Gravitational Lensing

Nelson Leung 

Kwok Hui Kin

Some History 

• In 1916, Albert Einstein’s General Relativity 

was published predicting light would be 

bend by gravity gravity. 

• Einstein himself thought that this 

iinformation f ti would ld bbe useless. 

l

• In 1919, 1919 this effect was confirmed 

by observation. 

• Einstein’s Einstein s General Relativity was 

correct again in its prediction. 

• But Einstein was wrong g in saying y g 

Gravitational Lensing is useless. 

• In this century since the first 

observational confirmation of 

Gravitational Lensing, it has 

became one of the most 

important astrophysical tool.

1. Deflection Angle 

↳ Deflection angle in Newtonian Mechanics 

↳ Review of General Relativity 

22. Introduction to Gravitational Lensing 

↳ Basic Theory 

‐ Lens Equation 

‐ Einstein’s Ring 

↳ Types of Lensing 

‐ Strong Lensing 

‐ Weak Lensing 

‐ Microlensing 

3. Applications of Gravitational Lensing 

↳ Observation of quasars 

↳ Discovery of Exoplanets 

‐ Caustic Crossing 

↳ Mass Profiles of Clusters 

‐ Reconstuction of mass distribution 

‐ Evidence of Dark Matter 

Outline

Deflection angle 

Introduction to Gravitational Lensing 

Applications 

Derive deflection angle by Newtonian mechanics 

By Hyperbolic orbital equation, 

When r = infinte, then cos(θ∞)= ‐1/ε, 

We can obtain the deflection angle by putting 

Energy of photon E=mc2 gy p /2, / , and momentum l = mc, , and θ∞= ∞ 90°-α 

This equation is analogous to 

the one derived from the 

General Relativity by a factor 

of 2!!

Deflection angle 

Introduction to Gravitational Lensing 

Review of GR 

Applications 

• Space‐time curved by mass 

• Light travels the shortest distance

D LS 

D L 

Deflection angle Introduction to Gravitational Lensing 

Applications 

The General Lens equation 

Adopted from : ‘Gravitational Lenses’, P.Schneider, J,Ehlers,E.E.Falco. 

ξ 

S L 

From Einstein’s deflection law, 

From geometry 

• Distance measured by redshift 

and the Hubble’s Law 

(1) 

(2) 

(3) 

(4)


Applications 

Multiple images from the same source 

Solve eq(4) for θ, we get 

θ ‐< < θ θE and θ +> > θ θE In 2D plane, we can observed 2 images 

from the same source, source one is inside and 

the other one is outside the Einstein ring.

Einstein Ring Applications 


g 

When the source, lens and observer are collinear, i.e. β(θ) = 0, 

• Ring‐like image with angular radius θ E can be observed. 

Object name: SDSS J162746.44‐005357.5


Types of Lensing 

Applications

Types of Lensing Applications 


Lens galaxy 

The gravitational lens G2237 + 0305 

Adopted from: 

http://hubblesite.org/newscenter/archive/releases/1 

990/20/image/a/ 

strong lensing 

Gravitational Lensing in the Galaxy Cluster Abell 2218 

 NASA / A. Fruchter / STScI 

• Lensing effect is strong enough to produce 

multiple images and giant arcs (red) 

• Confirm multiple images from same source by 

redshift, size, shape


Weak lensing 

Applications 

• Th The most common type of f 

gravitational lensing. 

• Line of sight is large. 

• The deflection of light is very slight. 

• No multiple images but distortion 

• No multiple images but distortion, 

shearing, rotation and magnification.


Microlensing 

Applications 

• The lens mass is too low, mass of a 

planet to a star, for the 

displacement of light to be 

observed easily. 

• The apparent brightening of the 

source may still ill bbe detected. 

d d


Applications 

Brightness g magnification g 

• Light flux F is the product of 

surface brightness g I and solid 

angle dw. Since I is contant, 

F is proportional to dw 

• dw = A/D 2 

• A S/D S 2 < AL/D L 2 => Fs

How cosmologists g and 

astrophysicists 

make use of these effects?

Deflection angle Introduction to Gravitational Lensing Applications 

Nature’s Nature s Telescope 

• Able to collect distant light to a point of focus, 

act as a telescope. p 

All b billi f li h 

• Allow us to observe billions of light‐years away 

object‐ e.g. Quasars.


Observation of Quasars 

What is Quasar? 

• Distant object determined by high redshift 

of electromagnetic energy, and hence 

appears faint 

• Extremely luminous 

• First discovered in late 1950s by radio 

telescope with no corresponding visible 

object 

• Until 1979 two quasar images are 

observed 

observed…


First Observation of Quasars 

0957+561 the first QSO image 

Adopted from : D.walsh, et.al. ‘0957+561 A, B: twin quasistellar object or gravitational lens?’, 

Nature Vol. 279,1979 

• 5.7 arcsecond separation 

• 1.405 redshift (87 billion light‐years) 

• Apparent magnitude m ~ 17 

Th Thanks k tto GL GL, we can now 

see the images of quasars!!


Spectra of observed object A and B. 

