July 28, 2009
Outline 1 History Historical Background
Outline 1 History Historical Background 2 to the project Theory: Deflection angle, lensing diagram, and equations
Outline 1 History Historical Background 2 to the project Theory: Deflection angle, lensing diagram, and equations 3
Outline 1 History Historical Background 2 to the project Theory: Deflection angle, lensing diagram, and equations 3 4 Concluding remarks
Historical Background History Historical Background 1919: Verification of Einstein s prediction of 1.74 deviation of grazing light rays by the sun.
Historical Background Historical Background 1919: Verification of Einstein s prediction of 1.74 deviation of grazing light rays by the sun. 1936: Einstein publishes a paper: Lens-like action of a star by the deviation of light in the gravitational field
Historical Background Historical Background 1919: Verification of Einstein s prediction of 1.74 deviation of grazing light 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 observational ideas
Historical Background Historical Background 1919: Verification of Einstein s prediction of 1.74 deviation of grazing light 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 observational ideas 1979: Walsh, Carswell, and Weymann observe the fist case of gravitational lensing - a quasar source lensed by a galaxy
Historical Background Historical Background 1919: Verification of Einstein s prediction of 1.74 deviation of grazing light 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 observational ideas 1979: Walsh, Carswell, and Weymann observe the fist case of gravitational lensing - a quasar source lensed by a galaxy Figure: First multiply imaged quasar Q0957+561 identified as a gravitationally lensed system.
Project Goals History to the project Theory: Deflection angle, lensing diagram, and equations Some of the project goals: Understand gravitational lensing
Project Goals History to the project Theory: Deflection angle, lensing diagram, and equations Some of the project goals: Understand gravitational lensing write tools for calculating and visualizing some gravitational lensing systems images of point and extended sources critical curves and caustics lightcurves
to the project Theory: Deflection angle, lensing diagram, and equations Project Goals Some of the project goals: Understand gravitational lensing write tools for calculating and visualizing some gravitational lensing systems images of point and extended sources critical curves and caustics lightcurves write tools for the ray-shooting method
to the project Theory: Deflection angle, lensing diagram, and equations Lensing Diagram I O S θ I θs α L S L Lens Plane Source P lane The Len s equation θ = θ S + D LS D OS α( θ)
to the project Theory: Deflection angle, lensing diagram, and equations Definitions and Approximations O θi S L Lens Plane The Len s equation θ = θ S + D LS D OS α( θ) θs α S L I Source Plane Thin lens approximation Small angle approximation Works for most practical purposes Distances are angular diameter distances
The deflection angle History to the project Theory: Deflection angle, lensing diagram, and equations For point mass M, light ray grazing at a distance R from the mass, Newtonian Calculation General Relativity α = 2GM c 2 1 R α = 4GM c 2 1 R
The simplest case History Point-mass lens of mass M
The simplest case Point-mass lens of mass M Lens equation: θ = θ S + D LS D OS 4GM D OL θ
The simplest case History Point-mass lens of mass M Lens equation: θ = θ S + D LS D OS 4GM D OL θ Two images: θ ± = 1 ( ) θ S ± θs 2 2 + 4θ2 E
The simplest case History Point-mass lens of mass M Lens equation: 1 θ = θ S + D LS D OS 4GM D OL θ Two images: θ ± = 1 ( ) θ S ± θs 2 2 + 4θ2 E y (in units of D OS θ E ) 0.5 0 0.5 Source disc Distorted images (arcs) 1 Einstein ring θ S = 0 = the image is a ring Magnification 1 0.5 0 0.5 1 x (in units of D OS θ E )
Magnification History Images are distorted and hence project different solid angle to our telescopes
Magnification Images are distorted and hence project different solid angle to our telescopes Unresolved images (Microlensing)
Magnification Images are distorted and hence project different solid angle to our telescopes Unresolved images (Microlensing) Total magnification as a function of time gives a typical lightcurve
Magnification Images are distorted and hence project different solid angle to our telescopes Unresolved images (Microlensing) Total magnification as a function of time gives a typical lightcurve
Star-planet type systems History A small mass (planet) near a larger mass (star)
Star-planet type systems A small mass (planet) near a larger mass (star) Expressions for α, lens equation, magnification get a bit more complicated.
Star-planet type systems History A small mass (planet) near a larger mass (star) Expressions for α, lens equation, magnification get a bit more complicated. Lightcurves are no more symmetric but show planetary perturbations The region where mag is (theoretically) is no more a point in the source plane = more interesting caustic formations (next slide) Total magnification(µ) 45 40 35 30 25 20 15 10 5 0.04 0 0.04 0 0.2 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x s ( in units of D OL θ E ) 10 7 4 0.2 Planetary perturbation (magnified above)
Image formation and caustics in planet-star type lensing Critical curve and image positions 0.2 Caustic and source positions y (in units of D OL θ E ) 1 0.5 0 0.5 1 Critical curve y (in units of D OS θ E ) 0.1 0 0.1 Point source outside caustic (3 images) Point source inside caustic (5 images) Caustic 1.5 1 0.5 0 0.5 1 1.5 x (in units of D OL θ E ) 0.2 0 0.1 0.2 0.3 x (in units of D OS θ E )
Example of planets detection OGLE-2006 BLG-109Lb,c A system similar to Jupiter / Saturn orbiting Sun about 5000 lys away
Ray-shooting History Standard method in use to generate magnification map for a lensing system
Ray-shooting History Standard method in use to generate magnification map for a lensing system Method: Populate a region of the image plane with large amount of photons (i.e. create a grid)
Ray-shooting History Standard method in use to generate magnification map for a lensing system 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 θ S = θ D OL D OS α( θ)
Ray-shooting History Standard method in use to generate magnification map for a lensing system 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 θ S = θ D OL D OS α( θ) the density of mapping is proportional to the magnification
Ray-shooting Standard method in use to generate magnification map for a lensing system 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 θ S = θ D OL D OS α( θ) the density of mapping is proportional to the magnification Wrote a C code to implement this basic idea
Ray-shooting example History 0.2 y (in units of D OS θ E ) 0.1 0 0.1 0.2 0.1 0 0.1 0.2 0.3 x (in units of D OS θ E )
Other models The ones already discussed are point mass models. There are other lensing models listed below, the ones that were studied during the project and 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
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 Program Dr. Weaver: I learned a lot during the summer Everyone! Questions??