CANDIDATE MICROLENSING EVENTS FROM M31 OBSERVATIONS WITH THE LOIANO TELESCOPE

The data have been collected at the 152 cm Cassini telescope located in Loiano (Bologna, Italy). We make use of a CCD EEV of 1340 × 1300 pixels of 058 for a total field of view of 13' × 126, with a gain of 1.0e⁻/ADU (this value has changed with respect to that of the first season because of some electronics problems) and low readout noise (3.5e⁻ pixel⁻¹). We have been monitoring two fields of view around the inner M31 region, centered respectively in R.A. = 0^h42^m50^s, decl. = 41°23'57'' (North) and R.A. = 0^h42^m50^s, decl. = 41°08'23'' (South; J2000), so to leave out the innermost (∼3') M31 bulge region, and with the CCD axes parallel to the south–north and east–west directions so to get the maximum field overlap with previous campaigns. This second-year campaign lasted 50 consecutive full nights, from 2007 November 11 to December 31, with a fraction of good weather of almost 60%. In order to test for achromaticity, data have been acquired in two bandpasses (similar to Cousins R and I), with exposure times up to 6 minutes per frame. Overall we collected about 410 (280) exposures per field over 31 nights in the R (I) band.⁹ Typical seeing values are ∼2'' (somewhat worse than during the first season). Sky flat frames were taken whenever possible so as to build a master flat image (per filter), and standard data reduction, including bias subtraction, was carried out using the IRAF package.¹⁰ We corrected I filter data for fringe effects. The analysis presented in this paper is based on the 2007 season data only.

As for the preliminary image analysis, we closely follow the strategy (the "pixel-photometry") adopted by the AGAPE group (Ansari et al. 1997; Calchi Novati et al. 2002), wherein each image is geometrically and photometrically aligned relative to a reference image. To account for seeing variations, we then substitute the flux of each pixel with that of the corresponding 5 pixel square "superpixel" centered on it (whose size is chosen so as to cover most of the average seeing disk) and then apply an empirical, linear, correction in the flux, again calibrating each image with respect to the reference image. The final expression for the flux error accounts both for the statistical error in the flux count and for the residual error linked to the seeing correction procedure. Finally, in order to increase the signal-to-noise ratio (S/N), we combine the images so to get one data point per night per filter.

We evaluate the calibration zero point for the instrumental magnitude versus standard (RI)_C magnitudes by using a sample of secondary reference stars (Massey et al. 2006). We find for R and I bands data C_R = 23.1 and C_I = 22.7, respectively (the reported values corresponding to the standard magnitude for an object with an instrumental magnitude of 1 ADU s⁻¹).

As for the first step, we closely follow the strategy outlined in Calchi Novati et al. (2003, 2005). To begin, we detect flux variations along light curves using the L estimator (we ask L > 50 to get rid of too small S/N variations). Each given flux variation enhances a signal over a few pixels (a "cluster") around the central one. We make use of the "Q" estimator to characterize the significance of the selected flux variations.¹¹ We fix a lower threshold Q_th = 50. At this stage, therefore, we have to shift from the light-curve analysis (the estimation of L and Q), to an analysis based on the spatial information across the CCD in which we have to distinguish, separate, and pick up the flux variations associated with each different cluster. This search is somewhat biased in favor of light curves showing a single variation (in particular, all short-period variables are in principle excluded). This first step is carried out using the (more numerous and less noisy) R-band data only.

The aim of the second step is to single out the variations whose shape is compatible with that of a Paczyński light curve. To this end, we use a series of selection criteria. As a starting point, we perform a seven-parameter Paczyński fit in the two bands simultaneously.¹² As an output, we retain the following parameters: the baseline levels; the time of maximum magnification, t₀; the full width at half-maximum duration, t_1/2 (proportional to the Einstein time, t_E, multiplied by a function of the impact parameter u₀); the flux deviation at maximum (with respect to the baseline), which we convert to magnitude and denote ΔR, together with the color of the variation, R − I (ΔR is proportional to the source flux, ϕ*, multiplied by a function of the impact parameter); and the reduced χ² of the fit. We make use of the full parametrization of the Paczyński fit, looking for the Einstein time, the magnification and the unlensed source flux values, even if the intrinsic parameter degeneracy (linked to the fact that the source is not resolved), in most cases, does not allow one to accurately estimate the single parameters. As a selection criterion, we ask χ²/dof < 10. This rather high threshold is motivated by the necessity to handle light curves contaminated by low-level noise of non-Gaussian nature that can be attributed in particular to nearby blended intrinsic variables.

