Properties of central caustics in planetary microlensing

To maximize the number of planet detections, current microlensing follow-up observations are focusing on high-magnification events that have a higher chance of being perturbed by central caustics. In this paper, we investigate the properties of central caustics and the perturbations that they induce. We derive analytic expressions for the location, size, and shape of the central caustic as a function of the star-planet separation, s, and the planet/star mass ratio, q, under the planetary perturbative approximation and compare the results with those based on numerical computations. While it has been known that the size of the planetary caustic is alpha q(1/2), we find from this work that the dependence of the size of the central caustic on q is linear, i.e., alpha q, implying that the central caustic shrinks much more rapidly with the decrease of q compared to the planetary caustic. The central caustic size also depends on the star-planet separation. If the size of the caustic is defined as the separation between the two cusps on the star-planet axis ( horizontal width), we find that the dependence of the central caustic size on the separation is alpha( s + s(-1)). While the size of the central caustic depends both on s and on q, its shape, defined as the vertical/horizontal width ratio, R-c, is solely dependent on the planetary separation, and we derive an analytic relation between R-c and s. Due to the smaller size of the central caustic, combined with a much more rapid decrease of its size with the decrease of q, the effect of finite source size on the perturbation induced by the central caustic is much more severe than the effect on the perturbation induced by the planetary caustic. As a result, we find that although giant planets with q greater than or similar to 10(-3) can be detected from the planet-search strategy of monitoring high-magnification events, detecting signals of Earth-mass planets with q similar to 10(-5) will be very difficult. Although the central caustics of a pair of planets with separations s and s(-1) are identical to linear order, we find that the magnification patterns induced by a pair of degenerate caustics of planets with q greater than or similar to 10(-3) are different to the level of being noticed in observations with less than or similar to 2% photometry. Considering that the majority of planets that would be detected by the strategy of monitoring high-magnification events are giant planets, we predict that the s <-> s(-1) degeneracy could be broken for a majority of planetary events from observations with good enough precision.