Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://lnfm1.sai.msu.ru/~rastor/Ralf_Bender/imprs_chapter_13_2007_large_scale_structure.pdf Äàòà èçìåíåíèÿ: Mon Mar 3 12:04:16 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 23:09:11 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: galaxy cluster

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I*M*P*R*S on ASTROPHYSICS at LMU Munich

Astrophysics Introductory Course
Lecture given by:

Ralf Bender and Roberto Saglia in collaboration with: Chris Botzler, Andre Crusius-WÄtzel, Niv Drory, Georg Feulner, Armin Gabasch, Ulrich Hopp, Claudia Maraston, Michael Matthias, Jan Snigula, Daniel Thomas
Powerpoint version with the help of Hanna Kotarba

Fall 2007
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Chapter 13 The Large­Scale Distribution of Galaxies

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13.1 The local galaxy distribution
Galaxies are not uniformly distributed in space. They rather form large filaments, sheets, and superclusters of galaxies, which surround regions with very low galaxy density (voids):

Center for Astrophysics (CFA, Harvard) Survey (from Peebles 1993)

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2dF Galaxy Redshift Survey
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Also in the vicinity of the Milky Way, galaxies are mostly found concentrated in a plane, the so-called Supergalactic Plane. The Supergalactic Coordinate System is a coordinate system with the Milky Way at its centre. The (X, Y ) plane is chosen to be identical with the Supergalactic Plane, the Y axis roughly points in the direction of the Virgo cluster. In the following plots (taken from Peebles 1993, Principles of Physical Cosmology) we show the Supergalactic Plane edge-on and face-on. The Milky Way is at the centre of the figures. The scale of the axes is given in cz (in units of km/s), making the width of each box 8 h-1 Mpc.

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Supergalactic Plane (edge-on)

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Supergalactic Plane (face-on)

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Typical Scales of Large­Scale Structure Galaxies Groups & Clusters Superclusters ~ 10 kpc ~ (0.3 . . . 5) Mpc ~ 50 Mpc

Superclusters of galaxies are the largest known structures in the universe. Distribution of different galaxy types Ellipticals and S0-galaxies prefer regions of high galaxy density, spirals and irregulars are found in lower denisty environment. Nevertheless all galaxy types cluster along filaments and in groups and galaxy clusters.

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Giovanelli et al. (1986) ApJ, 300, 77.
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13.2 The two-point correlation function of galaxies
The two-point correlation function (r) is a quantitative measure of galaxy clustering and is defined via the probability to find pairs of galaxies at a distance r:

dN

pair

= N o2 (1 + (r ))dV1dV2

where No is the mean background density and dV1 and dV2 are volume elements around the two positions under consideration. The two-point correlation function is related to the relative overdensity (x) = /o because we also have:

dN

pair

2 = ( x ) dV1 ( x + r ) dV2 = o (1 + ( x ))(1 + ( x + r ))dV1dV2

Averaging over a large volume removes the linear terms in (x) and we obtain:

< dN
and therefore:

pair

2 >= o (1+ < ( x ) ( x + r ) >)dV1dV2

(r ) =< ( x ) ( x + r ) >

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Observationally one obtains averaged over all galaxy types:

r (r ) = ro

-

with: = 1.8 and ro = 5/h Mpc (h = Ho/100km/s/Mpc) which is valid for scales from 100kpc to 10Mpc. Beyond 10Mpc the correlation function falls more rapidly.

angular correlation function of galaxies from APM survey Maddox et al. 1990, MN 242, 43

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13.3 The Local Group
The Milky Way belongs to a loose collection of galaxies called the Local Group. The brightest members of the Local Group Milky Way, and M33, three spiral galaxies. there are no elliptical galaxies found in the types in the Local Group members are the Magellanic Cloud) and dwarf ellipticals. are the Andromeda Galaxy (M31), the Apart from M32 (which is not very typical) Local Group. The most frequent galaxy irregulars (like the Large and the Small

The total number of galaxies known to belong to the Local Group is about 40, but there probably exists a number of dwarf galaxies which may have remained undetected (especially behind the Milky Way plane). All Local Group galaxies are gravitationally bound (M31 approaches the Milky Way with 120 km/s).

