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Thermodynamic Profiles of Galaxy Cluster Gas using a Multiwavelength Analysis of X-ray and Microwave Data

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Thermodynamic Profiles of Galaxy Cluster Gas using a Multiwavelength
Analysis of X-ray and Microwave Data
Jennifer Shitanishi
A dissertation presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In partial fulfillment of the
requirements for the the degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
August 9 2016
ProQuest Number: 10801141
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Contents
1 Introduction
6
1.1
Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Galaxy Clusters
2.1
7
History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.1
Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3
Physics of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3.1
2.4
Types of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Gas probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1
X-ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2
Sunyaev-Zel’dovich E↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5
Self-Similarity and Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6
Thermodynamic studies of galaxy cluster gas . . . . . . . . . . . . . . . . . . . . . . 17
3 The Instruments
3.1
X-ray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1
3.2
18
Chandra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
SZ/microwave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1
Bolocam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 The BOXSZ sample
20
5 Data Reduction
23
5.1
Chandra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2
Bolocam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6 Method
27
2
6.1
Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2
The Individual Cluster Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3
Fitting models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3.1
Onion shell deprojections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.3.2
Analytical profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7 Mock Clusters
32
7.1
Simulating the cluster emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.2
Check for Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Individual Cluster Results
35
8.1
Goodness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
8.2
Summary of individual-cluster fitting results . . . . . . . . . . . . . . . . . . . . . . . 36
9 Ensemble-cluster fitting
40
9.1
Previous Ensemble Profile Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.2
Mock cluster results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
9.3
BOXSZ results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
9.3.1
CC vs NCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.3.2
Disturbed vs Non-Disturbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.3.3
Spherical vs. Non-Spherical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.3.4
Temperature boundary T = 0keV
10 Conclusions
10.1 Summary & Discussion
. . . . . . . . . . . . . . . . . . . . . . . . 47
47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10.2 Expansion of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
11 Acknowledgements
49
12 References
51
13 Appendix A
56
3
14 Appendix B
14.0.1 Abell 2204
57
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
14.0.2 Abell 383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14.0.3 Abell 209 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14.0.4 Abell 1423
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14.0.5 Abell 963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
14.0.6 Abell 2261
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
14.0.7 Abell 2219
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
14.0.8 Abell 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
14.0.9 RX J2129.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
14.0.10 Abell 1835
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
14.0.11 Abell 697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
14.0.12 Abell 611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.0.13 MS2137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.0.14 MACS J1931.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.0.15 Abell S1063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
14.0.16 MACS J1115.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
14.0.17 Abell 370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
14.0.18 CL0024+17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
14.0.19 MACS J1532.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
14.0.20 MACS J0429.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
14.0.21 MACS J2211.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
14.0.22 MACS J1720.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
14.0.23 MACS J0416.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
14.0.24 MACS J0451.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
14.0.25 MACS J0417.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
14.0.26 MACS J1206.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
14.0.27 MACS J0329.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4
14.0.28 RX J1347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
14.0.29 MACS J1311.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
14.0.30 MACS J0257.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
14.0.31 MACS J0911.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
14.0.32 MACS J2214.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
14.0.33 CL0016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
14.0.34 MACS J1149.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
14.0.35 MACS J0717.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
14.0.36 MACS J1423.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
14.0.37 MS 0451.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
14.0.38 MACS J0025.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
14.0.39 MS2053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
14.0.40 MACS J0647.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.0.41 MACS J2129.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.0.42 MACS J0744.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.0.43 MS1054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
14.0.44 CL J0152.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
14.0.45 CL J1226.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5
1.
Introduction
1.1.
Motivation
This thesis focuses on the properties of the gas in galaxy clusters, which is of paramount
significance to both cosmology and cluster astrophysics. The gas (also known as the intracluster
medium, or ICM) is the dominant baryonic component of clusters, composed mostly of ionized
hydrogen, some helium, and trace amounts of heavier elements, and it constitutes approximately
5-10% of the mass of the cluster. Recovering ICM temperature profiles can lead to determining
cluster masses, which can be used to connect cluster probes to the underlying cosmology of the
universe. For example, the mass function of the galaxy cluster population provides constraints on
the matter power spectrum’s amplitude at the scale of the cluster. It also can provide constraints
on the matter density parameter ⌦m .
Isolated, relaxed clusters that have had time to cool are expected to be in hydrostatic
equilibrium (HSE). However, deviations from HSE can be seen in the cluster core, which exhibits
the contrasting e↵ects of radiative cooling and feedback mechanisms (e.g. active galactic nuclei),
and in the outskirts, where merging processes and the accretion of filaments onto the cluster can
be seen. The relative contributions of these physical processes to the cluster’s state is currently
not well known, but can be reflected in gas density, pressure, and temperature profiles.
The traditional method to find the temperature of clusters is through X-ray spectroscopy,
which is limited to high-density regions of the cluster. X-ray spectroscopy is also highly dependent
on calibration, so di↵erent instruments may yield conflicting results. The majority of literature
has explored the cluster core, being very bright and dense, while studying the fainter outer
regions is more of a challenge. Here, we will use a combination of X-ray surface brightness profiles
and microwave observations to bypass the need for X-ray spectroscopy. The microwave, in which
one can observe the Sunyaev-Zel’dovich (SZ) e↵ect, can probe the gas to much fainter regions and
further clusters. This study provides the largest sample of joint X-ray and SZ deprojected gas
profiles to date, as well as ensemble profiles, while constraining the intrinsic scatter between the
clusters. We will find temperature profiles out to large radii that have not been explored before.
1.2.
Overview
In this work, we combine SZ decrement maps from Bolocam, a bolometer array that was
stationed on the Caltech Sub-mm Observatory (CSO), with surface brightness maps from
Chandra, NASA’s X-ray satellite, to recover density and temperature profiles for the Bolocam
sample (BOXSZ) of 45 clusters. Markov Chain Monte Carlo (MCMC) is used to fit
non-parametric and parametric models on an individual cluster basis, as well as an average
non-parametric profile of all the clusters. Along with the average profiles, we fit for the intrinsic
scatter of the sample. First, an overview of galaxy clusters will be discussed in Section 2, and
instruments used to observe them in Section 3. In Section 4 we will discuss the cluster sample;
6
Section 5 will go over data reduction; and Section 6 will explain the modeling and fitting
methods. Section 7 will discuss mock cluster maps that were put through the pipeline; Section 8
will go over the individual cluster results; while Section 9 will include the joint-cluster fitting, and
we will end with our conclusions in Section 10.
2.
Galaxy Clusters
2.1.
History
Galaxy clusters were initially discovered in the optical waveband as a collection of galaxies
spatially close to one another on the plane of the sky. In 1784, Charles Messier discovered the
Virgo Cluster (the nearest cluster to us besides our Local Group) as several galaxies grouped
together. At the time, it was not known that they were extragalactic. Cepheid variable distance
measurements then allowed astronomers to learn about the 3-D nature of the universe, and that
galaxies, as well as the Virgo cluster, were not inside our galaxy.
2.1.1.
Dark Matter
The history of the knowledge of galaxy clusters is conflated with major dark matter
discoveries. They have provided some of the most compelling evidence of dark matter. In 1933
Fritz Zwicky studied Doppler shifts in the spectra of Coma cluster galaxies to find their velocities,
and using the virial theorem, he was able to estimate the mass of the cluster. The virial theorem
allows one to calculate the potential energy of the cluster from the average kinetic energy,
deducing the mass. The mass he found was several hundred times larger than the mass derived
from just the light from the galaxies (the total luminosity). The missing mass was called dark
matter.
The discovery of di↵use X-ray emission from galaxy clusters suggested a hot gas was
smoothly filling the cluster. While the mass of the ICM is the dominant baryonic component of
the cluster, it accounts for some, but not all, of the missing mass. The fact that the ICM emits
X-rays and that it fills clusters smoothly suggests the ICM is more or less in HSE. This implies a
much deeper gravitational potential well (i.e. a greater mass) than that of the galaxies and gas providing further evidence for missing, dark matter.
Lastly, gravitational lensing measures the bending of light of background objects due to the
massive amount of dark matter that exists within the foreground clusters. Whether by strong or
weak lensing, cluster mass distributions can be determined directly, without any assumptions of
HSE or other dynamics occurring within the cluster - it is only dependent on general relativity.
7
2.2.
Properties
Fig.
1.— Abell 383, a galaxy cluster as observed through the optical (white) and Xray (purple). The ICM is seen being roughly spherical, smoothly filling the cluster between the galaxies. This cluster also exhibits a large arc caused by gravitational lensing.
X-ray: NASA/CXC/Caltech/A.Newman et al/Tel Aviv/A.Morandi & M.Limousin; Optical:
NASA/STScI, ESO/VLT, SDSS
Galaxy clusters, being the largest virialized bodies in existence, are unique objects to study
in astrophysics and cosmology. They have masses ranging on scales of 1014 -1015 M , consisting of
tens to thousands of galaxies, hot gas of 107 to 108 K (around 5 keV) , and large amounts of dark
matter. Dark matter consists of most of the mass in a cluster (⇠90% of the total cluster mass),
while the rest is baryonic (gas and stars). Figure 1 shows a relaxed galaxy cluster in the optical
and X-ray (Newman et al. 2011, Morandi et al. 2011). Of the baryonic matter, in general the gas
dominates over the stars in the galaxies. The gas is composed mostly of hydrogen and helium.
The metallicity is approximately 0.3Z , where the solar abundance metallicity is 0.02% (the
percentage of the total mass that are metals - i.e. anything heavier than helium). The electron
number densities near the center of clusters are on the order of magnitude of 10 3 10 2 cm 3
(depending on the kind of cluster), and pressures are ⇠ 10 10 ergs/cm 3 . The densities and
pressures drop several orders of magnitude as they reach the outskirts.
Depending on the subfield, there are di↵erent conventions on radial distance units to use
when calculating radial profiles of the clusters. One can use R500 , an overdensity radius, which is
the radius at which the average density within is 500 times the critical density of the universe,
8
and is approximately 1 Mpc. Although there is no hard-edge to clusters, R500 roughly represents
the edge of the cluster. Other times studies use 200, or 2500 overdensities. In general,
R2500 ⇡ 0.3R200 , and R500 ⇡ 0.7R200 . The virial radius, Rvir , is the radius at which a sphere with
a constant density will collapse to become a galaxy cluster, and is approximately R200 . While
R200 is used due to its physical meaning, R2500 and R500 are around where X-ray observations
commonly reach out to, and where at most they can reach out to, respectively.
The mass, M500 , is the mass of the cluster within R500 . Other properties of the cluster can
also be denoted by R500 , such as n500 , T500 , or P500 , which are average density, temperature or
pressure within R500 , respectively. These quantities are important for this study because they are
useful for scaling clusters to be compared to each other, taking into account the evolution of the
universe. Section 9.2 details how these are calculated.
2.3.
Physics of clusters
There are several physical processes that take place inside a cluster, a↵ecting clusters to
di↵erent extents. Simulators often try to incorporate them in their work, but it becomes difficult
since there is not a lot known about these processes. Much of the time these have observable
e↵ects on the gas. So the more one can characterize the gas, the more we can constrain how these
processes interplay.
A first order e↵ect is gravitational heating, which describes how gas is stripped from
galaxies as they fall into the cluster. The extra energy of the gas is then converted into heat.
Radiative cooling is the process of gas falling into the center of the cluster, making the gas
increasingly dense there. Therefore, it emits more X-rays through thermal bremsstrahlung, which
cools the gas. In general, bremsstrahlung is the process by which radiation is released due to the
acceleration of a charged particle. In the scenario of clusters, we have thermal bremsstrahlung,
where electrons in a plasma in equilibrium scatter o↵ ions, causing the electrons to decelerate and
release X-ray photons. The cooling time of a cluster is the time it takes for the cluster to undergo
thermal bremsstrahlung until it runs out of energy.
Counteracting radiative cooling near the center are feedback mechanisms, which heat the
gas. If just radiative cooling existed along with gravitational physics, one would expect to observe
strong cooling flows in the center of clusters. However, this is not observed in reality, so there
must be forms of feedback that exist. These include active galactic nuclei (AGN), star formation,
and supernovae. AGN, which is the gas accretion onto black holes, is the leading possible source
of feedback (McNamara & Nulsen 2007, Puchwein et al. 2008). As gas falls into the black hole,
jets heat the gas, which slows down cooling. But once the accretion rate decreases, the jets shut
o↵. Burns (1990) found that cool cores of clusters host AGN, which can be seen as X-ray surface
brightness dips in the profile. Although star formation can counteract cooling, it does not provide
enough energy to be the primary source of feedback given cluster observations (Bregman & David
1989). However, they are still important to consider since there is often significant star formation
9
rates in the brightest cluster galaxies (BCGs) of relaxed clusters (Crawford et al. 1999).
Supernovae feedback, which heats the gas, is a minor e↵ect in cluster-scale feedback, although it
is important on galaxy scales.
Other e↵ects include shockwave pressure from gas moving within a cluster. Sloshing in
galaxy clusters can arise from a smaller cluster passing by, where ram pressure causes the gas of a
cluster to slow down (Ascasibar & Markevitch 2006), e.g. the Bullet Cluster. Eventually the gas
catches up, falling into the dark matter potential well. When it does this it can overshoot and
”slosh,” releasing energy.
On a larger scale, when two or more clusters merge together, it can cause disturbances
when the gas from each of the clusters collide together, heating the gas. It seems that cluster
formation is hierarchical, from the merging of more and more groups of galaxies and clusters,
along with the accretion of gas from surrounding filaments of the cosmic web that the cluster is
sitting in. One can see the e↵ects of merging especially in the outskirts.
2.3.1.
Types of clusters
Clusters that have cooling times less than the age of the universe have had enough time for
the gas to cool in the center. Usually this happens in relaxed, spherical clusters where there
haven’t been signs of merging in recent times that would disrupt the cool core (CC). In general
this results in lower redshift clusters having a higher probability of being cool core, while higher
redshift clusters (younger ones) have a higher chance of being a non-cool core (NCC) cluster. CC
clusters can have temperature drops in the center of around 50% of the peak temperature of the
cluster (or less of a drop, depending on the strength of the cooling flow). The CC vs NCC
description of a cluster is a major theme in galaxy cluster studies. There are di↵erent methods
with which CC cluster can be characterized, including: the cooling time, the cuspiness of the
density profile, or luminosity ratios. Hudson et al. (2010) compared the many ways a cool-core
can be defined and applied it to the HIghest X-ray FLUx Galaxy Cluster Sample (HIFLUGCS)
cluster sample. Essentially, there is some inconsistency between the di↵erent definitions. This
should be kept in mind when comparing di↵erent studies; what one study defines as a CC another
may not. In this dissertation we use the Mantz et al. (2010) method of luminosity cuts to keep
consistency with the Sayers et al. (2013) study.
Figure 2 shows typical X-ray maps of CC and NCC clusters, highlighting the di↵erences
between these two types of clusters and how the cluster physics is reflected in their density and
temperature profiles. A bright core can be seen in the CC cluster map but is not seen in the NCC
map. E↵ects of AGN or merging may have disrupted CC formation for the NCC cluster. The
density profile of the CC cluster requires two components; one for the core, and one for the
general cluster gas distribution. The NCC density profile would only show one density
component. The expected temperature profile for the CC cluster shows a temperature drop in the
center, while the NCC does not, but both would show a temperature drop in the outskirts.
10
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2.— (a) An X-ray map of a typical CC cluster (Abell 2204). (b) An X-ray map of a typical
NCC cluster (CL 0016). (c) and (e) show the expected density and temperature profiles for the
CC cluster, respectively, while (d) and (f) show the expected density and temperature profiles for
the NCC cluster, respectively.
Clusters can also be identified as disturbed or not-disturbed, due to the presence of AGN or
merging. There are also many di↵erent criteria for deciding if a cluster is disturbed or not. A few
11
include: obvious multiple peaks in the X-ray data, the o↵set between the BCG and the X-ray
peak or centroid. The method used in the BOXSZ sample is the X-ray centroid shift parameter,
w500 (Maughan et al. 2012), which describes the significance between the di↵erence between the
X-ray peak and the centroid locations.
These two basic properties of clusters will be useful when studying clusters on an individual
basis, but also were employed to create subsets of the cluster sample.
2.4.
Gas probes
2.4.1.
X-ray
The gas in galaxy clusters has foremost been studied through the X-ray, since that was how
the ICM was discovered.
A combination of X-ray imaging and spectroscopy can be used to find thermodynamic
profiles of density and temperature, respectively, and has been done routinely in the past. Surface
brightness is the number of photons per a given area per time, whereas spectroscopy looks at the
spectra of certain areas in the image. One can fit the spectra to emission models using di↵erent
packages such as XSPEC (Arnaud 1996) to constrain the temperature. Temperatures derived in
this way are called the spectroscopic temperature, Tspec . Mazzotta et al. (2004) had found that
Tspec can underestimate the mass-weighted temperatures (i.e. those derived from the SZ, since it
is dependent on the integrated pressure) by up to 40% due to the assumption of a single-phase
isothermal plasma.
Following the notation of Schneider (2010) on pg 243-244, the thermal bremsstrahlung has
an emissivity:
✏fv f
32⇡Z 2 e6 ne ni
=
3me c3
r
2⇡
e
3kB T me
hp v/kb T
gf f (T, v)
(1)
Z is the ion charge, e is the electron charge, the number densities are ne and ni for the
electrons and ions, T is the temperature of the gas, kB is Boltzmann’s constant, hp is Planck’s
constant, and me is the mass of the electrons. gf f is a factor to correct classical to quantum
mechanics:
3
9kB T
gf f ⇡ p ln
4hp v
⇡
(2)
However, the observed emissivity also includes contribution from line emission. Generally,
free-free emission is most important above around 1 keV, while below that line emission takes
over. ✏v is be the total emission from both sources, i.e. ✏v = ✏f f + ✏line .
12
Surface brightness is a projected quantity that can then be written as
Z 1
✏ (r)rdr
pv
SX (R) = 2
r 2 R2
R
(3)
Where R is the 2-D distance from the center according to the observer, and we integrate
over 3-D radius r.
This essentially reduces down to:
SX / n2e ⇤(Te )`
(4)
where ne is the electron density, ⇤(Te ) is the cooling function, Te is the electron
temperature, and ` represents the distance along the line of sight within the cluster. The gas
temperature is equal to the electron temperature (T = Te ) if the gas is in thermal equilibrium.
The frequency dependence as well as metallicity are incorporated into ⇤(Te ). The curve is
relatively flat and dominated by thermal bremsstrahlung above ⇠ 1 keV. Below of which, di↵erent
metal lines dominate. Cluster temperatures are usually above 1 keV, so, in the region of interest,
1/2
the cooling function has a relatively weak dependence on the temperature, approximately Te .
Refer to Figure 3 for a typical cooling function in the temperature ranges of interest.
Fig. 3.— A typical cooling function as a function of temperature, including only free-free emission.
The black line is the chosen energy range of interest (0.7-2 keV)to minimize temperature dependence
of the signal, while the red and blue lines correspond to the wide band (0.7-7 keV), and hard band
(2-7keV), respectively.
13
2.4.2.
Sunyaev-Zel’dovich E↵ect
The Sunyaev-Zel’dovich e↵ect is the inverse Compton scattering of low-energy photons from
the cosmic microwave background (CMB) scattering o↵ of the hot electrons in a galaxy cluster,
increasing the energy of the CMB photons. In particular we are interested in the thermal SZ
e↵ect (tSZ), where the e↵ect arises from the temperature of the gas. The kinematic SZ e↵ect
(kSZ) is a smaller e↵ect (on the order of 10% of the tSZ e↵ect) caused by the motion of the
cluster in reference with the CMB, but in this study we are only sensitive to the tSZ.
The shift in the energy of the photons from low to high (see Figure 4a) leaves behind a
temperature decrement and increment in frequency space (Figure 4b). This decrement is directly
related to the integrated pressure along the line of sight.
(a)
(b)
Fig. 4.— (a) The intensity before (dashed) and after (solid) a fictional cluster is added to the sky.
The shift for real clusters is smaller. The CMB radiation is altered by the the ICM. (b) The solid
line is the subtraction of the CMB intensity from the cluster (i.e. the two curves in (a)), but for a
realistic cluster mass. The maximum decrement is around 120 GHz. Figure credit: Carlstrom et
al., Annual Reviews of Astronomy and Astrophysics vol 40, 2002
SZ surveys have simultaneous frequencies that measure the increment and decrement. Once
a cluster has been detected, one can look at the actual strength and shape of the cluster by
looking at decrement maps. Studies normally look at the decrement because there is more
foreground contamination from the galaxy at the increment frequency.
