CLOSE/WIDE DEGENERACY IN CENTRAL PERTURBATIONS OF PLANETARY LENSING

We investigate the degeneracy in the pattern of central microlensing perturbations of a pair of planetary systems where the planets are located from the primary with projected separations in units of the Einstein radius s and $s^{-1}$, respectively. From this, we confirm the fact that although alike, the patterns of central perturbations induced by a close (s < 1) planet and a wide (s > 1) planet are not identical and the degree of difference depends on the planet/primary mass ratio and the planet-primary separation. We find that the difference can be greater than 5% for planetary systems with lensing parameters located in the parameter space of (1/1.8 < |s| < 1.8, q > $5{\times}10^{-3}$), (1/1.3 < |s| < 1.3, q > $1{\times}10^{-3}$), and (1/1.2 < |s| < 1.2, q > $5{\times}10^{-4}$), where q represents the planet/primary mass ratio. Although this range occupies a small fraction of the entire parameter space of planetary systems, we predict that the chance of resolving the close/wide degeneracy would not be meager considering that the planet detection efficiency is higher for planets with resonant separations (s $\sim$ 1) and heavier masses. We also find that the differences between the perturbation patterns are basically caused by the effect of the planetary caustic. This explains the tendency of the perturbation difference where (1) the difference increases as the planet/primary mass ratio increases and the separation approaches the Einstein radius, (2) the region of major difference is confined within the region around the line connecting the central and the planetary caustics, and (3) a wide (close) planetary system has a more extended central perturbation region toward the (opposite) direction of the planet.