Chandra analysis of a complete sample of galaxy clusters

Transcription

1 Chandra analysis of a complete sample of galaxy clusters Masterarbeit in Astrophysik von Gerrit Schellenberger angefertigt im Argelander-Institut für Astronomie vorgelegt der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn Oktober 2012

2 1. Gutachter: Prof. Dr. Thomas Reiprich 2. Gutachter: PD Dr. Jürgen Kerp

3

4 Contents 1. Introduction Clusters of galaxies Historical remarks Observational characterization of galaxy clusters HIFLUGCS X-ray Observations of galaxy clusters Important missions Cosmological context Basics Components of the Universe Inflation Cluster mass function Theoretical formalism Press-Schechter mass function Tinker mass function Observational mass function Data reduction Basics of X-ray data reduction CCDs Telescope ARF and RMF Pipeline Chandra - ACIS Background subtraction Data analysis Region selection Simulations Ellipse fitting algorithms Comparison of methods Profile fitting Surface brightness profile Temperature profile Total mass determination Gas mass determination Mean photon energy Cooling function Metallicity profile Deprojection analysis Gas components Results Observations Mass estimates Gas mass and total mass Example: Abell Isothermal cluster

5 Comparison with previous works Mass-Luminosity relation Cluster mass function Getting datapoints from masses Masses from ROSAT Masses from Chandra Gas mass fraction Ellipticities Discussion Total mass L x M tot relation Cluster mass function Absorption uncertainty Cross calibration with XMM-Newton Conclusion and outlook 87 A. Figures and Tables 89 A.1. List of Observations A.2. Neutral hydrogen column densities B. Cross calibration 95 B.1. XMM data reduction B.2. Selection B.3. Results B.3.1. Temperatures derived from Line-Ratio Bibliography 103

6 Abstract Since no larger virialized objects than galaxy clusters are known, they are perfect candidates to test cosmological models. The HIghest X-ray FLUx Galaxy Cluster Sample (HIFLUGCS) is a complete X-ray selected flux limited sample of 64 very bright and nearby galaxy clusters. I analyzed the existing Chandra data for all of these 64 clusters and modeled their temperature and surface brightness profiles with high precision. A proper background subtraction including a simultaneous fit to data from the ROSAT all-sky survey to model the components of the astrophysical background, was necessary. This gives individual total and gas mass estimates of a complete sample of galaxy clusters, which can be used to construct a cluster mass function. To obtain even more precise values, the clusters are not only treated as spherically symmetric objects, but also according to their apparent elliptical shape. The ellipticity was determined by using a non-iterative method from Buote and Canizares (1992), which calculates the moments of inertia. The total mass is then also calculated assuming a prolate ellipsoidal potential. After a deprojection analysis the total masses are used to construct an observational mass function, which is then fitted with the, from simulations constructed, Tinker halo-mass function to get cosmological parameters. The results are Ω ell m = for the elliptical regions and Ω sph m = for the spherical regions assuming σ 8 = 0.81 (Komatsu et al., 2011). The degeneracy between the two parameters increases with increasing Ω m, but it is also possible to constrain both parameters, e.g. for the elliptical non-deprojected regions: Ω m = σ 8 = A cross calibration using data from XMM-Newton of the HIFLUGCS clusters reveals that there exist systematic differences between the instruments and also the detectors which are most likely due to calibration problems. This introduces a bias for the cosmological parameters which is also quantified.

7

8 1. Introduction Galaxy clusters are the largest virialized objects in the Universe, so they are of high interest, especially for cosmology. Structure formation as well as the models of the whole Universe can be tested with these fascinating tools. So many questions are unsolved, like what is Dark Energy, how much Dark Matter is there, how will the Universe evolve in future? Looking at the largest objects out there is one of the most important steps to solve them. Currently roughly galaxy clusters are known, but even with best instruments currently available it is not possible to analyze all of them in detail. In this work I will focus on the 64 brightest and clearly visible clusters. Of course, they have already been analyzed in the past, but never all of them in such detail. This study is done in the X-ray regime, where the most massive and visible component of these giants, a very hot gas, is visible. The possibility of measuring the flux and temperature of this gas in such detail is now given, because high quality data of the Chandra satellite is available. The aim of this work is to analyze a complete sample of galaxy clusters and determine their total and gas masses individually. I will also account for the elliptical shape of each cluster individually and show the consequences for the resulting masses. Furthermore I will show how it is possible to construct a cluster mass function from this complete sample and constrain cosmological parameters. This chapter should give an overview about galaxy clusters and how they could be observed. In Chapter 2 the cosmoligcal context is explained and the details which are important for this work are investigated. Chapter 3 and 4 describe the methods, how I dealt with the data and which assumptions have to be made. The results are presented in Chapter 5 and being discussed in Chapter 6. The final conclusions are given in the last Chapter. The appendix consists not only of additional plots and tables, but also shows in Section B an important and detailed cross calibration with a different instrument, the XMM-Newton satellite. The results and consequences of this additional analysis are also explained in Chapter Clusters of galaxies As already mentioned galaxy clusters are the largest laboratories available. Their discovery, properties and impacts are explained within this Section and also the used sample is being introduced. But to stay in a chronological order, I will first start with the history of galaxy cluters Historical remarks In 1922 Ernst Öpik calculated the mass of and distance to the Andromeda galaxy using the velocity dispersion (Öpik, 1922). At this time, it was still known as the Andromeda nebula. The resulting distance of 455 kpc clearly identified this object as not belonging to the Milky Way. This can be regarded as the beginning of the extragalactic epoch of astronomy. But many years before this discovery, Max Wolf already found a nebulae concentration in the Perseus constellation (Wolf, 1906) and he studied this so called cluster in detail. Today, the Perseus Cluster is a well studied galaxy cluster, which consists of several hundreds of individual galaxies. A further important step was the publication of the Abell catalog (Abell, 1958), which lists 2712 galaxy clusters (1361 from the southern hemisphere added later) containing very bright galaxies. These clusters are still very important, because they are at low redshift and have a high richness (see Section 1.1.2). Apart from studies in the optical regime galaxy clusters were also detected by X-ray measurements. In 1966 diffuse emission around a galaxy in the Virgo Cluster (Byram et al., 1966) gave a first hint that galaxy clusters have to consist of more than only galaxies. Six years later, Uhuru, the first X-ray satellite, performed an all-sky survey and detected many sources outside the galactic plane, which were identified with known objects (Giacconi et al., 1972). From thereon galaxy clusters are regarded as bright X-ray sources with a spectrum corresponding to the emission of a very hot, i.e. ( ) K, optically thin gas. Until this time, some other important observations have changed the model of the Universe, like the cosmological redshift (Hubble, 1936), which implies an expanding Universe, and Fritz Zwicky s claim for Dark Matter by analyzing the velocity dispersion in galaxy clusters (Zwicky, 1933). Typical values for this velocity dispersion of σ v 10 3 km s, and thus crossing times of the order of 1 Gyr, lead to the assumption of relaxed clusters. This 5

9 CHAPTER 1. INTRODUCTION 1.1. CLUSTERS OF GALAXIES assumption is necessary for the calculation of the mass, e.g. with the virial equation, and so the introduction of additional matter components. As Smith (1936) suggested, this could also be internebular material. This problem was solved by analyzing X-ray observations of galaxy clusters. There one can estimate the mass of the intra cluster medium. It turned out that the dominant matter component is neither visible in the optical (i.e. emission of the individual galaxies) nor in X-rays, which means being not connected to the hot gas mentioned. So it is called Dark Matter ( Dunkle Materie ; Zwicky, 1933) and the origin and properties remained unknown. Only the gravitational interaction could and still can be measured and it was possible to quantify the amount of Dark Matter being one order of magnitude higher in mass than the hot gas and galaxies together. So to understand the evolution and dynamics of these objects, it is of extraordinary importance to have a special focus in every study on this Dark Matter component Observational characterization of galaxy clusters Galaxy clusters can be observed in nearly all wavelengths. Apart from the optical and X-ray regime already mentioned, they can be studied in the radio band, by analyzing so called Radio Lobes, Radio Halos and Radio Relics as well as AGNs, in the sub-millimeter band with the Sunyaev-Zeldovich effect and even in γ-rays by looking for Dark Matter annihilation or non-thermal emission. This makes it difficult to give a very clear definition of these objects. Usually they are called the largest virialized objects in the Universe. Superclusters are larger objects (second order clusters) and can also be bound, but not virialized, since the timescale to virialize is more than one order of magnitude higher ( yr) than for galaxy clusters. The definition from a cosmological point of view is discussed in Section 2, whereas here I focus on the observational aspects. Four main components are necessary to build a galaxy cluster (see e.g. Reiprich, 2001): Galaxies: Roughly big galaxies (without dwarf galaxies) are gravitationally bound to the cluster. An accumulation of much fewer galaxies is called galaxy group (Hickson, 1982). One very big galaxy is often located in the center of a cluster or a group. Intra cluster Medium (ICM): The ICM is the gas between the galaxies in the cluster. It is 5 10 times more massive than all the stars in all the cluster galaxies. It has a very high temperature of several 10 7 K, usually expressed in kev ( 1 kev k B K). More parameters of this medium are explained in Section Dark Matter: Adding up the mass of the galaxies and the ICM does not suffice to explain for example the velocity dispersions of galaxies in the cluster. So Zwicky was right when introducing an invisible matter component, although he didn t know about the ICM. Other measurements, like anisotropies in the Cosmic Microwave Background (CMB), support this arbitrary appearing solution. Apart from some candidates for a particle forming this matter component, still no explanation, of what Dark Matter is, was developed. Neutrinos are relativistic particles (Hot Dark Matter) and thus have a large free-streaming length, but they are ruled out as candidate because they predict a too large clustering scale of galaxies (White et al., 1983). Other popular candidates are Cold Dark Matter particles (e.g. Neutralino) only following gravitational and weak interactions. But all WIMP candidates haven t been observed yet. As galaxy clusters are apparently dominated by Dark Matter, they are perfect objects to study the nature of this phenomenon. Relativistic particles: These particles are characterized by a velocity only slightly lower than the speed of light. Radio observations show diffuse, large-scale synchrotron emission of relativistic electrons in galaxy clusters (Willson, 1970). These particles do not contribute significantly to the total mass, but they are still important to understand mixing processes in individual clusters. For this work, contributions from relativistic particles are neglected. Richness The richness is a parameter describing how many galaxies are associated with the cluster. Because not all galaxies can be counted, there exist some criteria used by different authors: Abell (1958) defined the richness as all galaxies that are not more than two magnitudes fainter than the third brightest galaxy and are within the (projected) Abell radius r A = 3h 1 50 Mpc. A few years later, Zwicky used a different definition which is, in contrast to the Abell definition, dependent on redshift. 6

10 1.1. CLUSTERS OF GALAXIES CHAPTER 1. INTRODUCTION Figure 1.1.: Example of a classification of X-ray sources with optical follow up information. Adopted from Giacconi and Burg (1993) Cool-core When one analyzes the central parts of a galaxy clusters, especially in X-rays, one will notice that due to the high density and metallicity of this region the gas is able to cool more efficiently. This is expressed for example in terms of the cooling time t cool = 3(n e + n i )kt, (1.1) ɛ where T is the electron temperature of the gas and n e and n i are the electron and ion number densitities, respectively. The emissivity ɛ is the luminosity per volume (see Equation 4.27). But observations have shown, that the gas in the center of galaxy clusters doesn t cool as far as expected from the cooling flow model introduced by Fabian (1994). It is now thought that there exists a heating mechanism that prevents the gas from cooling too much. A heating source could be AGNs inside the central galaxy of galaxy clusters, which are the most massive known black holes in the universe. These AGNs are also observed in the radio regime (Burns, 1990). For this analysis, the outer regions of galaxy clusters are more important, and therefore the effects of AGN feedback are not discussed in more detail. Finding galaxy clusters in X-rays Before one can create a sample of galaxy clusters for example selected in X-rays, catalogs have to be created using a specific algorithm. In contrast to optically selected samples, X-ray selected clusters are dynamically collapsed and can be detected as a single and extended object (see Giacconi and Burg, 1993). The decision tree in Figure 1.1 shows, what is necessary to classify a source from the ROSAT All-Sky survey (RASS) as a galaxy cluster: Source detection algorithms produced the RASS Faint Source Catalog (RASS-FSC) containing more than sources with a detection likelihood of at least 7 and at least 6 source photons (Voges et al., 2000). Almost bright sources (likelihood 15 and source counts 15) are listed in the RASS Bright Source Catalog (Voges et al., 1999). These sources are then reanalyzed (e.g. if they are extended). Extended sources with many optical counterparts are immediately classified as galaxy clusters (Fig. 1.1). But also point sources, i.e. sources, that cannot be resolved with the 1 point spread function of the RASS, can be galaxy clusters, if galaxies are known at that position HIFLUGCS The cluster sample used for the analysis is the HIghest X-ray FLUx Galaxy Cluster Sample (HIFLUGCS; Reiprich and Böhringer, 2002). It is X-ray selected from the ROSAT All-Sky Survey with a minimum flux of erg s 1 cm 2 in the ( ) kev band and a Galactic latitude b 20. Also the region of the Small and Large Magellanic Cloud as well as the Virgo Cluster region were excluded. In the end, the HIFLUGCS clusters were selected from deg 2, which means a sky coverage of about 65%. Because of the size and completeness, HIFLUGCS is the best currently available sample of local clusters. Nine catalogs with extended sources from the RASS were used to find cluster candidates. The final 64 clusters are from REFLEX (Böhringer et al., 2001), NORAS (Böhringer et al., 2000), NORAS II (Retzlaff, 2001) and BCS (Ebeling et al., 1998). For more details see (Reiprich, 2001, p ). After selecting candidates from these catalogs with a lower flux limit, all fluxes are redetermined with ROSAT PSPC pointing observations, where 7