Following the notation of Carlstrom et al. (2002), the SZ signal is given by:
14
TSZ
TCM B
= f (x)y
(5)
Where TCM B is the CMB temperature. y is the Compton y parameter, which describes the
photon’s gain in energy after passing through the cluster:
Z
k B Te
y=
(6)
T ne dl
me c2
Where Te is the electron temperature, ne is the electron density,
of an electron, and me is the electron mass.
T
is the Thompson cross-section
f (x) contains the frequency dependence, with x = hp v/kB TCM B :
f (x) = (x
SZ
ex + 1
ex 1
4)(1 +
SZ (x, Te ))
(7)
contains relativistic corrections. In the end, we can simplify the SZ emission as:
SSZ / ne Te `
(8)
` is the distance along the line of sight within the cluster. Since the SZ e↵ect is independent
of redshift, the signal does not diminish with distance - as it does in the optical and X-ray. The
CMB energy density increases with redshift as (1 + z)4 ; a factor of (1 + z) comes from the
frequency, and a factor of (1 + z)3 comes from the comoving number density of photons. This
exactly cancels the e↵ect of cosmological dimming of surface brightness, which decreases with
redshift as (1 + z) 4 . By definition, the flux is inversely proportional to the luminosity distance
squared, and the luminosity distance scales as (1 + z)2 . The luminosity distance is the distance
inferred from the di↵erence between the absolute and apparent magnitude of the cluster.
It is beneficial to forgo the use of X-ray spectroscopy, which requires long exposure times
and can be susceptible to calibration and modeling e↵ects depending on the instrument used.
Due to the di↵erent dependencies of the signals on density and temperature, we can combine
X-ray surface brightness and SZ maps to recover the density and temperature simultaneously.
While the density is mostly constrained by the X-ray, the pressure is constrained by the SZ, so
one can then find the temperature.
2.5.
Self-Similarity and Scaling Relations
Self-similarity is based on the simplifying scenario that clusters formed from a purely
gravitational spherical collapse (Kaiser 1986). Essentially, clusters can be scaled to take into
account the evolution of the universe (the changing average density of the universe, for example),
assuming self-similarity. Thus clusters are scaled versions of each other, having the same
properties, just scaled. By scaling the clusters by the di↵erent quantities, you can compare
15
close-by clusters with ones that are further away, and more massive clusters with smaller ones.
This way they can be compared on an equal basis.
The R500 and M500 values in this study were taken from the Mantz et al. (2010) study,
which depends on X-ray measurements. They determined R500 and M500 by using gas-mass
fraction profiles, which were based on cluster surface brightness profiles and average
temperatures. The gas-mass fraction, fgas is used to convert from the gas mass to total mass:
fgas = Mgas /Mtot . At R500 , fgas is based on simulations (with a bias factor of B). The equations
relating the quantities are:
M (r500 ) =
3
Mg (r500 )
4⇡500⇢crit (z)r500
=
(1 + B)fgas
3(1 + B)
(9)
There are several quantities by which a cluster can be scaled that are important in this
work. ne,500 , using Nagai et al. (2007), has no real mass dependence:
ne,500 = 500
⌦m ⇢crit
⌦b µe mp
(10)
⌦m is the total matter density, while ⌦b is the baryonic density, ⇢crit is the critical density
of the universe, µe is the mean molecular weight for electrons, and mp is the mass of the proton.
This factors in the conversion from the density of the total gas to the density of electrons.
The virial theorem relates the galaxy velocities in a cluster with the potential well, such
that M/r / v 2 . When the cluster is in equilibrium, and if we assume the ideal gas law, we can
then connect the average kinetic energy of the galaxies with that of the gas as: 12 mv 2 = 32 kB T .
Then the mass M can be related to the temperature: M/r / T , or M 2/3 / T . The normalization
factor we use to equate the two quantities come from Nagai et al. (2007):
T500 = 11.05 keV
✓
M500
15
10 h 1 M
◆2/3
E(z)2/3
(11)
E(z) represents the evolution of the Hubble parameter, which is dependent of the
p
cosmology: ⌦m (1 + z)3 + ⌦⇤ , assuming a flat universe, and h is the dimensionless Hubble
parameter. ⌦⇤ is the dark energy density.
If you study many clusters, you can observe a scatter away from self-similarity, which can
tell you the degree to which hydrostatic equilibrium is not observed and non-gravitational physics
occur. What has been found is that clusters are quite regular at intermediate radii, but deviate
from self-similarity at the center and in the outskirts, due to the e↵ects described in Section 2.3.
One way clusters can be studied is by observing trends between their observed quantities,
for example, the correlation between mass and temperature. Scaling relations can also be
established between SZ and X-ray quantities. Czakon et al. (2015) found scaling relations
16
between the integrated SZ and X-ray signal as well as mass for the BOXSZ sample. They found
di↵erences from self-similarity, but that it was likely due to X-ray calibration e↵ects.
2.6.
Thermodynamic studies of galaxy cluster gas
It is important to understand where this analysis fits in relation to the large literature of
galaxy cluster ICM studies.
Since the X-ray is the traditional method for observing the gas, it has the most volume and
includes the foundation of ICM research. We will note the most prominent and relevant studies.
Vikhlinin et al. (2006) used Chandra observations of 13 closeby clusters, and fit analytical density
and temperature profiles. These empirical profiles would further be used in countless future
papers, including ours. Cavagnolo et al. (2009) (denoted as the ACCEPT sample for future
reference) conducted a large sample of Chandra-observed clusters, using surface brightness and
spectroscopy. Their results for density, temperature, pressure, and entropy are publicly available
on the ACCEPT website. When comparing the results from this study, often we will compare to
the ACCEPT results as a representation of what X-ray only studies would find. Morandi et al.
(2015) used 320 clusters in the X-ray (Chandra) to stack emission measure (EM) profiles, where
R
EM = ne np dl. From this they were able to constrain gas density and slopes out to around R100 ,
and found that the gas density steepens after R500 . Arnaud et al. (2010) combined X-ray
observed clusters (using XMM-Newton) with simulations to find cluster pressure profiles.
Leccardi & Molendi (2008) used XMM-Newton X-ray spectroscopy to find mean temperature
profiles of about 50 galaxy clusters. Many more X-ray studies exist and are mentioned in the
individual cluster descriptions in the Appendix on a cluster-by-cluster basis. Reese et al. (2010)
tried three di↵erent calibration databases and reduction software (CIAO) for X-ray spectroscopy
using Chandra. They found that temperatures can change as much as 13% for their sample of 38
clusters, proving how sensitive X-ray spectroscopy is to the instrument. Thus it would be useful
to find a spectroscopic-independent method for finding temperatures.
Sayers et al. (2013a) (from now referred to as S13) did a pressure deprojection study on the
BOXSZ sample of clusters using only the SZ data to constrain pressure. Since this study is
essentially using the same data that S13 uses for the SZ, we will often compare our results to the
pressure deprojections. Planck Collaboration et al. (2013) studied 62 nearby clusters through the
SZ e↵ect to find pressure profiles, and they stacked pressure profiles to extend out to 3R500 .
Plagge et al. (2010) observed 15 galaxy clusters through the SZ e↵ect with the South Pole
Telescope (SPT). They fit pressure profiles to the clusters and found an average profile out to
Rvir .
There has been an emergence of joint X-ray and SZ fitting, however they often also include
X-ray spectroscopy. Bonamente et al. (2006) used a combination of X-ray and SZ data to find the
cosmic distance scale. Bonamente et al. (2012) did a study to compare the pressure profiles
derived from SZ data and X-ray data for 25 relaxed clusters, and found that the integrated
17
pressures were consistent. LaRoque et al. (2006) combined Chandra X-ray data with Owens
Valley Radio Observatory (OVRO)/Berkeley Illinois Maryland Array (BIMA) SZ data, and
jointly fit smooth profiles (density and temperature profiles of varying complexity). We will
compare our results with theirs for the 7 overlapping clusters. Eckert et al. (2012) combined
ROSAT gas density and Planck pressure profiles to look at gas behavior in the outskirts. They
found the density profile steepens after R500 , and explored how CC clusters have steeper profiles
in the outskirts than NCC clusters. Mahdavi et al. (2007) combined X-ray, SZ, and weak lensing
data to find the mass of a cluster, introducing ”Joint Analysis of Cluster Observations” (JACO).
Siegel et al. (in preparation) used JACO on the Cluster Lensing and Supernova survey with
Hubble (CLASH) clusters and found their mass and gas profiles.
There have been few studies that recovered the temperature profiles without X-ray
spectroscopy. Basu et al. (2010) conducted a non-parametric deprojection of one relaxed CC
cluster using only X-ray surface brightness and SZ imaging from the APEX-SZ bolometer, and
was able to find the temperature out to large radii, R200 . They use Abel’s integral inversion
method (Nord et al. 2009) to deproject the profiles, and found for the first time a temperature
profile that decreased in the outskirts to a significant level through the SZ, although the 1uncertainties become much larger with radius. They compared their results with spectroscopy
measurements and found theirs to be systematically higher, especially around R2500 . Another
notable work was Kay et al. (2008), who found SZ-only temperatures of simulations, which
requires observations at multiple SZ frequencies. They had found that SZ-derived temperatures
are higher than that of the X-ray. They postulated the di↵erences arise from the fact that the
X-ray is weighted by cool dense gas, while the SZ is weighted by hot dense gas.
There are many simulation studies that include the processes discussed in Section 2.3.
Roncarelli et al. (2006) did hydrodynamical numerical simulations of clusters, which included
cooling, star formation, and supernovae feedback. In general, their average density profiles are
steeper than what we find in real clusters, possibly due to the fact that AGN feedback is not
included. Battaglia et al. (2012) stacked average pressure profiles from smoothed particle
hydrodynamics (SPH) simulations, with di↵erent levels of AGN feedback, cooling, star formation,
and supernova feedback and found their e↵ect on scaling relations. Vazza et al. (2010) created
non-radiative simulations of 7 simulated clusters to find mixing properties of the ICM.
3.
The Instruments
3.1.
X-ray
X-ray telescopes are always space-based to be above the Earth’s atmosphere, since it blocks
light. X-ray wavelengths and other high-energy photons have high enough energy such that
photo-electric absorption by individual atoms occurs. Although the atmosphere is relatively thin,
because of its physical thickness in space, X-rays are absorbed due to the sheer number of atoms
they may encounter.
18
There has been a long lineup of X-ray observatories that have led us to our knowledge of
galaxy clusters today. Of notable interest are as follows: Uhuru (1970-1973) was the telescope
that discovered the X-rays originating from the ICM. It consisted of proportional counters, and
had a coarse resolution of 30’. Einstein (1978-1981) was the next large telescope that incorporated
mirrors, and made the first images of galaxy clusters. Its resolution greatly surpassed previous
telescopes with 3-5”. Roentgensatellite (ROSAT) (1990-1999) was the next major telescope with
3” resolution that made a large X-ray galaxy cluster survey (called REFLEX) still used today.
The two major X-ray observatories currently still in orbit are the The European Space Agency’s
X-Ray Multi-Mirror Mission - also known as XMM-Newton - and NASA’s Chandra Observatory
(CXO) (both 1999-now). XMM-Newton has 5” resolution spanning energy range of 0.1-12 keV,
whereas Chandra has sub-arcsecond resolution and an energy range of 0.1-10 keV. XMM-Newton
has the advantage of the option of spectroscopy/imaging with optical and ultraviolet observations
happening simultaneously, while Chandra has much higher resolution. Oftentimes the results of
these two instruments are compared due to their roughly compatible resolutions, and sometimes
di↵erences may arise from the instrumentation and calibration.
3.1.1.
Chandra
There are several di↵erent instruments on the CXO, including the High Resolution Camera
(HRC), Advanced CCD Imaging Spectrometer (ACIS), and the high resolution spectrometers.
The ACIS is of most use for this study because it has high spatial resolution, spectral resolution,
and large field-of-views (FoVs). Although we do not use spectroscopy for measuring spectroscopic
temperatures, some spectral information is necessary in the data reduction. Focal planes ACIS-I
and ACIS-S were both used. Each chip in these arrays have 1024x1024 pixels, with a spatial
resolution of 0.492”. The ACIS-I array consists of four chips making a square, with a FoV of
16.9’x16.9’, while the ACIS-S array is linear with 6 chips and a FoV of 8.3’x50.6’. The CCDs
work via the photoelectric e↵ect: the X-ray photon hits the detector and ejects an electron. The
position, energy, and time of the event are recorded.
The X-ray data was taken from the Chandra X-ray Observatory public archive. The
Chandra X-ray Center developed the Chandra Interactive Analysis of Observations (CIAO)
toolset to reduce Chandra data from the raw data, and is available online. The data reduction
mostly followed these procedures. Details are in Section 5.1.
3.2.
SZ/microwave
Observing the ICM through the SZ e↵ect is a more recent development, and many of the
instruments devised originally are still used today. The Sunyaev-Zel’dovich Array (SZA) is an
array of 8 radio telescopes, having 2 receivers with frequency bands of 26-36 GHz and 85-115
GHz. The South Pole Telescope (SPT) is able to measure signals from the microwave to sub-mm
19
wavelengths. It has the SPT-SZ camera consisting of 960-element array, observing at three
frequencies: 95 GHz, 150 GHz, and 220 GHz. It was designed for conducting SZ surveys, since it
can observe the decrement and increment. The ESA’s Planck satellite (2009-2013) had two
instruments, one of which (High Frequency Instrument, HFI) was able to probe the SZ e↵ect at
several bands.
3.2.1.
Bolocam
Bolocam is a 144-element bolometer array on the Caltech Submm Observatory (CSO) at
Mauna Kea, with a 8’ circular FOV. The frequency we used is centered at 143 GHz (2.1mm),
which measures the SZ temperature decrement. At this frequency, the resolution is 58”. The
resulting maps are 14’x14’, with 42x42 pixels. Although it has relatively low resolution, there is a
high signal-to-noise across a large FoV. The Bolocam team at Caltech formulated a pipeline to
reduce the data, which includes subtracting out atmospheric noise, modeling the noise, and
filtering using the transfer function and beam. See Section 5.2 for more details.
4.
The BOXSZ sample
The Bolocam X-ray/SZ (BOXSZ) sample consists of 45 clusters that were observed using
the Bolocam bolometer array, all of which had archival Chandra X-ray data available. Table 1
lists the important cluster characteristics, and Table 2 gives a summary of previous studies (See
Section 2.6) with the overlap of those studies with the BOXSZ sample. We refer the reader to S13
for more details on how the sample was picked, but essentially it was an ad hoc process, and the
selection function is not well-defined. The large redshift redshift range (from z = 0.15 to z = 0.89,
with a median redshift of z = 0.44) is key for studying the cluster outskirts. Figure 5 shows the
redshift distribution of the sample.
We di↵erentiate cool-core clusters (CC) from non-cool-core clusters (NCC) using an X-ray
luminosity ratio cut, which is the ratio of the luminosity in the central region to that of the whole
cluster (Mantz et al. 2010). Higher ratios correspond with cool-core clusters, as they are very
bright in the center due to their high density and low temperature. In general, CC clusters are
more prominent at low redshifts - see Figure 5 - since they are further along in their history, they
had more time to become relaxed. Clusters are classified as disturbed using the X-ray centroid
shift parameter, which describes the X-ray peak and centroid discrepancy. A relaxed, i.e.
non-disturbed, cluster will be mostly spherical, which means the peak in the brightness will be at
the center of the cluster, whereas a disturbed cluster can have o↵sets, due to merging or other
processes. The sample contains a sufficient variety of clusters: 17/45 clusters are cool-core
clusters, and 16/45 are disturbed. This will be useful for comparing ensemble characteristics to
see if there are di↵erences between the groups.
Since this analysis assumes spherical symmetry, it is useful to have a measure of the
20
ellipticity of the clusters. Czakon et al. (2015) fit elliptical models of increasing complexity to the
clusters, and resulted in finding the number of model parameters (MP) of the minimal model. We
call 1-2 fit model parameters as spherical clusters, while 3-4 as non-spherical. 26/45 clusters are
considered spherical, and 19/45 are consitered to be non-spherical.
The sample consists of clusters from two major catalogues; the Abell catalogue and the
MACS sample. Abell contains the brightest and closest clusters, established in the 1960’s. The
Abell catalogue consists of about 4,000 clusters at low redshift. George Abell constructed the
northern part of the survey by visually picking clusters from the Palomar Sky Survey (POSS). He
picked clusters based on how many galaxies were in the cluster, how compact they were, along
with other limits on relative distance from the galactic plane. The southern part of the survey
was completed later for the declinations below that of the POSS observations.
The other catalogue, the MAssive Cluster Survey (MACS) is based on X-ray luminous
clusters at higher redshifts (z > 0.3). Clusters were picked from the ROSAT Bright Source
Catalogue, with cuts based on X-ray flux and hardness, and were checked with optical surveys. It
contains 124 clusters.
Number of Clusters
10
8
6
4
2
0
0.0
0.2
0.4
0.6
0.8
1.0
z
Fig. 5.— Redshift distribution of the sample. The red solid line indicates the 28 NCC clusters
while the blue dotted line indicates the 17 CC clusters.
21
22
Table 1: Cluster Information
cluster name
z
A2204
A383
A1423
A209
A963
A2261
A2219
A267
RXJ2129.6
A1835
A697
A611
MS2137
MACSJ1931.8
AS1063
MACSJ1115.8
MACSJ1532.9
A370
CL0024P17
MACSJ1720.3
MACSJ0429.6
MACSJ2211.7
MACSJ0416.1
MACSJ0451.9
MACSJ0417.5
MACSJ1206.2
MACSJ0329.6
RXJ1347
MACSJ1311.0
MACSJ0257.1
MACSJ0911.2
MACSJ2214.9
CL0016
MACSJ1149.5
MACSJ0717.5
MACSJ1423.8
MS0451.6
MACSJ0025.4
MS2053
MACSJ0647.7
MACSJ2129.4
MACSJ0744.8
MS1054
CLJ0152.7
CLJ1226.9
0.15
0.19
0.21
0.21
0.21
0.22
0.23
0.23
0.24
0.25
0.28
0.29
0.31
0.35
0.35
0.36
0.36
0.38
0.39
0.39
0.4
0.4
0.42
0.43
0.44
0.44
0.45
0.45
0.49
0.5
0.5
0.5
0.54
0.54
0.55
0.55
0.55
0.58
0.58
0.59
0.59
0.69
0.83
0.83
0.89
R500
(kpc)
1460
1110
1350
1530
1350
1590
1740
1220
1280
1490
1650
1240
1060
1340
1760
1280
1310
1400
1000
1140
1100
1610
1270
1120
1690
1610
1190
1670
930
1200
1220
1390
1470
1530
1690
1090
1310
1120
820
1260
1250
1260
1070
970
1000
M500
1014 M
10.3
4.7
8.7
12.6
6.8
14.4
18.9
6.6
7.7
12.3
17.1
7.4
7.7
9.9
22.2
8.6
9.5
11.7
4.4
6.3
5.8
18.1
9.1
6.3
22.1
19.2
7.9
21.7
3.9
8.5
9.0
13.2
16.5
18.7
24.9
6.6
11.5
7.6
3.0
10.9
10.6
12.5
9.0
7.8
7.8
NCC/CC
CC
CC
NCC
NCC
NCC
CC
NCC
NCC
CC
CC
NCC
NCC
CC
CC
NCC
CC
CC
NCC
NCC
CC
CC
CC
NCC
NCC
CC
NCC
CC
CC
CC
NCC
NCC
NCC
NCC
NCC
NCC
CC
NCC
NCC
NCC
NCC
NCC
NCC
NCC
NCC
NCC
disturbed
spherical
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Table 2: Summary of Previous Studies
Study
Arnaud et al. (2010)
Basu et al. (2010)
Bonamente et al. (2012)
Cavagnolo et al. (2009)
Kay et al. (2008)
LaRoque et al. (2006)
Density Temperature Pressure Appendix* Notes
X
33 Clusters total
X
1 Cluster study (A2204)
X
14 Cluster overlap, 25 clusters total
X
X
X
X
36 Cluster overlap, 239 clusters total
X
Simulations
X
X
X
7 Cluster overlap, 37 clusters total
5 Cluster overlap, 28 clusters total,
Leccardi & Molendi (2008)
X
only mean profile
39 Cluster overlap, 320 Clusters total,
Morandi et al. 2015
X
only mean profile
1 Cluster overlap (AS1063),
Plagge et al. (2010)
X
15 Clusters total
Planck Collaboration et al. (2013)
X
5 Cluster overlap, 62 Clusters total
Sayers et al. (2013a)
X
X
45 Cluster overlap (BOXSZ sample)
1 Cluster overlap (A383),
Vikhlinin et al. (2006)
X
X
13 clusters total
Density, temperature, and pressure profiles from previous studies. *Study is included in individual cluster plots in the
Appendix.
