gravitational lensing

Hans Böhringer LMU Lecture Observational Cosmology II (§ 7) SS 2010 1

Lecture 7: Strong Gravitational Lensing

1.  Light Deflection by Masses 2.  The Lense Equation 3.  Lense Models and Geometric Lense Solutions 4.  Caustics and Critical Lines 5.  Gravitational Lensing by Galaxy Clusters


HST image of Abell 1689


First „Gravitational Arc“ discovered in the Galaxy Cluster A370

Abell 370 at z = 0.375

Background galaxy at z = 0.725


Discovery of Gravitational Lenses

•  Consideration of Newtonian light deflection of stars: Soldner (1804)

•  Deduced from equivalence principle: Einstein (1911); in GR Faktor 2 ! (1915)

•  Light deflection observed at the sun (eclipse) δ = 1.7“ Dyson et al. 1920 •  Zwicky 1937 suggests galaxies as gravitational lenses and lense-telescopes •  Refsdal 1964 suggestion measurement of H0 through the time difference in the light path of multiple images of lensed galaxies

•  First gravitational lense discovered (multiple imafes of a QSO) Walsh et al. 1979

QSO 0957+561 δ = 6“ •  Radio Einstein ring Hewitt et al. (1987)

•  First gravitationally lensing galaxy cluster discovered, A370 Soucail (1987)


Interest in Gravitational Lenses

•  Gravitational lenses can act as light amplifiers „gravitational lense telescopes“

•  An analysis of the lensing effect allows to determine the mass of the lense (galaxy, galaxy cluster, or microlensing object) •  Dependence of the lense equation on cosmologiocal parameters allows cosmological parameter constraints froma detailed lensing analysis ( (i) H0 from light path diffferences, Ωm and Λ from the cosmological model dependence of the diameter distance)

• Gravitational lensing studies of the LSS provides a „direct“ measure of the dark matter distribution


Light Deflection of a Point Mass

Classical treatment of the gravitational deflection of a massive particle with velocity v = c. We have already solved this problem in the consideration of two body relaxation :

The General Relativistic result is:

In GR the deflection angle is twice as large as in the classical case (this is due to the space curvature near the the deflecting mass)


Thin Mass Sheet Approximation

•  Light deflection by a massive object e.g. a galaxy cluster Simplification: the deflection happens only over a very short distance compared to the total light path of the light rays deflection considered in the „lense plane“

•  We only observe small deflection angles such that the process of multiple deflection is simply additive. •  The mass distribution of the lense is projected into the lense plane:

•  the deflection is then determined by the surface mass distribution in the lense plane:


Light Deflection of an Azimutally Symmetric Mass Distribution

For an azmutially symmetric mass distribution in the lense plane (e.g projected sperically symmetric distribution) the light deflection at impact parameter ξ is the same as for a point mass with the same mass value in the center of symmetry :


The Lense equation

observer lense

source

Dls Dol Dos

image


The Lense Equation II

Definition of the critical mass density :

Example : lense with constant surface mass density


Nomenclature


Defining a Deflection Potential Analogy to Electrostatics

Electrostatics in analogy (but 3dim. instead of 2dim. :


Image Distortion and Magnification


Magnification

The magnification of the image is given by the determinant of the magnification matrix :


Caustic and Critical Line for Spherically Symmetric Lense

Caustic (source plane) critical lines (image plane)

Tangential critical line (caustic) is yellow, radial critical line (caustic) is blue [from Hattori et al. 1999]


Caustic and Critical Line for Elliptical Lenses

The source near the cusp in the caustic gets strongly magnified


Caustic and Critical Line for Elliptical Lenses

Case 1:

Case 2:

From Hattori et al. 1999


Lense with Homogeneous Mass Distribution

observer source

The critical homogenous lense provides an ideal focus for a definite focal length - the observer only sees a very extended object


Einstein Ring

For symmetrical elnses:

for θs = 0


Prinzipal of the Gravitational Lensing


Lensing Equation for a Point Mass

Reformulation of the lense equation by means of θE :

Two images of the source :

Parity of the images: Positive in both cases

image1 source image2

1 2 1 2 1 2


Imaging of a Point Mass Lense

θE

x

B+

B-

Q


Magnification and Light Amplification

θ- θ+ Q M

Magnification + light amplification = „magnification“

The surface brightness is conserved in the lensing process

for µ is negative change of parity


Singular Isothermal Sphere

Mass distribution of the SIS :

Einstein Radius :

Multiple images are only: observed when the source lays inside the Einstein radius

With magnification :


Condition for Strong Lensing in Clusters

Einstein radius for galaxy clusters :

Critical density for strong lensing :


Goemetric Construction of the Imaging Process

Two determining equation have to be fullfiled:

α(θ)

θ

The solution is given by the intersection of the two curves

B1

Q B2


Refsdal & Surdej 1994, Rep. Prog. Phys. 56, 117

Geometrical construction for the Point Mass Lense


Construction for the SIS-Lense



For the Mass Distribution of a Spiral Galaxy



Construction for the Homogeneous Distribution



Model-Lensen for a Laboratory Experiment

(a) Point mass, (b) SIS potential, (c) spiral galaxy , (d) homogeneous disc (e) truncated homogeneous disc



Typical Lensed Images

a,b) slightly asymmetricSIS Lense (partial Einstein ring),

(c,d) two images merge into one

caustics images


Typical Images

(e,f) configuration with two images

(g,h) two images, where one is a merged image of originally three images of a four image lense


Lensing Effect

Effect of the Black Hole of the Mass of Jupiter in the middle of the Washington Mall


Abell 2218


A 2218 used as Gravitational Telescope

Spektrum einer sternbildenden Galaxie bei z= 5.6


Abell 2390


. Abell 370


The Giant Arc in Abell 370


Example: Lensing and X-ray Mass of the Cluster A2390

HST fom Kneib, Pello


Comparion of the Mass Determination for Abell 2390

From CHANDRA [Allen et al. 2000]

X-ray data from ROSAT & ASCA

gravitational lensing

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