### New calculation method of multiple gravitational lensing system

```New calculation method of
multiple gravitational lensing
system
F. 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
Lens plane
θ and β are normalized by
β → θ, j = 1, m
m: number of images
Source plane
Lensing equation
θy
βy
Image
θ
Source
Lens qi

?
β
Lensing equation
is difficult to solve
θｘ Single source
makes multiple
images
Observer
DL
DS
βｘ
Lensing equation
Lensing equation
Scalar potential
Lensing
Straight
projection
Jacobian matrix
Jacobian matrix
Jacobian determinant and magnification
Jacobian determinant
Magnification
θy
Magnification map on the lens plane
To get magnification map
on the source plane:
β → θ, j = 1, m
m: number of images
β =
j = 1  θ
θx
Linear expression
,
: infinitesimally small
Inverse matrix
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
Lens plane
Source plane
Lensing equation
θy
βy
Image
θ0
Source
θ21
Lens qi
β0

Observer
θｘ
β1
βt
βｘ
DL
DS
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!
```