1. New calculation method of multiple gravitational lensing system F. Abe Nagoya University 18 th International Conference on Gravitational Microlensing, Santa Barbara, 21 st Jan 2014

2. Contents Introduction Lensing equation Matrix expression Iteration Remaining problems Summary

3. Triple lens system (two planets, OGLE-2006-BLG-109)

4. Quasar microlensing (Garsden, Bate, Lewis, 2011, MNRAS 418, 1012) Multiple lenses cause complex magnification pattern!!

5. 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 10 th order polynomial equation (Rhie 2002) Inverse ray shooting (Schneider & Weiss 1987) Perturbation (Han 2005, Asada 2008)

6. Lensing configuration θyθy θｘθｘ βyβy βｘβｘ Observer Lens plane Source plane DLDL DSDS Source Image Lens q i Lensing equation ? Lensing equation is difficult to solve Single source makes multiple images and are normalized by

7. Lensing equation Scalar potential Straight projection Lensing

8. Jacobian matrix

9. Jacobian determinant and magnification Jacobian determinant Magnification

10. Magnification map on the lens plane θxθx θyθy To get magnification map on the source plane:

11. Linear expression Inverse matrix, : infinitesimally small

12. 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

13. Calculation of image position θyθy θｘθｘ βyβy βｘβｘ Observer Lens plane Source plane DLDL DSDS Source Image Lens q i Lensing equation

14. Iteration example

15. 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.

16. 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

17. Thank you!