clusters of galaxies the icm, mass measurements and statistical measures of clustering
TRANSCRIPT
![Page 1: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/1.jpg)
Clusters of galaxies
The ICM, mass measurements and statistical measures of
clustering
![Page 2: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/2.jpg)
Plan of this classThe intracluster medium, its origin, dynamics and
general propertiesEvidence of Dark Matter in clustersMasses derived by the virial theorem, x-rays and
gravitational lensingResults from studies of gravitational lensing in clustersStatistical measures of clustering
![Page 3: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/3.jpg)
The intracluster medium
![Page 4: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/4.jpg)
Clusters are among the most luminous X-ray sources in the sky. This X-ray emission comes from hot intracluster gas.
X-ray observations provide information on the amount, distribution, temperature and chemical composition of theIntracluster gas
![Page 5: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/5.jpg)
For comparison,
Cataclismic variables Lx = 1032 – 1038 erg/s
Milky Way, M31 Lx = 1039 erg/s
Clusters of galaxies Lx = 1043– 1045 erg/s
Only Seyferts, QSOs, and other AGN rival clusters in X-ray output
Clusters may emit nearly as much energy at X-ray wavelengths as visible
L(optical) = 100 L* galaxies = 1045 erg/s
![Page 6: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/6.jpg)
The Lx – σ correlation
![Page 7: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/7.jpg)
What is the origin of cluster X-ray emission?
Answer: hot (107 – 108 K) low-density (10-3 cm-3) gas, mostly hydrogen and helium, that fills space between galaxies. At these high temperatures the gas is fully ionized.
Two emission mechanisms:
1) Thermal bremsstrahlung (important for T > 4 x 107 K)
free electrons may be rapidly accelerated by the attractive force of atomic nuclei, resulting in photon emission
because the emission is due to Coulomb collisions, X-ray luminosity is a function of gas density and temperature
Lx = nelectron nion T1/2 = rho_gas2 T_gas1/2
2) Recombination of electrons with ions (important T < 4 x 107 K)
![Page 8: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/8.jpg)
Dynamics of the intracluster gas
The intracluster gas can be treated as:
An ideal fluid
In hydrostatic equilibrium
At a uniform temperature
![Page 9: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/9.jpg)
![Page 10: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/10.jpg)
![Page 11: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/11.jpg)
![Page 12: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/12.jpg)
![Page 13: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/13.jpg)
![Page 14: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/14.jpg)
X-ray spectra
Spectroscopy of the intracluster gas provides information on its temperature and composition
Observed spectra show exponential decrease at high-frequencies that is characteristic of bremsstrahlung.
Coma ClusterHughes et al. 93
![Page 15: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/15.jpg)
Emission lines due to Fe, Ni and other heavy elements are seen. This suggests that much of the intracluster gas must have been processed through stars.
Chemical abundance of the intracluster gas can be measured from the equivalent widths of these emission lines. It is found to be about 30-40% of solar abundance
If the galaxies and gas are both in thermal equilibrium in the cluster potential well, then one expects
m v(gal)2 = 3 kbTgas
Tgas proportional to v(gal) 2
![Page 16: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/16.jpg)
What is the origin of the intracluster gas?
Two possibilities: The intracluster gas once resided in galaxies and was later
removed.
- this would explain high metallicity of gas
- galaxies in the cores of rich clusters are
observed to be deficient in HI gas, which
suggests that stripping has occurred. The gas is primordial, originating at the time of cluster
formation.
- but since Mgas >> Mgal it is difficult to
understand how so much material could
have been stripped from galaxies
![Page 17: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/17.jpg)
How much gas is there in clusters?
![Page 18: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/18.jpg)
Cluster Mass estimates: X-ray gas
![Page 19: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/19.jpg)
The total gas mass in clusters exceeds the total galaxy mass. Gas contributes as much as 10-20% of
the total cluster mass.
