4 Resolution of the MACHO - LMC - 5 Puzzle : The Jerk - Parallax Microlens Degeneracy

By extending the constant-acceleration analysis of Smith, Mao, & Paczy´nski to include jerk, I show that microlens parallax measurements are subject to a four-fold discrete degeneracy. The new degeneracy is characterized by a projected velocity˜v j = −(3/4) csc β ec (cos 2 ψ sin 2 β ec + sin 2 ψ) 3/2 v ⊕ , where β ec is the ecliptic latitude, ψ is the phase of the Earth's orbit relative to opposition at the time of the event maximum, and v ⊕ = 30 km s −1 is the speed of the Earth. The degeneracy becomes important when the lens projected velocity˜v is of order˜v j. For events toward the Large Magellanic Cloud, ˜ v j ≃ (3/4)v ⊕ , so this degen-eracy is important primarily for lenses in the Milky Way disk. In particular, it solves the puzzle of MACHO-LMC-5, whose microlens parallax measurement had yielded mass and distance determinations for the lens that were inconsistent with photometric estimates. Toward the Galactic bulge, ˜ v j ranges from ∼ 0.2 km s −1 at the solstice to ∼ 200 km s −1 at the equinoxes, so the effect of the degeneracy depends strongly on the peak time of the event. The degeneracy applies mainly to events with Einstein timescales, t E yr/2π.