The Minimum Total Mass of MACHOs and Halo Models of the Galaxy

If the density distribution ρ(r) of MACHOs is spherically symmetric with respect to the Galactic center, it is shown that the minimal total mass M min of the MACHOs is 1.7×10M⊙τ −6.7 where τ −6.7 is the optical depth (τ) toward the Large Magellanic Cloud (LMC) in the unit of 2×10−7. If ρ(r) is a decreasing function of r, it is proved that M min is 5.6× 10M⊙τ −6.7 . Several spherical and axially symmetric halo models of the Galaxy with a few free parameters are also considered. It is found that M min ranges from 5.6 × 10M⊙τ −6.7 to ∼ 3 × 10M⊙τ −6.7 . For general case, the minimal column density Σ min of MACHOs is obtained as Σ min = 25M⊙pcτ −6.7 . If the clump of MACHOs exist only halfway between LMC and the sun, M min is 1.5 × 10M⊙. This shows that the total mass of MACHOs is smaller than 5×10M⊙ , i.e. ∼ 10% of the mass of the halo inside LMC, either if the density distribution of MACHOs is unusual or τ ≪ 2× 10. (11 October 1996, Accepted for publication in Apj. Letters.) Subject headings: dark matter — Galaxy: halo — Galaxy: structure — gravitational lensing