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The distribution of Local Group members in space (Cambridge Atlas of Astronomy Third Edition, Cambridge 1994)

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List of Local Group Members The following table gives the Names, the celestial coordinates and (for the equinox 2000), the Hubble type, the distance D (in kpc), the absolute V magnitude MV , and the radial velocity V0 (in km/s) of Local Group galaxies. The data were taken from M. Irvin's page on the local group (http://www.ast.cam.ac.uk/~mike/local members.html).

Name M31 Galaxy M33 LMC IC 10 NGC 6822 M32 NGC 205 SMC NGC 3109 NGC 185 IC 1613 NGC 147 Sextans A Sextans B WLM Sagittarius Fornax Pegasus NGC 224 NGC 598 00 17 01 05 00 19 00 00 00 10 00 01 00 10 09 23 18 02 23

Coordinates 40.0 42.4 31.1 24.0 17.7 42.1 40.0 37.6 51.0 00.8 36.2 02.2 30.5 08.6 57.4 59.4 51.9 37.8 26.1 +40 -28 +30 -69 +59 -14 +40 +41 -73 -25 +48 +01 +48 -04 +05 -15 -30 -34 +14 59 55 24 48 01 56 36 25 06 55 04 51 14 28 34 45 30 44 28

Type Sb Sbc Sc Irr Irr Irr E2 E5 Irr Irr E3 Irr E4 Irr Irr Irr dE7 dE3 Irr

D(kpc) 725 795 49 820 540 725 725 58 1260 620 765 589 1450 1300 940 24 131 759

Mv -21.1 -20.6 -18.9 -18.1 -17.6 -16.4 -16.4 -16.3 -16.2 -15.8 -15.3 -14.9 -14.8 -14.4 -14.3 -14.0 -14.0 -13.0 -12.7

Vo(km/s) -299 -180 270 -343 -49 -190 -239 163 403 -208 -236 -157 325 301 -116 140 53 -181

Name And VII Leo I Leo A And II And I And VI SagDIG Antlia Sculptor And III Leo II Cetus Sextans Phoenix LGS 3 Tucana Carina And V Ursa Minor Draco Cas Dw DDO 74 DDO 69 23 10 09 01 00 23 19 10 00 00 11 00 10 01 01 22 06 01 15 17

Coordinates 24.1 05.8 56.5 13.5 43.0 49.2 27.9 01.8 57.6 32.6 10.8 23.6 10.6 49.0 01.2 38.5 40.4 07.3 08.2 19.2 +50 +12 +30 +33 +37 +24 -17 -27 -33 +36 +22 -11 -01 -44 +21 -64 -50 +47 +67 +57 25 33 59 09 44 18 47 05 58 12 26 19 24 42 37 41 55 22 23 58

Type dE3 dE3 Irr dE3 dE0 dE3 Irr dE3 dE dE6 dE0 dE4 dE4 Irr Irr/dE dE5 dE4 dE dE5 dE3

D(kpc) 760 270 692 587 790 775 1150 1150 78 790 230 775 90 390 760 900 87 810 69 76

Mv -12.0 -12.0 -11.7 -11.7 -11.7 -11.3 -11.0 -10.7 -10.7 -10.2 -10.2 -10.1 -10.0 -9.9 -9.7 -9.6 -9.2 -9.1 -8.9 -8.6

Vo(km/s) 285 +26

DDO 209 NGC 221

Peg Dw

-79 361 107 76 224 56 -277 223 -250 -289

DDO 236 DDO DDO DDO DDO DDO 8 3 75 70 221

DDO 93

DDO 216

DDO 199 DDO 228

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The cumulative luminosity function of Local Group galaxies is consistent with a Schechter function. (van den Bergh 1992, A& A, 264, 75)

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13.4 Galaxy Clusters: overview