David, Jones andForman 95
![Page 20: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/20.jpg)
Evidence of Dark Matter (DM) in clusters
![Page 21: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/21.jpg)
Dark Matter in Clusters
A more accurate name for “clusters of galaxies” would be “clusters of dark matter”
Observational evidence suggests that 80-90% of the mass in clusters is in an invisible form
1) What evidence is there for dark matter?2) How much dark matter is there?
3) What is the distribution within clusters?
![Page 22: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/22.jpg)
Evidence of Dark Matter in clusters Virial mass estimates
If a cluster is in virial equilibrium then its mass can be estimated from Mvirial = R<v2>/G
Observations indicate that the total cluster mass exceeds the combined masses of all galaxies by factors of 10-20.
Example: the Coma Cluster
Mvirial = 1 x 1015 h-1 solar masses
Ltot = 4 x 1012 h-2 solar luminosities
Assuming a typical galaxy with M/L = 10
Then Mvirial/Mgalaxies = 25
![Page 23: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/23.jpg)
Typical mass to light ratios
Globular clusters 1-2 M/L Elliptical galaxies 5-10 h M/L Groups of galaxies 100-300 h M/L Rich clusters 300-500 h M/L
![Page 24: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/24.jpg)
Mass to light ratio of Coma
![Page 25: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/25.jpg)
Mass estimate using the Virial theorem
![Page 26: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/26.jpg)
X-ray mass estimates If the intracluster gas is in hydrostatic euilibrium in the cluster potential, then the cluster
mass can be determined from
![Page 27: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/27.jpg)
Gravitational lensing studies provide Gravitational lensing studies provide another independent evidence for DM another independent evidence for DM
in clustersin clusters
![Page 28: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/28.jpg)
Gravitational Lensing – some history
1913 – Einstein predicted that the gravitational field of massive objects can deflect light rays.
1919 – Eddington measured the deflection of starlight by the Sun, confirming Einstein’s prediction.
1937 – Zwicky suggested that galaxy clusters may produce observable lensing.
1987 – First evidence of “strong” gravitational lensing by clusters was found (Lynds/Petrosian, Soucail et al.)
1990 – “Weak” gravitational lensing by clusters was discovered (Tyson et a. 1990).
Today – Evidence of lensing has been found for several dozen clusters. New examples are being discovered all the time.
![Page 29: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/29.jpg)
![Page 30: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/30.jpg)
1986 – Lynds & Petrossian discover the first gravitational arcs in clusters of
galaxies
1987 – Soucail et al. determine the distance to the arc: twice the distance to the cluster that “contains” it.
STRONG LENSING
![Page 31: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/31.jpg)
Gravitational lensing: the basic ideas
![Page 32: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/32.jpg)
![Page 33: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/33.jpg)
Galaxy cluster
Background galaxy
Observer
Strong lens
Weak lens
![Page 34: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/34.jpg)
“Strong” lensing occurs when
Long arcs and multiple images are produced.
“Weak” lensing occurs when
Small arclets and distortions are produced.
![Page 35: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/35.jpg)
![Page 36: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/36.jpg)
![Page 37: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/37.jpg)
A 1451z = 0,199
Strong Lensing
![Page 38: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/38.jpg)
![Page 39: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/39.jpg)
A 1451z = 0,199
![Page 40: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/40.jpg)
Weak Gravitational Lensing
Mellier 99
![Page 41: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/41.jpg)
Why Weak Lensing ?
Allows the reconstruction of the surface mass density
Classical techniques (dynamics of the galaxies and X-ray emission of the hot intra-cluster gas) are based of the assumption of dynamical equilibrium
![Page 42: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/42.jpg)
Measuring Faint Galaxy Shapes
Cypriano et al. 2005
![Page 43: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/43.jpg)
Mass Light
Mass Light
A2029In 77% of the cases
the center of light and mass
distributions are consistent with each
other...
