3. Detection of RRab
3.1. The selection process
We will describe here the selection process of RRab stars. For the sake of homogeneity, we reprocessed the stars of the DUO field, using the same selection criteria as for the SAG field. The search for RRab in DUO has been performed through the BJ band.
A first selection was performed by calculating the about the mean magnitude () for each light curve. Stars with 8 were then searched for periodicity. This cut should select all variables with an amplitude 0.3 mag. A first estimate of the period was done with the string minimization method of Renson (1978). A more accurate period was then searched in a small window spanning 0.1 day around the first estimate, using a multi-harmonic periodogram method (Schwarzenberg-Czerny 1996). The next step was to fit a Fourier series (with up to five harmonics) to the folded light curve:
The about the fitted light curve was then calculated and all the stars for which have been selected as variable stars. At this step of the process the sample contained 7 000 variables.
The selection for RR Lyrae stars has been performed through the Fourier coefficients: for each variable we calculated the ratio of the amplitude of the first harmonic relative to the amplitude of the fundamental harmonic , and the phase difference . Fig. 2 shows a plot of versus for all stars satisfying and .
Several clumps lie in this figure. The most obvious one is located at 0.45, 0.7. This clump corresponds to RR Lyrae stars of Bailey type ab (hereafter RRab). For lower values of (i.e. for more symmetric light curves) we can distinguish two other clumps: one centred on (0.2, 3.2) and a shallower one at (0.15, 1.75), corresponding respectively to contact binaries and RR Lyrae of Bailey type c (RRc). A faint strip across the plot at is also visible and represents eclipsing binaries of Algol type. The selection of the RRab has been made with an ellipse centred on the clump (see Fig. 2) and finally a cut on periods () has been applied. The final sample contains 3 000 RRab.
The selected RRab may belong either to the MW or to the Sagittarius dwarf galaxy and we separated them through their distance modulus, assuming absolute magnitudes (Wesselink 1987) and MV=0.6 (Mateo et al. 1995). Furthermore, we take the mean color (V-I)0=0.460.06 after averaging over 27 RRab covering a wide range of metallicities from Table 1 of McNamara (1997). The apparent magnitude of each RRab has been estimated with the constant term of the Fourier series. Taking into account errors on the absolute magnitudes, apparent magnitudes, extinction and colors of RRab, the error on a single distance modulus is 0.3 mag in both field, the main source of uncertainty coming from extinction. Fig. 3 shows the histogram of distance modulus for both fields before and after correction for extinction.
The histograms were smoothed by estimating the mean magnitude every 0.1 mag in a 0.3 mag bin. Both histograms exhibit similar features: a broad bump centred on (8 kpc) corresponding to RRab of the MW, and a sharp bump centred on (24 kpc) representing RRab members of Sgr. According to current models of RRab densities in the Halo the Galactic contribution to the histograms for should be no more than 5-10 (Wetterer & McGraw 1996).
The 2D spatial distribution of all the RRab with a distance modulus greater than 16.3 is displayed on Fig. 4. This map includes 1 500 RRab.
The eastern and western box represents respectively the SAG field and the DUO field. The total area covered is about 50 deg2, and comprises the globular cluster M54 at which is associated to Sgr and located in its highest density region. The image of M54 is completely saturated until 1.5 half mass radius on our plates, thus we do not expect this globular cluster to contribute significantly to our RRab sample. The spatial distribution of RRab reveals a density gradient in the SE-NW direction. We also show in Fig. 4 the RRab discovered by the MACHO team (Alc97), confirming that these stars are the continuation of Sgr.
There are two steps where the completeness of the RRab sample might be affected: first, the detection of stars becomes difficult towards the Galactic Centre because of the increasing stellar density, and some RRab blended by a neighbouring stars are missed. Second, we might miss some RRab during the selection process.
3.2.1. Completeness of the extraction process
To quantify the loss induced by the first effect we simulated a set of 250 000 artificial stars with the same apparent magnitude than the detected RRab. These stars were then injected in small regions of 10 10´ uniformly spread over the fields and we tried to retrieve them with the same detection process as for the real stars. The lower panel of Fig. 5 displays the fraction of stars re-detected as a function of Galactic latitude.
