6. Clusters and groups of galaxies

The Milky Way (MW), M31 (Andromeda; see Fig. 6.5), and M33 (Fig. 6.6) are the three spiral galaxies in the Local Group, and they are also its most luminous members. The Andromeda galaxy is located at a distance of 770 kpc from us, M33 at about 850 kpc. The Local Group member next in luminosity is the Large Magellanic Cloud (LMC, see Fig. 6.7), which is orbiting around the Milky Way, together with the Small Magellanic Cloud (SMC), at a distance of ∼ 50 kpc ( ∼ 60 kpc, respectively, for the SMC). Both are satellite galaxies of the Milky Way and belong to the class of irregular galaxies (like about 11 other Local Group members). The other members of the Local Group are dwarf galaxies, which are very small and faint (see Fig. 6.8 for three examples). Because of their low luminosity and their low surface brightness, many of the known members of the Local Group were detected only fairly recently. For example, the Antlia galaxy, a dwarf spheroidal galaxy, was found in 1997. Its luminosity is about 10⁴ times smaller than that of the Milky Way.

Many of the dwarf galaxies are grouped around the Galaxy or around M31; these are known as satellite galaxies. Distributed around the Milky Way are the LMC, the SMC, and about 20 dwarf galaxies, several of them in the so-called Magellanic Stream (see Fig. 2.​19), a long, extended band of neutral hydrogen which was stripped from the Magellanic Clouds about 2 × 10⁸ yr ago by tidal interactions with the Milky Way. The Magellanic Stream contains about 3 × 10⁸ M _⊙ of neutral hydrogen.

The spatial distribution of satellite galaxies around the Milky Way shows a pronounced peculiarity, in that the 11 closest satellites form a highly flattened system. These satellites appear to lie essentially in a plane which is oriented perpendicular to the Galactic plane, concentrated along the minor axis of the disk. The satellites around M31 also seem to be distributed in an anisotropic way around their host. In fact, satellite galaxies around spirals seem to be preferentially located near the short axes of the projected light distribution, which has been termed the Holmberg effect, although the statistical significance of this alignment has been questioned, in particular in recent years. We will come back to this issue in Sect. 7.​8.

In fact, the Local Group is not a group of galaxies in the sense of this chapter; its spatial extent is too large for a group of this mass, and it is not dynamically relaxed. The bimodal distribution of galaxies in the Local Group seen in Fig. 6.4 instead suggests that two small galaxy groups—one centered on M31, the other one centered on the Milky Way—are in a process of merging.

We will present a simple mass estimate of the Local Group, from which we will find that it is considerably more massive than one would conclude from the observed luminosity of the associated galaxies.

M31 is one of the very few galaxies with a blueshifted spectrum. Hence, Andromeda and the Milky Way are approaching each other, at a relative velocity of v ≈ 120 km∕s. This value results from the velocity of M31 relative to the Sun of v ≈ 300 km∕s, and from the motion of the Sun around the Galactic center. Together with the distance to M31 of D ∼ 770 kpc, we conclude that both galaxies will collide on a time-scale of ∼ 6 × 10⁹ yr, if we disregard the transverse component of the relative velocity. From measurements of the proper motion of M31, one finds that its transverse velocity is small—thus a collision with the Milky Way will almost certainly occur.

The luminosity of the Local Group is dominated by the Milky Way and by M31, which together produce about 90 % of the total luminosity. If the mass density follows the light distribution, the dynamics of the Local Group should also be dominated by these two galaxies. Therefore, one can try to estimate the mass of the two galaxies from their relative motion, and with this also the mass of the Local Group.

