Properties of Central Caustics in Planetary Microlensing

To maximize the number of planet detections, current microlensing follow-upobservations are focusing on high-magnification events which have a higherchance of being perturbed by central caustics. In this paper, we investigatethe properties of central caustics and the perturbations induced by them. Wederive analytic expressions of the location, size, and shape of the centralcaustic as a function of the star-planet separation, $s$, and the planet/starmass ratio, $q$, under the planetary perturbative approximation and compare theresults with those based on numerical computations. While it has been knownthat the size of the planetary caustic is \propto \sqrt{q}, we find from thiswork that the dependence of the size of the central caustic on $q$ is linear,i.e., \propto q, implying that the central caustic shrinks much more rapidlywith the decrease of $q$ compared to the planetary caustic. The central-causticsize depends also on the star-planet separation. If the size of the caustic isdefined as the separation between the two cusps on the star-planet axis(horizontal width), we find that the dependence of the central-caustic size onthe separation is \propto (s+1/s). While the size of the central causticdepends both on $s$ and q, its shape defined as the vertical/horizontal widthratio, R_c, is solely dependent on the planetary separation and we derive ananalytic relation between R_c and s. Due to the smaller size of the centralcaustic combined with much more rapid decrease of its size with the decrease ofq, the effect of finite source size on the perturbation induced by the centralcaustic is much more severe than the effect on the perturbation induced by theplanetary caustic. Abridged.Comment: 5 pages, 4 figures, ApJ accepte