As a second test on the shape, we ask for the bump to be suitably sampled by the observed data points. We split our analysis on the basis of a more or a less demanding requirement, so that we are going to refer to set "A" and set "B" of candidates, respectively. The details are given in Appendix A.

As a final test on the shape, we look at the characteristics of the detected flux variations. The two relevant parameters are the event duration, t_1/2, and the flux deviation at maximum, ΔR. In order to appropriately delimit these parameter spaces, we have to balance for the efficiency, the expected event characteristics and the risk of contamination of the background of variable stars. We introduce a cut to exclude too faint variations, too noisy, and therefore difficult to distinguish against the variable contamination. As a selection criterion, we ask for ΔR < 21.5. As for the duration, we do not expect microlensing events to last more than 10–20 days (Section 4.1). Besides, we expect long-duration variations to be heavily contaminated by intrinsic variable signals. However, we prefer not to introduce any cut for this parameter allowing for long-duration candidates. As detailed below, all of these are rejected in the following steps of the analysis anyway.

Up to now the analysis is based on the light curve pixel-photometry only (besides the initial "cluster" analysis), in particular we do not make use of the PSF of the flux variations we are looking for. In this respect, our approach is completely different from that based on the difference image analysis. As it is, however, this approach suffers from a high risk of contamination by spurious variations. To reject them in an automated way, as a third step we perform an extremely rough difference image analysis. The underlying rationale is that, whenever we detect a variation on a light curve, in order to retain it we want to "see" a corresponding well-shaped PSF when we look at the image difference of the maximum magnification minus the baseline. Further details are given in Appendix B.

As a fourth and final step, we probe the surviving variations against the background of variable contamination. Our limited baseline, 50 days, does not allow us by itself to carry out this programme. For this reason, we make use of the three years baseline of the POINT-AGAPE data set.¹³ The rationale is that we want to reject flux variations that show variability along the much longer INT baseline with a comparable flux deviation to that detected on our OAB data. As a first step, given the OAB-detected variation, we look for the corresponding pixel within the POINT-AGAPE data set.¹⁴ To probe variability along the INT light curve, we use a Lomb periodogram analysis. As an estimator, we use the power "P_R" as defined in Numerical Recipes (Press et al. 1992). Second, we test whether the INT and the OAB flux deviations are compatible. To this end, given a relative flux calibration between the two data sets, first we rescale the OAB flux deviation; then we evaluate the difference of the observed flux deviations (the rescaled OAB minus the INT one), normalized by the INT error. We take this quantity, which we define δ(Δf), as our estimator. As a selection criterion, we ask for P_R < 20 or δ(Δf)>5.

We consider the bulge and the disk populations of M31 as sources, both M31 bulge and disk stars as lenses (we refer to these events as "M31 self-lensing"), and MACHOs in both the M31 and the Milky Way dark matter halos. We assume an M31 distance of 770 kpc.

As a model for the M31 luminous components we take the Kent (1989) bulge–disk decomposition, including the bulge ellipticity, and with the missing information of the vertical distribution of the disk modeled with a sech² law and scale height of 300. For both halos, we assume a spherical isothermal distribution with a core radius a = 5 kpc.

The peculiarity of M31 microlensing is that we look for flux variations of unresolved sources. The issue of estimating the number of available sources is therefore particularly delicate. To this end, we consider the value of the M31 surface brightness (as given by Kent 1989) and the underlying luminosity function. As has already been remarked (Ansari et al. 1997), it turns out that only luminous sources (about M_I < 2) are expected to give rise to detectable events. In the most crowded region, this may sum up to hundreds of available source stars per pixel (this can be considered as a completely blended situation with respect to, for instance, studies toward the LMC). As for the luminosity function that we use to characterize the sources, it is worth stressing that we are most interested in its bright end (even if we need the information over the complete magnitude range for the normalization), whereas, for the mass function that we need for the luminous lenses, we are rather interested into the opposite tail.