11 CHAPTER 1. INTRODUCTION 1.1. CLUSTERS OF GALAXIES Figure 1.2.: Map in galactic coordinates (in Aitoff projection) showing the position of the 63 HIFLUGCS clusters without RXJ 1504 (filled circles). 11 clusters above the flux limit but with b < 20 are also added (open triangles). Image taken from Reiprich and Böhringer (2002). feasible, and then the previously mentioned flux limit was applied to end up with the HIFLUGCS clusters. Originally it consisted of 63 clusters, but one cluster (RXJ 1504) was included later, because it was initially believed to be strongly contaminated by AGN emission and the total flux was only slightly above the flux limit. With high spatial resolution data from Chandra it became clear that this cluster has to be included, so the sample comprises now 64 clusters (see Table 1.1). Table 1.1.: List of the 64 HIFLUGCS clusters. Column a) defines the name of the cluster as it is used in this work, in column b) an alternative name is given. Columns c) and d) give the coordinates in J2000 as defined in Hudson et al. (2010) and e) is the redshift taken from Zhang et al. (2011). The angular diameter distance D A is calculated using the redshift assuming Ω Λ = 0.7 and Ω m = 0.3 (see Section 2.1 for details). Cluster Alternative RA DEC z D A Name Name [Mpc] (a) (b) (c) (d) (e) (f) 2A RBS h38m35.3s +09d57m55s ABELL h41m37.8s -09d20m33s ABELL h56m21.4s -01d15m47s ABELL h02m39.0s -21d57m15s ABELL h52m50.4s +36d08m46s ABELL h57m56.4s +13d00m59s ABELL h57m38.6s +06d02m00s ABELL h58m57.0s +13d34m56s ABELL h13m20.7s +10d28m35s ABELL h33m37.1s -13d14m46s ABELL h21m24.2s +55d44m20s ABELL h08m50.1s -09d38m12s ABELL h36m51.3s -27d31m35s ABELL h44m29.5s +19d50m21s ABELL h57m14.8s -17d21m13s ABELL h58m46.2s -01d45m11s ABELL h59m23.0s -04d11m10s ABELL 1656 Coma Cluster 12h59m48.7s +27d58m50s ABELL h26m52.2s -27d06m33s ABELL h49m00.5s +26d35m07s ABELL h10m56.0s +05d44m41s ABELL h16m45.5s +07d00m01s ABELL h23m01.9s +08d38m22s ABELL h22m42.6s +27d43m21s ABELL h58m16.1s +27d13m29s ABELL h02m17.2s +15d53m43s ABELL h15m34.1s -06d07m26s ABELL h28m38.5s +39d33m06s ABELL h32m45.7s +05d34m43s ABELL h02m44.0s +34d02m48s ABELL h12m31.0s +64d05m33s ABELL h03m43.5s +78d43m03s ABELL h24m00.5s +16d49m29s ABELL h25m18.0s -12d06m30s ABELL h38m18.4s +27d01m37s ABELL h44m51.0s +09d08m40s ABELL h17m52.4s -44d14m35s ABELL h42m39.6s -53d37m50s ABELL h31m11.9s -61d24m23s ABELL h00m43.6s -40d03m00s ABELL h26m15.4s -53d40m52s ABELL h27m31.1s -54d23m58s ABELL 3526 Centaurus Cluster 12h48m51.8s -41d18m21s ABELL h27m54.8s -31d29m32s ABELL h33m31.8s -31d40m23s ABELL h47m28.9s -32d51m57s continued on the next page 8

12 1.1. CLUSTERS OF GALAXIES CHAPTER 1. INTRODUCTION 12 Redshift distribution of the 64 HIFLUGCS clusters 10 Clusters per bin Redshift Figure 1.3.: Redshift distribution of the 64 HIFLUGCS clusters. The binwidth is 0.01 Cluster Alternative RA DEC z D A Name Name [Mpc] ABELL h07m27.5s -27d01m15s ABELL h12m30.1s -56d49m00s ABELL h47m37.0s -28d07m42s ABELL h56m40.7s -34d40m18s EXO 0422 RBS h25m50.7s -08d33m25s HydraA ABELL h18m30.4s -12d15m40s MKW 03s 15h21m51.9s +07d42m31s MKW 04 12h03m57.7s +01d53m18s MKW8 WBL h40m43.1s +03d27m11s NGC h23m40.0s +33d15m20s NGC 1399 Fornax 03h38m29.1s -35d27m03s NGC h19m37.9s +02d24m36s NGC h42m49.8s +02d41m16s NGC h15m24.0s -16d23m08s RXJ WHL J h04m07.5s -02d48m16s S1101 ABELL S h13m58.5s -42d43m39s Zw III h41m17.6s +15d23m44s ZwCl h17m41.4s +03d39m32s The redshift distribution is shown in Figure 1.3. From that one can see that about half of the clusters are below redshift 0.05, which makes this sample a local sample of galaxy clusters. Many studies analyzed the HIFLUGCS clusters to characterise and understand the cool core phenomenon (e.g. Chen et al., 2007; Hudson et al., 2010). Hudson et al. (2010) found that 72% of the clusters in HIFLUGCS are cool core clusters and the best way to characterize cool core clusters is for low redshift clusters the central cooling time t cool. The L X T vir scaling relation was determined by Mittal et al. (2011) also using HIFLUGCS by analyzing the effects of AGN heating and ICM cooling. Finally, Zhang et al. (2011) analyzed in detail the L σ, L M gas and r 500 σ scaling relations X-ray Observations of galaxy clusters As already mentioned in Chapter galaxy clusters consist of a hot intra cluster medium (ICM). The density of this gas is very low (n gas < 0.1 cm 3 ) and decreases to the cluster outskirts. The thermal free-free emission and line emission of highly ionized heavy elements inside this optically thin, collisionally ionized and diffuse gas are the most important emission mechanisms to understand the X-ray emission of galaxy clusters. Important quantities To describe the emission in an appropriate way some physical quantities should be mentioned before, because they will also be used later on (see Rybicki and Lightman, 1979). General assumptions are an isotropic radiating source at distance d and conservation of energy. The luminosity L X is the power of emitted photon energy, E, L X = de dt. (1.2) Restricting it to an X-ray energy band of (0.5 2) kev yields typical values of galaxy clusters of erg s 1. 9

13 CHAPTER 1. INTRODUCTION 1.1. CLUSTERS OF GALAXIES The flux in X-rays, f X, is the energy of photons E passing through a surface A during a time interval t. f X = de da dt Typical values for galaxy clusters in HIFLUGCS are several erg s 1 cm 2. Knowing the distance, d, to a source, one can calculate L X from the measured flux by multiplying with the surface area of the sphere of radius d around the source (see also Equation 2.13). By measuring the solid angle Ω of a source, one can also define the surface brightness: S X = de dt da dω For practical usage it is also useful not to use energy for the surface brightness but the number of photons (phts) in a given energy range. Multiplying with the mean energy one can go back to the usual surface brightness in (1.4). The emissivity ɛ is the luminosity divided by the volume V of the source, (1.3) (1.4) ɛ = dl X dv. (1.5) All these quantities can also be divided by the bandwith ν and are then called specific (e.g. specific flux f ν ). Free-free emission When a charged particle is accelerated, photon emission is induced. Since the ICM is a hot plasma, electrons are accelerated all the time (changing their directions due to electromagnetic interaction) and nuclei can be neglected because of the much higher mass. For the exact derivation a quantum treatment is necessary. The difference from a classical treatment is usually expressed in terms of gaunt factor g ff (ω, ν), which is a function of the energy of the electron ω and the frequency of the emission ν. Detailed formulae and tables are given, e.g. in Karzas and Latter (1961). Including this quantum correction term to the classically derived emissivity one ends up with ɛ ff ν = 25 πe 6 ( ) 0.5 ( 2π 3m e c 3 Z 2 n e n i T 1 2 exp hν ) g ff, 3m e k B k B T e }{{} (1.6) =A with A = erg K 1 2 cm 3 s 1 Hz 1, (1.7) where Z is the ion charge, m e the electron mass, e the absolute value of the electron charge, n e and n i the electron and ion number densities and T e the electron temperature. For simplicity when calculating the mean energy I assume the gaunt factor to be constant with energy, so that it vanishes in Eq Since I haven t used the gaunt factor any more, this is the only assumption to be made. Of course it is also used later on for the calculation of the cooling function, but therefore it is evaluated by the APEC model for each energy. As one can see in Figure 1.4 this assumption is valid, because it varies only slightly with energy. The most important parameter for a spectrum from free-free emission is T e. The transition from the power-law regime at lower energy and the exponential drop at high energies is called the cut-off. It is located where the photon energy is equal to the energy of the electrons k B T e (Fig. 1.5). So a very good way to determine T e is to look for the exponential cut-off. Line emission The other important radiation process, that has to be accounted for, is the line emission. For hydrogen the Lyman and Balmer lines, which lie in the optical and UV regime, are famous examples. But for heavy elements the transition energies are shifted to the X-ray band (see Figure 1.6). For example iron has some important ion configurations: The Fe-L complex (electron transitions to main quantum state 2) is located around 1 kev and the Fe-K complex (electron transition to main quantum state 1) at roughly 6.8 kev. The latter complex can be subdivided into transitions from helium-like iron ions (FeXXV) and hydrogen-like iron ions (FeXXVI). The line emission of the helium-like ones is dominated by a line at 6.7 kev and reaches its peak level at a plasma 10

14 1.1. CLUSTERS OF GALAXIES CHAPTER 1. INTRODUCTION Figure 1.4.: Gaunt factors for three different electron temperatures as a function of the photon energy for a plasma with only hydrogen in it. Taken from Kellogg et al. (1975). Figure 1.5.: Free-free spectrum with exponential cut-off. Taken from Rybicki and Lightman, 1979, p

15 CHAPTER 1. INTRODUCTION 1.1. CLUSTERS OF GALAXIES normalized counts s 1 kev Energy (kev) Figure 1.6.: Two simulated spectra convoled with the Chandra effective area. Line emission was included only for one of them, whereas the temperatures are equal. temperature of 5.43 kev. Hydrogen-like iron is dominant at higher temperatures and reaches its peak at a temperature of kev with a line emission at 6.97 kev (values taken from AtomDB 1 ). The energy resolution of current X-ray detectors is sufficient to separate the emission of the two ions. Absorption X-ray radiation is blocked by the Earth s atmosphere and has to be observed from space using satellites. But not only the atmosphere absorbs the extragalactic X-ray emission, but also the inter stellar medium (ISM) of the Milky Way. To correct for this absorption one has to know the relative abundances of the heavy elements, the cross sections σ(e) of the elements as a function of photon energy and a tracer for the absolute abundance of one element (e.g. Hydrogen). The abundance of neutral hydrogen can be determined from radio surveys, like the LAB survey (Kalberla et al., 2005). Since the radio data used here has a lower angular resolution, this value from the survey is only an estimate and the actual hydrogen column density N H can locally be higher, e.g. due to clumping. When one knows the hydrogen abundance, it is possible to determine the other elements abundances with an abundance table (for more details see Chapter 4.6). The still very often used is Anders & Grevesse table (Anders and Grevesse, 1989). These tables are created with data from meteorites and observations of the solar photosphere. Differences between the sun and meteorites are important especially for iron. The absorption can be estimated by I abs (E) = I 0 (E) exp n Hσ(E), (1.8) where the photo-electric cross section σ(e) is needed to correct for the absorption. These cross sections are also given in publications like Morrison and McCammon (1983), which are used for the common xspec wabs model. Unfortunately, this model automatically uses an older abundance table (Anders and Ebihara, 1982). Figure 1.7 shows the contribution of the different elements to the total cross section. Other processes There are further processes that contribute to the continuum radiation: The radiative and dielectronic recombination and the two photon decay. All of them are negligible, but when they are detected it s a good

16 1.2. IMPORTANT MISSIONS CHAPTER 1. INTRODUCTION Figure 1.7.: Photoelectric absorption cross section as a function of energy normalized to one hydrogen atom. The scaling σ E 3 is only for presentation reasons. Taken from Morrison and McCammon (1983) test for the theory. Radiative recombination is also called free-bound transition. A photon gets captured by an ion and produces a photon with the energy equal the kinetic energy of the electron plus the binding energy of the ion. Dielectronic recombination (DR) is much rarer than radiative recombination (RR). The difference between DR and RR is that in case of DR the captured electron does not release a photon, but puts another electron of the atom into an excited state. This doubly-excited state of the atom state of the atom is turned into a singly-excited state by emitting a photon. Since the energy of the original electron must match the excitation energy it is a resonant process. Two photon transitions: There are transitions in atoms that are absolute forbidden and therefore the transition can only occur with the emission of two photons. Important transitions are for Hydrogen-like atoms 2s 2 S 1/2 1s 2 S 1/2 and for Helium-like atoms 1s2s 1 S 0 1s 2 1 S 0. For X-ray astronomy, these processes are even less important than recombination. Figure 1.8 shows the two recombination methods described above compared with free-free emission. Above a plasma temperature of 500 ev free-free emission is dominant Important missions Apart from the Chandra satellite, which is mainly used for this analysis, there are also other very important X-ray missions to be mentioned here. The first X-ray satellite was Uhuru, which scanned 95% of the sky with a much higher sensitivity than previous rocket experiments and detected 339 sources. Then, after HEAO-I and HEAO-B (also known as Einstein Observatory) and EXOSAT, which could record long-duration light curves, ROSAT was launched on the 1st of June 1990 with twice the effective area of Einstein. On board was an X-ray telescope with position-sensitive proportional counters (PSPC) sensitive especially to low energies 13