![Page 44: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/44.jpg)
Mass Light...but there are exceptions
Mass Light
A3739
![Page 45: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/45.jpg)
Mass Light
Mass Light
A4010
![Page 46: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/46.jpg)
Mass Light
There is a strong alignment between the BGC and the dark
mater main axis
![Page 47: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/47.jpg)
Comparison with X-Rays
TX ~ TSIS,SIE
A2744
A1451A2163
![Page 48: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/48.jpg)
Comparison with the Velocity Dispertion
A1451
A2744A21
63
σv ~ σSIS,SIE
![Page 49: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/49.jpg)
The dynamical state of the clusters
Most of the clusters appears to be relaxed (lensing dynamical methods)
Cluster with TX > 8 keV (σv >1120 km/s) shows signs of dynamical activity
![Page 50: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/50.jpg)
The dynamical state of the clusters
A2744 – Virial mass> Lensing > X-rays
Girardi & Mezzetti (2001)
σσtotaltotal= 1777 = 1777 km/skm/s
σσAA = 1121 = 1121 km/skm/s
σσBB = 682 = 682 km/skm/s
Interpretation: There are two structures along the line of
sight
Chandra observations confirms fusion along the line of sight (Kempner & David 2004)
![Page 51: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/51.jpg)
Which method is the best one ?
Weak Lensing
Independent of the dynamical state
Needs good seeing
Reconstruct the 2-D potencial
Cannot separate components along the line of sight.
![Page 52: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/52.jpg)
Which method is the best one ?
X-Rays
All Sky Surveys (e.g. ROSAT) can provide large and homogeneous samples
Depend of thermal/dynamical state of the ICM
Cannot separate components along the line of sight.
![Page 53: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/53.jpg)
Which method is the best one ?
Dynamics of
galaxies
Depend on the dynamical state of the cluster galaxies (galaxies relaxes later than the ICM)
Can separate structures along the line of sight
Reliable results depends on a large number of galaxy velocities over a large area (e.g. Czoske et al. 2002)
No single method is perfect !
![Page 54: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/54.jpg)
What can we learn from gravitational lensing?
Gravitational lensing can be used to determine the amount and distribution of dark matter in clusters.
Unlike virial or X-ray mass determinations, lensing requires no assumptions about the dynamical state of the cluster!
The arc thickness is related to the cluster mass distribution. More concentrated mass distributions produce thinner arcs.
Modelling the positions and shapes of arcs and arclets allows the cluster potential to be mapped. Lensing models have become so good that in can predict the locations of faint additional arcs.
Gravitational lensing causes images to be magnified. Clusters of galaxies can be used as natural “telescopes”to study extremely distant galaxies that would be otherwise too faint to see.
Lensing can also be used to place cosmological constraints, because distances (Dos, Dol, Dls) depend on omega, Ho and lambda.
![Page 55: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/55.jpg)
z = 5.6
Ellis, Santos, Kneib & Kuijken (2001)
![Page 56: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/56.jpg)
What have we learned so far from gravitational lensing?
Samples of strong and weak gravitational lensing have been found in several dozen clusters.
Lensing mass estimates indicate large quantities of dark matter in clusters Lensing mass estimates agree with virial and X-ray masses (with a few
exceptions). The exceptions are probably clusters which are not in equilibrium. Hot clusters tend to present dynamical activity (major concern for
experiments designed to constrain cosmological parameters). Mass follows light in most cases. Cluster dark matter has a very steep radial distribution. Models of the cluster potential provide strong evidence of substructure in
the dark matter distribution. Gravitational lensing has been seen in clusters at z>1
![Page 57: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/57.jpg)
Clusters as Tracers of Large-scale Structure
![Page 58: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/58.jpg)
Why use clusters to map the large-scale structure of the universe?
Advantages
Clusters provide an efficient way of surveying a large volume of space
Cluster distribution provides information about conditions in the early universe
Clusters can be seen at great distances
Disadvantages
Their low space density makes clusters sparse tracers of the large scale structure
Results may depend on the chosen cluster sample
Redshifts of many clusters are still unmeasured
![Page 59: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/59.jpg)
Velocidade
![Page 60: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/60.jpg)
De Lapparent et al. 1988
Velocidade
d=v/Ho
Lei de Hubble
The Cfa Slice
![Page 61: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/61.jpg)
Velocidade
The Cfa Slice
![Page 62: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/62.jpg)
Velocidade
The Cfa Slice
![Page 63: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/63.jpg)
Large scale structure – 2dF
![Page 64: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/64.jpg)
Some history
1933 – Shapley noticed several binary and triple systems among the 25 clusters that he catalogued “it is possible that clusters are but nuclei or concentrations in a very extensive canopy of galaxies”.
