5 The self lensing problem

We discuss now the problem of the LMC self lensing and show how our results depend quite weakly on the systematics connected with it LMC. As discussed in the introduction, our analysis implicitly assumed that all the observed events are due to lenses belonging to our dark halo. Even if there are different evidencies against a dominant contribution of self lensing, it is still possible that two or three of the 13 events considered are due to LMC self lensing. This means that we have to repeat our analysis excluding from the sample those events which are not caused by MACHOs. Unfortunately, it is not really possible to establish precisely what are these events, due to the well known degeneracy in the lens parameters which does not allow to determine its distance with respect to the observer. However, we may qualitatevely correct our results excluding form the sample some events chosen according to which are more likely to be due to self lensing. To this aim we have used the results on the self lensing events timescale distribution obtained by Gyuk et al. (2000). Their analysis shows that the most likely duration of self lensing events is ${\sim} 100$d; so we repeated our analysis for different choices of the excluded events. As an example we discuss here the results obtained for the model A2c $(\mu_{\rm l} = 0.001, \mu_{\rm u} = 1.0, q = 0.8)$; similar results are obtained for the other models. As a first test, we have excluded from the sample the events labelled 7, 13 and 14 in Table 7 of Alcock et al. (2000a) which are the longest ones. We have then $< t_{\rm E} >_{\rm obsd}~= 68.78 \ {\rm d}$ and $N_{\rm ev}^{\rm obsd} = 10$; the uncertainties on these quantities increase to ${\sim} 32\%$ both on $< t_{\rm E} >_{\rm obsd}$ and on $N_{\rm ev}^{\rm obsd}$. We get

$$\alpha = 0.94 \div 1.67 ; \ f = 0.07 \div 0.24 ; \ (\alpha_0 , f_0) = (1.30, 0.14) .$$

Even if there is a little trend towards larger values of $\alpha$ and smaller values of f, these results are quite consistent with the ones given in Table 4. The central value for the slope $\alpha$ is slightly larger and f₀ slightly lower but the discrepancies are not significative. We have repeated the same analysis in other two cases: excluding events 5, 7 and 13 and exlcuding events 13, 21, 25 respectively. The results we get are consistent with each other and with the results for model A2c in Table 4. Finally we have also tried to exclude four events (events number 7, 13, 14 and 21) from the sample; the uncertainties are of course larger ( ${\sim} 33\%$ both on $< t_{\rm E} >_{\rm obsd}$ and on $N_{\rm ev}^{\rm obsd}$) while the mean duration is 66.07 d for 9 events. We obtain

$$\alpha = 0.99 \div 1.72; \ f = 0.06 \div 0.21; \ (\alpha_0 , f_0) = (1.34, 0.12) .$$

These results are still consistent with those obtained till now for model A2c. These qualitative tests are encouraging since they suggest that our analysis is not seriously affected by the systematics connected to the self lensing whose main effect seems to be to lower the statistics.