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The Cepheids of NGC1866: A Precise Benchmark for the Extragalactic Distance Scale and Stellar Evolution from Modern UBVI Photometry

We present the analysis of multiband time-series data for a sample of 24 Cepheids in the field of the Large Magellanic Cloud cluster NGC1866. Very accurate BVI VLT photometry is combined with archival UBVI data, covering a large temporal window, to obtain precise mean magnitudes and periods with typical errors of 1-2% and of 1 ppm, respectively. These results represent the first accurate and homogeneous dataset for a substantial sample of Cepheid variables belonging to a cluster and hence sharing common distance, age and original chemical composition. Comparisons of the resulting multiband Period-Luminosity and Wesenheit relations to both empirical and theoretical results for the Large Magellanic Cloud are presented and discussed to derive the distance of the cluster and to constrain the mass-luminosity relation of the Cepheids. The adopted theoretical scenario is also tested by comparison with independent calibrations of the Cepheid Wesenheit zero point based on trigonometric parallaxes and Baade-Wesselink techniques. Our analysis suggests that a mild overshooting and/or a moderate mass loss can affect intermediate-mass stellar evolution in this cluster and gives a distance modulus of 18.50 +- 0.01 mag. The obtained V,I color-magnitude diagram is also analysed and compared with both synthetic models and theoretical isochrones for a range of ages and metallicities and for different efficiencies of core overshooting. As a result, we find that the age of NGC1866 is about 140 Myr, assuming Z = 0.008 and the mild efficiency of overshooting suggested by the comparison with the pulsation models.


INTRODUCTION
Classical Cepheids play a fundamental role in the calibration of the extragalactic distance scale thanks to their characteristic period-luminosity (PL) and period-luminosity-color (PLC) relations. In particular, the application of a Large Magellanic Cloud (LMC)-based PL relation to external galaxies observed with the Hubble Space Telescope (HST) has led to the calibration of secondary distance indicators and in turn to an estimate of the Hubble constant (H0; see, two decades, on both observational and theoretical grounds, (see e.g. Gould 1994;Sasselov et al. 1997;Kennicutt et al. 1998;Fiorentino et al. 2002;Sakai et al. 2004;Marconi, Musella, & Fiorentino 2005;Bono et al. 2008;Romaniello et al. 2008;Marconi et al. 2010;Sandage et al. 2006;Freedman & Madore 2011;Gerke et al. 2011;Fausnaugh et al. 2015), with controversial results. No general consensus has been reached so far. Empirical calibrations of the LMC PL and PLC relations are generally based on field Cepheids, implying the presence of uncertainties due to the range in distance (depth effects van der Marel & Cioni 2001), metallicity (see e. g. Davies et al. 2015;Luck et al. 1998, and references therein) and extinction (differential reddening, see e.g. Haschke, Grebel, & Duffau 2011, and references therein).
On the other hand, any theoretical scenario for pulsation models aimed at providing robust support to empirical calibrations needs to be tested against observational constraints based on statistically significant samples of Cepheids with accurate light curves.
