in I, light curves in V were generated as scaled-up versions of the I light curves applying to A₁ the ${A}_{V}/{A}_{I}=1.7$ empirical factor given by Rodríguez et al. (2007). The period distribution for these simulated variables was taken as a normal distribution centered on 0.075 days and with a dispersion of 0.02 days following the results of Poleski et al. (2010) Finally, the modeled light curves were "observed" at one fixed time for each filter, that is, by taking, for each variable, a measurement from their light curves at a random phase and putting back these magnitudes into the CMDs.

The results from the modeling can be seen in Figure 2, where three combinations of incidence and maximum amplitude are shown. The most salient result is that even though stars can develop pulsations everywhere within the instability strip, the color and magnitude shift of stars in the upper MS are mostly parallel to the MS, incurring with little broadening of the MS width. On the contrary, δ Sct near the TO experience color and luminosity changes that are almost perpendicular to the TO artificially broadening the MSTO when observed at a random phase.

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In order to quantify the broadening of the MSTO area introduced by the δ Sct, following Goudfrooij et al. (2011) we construct a parallelogram where one pair of the sides is roughly parallel to the isochrones, while the other pair is approximately perpendicular (see the red boxes in Figure 2). The parallelogram is located where the difference in the isochrones is the largest, therefore the minimum distance to the axis that is parallel to the isochrones can be considered as a "pseudo-age," which in this case is introduced by the variable stars and does not represent a real age spread. The distribution of these pseudo-color/ages as a function of incidence with a fixed maximum amplitude A₁ = 0.1, can be seen in Figure 3, where generalized histograms have been produced using an Epanechnikov kernel (Silverman 1986). The models show that the density of stars near their parent isochrones decreases with increasing incidence, and the distributions developed extended wings departing from the Gaussian errors. The FWHM of the distribution can increase by up to 50% when an incidence of 50% is used.

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As seen in Figure 1, δ Sct will only affect the MSTO between ages ∼1 and 3 Gyr. For ages below 1 Gyr the TO point is too bright and δ Sct will only be produced in the MS. For ages older than 3 Gyr the redder edge of the instability strip is bluer than the MSTO, so no pulsations will be developed in the MS (only in some possible blue stragglers).

We quantify the influence of δ Sct as a function of age running our model with fixed incidence of 30% and maximum amplitude A₁ = 0.2, for ages from 0.75 to 2.75 Gyr with a step of 0.25 Gyr. The pseudo-age spread introduced at the MSTO is measured with the parallelogram approach introduced in the previous Section, centering it in each corresponding isochrone.

Niederhofer et al. (2016) used data from Goudfrooij et al. (2014) to argue that age spreads in the LMC clusters increase as a function of cluster age, reaching a maximum at 1.5–1.7 Gyr, and then decreasing after that age.

Figure 4 shows the change in the FWHM of the pseudo-age distributions as a function of model age. These are the mean values and standard deviations from 50 models per age. In general it shows a behavior similar to the one found by Niederhofer et al. (2016), although with a smaller amplitude. This behavior is easily explained as the MSTO region moves in and out of the instability strip causing the highest number of stars develop pulsations close to 2 Gyr. We stress, however, that absolute ages are model-dependent, and different sets of isochrones will show differences of a couple hundred Myr to define the edges of the instability strip.

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