5.
Data Reduction
Table 3 summarizes observation information for the X-ray and SZ data.
5.1.
Chandra
X-ray data was taken from the Chandra X-ray Observatory public archive. Observations
used were obtained from the Advanced CCD Imaging Spectrometer (ACIS) focal plane arrays
ACIS-I and ACIS-S. The data was reduced by the author according to standard processing using
the most recent calibration CIAO version 4.7.
First, the preprocessed data available on the public archive was processed to create new
level=2 event files by filtering level=1 event files on good GRADES. The GRADE is a number
assigned to every event that describes the charge distribution on the set of 3x3 pixels centered on
the event. There is a standard set of ACIS grades that is accepted, while bad grades are those
patterns that are caused by cosmic rays (charged-particle events) rather than actual X-rays.
Cosmic rays cause large blobs/streaks on the detector while X-rays are small and point-like.
Filtering out cosmic rays in this way optimizes the signal-to-background ratio. Many of the X-ray
observations were observed in the VFAINT mode, which means that the object is ”very faint,” so
there is a more thorough search for events, using a larger pixel area (5x5). An example of the
result of the point-source search of a cluster observation is shown in Figure 6.
Standard bad pixels and chip boundaries were removed. Standard Good Time Intervals
(GTIs) that are supplied by the pipeline were filtered for. These are the time periods where the
time line parameters were in acceptable ranges. Light curve filtering in the total 0.3-10 keV band
23
Fig. 6.— MACS J1206.2 X-ray event map, where the green ellipses are the detected cosmic rays,
and the red circles are where we extracted background rates for background subtraction.
of the background was performed to find any observation-specific bad pixels/columns and
background flares. The light curve is the count rate as a function of time. If the count rate
exceeded 3 above the quiescent (or calm) count rate, those data were excluded. Background
flares, caused by protons from the Earth’s magnetic field, have been shown to have significant
e↵ects on the temperature profiles, biasing it upwards by as much as a factor of 2 (Markevitch
2002), so it is important to filter out for these flares.
Afterwards, the data were filtered according to energy. Only the 0.7-2 keV band was chosen
so that the data were least a↵ected by the background, and also to minimize the temperature
dependence of the signal.
The background was calculated by using Chandra blank-sky observations that correspond
to the time of the observation took place, and is then renormalized to the flux measured in the
outskirts of the chips that did not include cluster signal. See Figure 6 for example background
extraction regions.
The images were binned such that each bin must include at least 100 total counts and 10
24
source (i.e. total - background) counts, with widths of at least 5”, so that the PSF e↵ects could
be ignored. The 10-count minimum on the source counts ensures that we can assume a Gaussian
likelihood for the X-ray data instead of a Poisson likelihood. Di↵erent studies have used di↵erent
cut-o↵s for when a Poisson distribution can be approximated by a Gaussian, but since we wish to
look at the dimmest parts of the cluster, 10 is roughly the minimum cuto↵ one could assume. To
convert counts into surface brightness, one also needs the area within the bin, which was
calculated without the point source extracted areas. One also needs an estimate of the e↵ective
area and exposure time. The exposure map included both of these e↵ects.
Figure 7 shows an example surface brightness profile resulting from the data reduction by
taking into account the total number of counts, background counts, area of the bin, and the
exposure map.
The cooling function was calculated using the Mekal plasma modeling code in XPEC
(Arnaud 1996). A constant metallicity was assumed, 0.3Z , and the nH column density number
was estimated using the online tool nH provided by HEASARC, which is dependent on the
cluster’s location. The resulting cooling function was also filtered for the energy range of interest,
0.7-2 keV, see Figure 3.
Fig. 7.— MACS J1931.8 surface brightness profile, where the green line represents the background
level, and the black points are the estimated surface brightness after subtracting the background.
25
Table 3: Observation Information
X-ray
Cluster
z
A2204
A383
A1423
A209
A963
A2261
A2219
A267
RXJ2129.6
A1835
A697
A611
MS2137
MACSJ1931.8
AS1063
MACSJ1115.8
MACSJ1532.9
A370
CL0024P17
MACSJ1720.3
MACSJ0429.6
MACSJ2211.7
MACSJ0416.1
MACSJ0451.9
MACSJ0417.5
MACSJ1206.2
MACSJ0329.6
RXJ1347
MACSJ1311.0
MACSJ0257.1
MACSJ0911.2
MACSJ2214.9
CL0016
MACSJ1149.5
MACSJ0717.5
MACSJ1423.8
MS0451.6
MACSJ0025.4
MS2053
MACSJ0647.7
MACSJ2129.4
MACSJ0744.8
MS1054
CLJ0152.7
CLJ1226.9
0.15
0.19
0.21
0.21
0.21
0.22
0.23
0.23
0.24
0.25
0.28
0.29
0.31
0.35
0.35
0.36
0.36
0.38
0.39
0.39
0.4
0.4
0.42
0.43
0.44
0.44
0.45
0.45
0.49
0.5
0.5
0.5
0.54
0.54
0.55
0.55
0.55
0.58
0.58
0.59
0.59
0.69
0.83
0.83
0.89
SZ
ObsID
Exposure Time
(ks)
S/N Peak
Obs Time
(hrs)
6104
2321
538
3579
903
5007
896
3580
552
7370
4217
3194
4974
9382
4966
9375
1649
7715
929
6107
3271
3284
10446
5815
11759
3277
3582
3592
6110
1654
5012
3259
520
3589
4200
4195
902
10413
1667
3584
3595
6111
512
913
5014
9.61
19.51
9.87
9.99
36.29
24.32
42.30
19.88
9.96
39.51
19.52
36.11
57.38
98.92
26.72
39.63
9.36
7.09
39.94
9.61
23.17
17.74
15.83
10.21
51.36
23.46
19.85
57.51
63.21
19.85
23.79
19.47
67.41
20.05
59.04
115.57
44.19
75.64
44.51
20.00
19.87
49.50
89.17
36.48
32.71
22.30
9.60
5.80
13.90
8.30
10.20
11.10
9.60
8.00
15.70
22.60
10.80
6.50
10.10
10.20
10.90
8.00
12.80
3.30
10.60
8.90
14.70
8.50
8.10
22.70
21.70
12.10
36.60
9.60
10.10
4.80
12.60
15.70
17.40
21.30
9.40
24.30
12.30
5.10
14.40
15.20
13.30
17.40
10.20
13.00
12.7
24.3
11.5
17.8
11.0
17.5
6.3
20.7
16.0
14.0
14.3
18.7
12.8
7.5
5.5
22.8
14.8
11.8
8.3
16.8
17.0
6.5
7.8
14.2
9.8
24.9
10.3
15.5
14.2
5.0
6.2
7.2
9.8
17.7
12.5
21.7
14.5
14.3
14.3
11.7
13.2
16.3
18.3
9.3
11.8
26
5.2.
Bolocam
Observations of the 45 clusters were made by the Bolocam team between November 2006
and March 2012. The SZ reduction technique was the implementation by Sayers et al. (2011).
See Table 3 for observation summary. Radio-sources were removed from the map; 6 clusters had
large radio sources that were close to the cluster center Sayers et al. (2013b). An iterative
procedure was adopted to calculate the proper transfer function per each cluster. When
comparing the data to the model, the model also is convolved with the beam and the transfer
function. See Figure 8 for a typical processed cluster map.
Fig. 8.— A processed SZ image of Abell 209 in flux units of mJy/beam.
6.
Method
A Markov Chain Monte Carlo (MCMC) method was used to maximize a joint-likelihood
function, which is the joint likelihood of X-ray and SZ likelihoods. Since the X-ray and SZ
emissions have di↵erent dependencies on density and temperature, a joint-likelihood allows us to
27
constrain both simultaneously. Two modeling methods are implemented; a nonparametric onion
shell deprojection, with constant density and temperatures within each shell, and a parametric fit
of smooth profiles for density and temperature.
The basis of the method applied here stems from Ameglio et al. (2007), with several key
di↵erences:
• First, the method is applied to real clusters, while Ameglio et al. (2007) had conducted the
method on simulated clusters.
• Second, there is no temperature regularization included in the deprojections, which had
acted to smooth out the temperature profiles from oscillations. The reasoning behind the
choice of no regularization was that it could take away information about possible
substructure, as well as also using analytical fits to find smoothed out profiles.
• The assumptions for the emission beyond the last shell are now more representative of
current knowledge.
• Radii of the bins of the image no longer need to be the same as the 3-D radii of the shells of
the deprojections.
• For the analytical smooth profiles, the integration of the surface brightness across bins
(instead of assuming a flat surface brightness across the bin) is employed, which can span
larger radii as one goes to the outskirts where that is important.
The details are outlined below.
6.1.
Markov Chain Monte Carlo
MCMC is a powerful tool that can be used in Bayesian statistics to determine many
parameters at once. Parameters are drawn randomly from a supplied distribution, and the
likelihood of those parameters given the data is calculated. The Metropolis-Hastings Algorithm
was used to decide how to accept or reject steps in the chain. If the next step of the chain has a
negative log-likelihood that is less than the previous step, it is automatically accepted. This would
correspond to an acceptance probability, ↵ of 1. However, it does not automatically reject steps
that have larger lnL. Instead, a random number is drawn from a uniform distribution between 0
and 1. If the ↵ is greater than the random number that was drawn, the step is accepted.
Once the burn-in of the parameters to their stable values is thrown out (which was done by
eye), one can then analyze the chain. To calculate the chain statistics, a program called GetDist
is used, which was formulated by Antony Lewis. It provides mean values, marginalized limits, and
best fits, along with the covariance matrix. It also calculates convergence statistics, such as the
split-rms test.
28
6.2.
The Individual Cluster Likelihood
We assume spherical symmetry to deproject 2-D maps into 3-D profiles. The X-ray data
maps are binned into radial bins, while the SZ is fit in 2-D. The reason for this is that the noise in
the SZ maps is heavily pixel-dependent, while the X-ray noise can be approximated as constant
within thin radial bins. The likelihood is just the product of the X-ray and SZ likelihoods:
L = LX
ray
· LSZ
(12)
The SZ observable is Gaussian, and has gaussian noise properties. The resulting
log-likelihood for a single cluster is given by:
ln(LSZ ) =
X
i
1 (Oi
2
Mi ) 2
(13)
2
i
Oi is the observation, Mi is the model converted into an observable, and
each pixel i.
is the error in
Technically, X-ray data are the result a Poisson process, so one could use the Poisson
likelihood. However, this likelihood is not as ready to give goodness-of-fit information directly
from the likelihood. By requiring that each radial bin has a sufficient number of source counts
(10), we approximate the Poisson likelihood with a Gaussian likelihood (where the variance is the
model value). Its corresponding log-likelihood is given as:
ln(LX
ray )
=
X
i
1 (Oi Mi )2
2
Mi
1
ln(2⇡Mi )
2
(14)
i is each radial bin. One needs the extra term since we are essentially fitting for the
variance (the model) as well, so the normalization cannot be ignored.
In practice, we are minimizing the negative log-likelihood, which is equivalent to
maximizing the likelihood.
6.3.
Fitting models
The two types of fitting we use are nonparametric onion shell deprojections, and analytical
fits. The former permits us to fit a model that is not necessarily smooth and not dependent on
functional forms, and allows us to see possible substructures, while the latter fits to a smooth
profile that one can easily compare with other studies.
29
6.3.1.
Onion shell deprojections
We assume an onion-skin model, which has a constant density and temperature assigned
within concentric shells (McLaughlin 1999). Refer to Figure 9 for the geometry. The X-ray signal
at a radial bin r is modeled as:
Fig. 9.— The onion shell model, adapted from McLaughlin 1999. The observer sees a 2D map
with radial bins denoted by r, while the 3D shells have radii Ri . `i is the distance along the line of
sight that radial shell i contributes to the 1D bin at radius r.
SX
ray (r)
= µe /[µH 4⇡(1 + z)4 ]
X
i=1
n2e,i ⇤(Te,i ) · li (R)
(15)
where µe and µH are the fractional electron and proton density values, respectively. This
equation accounts for the redshift dependence. i is the shell, ne,i is the electron density within the
shell, Te,i is the electron temperature within the shell, and li (R) is the length of the line of sight
that contributes from shell i to the bin of radius r.
The SZ signal at radial bin r is modeled as:
30
SSZ (r) = /me c2
X
i=1
where
ne,i kTe,i · li (R)
(16)
is the Thompson cross-section for the electron, me is the mass of the electron.
The 2-D X-ray data bins do not need to be the same radii as the 3-D model shells, with the
X-ray data bins being much smaller than the 3-D shells. For the individual clusters, we use 5
shells. The first shell is constrained to be half the approximate resolution of Bolocam, which is
⇠60” for the frequency range we are using, therefore 30” in radius. After this the shells are
spaced equally in logarithm to the cut-o↵ radius, rmax , to maximize the signal-to-noise in each
shell. We must also take into account the signal from outside the last bin.
We implement an iterative procedure to find the emission beyond the last shell. First the
deprojections are run assuming there is no emission beyond the last shell. Then we fit the
resulting deprojections to smooth profiles; the X-ray signal essentially comes from the density
deprojections, while the SZ signal comes from the pressure. We assume the shape of the profiles
to be constant and just fit for a normalization. The pressure profile we use is the S13 average
profile shape. The density profile shape used was taken from Pi↵aretti et al. (2011). The use of
di↵erent outer density profile shapes made negligible e↵ects on the reconstructed profiles. The
next iteration integrates the emission assuming those profiles with the previously fitted
normalizations. This is repeated until the profiles converge, which they do after 3-4 iterations.
We are assuming the signal is zero at 10r500 . The use of di↵erent known pressure profiles that
were fit with the deprojections extended out to large radii (Planck collaboration 2013, S13,
Arnaud et al. 2010) showed negligible e↵ects on the value on the last bin. Di↵erent density profile
shapes also made no significant di↵erence on the last bin values, since most density profiles have
relatively the same slope.
6.3.2.
Analytical profiles
For the analytical fits, we fit to smooth density and temperature profiles to the data.
Within the X-ray bin, we numerically integrate the surface brightness formula across the map bin
to compare with the observed X-ray counts.
We employ Vikhlinin et al. (2006) models, which are based on X-ray observations of a large
ensemble of clusters. The number of parameters we fit for was varied depending upon the type of
cluster being fitted for. The simplest model would be the single-beta density model with an
isothermal temperature profile:
ne (r) = n0,i (1 + (r/rc,i )2 )
Te (r) = T0
31
3 /2
(17)
(18)
n0,i and rc,i are the central density and the scale radius for the i’th component, respectively,
and is the slope parameter for the components.
Then, the next level of modeling would use a single-beta density profile and would allow for
a decrease in temperature in the outskirts. This is the most common fitting profile for typical
NCC clusters:
Te (r) = T0 1 + (r/rt )2
↵
(19)
rt is the radius scale parameter describing where the decrease occurs, and ↵ is the outer
slope.
Depending on the cluster, sometimes there are multiple components to the density profile,
which often happens in cool-core clusters, where there is a high density in the core of the cluster.
This double-beta density model is based on surface brightness equation, so the density
components are added in quadrature:
⇣
ne (r) = n20,1 (1 + (r/rc,1 )2 )
3
+ n20,2 (1 + (r/rc,2 )2 )
3
⌘1/2
(20)
where n0,i and rc,i are the density normalization and the scale radius for the i’th
component, respectively.
The temperature profile of a CC cluster fits for the temperature drop near the core of the
cluster:
Te (r) = T0
Tmin /T0 + (r/rcool )1.9
1 + (r/rt )2
1 + (r/rcool )1.9
↵
(21)
T0 is the normalization temperature, Tmin is the temperature at the center, and rcool is the
cool-core radius. 1.9 is the inner slope which was found in Vikhlinin et al. (2006).
The normalizations, scale radii, and slopes were fit for. The complexity of fitting was
increased until an optimal probability-to-exceed (PTE) was found.
7.
7.1.
Mock Clusters
Simulating the cluster emission
The analytical fit code was tested briefly on mock ball-of-gas clusters, with a single- or
double-beta modeled density profile and a constant temperature profile. The parameters used to
create the mock clusters were chosen based on real clusters in our sample. Four clusters were
chosen at various redshifts representative of our sample: Abell 1835, MACS J0417.5, MS 2053,
32
and MACS J0744.8, denoted by clusters A, B, C, and D, respectively. All the clusters were
checked to make sure they roughly fell on the luminosity-temperature relation. Only 100
realizations per cluster were created due to time constraints. The parameters of the clusters are
given in Table 4. Cluster C was the only cluster modeled with a single-beta profile, due to that
particular cluster being very dim in the X-ray.
X-ray maps were simulated using code provided by Elena Rasia (University of Michigan,
private communication), using XSPEC, with identical resolution as Chandra maps. Due to the
complexity of modeling multi-component temperatures in the X-ray, isothermal temperatures
were chosen. SZ decrement maps of identical Bolocam resolution were created by Tony
Mroczkowski (European Southern Observatory, private communication). The SZ transfer
functions that were used for each mock cluster were taken from those computed by their
real-cluster counterparts. 100 noise maps added to the cluster-only map were also taken from the
set of 1000 jacknife noise realizations per cluster for the real clusters.
The deprojection code was not tested on an individual cluster basis, but was instead tested
along with the joint-cluster fitting code (see Section 9.2).
Table 4: Mock Cluster Parameters
Cluster
Cluster
Cluster
Cluster
A
B
C
D
z
0.25
0.44
0.58
0.69
Real Counterpart
Abell 1835
MACS J0417.5
MS 2053
MACS J0744.8
M2500 (1014 M )
5.11
9.5
0.59
3.5
7.2.
T (keV)
7.0
9.5
5.0
8.0
ne0,1 (cm
0.2348
6.8E-2
9.0e-3
6.2e-2
3)
rc,1 (”)
11
9.8
16
6
ne0,2 (cm 3 )
2.13E-2
5.2E-3
1.05E-2
rc,2 (”)
48.6
80
26
Check for Biases
Once the maps were put through the individual cluster fitting code, MCMC chains were
produced. Instead of simply extracting the parameter values along with their uncertainties and
comparing them to the input parameters, we calculated the profiles produced by the parameters.
This is because there are some correlations between the parameters, so a set of parameters can be
slightly di↵erent than the input parameters but produce an identical profile within errors. From
the chains, the means of each parameter were found, and then used to calculate a density and
temperature profile as a function of radius representative of that realization. This was repeated
for the 100 realizations. From the set of 100 profiles, the median value for the density was found
as a function of radius as well as lower and upper 68% limits. Then the median value and the
upper and lower limits were compared with the input profile.
Table 5 shows the values for the temperature and Figure 10 shows the resulting biases in
density. In terms of the density, we only see a statistically significant bias in cluster A, with a bias
of -0.5%. Clusters B-D overall have output density profiles that represent what was put in. There
is deviation at small radii around 5”. However in general the X-ray maps are created such that
each bin is at least 5” in radial width, due to the need to make PSF e↵ects negligible. Cluster C
33
0.7
0.86
0.61
0.68
shows a slightly positive bias at ⇠200”. For the case of Cluster A, the actual parameter
estimation uncertainty for the density normalizations is around 0.4%, which may account for
some of the bias that we see. The clusters encompass di↵erent 68% limit widths due to their
varying distances, masses, and number of parameters that were fit for.
For the temperature profiles, the mean value for the temperature was calculated for each
realization, and the mean and standard deviation of the sample was calculated. We found no bias
in the temperature profiles. Therefore, we do not apply any bias correction factors when fitting
for the real clusters.
Table 5: Output Mock Temperatures
Cluster
A
B
C
D
Input T (keV)
7.0
9.5
5.0
8.0
Output T (keV)
6.43 ± 1.56
9.35 ± 1.30
4.69 ± 2.14
7.93 ± 1.15
(a)
(b)
(c)
(d)
Fig. 10.— Normalized di↵erences in the output and input density profiles. (a) Cluster A, the Abell
1835-like cluster, (b) Cluster B, the MACS J0417.5-like cluster, (c) Cluster C, the MS 2053-like
cluster, and (d) Cluster D, the MACS J0744.8-like cluster.
34
8.
