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

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9. Application to observed events

In this section I show the application of the method described here to the observed events towards the LMC. The first events have been claimed by EROS (Aubourg et al. 1993), namely EROS#1 and #2, and MACHO (Alcock et al. 1993), namely MACHO LMC#1, in 1993. The fit with a point-mass lens and point source for MACHO LMC#1 showed a discrepancy near the peak which has been solved with models involving a binary lens by Dominik & Hirshfeld (1994, 1996). The MACHO collaboration had found two other candidates, MACHO LMC#2 and #3, (Alcock et al. 1996) which have been meanwhile dismissed. In addition, they have claimed the existence of 7 additional events, MACHO LMC#4 to #10, (Pratt et al. 1996), where MACHO LMC#9 is due to a binary lens (Bennett et al. 1996). In the MACHO data taken from 1996 to March 1997, 5 additional LMC candidates showed up (Stubbs et al. 1997).

It has been shown that the EROS#2 event involves a periodic variable star (Ansari et al. 1995) and that EROS#1 involves an emission line Be type star (Beaulieu et al. 1995), so that both EROS#1 and EROS#2 involve a rare type of stars which makes these events suspect as microlensing candidates (e.g. Paczynski 1996). In addition, the MACHO LMC#10 event is likely to be a binary star (Pratt et al. 1996; Alcock et al. 1997).

By assuming that the lens is in the galactic halo and using the halo model of the last section, expectation values for the desired quantities can be obtained by inserting the fit parameters into Eqs. (88) to (91). Table 6 shows the expectation values for the Einstein radius and the mass for the events EROS#1 and EROS#2, MACHO LMC#4...#8 and #10, whereas the results for the binary lens events MACHO LMC#1 and MACHO LMC#9 are shown in Table 7. For MACHO LMC#1 six different binary lens models are shown (Dominik & Hirshfeld 1996; Dominik 1996). Note that the lens for MACHO LMC#9 probably resides in the LMC (Bennett et al. 1996).


Table 6. Expectation values for the point-source-point-mass-lens events towards the LMC. M # denotes MACHO events, while E # denotes EROS events.


Table 7. Expectation values of the physical parameters and used fit parameters for the 6 binary lens models for MACHO LMC#1, denoted by BL, BL1, BA, BA1, BA2, and BA3, and for MACHO LMC#9.

The parametrization used is that of Dominik & Hirshfeld (1996): [FORMULA] denotes the projected half-distance between the lens objects in the lens plane, where [FORMULA], [FORMULA] denotes the mass fraction in lens object 1. [FORMULA] denotes the Einstein radius corresponding to the mass of object 2 and [FORMULA] the characteristic time corresponding to [FORMULA]:


Similarly [FORMULA] denotes the projected half-separation in units of Einstein radii corresponding to the mass of object 1, so that


[FORMULA] and [FORMULA] denote the mass in units of [FORMULA] for objects 1 and 2 and the mass ratio r is given by


Since the true semimajor axis [FORMULA] is not yielded by the fit, T is estimated using [FORMULA], which corresponds to a minimal value [FORMULA], because [FORMULA] for any gravitationally bound system and [FORMULA].

The distribution of the physical quantities as well as symmetric intervals around the expectation value with probabilities of 68.3 % and 95.4 % are shown in the previous section.

The model BA3 has previously been omitted (Dominik & Hirshfeld 1996) due to the fact that the mass ratio between the lens objects is very extreme ([FORMULA]), which corresponds to unphysical values (see Table 7). However, the uncertainty of the mass ratio is very large for extreme mass ratios (Dominik & Hirshfeld 1996) due to the fact that the lens behaves nearly like a Chang-Refsdal lens. The same degenaracy has recently been rediscovered in the context of lensing by a star with a planet by Gaudi & Gould (1997). Table 8 shows the estimates for the 4 wide binary lens models (BA, BA1, BA2, BA3) where fit parameters at the upper 2- [FORMULA] -bounds of the mass ratio r have been used. It can be seen that the expectation values for the separation and the mass change dramatically for the fits with small values of r, leaving much room for speculations about the nature of the lens object.


Table 8. MACHO LMC#1: Estimates of physical parameters of the binary lens models using a mass ratio r at the upper 2- [FORMULA] -bound of r and corresponding values of [FORMULA] and [FORMULA].

Recently, Rhie & Bennett (1996) have speculated about a planetary companion in the MACHO LMC#1 event. However, as shown in this and the last section, there are two fundamental uncertainties (beyond the fact that there are only a few data points near the peak, which could have been solved by a denser sampling), namely the uncertainty in the mass ratio and the uncertainty due to the unknown lens position and its velocity. The planetary model of Rhie & Bennett (1996) corresponds to my model BA1, where the expectation value of the mass of the low-mass object is about [FORMULA]. Taking into account an uncertainty of a factor of 30, a value of 2 Jupiter masses is reached just at the 2- [FORMULA] -bound. As shown in Table 7, the expectation value for the mass of the low-mass object is about the same for all of my wide binary lens fits, so that there are the same prospects for a planet as the low-mass object for all of these models. However, the high-mass object will be different.

The estimates of physical quantities like the mass of the lens object(s), their separation and the rotation periods along with the uncertainties involved are needed to reveal the physical nature of the lens for each observed event. One can check whether the mass range is consistent with the assumption of a dark object, and of which nature the lens should be in this case (brown dwarf, white dwarf, neutron star,...). The determination of the mass range is crucial for claiming the existence of a planetary companion.

One can also check whether it is consistent to use a static binary lens model rather than one with rotating binary lens (Dominik 1997). Moreover, by comparing estimates for different lens populations (e.g. the galactic halo and the LMC halo) one may obtain indications to which population the lens belongs.

In contrast to estimates on the mass spectrum, there is no direct influence from the ensemble of observed events on the estimates for a specific event. However, there is an indirect influence, since the ensemble of events gives information about the mass spectrum, which in turn can be used to get a more accurate estimate for each observed event. However, a lot of events ([FORMULA]) are needed to get accurate information on the mass spectrum (Mao & Paczynski 1996). To be used for the estimates for a specific event, some of the higher mass moments should have been determined. Note that there will remain large uncertainties in the mass of a specific event even if the mass spectrum is known (unless it contains a sharp peak), since due to the broad distribution of the velocity, the range of timescales [FORMULA] for a certain mass is broad, so that the mass range for a specific event in turn may also be large.

While a white dwarf scenario is preferred by the LMC observations (e.g. Pratt et al. 1996), a brown-dwarf scenario is not ruled out (Spiro 1997) if one considers a flat halo. Though there are some restrictions on the average lens mass from the observed events, which will improve with more data, it will remain highly uncertain into which mass regime a certain event falls.

Since the ongoing microlensing observations constrain the galactic models and therefore give rise to more accurate determinations of the structure of the lens populations, the knowledge on specific events will also be improved by this.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998