4 Magnitude estimation

Figure 4: Luminosity functions in blue (full line) and in red (dashed line) at the epoch JD = 2448678.3. The upper histograms a) correspond to the magnitudes estimated for all the stars with DAOPHOT. The lower histograms b) correspond to the magnitudes estimated as described in Sect. 4.1 for the selected variable stars

Whereas the pixel method of analysis is able to detect variable stars beyond the crowding limit, it does not measure photometry - total flux - of these objects, that can be blended or even unresolved on part of the light curves. Obtaining their photometry would give a first indication of the type of the variable stars. Hence in this section, we associate a magnitude and colour to each flux measurement.

   
4.1 Pseudo-aperture photometry

As discussed in Paper II, the flux of the super-pixel is composed of the fraction of the flux of the star plus the local background (from sky and undetected stars). For our sample of variable stars, we can presume that there is a star within the corresponding super-pixel and that its flux significantly contributes to this super-pixel, at least at the maximum of the variation. Because of the crowding conditions, standard background estimates (circular annulus for example, see Stetson 1987) fail and cannot be used in an automatic way. Hence, we choose to perform a pseudo-aperture photometry as follows. For an image taken in the middle of the period of observation (JD 2448678.3) and with an average seeing, we use the PSF fitting procedure of DAOPHOT (Stetson 1987) to measure the fluxes of the resolved stars, and the background below them. This thus gives a local estimate of the background that is the less affected by the crowding of the field. Then for each selected super-pixel we look for the detected star that is closest. The background estimate associated with this star is supposed to be the same as the one present below the variable star (and is even identical if the variable stars are resolved on this reference frame). This background is subtracted from the super-pixel flux. This flux is then corrected for the seeing fraction and converted into a magnitude, corresponding to an isolated star.

Whereas this definition of the sky-background is rather robust to the crowding conditions, the magnitude estimation is not necessary so, as some additional flux (stellar background) could contribute to the super-pixel due to neighbours. We thus quantify the blending with the ratio $\phi _{\rm c}/\phi _0$ computed as follows: the flux $\phi_{\rm c}$ is the averaged value computed along the light curve of the central pixel of the super-pixel, and the flux $\phi_0$ is a similar average of the 8 surrounding pixels within the super-pixel. The behaviour of this parameter is described in the Appendix.

Optimised star detection with DAOPHOT allows to detect $135\,098$stars in blue and $142\,870$ stars in red. The corresponding luminosity function computed with DAOPHOT for all the stars present on the studied field, exhibited in Fig. 3a, shows that the star detection is in first approximation complete down to magnitude 18 in red and 19 in blue. Figure 3b shows that the magnitude distributions at the same epoch for the selected variable stars peak at the bright end, whereas the stars are redder than average with $B_{\rm EROS}-R_{\rm EROS} \simeq 2$. Whereas it is difficult to compute our detection efficiencies as no reliable theoretical distribution of variable stars is available, it is clear that we do not detect a population of variable stars unresolved at minimum, even though we do detect a tail of this distribution with very dim stars in at least one colour.

Figure 5: CMD at JD = 2448678.3: small dots correspond to the stars detected with DAOPHOT, symbols to the 631 selected variations. The different symbols correspond to different ranges of the $\phi _{\rm c}/\phi _0$ ratio: filled symbols correspond to stars with a high S/N ratio and not affected by crowding. The superimposed isochrones (full lines) are adapted from Bertelli et al. (1994) to the EROS system (Grison et al. 1995) with a foreground extinction $A_{B_{\rm EROS}}=0.54$ and $A_{R_{\rm EROS}}=0.37$, deduced from reddening E(B-V)=0.15 measured by Schwering $\&$ Israel (1991) with the extinction law from Cardelli et al. (1989). They correspond to LMC metalicity (Z=0.008), helium abundance (Y=0.25) with ages of $\log$(age) = 7.4, 8.4 and 9.4. The dashed lines show the uncertainties introduced on each isochrone by the photometric transformation

   
4.2 Colour-magnitude diagram

Figure 4 displays the position of the 631 variable stars in the CMD. This CMD has been obtained with the resolved stars detected with DAOPHOT in both colours. The conversion of the EROS magnitudes into a standard system has been studied by Grison et al. (1995), but introduces significant systematic errors especially for stars with $B_{\rm EROS}-R_{\rm EROS} > 1$. We hence choose to work in the EROS system. The crowding limit prevents the detection of stars with $R_{\rm EROS}>19$. The main sequence and red clump are clearly identified. Very little variations of the CMD are observed from one chip to another suggesting a uniform stellar population across this field. The vast majority of the detected variable stars lies in the red part. A visual inspection of the corresponding light curves shows that they are compatible with LT$\&$LPV. This catalogue offers the perspective to systematically study the later stages of stellar evolution within the LMC Bar. However, the isochrones superimposed in Fig. 4 illustrate the difficulty to classify variable stars with the sole CMD, as it is impossible to disentangle their age and mass for a given metalicity. The photometric system made it also inappropriate for further interpretation with this sole data. Due to their complexity and the difficulties to model them, the LT$\&$LPV are better characterised in the IR, as addressed in a subsequent paper. Some "bluer'' objects have also been detected and they will be classify when we study their periodicity.

As shown in Fig. 4, variable stars not significantly affected by crowding ( $\phi_{\rm c}/\phi_0>3$) lie in areas of the colour-magnitude diagram corresponding to stars expected variable. Those affected by blending and crowding will have to be treated with caution. In the CMD areas where the larger number of LT$\&$LPV have been detected, in the magnitude ( $R_{\rm EROS}=14.6-15.8$) and colour ( $B_{\rm EROS}-R_{\rm EROS}=1.7-2.3$) ranges, about 17% of the stars exhibit a variation detected with our analysis.

It is also clear that the vast majority of the detected variable stars are above the crowding limit. The few outliers that can be noticed correspond to stars unresolved in at least one colour, but their number does not exceed 5% of the total. In terms of microlensing, this means that events due to unresolved stars in the LMC will not be significantly contaminated by the bulk of variable stars. In further galaxies, like M31, variable stars will be a more troublesome affair, and will have to be carefully studied (e.g. Crotts & Tomaney 1996). However, high amplification microlensing events are far less likely to be mimicked by an intrinsic variation (Ansari et al. 1999), and will allow to probe possible biases introduced by variable stars.