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

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2. The pixel method

The photon flux of an individual star, [FORMULA], is spread among all pixels of the seeing disk and only part of this light, the seeing fraction f, reaches the pixel nearest to the centre of the star:

[EQUATION]

In a crowded field such as M 31, the light flux [FORMULA] on a pixel comes from the many stars in and around it, plus the sky background.

[EQUATION]

If the luminosity of a particular star is amplified by a factor A, the pixel flux increases by:

[EQUATION]

The amplification of the star luminosity can be detected if the flux on the pixel nearest to its centre rises sufficiently high above the rms fluctuation [FORMULA]:

[EQUATION]

Of course, to be detected, a lensing event should be visible on several exposures. One therefore typically requires that condition (4) be verified for at least 3 consecutive pictures with [FORMULA] and with [FORMULA] for at least one of the three.

Seeing variations induce unwanted fluctuations of the pixel fluxes. To minimise this problem, and to collect most of the light of any varying object, we replace each elementary pixel by a "super-pixel" centered on it. Each super-pixel is a square of [FORMULA] elementary pixels. The size of the square is chosen large enough to cover the whole seeing disk in most cases, but also not too large, to avoid dilution of a variable signal when it occurs. We have also tried to replace each pixel by an average of the neighbouring pixels weighted with the point spread function (PSF), as it is known to maximize the signal to noise ratio at the center of a star on a given image. However, for this very reason, it turns out that this procedure amplifies considerably the fluctuations in time due to seeing variations and therefore it is not appropriate for our method.

2.1. Microlensing tests

All of the classical tests can be applied to discriminate microlensing events against other sources of light variations.

Uniqueness The probability of a microlensing occurring twice on stars contributing to the same pixel is very weak, and it is safe to reject all non unique events.

Symmetry Except in the case of a multiple lens or star, the light curve should be symmetric in time around the maximum amplification.

Achromaticity Gravitational lensing is an achromatic phenomenon. However, the lensed star has not, in general, the same colour as the background and only the luminosity increase is achromatic (assuming constant seeing):

[EQUATION]

A specific signature: forward-backward asymmetry It has been pointed out by Crotts (1992) that M 31 provides a unique test of microlensing. As this galaxy is tilted with respect to our line of sight, the rate of microlensing should be higher for those regions of its disk which are on the far side, because they lie behind a larger fraction of the halo of M 31 and should undergo microlensing more often. Therefore, one expects a forward-backward asymmetry in the distribution of microlensing events, which cannot be faked by intrinsically variable objects.

2.2. Expected number of events

Most basic formulae can be found in Griest (1991) and De Rújula, Jetzer & Massó (1991). We only recall those few that we shall explicitly need.

The amplification A is related to the distance of the lens to the line of sight [FORMULA] ([FORMULA] is the Einstein radius) by the relation:

[EQUATION]

We detect the variation with the time t of this amplification when a lens passes near the line of sight with a transverse velocity [FORMULA]. Then

[EQUATION]

where [FORMULA] and [FORMULA] are the time and distance of maximum amplification, and the Einstein time, [FORMULA], is the time it takes for the lens to cover one Einstein radius.

The rate of events where the amplification is larger than a definite value A is proportional to the amplification radius [FORMULA] (obtained by inversion of Eq. (6))

[EQUATION]

where [FORMULA] is the rate of events for which the impact parameter gets smaller than the Einstein radius and the amplification exceeds 1.34. Note that the rate [FORMULA] is linear in the amplification radius [FORMULA], because it counts the number of stars that enter the area inside [FORMULA] per unit of time.

Lenses in the Milky Way halo The simple evaluations that follow can only be made for lenses in the halo of our Galaxy. We consider a "standard" spherical halo (Bahcall & Soneira 1980, Caldwell & Ostriker 1981)

[EQUATION]

cut at a distance of 100 kpc, where the density in the solar neighbourhood is [FORMULA] (Flores 1988), the core radius a ranges from 2 kpc (Bahcall & Soneira 1980) to 8 kpc (Caldwell & Ostriker 1981), and the distance from the sun to the galactic centre is [FORMULA] kpc. Assuming an isotropic distribution for the transverse velocity [FORMULA] of halo objects 1, the value of [FORMULA] in the direction of M 31 is:

[EQUATION]

taking into account only lenses of the halo of our galaxy.

