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HistoryIntroduction
Lensing systemsFinal words
REU Final Presentation
Saroj Adhikari (mentor: Dr. Weaver)
July 28, 2009
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Outline
1 HistoryHistorical Background
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Outline
1 HistoryHistorical Background
2 IntroductionIntroduction to the projectTheory: Deflection angle, lensing diagram, and equations
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Outline
1 HistoryHistorical Background
2 IntroductionIntroduction to the projectTheory: Deflection angle, lensing diagram, and equations
3 Lensing systemsSchwarzschild lensTwo-point-mass lensesRay-shooting method
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Outline
1 HistoryHistorical Background
2 IntroductionIntroduction to the projectTheory: Deflection angle, lensing diagram, and equations
3 Lensing systemsSchwarzschild lensTwo-point-mass lensesRay-shooting method
4 Final wordsConcluding remarks
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Historical Background
Historical Background
1919: Verification of Einstein’sprediction of 1.74” deviation of grazinglight rays by the sun.
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Historical Background
Historical Background
1919: Verification of Einstein’sprediction of 1.74” deviation of grazinglight rays by the sun.
1936: Einstein publishes a paper:Lens-like action of a star by the
deviation of light in the gravitational
field
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Historical Background
Historical Background
1919: Verification of Einstein’sprediction of 1.74” deviation of grazinglight rays by the sun.
1936: Einstein publishes a paper:Lens-like action of a star by the
deviation of light in the gravitational
field
Theoretical studies and observationalideas
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Historical Background
Historical Background
1919: Verification of Einstein’sprediction of 1.74” deviation of grazinglight rays by the sun.
1936: Einstein publishes a paper:Lens-like action of a star by the
deviation of light in the gravitational
field
Theoretical studies and observationalideas
1979: Walsh, Carswell, and Weymannobserve the fist case of gravitationallensing - a quasar source lensed by agalaxy
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Historical Background
Historical Background
1919: Verification of Einstein’sprediction of 1.74” deviation of grazinglight rays by the sun.
1936: Einstein publishes a paper:Lens-like action of a star by the
deviation of light in the gravitational
field
Theoretical studies and observationalideas
1979: Walsh, Carswell, and Weymannobserve the fist case of gravitationallensing - a quasar source lensed by agalaxy
Figure: First multiply imagedquasar Q0957+561 identified asa gravitationally lensed system.
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Introduction to the projectTheory: Deflection angle, lensing diagram, and equations
Project Goals
Some of the project goals:
Understand gravitational lensing
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Introduction to the projectTheory: Deflection angle, lensing diagram, and equations
Project Goals
Some of the project goals:
Understand gravitational lensing
write tools for calculating and visualizing some gravitational lensingsystems
images of point and extended sourcescritical curves and caustics
lightcurves
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Introduction to the projectTheory: Deflection angle, lensing diagram, and equations
Project Goals
Some of the project goals:
Understand gravitational lensing
write tools for calculating and visualizing some gravitational lensingsystems
images of point and extended sourcescritical curves and caustics
lightcurves
write tools for the ray-shooting method
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Introduction to the projectTheory: Deflection angle, lensing diagram, and equations
Lensing Diagram
~θI~θS
~α
b
O
b
L
b
L′
bS
bS ′
bI
LensPla
ne
Sourc
ePla
ne
The Len’s equation
~θ = ~θS +DLS
DOS
~α(~θ)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Introduction to the projectTheory: Deflection angle, lensing diagram, and equations
Definitions and Approximations
~θI~θS
~α
b
O
b
L
b
L′
bS
bS ′
bI
LensPla
ne
Sourc
ePla
ne
The Len’s equation
~θ = ~θS +DLS
DOS
~α(~θ)
Thin lensapproximation
Small angleapproximation
Works for most practicalpurposes
Distances are angulardiameter distances
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Introduction to the projectTheory: Deflection angle, lensing diagram, and equations
The deflection angle
For point mass M , light ray grazing at a distance R from the mass,
Newtonian Calculation
α =2GM
c2
1
R
General Relativity
α =4GM
c2
1
R
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
The simplest case
Point-mass lens of mass M
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
The simplest case
Point-mass lens of mass M
Lens equation:
θ = θS +DLS
DOS
4GM
DOLθ
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
The simplest case
Point-mass lens of mass M
Lens equation:
θ = θS +DLS
DOS
4GM
DOLθ
Two images:
θ± =1
2
(
θS ±
√
θ2S
+ 4θ2E
)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
The simplest case
Point-mass lens of mass M
Lens equation:
θ = θS +DLS
DOS
4GM
DOLθ
Two images:
θ± =1
2
(
θS ±
√
θ2S
+ 4θ2E
)
θS = 0 =⇒ the image is a ring
Magnification
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
x (in units of DOS
θE)
y (in
uni
ts o
f DO
Sθ E
)Einstein ring
Source disc
Distorted images(arcs)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Magnification
Images are distorted andhence project differentsolid angle to ourtelescopes
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Magnification
Images are distorted andhence project differentsolid angle to ourtelescopes
Unresolved images(Microlensing)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Magnification
Images are distorted andhence project differentsolid angle to ourtelescopes
Unresolved images(Microlensing)
Total magnification as afunction of time gives atypical lightcurve
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Magnification
Images are distorted andhence project differentsolid angle to ourtelescopes
Unresolved images(Microlensing)
Total magnification as afunction of time gives atypical lightcurve
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Star-planet type systems
A small mass (planet) near alarger mass (star)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Star-planet type systems
A small mass (planet) near alarger mass (star)
Expressions for ~α, lensequation, magnification get abit more complicated.
