Progress on the algorithm of multiple

lens analysisF. Abe

Nagoya University

20th Microlensing Workshop, IAP, Paris, 15th Jan 2016

Contents• Introduction• Lensing configuration and the problem• Matrix expression• Successive approximation• Flow of the calculation (Random number algorithm)• Demonstration• Summary

Reported at Santa Barbara 2014

Multiple lens systems

Two planets, OGLE-2006-BLG-109Gaudi, et al., 2008, Science 319, 927

Extrasolar moonLiebig and Wambsganss, 2010, A&A 520 A68

Binary+planet, OGLE-2013-BLG-341Gould, A., et al., 2014, Sci 345, 46

Future microlensing surveys from space will find a number of small anomalies because of the high photometric precisions!

Past attempts• Elegant algebraic approache• Binary (quintic equation, Witt & Mao 1995, Asada 2002)• Triple lens (10th order polynomial equation, Rhie 2002)• Difficult for more than fourfold lenses

• Brute-force numerical approach• Inverse-ray shooting (Schneider & Weise 1987)• Needs large computing power

• Approximate perturbative approache• Superposition of binary (Han 2005, Asada 2008)• Limitations (central caustics, no interference, …)

New method: Non-elegant successive numerical approach

Dead end?

Needs lots of money

Not enough for all purposes

Lensing configurationθy

θ ｘ

βy

β ｘ

Observer

Lens plane

Source plane

DL

DS

β⃑

SourceImage

θ⃑ Lens qi

�⃑�𝑖

Lensing equation

�⃑�= 𝑓 ( �⃑� )

Single source makes multiple imagesLensing equation is difficult to solve

θ⃑ β⃑and are normalized by

, j = 1, mm: number of images

=

Jacobian matrixScalar potential

Jacobian determinant and magnification

Jacobian determinant

Magnification of an image

= Total magnificationm : number of images

Linear approximation

Inverse matrix

, : infinitesimally small

Real source position

Traced source position from

Initial image position

Better image position

�⃑�1=�⃑�0+𝐶 ( �⃑�0 )+ ( �⃑�𝑡− �⃑� ( �⃑�0 ))Feed back We can get unlimited

precision by repeating feedback!

But we need to find all images

Random number trial

Host

Planet

Image

Planet

Large image

Small imageUniform trials sometimes loose those images

Denser trial around the planet and the host.

30 trials for each (planets and host) lensing zones

Grid trial : inefficientUniform random trial : loose small images

Flow of the calculation• Initial point: random selection

in a lensing zone (denser around the center)• Successive approximation to

get an image position• Repeat 30 times for a lensing zone• Repeat for all lensing zones

Select New θ (random in

lensing zones)w

Successive approximation

( < 20 steps)

Repeat

Images

Magnification

Magnification map can be produced in 10-15 minutes

Demonstration:Fourfold lensesThree planet system• If the source star is outside

of the caustics, five images are produced

• Three images are close to the planets Host

Source

Source trajectory

Planet 1 q = 0.005

Planet 2q = 0.003

Planet 3q = 0.006

Critical curves

Caustics

Images

~ 1 mas

Demonstration:Fourfold lensesThree planet system• A pair of images are produced

when the source star step into a caustic

• The images are disappeared when the source star go outside the caustic

Time (arbitrary unit)

The light curveM

agni

ficati

on

Summary• Using successive approximation and repeating random number trial,

images are found successfully for fourfold lens system• Basically there is no limitation on the number of lenses• What we need to do next are• Confirmations• Optimization of the algorithm• Analysis of real data

Thank you!