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

By extending the constant-acceleration analysis of Smith, Mao, & Paczyński 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)cscβ_ec(cos2ψsin2β_ec+sin2ψ)3/2v_⊕, 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_⊕=30kms-1 is the speed of the Earth. The degeneracy becomes important when the lens projected velocity v is of order vj. For events toward the Large Magellanic Cloud, v_j~=(3/4)v_⊕, so this degeneracy 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 summer 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, tE<~yr/2π.