Astron. Astrophys. 329, 522-537 (1998)

5. Analysis

In this section, we list the properties of the expected signal used to differentiate statistical fluctuations and variable stars from microlensing signals. Then we describe the parameters introduced to characterise a light curve, the simulated light curves and the selection criteria. Each season and each CCD are analysed independently.

5.1. Properties of the signal

The analysis is based on four properties of the signal :
- the magnification occurs simultaneously in both colours,
- a given star is magnified only once,
- the magnified stars are located representatively in the observed colour-magnitude diagram,
- the luminosity of a star is the same before and after the magnification.

In order to remain sensitive to events affected by finite source size or blending effects, generated by multiple source or deflector, the analysis is totally independent of the shape of the signal.

5.2. Definition of a variation

The luminosity m of a star is defined as where the flux is in ADU. We call a variation a set of consecutive points above, or under, the base luminosity. This base luminosity is computed as the most probable value of the luminosity histogram for the studied star.

In order to eliminate unreliable measurements, we apply a 3-point median filter. Let be the measurement error computed as described in Sect. 3.3. A variation begins with 1 point located at least 1 on one side of the base luminosity; it ends if 1 point is located at least 1 on the other side of the base, or if 3 consecutive points are at less than 1 on the same side of the base. This variation is validated if it contains at least 5 points. It is called positive if the points are above the base luminosity, negative if not. The definition is loose in order to detect very low or very short signals of any shape. Typically, a light curve contains between 50 and 150 variations per colour, mostly statistical fluctuations as almost all light curves are flat.

5.3. Characterisation of a light curve

For each variation, we compute with

where N is the number of measurements inside the variation, is the filtered luminosity at point i, its error, is the base luminosity and is the probability to have the given with N consecutive points. We take into account the fact the N points are located on the same side of the base with the factor .

We classify variations according to : the main variation has the largest and is the most improbable one.

We also define a correlation coefficient between red and blue light curves: where is the red luminosity and the blue one. We sum over all pairs of red and blue points satisfying the following conditions:
- the points are consecutive (i.e. with a temporal difference smaller than 20 minutes),
- measurements are taken outside the main two variations.

5.4. Simplified simulation

We superimpose a microlensing magnification on the observed light curves. We assume the theoretical shape of the signal generated by a point-like deflector and source. In this way, we take into account the time sampling, the luminosity distribution, the seeing effects, the sky background, the absorption variations, the statistical errors on luminosity and other systematic effects due to photometric reconstruction. For each magnified measurement, we re-compute the error accordingly.

A signal is characterised by the time of the maximum luminosity , the duration and the impact parameter . is taken randomly from the first day of the campaign minus 5 days to the last day of the campaign plus 5 days. With this overflow, we take into account edge effects of the considered time interval. is taken randomly between 0 and 2. An impact parameter of 2 corresponds to a maximum magnification of 6 % and we are not sensitive to smaller enhancements. is taken between 0.01 day (15 minutes) and 100 days. Because we ask for at least 5 points in a variation and we take an image every 20 minutes, we can not detect shorter events. 100 days is the typical duration of a signal generated by a deflector of 2 M : this is already beyond the sensitivity of the present search. To have a good accuracy on our efficiency to detect short events, we generate a uniform distribution in . Over one million events were simulated.

5.5. Selection criteria

The first required property of the signal (magnification in both colours simultaneously) is satisfied by the following requirements:
- the most significant variations in red and blue must correspond to an increase of the luminosity,
- overlap in time exists between the blue and red most significant variations.

The property of uniqueness (flat curve outside the main variations) is tested with two criteria:
- the of the main variation has to be at least three times larger than the of the second most significant one in both colours,
- the correlation coefficient has to be lower than 0.45.

In order to determine our detection efficiency, we have applied our selection procedure simultaneously on simulated and observed light curves.

Figs. 6 and 7 present the distributions of these parameters for observed and simulated events. The cut on is useful to eliminate statistical fluctuations, and is very efficient on cleaned data. On most of the eliminated simulated events, we had detected noise because the signal was too low, too short or occured at a badly sampled period. Not asking for any minimum value of the , we remain sensitive to the lowest ( 2.0) or shortest signals ( minutes) if they are well defined. The second cut is useful to eliminate intrinsically variable stars.

Fig. 6. Distribution of the ratio of red and blue ( of the second most important variation divided by the of the main variation). Up : data 91-95 (all stars, after previous cuts); down: simulated events (stars of 6 CCDs, 1 season, after previous cuts).

Fig. 7. The correlation coefficient after the previous cut. Up : data 91-95 (all stars, after previous cuts); down: simulated events (stars of 6 CCDs, 1 season, after previous cuts).

To test the third required property of the signal, we plot the distribution of microlensing candidates on a colour-magnitude diagram, from simulated light curves. We retain stars with:
-
- for SMC stars and for LMC stars
- or for LMC stars only.