1954 – Shane and Wirtanen’s galaxy maps showed “a strong tendency for clusters to occur in groups of two or more”.
1956 – Neyman, Scott and Shane’s pioneering statistical models of galaxy clustering included “second-order clusters”, I.e., superclusters.
1957 – Zwicky declared that “there is no evidence at all for any systematic clustering of clusters… clusters are distributed entirely at random.”
1958 – Abell examined the distribution of clusters in his catalogue, and concluded that “clusters of clusters of galaxies exist”
Today – No doubt that galaxy clusters are clustered. Instead, debate is about the SCALE of this clustering.
![Page 65: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/65.jpg)
Statistical measures of clustering
1) The two-point correlation function
2) The power-spectrum
3) Cluster alignments
![Page 66: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/66.jpg)
Probability of finding objects in dV1 and dV2 separated by distance r
![Page 67: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/67.jpg)
![Page 68: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/68.jpg)
Two-point correlation function for Abell clusters
Abell cluster correlation function has the same power-law form as that for galaxies
ξ (r) = A rγ =1 (r/r0) γ ξ (r) = 1 at r= r0
γ = - 1.8
r0 = 20-25 h-1 Mpc
Richer clusters are more strongly clustered than poorer clusters The Abell cluster correlation function has the same power-law
form as the galaxy correlation function, but with a 15 times greater amplitude (r0 = 5 h-1 Mpc for galaxies r0 = 20 h-1 Mpc for Abell clusters
Why is ξ (r) different for galaxies and clusters? Biasing! If Abell clusters have formed from rare high-density peaks
(ν > 3σ) in the matter distribution, then their clustering tendency will be enhanced by an amount ξcluster= ν2 ξmatter (Kaiser 1984).
![Page 69: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/69.jpg)
Two-point correlation function for other cluster samples
APM and EDCC clusters show a weaker clustering tendency than Abell clusters
r0 = 13-16 h-1 Mpc for both samples
ROSAT X-ray selected clusters
r0 = 14 h-1 Mpc
Why do different cluster samples give different results?
Three possibilities:
(a) The Abell catalogue is unreliable
(b) Richness-dependence of the cluster correlation function. Abell, APM and EDCC clusters are fundamentally different types of objects.
(c) X-ray selected samples are flux-limited rather than volume-limited. This means that any X-ray selected sample will contain a mixture of nearby poor clusters and distant rich clusters.
![Page 70: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/70.jpg)
Statistical measures: the power spectrum
![Page 71: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/71.jpg)
Statistical measures: the power spectrum
Although P(k) is more complicated to measure than the two-point correlation function it has two big advantages:
1) it can be more directly compared with theory 2) it is a more robust measure
ξ (r) + 1 = Npairs/Nrandom = Npairs/(n 4/3 π r3)
which is proportional to 1/n
Uncertainties in n produce large uncertainties in ξ when ξ << 1.
For P(k), each δk is proportional to n. Hence the shape of the power-spectrum is unaffected.
![Page 72: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/72.jpg)
Statistical measures: cluster alignments
Clusters are often embedded in large-scale filamentary features in the galaxy distribution.
Cluster major axes tend to point along these filaments towards neighbouring clusters, over scales of about 15 h-1 Mpc, perhaps up to 50 h-1 Mpc.
These cluster alignments may provide important clues about cluster formation and cosmology
![Page 73: Clusters of galaxies The ICM, mass measurements and statistical measures of clustering](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ccb5503460f94994ea9/html5/thumbnails/73.jpg)
Clusters as LSS tracers
Clusters of galaxies are efficient tracers of the large-scale structure of the universe.
There is strong evidence of structure on scales of over 100 h-1 Mpc in the cluster distribution.