Table 6: Goodness of Fit Details
A2204
A383
A1423
A209
A963
A2261
A2219
A267
RXJ2129.6
A1835
A697
A611
MS2137
MACSJ1931.8
AS1063
MACSJ1115.8
A370
CL0024P17
MACSJ1532.9
MACSJ0429.6
MACSJ2211.7
MACSJ1720.3
MACSJ0416.1
MACSJ0451.9
MACSJ0417.5
MACSJ1206.2
MACSJ0329.6
RXJ1347
MACSJ1311.0
MACSJ0257.1
MACSJ0911.2
MACSJ2214.9
CL0016
MACSJ1149.5
MACSJ0717.5
MACSJ1423.8
MS0451.6
MACSJ0025.4
MS2053
MACSJ0647.7
MACSJ2129.4
MACSJ0744.8
MS1054
CLJ0152.7
CLJ1226.9
X-ray data SZ data
77
1677
58
1388
44
1577
61
905
71
1518
76
1739
83
1566
45
772
33
484
92
1644
63
1649
63
1181
63
1186
52
537
77
1724
63
940
14
710
59
936
33
1352
25
424
43
1154
34
462
24
351
16
634
102
1730
42
733
23
400
61
780
24
185
28
500
27
516
28
560
58
703
34
622
58
700
61
732
56
634
31
199
11
88
23
539
21
459
28
254
25
127
15
155
14
117
2
XR
1239.55
742.58
170.90
903.42
1318.75
821.29
3319.62
190.82
215.22
2504.11
493.01
639.14
638.81
2473.80
1135.82
734.66
32.71
156.86
678.70
281.17
459.67
240.01
61.71
36.03
1767.26
256.18
138.60
1816.78
501.62
118.57
76.38
107.70
351.40
123.26
427.05
1383.17
424.26
81.72
17.72
81.28
58.28
246.58
80.63
24.51
84.30
Individual Cluster Results
Onion Shell Deprojections
2
2
2
dof
tot
SZ
red
1864.14 3103.69 1752 1.782
1453.50 2196.08 1444 1.531
1650.79 1821.69 1619 1.132
1046.86 1950.28 964 2.044
1574.04 2892.79 1587 1.834
1809.37 2630.66 1813 1.459
1473.09 4792.71 1647 2.928
723.02 913.84 815 1.135
536.87 752.09 515 1.489
2045.04 4549.15 1734 2.639
1992.64 2485.65 1710 1.462
1176.02 1815.16 1242 1.473
1186.75 1825.56 1247 1.476
546.03 3019.83 587 5.234
1804.71 2940.53 1799 1.644
1067.39 1802.05 1000 1.820
759.25 791.96 724 1.109
1006.99 1163.85 993 1.184
1456.80 2135.50 1383 1.555
523.59 804.76 447 1.842
1196.37 1656.04 1195 1.397
562.42 802.42 494 1.658
348.03 409.73 374 1.126
363.97 399.99 336 1.227
1878.09 3645.35 1830 2.003
757.47 1013.65 773 1.329
428.76 567.36 421 1.380
803.91 2620.69 839 3.161
165.11 666.73 207 3.384
476.42 594.99 526 1.153
561.55 637.93 542 1.199
704.66 812.35 673 1.225
738.41 1089.80 759 1.455
683.70 806.95 655 1.251
739.81 1166.86 756 1.564
704.84 2088.01 791 2.674
705.99 1130.25 688 1.667
230.13 311.85 228 1.431
71.80
89.53
97 1.029
547.38 628.66 560 1.143
508.27 566.55 478 1.211
287.72 534.30 280 1.979
143.74 224.37 150 1.603
218.91 243.41 170 1.521
134.16 218.46 129 1.836
35
PTE
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.004
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.022
0.000
0.000
0.000
0.000
0.000
0.049
0.003
0.000
0.000
0.000
0.000
0.000
0.009
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.405
0.011
0.001
0.000
0.000
0.000
0.000
2
XR
82.23
76.86
84.87
94.05
145.30
119.87
236.06
48.83
205.88
137.73
76.08
92.22
354.59
47.35
91.97
55.30
16.32
121.68
28.75
27.21
45.78
43.06
40.28
14.61
377.11
64.60
29.77
139.62
28.20
27.72
37.56
41.57
94.48
47.09
57.21
96.05
180.89
37.07
19.31
32.72
19.26
49.03
238.61
26.00
27.29
Analytical Profiles
2
dof
tot
2116.02 2198.25 1745
1481.79 1558.65 1440
1666.52 1751.39 1616
1044.52 1138.57 961
1580.28 1725.58 1582
1809.71 1929.58 1808
1503.62 1739.68 1644
731.71 780.54 812
543.01 748.90 513
2015.50 2153.23 1728
1980.35 2056.43 1707
1178.70 1270.92 1237
1218.05 1572.64 1243
541.69 589.04 581
1818.45 1910.42 1793
1062.86 1118.16 995
776.02 792.34 720
1010.19 1131.87 991
1456.93 1485.68 1379
524.98 552.20 441
1209.27 1255.05 1193
558.44 601.51 489
399.27 439.55 371
669.80 684.41 646
1887.78 2264.89 1823
766.82 831.43 767
431.00 460.77 415
811.80 951.43 832
179.26 207.46 203
493.29 521.01 524
566.92 604.49 538
658.15 699.72 582
772.25 866.73 755
685.74 732.82 652
783.68 840.90 751
702.76 798.82 787
762.28 943.17 686
302.52 339.59 226
72.84
92.16
95
557.32 590.04 558
570.72 589.98 476
294.17 343.20 275
148.60 387.21 148
245.55 271.55 166
125.47 152.76 126
2
SZ
2
red
1.260
1.082
1.084
1.185
1.091
1.067
1.058
0.961
1.460
1.246
1.205
1.027
1.265
1.014
1.065
1.124
1.100
1.142
1.077
1.252
1.052
1.230
1.185
1.059
1.242
1.084
1.110
1.144
1.022
0.994
1.124
1.202
1.148
1.124
1.120
1.015
1.375
1.503
0.970
1.057
1.239
1.248
2.616
1.636
1.212
PTE
0.000
0.015
0.010
0.000
0.006
0.023
0.050
0.781
0.000
0.000
0.000
0.245
0.000
0.400
0.027
0.004
0.031
0.001
0.023
0.000
0.103
0.000
0.008
0.143
0.000
0.053
0.060
0.002
0.400
0.529
0.024
0.001
0.003
0.015
0.012
0.377
0.000
0.000
0.564
0.168
0.000
0.003
0.000
0.000
0.053
8.1.
Goodness of fit
There are several methods to gauge the quality of the fit, usually denoted the ’goodness of
fit.’ We will use two: the reduced chi-squared value, and the p-value. The reduced chi-squared is
2
just: 2red = dof , with dof being the degrees of freedom, which is the number of data points
subtracted by the number of fitting parameters. While in general this is sufficient, one can also
compute the probability-to-exceed (PTE) of the result, which is a function of the chi-squared value
given the dof . The PTE is the probability that a random value from a chi-squared distribution
given the degrees of freedom, is greater than the given chi-squared value of the result. Therefore,
larger PTEs in general are ’better,’ since it roughly translates into the probability of random
numbers producing a larger chi-squared value (i.e. worse) than the given chi-squared value and
the noise. A significance level of ↵ = 0.05 will be used as a general guideline for the significance of
the fit. Table 6 lists the 2 values, the dof s, 2red values, and the PTEs for both fitting models.
8.2.
Summary of individual-cluster fitting results
Our results may be summarized as follows:
• We are able to extend our profiles out to or past R500 in 32/45 clusters. Our analysis
reaches out further than X-ray spectroscopy studies can do alone for all 45 clusters.
• For the clusters that have an overlap with the ACCEPT X-ray sample, about 8/36 (⇠22%)
show a strong disagreement in the density and pressure profiles. In nearly every one of those
cases, our densities and pressures are systematically higher than the ACCEPT profiles. This
is unusual and a path for further study since the densities are constrained by the surface
brightness profiles, which should be similar for the two samples.
• For the analytical fits, spherical clusters have better PTE values than the non-spherical
clusters, which is to be expected since we are assuming spherical symmetry. The average
PTE of the spherical set is 0.152 while the average PTE of the nonspherical set is 0.01. The
PTEs of the onion shell deprojections are mostly 0.0, driven by the X-ray data. A flat
density within the radial shell bounds is not the best representation of the data.
• We can resolve a cool-core in only 8/45 of the clusters, or 8/17 (47%) of the cool-core
subset. The larger the redshift, the more likely we are unable to resolve the cool-core, which
may be due to the limiting resolution of Bolocam (58”). Since the first deprojected bin is
required to be greater than half the Bolocam resolution, if a core is much smaller than that,
it may not be seen.
• 15/45 clusters have a clear temperature decrease in the outskirts of the clusters, either in
the analytical fits or deprojections. There are 12/45 clusters that show a significant increase
in the temperature profile in the outskirts, which is unexpected, especially since 4 of them
36
are CC, non-disturbed clusters. Temperatures in the outskirts are expected to decrease for
relaxed systems. Out of those 12 clusters, only one cluster has a pressure that is not within
errors of the S13 pressure profile, so our SZ observables are overall consistent with S13.
Comparing the density to Morandi et al. (2015)’s average density profile, 10/12 of those
clusters do not show any significant deviation from the average density profile slope, so it
does not seem that the density profiles are being mis-estimated, and consequently the
temperatures. 4/12 of the clusters were non-spherical, so there do not seem to be any
predominantly major asphericity issues causing the peculiar temperatures. The rest of the
clusters (18/45) have profiles that are consistent with isothermality. So, although the X-ray
(density) and the SZ (pressure) seem to agree with other studies, combining them gives way
to temperature profiles that have an increasing variability in temperature with radius.
Please refer to Appendix B for the individual cluster profiles as well as descriptions and
literature that exists for each cluster. The set of parameters for the analytical fits are in Table 7
while the deprojected profiles are in Table 8.
37
38
Cluster
A2204
A383
A1423
A209
A963
A2261
A2219
A267
RXJ2129.6
A1835
A697
A611
MS2137
MACSJ1931.8
AS1063
MACSJ1115.8
A370
CL0024P17
MACSJ1532.9
MACSJ0429.6
MACSJ2211.7
MACSJ1720.3
MACSJ0416.1
MACSJ0451.9
MACSJ0417.5
MACSJ1206.2
MACSJ0329.6
RXJ1347
MACSJ1311.0
MACSJ0257.1
MACSJ0911.2
MACSJ2214.9
CL0016
MACSJ1149.5
MACSJ0717.5
MACSJ1423.8
MS0451.6
MACSJ0025.4
MS2053
MACSJ0647.7
MACSJ2129.4
MACSJ0744.8
MS1054
CLJ0152.7
CLJ1226.9
n Profile
double-B
double-B
single-B
single-B
double-B
double-B
single-B
single-B
single-B
double-B
single-B
double-B
single-B
double-B
double-B
double-B
single-B
single-B
single-B
double-B
single-B
double-B
single-B
single-B
double-B
double-B
double-B
double-B
double-B
single-B
single-B
single-B
double-B
single-B
double-B
single-b
single-b
single-B
single-B
single-B
single-B
double-B
single-B
single-B
single-B
T Profile
ne0,1 (10 2 cm
CC-full
22.88+0.67
0.67
iso
13.50+0.61
0.62
NCC
2.61+0.20
0.20
+0.03
NCC
0.84 0.03
NCC
3.07+0.17
0.18
NCC
4.58+0.40
0.41
NCC
1.06+0.01
0.01
NCC
1.19+0.04
0.04
iso
7.17+0.33
0.33
CC-only-core
14.06+0.30
0.30
+0.03
NCC
1.03 0.03
NCC
14.55+5.67
5.11
CC-full
11.57+0.53
0.53
+0.86
CC-full
17.29 0.73
CC-only-core
3.49+0.55
0.58
CC-only-core
9.02+0.35
0.36
iso
0.58+0.04
0.04
iso
1.49+0.11
0.12
CC-full
12.35+0.40
0.39
CC-only-core
13.45+1.01
1.01
+0.08
iso
3.14 0.08
NCC
8.62+0.57
0.54
iso
0.58+0.02
0.03
+0.08
iso
1.04 0.08
CC-full
8.95+0.29
0.29
CC-only-core
3.88+0.39
0.39
CC-only-core
12.90+0.67
0.68
CC-full
31.60+2.87
2.65
iso
3.94+0.16
0.16
iso
2.74+0.14
0.14
NCC
0.84+0.06
0.06
CC-only-core
1.54+0.08
0.08
iso
1.61+0.58
0.58
+0.03
iso
0.68 0.03
NCC
1.36+0.14
0.14
CC-full
20.17+1.29
1.31
iso
1.79+0.04
0.04
iso
0.72+0.02
0.02
iso
1.14+0.12
0.12
iso
2.02+0.11
0.11
+0.11
iso
1.85 0.11
NCC
8.66+1.08
1.02
iso
0.68+0.01
0.01
+0.01
iso
0.26 0.01
NCC
65.87+2.24
2.29
Table 7: Analytical Fit Parameters
3)
rc,1 (”)
ne0,2 (10 3 cm
7.27+0.23
15.40+0.74
0.23
0.74
+0.31
5.58 0.30
19.73+1.73
17.98
14.42+1.00
0.99
51.36+2.06
2.08
+0.99
15.94+1.70
8.10
1.62
1.10
9.22+1.26
16.21+1.91
1.30
1.98
67.64+0.90
0.94
35.17+1.42
1.41
9.74+0.46
0.46
8.10+0.20
14.25+0.61
0.21
0.59
47.82+1.41
1.44
2.30+0.75
19.46+0.99
0.76
1.00
+0.13
3.25 0.13
+0.16
+0.84
6.17 0.16
14.01 0.84
9.88+2.59
22.14+1.39
2.48
1.46
+0.38
7.72 0.39
12.53+1.42
1.41
52.61+5.79
5.64
9.85+0.80
0.79
8.02+0.24
0.24
4.95+0.47
11.90+1.88
0.46
1.78
23.00+0.65
0.65
6.77+0.50
10.16+0.75
0.51
0.78
+8.09
79.34 7.41
+2.98
31.05 2.95
+0.17
7.84+0.27
6.58
0.27
0.17
12.62+2.12
9.80+1.66
2.10
1.65
5.78+0.37
8.99+0.67
0.36
0.68
3.44+0.26
48.02+2.86
0.28
2.83
13.30+1.22
7.00+0.92
1.12
5.99
14.60+0.79
0.79
26.05+2.12
2.14
22.78+1.29
1.27
9.14+3.74
10.19+0.46
3.96
9.66
52.45+2.89
2.93
20.09+2.55
6.39+0.21
2.54
0.20
+0.11
3.42 0.11
25.42+0.65
0.64
47.52+2.27
2.32
15.56+2.15
2.16
19.73+1.22
1.25
19.41+1.23
1.25
+3.13
4.69+0.78
16.39
0.76
3.20
+3.73
71.15 3.59
+11.95
137.72 12.49
9.95+0.65
0.70
3)
rc,2 (”)
T0
Tmin
50.06+2.01
0.641+0.008
16.82+3.93
3.51+1.89
2.07
0.008
3.40
2.01
+0.006
24.77+1.58
7.21+0.48
23.17 0.601 0.006
0.48
0.497+0.005
16.52+6.43
0.005
5.35
0.586+0.008
11.02+0.67
0.008
0.67
+0.014
+2.55
54.51+5.43
0.663
10.25
4.95
0.014
2.34
30.31+2.34
0.581+0.006
5.57+0.45
2.28
0.006
1.93
0.682+0.004
13.83+1.07
0.004
1.05
+0.69
0.639+0.010
10.04
0.010
0.69
+0.58
0.548+0.006
6.96
0.006
0.58
41.23+1.45
0.669+0.006
47.26+10.50
7.63+0.43
1.49
0.006
10.35
0.44
0.639+0.008
10.61+0.36
0.008
0.36
21.96+0.96
0.597+0.006
9.57+0.78
0.96
0.006
0.75
+0.002
0.491 0.002
30.84+5.62
1.60+1.05
5.71
1.04
30.93+1.37
0.689+0.010
34.99+16.06
3.03+1.97
1.35
0.010
13.39
1.99
33.64+1.55
0.676+0.007
63.05+18.82
12.13+1.02
1.47
0.007
19.24
1.03
30.09+2.59
0.647+0.012
32.21+22.38
4.85+1.29
2.51
0.012
16.21
1.37
0.708+0.043
13.17+0.82
0.043
0.78
+1.55
0.453+0.006
6.47
0.006
1.54
+3.59
+1.37
0.614+0.005
11.19
3.64
0.005
3.48
1.74
26.67+3.53
0.669+0.023
51.62+24.91
5.88+1.47
3.35
0.022
23.52
1.46
+0.007
+0.56
0.667 0.007
12.99 0.56
35.23+2.81
0.747+0.026
9.76+1.74
2.72
0.025
1.89
+0.111
1.104 0.100
7.98+0.78
0.77
0.683+0.032
7.06+0.68
0.032
0.68
+0.011
+7.06
+2.15
64.50+1.90
0.709
24.01
6.34
1.92
0.011
6.70
2.68
45.94+6.16
0.722+0.028
61.11+21.23
10.61+0.73
6.13
0.027
18.59
0.73
37.37+2.88
0.749+0.028
58.61+19.55
8.26+0.91
2.73
0.028
21.48
0.95
17.13+0.68
0.661+0.005
15.93+0.92
4.70+3.56
0.67
0.005
0.90
3.50
+5.23
+0.085
+0.71
40.10 34.50 0.925 0.071
7.00 0.72
0.584+0.009
10.85+0.94
0.009
0.94
0.557+0.015
6.51+1.31
0.015
1.31
+8.42
+2.36
0.600+0.012
28.19
6.90
0.012
8.25
2.47
+0.011
40.09+1.59
8.46+0.40
38.31 0.703 0.011
0.40
+0.023
+0.50
0.720 0.023
10.42 0.50
+0.036
+2.42
88.42+4.17
1.003
25.40
4.33
0.037
2.36
0.556+0.004
15.82+4.15
1.68+0.70
0.004
3.56
0.95
0.631+0.006
8.92+0.31
0.006
0.31
0.878+0.032
5.38+0.59
0.033
0.58
+0.035
0.604 0.036
4.01+1.13
1.14
0.636+0.015
11.76+0.52
0.015
0.53
0.620+0.014
11.05+0.53
0.014
0.53
+0.020
+1.75
20.06+3.07
0.622
10.66
3.04
0.020
1.82
1.168+0.065
6.38+0.28
0.063
0.29
1.717+0.282
6.14+0.66
0.200
0.66
+275.40
14.232+1.158
415.58
1.182
259.40
rcool
62.85+25.92
24.47
275.47+44.85
44.53
32.36+5.45
5.58
108.55+40.33
36.06
263.02+68.23
71.58
144.84+60.91
58.79
100.37+61.41
58.20
164.56+48.16
52.01
94.55+51.21
46.92
265.22+55.69
60.85
272.51+62.64
56.75
21.95+7.99