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Rich clusters of galaxies are the most massive virialized, high-overdensity systems known. In the optical light galaxy clusters have the following ranges of properties: Richness (number of cluster galaxies with luminosities 2 magnitudes dimmer than the third brightest cluster galaxies): 30-300 galaxies Radius (where the surface density of galaxies drops to 1% of the core density): 1-2 Mpc Radial velocity dispersion: 400-1400 km/s Mass (r<1.5 Mpc): 1014 -1015 M Optical B-Band luminosity: (r<1.5 Mpc): 1011 -1013 L Mass-to-light ratio: 300M / L 10-5 -10-6 Mpc-3 Cluster number density: Cluster correlation scale: 22 ± 4Mpc Fraction of galaxies with L>L* in clusters: ~5% Some important optical cluster catalogues are: By visual inspection of photographic plates: Abell (1958, ApJS 3, 211); Abell et al. (1989, ApJS, 70, 1); Zwicky et al. (1961-68) From (deep) CCD (multicolor) images: Postman et al. (1996, AJ, 111, 615), Gladders & Yee (2000, AJ, 120, 2148), Goto et al. (2002, AJ, 123, 1808, SLOAN)
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Two nearby clusters

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(Part of) the Virgo Cluster
IMPRS Astrophysics Introductory Course

central part of Coma cluster with two cDs.
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Two nearby clusters
Virgo Cluster:

Nearest large galaxy cluster with more than 2000 galaxies brighter than MB -14 (LB ~ 107.8L) Distance ~ 17Mpc (dependent on H0) Extend ~ 10° ^= 3Mpc â 3Mpc Irregular cluster, densest regions dominated by ellipticals Velocity dispersion of galaxies about 600km/s
Coma Cluster:

One of the most luminous clusters known Distance ~ 100Mpc (dependent on H0) Regular cluster with probably subcluster merging from SW Dominated by ellipticals and S0s, two central cDs and one in subcluster Velocity dispersion of galaxies about 1000km/s Strong X-ray source

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Spatial distribution and galaxy content
The King profile used for globular clusters or ellipticals describes well the cluster galay number density:

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r2 ng (r ) = n0 1 + 2 Rcore where r is the radial distance from the center, Rcore the core radius (0.1-0.25 Mpc) and n0 the central galaxy number density ( 103 Mpc -3 )
Cluster type Regular clusters Intermediate clusters Irregular clusters Field E 35% 20% 15% 10% S0 45% 50% 35% 20% Sp 20% 30% 50% 70%

-3/ 2

Morphology-density relation Evolution with redshift: Butcher-Oemler effect
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Morphology-Radius Relation:

Virgo

Fornax

Ferguson, Sandage (1989) ApJ, 346, L53. Ellipticals and S0's are more concentrated than spirals and irregulars.
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Distant clusters: EDisCS
CL1354.1-1231 CL1054.4-1245 z=0.76 CL1037.5-1243 z=0.75 z=0.58 CL1232.3-1250 CL1202.4-1224 z=0.54 z=0.42

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13.5 X-Ray Gas in Galaxy Clusters

Coma cluster (left: optical image, right: X-ray image)

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The baryonic gas is compressed in the deep cluster potential wells and shock-heated up to X-ray emitting temperatures T. The X-ray spectra show the characteristics of Bremsstrahlung of a 108K hot gas. Therefore the emissivity at frequency is: 32 Z 2 ne ni ( )= 3mec 3 2 -h exp 3kmeT kT g ff (T , )

where g ff is the Gaunt factor. Integrating over frequency one gets the volume emissivity:

= 2.4 10-27 T

-1/2

2 erg ne 3 cm s

The cooling time of the plasma is: t

cool

=

3nekT



1011 T ne

1/2

[s]

2 X-ray surface brightness distribution ( model): S(R)=S(0) 1+R 2 / rcore

(

)

-3 +1/ 2

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X-ray cluster properties for rich clusters:
Temperature: Luminosity: Core radius: Gas mass: Fe abundance: 2 - 14keV or 2 â 106 - 108 K 10
42.5

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- 10 erg / s
45

For the center of the Coma cluster:

0.1-0.2 Mpc 10-3 cm
-3

L 1044 erg / s ne 10-3 cm
-3



cool gas

1010 yrs 1013 M

Central electron density: ne M
gas

1013 - 1014 M

M

1/3 solar

Important X-ray catalogues: BÆhringer et al. (2001, 2004): clusters with z<0.45 from the ROSAT all-sky survey Rosati et al. (1998): clusters with z<1.2 from ROSAT pointed observations Most distant X-ray cluster: z=1.39 (Mullis et al. 2005, ApJ 623, 185)

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Masses of galaxy clusters From dynamics:

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Virial Theorem: rG < v 2 > M= G where rG =


i

mi
j


i` j

mm i rij

and < v >=
2


i

mi (vi - v )