17 CHAPTER 1. INTRODUCTION 1.2. IMPORTANT MISSIONS Figure 1.8.: Cooling rates for different plasma temperatures. B is free-free emission, L is inverse Compton effect, RR and DR are radiative and dielectronic recombination, respectively. Line emission is not included here. Image from Tucker and Gould (1966). < 2 kev and a field of view of 2. For six months ROSAT performed the first imaging all-sky survey, which is still commonly used, especially to analyze astrophysical background components. After that, some other missions were launched like ASCA, RXTE and BeppoSAX. A breakthrough was the launch of Chandra and XMM-Newton, both in 1999, because both of them (together with Suzaku) provide until today the best instruments to observe galaxy clusters in X-rays. Chandra (Weisskopf et al., 2000) has an angular resolution of and a large effective area. It was placed in a high Earth orbit with a high eccentricity, most of the time being out of the Earth s magnetosphere. The instruments on board are the Advanced CCD Imaging Spectrometer (ACIS) consisting of 10 CCD chips. The High Resolution Camera (HRC) is a microchannel plate with a larger field of view than ACIS. Finally two diffraction gratings, the Low Energy Transmission Grating (LETG) for energies of ( ) kev used with the HRC and the High Energy Transmission Grating (HETG) for energies of (0.4 10) kev used with the ACIS, can produce high resolution spectra. XMM-Newton (Jansen et al., 2001) has many mirrors, which gives an effective area four times that of Chandra. XMM is also in an elliptic orbit and has three imaging cameras (EPIC), where each of them has its own telesope. One of them (PN) uses a new type of CCD which has a higher quantum efficiency, especially at higher energies, for details see Strüder et al. (2001). Suzaku (Mitsuda et al., 2007) is a Japanese/U.S. mission launched in 2005 with imaging CCD detectors and a calorimeter which was damaged shortly after the launch and is now unusable. Suzaku is in a low-earth orbit and therefore has a low instrumental background. Important missions are summarized in Table

18 1.2. IMPORTANT MISSIONS CHAPTER 1. INTRODUCTION Mission Dates Main Energy range Maximum Best spatial Comment Detector effective area resolution kev cm 2 arcsec Uhuru PC First X-ray satellite HEAO A Flux-limited survey & X-ray Background study Einstein IPC first imaging satellite ROSAT PSPC All-sky survey ASCA SIS First CCD satellite & good spectral resolution Chandra 99- ACIS Best spatial resolution XMM 99- EPIC-PN High eff. area over large energy band Suzaku 05- XIS Low instrumental background Table 1.2.: Important missions and the properties of selected detectors on board. Adopted from Seward and Charles (2010). 15

19

20 2. Cosmological context Cosmology is a part of astronomy that deals with the Universe as a whole. It bases on Einstein s field equations and analyzing the large scale structure in the Universe. These theoretical formulations are important to understand the impact of the results achieved by the galaxy cluster analysis. In Section 2.1 the main ideas of the standard model of cosmology are summarized. Section 2.2 briefly describes the concept of inflation and Section 2.3 shows how to use and deal with the cluster mass function. Useful reviews are Peacock (1999) and Peebles and Ratra (2003) Basics Two important principles, which are commonly accepted as a basic building block for cosmology are homogeneity and isotropy. Isotropy means, that, on average, observational results do not depend on the direction. Homogeneity in a cosmological sense means, that the density is constant in space. Connecting isotropy with the Copernican principle (Earth is not a special location in the Universe) directly leads to homogeneity. But when considering expanding Universes homogeneity does not necessarily mean isotropy (density is constant but Universe expands differently into the different directions). Therefore the only expansion allowed in an isotropic Universe is the radial expansion. It is then possible to define a parameter H (called the Hubble constant) that describes the expansion at a certain distance without depending on the direction: ṙ = Hr. (2.1) The next important step is to measure distances in the Universe using a metric which is consistent with Einstein s field equations. This was done by Walker (1935) and Robertson (1935) and the resulting metric ds 2 = c 2 dt 2 a 2 (t) ( dω 2 + f 2 K(ω)(dθ 2 + sin 2 θdψ 2 ) ) (2.2) is called Robertson-Walker metric. t is the cosmic time, a(t) = r(t) r(t 0) is the scale factor normalized today s value (t 0 ) and ω is the comoving radial coordinate. f K is a function depending on the space curvature K: f K (ω) = 1 K sin ( Kω ) K > 0 ω K = 0 1 K sinh ( Kω ) K < 0, where K 1/2 is the curvature radius and the dimensions of K are length 2. K > 0 describes a closed Universe with finite size but without boundaries, K < 0 is the open case and K = 0 is a flat Universe with infinite size. Important to mention is that the redshift z, which is the relative change between the observed and emitted wavelength of a photon, can be translated to scale factor at the time when the photon was emitted (2.3) z = λ obs λ em λ em = 1 a(t em ) 1, (2.4) regardless what cosmological model for the Universe is used. The evolution of the scale factor a(t) is described by the Friedmann equations H(t) 2 = ȧ(t)2 a(t) 2 = 8πG Kc2 ρ(t) 3 a(t) 2 (2.5) ä(t) a(t) = 4πG 3c 2 (ρ(t)c2 + 3p(t)). (2.6) To arrive at (2.6), energy conservation was used, which should be fine locally. ρ denotes the energy density and p the pressure of all components. Equation 2.5 can be used to calculate a critical density ρ crit, which defines 17

21 CHAPTER 2. COSMOLOGICAL CONTEXT 2.1. BASICS Figure 2.1.: Illustration of properties for different models of Universes. Left: Evolution of the scale factor a with time for an open, closed and flat matter dominated Universe (R is a characteristic radius). Right: Certain combinations of Ω m0 and Ω vac0 build up different Universes. Both taken from Peacock (1999). the borderline between a collapsing and forever expanding Universe with only matter and radiation: ρ crit (z) = 3H(z)2 8πG = E(z)2 ρ crit,0, (2.7) where E(z) is the evolution of the Hubble parameter H(z) = H 0 E(z). According to WMAP 5-year data (Komatsu et al., 2009) H 0 has a value of roughly 70.5 ± 1.3 km s 1 Mpc 1 and since this value changed a lot in the past one usually writes H 0 = h 100 km s 1 Mpc 1 and expresses all quantities in terms of h. The density ρ is a combination of all contributors (pressure-less mass: m, radiation: r, vacuum energy: vac) and so one can assume that it can be split up. Ω(z) = ρ(z) ρ crit (z) = ρ m(z) + ρ r (z) + ρ vac (z) = ρ m(z) + ρ r (z) + ρ vac (z) ρ crit (z) E(z) 2 = Ω m (z) + Ω r (z) + Ω vac (z). (2.8) ρ crit,0 For practical usage, it is convenient to normalize the densities to the critical density of the Universe. All the quantities in (2.8) depend on redshift, but the evolution can be easily resolved: ρ m = ρ m,0 a 3 Ω m = Ω m0 a 3 E(z) 2, (2.9) a 4 ρ r = ρ r,0 a 4 Ω r = Ω r0 E(z) 2, (2.10) ρ vac = ρ vac,0 Ω vac = Ω vac0 1 E(z) 2, (2.11) where 0 denotes the value today. Using (2.5),(2.7) and (2.8) one finds Ω 0 1 = Kc2 H which can then all plugged o 2 into (2.5) to end up with the most common form of the first Friedmann equation, which shows the evolution of all the energy density parameters: H 2 (a) = H0 2 [( Ωvac + Ω m a 3 + Ω r a 4) E(z) 2 (Ω 0 1)a 2] = H0 2 ( Ωvac0 + Ω m0 a 3 + Ω r0 a 4 (Ω 0 1)a 2). (2.12) With this equation, models can be calculated and a few of them are shown in Figure 2.1. It becomes clear that independent of a Universe being spatially open, flat or closed, it is possible to expand forever depending on the Ω vac0 value. If Ω vac0 is negligible small compared to Ω m0 but still 0, it is also possible that the Universe recollapses. Only in a matter dominated Universe the open, flat or closed status determines the evolution of 18

22 2.1. BASICS CHAPTER 2. COSMOLOGICAL CONTEXT the Universe. Also different distance measurements are possible. On cosmological scales, the distances obtained by measuring a flux f and relating it to the luminosity L of the object (luminosity distance) L D L = 4πf, (2.13) or by comparing the apparent (angular) Θ and real extent A of an object perpendicular to the line of sight (angular diameter distance) D A = A Θ, (2.14) differ. The comoving distance is the distance between two objects without the effect of the expansion of the Universe and is found by setting ds = 0 in (2.2). For a flat Universe this simplifies to D C = z 2 z 1 c dz, (2.15) H(z) where z 1 and z 2 are the redshifts of the objects, whose distance is to be measured. The different distance measurements are related in a flat Universe as follows: D A = D C 1 + z, (2.16) D L = D C (1 + z), (2.17) D A = D L (1 + z) 2. (2.18) Figure 2.2 shows a comparison of the different distance measurements as a function of redshift for two different cosmologies. From now on for simplicity and consistency with literature the normalized density parameters Ω x refer to the values today Ω x0 (e.g. from now on Ω m means Ω m0 ) and redshift dependend density parameters are explicitly marked like Ω x (z) Components of the Universe One important equation to describe the components of the Universe is the fluid equation ( p ) ρ = 3ȧ a c 2 + ρ, (2.19) which can be derived by subtracting (2.6) from the time derivative of (2.5). For each component there can now be defined an equation of state (EOS) w = p c 2 ρ. (2.20) For radiation and relativistic particles w = 1 3, for pressureless matter w = 0. The dimensionless curvature term k = K R0, 2 where R 0 is the present not normalized scale factor, has a w = 1 3. It is important to note that the normalized density parameter for the curvature is defined as Ω k = kc2 H 2 0 r2 0 Plugging (2.20) into (2.19) one obtains the differential equation which can be solved easily for constant w: Equation (2.23) explains now the dependencies of (2.9) - (2.11). = Ω 0 1 (2.21) ρ ρ = 3ȧ (1 + w), (2.22) a ρ = ρ 0 a 3(1+w) (2.23) 19

23 CHAPTER 2. COSMOLOGICAL CONTEXT 2.1. BASICS Figure 2.2.: Comparison of different distance measurements as a function of redshift. Solid lines correspond to Ω m = 0.3 and Ω Λ = 0.7, dashed lines represent Ω m = 0.5 and Ω Λ = 0.5. Black lines (lowermost pair) are angular diameter distances, blue comoving (middle pair) distances and red (uppermost pair) luminosity distances. Created using icosmo 1.2, see Refregier et al.,

24 2.1. BASICS CHAPTER 2. COSMOLOGICAL CONTEXT Figure 2.3.: Fitted Dark Matter density profiles for a flat ΛCDM cosmology. Dashed line represents the high-mass halo, the solid line the low-mass one. Image adopted from Navarro et al. (1997). Dark Matter The ρ m term should represent all matter components without pressure in the Universe. Of course, this is the baryonic matter (mainly protons, neutrons). But the previously mentioned Dark Matter is also included: Ω m = Ω bary + Ω dm. Sometimes also baryonic matter that is not visible (MACHOS - massive compact halo objects (Alcock et al., 1993), e.g. brown dwarfs or black holes) is counted as a constituent of Dark Matter, but less important. Non-baryonic matter (e.g. WIMPS - weakly interacting massive particles) is currently the best answer to the question, what Dark Matter is. But again, this can be subdivided into hot Dark Matter (HDM), which means particles with relativistic velocities (e.g. neutrinos) and cold Dark Matter (CDM), see Blumenthal et al. (1984), with currently only hypothetical particles. CDM seems to be a much better candidate, since it is per definition non-relativistic and able to cool and collapse, while HDM smoothes out structures on small scales. Also HDM in terms of neutrinos (scalar and Dirac neutrinos within a certain mass range) are much too rare to explain the mass needed for the Dark Matter halos (see Freese, 1986). From N-body simulations of cold Dark Matter a simple density profile could be successfully fitted to the simulated data. This profile is now known as the NFW profile (see Navarro et al., 1996) ρ(r) δ c = ρ crit (r/r s )(1 + r/r s ) 2, (2.24) where r s = r 200 /c is a characteristic radius and ρ crit the critical density of the Universe and δ c = c 3 ln(1 + c) c 1+c, (2.25) where c is the concentration parameter. I will use c = 4 when applying the NFW profile. This profile shows a lower density for massive Dark Matter halos in the center then for low-mass ones (see Fig. 2.3). A very famous example for observational detection of Dark Matter is the bullet cluster (1E ) at redshift z 0.3, which shows a merging event of two clusters. As one can see in Fig. 2.4, the baryonic matter (black image) lacks behind the Dark Matter because of collisions, which is also indicated by the clearly visible cold front. The Dark Matter is collisionless and shows no shocks and directly passes through the other cluster. Dark Energy The previously used vacuum term could also be interpreted as a cosmological constant Λ, which is a kind of engine for the accelerated expansion of the Universe. A constant density for the Λ-term makes it necessary 21

25 CHAPTER 2. COSMOLOGICAL CONTEXT 2.2. INFLATION Figure 2.4.: Bullet cluster. Red contours are from weak lensing and traces the total matter. The black image is a Chandra observation and traces the visible matter. Image taken from Clowe et al. (2007). that w = 1. Models for the Λ-term with w 1 are no longer called cosmological constant, but Dark Energy. Equation (2.6) states that for an accelerated expansion ä > 0 and therefore w < 1 3. Since currently no one knows anything about the origin of the Dark Energy, it could also have a more general EOS: w = w(a). This would then mean: ρ DE = ρ DE0 exp 3 z w(z )d ln(1 + z ) (2.26) The consensus concordance cosmological model ΛCDM is commonly used and explains Dark Energy as a small vacuum energy which would be equivalent with a cosmological constant. A third explanation of Dark Energy would be, that Einstein s field equations are no longer valid on cosmological scales Inflation To develop a model for the formation of galaxy clusters, one has to go back to the epoch of inflation. Inflation was introduced to solve problems that arise with an expanding Universe: The horizon problem describes the temperature consistency of the cosmic microwave background (CMB) of large regions of the sky that were not in causal contact at this epoch (z 1100). When assuming Ω = 1 the flatness problem arises because this model is not stable to perturbations. Other problems (antimatter problem, structure problem,...) are well explained in Peacock (1999). A possible solution, especially to the first two problems, was suggested by Guth (1981). He (and some other authors like Sato, 1981) proposed a flat scalar field (inflaton) which should force an accelerated expansion of the Universe in the very early phase of the Universe by having an equation of state with negative pressure. So it would act like a Dark Energy. The expansion can be approximated by the de-sitter solution a e H Λt, (2.27) where H Λ is a constant. As a consequence of this expansion, primordial quantum fluctuations were turned into density fluctuations which grew up to galaxy clusters Cluster mass function With the theory of structure formation it is now possible to predict the number of galaxy clusters with a certain mass for given cosmological parameters. A very often used formalism is the Press-Schechter mass function (Press and Schechter, 1974). A summary of its usage is given in Reiprich (2001). 22

