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

6. Complete simulation and detection efficiency

In the previous section, we have used a simplified simulation in order to tune analysis cuts. We have seen however that finite source size and blending effects are non negligible, especially for low mass deflectors. Here we describe the simulation of these effects.

6.1. Finite source size

The finite size effect is characterised by the relevant parameter where is the source radius ( Sect. 2.2). The distribution of U depends on the halo model, through the distribution of stars along the line of sight. To get an efficiency taking into account finite size effects but which is independent of the halo model, we prefer to use the parameter () with

The efficiency takes naturally into account the distributions of stellar luminosities and radii because the simulation uses the observed light curves; is then a function of the two variables and only, which are independent of the model. These variables are correlated: a faint star in the main sequence is hardly detectable if magnified because of its large photometric error and is also little affected by the finite size effect because of its small radius.

The relevant parameter for the shape of the microlensing with source size effect -and thus for the detection efficiency- is U (). So the simulation is done generating and U. Our detection limit corresponds to a maximum magnification of 6 %, equivalent to in the point-like case (). As is shown on Fig. 12, this means that we are sensitive to signals with and . We are thus able to detect impact parameters larger than 2.0 when the projected radius of the source in the deflector plane is adapted.

Fig. 12. Simulation of the finite source size effect: the shaded () domain corresponds to maximal magnifications greater than 6 %.

We generate uniformly between 0 and 4.5 and U uniformly between 0 and 5.6. The stellar radius is computed from the B and V absolute magnitudes and the results of Allen, 1963; this raw estimate is sufficient here. In the same way we sample uniformly in to have a good sampling on the efficiency for the smallest durations , so that we simulate more events for stars with smaller radius. Fig. 13 presents the stellar radius distribution and the number of generated event per star as well. This number has been taken such that the efficiency has roughly the same statistical error in each bin of ().

Fig. 13. Distribution of the selected stars radii after simulation of the finite source size effect and only one generation per star. The numbers above the histogram indicate the number of generations per star which are used to compute the final efficiency.

Because of the dependence on , the finite source size effect of the source is negligible when the deflector is more massive than M . The source size effect produces a small increase of the efficiency when the deflector mass is between and M and a large decrease for lighter lenses.

6.2. Blending effect

In order to simulate the blending effect, we have produced synthetic images containing stars with luminosities up to a factor 15 below our detection threshold. We used luminosity functions from Elson et al, 1994, colour-magnitude diagram shapes from Hardy, 1984 and star densities from Vaucouleurs et al, 1970. Then we use our detection and photometry programmes on those simulated images. The detected stars of the LMC or SMC synthetic images have features comparable to that of the observed stars. A synthetic image with about 15,000 detected stars is obtained by including about 300,000 stars in the simulation.

Then, we study the apparent magnification as a function of the generated one. For each position where a star (with flux ) has been detected, we search for the most luminous generated star (star 1 with flux ) located less than 1 arcsec from . One arc-second corresponds to 2 pixels on the template image, i.e. one pixel on the standard images (the best seeing on our images is 1.3 arcsec). Likewise we search for the second most luminous generated star (star 2 with flux ) located in the same area. So we get with where represents the fraction of the star flux included in the measurement of . Fig. 14 presents the distributions of and of as a function of . The large dispersion at low flux is due to large photometric errors. The first distribution is not centred on zero, confirming that blending effects are more important at low luminosity. The second distribution is better centred, so we verify that it is sufficient to consider the two brightest generated stars close to the detected one and that . There remains a bias at the lowest flux.

Fig. 14. Distribution of (up) and of (down) as a function of ; is the reference flux of the detected star and is the flux of the generated star (we use here the red band pass luminosities).

If the main star is actually magnified by a factor A, the observed magnification, taking into account blending, is: So,

The fraction C only depends on the flux of the magnified star, and not on the magnification; this property has been tested for a large range of magnifications and seeings. So the blending is modeled in the same way for all images and all impact parameters.

We treat independently the cases of magnification on stars 1 or 2: efficiencies are first computed assuming that one of the two stars is magnified, then we add the two efficiencies. For each colour, the fraction C is modeled as a function of the type of the two generated stars (main sequence or red giant) and of the detected one.

The influence of blending on the width of the signal is illustrated in Fig. 15. For the first star, the fraction C is near 1 and is weakly affected. For the second star, the width of the signal is two to five times smaller than the generated signal and then will be hardly detectable.

Fig. 15. Distribution of the fraction ( / ) as a function of the fraction for stars 1 and 2 in the red passband. The reconstructed is obtained by fitting a point-like microlensing shape to the light curve.

The impact of blending on the detection efficiency is always low. If is larger than one day, the loss of efficiency on star 1 is more than compensated by the contribution of star 2. For smaller durations, we are rarely able to detect a magnification of the second star; the total loss of efficiency caused by blending is about five times lower than the loss caused by the finite source size.

6.3. Detection efficiency

The impact of a binary lens or a binary source on the efficiency is assumed to be negligible (Mao et al, 1991), (Griest et al, 1992). We compute the efficiency of our detection algorithm by the analysis of simulated microlensing light curves. Our simulation includes the finite source size and blending effects.

Fig. 16 presents the efficiency as a function of and of . The shape depends on the observed galaxy because the time sampling and the distributions of observed luminosities, colours and radii differ. Complete efficiencies useful for numerical computations are presented in the appendix.

Fig. 16. Detection efficiency as a function of and ; the finite size and blending effects are taken into account. The simulated light curves were generated with , , days, and normalised to .

© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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