The amplification A required for detection depends on the magnitude m of the star and on the surface magnitude µ of the background at the pixel position. The number of photoelectron/s actually counted by the CCD on our reference image, from a star of magnitude m is:

[EQUATION]

We measure [FORMULA] photoelectron/s with the Gunn r filter and [FORMULA] photoelectron/s for the Johnson B filter (see Eqs. (23 - 25) below, remembering that the gain of the CCD is 9.4).

To compare with other instruments, note that effective fluxes are related to photon fluxes [FORMULA] (in [FORMULA]) outside the atmosphere by:

[EQUATION]

Here [FORMULA] is the diameter of the telescope, [FORMULA] is the quantum efficiency of the CCD camera, and P is a variable loss factor, both atmospheric and instrumental, which is typically about 3.

Neglecting the night sky background (this is justified near the bulge of M 31), the number of photoelectron/s counted per square arcsecond from the background is:

[EQUATION]

Since the light of the galaxy is nothing but the integrated light of all stars, we get the very useful relation:

[EQUATION]

where [FORMULA] is the luminosity function of the galaxy (here defined as the number of stars of magnitude between m and [FORMULA] per arcsec2). When the star is microlensed, the signal in a pixel is, if the exposure time [FORMULA] remains small compared to [FORMULA]:

[EQUATION]

where the seeing fraction f is the fraction of the star flux that reaches the pixel. We estimate that our level of noise is approximately twice the statistical photon fluctuation (see Sect. 4.3):

[EQUATION]

where [FORMULA] is the angular surface of the pixel, in arcsec2. If one wants that the signal to noise ratio be larger than Q, then the lens must approach the line of sight of the lensed star nearer than

[EQUATION]

where we have used the fact that, when the amplification is large, [FORMULA]. We have neglected finite size effects, which would decrease the number of events for small mass lenses ([FORMULA]) in the halo of M 31. The total number of events with a signal to noise ratio above Q is then:

[EQUATION]

where [FORMULA] is the total duration of the observation, and [FORMULA] is the total solid angle covered. Taking into account Eqs. 14 and  17, the shape of the luminosity function [FORMULA] drops out and one finally gets:

[EQUATION]

Using Eq. 19 with [FORMULA], in the conditions of AGAPE (described below in Sect. 3), where the total observation period is 190 days, the total solid angle covered is [FORMULA], the super-pixel size is [FORMULA], and the mean surface magnitude lies around [FORMULA], we expect about 8 events from Milky Way lenses with a mass of [FORMULA]. However, this evaluation is an overestimate, because it only requires that one point of the light curve reaches a signal to noise ratio above 5, disregarding whatever happens at the preceeding and following points.

M 31 lenses Lenses in M 31 and its halo act on point-like sources in the same way as those of the Milky Way because, if one neglects the angular size of the source, the lensing phenomenon is symmetric between observer and source. The contribution of the lenses in M 31 cannot be evaluated in the same simple way, for two reasons. i) For low mass lenses in M 31 or in its halo, the angular Einstein radius is not much larger than the angular radius of most bright stars, which can no more be considered as point-like. As a result, the amplification is limited by finite size effects and seldom becomes large enough to be detectable. In fact lenses lighter than [FORMULA] around M 31 produce nearly no detectable microlensing. ii) On the contrary, for high masses, one expects lenses around M 31 to dominate, because M 31 is roughly twice as massive as the Milky Way, and because bulge-bulge lensing should be important in the central region we are looking at (Han & Gould 1996). The distribution of M 31 lenses, and therefore their contribution to the lensing rate, strongly depends on the region of the galaxy one considers. As a matter of fact, this is an advantage, because (Crotts 1992) it provides a signature of the lensing phenomenon, and it will allow to make a map of the distribution of M 31 lenses if one achieves enough statistics.