7.22
90.06+37.69
36.24
48.83+16.65
14.12
-
rt
248.42+127.01
114.84
315.38+141.83
147.02
68.52+32.42
34.69
147.73+93.48
90.58
114.98+26.31
26.72
244.03+96.37
92.29
222.96+68.19
68.88
484.83+514.25
375.45
53.73+12.18
12.11
358.62+239.51
205.91
76.51+32.91
38.89
435.05+326.61
226.51
390.96+292.08
263.10
38.14+6.13
6.51
122.48+44.68
44.42
96.02+57.60
56.96
315.38+141.83
147.02
R1
0 - 0.0537
0 - 0.0856
0 - 0.0762
0 - 0.0672
0 - 0.0762
0 - 0.067
0 - 0.0634
0 - 0.0904
0 - 0.0875
0 - 0.0787
0 - 0.0776
0 - 0.105
0 - 0.129
0 - 0.111
0 - 0.0842
0 - 0.118
0 - 0.112
0 - 0.159
0 - 0.115
0 - 0.147
0 - 0.1
0 - 0.139
0 - 0.131
0 - 0.15
0 - 0.101
0 - 0.106
0 - 0.145
0 - 0.104
0 - 0.195
0 - 0.153
0 - 0.15
0 - 0.132
0 - 0.13
0 - 0.125
0 - 0.114
0 - 0.176
0 - 0.147
0 - 0.176
0 - 0.241
0 - 0.158
0 - 0.159
0 - 0.169
0 - 0.213
0 - 0.235
0 - 0.233
R2
0.0537 - 0.109
0.0856 - 0.166
0.0762 - 0.151
0.0672 - 0.148
0.0762 - 0.15
0.067 - 0.138
0.0634 - 0.125
0.0904 - 0.162
0.0875 - 0.148
0.0787 - 0.158
0.0776 - 0.156
0.105 - 0.2
0.129 - 0.245
0.111 - 0.19
0.0842 - 0.173
0.118 - 0.217
0.112 - 0.198
0.159 - 0.292
0.115 - 0.222
0.147 - 0.244
0.1 - 0.189
0.139 - 0.235
0.131 - 0.213
0.15 - 0.242
0.101 - 0.208
0.106 - 0.189
0.145 - 0.241
0.104 - 0.186
0.195 - 0.293
0.153 - 0.26
0.15 - 0.256
0.132 - 0.232
0.13 - 0.23
0.125 - 0.218
0.114 - 0.202
0.176 - 0.315
0.147 - 0.258
0.176 - 0.267
0.241 - 0.332
0.158 - 0.271
0.159 - 0.268
0.169 - 0.264
0.213 - 0.305
0.235 - 0.346
0.233 - 0.331
R3
0.109 - 0.221
0.166 - 0.32
0.151 - 0.3
0.148 - 0.324
0.15 - 0.294
0.138 - 0.285
0.125 - 0.249
0.162 - 0.292
0.148 - 0.251
0.158 - 0.318
0.156 - 0.314
0.2 - 0.379
0.245 - 0.465
0.19 - 0.326
0.173 - 0.355
0.217 - 0.401
0.198 - 0.353
0.292 - 0.539
0.222 - 0.428
0.244 - 0.407
0.189 - 0.358
0.235 - 0.396
0.213 - 0.347
0.242 - 0.39
0.208 - 0.427
0.189 - 0.339
0.241 - 0.4
0.186 - 0.335
0.293 - 0.44
0.26 - 0.444
0.256 - 0.438
0.232 - 0.407
0.23 - 0.41
0.218 - 0.382
0.202 - 0.359
0.315 - 0.563
0.258 - 0.452
0.267 - 0.405
0.332 - 0.457
0.271 - 0.466
0.268 - 0.452
0.264 - 0.413
0.305 - 0.436
0.346 - 0.508
0.331 - 0.47
R4
0.221 - 0.448
0.32 - 0.619
0.3 - 0.595
0.324 - 0.711
0.294 - 0.578
0.285 - 0.588
0.249 - 0.492
0.292 - 0.524
0.251 - 0.425
0.318 - 0.638
0.314 - 0.631
0.379 - 0.718
0.465 - 0.882
0.326 - 0.559
0.355 - 0.727
0.401 - 0.738
0.353 - 0.628
0.539 - 0.993
0.428 - 0.825
0.407 - 0.677
0.358 - 0.678
0.396 - 0.667
0.347 - 0.565
0.39 - 0.629
0.427 - 0.878
0.339 - 0.606
0.4 - 0.664
0.335 - 0.603
0.44 - 0.662
0.444 - 0.758
0.438 - 0.749
0.407 - 0.716
0.41 - 0.728
0.382 - 0.668
0.359 - 0.637
0.563 - 1.01
0.452 - 0.794
0.405 - 0.614
0.457 - 0.63
0.466 - 0.801
0.452 - 0.761
0.413 - 0.645
0.436 - 0.624
0.508 - 0.747
0.47 - 0.667
R5
0.448 - 0.908
0.619 - 1.2
0.595 - 1.18
0.711 - 1.56
0.578 - 1.14
0.588 - 1.21
0.492 - 0.975
0.524 - 0.942
0.425 - 0.721
0.638 - 1.28
0.631 - 1.27
0.718 - 1.36
0.882 - 1.67
0.559 - 0.959
0.727 - 1.49
0.738 - 1.36
0.628 - 1.12
0.993 - 1.83
0.825 - 1.59
0.677 - 1.13
0.678 - 1.28
0.667 - 1.13
0.565 - 0.921
0.629 - 1.01
0.878 - 1.81
0.606 - 1.08
0.664 - 1.1
0.603 - 1.08
0.662 - 0.995
0.758 - 1.29
0.749 - 1.28
0.716 - 1.26
0.728 - 1.29
0.668 - 1.17
0.637 - 1.13
1.01 - 1.8
0.794 - 1.39
0.614 - 0.931
0.63 - 0.867
0.801 - 1.38
0.761 - 1.28
0.645 - 1.01
0.624 - 0.893
0.747 - 1.1
0.667 - 0.947
ne,1
65.07+1.71
1.66
33.4+0.85
0.87
20.43+0.86
0.81
14.98+0.87
0.87
25.69+0.92
1.1
28.22+0.4
0.41
18.52+0.68
0.7
16.78+0.61
0.63
29.19+0.65
0.64
42.27+0.95
0.96
15.86+0.56
0.55
19.78+0.48
0.47
14.24+0.46
0.4
27.78+0.83
0.65
28.93+0.41
0.38
23.51+0.4
0.37
7.55+0.67
0.66
7.38+0.24
0.23
31.84+0.35
0.35
16.67+0.37
0.37
26.99+0.57
0.64
15.73+0.42
0.42
8.38+0.46
0.47
10.74+0.51
0.53
18.41+0.39
0.39
18.81+0.46
0.48
14.15+0.41
0.39
32.63+0.39
0.43
12.65+0.21
0.21
13.82+0.34
0.34
7.71+0.34
0.34
12.19+0.4
0.39
10.46+0.29
0.28
7.97+0.39
0.39
9.37+0.27
0.28
13.19+0.6
0.34
14.42+0.25
0.26
6.96+0.23
0.23
4.82+0.29
0.27
11.49+0.56
0.49
11.13+0.39
0.39
10.85+0.64
0.49
4.63+0.17
0.21
+0.33
2.33 0.33
7.79+0.73
0.55
ne,2
22.52+0.45
0.45
12.12+0.19
0.2
7.79+0.31
0.31
8.95+0.23
0.23
11.42+0.27
0.24
11.92+0.12
0.12
13.77+0.34
0.38
9.55+0.23
0.23
12.45+0.37
0.38
13.72+0.21
0.19
9.82+0.16
0.16
7.91+0.16
0.15
4.01+0.13
0.13
9.62+0.21
0.19
12.32+0.19
0.2
8.33+0.17
0.15
5.55+0.31
0.32
3.26+0.11
0.1
+0.15
8.51 0.15
6.24+0.22
0.22
10.08+0.23
0.21
6.3+0.18
0.17
5.35+0.27
0.28
5.4+0.36
0.34
6.63+0.09
0.09
7.98+0.2
0.2
5.98+0.21
0.21
9.67+0.17
0.15
4.42+0.14
0.14
5.52+0.2
0.19
4.19+0.19
0.19
5.22+0.2
0.18
6.5+0.27
0.18
4.91+0.2
0.2
6.09+0.16
0.17
3.93+0.06
0.07
7.15+0.31
0.32
4.61+0.17
0.17
2.55+0.25
0.25
5.2+0.16
0.17
4.71+0.18
0.19
4.2+0.14
0.15
4.6+0.37
0.33
2.65+0.25
0.2
3.15+0.18
0.18
ne,3
8.36+0.09
0.09
4.21+0.06
0.06
3.28+0.09
0.09
4.2+0.07
0.07
4.57+0.06
0.06
4.04+0.04
0.04
6.98+0.07
0.07
4.16+0.09
0.09
5.76+0.14
0.13
4.83+0.04
0.04
4.08+0.05
0.05
3.16+0.05
0.05
1.3+0.04
0.04
4.08+0.07
0.07
4.17+0.04
0.04
3.11+0.05
0.05
2.68+0.12
0.12
1.25+0.05
0.05
+0.06
2.6 0.06
2.58+0.1
0.12
3.55+0.07
0.07
2.87+0.06
0.06
3.89+0.13
0.13
2.73+0.17
0.15
2.98+0.02
0.02
3.61+0.06
0.06
2.62+0.08
0.08
3.71+0.04
0.04
2.38+0.08
0.08
2.44+0.07
0.07
1.9+0.09
0.09
2.47+0.06
0.07
2.76+0.04
0.04
2.92+0.06
0.06
3.78+0.04
0.04
1.54+0.03
0.03
2.94+0.04
0.04
2.81+0.12
0.1
1.58+0.13
0.12
2.02+0.07
0.07
1.97+0.07
0.07
2.37+0.15
0.12
3.5+0.14
0.13
+0.07
1.2 0.07
2.13+0.16
0.16
ne,4
2.4+0.03
0.03
1.41+0.02
0.02
1.31+0.03
0.03
1.37+0.03
0.03
1.55+0.01
0.01
1.28+0.01
0.01
2.51+0.04
0.04
1.66+0.04
0.04
2.45+0.06
0.06
1.36+0.02
0.02
1.36+0.02
0.02
1+0.03
0.02
0.58+0.02
0.02
1.39+0.02
0.02
0.99+0.01
0.01
1.01+0.02
0.02
1.25+0.05
0.05
0.57+0.03
0.02
+0.03
0.77 0.03
1.14+0.04
0.04
1.16+0.03
0.03
1.08+0.03
0.03
1.7+0.05
0.04
1.39+0.07
0.08
0.85+0.01
0.01
1.3+0.03
0.03
1.07+0.05
0.04
1.25+0.02
0.02
0.97+0.04
0.04
0.95+0.04
0.04
0.84+0.04
0.04
1.01+0.03
0.03
0.99+0.02
0.02
1.1+0.03
0.03
1.47+0.03
0.03
0.55+0.01
0.02
0.93+0.02
0.02
1.31+0.03
0.03
1.14+0.05
0.06
0.73+0.03
0.03
0.9+0.03
0.03
1.03+0.04
0.04
1.43+0.07
0.06
+0.14
1.26 0.09
0.91+0.11
0.09
ne,5
0.64+0.01
0.01
0.44+0.02
0.02
0.42+0.01
0.01
0.41+0.01
0.01
0.42+0.01
0.01
0.35+0.01
0.01
0.72+0.02
0.01
0.53+0.02
0.02
0.78+0.04
0.03
0.33+0.01
0.01
0.37+0.01
0.01
0.35+0.01
0.01
0.36+0.01
0.01
0.48+0.01
0.01
0.26+0.01
0.01
0.32+0.01
0.01
0.32+0.03
0.03
0.35+0.01
0.01
+0.02
0.21 0.02
0.34+0.01
0.01
0.25+0.01
0.01
0.3+0.02
0.02
0.38+0.04
0.03
0.51+0.03
0.03
0.22+0.01
0.01
0.35+0.01
0.01
0.35+0.02
0.02
0.29+0.01
0.01
0.31+0.04
0.04
0.4+0.03
0.03
0.44+0.03
0.03
0.3+0.02
0.02
0.31+0.01
0.01
0.39+0.02
0.02
0.36+0.01
0.01
0.25+0.01
0.01
0.45+0.01
0.01
0.46+0.03
0.03
0.42+0.06
0.07
0.29+0.02
0.02
0.26+0.02
0.02
0.46+0.02
0.02
0.73+0.02
0.02
+0.03
0.52 0.02
0.38+0.08
0.07
Te,1
0.75+0.18
0.75
1.53+0.36
1.53
1.26+0.3
1.26
1.14+0.29
1.14
2+0.57
2
0.41+0.1
0.41
1.13+0.28
1.13
2.86+0.7
2.86
0.76+0.21
0.76
1.18+0.36
0.36
0.79+0.2
0.79
1.37+0.39
1.37
1.95+0.63
1.95
0.54+0.15
0.54
1.49+0.48
0.48
0.85+0.23
0.85
1.92+0.56
1.92
1.69+0.45
1.69
+0.08
0.33 0.33
1.1+0.23
1.1
1.23+0.44
0.46
1.79+0.45
1.79
1.36+0.39
1.36
1.14+0.32
1.14
0.75+0.27
0.28
0.96+0.22
0.96
0.93+0.27
0.93
1.08+0.22
0.22
1.03+0.28
1.03
2.41+0.88
0.89
1.03+0.27
1.03
0.8+0.24
0.8
0.59+0.18
0.59
0.86+0.24
0.86
1.07+0.29
1.07
0.7+0.19
0.7
1.27+0.32
0.34
1.22+0.35
1.22
1.22+0.31
1.22
0.57+0.14
0.57
1.1+0.33
1.1
0.37+0.08
0.37
0.99+0.25
0.99
+0.21
0.84 0.84
0.48+0.12
0.48
Te,2
1.16+0.3
0.3
0.6+0.16
0.6
1.39+0.35
1.39
0.68+0.27
0.28
0.88+0.26
0.88
0.42+0.14
0.14
0.89+0.33
0.35
0.89+0.25
0.89
1.07+0.29
1.07
0.61+0.24
0.25
0.52+0.18
0.18
0.94+0.28
0.94
2.02+0.57
2.02
0.99+0.25
0.99
0.53+0.24
0.25
0.6+0.17
0.6
1.58+0.4
1.58
1.57+0.41
1.57
+0.24
0.65 0.26
1.4+0.45
1.4
0.52+0.13
0.52
1.19+0.35
1.19
1.09+0.29
1.09
0.96+0.27
0.96
0.38+0.17
0.17
0.53+0.15
0.53
1+0.28
1
0.47+0.11
0.47
1.3+0.33
1.3
0.75+0.18
0.75
0.66+0.18
0.66
0.92+0.25
0.92
0.41+0.14
0.41
0.58+0.17
0.58
0.66+0.18
0.66
1.21+0.52
0.55
0.34+0.08
0.34
0.97+0.28
0.97
1.38+0.33
1.38
1.24+0.47
0.5
1.22+0.37
1.22
1.12+0.31
1.12
0.41+0.08
0.41
+0.06
0.39 0.39
1.14+0.3
1.14
Te,3
1.12+0.09
0.09
1.63+0.25
0.25
1.04+0.32
0.32
1.15+0.13
0.13
1.07+0.28
0.28
0.32+0.07
0.07
0.95+0.11
0.12
1.56+0.34
0.34
0.78+0.17
0.78
0.92+0.09
0.09
0.85+0.06
0.06
1.34+0.25
0.25
1.31+0.32
1.31
1.03+0.41
0.42
1.12+0.12
0.12
1.51+0.22
0.22
1.17+0.32
0.32
1.52+0.36
1.52
+0.17
0.71 0.17
1.46+0.39
1.46
1.11+0.17
0.17
0.67+0.18
0.67
1.08+0.46
0.47
1.06+0.27
1.06
0.78+0.05
0.05
0.92+0.12
0.12
1.52+0.43
0.44
0.94+0.1
0.1
1.35+0.37
1.35
0.56+0.13
0.56
0.55+0.14
0.55
1.34+0.24
0.25
0.81+0.11
0.12
0.67+0.12
0.12
0.91+0.09
0.09
0.97+0.35
0.36
0.9+0.12
0.12
0.61+0.16
0.61
1.3+0.27
1.3
1.09+0.3
0.3
0.82+0.32
0.32
0.59+0.21
0.59
0.46+0.2
0.17
+0.14
0.79 0.79
1.04+0.28
1.04
Te,4
0.72+0.13
0.13
1.93+0.3
0.3
0.81+0.3
0.3
0.96+0.24
0.24
0.19+0.04
0.19
0.12+0.03
0.12
0.71+0.17
0.17
1.49+0.35
0.34
1.17+0.2
0.2
1.25+0.16
0.16
0.35+0.09
0.1
0.51+0.11
0.51
1.71+0.6
0.61
1.93+0.25
0.25
1.62+0.3
0.3
1.8+0.33
0.33
1.56+0.22
0.22
2.47+1.01
1.04
+0.2
0.46 0.2
2.08+0.35
0.37
1.01+0.24
0.23
1.08+0.29
0.29
0.96+0.26
0.26
1.04+0.33
0.34
0.64+0.08
0.08
1.17+0.12
0.12
1.41+0.26
0.26
0.76+0.09
0.09
1.56+0.44
1.56
2.36+0.31
0.31
0.87+0.28
0.28
1.52+0.16
0.16
0.83+0.11
0.11
1.08+0.1
0.1
0.2+0.05
0.05
1.14+0.32
0.32
1.21+0.11
0.11
0.9+0.31
0.32
0.74+0.17
0.74
1.71+0.18
0.18
1.28+0.15
0.15
0.56+0.22
0.23
0.57+0.13
0.57
+0.19
0.66 0.66
1.33+0.36
1.33
The radial range spanning shell i are given by Ri in units of R500 , the density in shell i are given by ne,i and are in units of ne,500 . Finally, the temperatures of each shell i are given by
Te,i and are in units of T500 .
Cluster
A2204
A383
A1423
A209
A963
A2261
A2219
A267
RXJ2129.6
A1835
A697
A611
MS2137
MACSJ1931.8
AS1063
MACSJ1115.8
A370
CL0024P17
MACSJ1532.9
MACSJ0429.6
MACSJ2211.7
MACSJ1720.3
MACSJ0416.1
MACSJ0451.9
MACSJ0417.5
MACSJ1206.2
MACSJ0329.6
RXJ1347
MACSJ1311.0
MACSJ0257.1
MACSJ0911.2
MACSJ2214.9
CL0016
MACSJ1149.5
MACSJ0717.5
MACSJ1423.8
MS0451.6
MACSJ0025.4
MS2053
MACSJ0647.7
MACSJ2129.4
MACSJ0744.8
MS1054
CLJ0152.7
CLJ1226.9
Table 8: Onion Shell Deprojection Parameters
Te,5
0.23+0.05
0.23
0.52+0.11
0.52
0.34+0.07
0.34
0.48+0.1
0.48
0.63+0.15
0.63
0.17+0.04
0.17
0.66+0.27
0.27
1.19+0.31
1.19
1.19+0.44
0.45
1.63+0.34
0.34
0.26+0.06
0.26
0.32+0.07
0.32
0.48+0.1
0.48
1.85+0.54
0.54
2.26+0.62
0.64
2.27+0.6
0.6
2.45+0.64
0.64
2.03+0.44
2.03
0.41+0.09
0.41
3.82+0.67
0.72
1.03+0.24
1.03
1.17+0.29
1.17
2.53+0.61
0.62
1.23+0.41
0.41
0.23+0.05
0.23
1.45+0.29
0.29
2.18+0.45
0.45
0.75+0.23
0.23
4.03+0.57
0.45
1.35+0.52
0.53
0.49+0.12
0.49
2.18+0.39
0.39
0.92+0.22
0.22
1.07+0.2
0.2
0.06+0.01
0.06
0.73+0.19
0.73
0.86+0.17
0.17
0.77+0.17
0.77
0.96+0.18
0.96
1.49+0.32
0.32
1.59+0.34
0.34
0.11+0.03
0.11
0.96+0.16
0.16
+0.31
1.03 0.31
0.66+0.15
0.66
39
9.
Ensemble-cluster fitting
Cluster deprojections on an individual basis have large uncertainties due to the SZ data’s
limited constraining power on the pressure and therefore the temperature. However, if we jointly
fit the clusters with those errors, we can recover average profiles and an intrinsic scatter for
subsamples of clusters. This can tell us the overall behavior of clusters, how much the clusters
vary bin-by-bin, and how much they deviate from self-similarity. The clusters are individually
deprojected, as in Section 6.3.1, then those profiles are jointly fitted along with their covariance
matrices. The full details are given below:
The clusters are first scaled to R500 , which is X-ray derived, taken from Mantz et al. (2010).
These values have uncertainties of around 5%, which we do not propagate into the fitting code.
To be robust, one could include the e↵ect of the uncertainty in R500 through the use of
simulations. This was skipped for this analysis. The clusters are then binned to the same 5 radii
that are equally spaced in logarithm up to 1.25R500 , after the first bin which is set to 0.15R500 .
This first bin is greater than half the Bolocam resolution for most clusters. Due to the di↵ering
signal-to-noise of the X-ray maps, clusters may only contribute to 4 or lesser bins. Of the sample
of 45 clusters, 17 contribute to all 5 bins, 27 contribute to only the first 4 bins, and 1 cluster (RX
J2129.6) only contributes to 3 bins.
The clusters are then scaled according to the self-similarity equations given by Nagai et al.
(2007) (see section 2.5) to scale the densities to ne,500 , and the temperatures to T500 . The M500
values needed to calculate T500 were taken from the Mantz et al. (2010) study.
Once the clusters were individually deprojected, we followed a similar method to S13, in
which we assumed the density and temperatures distributions among clusters were Gaussian, and
we fit a log-likelihood to the deprojections and their resulting covariance matrices instead of to
the data. The log-likelihood used to constrain the model is given as:
lnL =
X
i
1⇥ T
x (S + Uj )
2 j
1
xj + ln|S + Uj |
⇤
(22)
where xj is a vector including n nj and T Tj values, the sum is over j clusters, Uj is the
individual cluster deprojected covariance matrices (which can be treated as a ’measurement’
covariance matrix), and S is the intrinsic scatter matrix that we fit for. See Figure 11 for a
typical Uj covariance matrix. We were unable to constrain all values of the S matrix using
MCMC due to the large number of parameters that we needed to fit for. To minimize the number
of free parameters, we assumed a diagonal-only S matrix, so there would be no covariances
between the intrinsic scatter values between bins and between densities and temperatures. This
resulted in fitting for 5 mean densities, 5 mean temperatures, 5 density intrinsic scatter elements,
and 5 temperature intrinsic scatter elements, for a total of 20 parameters.