The populous blue LMC cluster NGC1866 is already known to host an exceptionally rich sample of more then 20 Cepheids (Welch & Stetson 1993;Musella et al. 2006). One of these was identified by Musella et al. (2006) in a preliminary analysis of the proprietary BVI VLT data. It is unquestionable that such a unique sample of Cepheids-likely all members of the cluster and at the same distance, chemical composition and age-would constitute a milestone in our understanding of the Cepheid pulsational scenario. Indeed, it offers an unprecedented opportunity to investigate both empirical and theoretical estimates of the luminosity and color of the pulsating structures and their relation with the observed periods. For this reason, many authors have studied the NGC1866 Cepheids (Welch 1991;Welch & Stetson 1993;Gieren, Richtler, & Hilker 1994;Walker 1995;Gieren et al. 2000;Storm et al. 2005;Testa et al. 2007) in both the optical and near infrared bands, and have tested different methods to calibrate the PL relations in different filters. In Brocato et al. (2004) we already discussed the sample of the 23 known Cepheids in NGC1866, concluding that unfortunately only 4 to 6 Cepheids had light curves accurate enough to allow a meaningful determination of their luminosities and colors. On the basis of such a tantalizing situation, we took advantage of assigned observing time at the ESO Very Large Telescope to perform an accurate photometric investigation of the cluster field, with the aim of securing suitable data constraining the light curves of all the member Cepheids. Moreover, to get accurate information about radial velocities and chemical abundances of the stars in NGC1866, we have performed FLAMES@VLT spectroscopic observations for 30 stars (19 belonging to the cluster and 11 to the LMC field), including 3 Cepheids (Mucciarelli et al. 2011;Molinaro et al. 2012). Mucciarelli et al. (2011) found that, as far as the chemical composition is concerned, the cluster stars are reasonably homogeneous. Indeed, they appear to share the same abundances within the uncertainties, and this property is independent of the evolutionary status.  Wesselink method (Ripepi et al. 1997) to a sample of 11 Cepheids, using radial velocities obtained both from our FLAMES investigation and from literature data (see references therein), and light curves based on a part of the UBVI data used in this paper complemented with K data by Testa et al. (2007). In this way, they obtained a direct estimate of the distance modulus of NGC1866, µ0 = 18.51 ± 0.03 mag (see Molinaro et al. 2012, for details).
In this paper, we analyze the properties of NGC1866 Cepheids, relying on an extensive multifilter dataset not only to derive information about the distance, but also to test pulsational and evolutionary theoretical models.
The paper is organized as follows. We present observations and data reduction in section 2 and Cepheid properties in section 3. The adopted pulsational models are briefly described in section 4. To derive the distance of NGC1866 we have applied in section 5 the theoretical period-luminosity and Wesenheit relations, and in section 6 two empirical calibrations based on trigonometric parallaxes and the Baade-Wesselink method. In section 7, a deep and accurate V,V-I color-magnitude diagram is presented and compared with theoretical isochrones to derive additional information on the cluster properties. The conclusions close the paper.

OBSERVATIONS AND DATA REDUCTION.
Our photometric database comprises 1471 CCD images (see Table 1).
The median seeing of the images was 1.8 arcsec, but it is worth noting that for the B , V , and I filters, we also have a set of high precision photometric proprietary data obtained with FORS1@VLT. For this data set, the detector was a 2048x2048 Tektronix CCD with 24µ × 24µ pixels. Projected on the sky, the scale is 0.2 arcsec/pixel for a total field of view of 6.8 ×6.8 . We have obtained time-series photometry on one pointing centered on NGC1866 in the BVI filters with exposures of 60 s in each filter. The observations were efficiently carried out in service mode, with the constraint of seeing better than 0.7 because the target cluster is very crowded and many Cepheids are located near the cluster center. The reduction and calibration of all the photometric data was carried out with the DAOPHOT/ALLFRAME packages (Stetson 1987(Stetson , 1994)-which combine excellent precision with a large degree of automation-and using standard methodologies as described in, for example, Stetson (2000) and Stetson, Catelan, & Smith (2005). Special care has been taken in deriving an accurate PSF for each image because of the high degree of crowding not only in the central regions of the cluster but also in its outskirts.
The calibration was based on local standards in NGC1866. The median internal photometric precision based on these local standards is 0.01, 0.001, 0.0008 and 0.002 mag for U , B , V and I bands, respectively, using all the data sets, and 0.0006, 0.0004 and 0.0008 mag for the B , V and I bands, respectively, considering only the VLT data. Concerning the external accuracy of our photometry, the root-mean-square differences of our mean magnitudes for Landolt's standards (Landolt 1992), on a star-by-star basis, is of ∼ 0.029, ∼ 0.016, ∼ 0.013 and ∼ 0.016 mag in the U , B , V and I bands, respectively. These differences represent 180, 378, 400, and 250 individual Landolt standards, so our mean photometric system differs from Landolt's mean system by not less than ∼ 0.001 mag in B ,V and I , and ∼ 0.002 mag in U . This is probably a fundamental limit for the external, absolute accuracy that can be obtained with CCD measurements, given that we use different filters and detectors than Landolt used.