Numerical simulation To give ourselves the possibility: i) to take into account the lenses of M 31, ii) to put into our evaluations the real event selection criteria and to change them, iii) to work with the true observation conditions, such as the varying seeing and the real distribution in time of the observation nights, iv) to play with the distributions, still poorly constrained, of the lenses and source stars both in the Milky Way and in M 31, we have built a Monte-Carlo simulation. Typical inputs for the simulation are as follows. The halo of our galaxy is taken "standard" (Eq. 9) with a core radius a of 5 kpc, the halo of M 31 is taken twice as large. An event is called detected if the light curve shows a series of at least three consecutive points with a signal to noise ratio above 3 and above 5 for one of these points. With these assumptions, the number of expected events is about 3 from the Milky Way halo, and 8 from the M 31 halo. Bulge-bulge lensing in M 31 has not yet been included in our simulations but, according to Han & Gould (1996), should contribute as much as lensing by the M 31 halo. One must,however, emphasise that the number of events one expects depends on the detailed process of analysis and on the event selection, which are not settled at this stage. It is interesting to compare qualitatively the Monte-Carlo simulations with the analytic expressions above which, although crude numerically, show some interesting features.

  1. As can be seen from Eq. 19, the lensing rate does not depend on the shape of the luminosity function [FORMULA] of M 31. This is quite welcome since this function is largely unknown (except for the brightest resolved stars) and moreover it changes from the centre to the outskirts of M 31. Our Monte-Carlo simulation indeed confirms that the rate depends only weakly on the shape [FORMULA]
  2. The lensing rate scales with the galactic surface brightness as 10 [FORMULA] as a result of the competition between the number of source stars and the photon noise. Our Monte-Carlo simulation confirms this behaviour. This scaling in µ is related to the statistical nature of the fluctuations, which is proportional to the square root of the number of photons. It is certainly wrong when the statistical error is very small, then we know that other sources of fluctuations, such as the Tonry-Schneider surface brightness fluctuations (Tonry & Schneider  1988), and the residuals of the geometric alignment, take over. We take into account this fact in our Monte-Carlo simulations by setting a lower bound on the relative fluctuation. As we shall see in Sect. 4.3, this bound is not higher than 0.1% in our data. This lowest level of fluctuation is of crucial importance: if we were only able to reach 0.2%, the expected number of events would drop by a factor of 3.

The monte-carlo simulation allows to predict the distribution of various quantities that characterise microlensing events. In Fig. 1 the distributions of two time scales are compared: i) the effective duration of the events [FORMULA], i.e. the time during which an event is effectively detected with a signal to noise ratio higher than 3; ii) twice the Einstein time [FORMULA] (twice because, in comparing with the effective time, the diameter rather than the radius of the Einstein ring is relevant to the total duration of an event). The two distributions are very different. The absence of events with an effective duration [FORMULA] between 100 and 240 days is related to the distribution of our observation periods: first 60 days in 1994, then a 240 days gap, and finally 150 days in 1995.

[FIGURE] Fig. 1. Simulated distributions of the effective duration [FORMULA] and twice the Einstein time [FORMULA], for "detected events".

Fig. 2 displays the distributions of the absolute V magnitude of lensed stars, and of the amplification at maximum in the conditions of the real observation. As expected, the stars involved in detectable microlensing events are giants, and the amplifications are high, with a mean value of about 13.

[FIGURE] Fig. 2. Simulated distributions of the absolute V magnitude of the lensed star and of the maximum amplification, for "detected" events.

In Fig. 3 are displayed super-pixel light curves of simulated microlensing events satisfying our detection criteria, in the real observation conditions.

[FIGURE] Fig. 3a-c. Light curves of simulated "detected" microlensing events. The solid lines are the theoretical Paczyski curves.
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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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