Observable optical depth and halo lens mass function

We present a simple definition of observable optical depth τoble which depends on the detection efficiency as a function of the MACHOmass μ and on the lensing objects Mass Function (MF). The halo mass fraction f is then correctly evaluated as τobsd/τoble, being τobsd the directly observed optical depth. We assume, then, a homogenous power law MF for the dark halo and estimate the exponent α using the inverse problem method and the hypothesis f = 1. The result we get is α = 1.11. Next, we test this value calculating some microlensing quantities towards LMC and comparing our MF to the other ones cited in literature. In both cases we find a good agreement. As a further test, we repeat the same kind of work for different values of the lower mass limit μl and f , obtaining a series of values for α which we test in the same way as before. The only surviving model is the one with μl = 10 , f = 1 and α = 1.11. Then, we conclude that this model of full MACHO halo with a homogenous power law MF may reconcile data and theory, but the nature of MACHOs remains still unclear.