2


i

mi

From X-rays:

Hydrostatic equilibrium: GM (< r ) 1 dP =- g dr r2 As for elliptical galaxies: kT(r)r 2 d ln g (r ) d ln T (r ) + M(
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The following correlations exist between the different components of galaxy clusters: · The central galaxy density is higher for higher LX. · The fraction of spirals is lower for higher LX. · The temperature T is proportional to LX and typically 108K. · The gas metallicity is lower for higher T and typically 1/3 of solar. · The ratio of gas-mass to galaxy-mass increases with T up to 5 or more. · The dominant component in all clusters is dark matter. This follows consistently from the dynamics of galaxies, the hydrostatic equilibrium of the X-ray gas and from gravitational lensing. The typical mass ratios are: galaxies : X-ray-gas : dark-matter 1 : 5 : 25

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Clusters in microwaves and the Sunyaev & Zel`dovic effect

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The photons of the CBR suffer Inverse Compton scattering against the hot electrons of the intracluster medium, preferentially gaining energy. The CMB spectrum gets shifted to higher frequencies: at wavelengths <1.4 mm the clusters appear as bright pacthes in the CMB. To first order, the CMB distortion is proportional to the integral along the line of sight of the electron density times its thermal energy:

I kT dl = 2 y, y = ne T 2 I me c
Carlstrom, Holder & Reese, 2002, ARAA, 40, 643
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The SZ effect is independent of Redshift, unlike the optical or X-ray surface brightness that suffer from cosmological dimming (see cosmology lessons) (1 + z ) 4 Therefore it is a powerful method to detect clusters at high redshifts. Cluster surveys are starting with e.g. APEX (Atacama Pathfinder Experiment) If the cluster is not resolved, the SZ signal measures the thermal energy of the electrons:

Y = ydA neTdV M gas T
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13.6 Masses of galaxy clusters from gravitational lensing

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13.6.1 Basics of Gravitational Lensing
One consequence of Einstein's Theory of Relativity is that light rays are deflected by gravity. Einstein calculated the magnitude of the deflection that is caused by the sun. Since the potential and the velocity of the deflecting mass are small (v « c and « c2) the deviation angle is expected to be small as well. According to Einstein's formula, a light ray passing the surface of the Sun tangentially is deflected by 1.7". This deflection angle has in the mean time been confirmed with a very high accuracy (0.1 %). For further information see also: R. Narayan, M. Bartelman: Lectures on Gravitational Lensing; in: Formation of Structure in the Universe Edited by Avishai Dekel and Jeremiah P. Ostriker. Cambridge: Cambridge University Press, 1999., p.360 Schneider, Ehlers, Falco: Gravitational Lenses Springer Verlag

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The light path from the source to the observer can then be broken up into three distinct zones: 1. Light travels from the source to a point close to the lens through unperturbed spacetime, since b « Dd. Near the lens the light is deflected. Light travels to the observer through unperturbed spacetime, since b « Dds.

2. 3.

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In a naive Newtonian approximation one would derive:

=

vz 1 d 1 d = dt = 2 dl c c dz c dz


*: acceleration in z direction; because the acceleration doesn't depend on the energy of the photons, gravitational lenses are achromatic.

This result differs only by a factor of two from the correct general relativistic result: 2 G.R. = 2 dl c where the deflection angle , written as vector perpendicular to the light propagation l, is the integral of the potential gradient perpendicular to the light propagation.
For a point mass the potential can be written as:

(l , z ) =
Therefore:

-GM (l 2 + z 2 )1

/2

+GMz d =2 dz (l + z 2 )3
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= ( )
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After integration:

2 = 2 c



-



GMz (l 2 + z 2 )

3/ 2

4GMz dl dl = c 2 (l 2 + z 2 ) 0



3/ 2

4GMz = c2

l z 2 (l 2 + z 2 )1/ 2



0

Thus the deflection angle for a light ray with impact parameter b = z near the point mass M becomes:

=

4GM 2 RS = c 2b b

where RS = 2GM/c2 is the Schwarzschild radius of the mass M, i.e. the radius of the black hole belonging to the mass M. Therefore for the sun (M 2 · 1033 g RS 3.0 km) we get a deflection angle at the Radius of the sun ( 700000km) of:



,R

1.7 ''

