On the microlensing optical depth of the Galactic bar

The microlensing probability (optical depth τ) towards the Galactic Centre carries information about the mass distribution of the Galatic bulge/bar, and so it can be used to constrain the very uncertain shape parameters of the bar. We find τ depends on the bar mass, radial profile, angle, axis scalelengths and boxyness by a few simple analytical formulae, which show: (1) τ is largest if the angle α and the axial ratio y₀/x₀ conspire so that y₀/x₀ = tan α. (2) At a fixed field on the minor axis but away from the centre, boxy bars with a flat density profile tend to give a larger optical depth than ellipsoidal bars with a steep profile. (3) Main-sequence sources should have a significantly lower (20-50 per cent lower) optical depth than red clump giants if main-sequence stars are not observed as deep as the bright clump giants. An application to four models constrained by the COBE maps shows most models produce optical depth 2σ lower than MACHO and OGLE observed values even with both a massive bar, 2.8 × 10¹⁰ M_⊙, and a full disc. The high τ argues for a massive ( > 2 × 10¹⁰ M_̇) boxy bar with y₀/x₀ ≈ tan α and α < 20° and with a flat radial profile up to corotation.