40
Fig. 11.— A typical covariance matrix of an onion-shell deprojection (the cluster is Abell 697).
ni denotes the i’th density shell while Ti denotes the i’th temperature shell. The units for the
covariances are in scaled units of ne,500 and Te,500 .
9.1.
Previous Ensemble Profile Studies
There have been several studies on both the X-ray- and SZ- side that have characterized
average profiles of clusters.
Vikhlinin et al. (2006) used Chandra data of 13 relaxed clusters with a maximum redshift
of 0.232 to find average density and temperature profiles (the same temperature profile adopted in
the analytical fit analysis). They were able to find self-similarity with a scatter of 7-12% in the
density profile near R500 . Leccardi & Molendi (2008) found radial temperature profiles using
41
spectroscopy for a sample of 48 galaxy clusters with XMM-Newton, including a mixture of CC
and NCC clusters in a redshift range of 0.1 < z < 0.3. Excluding the core, they were able to find
an intrinsic scatter of the temperature profiles of around 6%. The radial range went up to
⇠ 0.6R180 , or approximately 0.8R500 . However, their sample did not include merging or disturbed
clusters. Eckert et al. (2012) used ROSAT to study 31 clusters in 0.04 < z < 0.2, half CC clusters
and the other half NCC. They found a 25% scatter at R200 ⇡ 1.5R500 . Morandi et al. (2015)
stacked Chandra emission measure profiles of 320 clusters with a redshift range of
0.056 < z < 1.24 to find an average density and intrinsic scatter. They found a scatter in the
density of 20% at R500 , and a scatter of 30% at R200 (or ⇠ 1.5R500 ). Mantz et al. (2016) found
the mean density and temperature profiles and intrinsic scatters of his sample of 40 relaxed, CC
clusters using only X-ray data. The temperature scatter is around 10% for most bins.
Arnaud et al. (2010) analyzed the REXCESS data (31 clusters, with XMM-observed
density profiles and simulations) to find pressure intrinsic scatters varying from 20-40%. Outside
of r > 0.2R500 they found the maximum scatter of 25% up to R500 , with the scatter increasing
near the center. S13 found pressures and scatters for the BOXSZ sample and found consistent
results with Arnaud et al. (2010). The pressure profile goes out much further, however, to 2R500
with an intrinsic scatter reaching 60%. At R500 they find 40% scatter in the pressure.
9.2.
Mock cluster results
The multiple-cluster fitting code was tested on mock cluster samples with known input
average profiles and intrinsic scatter matrices. As in the individual cluster fitting simulations, the
same 4 cluster samples at di↵erent redshifts were created. The maps were created similarly to the
individual cluster mock maps. The density intrinsic scatter was set to zero due to the difficulty of
comparing the deprojected onion shell scatters with those from a smooth profile. The
temperature profile was still assumed to be isothermal, however they were scattered by an input
intrinsic scatter per realization. 40 clusters were created per sample, and individual deprojections
were computed for each. Afterward the joint fitting code was fit on those 40 deprojections. 40
simulations were chosen to reflect our true sample size of 45 clusters, so that we can see typical
error bars on the scatters. The intrinsic scatters for the cluster samples at redshifts 0.25, 0.44,
0.58, and 0.69 were 25, 25, 30, and 25%, respectively.
Figure 12 shows the output fractional intrinsic scatters from several methods. First, the
original method that will be applied on the real clusters with zero o↵-diagonal intrinsic scatter
element was used. Overall the output profiles recovered the input scatters, within errors. The
only cluster set that showed some bias was the cluster sample D at large redshift. However, the
non-zero o↵-diagonals do not represent the technique of the creation of the mock cluster samples.
Instead, since the clusters were created such that the profile shifted away from the mean value by
the same amount in all bins (isothermal mock maps), we also set non-zero o↵-diagonal intrinsic
scatters corresponding to temperature bins with correlations of 1. However, in general this
42
yielded greater deviations from the input profile. The Mantz (2016) publicly-available Gibbs
Sampler ”LRGS” was also used on the data, and in general fit for similar results as our own
technique. The Gibbs Sampler instead fits for all elements of the intrinsic scatter matrix at once.
In the end, since we do not know exactly how the bins are correlated in the real clusters, we will
continue with the 0 o↵-diagonal method.
Fig. 12.— Output intrinsic scatters with errors for the four mock cluster samples. The dashed line
is the input intrinsic scatter.
9.3.
BOXSZ results
The multiple cluster fitting code was conducted on: the whole sample, CC/NCC,
disturbed/non-disturbed, and spherical/non-spherical subsets.
The mean profiles are plotted with the uncertainty on the mean in Figure 13. The intrinsic
scatter is plotted separately in Figure 14. The intrinsic scatters are plotted as the fractional
intrinsic scatter, which is the scatter divided by the mean values. This is done to see the relative
scatter in the profiles; the central density shell deviates by much more than the rest of the cluster
in an absolute sense, but if the log-scale nature of the density profile is considered, the relative
43
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 13.— Mean density and temperature profiles with their associated 68% limits. A solid line is
drawn for the full sample to aid the eye. The di↵erent subsets within a plot were o↵set in the radial
direction for ease of viewing.The top row (a and b) are the profiles for the CC vs. NCC subsets,
the middle row (c and d) are for the disturbed vs. non-disturbed subsets, and the bottom row (e
and f) are for the spherical vs. non-spherical subsets.
44
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 14.— Boxplots for the fractional intrinsic scatter profiles for density and temperature. The
di↵erent subsets within a plot were o↵set in the x direction for ease of viewing. The temperatures
were cut o↵ so that one could still see the other bins clearly. The top row (a and b) are the profiles
for the CC vs. NCC subsets, the middle row (c and d) are for the disturbed vs. non-disturbed
subsets, and the bottom row (e and f) are for the spherical vs. non-spherical subsets.
45
scatter is comparable to the other density shells. Due to the distribution of the fractional intrinsic
scatter often being non-symmetric, box-plots were used. The box represents the inner quartile
range, divided by the median. The whiskers represent either: the minimum (maximum) value
from the distribution, or, 1.5 times the inner quartile range, whichever has the smaller value from
the median.
Overall, at larger radii, the mean temperatures found are relatively flat, unlike
clearly-decreasing temperature profiles that others had found (see Section 9.1).
9.3.1.
CC vs NCC
A higher density in the inner region is found for the CC over the NCC clusters, and a
steeper temperature slope in the center for the CC subset. This is expected due to the presence of
the dense cool-core. Surprisingly, the largest density intrinsic scatter occurs in the CC subset in
the last radial shell. The temperature scatters are less constrained, especially in the last shell.
9.3.2.
Disturbed vs Non-Disturbed
The density profiles of these subsets reflect the CC vs NCC comparison, however, the
samples are slightly di↵erent. In general CC clusters are non-disturbed, and most of the disturbed
sample include NCC clusters, but there are exceptions. The disturbed clusters have higher
density scatters in all bins, especially in the last density bin. The temperature scatters show
smaller observable di↵erences between the subsets. In the central bin there is more temperature
scatter for the disturbed sample than the non-disturbed, but at the largest radii the
non-disturbed sample actually has the larger scatter (but also weaker constraints).
9.3.3.
Spherical vs. Non-Spherical
Sphericity does not seem to a↵ect the overall density profiles, and the density scatters are
all within errors of each other. There is the least constraint on temperature in the last shell when
dealing with non-spherical clusters. The scatter in temperature is smaller when comparing
non-spherical clusters to spherical ones in the 2nd and last shell. The first shell shows a higher
temperature scatter which may be due to CC clusters in the spherical sample, while the
intermediate-shells (3 and 4) do not show a di↵erence in spherical vs. non-spherical
46
9.3.4.
Temperature boundary T = 0keV
As one can observe in the Appendix in the individual profiles, sometimes the deprojected
temperature values would have a non-negligible distribution at the boundary condition of T = 0.
However, the SZ data has noise oscillations such that a ’negative’ temperature is possible. To see
if this gave any bias, the individual deprojections were done both for the case of a temperature
boundary condition of T > 0 and also for no lower bound to the temperature. When comparing
the results of both methods, there is a negligible di↵erence in the fractional intrinsic scatters.
10.
10.1.
Conclusions
Summary & Discussion
Individual cluster density, temperature, and pressure profiles out to the cluster outskirts
were found for the BOXSZ sample of 45 clusters using a combination of X-ray and SZ
measurements. We were able to extend our analysis farther into the outskirts than typical X-ray
experiments for all the clusters, and were able to reach out to or past R500 for 71% of the sample.
This method provides spectroscopic-independent derivations of cluster temperatures. 33 and 27%
of the sample displayed a temperature decrease and increase in the outskirts, respectively, while
the rest of the clusters were consistent with an isothermal profile.
There were significant di↵erences in the recovered profiles in a substantial number of
clusters from other X-ray only studies. While some of the disagreement may be due to noise
fluctuations in the SZ, there could be other processes at play. There are known biases between
spectroscopic temperatures and the mass-weighted temperatures that we are constraining, such
that the spectroscopic-derived temperatures can underestimate the mass-weighted temperatures
by up to 40% (Mazzotta et al. 2004).
Another possibility is that the actual X-ray and SZ data used together in this analysis could
be in tension. The assumption of spherical symmetry on a cluster that is, in reality, non-spherical
can cause di↵erent measurements of SZ and X-ray. If a cluster is elongated along the line of sight,
the X-ray surface brightness would be underestimated compared to the SZ (Cooray 2000), due to
the di↵erent density dependences. Puy et al (2000) explored how typical axis ratios of elliptical
clusters would a↵ect X-ray and SZ observables, possibly adding 25% and 10% errors, respectively.
Bonamente et al. (2005) used XMM-Newton to study soft excess X-ray emission that does not
originate from the ICM for a particular cluster. They found that most likely, the soft excess could
be caused by ’warm’ gas that is mixed in with the hot gas. This would explain cases where the
X-ray pressure is greater than the SZ, but, in general we see an opposite e↵ect, where the SZ
pressure is greater than the X-ray. Another source of X-ray/SZ discrepancies is radio source
contamination that alters the SZ signal. The SZ data in this work had radio sources removed
(S13), however if there was an improper subtraction of the radio galaxy contribution, the SZ
signal from the cluster could become biased. Bonamente et al. (2011), however, compared
47
pressure profiles of 25 relaxed galaxy clusters using the X-ray and SZ and found that their
integrated pressures were consistent around R500 . Others have also found that there is no
significant di↵erence between SZ and X-ray observations for well-behaved clusters (Melin et al.
2011, Planck et al. 2011). The di↵erence here, however, is that our sample includes disturbed,
and unrelaxed clusters.
An ensemble-cluster analysis of scaled deprojections at common radii out to 1.25R500
yielded mean cluster density and temperature profiles. The scatter of the clusters from the mean
was also found. The mean profiles reflected most expectations: CC clusters have higher densities
in the inner regions than NCC clusters, and CC clusters exhibit a lower temperature in the center
of the cluster than NCC clusters. We detect only an overall slight decrease with radius in the
outer regions when considering all the clusters. A common theme found in the fractional intrinsic
scatter profiles is that the scatter is high in the center of the cluster where non-gravitational
processes occur, lowest in the intermediate regions where the cluster is regular, and highest at
larger radii (R500 ) where the clusters merge with other clusters or with the cosmic web. The
density fractional intrinsic scatter profile is consistent with previous studies, ranging between
⇠ 10 20%, but the 30 60% fractional temperature intrinsic scatter profile we find is much
larger than what has been observed before (⇠ 6 10%). Even amongst the CC subset of clusters,
the scatter in temperature is larger than what we would expect. It is interesting that, although
we find similar density scatters, the temperature scatters are much larger - though they do have
larger uncertainties on the scatters. This possibly suggests that the source of the di↵erences is
coming from the SZ, or the combination of SZ and X-ray data. As described before, the
mass-weighted measurements of the SZ-driven temperatures can be higher than the X-ray
spectroscopic temperatures. Our large temperature scatters compared to the smaller X-ray
spectroscopic scatters may suggest that clusters have varying degrees of multi-phase temperature
components, where gas of di↵erent temperatures are mixing within the same volume. The
outermost radius in general had the least constraining power, possibly due to a combination of
the smaller number of clusters contributing to that radial shell, or the quality of the data there.
It is also important to note that the cluster sample did not have a clear selection function;
clusters were selected adhoc. So although we have a large mixture of clusters in a large redshift
range, it may not be the best group to represent galaxy clusters as a whole. Despite this, our
analysis provides a useful method to constrain spectroscopic-independent temperatures for galaxy
clusters. Not only is it independent of spectroscopy calibration, the SZ allows us to reach out to
large radii even for far and faint clusters.
10.2.
Expansion of this Work
There were several ways that this analysis could be revised. The simulated clusters in this
analysis were actually ”mock” clusters consisting of a single-beta density and isothermal ball of
gas. It may be helpful to test the method on actual simulated clusters, to see the e↵ect of
48
clumping, AGN, etc on the profiles. It might also be useful, although difficult, to add temperature
variations with respect to radius in the simulations.
Throughout this analysis, spherical symmetry was assumed. However it is known that most
clusters are not actually spherical but rather, prolate (Cooray 2000). Extending this work to a
triaxial analysis would be beneficial since some of the peculiar behavior in the profiles could be
accounted to asphericity. LaRoque et al. (2006) calculated that asphericity could possibly
contribute up to 20% uncertainty in the resulting profiles. On another level, further studies could
explore excising known substructures, or to do a more advanced fitting involving
azimuthally-dependent measurements. Or, for example, if a cluster had a clear binary merger
system, one would need to model the gas for those two subclusters separately with di↵erent
centers. In the onion shell deprojections, a constant density and temperature within such large
shells is likely not the best representation of reality. It may be useful to interpolate within each
shell, requiring the boundaries of the shells to be continuous.
In the ensemble-cluster fitting technique, we had chosen non-zero o↵-diagonal intrinsic
scatter elements, but in reality, adjacent shells are correlated to a certain degree, as well as a
strong correlation between density and temperature within a shell. Covering all the possible
aspects expanding this study would have exceed the time and scope for the current research
project.
This technique will be particularly useful when applied to higher-resolution SZ instruments,
such as MUltiplexed SQUID/TES Array at Ninety GHz (MUSTANG) at the Green Bank
Telescope (GBT) (with 10-18” resolution), or the upcoming New IRAM KID Arrays (NIKA)
Camera on the Institute for Radio Astronomy in the Millimeter Range (IRAM) 30-m telescope,
which has a ⇠20” resolution at the SZ decrement. Finally, the temperatures found here can be
used to calculate masses that are spectroscopic-independent.
11.
Acknowledgements
I would like to thank my advisor Elena Pierpaoli and the Bolocam team at Caltech (Jack
Sayers, Sunil Golwala, Nicole Czakon) for their guidance. Also, a big thank you to Elena Rasia
and Tony Mroczkowski for helping me create the mock clusters for the X-ray and SZ maps, and
the USC cosmology group for useful discussions. I would also like to thank my family and Nick
Guggemos for their support.
Thank you to the Women in Science and Engineering (WiSE) Progam at USC for providing
a Merit Fellowship that partially funded my research in 2014-2015 year.
Computation for the work described in this paper was supported by the University of
Southern Californias Center for High-Performance Computing (hpc.usc.edu). This research has
made use of data and/or software provided by the High Energy Astrophysics Science Archive
Research Center (HEASARC), which is a service of the Astrophysics Science Division at
49
NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical
Observatory.
50
12.
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This preprint was prepared with the AAS LATEX macros v5.2.
55
13.
Appendix A
Tables 9 and 10 are lists of useful symbols and acronyms used in this thesis, respectively.
Table 9: Symbol Guide
Symbol
⌦m
⌦b
M
Z
Symbol
hp
gf f
✏v
✏fv f
Meaning
Planck constant
Gaunt factor
total X-ray emission
free-free emission
✏line
line emission
v
frequency
Rvir
M500
Meaning
matter density parameter
baryonic matter density parameter
solar mass
solar abundance metallicity
radius at which the mean density within is
500 times the critical density of the universe
radius at which the mean density within is
X times the critical density of the universe
virial radius
mass within R500
SX
n500
mean electron density within R500
`
T500
P500
w500
Tspec
Te
T
ne
ni
me
kB
z
E(z)
h
⌦⇤
Mgas
Mtot
fgas
mean electron temperature within R500
mean electron pressure within R500
X-ray centroid shift parameter
spectroscopic temperature
electron temperature
gas temperature
electron density
ion density
electron mass
Boltzmann constant
redshift
evolution of the Hubble parameter
dimensionless Hubble parameter
dark energy density
gas mass
total mass
gass-mass fraction, Mgas /Mtot
chi-squared value
reduced chi-squared value
y
TCM B
probability to exceed
xj
X-ray surface brightness
cooling function
distance along the line of sight within
a cluster
Compton y parameter
Cosmic microwave background temperature
Thompson cross-section for an electron
relativistic correction to the SZ e↵ect
hydrogen column density
likelihood
observable, i.e. data
model in data units
measurement uncertainty
fractional proton density
fraction electron density
density normalization for component i
scale radius for density component i
density slope parameter
temperature normalization
central temperature
cool-core temperature scale radius
outer temperature profile scale radius
outer temperature profile slope
vector of di↵erence of mean densities, temperatures
and cluster j densities, temperatures
covariance matrix for individual cluster j
R500
RX
2
2
red
PTE
dof
S
T
SZ
nH
L
Oi
Mi
i
µH
µe
n0,i
rc,i
T0
Tmin
rcool
rt
↵
degrees of freedom
Uj
intrinsic scatter covariance matrix
Listed in order of appearance and grouped with relevance.
56
Table 10: Acronym Guide
Acronym
ACCEPT
AGN
ACIS
Meaning
Archive of Chandra Cluster Entropy Profile Tables
Active Galactic Nuclei
The Atacama Pathfinder EXperiment
SunyaevZeldovich Instrument
Chandra’s Advanced CCD Imaging Spectrometer
BCG
Brightest Cluster Galaxy
BOXSZ
CC
CIAO
Bolocam X-ray Sample
Cool-Core
Chandra Interactive Analysis of Observations
CMB
Cosmic Microwave Background
APEX-SZ
CXO
Chandra X-ray Observatory
FoV
Field of View
HIFLUGCS
HIghest X-ray FLUx Galaxy Cluster Sample
HRC
CXO’s High Resolution Camera
HSE
hydrostatic equilibrium
ICM
Intracluster Medium
kSZ
kinetic SZ e↵ect
Listed in alphabetical order.
14.
Acronym
MACS
MCMC
Meaning
MAssive Cluster Survey
Markov Chain Monte Carlo
MP
Model Parameters
NCC
Non-Cool-Core
Owens Valley Radio Observatory/
OVRO/BIMA
Berkeley Illinois Maryland Array
POSS
Palomar Sky Survey
PSF
Point Source Function
REFLEX
ROSAT-ESO Flux-Limited X-ray survey
The Representative XMM-Newton Cluster
REXCESS
Structure Survey
ROSAT
Roentgensatellite (an X-ray satellite)
SPH
smoothed particle hydrodynamics
SPH
smoothed particle hydrodynamics
SPT
South Pole Telescope
SZ
Sunyaev-Zel’dovich E↵ect
XMM-Newton The European Space Agency’s X-Ray Multi-Mirror Mission
Appendix B
The individual cluster descriptions and their profiles are listed iin order of increasing
redshift. In general as the redshift becomes higher there are less studies done on the object.
14.0.1.
Abell 2204
Abell 2204 is a relaxed cool-core-cluster. It has been studied extensively due to its high
luminosity and close distance. The deprojections and analytical profile distinctly resolve the cool
core, while also seeing a temperature drop to the outskirts. The density profile is well fit by a
double-beta profile. Overall, our results matches ACCEPT density and temperature profile
shapes, and the S13 pressure profile.
Sanders et al. (2005), using X-ray data, found a complicated core consisting of several cold
fronts that may be attributed to the cluster recovering from a merger. Using spectroscopy, the
temperature oscillates and has an axial dependence, but in general peaks around 100-200 kpc
(40-80”) before sloping downward in the outskirts, consistent with our results. Gu et al. (2009)
also found such temperature substructures, and speculated the cause being AGN. Sanders et al.
(2009), again using X-ray data, found dips in the surface brightness that may be caused by radio
bubble formation in the core of the cluster. Reipreich et al. (2009) used Suzaku X-ray
measurements to probe the cluster out to large radii, around 12’, where the temperature dropped
down to 4keV. They also compared the outer temperature profile with the hydrodynamical
57
simulations of Roncarelli et al. (2006), and found the data matched the predictions. Many other
studies have similarly found temperature profiles out to large radii for this cluster due to its
proximity and found consistent results (Hasler et al. 2012, Bonamente et al. 2006).
14.0.2.