26 2.3. CLUSTER MASS FUNCTION CHAPTER 2. COSMOLOGICAL CONTEXT Theoretical formalism The underlying idea is that very small perturbations to an initial density field are produced at very early times. If the perturbation is large enough, gravity forces it to collapse. The earliest imprint of these density fluctuations are the inhomogeneities of the Cosmic Microwave Background (CMB). Density fluctuations measured by the density contrast ρ( r, t) δ( r, t) =, (2.28) ρ(t) where ρ is the mean matter density, can be linearized as long as δ( r) is small ( 1). Starting from the conservation of matter (continuity equation), the description of motions in fluids (Euler equation) and the Poisson equation, one can derive a differential equation for the density contrast 2 δ( r, t) t 2 + 2ȧ a The general solution can be decomposed into δ( r, t) t 3H2 0 Ω m 2a 3 δ( r, t) = 0. (2.29) δ( r, t) = D + (t) + ( r) + D (t) ( r), (2.30) where D ± (t) describe the two time dependent solutions. D (t) is decaying (in an Einstein-de Sitter Universe D = a 3 2 ) and not important for later times. For a Dark Energy EOS with w = 1, D + (t) can be solved analytically: D + (t) 5 2 Ω mh(a)h 2 0 a 0 1 (a H(a )) 3 da. (2.31) I define the constant in (2.31) such that D + (t 0 ) = 1 (see also Schneider, 2009), where t 0 denotes today: a H(a) 0 D + (t) = 1 H (a H(a )) 3 da (2.32) 1 da (a H(a )) 3 To see how a linear overdensity δ v c would grow from the time of virialization until today one simply neglects D (t) in (2.30) and identifies + ( r) with the size of the overdensity today δ c : δ c (z) = δv c (z) D + (z). (2.33) Solving (2.32) numerically leads to the result, that the growth factor also depends on the nature of dark energy, which can be seen in Fig A smaller w parameter for the Dark Energy equation of state results in a faster growth of structure. This is easily understandable since a lower value of w means a higher importance of the component today and makes it less important at early times. For a flat universe this results in a larger Ω m at early times and the expansion of the Universe is slowed down at early times (due to a high w). Because larger Ω m and the smaller Ω DE (at early times) structures could grow faster. The right plot of Fig. 2.5 shows that for a cosmological constant nature of Dark Energy structures formed at z = 1100 grew approximately by a factor of It is important to note here that the density fluctuations would have grown until δ without Dark Matter, since we observe fluctuations in the baryon density in the CMB of the order of at z = So, within this theory, structure formation until today without Dark Matter would not be possible. The next important quantity to be determined is the linear overdensity δc v of a structure at the time of virialization. Linear theory predicts (e.g. see Kitayama and Suto, 1996) that δ v c = { 3(12π) 2/ , for Ω m = 1 and Ω 0 = 1 2(12π) 2/3 20 [ log 10 Ω f (z)], for Ω m < 1 and Ω 0 = 1, (2.34) where Ω f (z) = Ω m (1 + z) 3 Ω m (1 + z) 3 + (1 Ω m Ω Λ )(1 + z) 2 + Ω Λ. (2.35) 23

27 CHAPTER 2. COSMOLOGICAL CONTEXT 2.3. CLUSTER MASS FUNCTION D + (z) Growth Factor Ω DE = 0.7 Ω M = 0.3 H 0 = 71 km/s/mpc σ 8 = 0.8 w = -1 w = -4/3 w = -1/3 D + (z) Ω DE = 0.7 Ω M = 0.3 Growth Factor w = H 0 = 71 km/s/mpc 0.2 σ 8 = z z Figure 2.5.: Normalized growth factor as a function of redshift. Left: Comparison for different EOS of Dark Energy. Right: Growth of structures from CMB creation to today for a cosmological constant cosmology linear overdensity for virialization δ c v (z) σ 8 = H 0 = 71 km/s/mpc Ω M = 0.3, Ω Λ = 0.7 Ω M = 1, Ω Λ = 0 Ω M = 0.5, Ω Λ = z Figure 2.6.: Linear overdensity at the time of virialization: Numerical solution to (2.34) plotted as a function of redshift for different cosmologies. 24

28 2.3. CLUSTER MASS FUNCTION CHAPTER 2. COSMOLOGICAL CONTEXT Figure 2.7.: Comparison of Transfer functions for different models. The T k 1 behaviour is only present for adiabatic models. Taken from Peacock, The mean non-linear density inside this virialized sphere is 178 times the mean density for a flat universe with Ω m = 1. This value (or 200 as an approximation) is often used to define the virial radius of an object like a galaxy cluster. The cosmological parameters have only a minor effect on the value of δc v, as it can be seen in Fig Finally it is important to know how many clusters can develop from the initial density field within a sphere of radius R. This is simply counting how many fluctuations in the field exceed the linear density contrast threshold and therefore collapse. Assuming the density field to be Gaussian with a probability density one can use a top hat filter which is in k space p = 1 2πσ(R, t) exp W (x) = W (kr) = { 3 4πR 3 ( δ 2 ) 2σ 2 (R, t), for R < x 0, for R x, (2.36) (2.37) 3(sin(kR) kr cos(kr)) (kr) 3, (2.38) to select the amplitude of density fluctuations. Present density fluctuations are often parameterized by the RMS with a top hat filter of 8h 1 Mpc σ 8 = σ(8h 1 Mpc) 1, (2.39) where σ(r) 2 is the variance of the smoothed density field σ(r) 2 1 = W (kr) 2 P (k) d 3 (2π) 3 k, (2.40) with a power spectrum P (k). The power spectrum itself is proportional to the square of the transfer function T (k) 2 times the wavenumber k. The transfer function corrects for the different evolution of sub and super horizon perturbations. After entering the horizon during the radiation dominated phase, perturbations didn t grow any more until the matter dominated epoch started, whereas super horizon perturbations always grow. The Dark Matter models in Fig. 2.7 show that small perturbations (large k) are suppressed during the radiation dominated phase of the Universe. The turnover point corresponds to the horizon size when matter and radiation were equally dominant. Equation (2.40) can be rewritten in terms of σ 8 0 σ(m) 2 = σ8 2 k 2+n T (k) 2 W (kr(m) 2 dk k 0 2+n T (k) 2 W (k h 1 8 Mpc) 2 dk. (2.41) 25

29 CHAPTER 2. COSMOLOGICAL CONTEXT 2.3. CLUSTER MASS FUNCTION 1e-05 1e-06 1e-07 Power Spectrum n = 1 n = 1.05 n = e-08 P(k) 1e-09 1e-10 1e-11 1e-12 T 0 = K Ω M = 0.3 H 0 = 71 km/s/mpc 1e k Figure 2.8.: Power spectrum for different spectral indices. The power spectrum P (k) was assumed to have a primordial spectral index n = 1, which means P (k) kt (k) 2, (2.42) and is known as the Harrison-Zeldovich spectrum. It is plotted in Fig Other spectral indices slightly scale the power spectrum for larger k. For the transfer function the fitting formula from Bardeen et al. (1986) was used ln( q) [ T (q(k)) = q + (16.1q) 2 + (5.46q) 3 + (6.71q) 4] 1/4, (2.43) 2.34q with ) 2 k ( T K q(k) = ( h 2 Ω m exp Ω b h Ω b 0.5 Ω m ) (2.44) taken from Sugiyama (1995). T 0 was set to K and the baryon density Ω b h 2 = taken from Burles and Tytler (1998). A more detailed transfer function was derived by Eisenstein and Hu (1998) including the effects of baryonic acoustic oscillations, Compton drag, velocity overshoot, baryon infall, adiabatic damping, Silk damping and cold Dark Matter growth suppression. For a non-zero Ω baryon the baryonic oscillations can be clearly identified in this model Press-Schechter mass function According to Press and Schechter (1974) the cluster mass function, the number of clusters per comoving volume and per mass, can now be established ( dn(m) 2 dm = ρ 0 δ c (z) dσ(m) π M σ(m) 2 dm exp δ c(z) 2 ) 2σ(M) 2. (2.45) A factor of 2 is already introduced to normalize it to 1. ρ 0 is the mean matter density of the Universe today ρ 0 = Ω m h M Mpc 3, (2.46) which is equal to the critical density today ρ crit,0 for a flat matter dominated Universe. Some examples in Fig. 2.9 show the calculated cluster mass functions for different cosmological parameter values and redshifts. The upper left plot in Fig. 2.9 shows the density of clusters as a function of mass in a ΛCDM Universe, whereas the upper right plot shows the same in an EdS Universe. One can see that in an EdS Universe the abundance of high mass clusters is much higher than in a ΛCDM Universe for the same σ 8. Modifications of the EOS of the Dark Energy have a very minor effect on the cluster mass function in the local universe (lower left plot). Finally the lower right plot illustrates the dependence of the cluster mass function on σ 8 : A higher value of σ 8 means a wider spread of density fluctuations and therefore more clusters with a higher mass but less with a lower mass. 26

30 2.3. CLUSTER MASS FUNCTION CHAPTER 2. COSMOLOGICAL CONTEXT Cluster / Mpc 3 / M 1e-16 1e-18 1e-20 1e-22 1e-24 1e-26 1e-28 Ω Λ = 0.7 Ω M = 0.3 H 0 = 71 km/s/mpc σ 8 = 0.8 CMF z=0 z=0.01 z=0.1 z=0.5 z=1 Cluster / Mpc 3 / M 1e-15 1e-16 1e-17 1e-18 1e-19 1e-20 1e-21 1e-22 Ω Λ = 0 Ω M = 1 H 0 = 71 km/s/mpc σ 8 = 0.8 CMF z=0 z=0.01 z=0.1 z=0.5 z=1 1e-30 1e+13 1e+14 1e+15 1e-23 1e+13 1e+14 1e+15 Mass in M Mass in M 1e-16 1e-17 CMF w = -1 w = -1/3 w = -4/3 1e-16 1e-17 1e-18 CMF σ 8 = 0.8, z = 0.01 σ 8 = 0.9, z = 0.01 σ 8 = 0.8, z = 0.5 σ 8 = 0.9, z = 0.5 Cluster / Mpc 3 / M 1e-18 1e-19 1e-20 1e-21 1e-22 Ω DE = 0.7 Ω M = 0.3 H 0 = 71 km/s/mpc σ 8 = 0.8 z = 0.1 Cluster / Mpc 3 / M 1e-19 1e-20 1e-21 1e-22 1e-23 1e-24 1e-25 Ω Λ = 0.7 Ω M = 0.3 H 0 = 71 km/s/mpc 1e-23 1e+13 1e+14 1e+15 1e-26 1e+13 1e+14 1e+15 Mass in M Mass in M Figure 2.9.: Cluster mass functions for different cosmological parameters and redshifts. The Dark Energy EOS w is 1 if not stated otherwise (lower left plot). 27

31 CHAPTER 2. COSMOLOGICAL CONTEXT 2.3. CLUSTER MASS FUNCTION Tinker mass function But there also exist other approaches to evaluate the cluster mass function for a given cosmology, like the Tinker mass function described in Tinker et al. (2008). This mass function is established from a fit to distributions of Dark Matter halos detected in collisionless numerical simulations. The cluster mass function of the form with dn(m) dm = f(σ) ρ 0 dσ(m) M dm, (2.47) [ (σ ) ] a ( f(σ) = A + 1 exp c ) b σ 2, (2.48) was used. The parameters A,a,b and c, their second derivatives with respect to σ and their redshift evolution are given in tables for certain overdensities. These parameters are interpolated using the spline interpolation given in Press et al. (1992) for other overdensities than the 9 given in Tinker et al. (2008). The Tinker mass function provides a much better description than the Press-Schechter one, because it is evaluated from WMAP1 simulations Observational mass function To construct a mass function from the observed total masses, one has know a specific comoving volume V max,i of each cluster. This volume, V max, (L) = ω ( ) 3 DL,max, (2.49) z max is the maximum volume inside which a cluster with a given luminosity L (or mass) can be found within a survey with a given flux limit and sky coverage ω. D L,max is the maximum luminosity distance D L,max = L SF [( ) kev] 4πK(T gas, z max ) f OF lim( ) kev (2.50) and z max the corresponding redshift. K(T gas, z) is the K-correction term to correct for the different luminosities L in the observer (OF) and source rest frame (SF), and f OF constructed, lim( ) kev K(T gas, z) = LSF [( ) kev] L OF [( ) kev], (2.51) is the survey flux limit in the observer frame. Then, the cluster mass function can be dn(m) dm = 1 M N i=1 1 V max,i. (2.52) For more details see Reiprich (2001); Schechter (1976). For the HIFLUGCS sample the volumes V max are taken from Reiprich (2001) (for a cosmology with Ω m = 1, Ω Λ = 0 and h = 0.5) and listed in Tab Using z max the maximum volume for a new cosmology can be calculated [ ( 4πD 3 H D C D 2Ω k D H 1 + Ω 2 Ωk ) ] C k 1 D D 2 C H Ωk D H 4π V max = 3 D3 C, for Ω k > 0, for Ω k = 0 (2.53) [ 4πD 3 H D C 2Ω k D H D 1 + Ω 2 C k 1 D 2 H ( Ωk ) ] arcsin D C Ωk D H, for Ω k < 0, where D C is the comoving distance as defined in 2.15 for the maximum redshift and D H = c distance, see Carroll et al. (1992). H 0 is the Hubble 28

32 2.3. CLUSTER MASS FUNCTION CHAPTER 2. COSMOLOGICAL CONTEXT Cluster V max z max Cluster V max z max Name [10 8 Mpc 3 ] Name [10 8 Mpc 3 ] 2A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A EXO A HydraA A IIIZw A MKW A MKW A MKW A NGC A NGC A NGC A NGC A NGC A RXCJ A S A ZwCl Table 2.1.: Maximum comoving volume for a EdS universe with h = 0.5 for the HIFLUGCS clusters. The V max values are taken from Reiprich (2001) and z max is calculated. 29