We have made use of the IAC-star software (Aparicio & Gallart 2004) to build a synthetic luminosity function for the M31 bulge, following the procedure outlined in Bozza et al. (2008), in particular as for the metallicity distribution (Sarajedini & Jablonka 2005), with the difference that we have now used, as a mass function, a power law ξ(m) ∝ m^−α with index α = 1.33 up to 1 M_☉ and α = 2 above. For bulge lenses we have for consistency used the same mass function with upper bound fixed at one solar mass. For the disk luminosity function, as in Calchi Novati et al. (2005), we make use of the local neighborhood data obtained by Hipparcos corrected at the bright end (Perryman et al. 1997; Jahreiß & Wielen 1997). For the disk lens mass function, we follow as well the local determination (Kroupa 2007) with upper bound fixed at 10 M_☉. For MACHO masses, we try a set of single values ranging from 10⁻³ to 1 M_☉.

We fix the total mass of the bulge to 4 × 10¹⁰ M_☉ (Kent 1989; this can be considered a "safe" value for microlensing analyses, for the purpose of evaluating the expected self-lensing signal, as it is likely to be an upper limit, Riffeser et al. 2006, for this quantity), and that of the disk to 3 × 10¹⁰ M_☉ (Kerins et al. 2001; Riffeser et al. 2006). Looking for microlensing effects, the value we are actually interested in is the stellar mass, and indeed our overall value for this quantity agrees well with the analysis of Tamm et al. (2007). We consider a uniform extinction across the field, both foreground, E(B − V) = 0.062 (Schlegel et al. 1998), and intrinsic (for this second term, this hypothesis should of course be taken only as a first-order approximation), ext_R = 0.19 for the bulge (Han 1996; Riffeser et al. 2006), and E(B − V) = 0.22 for the disk (Stephens et al. 2003; Riffeser et al. 2006). Together with the M31 color (not corrected for extinction) B − r = 1.3 (Kent 1989), B − V = 1.0, and V − R = 0.8 (Walterbos & Kennicutt 1987), this translates into the values (corrected for extinction) M/L_R = 3.1 and 1.1 for the bulge and disk, respectively.

The bulge velocity distribution is dominated by its dispersive component, with line-of-sight velocity dispersion of σ = 120 km s⁻¹. For the disk, we take into account both the dispersive motion, σ = 60 km s⁻¹ (a value that can be taken as un upper limit), and a circular bulk motion, with disk circular velocity v = 250 km s⁻¹ (Carignan et al. 2006). M31 proper motion is set according to van der Marel & Guhathakurta (2008).

Given the values of the circular velocity, 220 km s⁻¹ and 250 km s⁻¹ for the Milky Way and M31 respectively, we fix accordingly the central densities and therefore the overall masses of the halos and the values of the one-dimensional velocity dispersions. As for the total dark matter halo mass value, within a truncation radius of 100 kpc and 130 kpc for the Milky Way and M31 (we estimate the ratio of these values from those of the circular velocities), we have, respectively, 1.0 × 10¹² M_☉ (in good agreement with previous determination, e.g., Vallenari et al. 2006, but somewhat in excess with respect to the recent determination by Xue et al. 2008) and 1.8 × 10¹² M_☉.

Within the Monte Carlo simulation, given the astrophysical model, we generate microlensing signals with Paczyński magnification corrected for finite size effects of the sources,¹⁵ build the corresponding light curves and carry out a first rough selection, paying attention not to reject any light curve that could be detected through our pipeline. In particular, for a variation to be selected, as a unique criterion we ask for at least three consecutive points (with one point per night and reproducing the observed sampling) 3σ above the background level (L > 0 according to the estimator introduced in Section 3). On the other hand, we are aware that we cannot reproduce within the Monte Carlo, where we only deal with light curves, all of the actual conditions of the pipeline we carry out in the real data set (where the analysis starts from the images). Amongst other effects, we most prominently cannot reproduce crowding effects, the underlying variable signals and the sources of non-Gaussian noise. Furthermore, we cannot run the first essential step of bump "cluster" detection. To account for these effects, we simulate those light curves that are selected within the Monte Carlo on the images (before the geometrical and all photometric corrections, namely, on the astronomical images just after the basic bias-flat fielding reductions) and then we run from scratch our pipeline. Conceptually, this is just a last step in the Monte Carlo that allows us to accurately evaluate the efficiency of our pipeline. The "efficiency," hereafter, should therefore be intended as that relative to the light curves selected within the Monte Carlo.