Abell 383
Abell 383 is specified as a CC cluster, however the cool core is not resolved in this analysis.
Density and temperature match with ACCEPT up until ACCEPT’s reach, after that there is an
increase in the temperature and then an immediate drop at around R500 . Due to the bumpiness
of the temperature profile, it is best fit by an isothermal profile. The high temperature in the
outskirts could be due to it being non-spherical.
Abell 383 is a popular lensing cluster (Sand et al (2008) used the Hubble Space Telescope
with strong and weak lensing, Zitrin et al. (2012) used Subaru, and Huang et al. (2011) used
weak lensing using CFHT + Subaru). Morandi & Limousin (2012) conducted a lensing and X-ray
study (imaging and spectroscopy, respectively). They found an X-ray - lensing mass discrepancy
that goes away if a triaxial geometry is considered. Although it is in a relaxed state, the
asphericity may contribute to the irregularities seen in our profiles.
14.0.3.
Abell 209
This cluster is categorized as a nearby, cool-core, non-disturbed, and non-spherical cluster.
We are unable to resolve the cool core but see a temperature drop in the outskirts both in the
analytical fit and deprojections before r500 . Mercurio et al. (2003) found a slightly elongated
structure in the X-ray map arising from two X-ray peaks, which our maps could not confirm. To
add to the suspected disturbed nature of the cluster, Paulin-Henriksson et al. (2007) also found
elongation in the mass distribution through weak lensing. Compared to ACCEPT, we have higher
density, and pressure profiles. Mercurio et al. (2003) found a temperature of 10.2+1.4
1.2 keV using
X-ray spectroscopy within 3’, which is consistent with our results.
14.0.4.
Abell 1423
This cluster is a non-cool-core, non-disturbed cluster according to our metric. The central
region is not very well constrained in either the deprojections or the analytical fits, but a slow
decrease in the temperature profile with increasing radius can be seen.
AMI Consortium et al. (2012) found the gas temperature from SZ data and not from X-ray
spectroscopy, but assumed an isothermal temperature, in which they found 3.0 +/- 0.8 keV.
Comparing it with large-radius X-ray temperature (which is 5.2 keV from ACCEPT), our results
find a temperature closer to ACCEPT’s. We find a slightly higher density and pressure profile
58
than the X-ray ACCEPT data, and consistent pressure profile compared with S13. It seems that
the SZ-derived pressure is significantly higher than the X-ray especially in the center of the
cluster. This cluster is also a CLASH cluster, which is a selection of 25 massive clusters observed
with the HST to derive gravitational lensing mass distributions (Postman et al. 2012).
14.0.5.
Abell 963
Abell 963 is a non-cool-core, non-disturbed cluster, which our profiles indeed reflect.
Although it is a NCC cluster, a double-beta profile fit better than a single-beta profile. The
pressure profile is consistent with the S13 pressure profile, and a clear decrease in the temperature
profile with radius is observed.
Sayers et al. (2013b) describes how a prominent radio source in the SZ was removed,
located only 2.5’ from the center of the cluster. Whenever the source is close to the center of the
cluster where the signal is the highest, removal can be difficult. We do not consider it a cool-core
cluster, but others have found a slight temperature decrease in the center (Baldi et al 2007,
Cavagnolo 2006). Leccardi et al. (2008), using XMM-Newton, specifically made note of this and
identified it as an ’intermediate’ cluster since the temperature decrease in the center is not large.
This cluster is also a CLASH cluster (Postman et al. 2012).
14.0.6.
Abell 2261
Although this cluster is identified as a CC cluster, we do not see the cool core. The
temperature is systematically and significantly lower than ACCEPT. The pressure profile is
consistent with S13.
The cluster had a prominent radio source in the SZ that was removed (Sayers et al. 2013b).
It is a CLASH cluster (Postman et al 2012), in which Coe et al 2012 did a detailed studied on the
mass measurement, also deducing that the DM halo is elongated along the line of sight. ACCEPT
(Cavagnolo 2009) sees a temperature drop in the center. Several studies have observed that this
cluster does not have a strong cool-core. Coe et al 2012 found a density profile slope that is
shallower than the usual slope at small radii for cool-core clusters, so they defined it as a
”borderline” relaxed and cool core cluster. Baldi et al 2007 found only a slight cool-core, with a
central temperature of about 7.7 +/- 0.4 keV and peak of 9.0 +/- 0.4 keV. Bonamente 2006 used
X-ray spectroscopy, and found a temperature profile out to 150”, which also matches with the
ACCEPT profile. It seems that the SZ data is what is driving the di↵erence between our
temperature profiles and the above described X-ray profiles.
59
14.0.7.
Abell 2219
Our fits for this cluster show no cool core but a decrease in temperature in the outskirts.
We find a very good agreement between our temperature and pressure profiles with ACCEPT and
S13, however our density profiles are larger than ACCEPT. It is identified as a NCC, disturbed
cluster. Million & Allen (2009) found a shock front, >16keV gas 2’ NW of cluster. The cluster is
possibly undergoing a large merger event (Boschin et al. 2004), and is elongated. Canning et al.
(2015) did a detailed Chandra study, identifying numerous cold and shock fronts, and a large
temperature spike ⇠25” at the core of the cluster. Also seen is a spike in temperature around
130” in the NW direction (as in Allen et al. 2009). This contrasts our analysis, where only little
temperature fluctuations can be seen.
14.0.8.
Abell 267
This is a NCC, disturbed cluster. Our deprojections and analytical fits show a mostly flat
temperature profile. In the analytical fits, including a temperature decrease in the outskirts
showed to be a better fit than an isothermal profile, but the deprojections do not constrain this
temperature drop. Systematically higher density, and pressure profiles than ACCEPT, but
pressures consistent with S13. Overall the temperature is consistent with ACCEPT’s relatively
flat temperature profile.
This is another cluster that had a strong radio source in the SZ that was removed (Sayers et
al. 2013b). Several studies suggest di↵erent reasons for the disturbed nature of the cluster.
Jimenez-Bailon et al. (2013) used XMM-Newton spectroscopy to find a deprojected temperature
profile, and found a flat profile ⇠6 keV. They consider A267 a ’fossil’ system, which is dominated
by a massive elliptical galaxy. Fossil systems in general have not merged recently, expecting to
host a cool-core, which we do not see for this cluster. One possibility is the core being heated by
AGN. Zhang et al. (2008) found an irregular shape of the cluster, which would suggest a merger,
instead. This is another cluster selected in the CLASH sample (Postman et al. 2012).
14.0.9.
RX J2129.6
This is a non-disturbed CC cluster, but we don’t constrain a cool core in our analysis. The
profile was best fit with a single-beta density profile, and an isothermal temperature profile.
These are relatively consistent with ACCEPT profiles, but they found a slight cool-core. There is
some deviation between our deprojections and the analytical fit near the center. It may suggest
that the single-beta profile is not the best description of the cluster, but a double-beta profile was
unable to be constrained. This is one of the few clusters in the sample that does not reach out to
R500 , due to the poor quality of the X-ray data.
Landry et al. (2013) used Chandra observations to identify this cluster as a relaxed cluster,
60
based on the centroid shift (distance between x-ray emission peak and centroid). Using the
temperature profiles defined by Vikhlinin et al. (2006) and Bulbul et al. (2010), they found a
central temperature of 4 keV, peak temperature of 8 keV at around 90” and decrease temperature
back down to 4 keV in the outskirts (around 240”). Our temperature profile is consistent with
these results except for the decrease in the outskirts.
14.0.10.
Abell 1835
Abell 1835 is a well-studied CC cluster. We can clearly resolve a CC, but don’t see drop in
temperature in outskirts in either the analytical fits or smooth profiles. In the outer temperature
deprojected bins we see deviations from our own analytical fits, but this may have been due to
the fact that the inner temperature slope was taken from the average value from the Vikhlinin et
al 2006 study. It may be the case that this cluster is not as steep in the center. The cluster was
well-fit by a double-beta density profile. Our pressure deprojections are consistent with Sayers
2013, but the outer pressure slope is significantly di↵erent than the LaRoque 2006 joint X-ray/SZ
study.
Many studies have found this cluster to be spherically symmetric without any significant
substructure through the SZ or X-ray (Korngut et al 2011, Ichikawa et al 2013). Ichikawa et al.
(2013) used Suzaku X-ray spectroscopy observations to study A1835 in the outskirts, and they
found that the temperature slowly decreases from 8keV in the inner region to around 2keV near
the virial radius. Li et al. (2012) did a deprojected analysis comparing XMM Newton and
Chandra spectroscopy, and found lower temperatures from XMM-Newton than Chandra, as well
as lower density profile from XMM-Newton than Chandra. This supports the point that even in a
well-behaved, bright cluster, di↵erent spectroscopy measurements can yield results that are in
tension with each other. Bonamente et al. (2013) found surface brightness emission out to 10’, as
well as a temperature drop of a factor of 10 from the peak. Baldi et al 2007 also found
temperature drop of a factor of 2 in core.
14.0.11.
Abell 697
This is a NCC cluster, classified as non-disturbed. See a relatively flat profile (besides 2nd
deprojected bin) and a decrease in temperature in the outskirts out to R500 . Pressures match
with S13, as well as with the joint X-ray/SZ LaRoque et al. (2006) study. There is good
agreement in density, temperature, and pressure with ACCEPT profiles, where a slight decrease
in temperature with radius is seen.
Although we characterized the cluster as non-disturbed, several other studies have actually
shown that the cluster has undergone recent merging. Girardi et al. (2014) did a multi
wavelength study using the optical and X-ray, where line of sight galaxy velocity dispersions and
spectroscopic temperatures were calculated, respectively and found that A697 is not relaxed.
61
With elongated X-ray emission and substructures near the center, A697 likely went through many
mergers. Through lensing, Metzger et al. (2000) also concluded that the cluster had undergone a
recent merger.
14.0.12.
Abell 611
This is a NCC, non-disturbed cluster. We see a relatively flat profile in the center and
smaller temperatures in the outskirts. We see significantly higher densities and pressures
compared to the ACCEPT study, and slightly higher density profile than the LaRoque et al.
(2006) joint X-ray/SZ study.
This cluster has many lensing studies. In terms of ICM studies, Donnaruma et al. (2011)
did an analysis with X-ray using Chandra and lensing using HST. Unlike our analysis, they
identify it as a cool-core cluster (unlike us), and found a central temperature of 6keV, peak of 8
keV at 100 kpc (20”), and a slow drop in temperature to 5keV at 600 kpc (150”). We do not
constrain a cool core, although our inner temperature bins have lower limits consistent with zero.
Bonamente et al. (2006) also finds a relatively flat temperature profile out to 130”, using
spectroscopic temperatures from the X-ray.
14.0.13.
MS2137
This is a non-disturbed CC cluster. We can clearly resolve the CC and the temperature
drop in the outskirts in the analytical fits, however the deprojections do reflect these features. It
is not well constrained by the deprojections, which is probably due to relatively low
signal-to-noise in the SZ data. Although it is a CC cluster, a single-beta density profile was a
better fit to the data. Consistent density, temperature, and pressure profiles with ACCEPT, and
pressure profile with S13.
Many lensing studies have been done on this cluster (Gavazzi 2003, Gavazzi et al. 2005,
Sand 2008). Donnarumma et al. (2009) found spectroscopic temperature profile using Chandra
with a 4 keV central temperature, a peak of about 5.5 keV at 100 kpc (around 20”), and
decreasing temperature to 3.5 keV in the outskirts at 500 kpc (around 110”). Our profiles are
mostly consistent with these results except that the temperature profile peaks at a higher
temperature.
14.0.14.
MACS J1931.8
A CC, non-disturbed cluster, MACS J1931.8 was best-fit by a double-beta density profile
and a full cool-core temperature profile. Our profiles are consistent with ACCEPT density profile
and S13 pressure profile. In terms of the temperature, there is consistency with ACCEPT
62
temperature profile out to where ACCEPT has data, then the temperature increases around 100”,
and then we do not see a significant drop in the temperature in the outskirts.
Santos et al. (2016) used the Herschel telescope and Chandra data to investigate star
formation in this cluster and found that there is a relatively high star formation rate even with
conservative estimates for AGN activity. In general, BCGs with active star formation exist in CC
clusters (Ho↵er et al. 2012). A very detailed multi-wavelength study was done by Ehlert et al.
(2011) using X-ray (Chandra), optical (Subaru), and radio (Very Large Array) data. They found
a cool core with AGN feedback (seen in the X-ray and in the radio), along with evidence of
merging, suggesting a cool core that is currently being destroyed. They created a temperature
map of the cluster from X-ray spectroscopy out to radii of 500 kpc (100”), and found an
asymmetric temperature distribution. In particular, a spiral of cool gas around the core at ⇠200
kpc. There is also evidence of merging at small radii (50 kpc). Their azimuthally averaged
temperature profile had a central temperature of 5 keV and peak of 10keV. Their optical data
also suggested a merger. It is a possibility that the merging processes described by this study
could contribute to why we do not see the cluster temperature decrease in the outskirts.
14.0.15.
Abell S1063
Although this cluster is NCC, and non-disturbed, it has the most peculiar temperature
profile in the sample. We see an increasing temperature profile in the outskirts, much larger than
ACCEPT, and nearly unphysical temperature in the last temperature bin (around 40keV).
However, the pressure profile matches with S13, which is also larger than the ACCEPT pressure
profile.
Gomez et al. (2012) found evidence of merging through both X-ray and optical
observations. They found the X-ray emission to be elongated in the same direction as two high
density galaxy regions. They found high cluster temperatures of 12-17 keV from the center to
⇠800 kpc (160”), seemingly isothermal.
14.0.16.
MACS J1115.8
This is a CC, non-disturbed cluster. As in AS1063, we find a possible cool core followed by
an increase in the temperature at large radii. The pressure profile is consistent with S13. The
density and pressure profiles are higher than ACCEPT density profile by factor of 2.
Temperature profile consistent with ACCEPT temperature profile at intermediate radii.
Although it is a CLASH cluster (Postman et al. 2012), not much has been done in the
literature to study this cluster individually. Donahue et al. (2014) found temperature profiles
from X-ray only (XMM Newton and Chandra), showing a central temperature of around 3 keV
and rising to 8-9 keV. The XMM Newton then shows a decrease in the temperature profile in a
63
bin centered at 800 kpc (160”), which is in definite contradiction with our results.
14.0.17.
Abell 370
Abell 370 is a NCC, disturbed cluster. We find consistent profiles with ACCEPT and S13.
The temperature profile shows little trend with radius, with a slight increase in the outskirts, but
an isothermal profile best fit the data. The disturbed nature of this cluster could explain the
peculiar temperature profile.
Several studies through di↵erent wavelengths have found that this cluster is not spherical.
This is a very popular lensing cluster, and many studies have been done on the large arcs present
in the optical as well as weak lensing (Medezinski et al. 2010 with Subaru). De Filippis et al.
(2005) combined X-ray and SZ data to find the 3d shapes of galaxy clusters, including Abell 370.
It is found to have a triaxial morphology, elongated along the line of sight. Richard et al. (2010)
did a strong lensing analysis based on HST/ACS observations and reconstructed the mass
distribution, which resulted as elongated, bimodal, and aligned with the Chandra X-ray
luminosity maps. The galaxy distribution also shows bimodality, suggesting a merging cluster.
Grego et al. (2000) had OVRO SZ observations of this cluster, also showing a smooth but
non-spherical distribution.
14.0.18.
CL0024+17
This cluster is a NCC, disturbed cluster. The temperature profile is consistent with
isothermal, and we do not see decrease in the outskirts.
The cluster has been well-studied through gravitational lensing due to its multiple arcs, as
well as in the X-ray. Zhang et al. (2005) did a XMM-Newton study with imaging and
spectroscopy, and found a temperature decrease at large radii (1.3-3’) from the isothermal ⇠4keV
central region. We do not see such a drop. They also found an elongation in the X-ray hard ratio
map on large scales, and substructure at large radii. Ota et al. (2004) used Chandra spectroscopy
and found a nearly isothermal profile of ⇠4.5 keV out to 600 kpc (110”), which is lower than our
results. Bohringer et al. (2000) used ROSAT to look at the cluster’s X-ray morphology, and found
a very small core radius in the surface brightness. They also found an elongation in the X-ray
emission, however they claim it is consistent with a spherical model. Tyson et al 1998 found a
mass map using strong lensing, and found a relaxed distribution. Umetsu et al. (2010) did a
full-lensing analysis, including Subaru and ACS/NIC3 observations, and looked at X-ray data
with simulations. They suggested that the cluster is in a post-collisional state, with two clusters
at the same line of sight, as well as finding the mass profile of the cluster. Jee et al. (2007) noted
that the X-ray surface brightness from Chandra is better fit by two isothermal beta models,
suggesting that possibly one could be seeing two systems that are along the same line of sight.
64
14.0.19.
MACS J1532.9
This is a CC, nondisturbed cluster. We can resolve the cool core, but it seems our lower
limits are consistent with zero. The pressures are consistent with S13.
This is a CLASH cluster (Postman et al. 2012). Donahue et al. (2014) found temperature
profiles from X-ray-only data (XMM Newton and Chandra) with the temperature profile showing
a central temperature of 3-4 keV and rising to 8 keV. The XMM Newton profile decreased down
to ⇠4keV out to 900 kpc (⇠180”), while the Chandra profile does not indicate the temperature
decrease, instead it plateaus out at ⇠9keV. Hlavacek-Larrondo et al. (2013) did a Chandra,
XMM-Newton, VLA, and HST analysis. They observed AGN feedback evidence, a cold front, and
X-ray cavities. They note a di↵erence in the temperature map from the east and the west side of
the cluster. They find a central temperature of 4 keV rising to 9 keV at 250 kpc (⇠50”), and
detected slight di↵erences in the temperature profile at small radii (<25”) in di↵erent directions,
with higher temperatures in the south and west directions.
14.0.20.
MACS J0429.6
This is a CC, relaxed cluster, but our results show a large temperature increase in the
outskirts. Since the analytical fits also show a high temperature at large radii, we can rule out the
possibility that it is due to temperature bin correlations/bumpiness of the deprojections.
Donahue et al. (2014) found temperature profiles from X-ray-only data (XMM Newton and
Chandra) with the temperature profile showing a central temperature of 4keV then jumping up to
9 keV relatively quickly at 200 kpc (40”). Using XMM Newton they detect a flat profile at large
radii at 4 keV, which is not consistent with the large temperatures we see in our profiles. Not
much else has really been done on this cluster on an individual-cluster basis. This cluster is a
CLASH cluster (Postman et al. 2012).
14.0.21.
MACS J2211.7
This is a CC, non-disturbed cluster. Our densities and pressures are higher than ACCEPT,
but consistent with S13. Temperatures are consistent with ACCEPT up to the point ACCEPT
has data. This cluster has relatively hot values of 10-16 keV throughout. We cannot resolve the
cool-core with the analytical fit. The best fit was an isothermal temperature with a single-beta
density profile. Not much has been done on this cluster on an individual-cluster basis.
65
14.0.22.
MACS J1720.3
MACS J1720.3 is a CC, non-disturbed cluster. Our profiles match well with ACCEPT and
S13, but cannot constrain the cool core, which may be due to the limiting resolution of Bolocam.
A decrease in the temperature in the outskirts can be seen in the analytical fits. ACCEPT is able
to see evidence of a slight cool-core.
Donahue et al. (2014) found temperature profiles from X-ray-only data (XMM-Newton and
Chandra), and found that the temperature profiles from the two instrument have di↵erent shapes.
Both have a central temperature of ⇠3.5 keV. XMM Newton jumps to 10 keV then slowly
decreases down to ⇠2 keV at 900 kpc (170”). Chandra slowly reaches 10 keV but then drops
down to temperatures close to zero around 400kpc (70”). Essentially, the outer temperature slope
is steeper for Chandra, but the inner temperature slope is steeper for XMM Newton. Maughan et
al. (2008) used Chandra imaging and spectra and assumed isothermal profiles. Found a
temperature of ⇠6.1 keV for radii < R500 , but if one excised the central region 0.15 < r < 1R500 ,
the temperature found was 7.8 keV, both consistent with our results. This cluster was also
selected as a CLASH cluster (Postman et al. 2012).
14.0.23.
MACS J0416.1
MACS J0416.1 is a NCC, disturbed cluster. We can see a dip in the density profile, seen as
the second deprojected density bin, which may be due to the disturbed nature of the cluster. The
pressure profile is consistent with S13, and we see a relatively flat temperature profile. This
cluster was not included in the ACCEPT sample.
Several studies have already identified this cluster as being a binary merger system. Mann
& Ebeling (2012) identified through the optical and X-ray that this is a currently merging cluster,
and could possibly be a binary head-on collision type, just after the first collision. Jauzac et al.