Adopted from : D.walsh, et.al. ‘0957+561 A, B: twin quasistellar 

object or gravitational lens?’, Nature Vol. 279,1979 

• Identical relativistic jet, powerful 

jets of plasma emerging from 

massive objects 

• Similar redshift 

• Similar Visible light spectrum 

• Magnification of images by GL 

make it possible to see the images 

• Very unlikely to have identical 

quasars that close to each other. 

Concluding they are multiple 

images g from a same quasar q 

produced by strong gravitational 

lensing.


Caustic Curve 

• Magnification increases when source is approaching 

the h caustic i li line (red ( d li line) ) 

• Microlensing in multiple lens system have caustic 

curve, where the magnification is very high. 

• When the source is inside the caustic contour, more 

image will produce


Discovery y of Exoplanets p 

Light curve of OGLE 2003‐BLG‐235/MOA 2003‐BLG‐53 

Adopted from: www.nd.edu/~bennett/ 

• Microlensing is the only method of 

dt detecting ti planets l t in i the th other th galaxies l i 

until now 

• Without a planet, planet we can only observe hill hill‐ 

shaped curve due to the star (single‐lens 

system) 

• A planet p with star pproduces two spikes p on 

the curve (two‐lens system) 

• Graph analysis shows the existence of 

small mass planet


Adopted from: www.nd.edu/~bennett/ 

Advantage of this method 

• The only y method to find exoplanet p of other 

galaxies. 

• Does not require planets whose orbits happen 

to be perfectly aligned from the astronomers' 

vantage point. (Transit Method—If a planet 

crosses in front of its parent’s star disk, the 

observed brightness of the star drops by a 

small amount depending on the relative size of 

the planet) 

• Relatively short period of observation time, at 

most weeks. (Astrometry ‐ detecting 

extroplanets through observation of the 

fluctuation of motion of star due to planets ‐ 

requires years or even decades) 

• Able to find small planets directly


Mass Profile of Galaxy Clusters 

Reconstruction of mass distribution 

• Mass mapping by GL is valuable because does not 

depend on dynamic state of the cluster/galaxy 

(h (the tradition method h for f searching h mass 

distribution is by their dynamic state) 

• Using strong and Weak Lensing to map the mass 

distribution of the Lens 

• Multiple images of Strong Lensing enable us to 

construct the lens lens’ mass using the lens equation 

with numerical calculations 

• Weak Lens’ shearing effect is the other parameter 

for mass mapping 

• In real situations, parametric models of the 

source and the lens are complicated We 

construct mass model to match observed images 

and find best‐fit parameters 

Adopted d dffrom: Annu. Rev. Astron. Astrophys. h 1999. 37: 127‐189 

source and the lens are complicated. We


Mapping of mass using Strong Lensing 

• Recall this equation equation. In 2D case case, if the double images from 

a same source is identified, this equation can be solved to 

find the lens’ mass. 

• In real situation, where there are many different sources 

producing multiple images, a mass model is proposed and 

fitted to produced the corresponding multiple images.


Mapping of mass using Weak Lensing 

This is called the Mapped potential of the lens. 

pp p 

The integral is the three‐dimensional Newtonian gravitational potential of the 

lensing object projected on the coordinate along the line of sight 

L


Mass mapping by Shearing effect 

shearing 

• The orientation of intrinsic ellipticities of galaxies 

should be almost entirely random 

• Observing shearing of the images images, we can get the 

mass distribution 

shear convergence 

θ = Angular position 

Potential mapped on the coordinate of line 

of sight g


Evidence of Dark Matter 

Adopted from: J. Tyson, G. Kochanski and I. Dell’Antonio, Ap. J. Lett. 498 107 (1998) 

• Combine weak and strong lensing analysis analysis, we can map the mass distribution of 

a cluster. Galaxy cluster CL0024+1654 is an example. 

• Spikes are due to the visible galaxy 

• A smooth background mass density is observed in the region without any 

luminous matter

More Applications… 

Applications 

• Finding Hubble constant 

• Black Hole hunting 

• ….

Conclusion 

In recent years, gravitational lensing has become 

more and more important p in astrophysics p y and 

cosmology. We hope through our presentation 

you could get the basic ideas and understand its 

usefulness in studying our universe.

Thank you

References 

• D.walsh, et.al. ‘0957+561 A, B: twin quasistellar object or gravitational lens?’, Nature Vol. 279,1979. 

• PSchneider P.Schneider, J Ehlers Ehlers, EE E.E. Falco Falco, ‘Gravitational Gravitational Lenses’ Lenses , Springer Springer, 1999 1999. 

• A.Abdo, ‘Gravitational Lensing’, department of physics and astronomy, Michigan State University. 

• Wambsganss, J., ‘Gravitational Lensing in Astronomy’, Living Rev. Relativity 1, (1998), 12. 

• Frittelli Frittelli, SS., Newman Newman, E., E ‘ An Exact Gravitational Lensing equation equation’ arXiv:gr‐qc/9810017v1 arXiv:gr qc/9810017v1, 5 Oct 

1998. 

• www.googlescholar.com/ 

• http://articles.adsabs.harvard.edu/full/1989A%26A...221....1K 

• http://www.aanda.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles 

/aa/full/2002/30/aah3555/aah3555.right.html 

• http://ned.ipac.caltech.edu/level5/Blandford/frames.html 

• http://relativity.livingreviews.org/open?pubNo=lrr‐1998‐12&page=articlese1.html

Gravitational Lensing

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?