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Star-planet type systems
A small mass (planet) near alarger mass (star)
Expressions for ~α, lensequation, magnification get abit more complicated.
Lightcurves are no moresymmetric but show planetaryperturbations
The region where mag is ∞
(theoretically) is no more apoint in the source plane =⇒more interesting causticformations (next slide)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
35
40
45
xs ( in units of D
OLθ
E)
Tot
al m
agni
ficat
ion(
µ)
0.2
47
10
0 0.2−0.04
0
0.04
Planetary perturbation(magnified above)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Image formation and caustics in planet-star type lensing
0 0.1 0.2 0.3−0.2
−0.1
0
0.1
0.2
x (in units of DOS
θE)
y (in
uni
ts o
f DO
Sθ E
)
Caustic and source positions
x (in units of DOL
θE)
Critical curve and image positions
y (in
uni
ts o
f DO
Lθ E)
−1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5
0
0.5
1Point source insidecaustic (5 images)
Caustic
Point source outsidecaustic (3 images)
Critical curve
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Example of planets detection
OGLE-2006BLG-109Lb,c
A system similarto Jupiter /Saturn orbitingSun
about 5000 lysaway
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Ray-shooting
Standard method in use to generate magnification map for a lensingsystem
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Ray-shooting
Standard method in use to generate magnification map for a lensingsystem
Method:
Populate a region of the image plane with large amount of photons(i.e. create a grid)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Ray-shooting
Standard method in use to generate magnification map for a lensingsystem
Method:
Populate a region of the image plane with large amount of photons(i.e. create a grid)calculate the mapping from the image plane to the source plane~θS = ~θ −
DOL
DOS~α(~θ)
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Ray-shooting
Standard method in use to generate magnification map for a lensingsystem
Method:
Populate a region of the image plane with large amount of photons(i.e. create a grid)calculate the mapping from the image plane to the source plane~θS = ~θ −
DOL
DOS~α(~θ)
the density of mapping is proportional to the magnification
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Ray-shooting
Standard method in use to generate magnification map for a lensingsystem
Method:
Populate a region of the image plane with large amount of photons(i.e. create a grid)calculate the mapping from the image plane to the source plane~θS = ~θ −
DOL
DOS~α(~θ)
the density of mapping is proportional to the magnification
Wrote a C code to implement this basic idea
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Ray-shooting example
x (in units of DOS
θE)
y (in
uni
ts o
f DO
Sθ E
)
−0.1 0 0.1 0.2 0.3
−0.2
−0.1
0
0.1
0.2
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Schwarzschild lensTwo-point-mass lensesRay-shooting method
Other models
The ones already discussed are point mass models. There are otherlensing models listed below, the ones that were studied during the projectand codes written to calculate and visualize the models are in bold face.
Singular isothermal sphere (SIS)
Non-singular isothermal sphere (NSIS)
Uniform sheet
Elliptical potential (both singular/non-singular)
Isothermal ellipsoids
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation
HistoryIntroduction
Lensing systemsFinal words
Concluding remarks
Finally..
My basic references:
Gravitational Lensing : Schneider et.al.Singularity theory and gravitational lensing: Petters et al.Many papers on (arxiv, A&A, Phys Rev et.al)
Acknowledgements
KSU REU ProgramDr. Weaver: I learned a lot during the summerEveryone!
Questions??
Saroj Adhikari (mentor: Dr. Weaver) REU Final Presentation