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New calculation method of multiple gravitational lensing system

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

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New calculation method of multiple gravitational lensing system

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  1. New calculation method of multiple gravitational lensing system F. Abe Nagoya University 18th International Conference on Gravitational Microlensing, Santa Barbara, 21st 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 • 10th order polynomial equation (Rhie 2002) • Inverse ray shooting (Schneider & Weiss 1987) • Perturbation (Han 2005, Asada 2008)

  6. Lensing configuration , j = 1, m m: number of images Lensplane Sourceplane and are normalized by Lensing equation θy βy Image Source Lensqi ? Lensing equation is difficult to solve Single source makes multiple images Observer θｘ βｘ DL DS

  7. Lensing equation Lensing equation Scalar potential Lensing Straight projection

  8. Jacobian matrix Jacobian matrix

  9. Jacobian determinant and magnification Jacobian determinant Magnification

  10. Magnification map on the lens plane θy To get magnification map on the source plane: , j = 1, m m: number of images = θx

  11. Linear expression : infinitesimally small , Inverse matrix

  12. Calculation of image position : initial point on the source plane exactly traced from a point on the lensing plane : atarget 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 Lensplane Sourceplane Lensing equation θy βy Image Source 1 0 Lensqi 1 2 t Observer θｘ βｘ DL DS

  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!

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