Microlensing Parallax for Observers in Heliocentric Motion

Motivated by the ongoing Spitzer observational campaign, and the forecoming K2 one, we revisit, working in an heliocentric reference frame, the geometrical foundation for the analysis of the microlensing parallax, as measured with the simultaneous observation of the same microlensing event from two observers with relative distance of order AU. For the case of observers at rest we discuss the well known fourfold microlensing parallax degeneracy and determine an equation for the degenerate directions of the lens trajectory. For the case of observers in motion, we write down an extension of the Gould (1994) relationship between the microlensing parallax and the observable quantities and, at the same time, we highlight the functional dependence of these same quantities from the timescale of the underlying microlensing event. Furthermore, through a series of examples, we show the importance of taking into account the motion of the observers to correctly recover the parameters of the underlying microlensing event. In particular we discuss the cases of the amplitude of the microlensing parallax and that of the difference of the timescales between the observed microlensing events, key to understand the breaking of the microlensing parallax degeneracy. Finally, we consider the case of the simultaneous observation of the same microlensing event from ground and two satellites, a case relevant for the expected joint K2 and Spitzer observational programs in 2016. Subject headings: gravitational lensing: micro NASA Exoplanet Science Institute, MS 100-22, California Institute of Technology, Pasadena, CA 91125, USA Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy Istituto Internazionale per gli Alti Studi Scientifici (IIASS), Via G. Pellegrino 19, 84019 Vietri Sul Mare (SA), Italy Sagan Visiting Fellow