New calculation method of multiple gravitational

lensing systemF. 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

θy

θ ｘ

βy

β ｘObserver

Lens plane Source plane

DL

DS

β⃑

SourceImage

θ⃑ Lens qi

�⃑�𝑖

Lensing equation

?

Lensing equation is difficult to solveSingle source makes multiple images

θ⃑ β⃑and are normalized by

, j = 1, mm: number of images

Lensing equation

Lensing equation

Scalar potential

Straight projection

Lensing

Jacobian matrix

Jacobian matrix

Jacobian determinant and magnification

Jacobian determinant

Magnification

Magnification map on the lens plane

, j = 1, mm: number of images =

θx

θ y

To get magnification map on the source plane:

Linear expression

Inverse matrix

, : infinitesimally small

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

θy

θ ｘ

βy

β ｘObserver

Lens plane Source plane

DL

DS

β⃑ 0

SourceImage

0 Lens qi

�⃑�𝑖

Lensing equation

t

1

1

2

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!