## 2. The pixel methodThe photon flux of an individual star, , is
spread among all pixels of the seeing disk and only part of this
light, the seeing fraction In a crowded field such as M 31, the light flux on a pixel comes from the many stars in and around it, plus the sky background. If the luminosity of a particular star is amplified by a factor
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 : 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 and with 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 ## 2.1. Microlensing testsAll of the classical tests can be applied to discriminate microlensing events against other sources of light variations.
## 2.2. Expected number of eventsMost 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 We detect the variation with the time where and are the time and distance of maximum amplification, and the Einstein time, , 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 where 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 is linear in the amplification radius , because it counts the number of stars that enter the area inside per unit of time.
cut at a distance of 100 kpc, where the density in the solar
neighbourhood is (Flores 1988), the core radius
taking into account only lenses of the halo of our galaxy. The amplification We measure photoelectron/s with the Gunn r filter and 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 (in ) outside the atmosphere by: Here is the diameter of the telescope,
is the quantum efficiency of the CCD camera,
and 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: Since the light of the galaxy is nothing but the integrated light of all stars, we get the very useful relation: where is the luminosity function of the
galaxy (here defined as the number of stars of magnitude between
where the seeing fraction where is the angular surface of the pixel,
in arcsec where we have used the fact that, when the amplification is large, . We have neglected finite size effects, which would decrease the number of events for small mass lenses () in the halo of M 31. The total number of events with a signal to noise ratio above Q is then: where is the total duration of the observation, and is the total solid angle covered. Taking into account Eqs. 14 and 17, the shape of the luminosity function drops out and one finally gets: Using Eq. 19 with , in the conditions of AGAPE (described below in Sect. 3), where the total observation period is 190 days, the total solid angle covered is , the super-pixel size is , and the mean surface magnitude lies around , we expect about 8 events from Milky Way lenses with a mass of . 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.
- As can be seen from Eq. 19, the lensing rate does not depend on the shape of the luminosity function 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
- The lensing rate scales with the galactic surface brightness as 10
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 , 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 (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 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.
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.
In Fig. 3 are displayed super-pixel light curves of
© European Southern Observatory (ESO) 1997 Online publication: May 5, 1998 |