NGC1866 CEPHEIDS
The position and the light curves in the UBVI bands of the Cepheid identified in Musella et al. (2006) are reported in Figs. 1 and 2, respectively. The coordinates of this Cepheid are α = 05:13:20.79 and δ = −65:24:57.8 (J2000) at about 3.5 arcmin from the cluster center. The light curves of all other previously known variables are plotted in Fig. 3 In particular, grey dots are data from the MACHO project, red dots are VLT data and open circles are all the other archival data (see Table 1).
The location in the extreme cluster periphery of the Cepheid identified in Musella et al. (2006) may raise serious doubts about its membership. Lupton et al. (1989) suggest that NGC1866 is not tidally limited but is embedded in an unbound halo. Noyola & Gebhardt (2007), imposing the existence of a tidal radius, find an half-light radius of 49.7 arcsec: much less than the distance of this Cepheid (∼ 3.4 arcmin). However, in the following we will demonstrate that both the location in the color-magnitude diagram (CMD) and the pulsational properties of this variable appear in close agreement with the behaviour of the other cluster Cepheids. Obviously, to have a firmer conclusion about the membership of this distant Cepheid, the mean radial velocity should also be measured through accurate spectroscopic observations.
The obtained light curves show the superior accuracy of the VLT photometry when compared with the other datasets. In particular, for the MACHO data in the V band, it was necessary to reject several scattered phase points. We use all datasets, spanning a large time window, to determine accurate periods using Period04 (Lenz & Breger 2005). This program allows us to determine also the error on the derived period via Monte Carlo simulations. On the other hand, to obtain mean (intensity-averaged) magnitudes and colors as accurately as possible, we fitted the light curves in all available bands with a smoothing spline obtained with a C code written by one of the authors (R. M). In particular, for the B , V and I light curves we used only the VLT data, adopting the other datasets only for the U filter. All the derived periods, UBVI mean magnitudes and amplitudes A λ (for each λ filter) are reported in Table 2, but we will not use the U band in the following analysis. As an estimate of the uncertainty in the calculated mean magnitudes, we report in Table 2 the r.m.s. of the residuals of the data around the fitted curves; of course, these residuals will not fully represent consistent photometric errors resulting from the blending conditions of individual stars. Table 2 also reports the complementary K band data obtained by Testa et al. (2007).
Analyzing the Cepheid light curves in Fig. 3, we note that We5, We7 and WS9 appear to be affected by more noise than the other objects, in particular in the B and V bands, and we can see similar but less evident scatter also for We4 and WS5. This is likely the result of blending due to the location of these variables in the central very crowded region, combined with varying seeing conditions. Moreover, a timeseries analysis for double-mode behaviour for such crowded stars is likely inconclusive given the long intervals between observational epochs and the relatively short periods with excellent seeing. The Cepheids in the external region have well defined light curves. V5, V6 and V8 have periods and light curves typical of First Overtone (FO) Cepheids. In particular, V5 is the brightest Cepheid and has a very blue color but, as pointed out by Welch (1991), its membership seems to be excluded according to its mean radial velocity and large distance from the center of the cluster. Welch (1991), on the basis of the radial velocity, also ruled out membership for HV12204.

PULSATION MODELS
For the interpretation of the observed properties of the variable stars in NGC1866, we adopt the theoretical pulsation scenario developed by our team (see e.g. Marconi, Musella, & Fiorentino 2005;Marconi et al. 2010Marconi et al. , 2013, and references therein) and based on nonlinear convective pulsation models for different assumptions on the metallicity. For each assumed fixed chemical composition, a wide range of model masses is explored and the luminosities are selected according to evolutionary predictions for the mass-luminosity (ML) relation of central heliumburning intermediate-mass stars. In the "canonical" scenario both mass loss and convective overshooting are neglected, whereas in the "noncanonical" assumption an overluminosity of 0.25 dex at fixed mass is expected as a result of mild core overshooting and/or moderate mass loss (see e.g. Chiosi et al. 1993;, for details). According to this theoretical scenario the effect of variations in both the metallicity and the helium content on the predicted coefficients of the PL and PLC relations is not negligible Caputo et al. 2000;Marconi, Musella, & Fiorentino 2005). On the other hand, we can adopt the Wesenheit (WPL) relations (Madore 1982 is the ratio of total to selective absorption (see e.g. Cardelli et al. 1989); this represents a reddening-free formulation of the PL relation with a reduced intrinsic scatter. In particular, Caputo et al. (2000) showed that the intrinsic scatter and the dependence on the metallicity of the WPL(V,I ) relation is almost negligible.