Fig. 8 presents these diagrams for the LMC and the SMC. We observe that candidates are concentrated in two regions of very low stellar density. The first one corresponds to the area of supergiant stars. Long period variable stars are concentrated here and can exhibit a unique variation in one season, similar to the light curve presented in the lower part of Fig. 9. Moreover, these eliminated stars present a very long main variation ( a few months) compared to the sensitivity range of the CCD programme.

Fig. 8a and b. Up: the colour-magnitude diagram of the LMC. Stars which satisfy previous criteria are symbolised by points (simulated signals, 1 CCD, 1 season) or by stars (data, all CCD, all seasons). Only stars inside the area limited by the dotted line are kept. Down: same plot for the SMC.

Fig. 9. Light curves of stars eliminated by the cut on the the colour-magnitude diagram. Up: a star located in the upper part of the main sequence; down: a star located in the super giant area.

The second area is the upper part of the main sequence. The light curve of one candidate, presented in the upper part of Fig. 9, is a priori compatible with the expected signal in spite of an asymmetric shape. Only from the fact that we get a tenth of candidate stars with comparable light curves and similar positions on the colour-magnitude diagram are we able to eliminate them. This new type of variable stars has also been observed by our competitors (MACHO group: Alcock et al, 1996b).

To test the last required property of the signal, we need at the very minimum one point before and one after the main variation. This cut strongly affects long duration events, but we do not expect to have any sensitivity for such long events.

Table 3 summarises the efficiency of the cuts on simulated curves and the corresponding rejection on data.

Table 3. Efficiency of the cuts applied to select microlensing events: 1) positive variations and temporal overlap, 2) cut on the , 3) correlation coefficient, 4) colour-magnitude diagram and 5) at least one point before and after the variation. For the data, we give the number of stars remaining after the cut indicated in the row and the corresponding percentage of stars verifying this criteria. For the simulation, we give the percentage of stars verifying all previous cuts and the bracketed percentage is the efficiency of the cut indicated in the row. The data correspond to all observed light curves, all seasons. For the simulation, we use the 1,074,401 generated events with , , ).

A global efficiency of 16 % on the simulated curves can seem low, but a large majority of the simulated signals have a duration lower than a few hours. So they have a high probability to occur during the day or a badly sampled period. For the SMC observations, the detection efficiency is near 40 % for events of 3 days and lower than 5 % for events shorter than 2.4 hours (with ). Moreover, signals with large impact parameters are detectable essentially on the less numerous bright stars which have more accurate photometric measurements.

At this stage, we end the automatic analysis. We remain with 35 light curves, well distributed on the colour-magnitude diagram; 34 have a signal of a few hours and one exhibits a long variation.

5.6. Individual inspection of remaining light curves

Fig. 10 shows few typical events chosen amongst the 34 first candidates. They coincide marginally in time, are achromatic, highly asymmetric and occur during the one night.

Fig. 10. Light curves of stars selected by the automatic analysis with a short signal. Error bars are omitted to improve readibility.

Red and blue images are taken with the same CCD camera and the flat fields and offsets taken in the afternoon are used to reduce images of all the following night. All the flat fields or offsets used for the night of a given detected signal contain pixels with abnormal high or low values. These pixels are located exactly at the position of the candidate at the maximum of the detected luminosity increase. During a night, the source moves on the CCD because of differential flexions between the telescope and the guiding system. This movement simulates more or less a microlensing signal when the trajectory of a star crosses one or several bad pixels. Because of the under-sampling of the images, it is difficult to detect this problem by comparaison with the shape of the PSF. We found that all the 34 short signals have been caused by electronic problems and have rejected them.

The last candidate is centered on the clump of the red giants and has a long variation during the first season. But the same star has also a variable light curve during the second season and can thus be identified as a long period variable: this excludes a gravitational magnification. Fig. 11 presents its light curves.

Fig. 11. Light curves in 91-92 (top) and 92-93 (bottom) of the candidate with a long variation. Error bars are omitted to improve readibility.

5.7. Number of detected events

None of the 360,000 light curves verifies the properties of a gravitational magnification. The analysis contains no hypothesis on the shape or the duration of the signal. The identification of events generated by flat field or offset problems (Fig. 10) exhibits a contrario our sensitivity to very short, very low or very irregular signals.

Other analyses have been attempted. One uses light curves available in one colour only: the corresponding source stars have a low luminosity. We ask for a signal longer than one night in order to prevent flat field or offset problems. Another one used the mean luminosity per night. We are then more sensitive to quite low but long magnifications. None of these analyses exhibits a microlensing candidate. These last two analyses have not been used to compute our experimental limits on the halo because their contribution to our sensitivity would remain small.

© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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