Within the Monte Carlo, each simulated event carries a (different) "weight," w_i (where i is the index of the simulated events) that is linked in part to the drawing process and in part to the quantity we are evaluating (the microlensing rate), and is completely independent of the selection process.¹⁶ The expected number of events is therefore the sum of the weights, n_exp = ∑_iw_i, where the sum runs over the events selected within the Monte Carlo (correspondingly we can estimate the associated statistical error based on Poisson statistics). Accordingly, by "efficiency," _w, we mean the ratio of selected over simulated events, where the number we refer to is always given by the sum of the weights. This is usually different (both numerically and from a logical point of view) from the actual ratio of selected over simulated events, a quantity that is not used even if in some cases it may be useful to be looked at. In particular, we do not expect, and in fact we do not find, these two values to be too different. Indeed, such a result should be taken as a hint of the presence of some bias in the way the event weights are distributed with respect to the selection process within the simulation.

For each lens population, we simulate up to a few thousands events per field, with 500 events per field per simulation in order to avoid overlap problems. Indeed, in particular in the inner M31 region, where we expect most of the events, simulated events may overlap (we draw at random from the Monte Carlo the events we simulate, with all their characteristics, including the line of sight) and thus lead us to bias the estimate of the efficiency of our pipeline. To test for this effect, for a fixed set of 500 events per field for which we had already evaluated the efficiency, we have carried out as many different simulations as needed, taking care to leave a minimum distance of at least 20 pixels among any couple of generated events, so to exclude overlap problems. As a result, we have found no significant differences in the two analyses. Overall, we have simulated 12,000 light curves to evaluate self-lensing efficiency, and 8000 for each value of the mass for MACHO lensing.

According to the selection pipeline, in which we require for the flux variations to be large enough, ΔR < 21.5, out of the Monte Carlo we extract, and then simulate on the images, selected events with flux deviation at maximum down to¹⁷ ΔR = 21.8. This limit is used accordingly when we evaluate the number of expected events. This way we allow for the observed rms of the evaluated flux deviations versus the input values. The exact value of this threshold is not, however, essential, as long as we keep the selection criterion on the flux deviation fixed at ΔR < 21.5 coherently with the selection pipeline. A brighter threshold would translate into a larger value for the efficiency and, at the same time, a smaller number of expected events (not corrected for the efficiency), and vice versa. These two effects balance when we evaluate the number of expected events corrected for the efficiency.

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As for the expected characteristics of the observed events, in Figure 1 we show the resulting distributions for the distance from the M31 center (both self-lensing and MACHO lensing), and, for self-lensing, the distribution for the durations (the full width at half-maximum, t_1/2) and the flux deviations at maximum (ΔR).

Finally, it is worth stressing that we simulate microlensing events only. Therefore, the simulation is restricted to saying whether and how our pipeline is going to select microlensing signals but it can say nothing about whether it might select as a microlensing a variable signal of different origin.

In Table 1, we report the results of the selection pipeline analysis together with the results for the efficiency of the corresponding analysis carried out on self-lensing simulated events.

Table 1. Microlensing Selection Pipeline: The Results

	Selection No. of Events		Simulation Efficiency w (%)
Criterion	Set A	Set B	Set A	Set B
Bump detection	4200		41.2 ± 3.5
L1 > 50	3033		35.5 ± 3.2
Shape analysis
χ2/dof < 10	2901		32.7 ± 3.0
Sampling	174	241	18.3 ± 2.0	21.9 ± 2.4
ΔR < 21.5	75	108	13.0 ± 1.5	20.4 ± 2.3
PSF	14	23	13.0 ± 1.5	15.7 ± 1.9
Variable	1	2	11.9 ± 1.4	14.6 ± 1.8

Notes. The results of the selection pipeline for microlensing light curves: analysis and simulation. For each step we report the number of selected light curves and the efficiency of the pipeline (%) for the expected self-lensing signal. According to the choice of the sampling criterion, we have split our selection results in set A and set B (left and right columns, respectively).