The coefficients of the theoretical PL and WPL relations for the chemical composition of NGC1866 (Z = 0.008, Y = 0.25), derived in the period range from log P = 0.4 (corresponding to the minimum fundamental period in NGC1866 1 ) to log P = 1.5 (corresponding to the maximum period in the OGLE sample), are those by Fiorentino et al. (2002) and Fiorentino et al. (2007), respectively. In particular the WPL relation was also derived with an explicit dependence on both the metallicity and the excess luminosity log L/Lc, where Lc is the canonical luminosity level. This allows us to check the theoretical models both for canonical (log L/Lc = 0) and noncanonical (log L/Lc = 0.25 dex) assumptions (see Table 3 in Fiorentino et al. 2007). For the V , I filter combination, moving from the canonical to the non canonical assumption, the WPL(V,I ) relation gets fainter by about 0.21 mag.     Welch (1991) and they are not considered in the following analysis.

THE PL AND PERIOD-WESENHEIT RELATIONS
talised periods 2 (filled red symbols). It is worth noting that at fixed period, even taking into account only the reliable Cepheids, the scatter in magnitude is significant (0.12, 0.09 and 0.07 mag in the B, V , and I bands, respectively). Given that our targets share the same age and chemical composition, this can only be ascribed to i) the finite width of the instability strip (see Caputo et al. 2000, for a detailed discussion) and ii) a possible mass-loss effect. For the V and I PL relations (middle and bottom panel of Fig. 4), we compared our sample with the OGLE one (grey crosses are first overtone Cepheids and grey filled circles the fundamental ones). The black straight lines represent the mean OGLE PL relations. It is evident that the NGC1866 sample appears to be brighter than the OGLE III one (Soszynski et al. 2008). Indeed, when we overlay the OGLE III V and I PL slopes on the Cepheid population of NGC1866 3 we obtain a shift of ∆µV = −0.12 ± 0.02 mag and ∆µI = −0.08 ± 0.02 mag.
The black dotted lines and the red dashed lines in Fig.  4 represent the linear regression of the OGLE data and of our Cepheid sample, respectively, adopting the theoretical slopes for Z = 0.008 in the canonical assumption and in the same period range covered by OGLE (0.4 < log P < 1.5) 2 The period the variable would have if it were a fundamental one, obtained as log P f un = log P + 0.127. 3 We notice that in all our fits of the NGC1866 sample, we include only the reliable fundamental Cepheids and the two first overtone V6 and V8, adopting their fundamentalised periods reported in Table 6  .02 mag (the adopted errors are the standard deviation of the mean). To derive the absolute distance moduli, we need to know the color excess. One of the most used values for the color excess of NGC1866 and LMC is E(B − V ) = 0.06 mag (see e.g. Storm et al. 2005;Schlegel et al. 1998). However, we can also derive this value from the difference of the apparent moduli in two different bands. In particular E(B − V ) = µB − µV and E(V − I) = µV − µI . Adopting the inferred apparent distance moduli, we obtain E(V − I)OGLE = 0.078 ± 0.009 mag-a value that, combined with the extinction law by Cardelli et al. (1989), gives E(B −V )OGLE = 0.064±0.007 mag. From the apparent distance moduli of NGC1866, we obtain E(B − V )NGC1866 = 0.02±0.04 mag and E(V −I)NGC1866 = 0.03±0.03 mag. The estimate for the LMC agrees quite well with the aforementioned value E(B −V ) = 0.06 mag. However, the values that we obtain for NGC1866 are significantly lower than the ex- pected ones, but the distribution of the Cepheid individual color excesses is not Gaussian and if we compute the medians, we obtain E(B − V )NGC1866 = 0.04 ± 0.03 mag and E(V −I)NGC1866 = 0.04±0.02 mag. Moreover, it is worth remembering that all the other NGC1866 stars that are bright enough to be significant when contaminating the Cepheids' photometry (either as binaries or due to coincidence along the line of sight) are going to be bluer than the Cepheids, thus affecting B more than V and V more than I . Perhaps only the very outer stars could be contaminated by redder field stars. For this reason, the reddening values determined by the apparent moduli differences are systematically underestimated. Due to the strength of the literature color excess determination, we adopt for both the samples the value E(B −V ) = 0.06 mag together with a ratio between AV and E(B −V ) given by RV = 3.1 (Cardelli et al. 1989), obtaining µ OGLE 0 = 18.718 mag and µ N GC1866 0 = 18.58 mag.
