New calculation method of multiple gravitational lensing system F. Abe Nagoya University 18th International Conference on Gravitational Microlensing, Santa Barbara, 21st Jan 2014 Contents • Introduction • Lensing equation • Matrix expression • Iteration • Remaining problems • Summary Triple lens system (two planets, OGLE-2006-BLG-109) Quasar microlensing (Garsden, Bate, Lewis, 2011, MNRAS 418, 1012) Multiple lenses cause complex magnification pattern!! Calculation methods • Single lens • Simple quadratic equation (Liebes 1964) • Binary lens • Quintic equation (Witt & Mao 1995, Asada 2002) • Inverse ray shooting (Schneider & Weiss 1987) • Triple lens and more • 10th order polynomial equation (Rhie 2002) • Inverse ray shooting (Schneider & Weiss 1987) • Perturbation (Han 2005, Asada 2008) Lensing configuration Lens plane θ and β are normalized by β → θ, j = 1, m m: number of images Source plane Lensing equation θy βy Image θ Source Lens qi ? β Lensing equation is difficult to solve θｘ Single source makes multiple images Observer DL DS βｘ Lensing equation Lensing equation Scalar potential Lensing Straight projection Jacobian matrix Jacobian matrix Jacobian determinant and magnification Jacobian determinant Magnification θy Magnification map on the lens plane To get magnification map on the source plane: β → θ, j = 1, m m: number of images β = j = 1 θ θx Linear expression , : infinitesimally small Inverse matrix Calculation of image position : initial point on the source plane exactly traced from a point on the lensing plane : a target point on the source plane close to : first approximation of the image position corresponding to : second approximation of the image position corresponding to Iteration Calculation of image position Lens plane Source plane Lensing equation θy βy Image θ0 Source θ21 Lens qi β0 Observer θｘ β1 βt βｘ DL DS Iteration example Problems in and • This method only finds an image close to . • To find other images, we must try other . • If steps over caustic, calculation become divergent. So we need to select other . Summary • Analytic form of Jacobian matrix is derived for general multiple lens system • Using Jacobian determinant, magnification on the lens plane can be calculated • Approximate image position can be calculated from a close reference source point which is exactly traced from lens plane • Calculation to get image position converges in 3-5 times iteration • Although there are problems to get reference point, this method may be useful for future multiple lens analyses Thank you!