Filled circles are stars injected onto the SAG field whereas crosses are stars simulated in the DUO field. The dispersion reflects mainly the dependence on Galactic longitude. One can see that while the fraction of stars re-detected in SAG stays at a high level (above 90) and varies slowly, this is not the case in DUO where this fraction drops abruptly to reach 40 at . The reason for the higher variation rate in DUO is that the stellar density gradient increases at a higher rate towards the Galactic Centre (the mean density gradient is 15 stars.arcmin-2.deg-1 in SAG and 25 stars.arcmin-2.deg-1 in DUO ). Another feature visible on Fig. 5b is that the loss induced by crowding is intrinsically higher in DUO than in SAG as can be seen in the range (10% offset). This can be explained by the lower resolution of the IIIaJ emulsion in DUO relative to the finer grained 4415 emulsion in SAG (Parker & Malin 1999). Furthermore, the lower extinction in DUO results in a higher number of stars detected ( in the overlap), increasing by this way the crowding.
3.2.2. Completeness of the selection process
Our selection process might not be able to detect variable of low amplitude. To check the dependency of completeness on amplitude we simulated a set of 1 000 RRab light curves with the same time sampling as the real ones, the Fourier coefficients have been taken from Simon & Teays (1982). The distributions in amplitude, magnitude and period (excluding integer fractions of a day) of the simulated light curves were chosen in a way to match the actual distributions of the detected RRab, and the phasing was uniformly distributed between 0 and . This set of simulated light curves was then injected in 100 regions of (uniformly distributed over the fields) from which we took the errors to deteriorate the light curves. These light curves were then reduced in the same way as the real RRab. Fig. 5a shows the completeness levels of our selection process as a function of amplitude for the two fields, averaged over 50 000 simulated light curves. The shapes of the completeness curves are nearly identical for both fields and the difference is not significant. Fig. 5a shows that the detection rate stays above 95 for amplitude 0.8 mag, and then drops abruptly down to 20 at amplitude=0.5 mag. These results signify that our selection process would detect almost all the RRab with an amplitude above 0.8 mag if these were point sources.
However, the completeness levels will differ between SAG and DUO because the amplitudes of the RRab measured in each filter are different, being more important on average for SAG than for DUO . This difference occurs because the color band of SAG peaks at shorter wavelength than the color band used for DUO whereas the amplitude of RRab decreases with increasing wavelength (Smith 1995). A least square fit between the amplitudes of 30 RRab in common in the overlap yielded the relation
where ADUO and ASAG represent respectively the amplitudes measured in DUO and in SAG . In order to construct a consistent density map, we will consider in the remainder of this paper only those RRab satisfying amplitude mag in SAG and amplitude mag in DUO , where 0.54 has been derived from the above relation (1). These cuts have been chosen both to ensure the largest sample as possible and to keep the completeness corrections at a manageable level. The corresponding corrections are 3.7% in SAG and 12.3% in DUO .
Periods: Some RRab are missed because their periods are close to an integer fraction of a day, this causes points of the folded light curve to accumulate in a narrow phase range. The fitted Fourier series is then poorly constrained over a large fraction of the light curve and some of these stars might lie outside the ellipse of our selection process (see Fig. 2). Monte-carlo simulations shows that we miss about of the RRab within the range 0.49d to 0.51d for both fields, corresponding to a total loss of .
3.3. Homogeneity between the fields
An important point to inspect for checking the consistency between the two fields is the overlap (Fig. 6). Open circles correspond to RRab detected in the SAG field with an amplitude(Bi) 0.60 mag. while crosses represent RRab detected in the DUO field and having an amplitude. The dashed lines indicate the limit of each plate, their inclination is due to a slight tilt between the two plates. Note that this overlap applies to the reference frames and is not necessarily constant from plate to plate. Fig. 6 reveals that most of the RRab are detected independently in the SAG field and in the DUO field. However, some RRab are not detected twice and it is important to understand the reasons why these stars are missed by one of the fields:
Most of the missed RRab will therefore have no statistical incidence and should not bias the density map. The only concern is for the greater sensitivity of the DUO field to crowding. However this effect should be lowered by the crowding correction. Turning now to the western edge of DUO we re-detect 7 RRab out of the 8 detected by the MACHO team in our field (disregarding two RRab located close to the edge). This is a satisfactory result.
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000