(2015) conducted a joint X-ray and optical study, combining both strong and weak lensing, along
with Chandra observations. They saw a large structure associated with a line-of-sight filament
that could not be seen in the X-ray. A large o↵set in the radial velocity between two subclusters
was found, along with their temperatures of 10 and 13.6 keV. They tried to fit an isothermal
model, but unlike our results, it yielded unphysical results (such as high temperatures). Ogrean et
al. (2015) created temperature, pressure, and entropy maps using a multiwavelength analysis of
Chandra, the Jansky Very Large Array, the Giant Metrewave Radio Telescope, and the HST. One
can clearly see an elongation in the temperature map, as well as an overall relatively high
temperature (mean temperature of 10 keV). The radio halo is also elongated along the same
direction. The study distinguishes between the two scenarios; before merging or after merging,
but find that there is more evidence for it being a pre-merging system. This cluster is also a
CLASH cluster (Postman et al. 2012).
66
14.0.24.
MACS J0451.9
This is a NCC, disturbed cluster. The profiles are isothermal in the deprojections, and is
best fit with an isothermal smooth profile. This cluster does not have data in ACCEPT sample.
Mann & Ebeling (2012) found this cluster to have highly irregular morphology, but was not
an extreme or active merger due to its BCG-Xray peak separation and BCG-Xray center
separations not being large enough. Maughan et al. (2008) used Chandra imaging and spectra
and assumed isothermal profiles, and found a temperature of ⇠5.6 keV for radii < R500 ,
consistent with our results, but if one excised the central region r < 0.15R500 , the temperature
found was 4.8 keV.
14.0.25.
MACS J0417.5
This cluster is one of the rarer CC, yet disturbed clusters. The density is best fit by a
double-beta profile, and we can resolve a cool-core in the analytical temperature profile. The first
deprojected bin in the temperature may be due to the oscillating nature of the temperature
deprojections (due to the bins being highly correlated) or could be physical, possibly a reflection
of the cluster being disturbed. We can see a decrease in temperature in the outskirts, but the
slope of the analytical fit is shallower than what we see in the deprojections, which might be due
to the fact that we used the average values for the slope from Vikhlinin et al. (2006). Overall our
results match with that of ACCEPT and S13.
Mann & Ebeling (2012) identified this cluster as a primary candidate for a binary, head-on
collision type merger. In the optical two cores are visible. The X-ray core aligns with one of the
optical cores, but the X-ray emission bleeds into the outskirts and meets with the second of the
optical cores.
14.0.26.
MACS J1206.2
This cluster is a NCC, non-disturbed cluster. The pressures we find are slightly higher than
ACCEPT but consistent with S13. The temperature is consistent with ACCEPT but then
increases in temperature around 100”.
This cluster has been studied extensively in recent years, but not so much in terms of
temperature profiles. Donahue et al. (2014) found temperature profiles from X-ray-only data
(XMM Newton and Chandra). The Chandra profile starts with a large temperature peak in the
center at 15 keV, going down to a nearly isothermal profile at 10 keV until where it has data at
1000 kpc (180”). The XMM Newton profile on the other hand has a cool core of ⇠7keV, then
increased temperature profile of 10 keV until the outskirts where the temperature profile
decreases to 5 keV. Both of these results are in tension with the outskirt temperature profile in
67
our study. Gilmour et al. (2009) found this cluster to be very relaxed in the X-ray. Mann &
Ebeling (2012) found this cluster to be relaxed with good X-ray and optical agreement. Young et
al. (2015) obtained high-resolution SZE maps using MUSTANG, detecting the central AGN.
There seems to be a residual decrement in the North-East, which may be due to a small group of
galaxies. This could possibly explain how although the X-ray density profiles are consistent with
ours, the temperature profiles disagree (driven by an excess SZ signal). Umetsu et al. (2012) did
a full-lensing analysis, also finding that cluster is relaxed. This cluster is also a CLASH cluster
(Postman et al. 2012).
14.0.27.
MACS J0329.6
This cluster is another one of the CC but disturbed cluster. There is very slight evidence
for a cool core with an increase in temperature in the outskirts. The pressures are consistent with
S13 pressures except for the last bin. The densities are consistent with that of ACCEPT.
Mann & Ebeling (2012) find this to be a relaxed cluster according to X-ray and optical
alignment. Maughan et al. (2008) found a temperature of 4.5 keV for r < R500 , and 4.4 keV for
radii within 0.15 < r < 1R500 . Kotov et al. (2006) used Chandra/XMM Newton to find that it
followed a standard cool-core temperature profile, except for a dip in temperature at intermediate
radii (around 100 kpc), which could hint at some substructure. Giacintucci et al. (2014) found a
possible radio minihalo centered in the cluster using VLA data, filling out much of the core to 150
kpc. This cluster is also included in the CLASH sample (Postman et al 2012).
14.0.28.
RX J1347
RX J1347 is a well-studied, CC, non-disturbed cluster, since it is one of the most X-ray
luminous clusters. We can see a cool core in the smooth profiles, but no decrease in the outskirts.
Otherwise, it is consistent with ACCEPT and P13.
Donahue et al. (2014) temperature profiles show central temperature of 5 keV, peak up to
17 keV, and decreasing in the outskirts. However the XMM profile decreases slower than the
Chandra profile. Maughan et al. (2008) used Chandra imaging and spectra and assumed
isothermal profiles, finding 12.2 keV for r < R500 , and 11.7 keV for 0.15 < r < 1R500 . Lu et al.
(2010) studied this cluster using the optical waveband, including spectroscopy and photometry of
the galaxies within the cluster, and weak lensing. They find another nearby cluster 7 Mpc away,
and, from velocity dispersions, an excess of galaxies between the two clusters, which could be
explained by a filament connecting the clusters.
This cluster has many SZ-focused studies devoted to it. Ferrari et al. (2011) compared
radio GMRT data with MUSTANG, Chandra, and XMM Newton X-ray data, and found a
correlation between the intracluster radio emission and X-ray and SZ emission. Korngut et al.
68
(2011) made a high resolution SZ map using MUSTANG, and was able to find substructure 20”
from the center in the form of gas that has been heated through shocks caused by mergers.This
may explain the irregularities in the inner part of our temperature profile (first 2 shells).
Kitayama et al. (2004) combine X-ray and SZ data to study the cluster, also finding substructure
near the core. They also apply an X-ray spectroscopic-independent deprojection using the X-ray
and SZ to constrain a temperature profile out to 80”, which corresponds to only the first 3
deprojected shells in our analysis. It is difficult to compare since the binning is di↵erent, but their
temperature profile starts to dip where our profile does, and in general they also find very hot gas
of up to 20 keV. Komatsu et al. (2001) mapped the SZ signal at 150 GHz with 13” resolution
using the Nobeyama telescope, and also detected the excess SZ emission at 20”. Pointecouteau et
al. (2001) on the other hand, made a similarly resolved map, and could not constrain
substructure, although they did observe that the signal was not completely spherical. Most
recently, Sayers et al. (2016) was able to constrain the peculiar velocity of the cluster through 4
SZ bands using MUSIC and one from Bolocam.
14.0.29.
MACS J1311.0
This is a CC, nondisturbed cluster. However, our temperature profile was best fit with an
isothermal profile, while the density was best fit with a double-beta profile. We see an overall
consistency with S13 pressure profiles, as well as ACCEPT density profiles. The last deprojected
temperature bin is much higher than the rest of the bins, and the last deprojected pressure bin is
higher than the S13 pressure profile.
14.0.30.
MACS J0257.1
This cluster is a NCC, non-disturbed cluster. We find higher densities and pressures than
ACCEPT, but consistent with S13 pressures. Single-beta density and isothermal profiles fit the
data best, although the temperature profile is highly irregular in the deprojections (which is also
reflected in the pressure deprojections).
Through the optical, Kartaltepe et al. (2008) found that on large scales, this cluster has
visible systems falling into the cluster via filaments. Hlavacek-Larrando et al. (2012) used X-ray
spectra to find the temperature but could not constrain a cool core.
14.0.31.
MACS J0911.2
This cluster is a NCC, non-disturbed cluster. Our pressures are consistent with S13
pressures. We see a slight downward trend in the analytical temperature profile, however it could
have been also well fit by an isothermal. A single-beta density profile was sufficient to describe
69
the density profile. No ACCEPT data is available for this cluster.
Kartaltepe et al. (2008) found that through the optical, this cluster actually consists of two
subclusters with disparate masses separated by 1 Mpc. The X-ray centroid was found to be
significantly o↵set from the galaxy surface density in the smaller mass case, suggesting that this
system has undergone a merger.
14.0.32.
MACS J2214.9
This cluster is a NCC, disturbed cluster. We find densities and pressures higher than
ACCEPT, but pressures that match with S13. The temperatures match with ACCEPT up until
60”, where it deviates and turns upward instead of sloping downward. This could possibly be due
to the cluster being disturbed.
This cluster had a prominent radio source in the SZ that was removed (Sayers et al. 2013b).
Mann & Ebeling (2012) found this cluster to have the most relaxed morphological denotation
with a prominent cool core and the X-ray peak and BCG are perfectly aligned. This conflicts
with our disturbed denotation for this cluster.
14.0.33.
CL0016
This is a NCC, nondisturbed cluster. Densities, pressures, and temperatures are consistent
with ACCEPT and LaRoque et al. (2006) profiles. Pressure is also consistent with S13. The last
bin has little constraining power. A flat temperature profile is found in our analysis
This cluster has been studied extensively in the past, due to the cluster’s mass and high
luminosity. Combined with its relatively high redshift, it makes it an interesting object. Several
studies had found this cluster to be relaxed (Kotov and Vikhlinin 2005), however, Solovyeva et al.
(2007) found evidence for it to be undergoing a merger near the center of the cluster through
XMM-Newton measurements. They also found a temperature profile out to R200 , showing a clear
decrease in the outskirts, around 4keV. Worrall and Birkinshaw (2003) found no evidence of a
cool core, but also found evidence for a merger in the center of the cluster due to the nonspherical
X-ray shape of the cluster in the center of the cluster. Overall they find a temperature of ⇠9.13
keV. Reese et al. (2000) did a joint fit of X-ray and SZE data to find a single-B isothermal profile,
which they then also used to calculate the distance to the cluster. They found that a single-B
isothermal was a good fit to their data.
70
14.0.34.
MACS J1149.5
This is a NCC, disturbed cluster. Our densities are consistent with ACCEPT, but pressures
are higher, though consistent with S13. An isothermal profile best-fit the data, but in the
deprojections again an increase in the outskirts can be seen.
This massive cluster has been mostly studied through gravitational lensing, since it exhibits
strong lensing of a spiral galaxy. Using HST data from Smith et al. (2009) and others,
Mohammed et al. (2016) studied the mass distribution of this cluster and found several main
peaks and clear non-sphericity, suggesting this is a merging cluster.
14.0.35.
MACS J0717.5
This is a NCC, disturbed cluster that reaches out to R500 . We see a very clear temperature
decrease in the outskirts, with probably the fastest decrease with radius out of the sample. Our
profiles match ACCEPT profiles and S13 pressure profiles.
This cluster has been studied extensively in all di↵erent wavelengths. Ma et al. (2009) found
that the cluster consists of four large parts using X-ray data and the positions of the galaxies.
Through lensing (Umetsu et al. 2016, Limousin et al. 2016, Diego et al. 2015, Medezinski et al.
2013), four large masses are also clearly found, reflecting that this cluster is a complex merger. It
has a very strong radio halo (Edge et al. 2003). One can also see sub-components through the SZ
e↵ect. Mroczkowski et al. (2012) studied this cluster through the SZ e↵ect, X-ray and lensing
observations. They also found very hot gas near 30keV in some regions of the cluster. Sayers et
al. (2016) measured the kinetic SZ signal coming from one of the subclusters.
14.0.36.
MACS J1423.8
This cluster is a CC, nondisturbed cluster Our profile can see the cool core, and possible
substructure in 3rd bin of temperature deprojections, along with a slight decrease in temperature
in the outskirts. Our profiles match with S13 and ACCEPT.
Adam et al. (2016) did a multi-wavelength analysis of the cluster, including X-ray surface
brightness and spectroscopy and high resolution SZ data to jointly constrain smooth ICM profiles,
comparing X-ray only results with the joint fit, and found a good agreement. This agreement can
be seen in our own study. They also explored the e↵ect of sub-mm and radio point sources on the
ICM profiles. Kotov et al. (2006) used Chandra X-ray spectroscopy to constrain the temperature
profile, which is roughly consistent with our results. It starts with a core temperature of 4 kev
and peaks at 8 keV before sloping down to 4keV again near R500 . Similarity, Morandi et al.
(2010) used Chandra data, and found a cool-core (3 keV core and 7keV peak at 300kpc).
LaRoque et al. (2003) used OVRO/BIMA SZ measurements, and calculated a SZ-derived
71
temperature for this cluster of about 13.6 keV +8.2,-2.5. The cluster has been found to be
slightly elongated and relaxed, with little substructure (Limousin et al. (2010) using lensing and
optical, Zitrin et al. (2011) using strong lensing).
14.0.37.
MS 0451.6
This is a NCC, disturbed cluster. Our densities and pressures are higher than ACCEPT by
nearly a factor of 10. The pressure profile is consistent with S13 pressure except for the last bin.
The temperature profile has some oscillations. An isothermal single-beta model best fit the data.
Donahue et al. (2003) used Chandra X-ray spectroscopy to find a temperature of 10.2 keV,
assuming an constant global temperature, which is within error of our isothermal temperature.
They also found an elliptical cluster morphology, and a shift in the BCG location and X-ray
centroid. Although they believe the cluster is not quite in hydrostatic equilibrium, it does not
display strong merger characteristics.
14.0.38.
MACS J0025.4
This is a NCC, non-disturbed cluster. The cluster has a relatively flat deprojected
temperature profile consistent with isothermal. The pressures are consistent with S13.
Although we classify this cluster as non-disturbed, other studies find that it is clearly major
merger (Bradac et al. 2008, Karaltepe et al. 2009), with two groups of galaxies of relatively the
same size. Bradac et al. (2008) found that the cluster mass distribution reflects this, and that one
can see that it is not in line with the gas distribution (seen from the X-ray), similar to the Bullet
Cluster. The deprojected temperature profile of the cluster ranges from 4keV in the center, to 8
keV at 80”, with some variation in-between. We see a similar dip in the temperature profile at the
same radius of 50”. Ma et al. (2010) did an optical study using Keck spectroscopy of the galaxies
and HST imagining to determine galaxy spectral type to describe the cluster.
14.0.39.
MS2053
This cluster is a NCC, disturbed cluster. We find a flat temperature profile ⇠4keV, and a
bumpy density profile in the deprojections.
It is the least massive cluster in our sample, with a M500 ⇡ 3 ⇥ 1014 M , and one of the
dimmest cluster in the SZ. The low mass coupled with distance makes the X-ray data very weak.
This makes it one of the few clusters that could not reach out to R500 . Verdugo et al. (2007) used
HST to find the mass distribution for MS 2053, and found a bimodal distribution indicating a
merger. They also find a elongation of the overall mass distribution.
72
14.0.40.
MACS J0647.7
This is a NCC, non-disturbed cluster. We have a good match with ACCEPT and S13
profiles, except for the first and last deprojected bins. We do not see a decrease in the
temperature at large radii. Our density profile is consistently higher than LaRoque et al. (2006).
Young et al. (2015) did high-resolution SZ measurements using MUSTANG. They found
that the MUSTANG image is consistent with that of Bolocam and X-ray images.
14.0.41.
MACS J2129.4
This is a NCC, disturbed cluster. In the temperature profile there seems to be a hint of a
cool-core in the deprojections. Our pressures are consistent with S13. Mann& Ebeling (2012)
conducted a classification of clusters based on morphology, using a combination of X-ray and
optical data. For this cluster they deduced it was a merger because of the X-ray centroid location
was significantly di↵erent from the BCG.
14.0.42.
MACS J0744.8
This cluster is a NCC, disturbed cluster. We do see a definite drop in temperature in the
outskirts. It matches well with both ACCEPT and S13. LaRoque et al. (2003) studied this
cluster using OVRO/BIMA and found an average temperature of around 17.9 keV. Korngut et al.
(2011) studied this cluster through the high resolution SZ images from MUSTANG, and compared
it with X-ray data. They found evidence of this cluster having a shock-front created by merging.
14.0.43.
MS1054
This is a NCC, disturbed cluster. There is an interesting dip in the temperature in the
second bin, along with last density bin being higher than LaRoque et al. (2006). We have good
agreement with S13, but a shallower slope than LaRoque et al. (2006) pressure in the outskirts.
Neumann et al. (2000) studied this cluster using the X-ray (ROSAT), finding substructure
and other evidences for recent merging processes (BCG and X-ray peak o↵set, X-ray centroid and
peak o↵set). Gioia et al. (2004) used XMM-Newton to find an average temperature of ⇠ 7keV
(perfectly consistent with our analysis), as well as a significant substructure of 2 clumps with
distinctive temperatures.
73
14.0.44.
CL J0152.7
This is a NCC, disturbed cluster. We find a flat temperature profile, and pressures
consistent with P13. The deprojected density profile is rather irregular, possibly due to the
disturbed nature of the cluster.
Huo et al. (2004) completed a joint optical and X-ray analysis of using Keck and Chandra,
respectively. In both bands they find two subclusters. They also find that the cluster follows
scaling relations for low-redshift clusters. Girardi et al. (2005) studied the velocity dispersion of
the galaxy cluster members, finding that the cluster must be merging due to the higher velocity
dispersion. Massardi et al (2010) used high resolution SZ data to find that the peaks of the SZ
and X-ray emission are o↵set, suggesting a merger. Maughan et al. (2006) used XMM Newton to
study the X-ray emission distribution in detail, finding many substructures and smaller groups,
concluding that the cluster has recently undergone many mergers.
14.0.45.
CL J1226.9
This is a NCC, non disturbed cluster. Our profiles provide a good match with ACCEPT
and S13 profiles. We found a flat temperature profile, although the central shell seems to be lower
than the rest of the cluster, however, the data seems to only robustly provide upper limits to the
profile.
XMM-Newton temperature maps of the cluster show departures from spherical symmetry,
suggesting merging (Maughan et al. 2007). Jee & Tyson (2009) additionally found evidence of
subclumps through weak lensing. Korngut et al. (2011) used the high resolution MUSTANG
instrument to find multiple peaks in the SZ map, further providing evidence that this cluster is in
fact a merger, despite our NCC denotation.
74
75
Fig. 15.— Density, temperature, and pressure profiles for Abell 2204. Black lines are the deprojected results, while blue lines are
the analytical fits +/- 1 . Green lines are ACCEPT X-ray studies (Cavagnolo 2009) and red lines are the Sayers 2013 pressure
deprojections. In each plot a vertical line denotes R500 for that cluster. For subsequent figures, apply this description.
76
Fig. 16.— Abell 383
77
Fig. 17.— Abell 209
78
Fig. 18.— Abell 1423
79
Fig. 19.— Abell 963
80
Fig. 20.— Abell 2261
81
Fig. 21.— Abell 2219
82
Fig. 22.— Abell 267
83
Fig. 23.— RX J2129.6
84
Fig. 24.— Abell 1835
85
Fig. 25.— Abell 697
86
Fig. 26.— Abell 611
87
Fig. 27.— MS2137
88
Fig. 28.— MACS J1931.8
89
Fig. 29.— AS1063
90
Fig. 30.— MACS J1115.8
91
Fig. 31.— Abell 370
92
Fig. 32.— CL 0024+17
93
Fig. 33.— MACS J1532.9
94
Fig. 34.— MACS J0429.6
95
Fig. 35.— MACS J2211.7
96
Fig. 36.— MACS J1720.3
97
Fig. 37.— MACS J0416.1
98
Fig. 38.— MACS J0451.9
99
Fig. 39.— MACS J0417.5
100
Fig. 40.— MACS J1206.2
101
Fig. 41.— MACS J0329.6
102
Fig. 42.— RX J1347
103
Fig. 43.— MACS J1311.0
104
Fig. 44.— MACS J0257.1
105
Fig. 45.— MACS J0911.2
106
Fig. 46.— MACS J2214.9
107
Fig. 47.— CL 0016
108
Fig. 48.— MACS J1149.5
109
Fig. 49.— MACS J0717.5
110
Fig. 50.— MACS J1423.8
111
Fig. 51.— MS 0451.6
112
Fig. 52.— MACS J0025.4
113
Fig. 53.— MS 2053
114
Fig. 54.— MACS J0647.7
115
Fig. 55.— MACS J2129.4
116
Fig. 56.— MACS J0744.8
117
Fig. 57.— MS 1054
118
Fig. 58.— CL J0152.7
119
Fig. 59.— CL J1226.9
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