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We start the analysis working over the complete set of pixels, namely 2 × (1340 × 1300) light curves. The initial sample of selected flux variations consists of ∼4000 light curves. Within the shape analysis, the sampling cut severely reduces this initial set, and then the flux deviation cut leaves us with ∼200 light curves, most of which, according to the PSF analysis, are to be attributed to spurious variations. Finally, we are left with ∼40 flux variations, divided into sets A and B (according to the sampling criterion), most of which we expect to be intrinsic variables whose single-bump appearance is to be attributed to our short (50 days) baseline. (As outlined in Appendix A, set A flux variations are not a subsample of set B: among those surviving the PSF cut, 14 and 23 respectively, only two flux variations are in common between the two data sets, out of which one also survives the last cut.) The results of the last cut analysis are shown in the bottom panel of Figure 2 in the parameter space P_R–δ(Δf). We find, as expected, the flux deviation of most OAB variations to be compatible with the corresponding INT variations (small values of δ(Δf)), for which, at the same time, we find a clear sign of variability (large value of P_R). As an example, in Figure 3, we show the INT extension of four OAB-selected light curves. Only for two selected OAB flux variations, instead, we find the corresponding INT extension to be flat. Therefore, the selection pipeline finally leaves us with two microlensing candidates (one belonging to set B only). The same sample of light curves is represented in the ΔR_max–t_1/2 parameter space (top panel of Figure 2). We find most of the variations located in the upper part (corresponding to small flux deviations), with set B variations (empty symbols) biased toward short durations. In particular, we find the set A microlensing candidate (filled square symbol) located in a parameter space region where the contamination by the intrinsic variable signals is large. The only clear outlier is the bright and short set B microlensing candidate. We discuss the selected candidates in detail in Section 5.2.

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Comparing with the efficiency simulation analysis, a few points are worth being mentioned. First, it may look as if the "bump detection" step alone severely reduces the overall efficiency. However, in fact, only a very small fraction of light curves that are not selected at this point would pass all the other criteria. According to the same principle, the single cut that excludes most of the simulated light curves is the sampling criterion.¹⁸ This is also the reason why we have split our selection pipeline at this level. As for the simulation, we stress that this simply reflects the choice we have made in the Monte Carlo to select light curves on the basis of the L > 0 criterion only. Next, the PSF analysis proves to be a rather efficient criterion. Indeed it results that almost 50% of the simulated light curves fulfill this criterion and that this fraction rises to about 80% if we consider the subset of light curves that have already passed the bump detection and the shape analysis cuts. A usual reason that may cause the PSF Gaussian fit to fail is the presence, near the simulated event, of some other resolved object. Often enough, however, in these cases also the pixel photometry we use may have problems. (For both of these related aspects, a proper difference image analysis approach would be of course expected to give better results.) Poorly sampled light curves, on the other hand, simply do not have enough points near maximum magnification, so that it turns out to be usually not possible to carry out a good enough PSF fit. Finally, as for the analysis on the POINT-AGAPE extension to check for variable signals, we have seen this cut to be essential to get rid of otherwise dangerous contaminating flux variations. At the same time, this shows to be an extremely efficient criterion. Out of the complete set of simulated light curves, only about 13% of the INT light curves are clearly variable (P_R > 20) and this fraction falls to 7% when we add the demand for the INT flux deviation to be compatible with the OAB simulated one.

In Table 2, we report the results for the efficiency of the simulations for MACHO lenses. With respect to self-lensing events, there is a (rather small) effect linked to the different spatial distribution. The main effect is, however, due to the value of the mass, which is linked to the event duration, with smaller values of the efficiency corresponding to decreasing values of the MACHO lens mass.

