Rapidly rotating lenses: repeating features in the light curves of short-period binary microlenses

Matthew T. Penny 1 Eamonn Kerins 1 Shude Mao 0 1 0 National Astronomical Observatories, Chinese Academy of Sciences , A20 Datun Road, Chaoyang District, Beijing 100012 , China 1 Jodrell Bank Centre for Astrophysics , The Alan Turing Building , School of Physics and Astronomy, The University of Manchester , Oxford Road, Manchester M13 9PL A B S T R A C T Microlensing is most sensitive to binary lenses with relatively large orbital separations, and as such, typical binary microlensing events show little or no orbital motion during the event. However, despite the strength of binary microlensing features falling off rapidly as the lens separation decreases, we show that it is possible to detect repeating features in the light curve of binary microlenses that complete several orbits during the microlensing event. We investigate the light-curve features of such rapidly rotating lens (RRL) events. We derive analytical limits on the range of parameters where these effects are detectable, and confirm these numerically. Using a population synthesis Galactic model, we estimate the RRL event rate for a ground-based and a space-based microlensing survey to be 0.32f b and 7.8f b events per year, respectively, assuming year-round monitoring, where f b is the binary fraction. We detail how RRL event parameters can be quickly estimated from their light curves, and suggest a method to model RRL events using timing measurements of light-curve features. Modelling RRL light curves will yield the lens orbital period and possibly measurements of all orbital elements, including the inclination and eccentricity. Measurement of the period from the light curve allows a mass-distance relation to be defined, which when combined with a measurement of microlens parallax or finite-source effects can yield a mass measurement to a twofold degeneracy. With sub-per cent accuracy photometry, it is possible to detect planetary companions, but the likelihood of this is very small. - By monitoring hundreds of millions of stars towards the Galactic bulge and Magellanic Clouds, gravitational microlensing surveys such as the OGLE (Udalski 2003) and MOA (Hearnshaw et al. 2005) detect 1000 microlensing events per year. The light curves of most microlensing events follow the typical Paczy nski (1986) form of a point-mass lens with a point source. However, many event light curves have a more complex form due to the effects of a binary or planetary companion to the lens (Mao & Paczynski 1991; Gould & Loeb 1992), a binary companion to the source (Griest & Hu 1992), microlens parallax (Refsdal 1966; Gould 1992), finite-source size (Gould 1994; Nemiroff & Wickramasinghe 1994; Witt & Mao 1994), or a combination of such. These effects provide additional information about the lens, such as the mass ratio and projected separation in binary or planetary lens events, or massdistance relationships with parallax or finite-source effects. Measurement of both the finite-source size and microlensing parallax allows the lens mass to be solved for uniquely (Gould 1992), and if these occur in a binary or planetary lensing event, then it also allows for the measurement of the companion mass. The complexities of microlensing light curves, to a greater or lesser extent, can all be considered as deviations from the singlelens Paczynski form. The deviations may be relatively minor and can cover the entire light curve, as in most parallax events (e.g. Smith, Mao & Wozniak 2002a), or they can be large and cover only a small fraction of the light curve, as in many binary-lens events (e.g. Kubas et al. 2005; Beaulieu et al. 2006). In binarylens events, these deviations from the single-lens form are caused by a difference in the magnification pattern of the lens. The most prominent features of the binary-lens magnification pattern are caustics, where the magnification of a point source diverges (see Fig. 1). A source passing over a caustic will show a sharp rise in its Figure 1. The magnification pattern of a close topology microlens. The dots denote the lens positions, with the primary lens at negative x. The lens has a mass ratio q = 0.3 and projected separation s = 0.6. Notable features of the magnification pattern are labelled. magnification as it enters and a sharp fall as it leaves. Other, more smooth magnification pattern features can be associated with the caustics in some way. The caustic features of a binary-lens magnification pattern are a natural feature of the binary-lens mapping, z z1 + z z2 which maps the angular position of the source to image positions under the influence of the lens, and where we have used complex coordinates (e.g. z = x + iy; Witt 1990); bars represent complex conjugation, zs is the position of the source, z is the position of the image, z1 and z2 are the positions of the primary and secondary lens components, respectively, and q = M2/M1 is the mass ratio of the lens components. All angles have been normalized to the Einstein ring radius E = DrEl = D1l 4cG2 x(1 x)DsM, (2) where rE is the physical Einstein radius, Dl and Ds are the distance to the lens and source, respectively, M = M1 + M2 is the total lens mass, x = Dl/Ds is the fractional lens distance, and G and c are the gravitational constant and speed of light, respectively. The magnification of an image is given by the determinant of the Jacobian of the lens mapping The magnification diverges when det J = 0, and the solutions of this equation form smooth, closed curves in the image plane, called critical curves, which when mapped back to the source plane form closed, cuspy curves, the caustics (see Figs 1 and 8). In a binary-lens event, the caustics are largest when the projected lens separation s = |z2 z1| 1, that is, the lens components orbit with semimajor axis a rE 23 au. At these separations, there is only a single, so-called resonant, caustic. Lenses with wider separations have two caustics, while those with closer separations have three (Schneider & Weiss 1986). The caustics of close and wide systems are smaller than those of resonant systems, and so the source is less likely to encounter them. Far from the caustics, the binary-lens features tend to be weak and therefore the light curves of binary lenses with very close or very wide orbits (which have correspondingly very small caustics), in most cases, will be indistinguishable from those of a single lens (e.g. Gaudi & Gould 1997). The binary lenses with the strongest light-curve features thus have orbital periods T 1000 d, much longer than the microlensing-event time-scale for a typical Galactic microlensing event, where vt is the relative lenssource transverse velocity (source velocity). The lenses therefore complete only a small fraction of their orbit during the course of the microlensing event, and only a fraction of the events are expected to show detectable signs of orbital motion in their light curves (Gaudi & Gould 1997; Dominik 1998; Ioka, Nishi & Kan-Ya 1999; Konn (...truncated)