1. Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016

2. Contents Introduction Lensing configuration and the problem Matrix expression Successive approximation Flow of the calculation (Random number algorithm) Demonstration Summary Reported at Santa Barbara 2014

3. Multiple lens systems Two planets, OGLE-2006-BLG-109 Gaudi, et al., 2008, Science 319, 927 Extrasolar moon Liebig and Wambsganss, 2010, A&A 520 A68 Binary+planet, OGLE-2013-BLG-341 Gould, A., et al., 2014, Sci 345, 46 Future microlensing surveys from space will find a number of small anomalies because of the high photometric precisions!

4. Past attempts Elegant algebraic approache Binary (quintic equation, Witt & Mao 1995, Asada 2002) Triple lens (10 th order polynomial equation, Rhie 2002) Difficult for more than fourfold lenses Brute-force numerical approach Inverse-ray shooting (Schneider & Weise 1987) Needs large computing power Approximate perturbative approache Superposition of binary (Han 2005, Asada 2008) Limitations (central caustics, no interference, …) New method: Non-elegant successive numerical approach Dead end? Needs lots of money Not enough for all purposes

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

6. Jacobian matrix Scalar potential

7. Jacobian determinant and magnification Jacobian determinant Magnification of an image Total magnification m : number of images

8. Linear approximation Inverse matrix, : infinitesimally small Real source position Initial image position Better image position Feed back We can get unlimited precision by repeating feedback! But we need to find all images

9. Random number trial Host Planet Image Planet Large image Small image Uniform trials sometimes loose those images Denser trial around the planet and the host. 30 trials for each (planets and host) lensing zones Grid trial : inefficient Uniform random trial : loose small images

10. Flow of the calculation Select New θ (random in lensing zones) w Successive approximation ( < 20 steps) Repeat Images Magnification Magnification map can be produced in 10-15 minutes

11. Demonstration: Fourfold lenses Three planet system If the source star is outside of the caustics, five images are produced Three images are close to the planets Host Source Source trajectory Planet 1 q = 0.005 Planet 2 q = 0.003 Planet 3 q = 0.006 Critical curves Caustics Images ~ 1 mas

12. Demonstration: Fourfold lenses Three planet system A pair of images are produced when the source star step into a caustic The images are disappeared when the source star go outside the caustic

13. Time (arbitrary unit) The light curve Magnification

14. Summary Using successive approximation and repeating random number trial, images are found successfully for fourfold lens system Basically there is no limitation on the number of lenses What we need to do next are Confirmations Optimization of the algorithm Analysis of real data

15. Thank you!