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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 Magnification map on the lens plane Transfer matrix

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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 • Magnification map on the lens plane • Transfer matrix • Approximation • Iteration • Problems • Summary

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

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

  5. Calculation methods • Single lens • Simple quadratic equation (Lieb 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 ? Single source makes multiple images Lensing equation is difficult to solve 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 , j = 1, m m: number of images = θx

  11. Linear expression : infinitesimally small , Inverse matrix

  12. Calculation of image position : reference point on the source plane exactly traced from a point on the lensing plane : a 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 . • One of the methods is to calculate and for mesh points to use as and . • The other method is to solve polynomial equation for similar configuration. • 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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