Table 2. Efficiency Analysis: The Results for MACHO Lenses

	Efficiency w (%)
Mass (M☉)	Set A	Set B
1	16.5 ± 1.2	18.7 ± 0.9
0.5	18.5 ± 1.1	21.3 ± 1.2
0.1	16.5 ± 1.2	19.9 ± 1.4
10−2	13.8 ± 1.2	16.5 ± 1.3
10−3	11.1 ± 1.3	13.2 ± 1.4

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In Tables 1 and 2, we have given the results for the efficiency as a single value for the overall set of simulated events. On the other hand, the efficiency does vary quite significantly for data binned, for instance, in the distance from the M31 center and/or in the flux deviation at maximum. We find the larger values for the efficiency, up to 30% or more, for bright events in the outer regions of M31. On the other hand, the expected number of events, not corrected for the efficiency, is larger near the M31 center for faint events (Figure 1), namely, right where the efficiency is smaller (down to below 5%, depending on the choice of the binning). Overall, however, we find the expected number of events corrected for the efficiency to be rather insensitive, within the statistical error of the simulation, to any binning scheme, and this motivates our choice for the way to present our results.

The selection pipeline described in the previous section leaves us with two candidate microlensing events, which we name OAB-N1 and OAB-N2 ("N" indicates that they are both located in our "North" field). Their characteristics are summarized in Table 3, and their position within our field of view is shown in Figure 4.¹⁹ As for the sampling criterion, OAB-N1 fulfills both sets A and B demands, while OAB-N2 only the set B one.

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OAB-N1. This is a relatively large flux variation, with significance bump estimators equal to L = 183 and Q = 108; quite short, t_1/2 = 7.1 days, and not too bright, ΔR = 21.1. The OAB-N1 light curve is shown in Figure 5, together with its INT extension and the image difference around the candidate position upon which is based our PSF analysis. The INT data extension of the OAB-N1 light curve appears to be flat (small Lomb periodogram power, P_R = 6.3, and significantly large value for the difference of flux deviation, δ(Δf) = 10.0). However, the quality of the Paczyński fit is not good. A few points before the bump deviate significantly from the expected shape, and this is reflected in the rather poor value of χ². This might be attributed to underlying nearby variables, but we cannot exclude it to be a sign of an intrinsic nonmicrolensing nature. Indeed, we have shown that OAB-N1 is located in a part of the t_1/2–ΔR parameter space where the background of variable stars is large (top panel, Figure 2). A possible contamination, compatible with its "one bump" nature, besides a possible very long period variable, might come from some kind of eruptive variable (even if Novaecan be excluded because the flux variation is far too small). On the other hand, the descent is fairly well sampled and matches nicely enough the fitted Paczyński shape. In the top-right part of Figure 5, we show the "color" light curve, namely the ratio R over I band of the difference of the light curve flux, along the bump, and the background level, that we expect to be constant for microlensing.

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Through the pipeline, and in particular for OAB-N1, as an initial condition for the time of maximum magnification in the Paczyński fit we choose the value of the time corresponding to the data point with the maximum flux value, in this case t = 4435.4 (JD−2450000.0). During the fit procedure, it is difficult for this parameter to exit from the χ² minimum well around this value (whose bounds are usually set by the sampling) and indeed as a result we find t₀ = 4433.8 ± 0.2 (JD−2450000.0). Motivated by the OAB-N1 light curve appearance, irregular sampling, and noisy aspect, we have therefore carried out a search for other χ² minima in different regions of the t₀ space. As a result, we have found a new, well isolated, minimum with a lower value of the reduced χ² (χ²/dof = 3.5 to be compared with χ²/dof = 3.9 found in the previous case) for t₀ = 4430.6 ± 0.4 (JD−2450000.0). Correspondingly, we find new values for the duration and the flux deviation at maximum, t_1/2 = 18 days and ΔR = 21.5 (this last value being at the limit of our threshold cut), namely, a rather different result from the previous one. Such a light curve would have still been selected as a microlensing candidate within our pipeline. However, the longer duration would have further weakened its microlensing interpretation (Section 5.4). The coexistence of these two minima might be suggestive, accepting the microlensing hypothesis, of a binary lens or binary source solution. The available data, however, do not allow us to robustly test this hypothesis.²⁰