In order to calculate the deflection angle caused by an arbitrary mass distribution (e.g. a galaxy cluster) we use the fact that the extent of the mass distribution is very small compared to the distances between source, lens and observer: l « Dds and l « Dd
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Therefore, the mass distribution of the lens can be treated as if it were an infinitely thin mass sheet perpendicular to the line-of-sight. The surface mass density is simply obtained by projection. The plane of the mass sheet is called the lens plane. The mass sheet is characterized by its surface mass density


dm =

( ) = ( , l )dl
l

The deflection of a light ray passing the lens plane at by a mass element



( ')d 2 ' at ' is:

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d =

4Gdm c2 | - ' |

To get the deflection caused by all mass elements, we have to integrate over the whole surface. Doing this we must take into account that, e.g., the deflection caused by mass elements lying on opposite sides of the light ray may cancel out. Therefore we must add the deflection angles as vectors:

4G ( - ') ( ') 2 d ' ( ) = 2 c | - ' |2
Special case: For a spherical mass distribution the lensing problem can be reduced to one dimension. The deflection angle then points toward the center of symmetry and we get:

( ) =

4GM (< ) c 2

where is the distance from the lens center and M(< ) is the mass enclosed within radius ,

M (< ) = 2


0



( ') ' d '

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13.6.2 Lensing Geometry and Lens Equation

Important relations:

^ Dds = Ds ^ Ds = Ds + Dd

(13.1)
s

(13.2)

Note: The distances D are angular diameter distances.

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Using the previous two equations one obtains the so called lens equation:

= - = -

Dds ^ Ds

(13.3)

The lens equation relates the real position (angle) of the source (without a lens) with the position of the lensed image. Important note: only angular distances are needed for deriving the lens equation. In general, i.e. over cosmological distances: Dds Ds - Dd.

13.6.3 Einstein radius and critical surface density
Consider now a circularly symmetric lens with an arbitrary mass profile. Due to the rotational symmetry of the lens system, a source, which lies exactly on the optical axis ( = = 0 ) is imaged as a ring. This ring is the so called Einstein ring:

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=0

=
= = = = Dds ^ Ds Dds 4G M (< ) Ds c 2 Dds 4 G M (< ) 2 2 Ds c Dds 4 G 2 cr Dd Ds c

(13.4) (13.5) (13.6) (13.7) (13.8)

Therefore the critical surface density to observe an Einstein ring is:

Ds c2 g D 1Gpc cr = = 0.3 2 s cm Dds Dd 4 G Dds Dd

critical surface density

Note: The critical surface density depends only on the angular distances between source, lens and observer. The radius of the Einstein ring can be calculated using the previous equation and = Dd:
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2 E =

Dds 4G 2 M Ds Dd c

<

E

Einstein radius

where M


is the projected mass within E.

If the surface mass density has the value cr and is constant in , we get a ideal convex lens. All light rays would then be focused in the point of observation:

For a typical gravitational lens decreases as a function of the radius. Therefore only at a certain radius the condition for a circular image is fulfilled:

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Furthermore gravitational lenses are hardly ever spherically symmetric. For an elliptical mass distribution one observes only parts of the ring, the so called arcs. Examples of Einstein angles E
1. Galaxy clusters: typical mass: M typical distances: This leads to: 1014 M 1 Gpc


where D = Ds Dd . Dds

E

M 10 '' 13 10 M

`1 / 2

D Gcp

-1/ 2

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Thus for massive galaxy cluster (M > 1014M within a few hundreds of kpc) we get observable angles in the order of ten arcsecs. 2. Stars (or similar objects) in the Milky Way:



E

M 0.001'' M

1/ 2

D 10kpc

-1/ 2

Such a tiny angle cannot be directly observed, but sometimes it is possible to detect the amplification it causes.

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Galaxy Cluster Cl0024+1645, strong lensing reconstruction (left, courtesy S. Seitz) of HST image (right, Colless et al.); light blue = caustic structure, bold green = critical lines of 'infinite' amplification, squares = observed positions of multiple imaged source (A,B,C,D,E in color image), yellow crosses = predicted position of the lensmodel, yellow circle = position of source in source plane, red crosses = mass centers used for the lens model. The caustics are obtained by mapping the critical lines into the source plane. IMPRS Astrophysics Introductory Course Fall 2007