new calculation method of multiple gravitational lensing system f. abe nagoya university 18 th...

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!