OAB-N2. This looks (Figure 6) like an extremely bright and short flux variation (t_1/2 = 2.6 days, ΔR = 19.1), located near the M31 center (at a distance d = 28), with a completely flat INT extension (P_R = 8, δ(Δf) = 25). The sampling along the bump is, however, extremely poor (indeed, both the time of maximum amplification and the flux deviation at maximum are not strongly constrained; Table 3). In particular, the lack of data points in the descent prevents us from probing the expected microlensing symmetric shape, so that it is difficult to draw definitive conclusions about its nature. On the other hand, the rising part of OAB-N2 looks achromatic, and is rather well constrained, and this strengthens the microlensing hypothesis. Indeed, both the shape and the achromaticity are hardly compatible with the only possible kind of contamination for such a kind of flux variation, if not microlensing, namely some sort of eruptivelike object. Furthermore, though few, the points along the bump allow us to get to a reasonable fit for all the parameters of the microlensing event. At the 1σ level, the χ² analysis gives us best values and lower bounds for the impact parameter and the source flux and best value and upper bound for the Einstein time (with, however, a rather large relative 1σ error, even exceeding 50%; Table 3). Finally, the guess for the flux source value allows us to carry out a more constrained color analysis (as for OAB-N1, but subtracting the source flux from the baseline, bottom-left panel of Figure 6). As for the value of the source flux, the results of the Paczyński fit have been confirmed by the cross-identification of the possible source star in the catalog of Massey et al. (2006). Indeed, within 1 pixel of our evaluated position, we find a typical red giant with R_C = 20.9 and R − I = 1.2 (values fully compatible with those evaluated from our data set alone). Knowledge of the source flux allows us to better constrain the fit parameters; in particular, the time of maximum amplification and the flux deviation at maximum (in fact t₀ shifts back to the position of the last observed data point so that, accordingly, ΔR gets fainter). Furthermore, we may now completely break the degeneracy among the amplification parameters. Indeed, we estimate t_E = 4.2^+0.6_−0.4 days and u₀ = 0.22^+0.01_−0.03 (A_max = 4.5).

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We have also searched for X-ray counterparts in the XMM-Newton archive²¹ and verified that neither of the microlensing candidates' coordinates correspond to any of the identified sources in the EPIC images (Strüder et al. 2001; Turner et al. 2001; Shirey et al. 2001). In the case of OAB-N2 there is an X-ray counterpart, but only within ∼30'', so that we can safely rule out the identification.

The Monte Carlo simulation, completed by the efficiency analysis, allows us to estimate the expected number of microlensing events for our experimental setup. In Table 4, we report the resulting values, already corrected for the efficiency. The results we obtain using the set B criteria are larger by up to about 30% with respect to set A. This is a combined effect of the larger value of the efficiency and of the effective baseline length increase.

Table 5. Monte Carlo Parameters: Distributions Function and Limits for Each Lens/Source Population

	MW Halo	M31 Halo	M31 Bulge	M31 Disk	Limit
θ	Uniform	Uniform	K89	K89+f(ζ)	Field of view
ρ	Uniform	$\rho _\mathrm{HM31}(\vec{r})$	K89	K89+f(ζ)	Field of view
Dl	$\rho _\mathrm{HMW}(\vec{r})$	$\rho _\mathrm{HM31}(\vec{r})$	K89	K89+f(ζ)	Model
μl	δ(μl)	δ(μl)	μ−α	μ−α	Model
Ds	⋅⋅⋅	⋅⋅⋅	K89	K89+f(ζ)	Model
ϕs	⋅⋅⋅	⋅⋅⋅	IAC	Hipparcos	MI < 2
vr	Gaussian	Gaussian	Gaussian	Gaussian	0, ∞
ψv	Uniform	Uniform	Uniform	Uniform	0, 2π
u0	Uniform	Uniform	Uniform	Uniform	0, uMAX(=2)
t0	Uniform	Uniform	Uniform	Uniform	0, ΔTobs

Notes. The parameters have been introduced in Appendix C. The details for the density function of the Milky Way and the M31 halos and for the source luminosity functions are given in Section 4.1. K89 stands for the Kent (1989) bulge–disk decomposition. For the disk, as only a two-dimensional distribution on the plane is given, we consider a vertical distribution f(ζ) = cosh(−(ζ/ζ_h))², where ζ_h is the vertical scale height. The index for the power-law mass function is given in Section 4.1. In the last column, we report the range of variability for each parameter: for θ, ρ "field of view" indicates the two observed fields, "Nord" and "Sud" (Section 2.1); for the distances we take the physical boundaries of the component considered, in particular for the halos we use a truncation radius as specified in Section 4.1, for the bulge ∼7 kpc around the M31 distance, while for the disk vertical component we draw directly from the given distribution (for a fixed line of sight, this allows us to evaluate the physical distances); for bulge and disk lenses, μ_l is drawn with a lower limit of 0.08 M_☉ and upper limit of 1.0 M_☉ and 10.0 M_☉, respectively.