To reduce the problems due to the determination of the color excess, we adopt the reddening free WPL relations. It is worth noting that the definition of the WPL relation is not unique: it depends on the adopted reddening law from which we derive the color term. As theoretical WPL relations, we adopt those, dependent on metallicity Z and luminosity level log L/L , obtained by Fiorentino et al. (2007) (see their Table 3) using the absorption to reddening ratio from Dean, Warren, & Cousins (1978) and Laney & Stobie (1993).  term and compare the data with the empirical relation by OGLE. Symbols are the same as in Fig. 4. If we adopt the OGLE WPL(V,I ) relation, the difference between the distance moduli of NGC1866 and LMC is ∆µ = −0.018 ± 0.014 mag. Moreover, using the slope of the plotted canonical WPL(V,I ) relation, we obtain µ OGLE 0 = 18.717 ± 0.003 mag and µ N GC1866 0 = 18.71 ± 0.01 mag. This result implies that removing the reddening problem, the inferred LMC and NGC 1866 moduli are in good agreement. The values based on the predicted canonical WPL relation seem to favor the "long distance scale" for the LMC. On the other hand, as noted above, in the noncanonical assumption (log L/Lc = 0.25 dex), the predicted Wesenheit function is fainter by about 0.21 mag and in turn the inferred distance moduli are µ OGLE 0 = 18.507 ± 0.003 mag and µ N GC1866 0 = 18.50 ± 0.01 mag.

THE EMPIRICAL ROUTE TO THE DISTANCE OF NGC1866
To test the results obtained with the theoretical relation, we discuss two empirical calibrations of Cepheid distances based on two direct methods: i) the trigonometric parallaxes available for a sample of Galactic Cepheids and obtained with the Hipparcos satellite (van Leeuwen et al. 2007) and the HST (Benedict et al. 2007;Riess et al. 2014;Casertano et al. 2015); ii) the Baade-Wesselink method applied to 70 Galactic (Storm et al. 2011a) and 36 LMC Cepheids (Storm et al. 2011b). Following the same procedure adopted in Ripepi et al. (2012), their sample of MW Cepheids with distances from trigonometric parallaxes by van Leeuwen et al. (2007) and Benedict et al. (2007) has been selected (see their Table 6) retaining only the variables with δπ/π 0.2. For variables present in both the Hipparcos and the HST sets, Ripepi et al. (2012) have used a weighted average of the two parallaxes. Moreover, we add two Cepheids, SY Aur and SS CMa, with very accurate parallaxes obtained by Riess et al. (2014) and Casertano et al. (2015), respectively. In Table 3, we report the parallax, the V and I magnitudes (taken from Table  A1 in van Leeuwen et al. 2007) and the Lutz-Kelker (LK) correction (calculated according to Benedict et al. 2007) for the Galactic Cepheids used in our analysis.
In the upper panel of Fig. 6, we show WPL(V,I ) (with the theoretical color coefficient of 1.52, see above) for these Galactic Cepheids (green symbols) and for our sample (black and red symbols). The green line represents the relation derived from a weighted fit of the Galactic Cepheids absolute magnitudes, W P Lpar(V, I) = −3.35 log P − 2.57 (σ = 0.05), whereas the black line is the relation obtained assuming the slope of the theoretical WPL(V,I ) relation for Z = 0.008 discussed in the previous section. From the former relation we obtain µ N GC1866 0 = 18.49 ± 0.05 mag, whereas the theoretical relation provides µ N GC1866 0 = 18.73 ± 0.04 mag in the canonical assumption and 18.52 ± 0.04 mag in the noncanonical one. We note that in spite of the difference between the slope obtained for the Galactic Cepheids (−3.35) and the theoretical one (−3.30), when the theoretical slope is applied to the Galactic Cepheids, the zero point obtained (−2.62 ± 0.04) is in good agreement with the theoretical noncanonical one (−2.63).
