The Minimum Total Mass of MACHOs and Halo Models of the Galaxy
  If the density distribution \rho (r) of MACHOs is spherically symmetric with respect to the Galactic center, it is shown that the minimal total mass M_{min}^{{ MACHO}} of the MACHOs is 1.7\times 10^{10}\sol \tau_{-6.7}^{{ LMC}} where \tau_{-6.7}^{{ LMC}} is the optical depth (\tau^{{ LMC}}) toward the Large Magellanic Cloud (LMC) in the unit of 2\times 10^{-7}. If \rho (r) is a decreasing function of r, it is proved that M_{min}^{{ MACHO}} is 5.6\times 10^{10}\sol \tau_{-6.7}^{{ LMC}}. Several spherical and axially symmetric halo models of the Galaxy with a few free parameters are also considered. It is found that M_{min}^{{ MACHO}} ranges from 5.6\times 10^{10}\sol \tau_{-6.7}^{{ LMC}} to \sim 3 \times 10^{11}\sol \tau_{-6.7}^{{ LMC}}. For general case, the minimal column density \Sigma_{min}^{{ MACHO}} of MACHOs is obtained as \Sigma_{min}^{{ MACHO}} =25 \sol { pc}^{-2}\tau_{-6.7}^{{ LMC}}. If the clump of MACHOs exist only halfway between LMC and the sun, M_{min}^{{ MACHO}} is 1.5\times 10^9\sol. This shows that the total mass of MACHOs is smaller than 5 \times 10^{10}\sol , i.e. \sim 10\% of the mass of the halo inside LMC, either if the density distribution of MACHOs is unusual or \tau^{{ LMC}}\ll 2\times 10^{-7}. 
T. Nakamura, Y. Kan-ya and R. Nishi