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As for self-lensing events, it turns out that most of them, ∼50%, are due to bulge–bulge events (lens–source, respectively), with the remaining distributed almost equally between disk–bulge and bulge–disk events. The second configuration is enhanced in the South field because in this case we see the bulge in front of the disk. It is also worth noting that, according to our model, almost half of the overall bulge mass is located within our cone of view, but only about 1/7 of that of the M31 disk.

As for MACHO lensing, about 2/3 of the events are expected to belong to the M31 halo. Actually, the overall mass within our cone of view of the M31 halo is larger than the Galactic one by a factor of a few thousands, but Milky Way halo lensing is strongly enhanced by the much larger size of the Einstein radius. We expect the largest number of MACHO lensing events from 10⁻¹ to 10⁻² M_☉ objects, as our efficiency dramatically drops for smaller MACHO masses.

Together with the expected event number we report the statistical error associated with the simulation on the basis of Poisson statistics. The uncertainties intrinsic in the model (whose detailed discussion is beyond the scope of the present paper) make, however, the corresponding systematic error in principle even larger.

We have designed our (fully automated) pipeline so as to get rid of most forms of stellar variability, while preserving genuine microlensing. As a result, therefore, all survivors could in principle be variables but with varying (hopefully large) degrees of confidence that they are not. Coming to the present data set, we have selected two flux variations compatible with a microlensing signal. While discussing the possible physical meaning of this result, however, we have to keep in mind that, although with different degrees of confidence, we have no strong evidence in favor of the microlensing hypothesis for either candidate: OAB-N1 because of both the light curve appearance (reflected in the χ² value) and its position in the duration/flux-deviation parameter space, and OAB-N2 because of its extremely poor sampling (that has not prevented us, however, from finding a convincing microlensing Paczyński fit further confirmed by the possible identification of the source). In conclusion, looking at the candidate light curves, we might be tempted to classify OAB-N1 as a very "poor" candidate, and OAB-N2 as a "good" one, keeping in mind, however, that this might simply reflect the bias induced by the observational sampling.

With the care suggested by the above discussion, we come now to the comparison of the observed events with the expected signal. As for the event characteristics, we may take as a reference the distributions shown in Figure 1. Given the caveat that these distributions cannot be compared directly with the results of the analysis because of the correlation of the pipeline efficiency with the event characteristics, both candidates appear compatible with the expected signal with respect to the duration and the distance from the M31 center (being both shorter and nearer to the M31 center, OAB-N2 appears, also in this respect, a stronger candidate than OAB-N1). Coming to the number of events, we must compare our observed candidates with the results reported in Table 4. First, we are bound to consider the case that MACHO lensing does not contribute at all, so that we are left with the expected lensing signal from the luminous populations alone, with an expected number of events for self-lensing somewhat smaller than 1. Even allowing for both our candidates to be genuine microlensing, this is still, according to Poisson statistic, fully compatible with the observations.

On the other hand, the expected number of MACHO lensing events is not large if compared to that of self-lensing. Even for a full MACHO halo we would expect about as many self-lensing events as MACHO lensing. It is clear, therefore, that on the basis of the number of events alone our statistics are still too small to draw conclusions on this contribution.

It may be asked, then, how we might disentangle the two signals: self-lensing from MACHO lensing. First, an improvement in the event statistics might help us to further exploit the expected differences in the spatial distributions (top panels, Figure 1). Second, a good enough sampling (such as was not the case for the present selection) might allow a sufficient characterization of the selected flux variations. In turn, this could give possible hints on the nature of the lens, as was the case for the detailed analysis linked to the finite source size effect for the PA-S3 event carried out by Riffeser et al. (2008). Both these reasons motivate the need for further observations.