The middle and bottom panels of Fig. 6 show our NGC1866 data (black and red symbols) together with MW (cyan symbols) and LMC (violet symbols) Cepheid samples with Baade-Wesselink (BW) distance estimates by Storm et al. (2011a) and Storm et al. (2011b), respectively. In these papers, the authors adopt 1.54 as color term in the WPL(V,I ). For the Galactic sample, Storm et al. (2011a) give a WPL slope of −3.26 and a zero point of −2.7. Adopting 1.52 as color term, we obtain the same slope and a slightly different zero point of −2.68 (instead of the values −3.38 and −2.54 obtained above from the weighted fit of the MW Cepheid with parallaxes). The BW slope (−3.26) and the theoretical one (−3.30) are very similar and when ap-  For the LMC BW Cepheids, Storm et al. (2011a) give a WPL slope of −3.41 and a zero point of −2.46. Adopting a color term of 1.52, we obtain the same slope and −2.44 as zero point. Applying this relation, we obtain a distance modulus of 18.39 ± 0.03 mag for NGC1866. The shorter distance scale obtained from the BW calibration supports previous determinations based on this method in the literature (see, e.g., the discussion in Molinaro et al. 2012, and references therein). On the other hand the application of the CORS version of the BW method by Molinaro et al. (2012) to the NGC1866 Cepheids used in their work gives a distance modulus of 18.51 ± 0.03 mag. The uncertainties on the distance estimates based on BW techniques depend on the controversial issue of the adopted projection factor which converts the measured radial velocity into the pulsational one (see e.g. Barnes 2009;Storm et al. 2011b;Molinaro et al. 2012, and references therein). In Fig. 7, we plot the differences between the individual moduli obtained by Molinaro et al. (2012) for a sample of 9 fundamental NGC1866 Cepheids and those obtained for the same variables adopting the theoretical slope and zero point of the WPL for Z = 0.008, both in the canonical (black symbols) and in the non-canonical (red symbols) assumption. In this figure we report also a black line for ∆µ = 0 mag corresponding to perfect agreement.
Inspection of these values and of Fig. 7 indicates that Figure 7. The differences between the individual moduli obtained by Molinaro et al. (2012) for a sample of 9 fundamental NGC1866 Cepheids (see their Table 5) and those obtained for the same variables, adopting the theoretical slope and zero point of the WPL for Z = 0.008, both in the canonical (black symbols) and in the non-canonical (red symbols) assumption. The black line corresponds to ∆µ = 0. the adopted theoretical scenario in the noncanonical assumption provides better agreement with the most recent and widely assumed distance estimates for NGC1866. Indeed, for the LMC, being near to our Galaxy and hosting several stellar distance indicators, we have hundreds of different and independent distance measurements. For example, a recent interesting measurement was obtained by the Carnegie Hubble Program (Monson et al. 2012) to calibrate the extragalactic distance scale, with the final goal to obtain H0 with an accuracy of 2%. For this purpose, they observed Cepheids in the Milky Way and nearby galaxies in the mid-infrared bands and found for the LMC a modulus of 18.48 ± 0.04 mag. The same value (with an error of ±0.05 mag) was obtained by Walker (2012) by the analysis of LMC distances based on different stellar distance indicators (Cepheids, RR Lyrae, Red Variables, Red Clump Stars and Eclipsing Variables). Finally it is worth noting that Pietrzyński et al. (2013) give a very accurate (within 2%) LMC distance measurement based on the observations of 8 late-type eclipsing binaries in this galaxy, finding a value of 18.493 ± 0.008(statistical)±0.047(systematic) mag. This conclusion seems to favor the hypothesis of a moderate overluminosity of NGC1866 Cepheids at fixed mass, possibly due to a combination of core overshooting and mass loss effects, in agreement with the results by Brocato et al. (2004).

