Dark mass moments

Paczyński has shown how to detect the ‘dark’ constituents of the halo of our Galaxy, should their mass lie in the approximate domain $$10^{-6} \lt M/M_\odot \lt 10^{-1}$$. (This range snugly and auspiciously overlaps with that of H/He aggregates heavy enough not to evaporate, but too light to ignite or to be visible.) ‘Massive Halo Objects’ (MHOs), when crossing close to the line-of-sight of a more distant and ‘fixed’ star, would gravitationally ‘microlense’ its image, resulting in a time-dependent achromatic magnification with a predictable shape. Observations now in preparation will monitor ∼ 10⁶ stars in the Large Magellanic Cloud (LMC) during ∼ 1 yr. Barring unsuspected backgrounds, these observations will establish or exclude a halo dark mass constituency in the quoted mass range. In the event of a discovery, the next most crucial question will concern the MHO mass distribution, well concealed in the raw data by convolutions with the distributions of various unobservable quantities, that describe the motion and position of individual MHOs. Wc discuss the extraction from the data of this MHO mass function and prove that its moments are explicitly related to the moments of the observed distribution of durations of the microlensing events.