Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing...
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![Page 1: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/1.jpg)
Progress on the algorithm of multiple
lens analysisF. Abe
Nagoya University
20th Microlensing Workshop, IAP, Paris, 15th Jan 2016
![Page 2: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/2.jpg)
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
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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!
![Page 4: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/4.jpg)
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
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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
=
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Jacobian matrixScalar potential
![Page 7: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/7.jpg)
Jacobian determinant and magnification
Jacobian determinant
Magnification of an image
= Total magnificationm : number of images
![Page 8: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/8.jpg)
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
![Page 9: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/9.jpg)
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
![Page 10: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/10.jpg)
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
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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
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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
![Page 13: Progress on the algorithm of multiple lens analysis F. Abe Nagoya University 20th Microlensing Workshop, IAP, Paris, 15th Jan 2016.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b337f8b9ab05999bfc0/html5/thumbnails/13.jpg)
Time (arbitrary unit)
The light curveM
agni
ficati
on
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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
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Thank you!