33

34 3. Data reduction This Chapter summarizes the important steps for the X-ray data reduction, which are necessary for this analysis. I first have to explain the main properties of the telescope and the detector and will then deal with the treatment of the data products themselves. One of the most important steps, the background subtraction, is described at the end of this Chapter Basics of X-ray data reduction Current X-ray missions use Charge Coupled Devices (CCDs) to simultaneously detect the position, energy and the arrival time (within the uncertainty of the readout time) of an incoming photon. It is therefore necessary to first explain the main properties of these instruments CCDs In optical astronomy CCDs are used since the 1980s, because they are much more efficient than previous devices. Some X-ray CCDs are very similar to the ones used in the optical wavelengths, but with a thicker depletion layer to detect higher energy photons. When a photon hits a pixel of the CCD a charge, comparable to the photon energy, is deposited there. After a certain time the chip is read out by shifting the charges over the detector. Since X-ray photons are much rarer than optical photons and because the readout time is short, one can assume that in each pixel during each read out at maximum only one photon was detected. So the charge can be translated into the photon energy. Of course, a correction for the so called charge transfer inefficiency (CTI) has to be made, which is also dependent on the detector position (see Fig. 3.1). For bright (point) sources there are sometimes more than one photon per pixel detected during the readout intervall, but only recognized as a single event with a very high energy. This effect is called pile-up and can be accounted for in the final spectral analysis. Usually pile-up effects are not present for source rates below 0.01 cnts px 1 s 1 in case of ACIS of Chandra. A true spectrum (solid line) and a measured spectrum (dotted line) affected by the pile-up effect is shown in Fig As expected, higher energies are reduced and the low energy flux is raised. But for this work, pile-up effects should not be a problem since I m dealing with extended sources Telescope To focus X-ray photons onto the detector, mirrors (metal surfaces) are placed that the photons hit them in a small angle. Below Θ c arcmin 5.6 λ ρ Å g/cm 3, (3.1) where λ is the photon wavelength and ρ the density of the reflecting material, photons are reflect well from a perfectly smooth surface. Usually Θ is of order 1. The Chandra mirrors are arranged that photons are reflected twice, first by a parabolic mirror shell and then by a hyperbolic shell, which is then called Wolter type 1. The effective area is now the equivalent area, a pixel would need, to measure the same flux as a pixel on the detector with the mirror assembly measures. It is energy dependent and varies with the position on the detector, as it can be seen in Fig Here the quantum efficiency is already included. The quantum efficiency is the product of the probability for absorbing a photon in depletion layer, which is a smooth function of energy, and the filter transmission probability of photons, which is characterized by absorption edges. The attenuation with off-axis angle is also called vignetting effect and automatically corrected when accounting for the effective area. A further effect has to be accounted for when analyzing data: The finite resolution in terms of energy and spatial resolution: The energy resolution is limited by statistical fluctuations of the number of conductive electrons N produced by an incoming photon. N has a Poissonian distribution and a standard deviation of N. See 31

35 CHAPTER 3. DATA REDUCTION 3.1. BASICS OF X-RAY DATA REDUCTION Figure 3.1.: Effect of the CTI on the energy resolution for a front illuminated ACIS chip (I3) and a back illuminated one (S3). Taken from Chandra X-ray Center Chandra Project Science (2010). Figure 3.2.: Pileup effect shown for the ACIS instrument. The solid line is the true spectrum, the dotted line the measurement. Taken from Chandra X-ray Center Chandra Project Science (2010). 32

36 3.2. PIPELINE CHAPTER 3. DATA REDUCTION Figure 3.3.: Chandra effective area as a function of energy. Left: Absolute on-axis value for BI and FI chips. Right: Relative change to the on-axis value for different off-axis angles (in arcminutes), which is also called vignetting effect. Taken from Chandra X-ray Center Chandra Project Science (2010). Fig. 3.4 (Left). The back illuminated chips have a worse energy resolution than front illuminated chips only for very small row numbers (see Figure 3.1). Since the mirror optics are not perfect (e.g. have a non-flat focal plane) a point source is mapped as a bright core with a faint halo around. This brightness distribution of a point source is called the point spread function (PSF). The PSF usually becomes larger and asymmetric for larger off-axis angles. See Fig. 3.4 (Right). Unfortunately there is a slow and continuous reduction of the effective area over the Chandra telescope lifetime. This is due to a contamination of the ACIS filters with gassed material that condenses. It mainly affects low energies. Since CALDB a model corrects for this effect and with new CALDB versions often updated contamination models are provided (for details see Arnaud et al., 2011) ARF and RMF Some of the effects described above are not corrected directly, but the information is written to specific files which are accounted for during the spectral fitting process. One usually defines the observed counts C as C(P I) = T RMF(P I, E) ARF(E) S(E) de, (3.2) where the response is split up into a unitless matrix (RMF) and a vector (ARF) in units of area, S is the source flux and T the total observing time. P I is the detector channel. The RMF only depends on the detector coordinates, the photon energy E and the channel P I. The ARF is variable with detector coordinates and photon energy. Of course all responses are variable with time. In easy words, the RMF gives the probability of a photon with a given energy to be detected in a certain channel as shown in Fig Idealy this would be a diagonal matrix. The ARF is then the effective area at a given position and energy (e.g. Figure 3.3) Pipeline Now I will describe how the available software was used to get the spectral information. Automating this process and writing it into a script that can easily be run is the creation of a pipeline. 33

37 CHAPTER 3. DATA REDUCTION 3.2. PIPELINE Figure 3.4.: Finite resolution of Chandra: Left: Energy resolution as a function of energy. Right: Spatial resolution shown as the angle necessary to enclose a certain fraction of energy of a point source as a function of the off-axis angle. Taken from Chandra X-ray Center Chandra Project Science (2010). Figure 3.5.: Visualization of the RMF. The x-axis is the detector channel linearly from 1 to 1024, the y-axis the photon energy linearly from (0.3 11) kev. The colorbar represents the probability (in logarithmic scale). 34

38 3.2. PIPELINE CHAPTER 3. DATA REDUCTION Chandra - ACIS For the reduction of the Chandra ACIS data I first used the Chandra Analysis Pipeline written by a previous group member (Danny Hudson, edited by Helen Eckmiller and Vera Jaritz). I changed some steps to use it for the latest CIAO version (CIAO 4.4, CALDB 4.4.7). CIAO is the standard software package for Chandra data analysis and developed by the CXC 1. The pipeline mentioned was not written for the latest CIAO version, but for an older one and adapted to CIAO 4.2. So to use the latest CIAO tasks and scripts, which are corrected for bugs and work fine with the current calibration, I wrote my own pipeline, which I will describe in the following. x Basic setup To run the new pipeline, one has only to set three parameters: The observation ID, the cluster name and the directory number for the parameter files (important when running simultaneously). The first step is to download the data, which is automatically achieved by the task download_chandra_obsid with only given the observation ID. The next step is to create the level 2 event files out of the raw data. For this, the new task chandra_repro is applied. The following processes are run (see Cosmic ray interactions produce charge in the CCD that is also detected in a few of the following frames. This effect is called afterglow. In previous versions the delivered level 2 event files are cleaned from afterglows such that also a small fraction of source photons was rejected. If the downloaded event files (also level 1) are corrected in that way, the correction has to be removed and the new, more precise detection algorithm acis_run_hotpix is executed. This tool also detects bad pixels. Creation of new level 2 event files by running the acis_process_events task, which updates the time, coordinates, pulse-height and status information from the raw file to the new calibration. Also a CTI (charge transfer inefficiency) correction is made, which accounts for variations of the focal plane temperature. If the correction is done, a header keyword is added (CTI_APP), which is needed by some of the following programs. So after the level 2 event files are created the header keyword is checked again by the script (check_ctiapp.sh). Setting observation-specific badpixel files which are used later on. Note that the bad pixel events are not removed, but the information is stored in a file accounted for in the following analysis. The next step is to get the regions where the spectra should be extracted and which point sources should be excluded. The coordinates of the emission peak center are taken from Hudson et al. (2010). The coordinates are then transformed using dmcoords (from celestial to Chandra-Sky coordinates) and saved in a region file. Lightcurve cleaning Then the needed chips (for I-observations, I-Chips are combined) are cleaned from solar flares. The filtering is done in the main energy band (0.3 12) kev. A time binning of s is used for the front illuminated chips (i.e. all except S1 and S3). For the two back illuminated chips the flare detection is performed in the (2.5 7) kev band with a bin size of s. All these values are suggested in the Markevitch Cookbook 2. Finally the cleaning is done by deflare using the lc_clean algorithm and the good time intervals (GTI) are determined and applied in the following. Note that the lc_clean method uses a sigma clipping (3σ) to calculate the mean rate. Then all events are accepted, which correspond to a time interval bin having a rate at maximum 20% above or below the mean rate. All time bins which are marked green in Figure 3.6 would be accepted. Background file processing Since the cleaning of the source event files is now finished, the background files then have to be prepared. The first step is to locate the correct files which are provided in the CALDB. This is also now very easy by using the acis_bkgrnd_lookup script. It automatically finds the blank-sky background files matching a specific observation and chip. If it is an I-Pointing observation, the 4 I-Chips of the background observation will be merged with dmmerge. These blank-sky background files are only used for a first analysis of the inner region of the cluster, where the background is not dominant (see Chapter B), and for the calculation of the signal to noise ratio

39 CHAPTER 3. DATA REDUCTION 3.2. PIPELINE Hydra A 3 mean rate= s 1 obsid=4969 Count Rate (s 1 ) D Time (ks) 10 Number histo_7.fits Count Rate (s 1 ) lc_clean Figure 3.6.: Lightcurve of the HydraA observation 4969 (S3 Chip). Red regions in the top panel mark time intervals that were excluded. The green region in the histogram (bottom) shows the count rate distribution of accepted intervals. (S/N), which is used later on to determine the size of the regions. For the proper background subtraction, the stowed observations are selected manually from the CALDB (see Chapter 3.3 for the background subtraction). Since the background files might be created with a different gainfile and/or pointing, this has to be checked and accounted for. In the first case, the background files are reprocessed (acis_process_events) with the gainfile of the observation. To correct for different pointing, the header keywords from the source observation file are simply copied to the background file header. The old task make_acisbg is now replaced by reproject_events, which matches the background file to the events file by transforming the events into the new coordinates. As some observations are done in the very faint (VFAINT) mode, the background files still contain some status 1 events which have to be filtered out by the dmcopy task. The last correction before extracting the spectrum is the different high energy count rate between the background and the source observation. By computing a normalization factor, which is later multiplied to the backscale value of the background spectrum, this can be handled very easily. The meaning of this parameter is explained in Section 3.3. Spectrum extraction For selecting the regions (of circular or elliptical shape), I applied a S/N threshold of S/N = source counts source + background counts > 70, (3.3) where the background counts are estimated from the blank sky files. The final steps, the spectrum extraction, creation of a weighted ARF (out of a weighted map) and RMF and the grouping of both spectra is done by one task called specextract. As described in the science threads 3, the weighted map should be created by the sky2det tool (included in the specextract script) without any binning. Therefore creating the arf takes a lot of cpu time, especially for large regions. The fitting procedure is done in the same way as with the tool xspec using an APEC model for the cluster emission combined with the wabs model to account for the absorption

40 3.3. BACKGROUND SUBTRACTION CHAPTER 3. DATA REDUCTION Temperature old Pipeline Metallicity old Pipeline Temperature new Pipeline Metallicity new Pipeline (a) Best fit temperatures (b) Best fit metallicity Figure 3.7.: Comparison of the old and new pipeline best fit temperatures measured in kev. The line indicates the agreement of the two temperatures/abundances. A systematic difference can not be detected. Comparison First it should be checked whether the two data reduction methods (the pipeline written by D. Hudson and the new one described here) end up with the same results and, if there are differences, which results are more reliable. Figure 3.7 shows the best-fit temperatures (and abundances) of the two pipelines. As there seems to exist no systematic difference one can conclude a good agreement between the two methods. Only at high temperatures (above 6 kev) where some scattering starts the two temperatures do not always agree very well. After looking to the regarding clusters (mainly A2204, A2256, A1656) it turns out that there might be some flares left in during the data reduction with the new pipeline.these cases are then filtered manually by setting the mean countrate for the lightcurve filtering algorithm to a reasonable value. In general, all lightcurves are checked by eye, whether the algorithm worked fine Background subtraction Background components As for every measurement also X-ray observations do not only record the signal of the object one is observing, but also disturbing noise. This so called background can be devided into the instrumental background, the solar wind charge exchange (SWCX) and the astrophysical background. The instrumental background is different for every detector and satellite. Since CCD/CMOS detectors produce noise (thermal noise, shot noise, 1/f-noise) they are cooled down (to 120 in case of ACIS) that the important thermal noise does not contribute significantly. Other important components of the instrumental background are high energetic particles that hit the CCD or interact with the material of the satellite and produce fluorescent lines and continuum emission. This component is constant on short timescales and not folded by the instrumental effective area. Unfortunately it is spatially variable over the detector which has to be taken into account for the background subtraction, a highly time variable flux of soft protons which hits the CCD and produces a power-law continuum spectrum. The solar wind charge exchange is produced by a highly charged solar wind, that collides with neutral gas of the geocorona. Electrons from the neutral gas are transfered to the ions and excited and emit X-rays after decaying (Wargelin et al., 2004). This component dominates when a satellite on an elliptic orbit enters the Earths radiation belt. Sometimes the SWCX is also regarded as an astrophysical background source, but more common are 37