THE COLOR-MAGNITUDE DIAGRAM
As a by-product of the accurate time-series multiband observations, we are able to build new CMDs. Figs. 8 and 9 show in the bottom panels the V,V-I CMD of NGC 1866 and in the top panels a zoom of the region around the Cepheid IS. To derive the mean magnitudes for the stars identified in the observed field, we use only the VLT data that, as described in Sect. 2, are much more precise than the other ones. Thanks to the robustness of estimated mean magnitudes and colors, most of the reliable fundamental Cepheids (black filled circles) are located in a very restricted region of the diagram, corresponding to the tip of the blue loop of He burning giants.
To compare the observed CMD and the Cepheid location with evolutionary prescriptions, in Fig. 8 we considered the synthetic CMDs computed with the stellar population synthesis code SPoT 4 by Brocato et al. (2003), based on two sets of stellar evolutionary tracks calculated for this purpose with the Pisa evolutionary code , with canonical (no overshooting) and non-canonical (moderate overshooting, 0.25 dex) assumptions. In particular, we considered Z = 0.007 and ages of 140 Myr and 180 Myr for comparison with the results by Brocato et al. (2003Brocato et al. ( , 2004) (see those papers for more details). Moreover, we assumed a distance modulus of 18.5 mag and a color excess E(B-V ) = 0.06 mag. The left panel of Fig. 8 shows the comparison between canonical and noncanonical synthetic CMDs for an age of 140 Myr with the observed data. We note that the location of the Cepheids is intermediate between the blue loops obtained in the two assumptions. This suggests a possibly lower overshooting efficiency compared to that adopted, or a slightly lower cluster metallicity. In the right panel, we use the same canonical synthetic CMD, but increase the age of the noncanonical one to 180 Myr. We note that, for the higher age, the luminosity level predicted in the noncanonical assumption seems to reproduce the observed one, but the blue loop extension is too short (see also discussion below).
We also adopted the isochrones from the BaSTI (Bag of Stellar Tracks and Isochrones) database (Pietrinferni et al. 2004(Pietrinferni et al. , 2006, which also allows us to compare models based on the canonical assumptions and models including core overshooting at different chemical compositions. In the left panel of Fig. 9, at fixed metallicity Z = 0.008 and age of 140 Myr, we compare the canonical isochrone with the non-canonical one. As expected, the canonical blue loop is fainter than the observed Cepheids, whereas the noncanonical one is too bright. We note that the overshooting efficiency adopted in the BASTI database is almost twice that adopted in the pulsational scenario, so that the result obtained seems to confirm our finding based on the pulsational analysis that Cepheids in NGC1866 might be affected by mild overshooting of 0.2 dex and/or a small amount of mass loss. At fixed age, the loop luminosity is also affected by metallicity, thus in the central panel we compare the canonical isochrone at 140 Myr for Z = 0.008 with the one for Z = 0.004. The latter isochrone seems to better match the observed Cepheid location, but it does not properly reproduce the Main Sequence (MS). However, we cannot exclude an intermediate metallicity close to Z = 0.006. Finally, in the right panel, we try to reproduce the luminosity of the observed Cepheids by varying the isochrone age in the two scenarios, assuming a metallicity of Z = 0.008. We find that in the canonical assumption the predicted age is close to 120 Myr and in the non-canonical one it is about 170 Myr. The overshooting efficiency in the non-canonical isochrone is higher than expected according to the pulsational results. In the case of mild overshooting, an intermediate age of about 140 Myr is predicted. We note that the theoretical blue loops for Z = 0.008 appear too short at least for some Figure 8. In this figure we show, in the bottom panels, the V,V-I CMD of NGC1866 obtained using only the very precise VLT data compared with the synthetic CMDs computed with the stellar population synthesis code SPoT by Brocato et al. (2003). The symbols used for NGC1866 Cepheids are the same of Fig. 4. The comparison with canonical (blue dots) and noncanonical (green dots and labeled as NC) synthetic CMDs are shown. The age adopted for the canonical synthetic CMD is 140 Myr, and that adopted for the noncanonical one is 140 Myr in the left panel and 180 Myr in the right one. In the top panels, there is a zoom of the region around the Cepheid IS.
ages and this discrepancy can be due to different numerical and physical assumptions in the theoretical scenario (Bono et al. 2000b;Castellani et al. 1990;Valle et al. 2009;Walmswell et al. 2015), even if-as noted above-a slightly lower metal content cannot be excluded.

