The Minimum Total Mass of MACHO[CLC]s[/CLC] and Halo Models of the Galaxy

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Abstract
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 MminMACHO of the MACHOs is 1.7 × 1010 M☉ τ−6.7LMC, where τ−6.7LMC is the optical depth (τLMC) toward the Large Magellanic Cloud (LMC) in units of 2 × 10-7. If ρ(r) is a decreasing function of r, it is proved that MminMACHO is 5.6 × 1010 M☉ τ−6.7LMC. Several spherical and axially symmetric halo models of the Galaxy with a few free parameters are also considered. It is found that MminMACHO ranges from 5.6 × 1010 to ~3 × 1011 M☉ τ−6.7LMC. For a general case, the minimal column density ∑minMACHO of MACHOs is obtained as ∑minMACHO = 25 M☉ pc-2 τ−6.7LMC. If the clump of MACHOs exist only halfway between the LMC and the Sun, MminMACHO is 1.5 × 109 M☉. This shows that the total mass of MACHOs is smaller than 5 × 1010 M☉, i.e., ~10% of the mass of the halo inside the LMC, if either the density distribution of MACHOs is unusual or τLMC 2 × 10-7.
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