41 CHAPTER 3. DATA REDUCTION 3.3. BACKGROUND SUBTRACTION Figure 3.8.: ROSAT All-sky map of the ( ) kev band in Aitoff-Hammer projection centered on the Galactic center and the Galactic longitude increasing to the left. The colorbar is in cnts s 1 arcmin 2. Taken from Snowden et al. (1997). the cosmic X-ray background (CXB), which is a diffuse background of unresolved point sources (most likely AGNs) which produce a powerlaw spectrum E 1.41 (Kushino et al., 2002). This is the dominant astrophysical background component above 1 kev. The thermal emission of the hot plasma of the galactic halo with a temperature of ( ) kev, which is sometimes also modeled with two absorbed thermal models like in Snowden et al. (2008) to account also for the cooler component of the galactic halo. In this work the spectral fits to the Chandra data are started at 0.7 kev, so the cooler components are negligible (see also Fig. 3.9) and only one thermal model is used for the galactic halo emission. The local hot bubble emission is also an astrophysical background component. It is still under debate, whether this emission is really the result of a former supernova in our galactic neighborhood or maybe due to heliospheric SWCX, but nevertheless, the spectrum can be described by a thermal model with a temperature of about 0.1 kev. The characteristics of helispheric SWCX is line emission, but a cool (0.1 kev) thermal component is also dominated by line emission. A way to distinguish them would be to look at the strength of a forbidden line, which is favoured by the SWCX. Unfortunately the energy resolution of current instruments is too low to resolve these lines (see e.g. Koutroumpa et al., 2011). Of course, the model will also include (at least partly) the cool component of the galactic halo. In some rare cases the temperature determined for the galactic halo emission is not consistent with the virial temperature of the Milky Way. This can be seen as statistical fluctuations of the measurement. The galactic halo and the local hot bubble emission are highly variable across the sky, as it can be seen in Fig. 3.8, which mainly shows the galactic halo component. Of course, when accounting for the absorption, the gradient towards the galactic plane in Fig. 3.8 is lowered. But still the spatial variation is high. Figure 3.9 illustrates the contribution of the individual background components. The datapoints (crosses with solid fit) are on top and already particle background subtracted. In this case, the particle background (double crosses) is dominating over the cluster emission above 6.5 kev. The lower solid line is the sum of all three astrophysical sources, shown as three dotted lines. This illustrates the need of a proper background subtraction. For example if the particle background is subtracted wrong, the resulting temperature of a spectral fit is changed significantly, especially in outer regions, where the cluster emission is much fainter. Methods for background subtraction Several methods are possible to subtract the different background components. A widely used method to handle the time variability of the particle background and spatial variability of the particle induced background 38

42 3.3. BACKGROUND SUBTRACTION CHAPTER 3. DATA REDUCTION Figure 3.9.: Data showing the contribution of the different background components. The best fit cluster temperature was 8.44 kev, the relative abundance See text for details. components, is to use outer regions without cluster emission of the same observation. The problem with this method is, that the background components are also variable across the detector (e.g. due to the fluorescent lines emitted by the material behind the detector) and so using outer regions of the same observation can be problematic. Also the HIFLUGCS clusters are nearby and very extended clusters, which makes it impossible to find large enough regions without cluster emission within the Chandra field of view. One could think of using pointed observations near the cluster to estimate the background, but this introduces inconsistency as this is not possible for all the HIFLUGCS clusters and they are observed at different times and so this does not account for the time variability. Another possibility is to use so called blank-sky background files. This are exposures of the sky where no bright sources are located. These events files are already reduced, cleaned from solar flares and distributed together with the official calibration files (CALDB). So it is possible to extract spectra from blank-sky files of the same area of the detector that have been used for the actual observation. Of course one misses both the spatial and time variability of the background. To correct for the time variability of the particle background, one re-scales the spectral count rate of the blank-sky with a constant factor to have the same count rate as spectrum in a high energy band, e.g. (9.5 12) kev, to the cluster observation. This high energy band should have a negligible effective area, which means no X-ray photons but only particle events are detected. This scaling method assumes that the time variability affects the whole energy band equally, which is not true in general. But in practise this assumption is not too bad. The spatial variability of the foreground components cannot be corrected easily. The third method is to treat particle and astrophysical background independently. Like the blank-sky files, there also exist event files in the CALDB which only consist of particle events. These files can be created, when the detector is in the stowed position and not exposed to the sky. In this case the effective area is again 0. The spectrum from these stowed background files can be re-scaled in the same way as the blank-sky files. The other background components can for example be estimated from data of a region around the cluster (e.g. from the ROSAT all-sky survey). For my analysis I decided to use the last method and independently subtract particle and astrophysical background, because this seems to be a good method to subtract the background of HIFLUGCS clusters in Chandra observations. In the following I describe in more detail how I realized this method: Stowed-observation files from the CALDB were used to estimate the particle background. The steps which were performed include 39

43 CHAPTER 3. DATA REDUCTION 3.3. BACKGROUND SUBTRACTION choosing the correct file from a period (2001 or 2005) which is closer to the observation date filtering the events with status 0 if the observation was taken in VFAINT mode calculating the high energy count rate using specextract to extract background spectra of the same region on the detector and multiplying the backscal value with the ratio of the high energy count rates setting the areascal keyword to the extraction area in units of arcmin 2. The backscale value is set by the CIAO software to the number of pixels used for the spectrum dived by is the total number of pixels on 8 chips and is used by the CIAO software to normalize, although the number of CCD chips used for an observation is lower. Multiplying the unscaled backscale value with yields the extraction area in arcmin 2. The spectra from the ROSAT all-sky survey are downloaded using the command line tool from S. Snowden 4. The extraction region is a circular annulus from 0.7 to 1 around the cluster center. The spectra consist of seven energy bins and from (0.1 2) kev and are already scaled to 1 arcmin 2. A simultaneous fit using an absorbed thermal model (APEC*wabs) for the cluster emission is performed using the spectra from all regions with the stowed spectra defined as background file and using a model for the astrophysical background consisting of two (one absorbed) thermal APEC models with solar metal abundances (representing the local hot bubble or SWCX and galactic halo emission) and an absorbed power-law with photon index of Since the N H is fixed, these background models have five free parameters (three normalizations and two temperatures) and can in principle be fitted by the ROSAT spectra (with seven datapoints) alone. But it is more accurate to do a simultaneous fit to all extracted regions linking the five background parameters, not only because of the better energy resolution of Chandra, but in the outer cluster regions a large fraction of the emission comes from the background and has to be accounted for

44 4. Data analysis After the general treatment of the X-ray data was explained in the previous Chapter, I will now focus on how to get scientific results (e.g. mass estimates) and what kind of problems arise. This starts with the selection of the regions for the temperature and surface brightness profiles, how these profiles are then used and on what steps care has to be taken of when deriving the masses Region selection Observations and simulations show, that galaxy clusters are not perfectly spherical symmetric. A better approximation is to treat them as ellipsoids by extracting the surface brightness and the temperature from elliptical annuli. This Section deals with the methods to determine the elliptic regions. The effects on the formulas for the mass determination are discussed in Section Simulations The ellipticity ɛ of a projected ellipsoid is defined in the literature in different ways, here I will use simply the ratio of the projected major axis a and the projected minor axis b as ellipticity: ɛ = a b. (4.1) This means ɛ 1 and ɛ = 1 refers to a sphere. The shape along the z-axis (line of sight) and the two inclination angles cannot be measured with X-rays alone. So there have to be made some assumptions. For simplicity I first assume that the inclination angles α = β = 0, which means the line of sight is perpendicular to two of the real axes of the ellipsoid and parallel to the third. Two extreme cases of shapes of ellipsoids can then be characterized: Prolate ellipsoids have a third axis c that is equal to the minor axis b, whereas oblate ellipsoids have c = a. It should be mentioned that the prolate and oblate shape is a general characterization of ellipsoids and does not depend on the alignment. Many theoretical papers agree that Dark Matter halos can be approximated by an ellipsoid (e.g. Thomas et al., 1998; Tormen, 1997; Jing et al., 1995; Dubinski, 1994; Warren et al., 1992). One reason for the ellipsoid shape of galaxy clusters is pointed out in Tormen (1997). The infall of satellite Dark Matter haloes onto the cluster has a very anisotropic distribution and therefore the final shape of the cluster is not shaped spherically symmetric. A typical axis ratio found by the named authors is 0.5. More recent publications include the effects of baryons in their simulations and found axis ratios tending to be more spherical (Bailin et al., 2005; Kazantzidis et al., 2004; Springel et al., 2004). Especially in Springel et al. (2004) it was shown that ɛ is lower when adding the effects of adiabatic gas to the collisionless Dark Matter in the simulations and even lower with cooling and star formation effects. Also the dependencies of the mean ellipticity on mass and redshift is shown in Allgood et al. (2006) (ɛ is decreasing with mass and redshift). Observations also detect these predicted ellipticities. In Kolokotronis et al. (2001) a comparison between an X-ray (ROSAT) and optical (APM) analysis of 22 Abell clusters was done to measure the shape in both wavelengths. Good agreement was found concerning the measured ellipticity and position angle. Also a correlation between the center of mass shift (a proxy for substructure) and the ellipticity was found. A further analysis with XMM-Newton was done by Kawahara (2010). The result shows that most clusters have ɛ 1.2 (see Fig. 4.1) Ellipse fitting algorithms Now I will discuss some methods how to determine the ellipticity of galaxy clusters from an X-ray image. The basic principle is the same as analyzing ellipticities of galaxies in the optical. In Carter and Metcalfe (1980) a method for finding the parameters of a best fitting ellipse to a distribution of galaxies (points) was developed. This method is based on the calculation of the moments of inertia described in Trumpler and Weaver (1953). McMillan et al. (1989) extended this method for a binned image, in which pixel values can also have values 41

45 CHAPTER 4. DATA ANALYSIS 4.1. REGION SELECTION Figure 4.1.: Propability distribution of ellipticities of simulated clusters are shown as solid lines and compared with observations (datapoints). The x-axis shows ɛ 1 from 0 to 1 in linear steps of 0.2. Figure taken from Kawahara (2010). larger than 1 (and not only 0 and 1 as in Carter and Metcalfe, 1980). The general moments of the coordinates x and y with the weighting function f(x, y) is given by (Trumpler and Weaver, 1953, Eq. 1.66): µ i,k = µ i,k = x i y k f(x, y) dx dy, i, k N 0 (4.2) (x x 0 ) i (y y 0 ) k f(x, y) dx dy, i, k N 0 (4.3) µ i,k (4.3) are called the central moments around the weighted center x 0, y 0 µ 10 = 1 x f(x, y) = x 0 N x,y (4.4) µ 01 = 1 y f(x, y) = y 0 N (4.5) x,y of the distribution. For discrete points x, y N and interpreting f(x, y) as the pixel value at (x, y) and after normalizing to the total number of counts N = x,y f(x, y) one ends up with first five and important central moments µ 10 = 1 (x x 0 ) f(x, y) = 0 (4.6) N x,y µ 01 = 1 (y y 0 ) f(x, y) = 0 (4.7) N x,y µ 11 = 1 x y f(x, y) x 0 y 0 (4.8) N x,y µ 20 = 1 x 2 f(x, y) x 2 0 (4.9) N x,y µ 02 = 1 y 2 f(x, y) y0 2 (4.10) N x,y One can now establish the matrix of moments µ 20 µ 11 µ 11 µ 02 (4.11) 42

46 4.1. REGION SELECTION CHAPTER 4. DATA ANALYSIS and interpret the eigenvalues Λ 2 + and Λ 2 < Λ 2 + as the square of the principal axes (for details see Trumpler and Weaver, 1953 and Carter and Metcalfe, 1980). 0 = (µ 20 Λ 2 )(µ 02 Λ 2 ) µ 11 (4.12) ɛ = Λ + (4.13) Λ ( Λ Θ = cot 1 2 ) + µ 02 + π (4.14) µ 11 2 Θ is the orientation angle of the ellipse and Λ ± are the positive roots of the eigenvalues. This method is an iterative approach, which means, one starts with a circular aperture inside which ɛ and Θ are calculated. With these new values a new (now elliptic) aperture is defined, which is used again for the new calculation until the values converge. For a really elliptical distribution, ɛ and Θ are good estimates, but if the shape is much more complex the outer parts of the aperture are overweighted. The uncertainty of the final values can be calculated by redistributing each value of the pixels (e.g. in the surface brightness map) using a Poissonian random number generator and calculate new values for the ellipse. Doing this 1000 times one gets estimates for the uncertainty. One modification to that method is not to use a circle in the beginning, but to use an annulus and vary the shape of that annulus. In Carter and Metcalfe (1980) this modification is claimed to theoretically produce better results and when applying it to X-ray images the disturbing effects of the central galaxy in the cluster can be excluded. A non-iterative modification is shown in Buote and Canizares (1992): The aperture inside which the pixel values are used for the calculation is a circle around the emission weighted center. Then one slightly modifies the equations 4.13 and 4.14: ɛ = Λ2 + Λ 2 Θ = tan 1 ( µ11 µ 02 Λ 2 + ) (4.15) (4.16) With these equations the shape of the ellipse can be calculated without iterating. Testing this with simulated elliptical distributions described by a beta-model shows a dependence of the agreement between measured and initial ɛ on the value of β (see Fig. 4.2). The exponent r n and β are given by (4.17). β = n (4.17) The simple simulation shows that for β [0.33, 1] the modification is acceptable, especially for β = 0.5 perfect agreement is achieved. Another method to find the shape of a galaxy cluster X-ray image is provided by the CIAO tool dmellipse. The tool varies the angle, ellipticity and centroid of an ellipse and reduces iteratively the radius until a certain fraction of the total counts is inside the ellipse. Unfortunately the exact algorithm is not described and the tool is very slow compared to the other methods, which makes an uncertainty estimate very difficult Comparison of methods For a comparison of the three methods described above I used Abell 2244, a weak cool core cluster with a temperature of about 5.5 kev, and derived ɛ for different radii. The result is shown in Fig. 4.3, where CM (Carter & Metcalfe) corresponds to the iterative method described first. This can be subdivided in two cases, using a circle (for the following iterations full ellipses) as aperture or excluding an inner part by using annuli. The Buote method is the non-iterative modification from Buote and Canizares (1992). dmellipse is the CIAO tool, for which only a few radii could be calculated because of the slow algorithm. All errors are estimated from 100 random realizations of the input image. In practise, first a binned image is created. The detected point sources (detected by wavdetect) are removed and filled up with Poisson noise from the surrounding. Then the produced image is smoothed by a small Gaussian (3 ). Although it is a symmetric gaussian smoothing, it should not change the elliptic shape of the image, because the cluster shape is determined on much larger scales than the smoothing scales. The results, presented in Fig. 4.3, show consistency of the different methods at radii 43