CONCLUSIONS.
We have investigated the properties of a sample of Cepheids in NGC1866 on the basis of both archival and very precise VLT data.
As a result of the time series analysis, accurate light curves are obtained for 21 fundamental and three first overtone pulsators. We also derive the mean magnitudes and colors for each pulsator and compare their behavior with both empirical (OGLE) and predicted LMC PL and WPL(V,I ) relations. The results from the PL are affected both by the intrinsic dispersion and by reddening uncertainties, whereas the WPL(V,I ) relation gives more reli- Figure 9. In the three bottom panels of this figure, we show the same observational CMD of Fig. 8 compared with the isochrones from the BaSTI (Bag of Stellar Tracks and Isochrones) database (Pietrinferni et al. 2004(Pietrinferni et al. , 2006 for different assumptions of overshooting efficiency, metallicities and ages (see text for details). In the top panels, there is a zoom of the region around the Cepheid IS.
able results. Adopting the slope of the theoretical canonical WPL(V,I ) relation, we obtain µ OGLE 0 = 18.717 ± 0.003 mag and µ N GC1866 0 = 18.71 ± 0.01 mag, for the OGLE LMC field sample and our NGC1866 data, respectively. On the other hand in the noncanonical assumption (log L/Lc = 0.25 dex), the inferred distance moduli are µ OGLE 0 = 18.57 ± 0.003 and µ N GC1866 0 = 18.50 ± 0.01 mag. As a test of our theoretical approach, we also considered two empirical calibrations of Cepheid distances based on trigonometric parallaxes and BW measurements. Even if the slope of the WPL(V,I ) relation obtained for the Galactic Cepheids with parallaxes is steeper by 0.06 than the theoretical one, its application to the NGC1866 sample gives a distance modulus of 18.49 ± 0.07 mag, in very good agreement with that obtained applying the theoretical noncanonical slope (18.52 ± 0.04 mag).
On the other hand, using Galactic Cepheids with BW measurements, the slope of the WPL(V,I ) relation is very similar to the theoretical one. Both the slopes give the same distance modulus of 18.54 mag for NGC1866, adopting the corresponding zero point. Using the LMC BW sample and adopting the Storm et al. (2011b) WPL(V,I relation (recomputed using 1.52 as color term), we obtain for NGC1866 a distance of 18.39 ± 0.03 mag-significantly shorter than the values derived above. This shorter distance scale is in agreement with previous determinations based on the BW method and its deviation from most of the other recent results for the LMC has already been debated in the literature (see e.g. Molinaro et al. 2012). Therefore, our investigation seems to favor a value close to 18.5 mag for the distance modulus of NGC1866, as also obtained by the theoretical noncanonical WPL(V,I ) rela-tion, thus suggesting that mild overshooting and/or moderate mass loss can affect intermediate-mass stellar evolution in this cluster.
To conclude, we have compared the V,V-I CMD of NGC1866 with the predictions of evolutionary and synthetic models, adopting µ0 = 18.5 mag, E(B −V ) = 0.06 mag. The main conclusions of this comparison are: • for the typical metallicity adopted for NGC1866 (Z = 0.007 ÷ 0.008), an age close to 120 Myr (170 Myr) is suggested by the comparison with canonical (noncanonical) isochrones. As the noncanonical isochrones are computed for an overshooting efficiency almost twice that assumed in the pulsational analysis, we expect an age around 140 Myr in the mild overshooting and/or moderate mass loss scenario; • decreasing the metallicity of the canonical isochrone at 140 Myr from Z = 0.008 to Z = 0.004, we find better agreement with the Cepheid distribution, but a worse fit for the main sequence phase. On this basis, an intermediate metallicity close to Z = 0.006 cannot be ruled out.
Finally, at Z = 0.008, taking into account the moderate overluminosity required by pulsational models to find a distance modulus in agreement with most empirical calibrations, we can conclude that our age determination for NGC1866 is close to 140 Myr.