Astron. Astrophys. 330, 963-974 (1998)

## 1. Introduction

Some attempts have been made to obtain information about the mass of the lens from observed microlensing events. Unfortunately, the mass cannot be inferred directly. Instead, the only relevant information directly available from a fit of a light curve to the observed data points is the timescale , and the mass depends on this timescale () as well as on the position of the lens and the transverse velocity of the lens relative to the line-of-sight. Since the latter parameters are both not observable (except for extraordinary cases where some additional information can be obtained), one can only obtain statistical information on the lens mass assuming distributions of the lens position and the transverse lens velocity. Griest (1991), cited as GRI in the following, argued that the most likely mass distribution is that which yields the best fit to the distribution of timescales of the observed events. De Rújula et al. (1991), cited as RJM in the following, Jetzer & Massó (1994) and Jetzer (1994) have shown that one can extract statistical moments of the lens mass distribution from the moments of the distribution of the timescales.

While these attempts use the distribution of timescales to obtain information about the mass spectrum of the lenses as realized in nature, I will discuss another topic in this paper. For a given observed event, I investigate the probability distributions of the lens mass and other physical quantities like the Einstein radius and the separation and rotation period of the lens objects for a binary lens. These probability distributions give the answer to the question, how probable certain ranges of the considered physical quantity are for the given lens system having produced the event. Note that this question is not answered by applying the methods of GRI or RJM It is however of special importance for planetary systems. Also note that additional parameters will enter the calculation of the event rate if one considers events which deviate from the point-source-point-mass-lens model. A binary lens will give serious problems for determining the mass spectrum from the timescale distribution since one measures the agglomeration of two objects each from the mass spectrum for one event rather than a single object from the mass spectrum. In contrast to this, the probability distributions presented in this paper also give meaningful results for `anomalous' events.

This paper is organized as follows. In Sect. 2, it is shown which information directly results from a fit of a galactic microlensing light curve. In Sect. 3, the relation between the event rate and the mass spectrum, the spatial distribution of the lenses, and the distribution of the relative velocity is given. Sect. 4 shows how the probability distribution for the lens mass can be derived. Sect. 5 gives results for the moments of further physical quantities and the probability distribution of these quantities around their expectation value. In Sect. 6, it is shown that the expectation value for the mass coincides with the value obtained by applying the mass moment method of RJM to one event. In Sect. 7, two special forms of the velocity distribution are discussed in more detail: a Maxwellian distribution and a fixed velocity. In Sect. 8, the expectation values and the probability distributions for the lens mass, the Einstein radius, and the separation and the rotation period for binary lenses are shown for a simple model of the galactic halo. In addition, intervals corresponding to probabilites of 68.3 % and 95.4 % are given for these quantities. In Sect. 9, the implications for the observed events towards the LMC (Alcock et al. 1993; Auborg et al. 1993; Dominik & Hirshfeld 1996; Alcock et al. 1996, 1997; Pratt et al. 1996; Bennett et al. 1996) are discussed explicitly. I discuss what information can be extracted and how the observation of more events influences the results.

© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998