47 CHAPTER 4. DATA ANALYSIS 4.1. REGION SELECTION Figure 4.2.: Comparison of simulated elliptical distributions ɛ 0 and the determined ellipticity with the method described in Buote and Canizares (1992) (image taken from the paper). The best agreement with r 2 corresponds to β = A2244 Ellipticity dmellipse Buote CM annulus CM circle 1.3 ellipticity Radius [arcmin] Figure 4.3.: Comparison of the different methods to determine the ellipticity applied for Abell

48 4.2. PROFILE FITTING CHAPTER 4. DATA ANALYSIS r > 72. Larger radii than shown in Fig. 4.3 cannot be used because the aperture includes regions outside the ACIS chips. Since in general the ellipticity does not change for larger radii, the values are determined for all the HIFLUGCS clusters at r 100 using the Buote method Profile fitting After the regions, circular or elliptical annuli, are selected and background cleaned spectra are fitted, it is necessary to fit the derived profile with a model. The Chandra FOV is in most cases not large enough to measure out to r 500 (a radius, within which the density is 500 times the critical density of the universe). So an approximation to larger radii is necessary, which can be achieved by fitting models to the data. At this point it is possible to combine different observations of the same cluster to one temperature or surface brightness profile, because the datapoints in the profiles are statistically independent. The advantage of this combination is that no simulatneous fit has to be performed, which would make it necessary that regions are equal for all observations Surface brightness profile An often used surface brightness model is the so called β-model (Cavaliere and Fusco-Femiano, 1976). It can be derived from the King-model density profile ) 3 ρ gas (r) = ρ gas (0) (1 + r2 2 β, (4.18) where r 0 is the core radius and β is a dimensionless parameter. This King-model can be derived analytically assuming also an isothermal medium. The surface brightness profile can be obtained by integrating the emissivity, which is proportional to n 2 e (for T > 2 kev), and one ends up with r 2 0 SBR(r) = SBR(0) (1 + r2 r 2 0 ) 3β (4.19) Since at least the isothermal approximation is not valid, this should only be used as a fitting formula. In this work, I use a triple-β-model (Equation 4.19 summed up three times with different parameters) plus adding a constant to account for the background. Of course, the background level is different for back-illuminated chips, but since the background level is only important at the large radii (i.e. for S observations not on the BI chip), this variations of the background level of the surface brightness profile can be assumed to be negligible. The high quality Chandra data is able to resolve variations, which would not be fitted correctly by only using a double-β-model (see Fig. 4.4). One can see in Figure 4.4 that the extrapolation works much better using a triple-β-model, especially when the background cannot be measured out to large radii, like in this example Temperature profile The temperature profiles are fitted by a simple fitting, which consists of two smoothed power-laws, (( ) s ( 1 T (r) = T 1 + ( T r ) p T 2 ( r 100 ) p2 ) s ) 1 s, (4.20) where T 1, T 2, p 1, p 2 and s are free parameters. T 1 and T 2 are some temperature parameters, p 1 and p 2 describe the increasing and decreasing slope and s is parameter characterizing the smoothness. The normalization of 100 is more or less arbitrary. A similar formula was also used for example in Gastaldello et al. (2007). An example for the fit can be seen in Figure 4.5 for Abell Total mass determination The main topic of this work is the mass determination of galaxy clusters from X-ray observations. The radiation one can observe comes from the visible component of matter. But the Dark Matter also influences the 45

49 CHAPTER 4. DATA ANALYSIS 4.3. TOTAL MASS DETERMINATION Figure 4.4.: Comparison of the double- and triple-β-model for the surface brightness profile, shown for Abell Figure 4.5.: Fitted temperature profile for the elliptical and spherical regions of Abell For the elliptical regions, radius means the major axis. 46

50 4.4. GAS MASS DETERMINATION CHAPTER 4. DATA ANALYSIS observations. To examine this in more detail, one has to look at the equation, that describes the gas dynamics, the Navier-Stokes equation (NSE) ( ) v ρ + ( v ) v = F t ext p + η v. (4.21) The NSE is a special case of the continuity equation and requires conservation of mass, energy and momentum. The left hand side of (4.21) represents the total derivative of the velocity field v. The right hand side consists of the external force F ext and the pressure force p (both per volume) and a further force η v, which represents friction. Without friction (η = 0) the NSE becomes the Euler Equation ( ) v ρ + ( v ) v = F t ext p, (4.22) which can be further simplified by considering a static case (left hand side becomes 0). In the case of a hot gas in a galaxy cluster the following statements can be made: The assumption of negligible friction is valid since the mean free path of the particles is roughly > 20 kpc (Sarazin, 1988). On the other side, the large mean free path, especially in the outer regions of clusters, may affect that the fluid approximation is no longer valid. The static case is valid, because the sound speed of the ICM is large enough. Of course merging events are not included in this consideration. F ext can be interpreted as the gravitational force. All other forces (magnetic) are not considered and are small compared to F grav = ρ Φ = ρ GM r (force per volume). 2 Assuming the ideal gas equation holds, the pressure can be rewritten p = k BρT µm p, where µ is the mean molecular weight and m p the proton mass. At first one can assume spherical symmetry, so spatial derivatives only have a radial component now. All this can be used to simplify (4.22) and derive the hydrostatic equation M tot (< r) = k ( BT gas r d ln ρgas + d ln T ) gas µgm p d ln r d ln r. (4.23) It is important to mention that using this formula it is possible to calculate the total mass of a cluster inside a certain radius r. Only the properties of the cluster at the given radius are necessary (temperature and gradients), not the structure inside the radius. How to modify the hydrostatic equation for ellipsoidal potentials was shown in detail in Buote and Humphrey (2012) with M(< a v ) = η(p v, q v ) a2 v G η(p v, q v ) = p vq v 3 ( p ) v qv 2 dφ da v, (4.24), (4.25) where a v is the ellipsoidal radius (major axis), p v = b a and q v = c a with the principal axis a,b and c. Simplifying (4.24) gives M tot (< a v ) = η(p v, q v ) k ( BT gas a v d ln ρgas + d ln T ) gas. (4.26) µgm p d ln a v d ln a v 4.4. Gas mass determination The X-ray emission traces the baryonic matter of the cluster, because for an optically thin, hot plasma the luminosity is proportional to the electron density squared (see Eq. 1.6). In general, line emission makes it necessary to calculate the cooling function Λ numerically Λ = ɛ n e n H, (4.27) 47

51 CHAPTER 4. DATA ANALYSIS 4.4. GAS MASS DETERMINATION Figure 4.6.: Cooling function for collisional ionization equilibrium (CIE). Free-free emission dominates above 1 kev. The main contributor (in the X-ray band) to the line emission are the Iron resonance lines. Adopted from Sutherland and Dopita, where n e is the electron number density and n H is the number density of Hydrogen atoms in the gas. Figure 4.6 shows the cooling function and the main contributors. Above 1 kev free-free emission dominates, whereas below 1 kev the Iron resonance lines are dominant. The cooling function can now be simulated using xspec and the AtomDB for each cluster using the corresponding redshift. Since it is metallicity dependent, a matrix has to be created which includes the values of Λ(T, A). The observable in (4.27) is the emissivity ɛ. It can be obtained using the surface brightness (SBR) profile. The SBR profile shows the number of photons in a defined energy band, e.g. (0.5 2) kev, per second and detector area and solid angle [SBR] = cnts s cm 2 arcmin 2, (4.28) as a function of the radius. In the following the assumption is made, that the volume which contributes to the surface brightness is the same as the volume corresponding to an annulus (inside which a constant ɛ is assumed. This is of course not correct, but can be seen as a very rough approximation. A more proper way to do that is explained in Chapter 4.5. To get the emissivity one then has to multiply the SBR with the mean photon energy (see Chapter 4.4.1), the surface of the sphere around the cluster with r = D L, the solid angle Ω which the cluster occupies and divide by the physical volume V of the cluster shell (with the outer radius R out and inner radius R in ) which corresponds to the solid angle 4πDL 2 ɛ [0.5 2] = SBR [0.5 2] E Ω mean 4 3 π(r3 out Rin 3 ). (4.29) In the following the derived gas mass according to this section is called projected gas mass, where as the result of the proper derivation is called deprojected gas mass Mean photon energy The mean photon energy can be approximated by assuming free-free emission E mean E mean,ff = 2 kev 0.5 kev hν g ff e hν 2 kev 0.5 kev g ff e hν k B T dν k B T dν. (4.30) The gaunt factor g ff is assumed to be constant and cancels out. The redshift was ignored, because as it turns out a higher redshift pushes the mean photon energy of free-free emission towards lower values. But because of the missing line emission, values are usually already underestimated. Simulating free-free emission and free-free plus line emission and calculating each time the mean energy shows that only for very low temperature plasmas a difference is present (see Fig. 4.7 and 4.8). At 1 kev the difference is at maximum 8% for a metallicity of 0.5 and 6% for metallicity of 0.1. The largest fraction of the complete gas mass is located at regions where 48

52 4.4. GAS MASS DETERMINATION CHAPTER 4. DATA ANALYSIS E mean,apec / E mean,ff z = 0 z = 0.05 z = 0.1 z = 0.15 z = 0.2 E mean,apec / E mean,ff z = 0 z = 0.05 z = 0.1 z = 0.15 z = Temperature in kev Temperature in kev Figure 4.7.: Ratio of the mean photon energy with line emission (APEC model) and without (Equation 4.30) as a function of cluster temperature for different redshifts. The observer frame energy band is (0.5 2) kev. Solid lines represent a metallicity of 0.1, dotted lines 0.5. E mean,apec / E mean,ff z = 0 z = 0.05 z = 0.1 z = 0.15 z = 0.2 E mean,apec / E mean,ff z = 0 z = 0.05 z = 0.1 z = 0.15 z = Temperature in kev Temperature in kev Figure 4.8.: Same as Fig. 4.7 except that all lines refer to metallicity of 0.5, but the dotted lines show the ratio using a redshifted free-free emission. the metallicity is lower than 0.5. But most of the HIFLUGCS clusters have a higher temperature. Above 4 kev the relative difference between the correct and free-free mean energy is below 1% and for a redshift of 0.1 even below 0.3%. According to (4.30) and (4.27) an underestimation of E mean of 1% would result in an underestimation of the density of 0.5%. Figure 4.8 shows the effects of a redshifted free-free emission as dotted lines. For the redshift range of the HIFLUGCS clusters it is better to use the not redshifted free-free emission (4.30) Cooling function The cooling function can be simulated using a plasma emission code like APEC or MEKAL. These models usually have a normalization Norm = πDL 2 n e n H dv. (4.31) With the definition for the observer frame flux f OF = L OF 4πD 2 L, (4.32) 49

53 CHAPTER 4. DATA ANALYSIS 4.4. GAS MASS DETERMINATION Λ [0.5-2] in cm 3 erg / s 2e-23 1e-23 7e-24 Z = 0.1 Z = 0.2 Z = 0.3 Z = 0.4 Z = 0.5 Z = 0.6 Z = 0.7 Z = 0.8 Z = 0.9 5e-24 4e Temperature in kev Figure 4.9.: Cooling function Λ as a function of cluster temperature for different metallicities. with the observer frame luminosity L OF and the corresponding luminosity distance D L (the definition for source frame, SF, is analogous), one can rewrite the cooling function assuming constant cooling function inside the volume V Λ = ɛ = L SF f SF 4πDL 2 = n e n H V n e n H πDL 2 Norm = K f OF (4.33) Norm Note that all fluxes and luminosities are restricted to the same energy band and the so called K-correction, a correction between source and observer rest frame K = f SF f OF, (4.34) is small due to the small redshift range of HIFLUGCS. In general Λ is a function of the temperature T, the relative abundance of metals Z, the redshift z and of course of the energy band which is used. For (0.5 2) kev typical values are a few erg s 1 cm 3 (see Fig. 4.9). This plot also shows that efficient cooling is mainly done by line emission, which is strongly dependent on the relative abundance of metals and suppressed at high temperatures. So the cooling is most efficient at low temperatures and high relative abundances Metallicity profile In this work the metallicity profile of a galaxy cluster is measured at the same radii as the temperatures. For calculating the gas density the metallicity enters in terms of the cooling function (see Fig. 4.9). In practise, cooling functions are simulated for the cluster redshift but for a large variety of temperatures and metallicities and then stored in a fits table. Temperatures and metallicities are changed in each Monte Carlo simulation (an algorithm that repeats an operation with random numbers with a specific distribution) and then the best cooling function can be chosen. But unlike the temperatures, the metallicities are not fitted by a model, which results in an unsmooth density profile (see left part of Fig. 4.11). To avoid this, the metallicity profile should be smoothed. One possibility would be to interpolate. This requires that the resulting profile goes exactly through each data point. In some cases, e.g. for multiple observations which are measured at exactly the same radii, this is not possible. Another method is to fit a basic spline (B-spline) function, which not necessarily goes through each data point. The smooth B-spline function n 1 f(x) = c i B i,k (x) (4.35) i=0 50

54 4.4. GAS MASS DETERMINATION CHAPTER 4. DATA ANALYSIS Data BSpline 0.6 relative Abundance Radius in px Figure 4.10.: Profile of the relative abundance of Abell 2199 (1 px corresponds to arcsec). Four observations are combined in this plot and the red curve represents the smoothing bspline function. See text for details. 1e+06 Abell 2199 Density Profile Projected Density Deprojected Density 1e+06 Abell 2199 Density Profile Projected Density Deprojected Density particles per m particles per m Radius in pixel Radius in pixel Figure 4.11.: Gas density profiles created with a gsl Basic spline smoothed metallicity profile (right) and without (left). Density profiles calculated including a deprojection analysis are shown in green. The red lines are calculated following the description in Chapter

