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Astron. Astrophys. 324, 843-856 (1997)

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5. Light curves

The pixel method relies on the inspection of pixel light curves. Light curves are graphs of the variation of pixel intensities. Elementary pixels are small ([FORMULA]), which is very useful to get a good geometric alignment. However, elementary pixels undergo strong fluctuations due to seeing variations that hamper detection of truly variable stellar objects. For this reason we replace each pixel by a super-pixel, as explained in Sect. 2. A convenient size for the super-pixel, in vue of the average seeing of [FORMULA], turns out to be [FORMULA], wich corresponds to super-pixels built with [FORMULA] elementary pixels.

Using super-pixels provides a substantial gain in stability. Fig. 9 shows maps of the relative fluctuation along the light curve of elementary [FORMULA] wide pixels, and of [FORMULA] wide [FORMULA] super-pixels of field A (Notice that there are as many super-pixels as elementary pixels). On elementary pixels, the dispersion is below 1% on most of the field. For super-pixels, the dispersion drops down to 0.3% in average and even reaches a level below 0.1% in the most stable regions, as announced earlier. It remains everywhere around twice the photon noise.

[FIGURE] Fig. 9a and b. Maps of the relative fluctuation on field A for [FORMULA] elementary pixels (upper map), and [FORMULA] super-pixels (lower map) .

To compare in more detail the super-pixel fluctuation to the photon noise, we have computed along the light curve of each super-pixel the [FORMULA] of the difference between the intensity on the current image and its average in time. In Fig. 10, we display the distribution of this [FORMULA] for the super-pixels of field A, using two different seeing selections. The error [FORMULA] entering the [FORMULA] is chosen in such a way that the maximum of the distribution of the [FORMULA] coincides with that of the ideal Poisson law. This is achieved for [FORMULA] where [FORMULA] is the statistical photon noise. The true distribution shows non-poissonian tails. Clearly there are non-statistical contributions to the fluctuations and a comparison between Fig. 10 a and Fig. 10 b shows that they are largely due to seeing variations. Further work is in progress to cope with the latter. This non poissonian behaviour is also responsible for the fact that, in going from pixels to super-pixels, one gains less than the factor 7 expected if fluctuations were of pure statistical origin.

[FIGURE] Fig. 10. The distribution of [FORMULA] along light curves of field A

We have made the same study replacing super-pixels by a PSF weighted average. The fluctuation is twice larger than with super-pixels, and the tails due to seeing variations in the [FORMULA] distribution are much larger.

Fig. 11 illustrates the considerations above with the light curve of a stable super-pixel, keeping only the frames with seeing between [FORMULA] and [FORMULA]. Super-pixel intensities are in ADU/s (1 ADU/s on a 2.1 [FORMULA] super-pixel corresponds to a surface magnitude [FORMULA]). The R.M.S fluctuation along the light curve is 0.045 ADU/s, to be compared with the average photon noise which is around 0.04 ADU/s. If one keeps all points, irrespective of the seeing, the RMS fluctuation becomes 0.065 ADU/s. The error bars correspond to 1.7 [FORMULA], that is around 0.07 ADU/s in average.

[FIGURE] Fig. 11. The light curve of a stable [FORMULA] super-pixel. The intensity per super-pixel is in ADU/s ([FORMULA]).

With this level of stability, we are able to clearly see variations at the level of a few percent as is apparent from Fig. 12. Let us stress the following features of this figure.

[FIGURE] Fig. 12. An example of a variable object.
  1. This light curve shows two clear variations, it is a variable star, not a microlensing.
  2. On graph (b), only points corresponding to a seeing between [FORMULA] and [FORMULA] have been retained and the light curve appears much smoother than on graph (a).
  3. After seeing selection (graph (b)), the first variation can clearly be seen, because of its coherence in time, although it is only about 0.5 ADU/s, that is about 5 times the average error bar in this period.

We see that the selection criteria we have introduced in our Monte-Carlo simulation in Sect. 2.2 (3 points above  [FORMULA] and one of them above [FORMULA]) are indeed realistic. However our present thresholds are much higher, because these criteria would be sufficient if microlensing were the only possible source of variations. This is of course not the case and variable stars are far more numerous. If we used only the criteria of our simulation, we would be swamped by variable objects. Therefore, to isolate microlensing events we have to build filters which reject most of the variable objects but not the microlensing events satisfying our criteria. There are many conditions that can be added, such as:

  • the usual conditions of unicity, symmetry and achromaticity
  • the quality of fits by a Paczyski curve
  • limits on the duration of events expected from MACHO's with reasonable masses compared with what is expected from simulations (see Fig. 1).

We are working on that. We will be in a better position after the 30 observation nights we shall have in autumn 1996. Although these nights will be too few and too scattered to allow detection of new events, they will allow to constrain efficiently fits of events that occured in 1994 and 1995

Even events that overshoot by far our criteria would have been extremely difficult to detect by monitoring resolved stars. This is illustrated in Fig. 13. The two dimensional surface plots (a) and (b) map the intensities of elementary pixels around the centre of a detected variation. Plot (a) corresponds to the minimum of the light curve and plot (b) to the maximum. Most structures appear similar on the two plots, which means that they correspond to real structures of M 31. They are the surface brightness fluctuations of Tonry & Schneider (1988). At the centre however, a tiny bump, barely visible on graph (a), has grown into a clear PSF-shaped peak on (b). This tells us that we are really looking at a varying stellar object, barely detectable as a resolved star.

[FIGURE] Fig. 13. Appearance of a star. The vertical scale is the intensity per elementary pixel, in ADU/s

Variable stars are interesting in their own right. Numerous variable objects such as the preceding ones have been detected, but we are only beginning to analyse their nature. Fig. 14 shows the light curves of two objects, one of which is probably a cepheid, and the other a nova. We have a host of other cepheid candidates and five novae with peak magnitude and rate of decrease similar to the one shown on Fig. 14, and very similar to the M 31 novae quoted in (Hodge 1992).

[FIGURE] Fig. 14. Likely cepheid and nova.

We also see variations compatible with microlensing (about 20). However at this stage, we are not in a position to claim that we have seen microlensing events for several reasons. First, our lever arm in time is not sufficient to be sure that the variations do not repeat, and even in some cases, to be sure that events are really symmetric. The situation will improve with the 30 nights we expect in autumn 1996. Second, we have not yet analyzed the blue light curves, therefore we cannot yet test achromaticity. Fig. 15 shows one of these light curves. The Paczisky curve on Fig. 15 corresponds to a star of absolute magnitude [FORMULA] amplified by a factor 6 at maximum and with an Einstein time scale [FORMULA] days. These number are not well determined because of a parameter degeneracy for high amplification events (see for instance Gould 1995), which is the case of most events we can detect. A time scale and a maximum amplification twice as large associated with a star twice fainter would fit just as well. However the time scale cannot be much shorter, because the star should be brighter and would be seen more clearly before the lensing begins. The effective time [FORMULA] is 19 days if one measures it between real points of observation where the signal to noise ratio is higher than 3, and 40 days if one refers to the time during which the Paczisky curve remains at 3 [FORMULA] above the background. This effective time is a powerful mean to eliminate fake microlensing events: our simulation tells us that [FORMULA] should be smaller than 60 days for lenses with masses around 0.08 [FORMULA]. The numerous time gaps we have in the observations make it difficult to find short events, and it is important to our approach to have as few gaps as possible in the time sampling.

[FIGURE] Fig. 15. A possible microlensing event. The solid horizontal line is the basis level of the super-pixel intensity (in ADU/s), the dashed line lies [FORMULA] above and the dotted line [FORMULA] above.
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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998