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

By extending the constant-acceleration analysis of Smith, Mao, & Paczynski 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 \tilde v_j = -(3/4)\csc\beta_\ec(\cos^2\psi\sin^2\beta_\ec+\sin^2\psi)^{3/2}v_earth, where \beta_\ec is the ecliptic latitude, \psi is the phase of the Earth’s orbit relative to opposition at the time of the event maximum, and v_earth=30 km/s is the speed of the Earth. The degeneracy becomes important when the lens projected velocity \tilde v is of order \tilde v_j. For events toward the Large Magellanic Cloud, \tilde v_j\simeq (3/4)v_earth, 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, \tilde v_j ranges from about 0.2 km/s at the solstice to about 200 km/s 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/2pi.