55 CHAPTER 4. DATA ANALYSIS 4.5. DEPROJECTION ANALYSIS Figure 4.12.: Projection parameters for spherical symmetry. Figure taken from Ciotti (2000). consists of the coefficients c i, which are determined by a least-squares fit, and the splines B i,1 (x) = B i,k (x) = { 1 for t i x < t i+1 0 else [ x ti t i+k 1 t i ] (4.36) [ ] ti+k x B i,k 1 (x) + t i+k B i+1,k 1 (x). (4.37) t i+k Since cubic splines are used, k is set to 4. t i are the n + k knot points, where n is the number of data points. The functions are implemented within the gsl libraries 1. The result of the fit for a cluster metallicity profile can be seen in Fig If data is needed at larger radii than the last measurement, the metallicity is assumed to decrease as r 1, since metal abundances typically decrease to the cluster outskirts (see e.g. Leccardi and Molendi, 2008). Unfortunately, this produces a non-differentiable location in the metallicity profile (in the left part of Figure 4.10 at 1200 px). But this has no effects on the final density profile Deprojection analysis When observing the optically thin ICM in X-rays one has to deal with the projection effects, which means I cannot determine the three-dimensional structure but only the two-dimensional distribution with a summation along the line of sight z. This introduces a degeneracy, which many three-dimensional distributions would be possible to end up with the observed data. This degeneracy can only be solved assuming for example a certain gas distribution (e.g. spherical or elliptical). In this case the projection can be reversed and original structure can be reconstructed. If the reconstruction produces unphysical values (e.g. negative emissivity) the assumption was apparently wrong. I introduce a new quantity κ which is simply the brightness per volume (see Eq. 4.38). At first I assume spherical symmetry, which means κ depends only on the radius r (see Fig. 4.12). Then the measured projected surface brightness I(s) is given by I(s) = κ(r) dz = 2 R κ(r)r dr. (4.38) r2 s

56 4.5. DEPROJECTION ANALYSIS CHAPTER 4. DATA ANALYSIS Figure 4.13.: Schematic description of an elliptic density distribution. The telescope plane is indicated by Film. Taken from Gueron and Deutsch (1996). Using the Abel integral equation (see Binney and Tremaine, 1987) f(x) = x g(t) = sin(απ) π with 0 < α < 1 Equation (4.38) can be solved for κ κ(r) = 1 π g(t) dt (4.39) (t x) α r t df 1 dx, (4.40) dx (x t) 1 α di 1 ds. (4.41) ds s2 r2 Especially for the deprojection of the surface brightness this is a powerful technique. For elliptical distributions the substitution has to be changed a little bit. If the ellipticity can be described by a triaxial model x 2 a 2 + y2 b 2 = 1, (4.42) where x and y lie in the projection plane and a and b are the semi-major and semi-minor axis respectively, then, following Kreye et al. (1993), the emissivity can be described by κ(a) = a 0 1 b 0 π a di 1 ds, (4.43) ds s2 a2 where a 0 and b 0 are the major and minor axis, respectively, and a is the projected major axis. This refers only to the oblate case. A very general treatment of this problem is discussed in detail in Gueron and Deutsch (1996). There, the final solution for an elliptic distribution is not derived using the Abel inversion, but a derivative-free solution is presented. But since the deprojection of a general elliptic distribution with an angle 90 between the line of sight and the elliptic plane of the cluster (see Fig. 4.13) is presented there, I will use their derivation and do an Abel inversion. The observed surface brightness with the new parametrization x 2 a 2 + y2 b 2 = k2, 0 < k 1 (4.44) 53

57 CHAPTER 4. DATA ANALYSIS 4.5. DEPROJECTION ANALYSIS Figure 4.14.: Comparison of the xspec model projct and the dsdeprojct code using simulated data of two plasma components (hot and dens, cooler and lower density). Image taken from Russell et al. (2008). is given by 1 k I(D) = 2a m b m β κ(k)dk, (4.45) k2 D2 D where D = rβ, β = 1+m 2 a 2 m m2 +b 2 m. m is the slope as indicated in Fig and k a parameter describing the shells. a m and b m are the maximum semi-major and semi-minor axis (see Eq. 4.44). The final solution using Abel inversion is then κ(k) = 1 1 di 1 πabβ dk D2 k. (4.46) 2 k The results (4.43) or (4.46) can now be used to calculate the projection corrected density profile of the gas mass. Therefore only the surface brightness (SBR) in Eq needs to be replaced by κ and the volume in the denominator is changed to an area: ɛ depro = κ E mean 4πD 2 L Ω π(r 2 out R 2 in ). (4.47) To arrive at the gas density one also needs the temperature (for the cooling function). This quantity also needs to be deprojected. Several methods exist to do this: An xspec model projct exists, to deproject the models fitted to the spectra. For this, one needs to fit all the regions of a cluster simultaneously and the radii have to be submitted to xspec. This method produces, especially for high quality data, oscillations in the deprojected temperature and density profile (see Russell et al., 2008 and Fig. 4.14). It is also very time-consuming because depending on the cluster there can be up to 100 free parameters to be fitted. As indicated, there exists a direct, model independent deprojection code (DSDeproj) by Sanders and Fabian (2007), which regroups the spectra and directly subtracts them according to the projection volume. This code was tested in detail but did not produce satisfying results, like a density peak at the outermost radius (Fig. 4.14). 54

58 4.5. DEPROJECTION ANALYSIS CHAPTER 4. DATA ANALYSIS 8 7 A85 Temperature Profile Data Fit Deprojected 6 kt in kev Radius in arcmin Figure 4.15.: Deprojected temperature profile of Abell 85 following the method described in Lima Neto (2005). The same method used for the surface brightness deprojection can be adopted to deproject the temperature profile. This is described in Lima Neto (2005) using only free-free emission for the deprojection, which is a good assumption for hot objects like galaxy clusters. In the following, I will use this method for the temperature deprojection. Apart from the assumption of only free-free emission contributing to the emissivity ɛ(r) = Kn 2 (r)t 1 2 (r), (4.48) where K is a constant, one also has to assume that the projected temperature is the average temperature along the line of sight weighted by the emissivity T proj (r) = T (x)n 2 (x)t 1 2 (x) r x dx x 2 r 2. (4.49) n 2 (x)t 1 x 2 (x) dx x2 r 2 r This assumption is valid for an energy independent effective area. A generalization for multitemperature plasmas is presented in Vikhlinin (2006). Inserting the definition of the surface brightness for free-free emission SBR ff (r) = 2K and using the Abel inversion, one ends up with T (r) = r r n 2 (x)t 1 x 2 (x) dx (4.50) x2 r2 d dx (SBR 1 ff(x) T proj (x)) dx x2 r 2 r d dx (SBR 1 ff(x)) dx x2 r 2. (4.51) An example is shown in Fig The behavior of the deprojected temperature profile is as expected from Lima Neto (2005), which means a temperature decrease in the cool core region and a slightly higher temperature at larger radii, which converges to the measured temperature. For the elliptic case the deprojection of the temperature stays the same, as the coefficients are vanishing (see Eq. 4.49) and the major axis is used as radial coordinate. 55

59 CHAPTER 4. DATA ANALYSIS 4.6. GAS COMPONENTS Figure 4.16.: Relative difference to hydrogen from different tables divided by the Angr value. Aspl: (Asplund et al., 2009), Lodd: (Lodders, 2003), Wilm: (Wilms et al., 2000), Grsa: (Grevesse and Sauval, 1998), Feld: (Feldman, 1992), Angr: (Anders and Grevesse, 1989), Aneb: (Anders and Ebihara, 1982) Gas components Except protons and electrons, there are also ions of heavier elements located in the ICM. One has to accounted for that when calculating µ for the total mass and the electron to gas density ratio for the gas mass. First one has to look at some general definitions for particle (number) densities Metals n e = n H + 2 n He + Zi n i (4.52) Metals n gas = n e + n H + n He + ni (4.53) n ion = n gas n e (4.54) ( ) Metals ρ gas = n H m p + m p 4n He + 2Zi n i (4.55) η n gas n e (4.56) τ n e n H, (4.57) where m p is the proton mass, n e is the electron number density, n gas is the number density of all particles, n ion is the number density of the ions in the gas and ρ gas is the mass density of the gas (neglecting electrons). All other number densities n X refer to the named atom X. The assumptions used further on are the gas is fully ionized. Helium is always assumed to have a relative abundance of (independent of the value in the abundance table) and not rescaled. This is the primordial value in a standard ΛCDM universe. Note that the abundance value used for Helium changes µ more than the different metals. Finally all ions (except Hydrogen) are assumed to have equal numbers of protons and neutrons. Equation (4.27) can now be written as Λ = ɛτ n 2 e. (4.58) The mean molecular weight µ, which is necessary for the total mass (see Eq. 4.23), can also directly be calculated µ = ρ gas = 2τ 1. (4.59) m p n gas τη µ usually has values around 0.6. The individual number densities n i (e.g. n He, n Fe,...) can be calculated using a relative abundance table and a measured metallicity A. In this work, µ is calculated using the abundance 56

60 4.6. GAS COMPONENTS CHAPTER 4. DATA ANALYSIS table of Anders and Grevesse (1989). The same table is used for the analysis (e.g. within the APEC model), except for the Galactic absorption correction, as described in Section In Fig and Tab. 4.1 can be seen that it does not make a big difference for the calculation of µ to use a different abundance table. The relative difference of µ values for the different abundance tables for solar metallicity can be seen in the last row of Tab For a relative abundance of 0.5 the relative difference of µ for different tables is reduced by a factor of 2. Up to now, the abundance table from Asplund et al. (2009) seems to have reliable values, since there is consistency between measurements of meteorites and the solar photosphere. Unfortunately, many publications used the abundances from Anders and Grevesse (1989), so it might be useful to still use them for comparison. Table 4.1.: Abundances of the elements relative to Hydrogen for seven different references. The second last row shows the µ value (minus 0.6) derived with the corresponding table for solar metallicity, last row shows the relative difference of µ to the Angr table. Table references can be found in the caption of Fig Element Angr Feld Aneb Grsa Wilm Lodd Aspl H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn µ µ angr µ i µ i 0% 0.02% 0.07% 0.21% 0.52% 0.48% 0.46% 57

61

62 5. Results I will now summarize the results obtained by the procedure described above. First of all I will give an overview about all the Chandra observations used for the analysis. Then I describe the results for the masses and show one example in more detail. Two clusters could not be analyzed by the standardized procedure, so they are treated as isothermal objects. Since these clusters have already been analyzed by Reiprich and Böhringer (2002), I will compare my results to the old ones. Also the results for the cosmological parameters using the application to cluster mass function are compared. Finally the results for the gas mass and ellipticity of the clusters are presented Observations For this work 119 observations have been analyzed with a total uncleaned exposure time of 4.76 Ms. The details to the individual clusters are summarized in Table A.1 in the appendix. Note that if a cluster has more than one observation, the ones with less than 10 ks are excluded because there would be only a minor improvement. Also very old observations (1999) are not used because of calibration uncertainties. For Chandra usually a low fraction of the total exposure time is affected by soft proton flares, which marginally reduces the used exposure time Mass estimates Gas mass and total mass This chapter summarizes the results for the masses of the cluster, which were achieved with the methods described above. For each cluster there exist 32 total masses and 32 gas masses, because I analyzed the cluster masses for 8 overdensities (15000, 10000, 5000, 2500, 2000, 1500, 1000 an 500), each of them in elliptical and spherical regions and again, each of them projected and deprojected. The assumption for the elliptical case is an prolate ellipsoid without inclination against the line of sight. The errorbars are 68 % confidence level from 100 realizations (for each mass) of a Monte Carlo Simulation, varying the best fit temperatures and surface brightness values assuming a Gaussian distribution. The masses were obtained using Ω m = 0.3, Ω Λ = 0.7 and h = In Section a detailed mass determination is shown as an example for the cluster Abell Example: Abell 2029 Abell 2029 is a cool core cluster with a relatively high temperature around 8 kev and a redshift of z = (Zhang et al., 2011). Three Chandra observations are available for this cluster (OBSID: 891, 4977, 6101). An old 20 ks exposure from April 2000 using the ACIS-S chips (OBSID 891), a long 80 ks ACIS-S exposure (OBSID 4977) and a shorter ACIS-I observation (10 ks, OBSID 6101). The last two are both from 2004 and the ACIS-I observation is in VFAINT datamode. The alignment of the three observations is shown in Fig The cleaned exposure time is ks for the chips with the pointing (since S-chips and in S-observations also I-chips are reduced separately, this number depends on the chip). The determined ellipticity is ɛ = 1.26, which is well constrained by all three observations (see Figure 5.2). The temperatures determined in elliptical regions are consistent for the three observations (right plot in Figure 5.3), except the outermost region in the short I-observation shows a higher temperature. Since this observation is just at the exposure time lower limit (no observations are analyzed below 10 ks), not all point sources are clearly subtracted. When examining this observation by eye, it becomes clear that there exist some point or slightly extended sources in the last bin which are not subtracted. In the longer S-observation, this area is not covered. However, the point sources are subtracted in observation 891. This explains the temperature increase for the I-observation. The surface brightness (SBR) profiles are shown in Fig The SBR values of the different observations 59

63 CHAPTER 5. RESULTS 5.2. MASS ESTIMATES 05:00.0 6:00: :00.0 Declination 50: : : :00.0 5:30: : : :10: :30.0 Right ascension Figure 5.1.: Alignment of the three Chandra observations of Abell Red is observation 4977, green is 6101 and blue is A2029 Epsilon Angle 1/ellipticity // angle in PI OBSID Figure 5.2.: Inverse ellipticity and rotation angle for Abell 2029 from three observations. For the angle, the y- axis is in units of π. Horizontal lines represent the average values and the larger errorbars take also into account the uncertainty when determining the ellipticity within a radius of (50 170) arcsec. The redistribution of the pixel values at a fixed aperture (as described in Chapter 4.1.2) results in the smaller errorbars. 60