In the early phases of the Universe, the Galaxy and M31 were close together and both took part in the Hubble expansion. By their mutual gravitational attraction, their relative motion was decelerated until it came to a halt—at a time t _max at which the two galaxies had their maximum separation r _max from each other. From this time on, they have been moving towards each other. The relative velocity v(t) and the separation r(t) follow from the conservation of energy, where M is the sum of the masses of the Milky Way and M31, and C is an integration constant, related to the total energy of the M31/MW-system. This constant can be determined by considering (6.1) at the time of maximum separation, when r = r _max and v = 0. With this, follows immediately. Since , (6.1) is a differential equation for r(t),

It can be solved using the initial condition r = 0 at t = 0. For our purpose, an approximate consideration is sufficient. Solving the equation for dt we obtain, by integration, a relation between r _max and t _max, Since the differential equation is symmetric with respect to changing v → −v, the collision will happen at 2t _max. Estimating the time from today to the collision, by assuming the relative velocity to be constant during this time, then yields , and one obtains , or where t ₀ ≈ 14 × 10⁹ yr is the current age of the Universe. Hence, together with (6.2) this yields Now by inserting the values r(t ₀) = D and v = v(t ₀), we obtain the mass M, This mass is much larger than the mass of the two galaxies as observed in stars and gas. The mass estimate yields a mass-to-light ratio for the Local Group of , much larger than that of any known stellar population. This is therefore another indication of the presence of dark matter because we can see only about 5 % of the estimated mass in the Milky Way and Andromeda. Another mass estimate follows from the kinematics of the Magellanic Stream, which also yields .

Tidal streams. One of the most interesting galaxies in the Local Group is the Sagittarius dwarf galaxy which was only discovered in 1994. Since it is located in the direction of the Galactic bulge, it is barely visible on optical images, if at all, as an overdensity of stars. Furthermore, it has a very low surface brightness. It was discovered in an analysis of stellar kinematics in the direction of the bulge, in which a coherent group of stars was found with a velocity distinctly different from that of bulge stars. In addition, the stars belonging to this overdensity have a much lower metallicity, reflected in their colors. The Sagittarius dwarf galaxy is located close to the Galactic plane, at a distance of about 16 kpc from the Galactic center and nearly in the direct extension of our line-of-sight to the GC. This proximity implies that it must be experiencing strong tidal gravitational forces on its orbit around the Milky Way; over the course of time, these will have the effect that the Sagittarius dwarf galaxy will be slowly disrupted. In fact, in recent years a relatively narrow band of stars was found around the Milky Way. These stars are located along the orbit of the Sagittarius galaxy (see Fig. 2.​18). Their chemical composition supports the interpretation that they are stars stripped from the Sagittarius dwarf galaxy by tidal forces. In addition, globular clusters were identified which presumably once belonged to the Sagittarius dwarf galaxy, but which were also removed from it by tidal forces and are now part of the globular cluster population in the Galactic halo. Indeed, more tidal streams have been discovered recently, both in the Milky Way and in Andromeda, as well as in other neighboring galaxies.

The neighborhood of the Local Group. The Local Group is indeed a concentration of galaxies: while it contains more than 50 members within ∼ 1 Mpc, the next neighboring galaxies are found only in the Sculptor Group, which contains about six members¹ and is located at a distance of D ∼ 1. 8 Mpc. The next galaxy group after this is the M81-group of ∼ 8 galaxies at D ∼ 3. 1 Mpc, the two most prominent galaxies of which are displayed in Fig. 6.9.

The other nearby associations of galaxies within 10 Mpc from us shall also be mentioned: the Centaurus group with 17 members and D ∼ 3. 5 Mpc, the M101-group with 5 members and D ∼ 7. 7 Mpc, the M66- and M96-group with together 10 members located at D ∼ 9. 4 Mpc, and the NGC 1023-group with 6 members at D = 9. 6 Mpc.

Most galaxies are members of a group. Many more dwarf galaxies exist than luminous galaxies, and dwarf galaxies are located preferentially in the vicinity of larger galaxies. Some members of the Local Group are so under-luminous that they would hardly be observable outside the Local Group.

One large concentration of galaxies was already known in the eighteenth century (W. Herschel)—the Virgo cluster. Its galaxies extend over a region of about 10^∘× 10^∘ in the sky, and its distance is D ∼ 16 Mpc. The Virgo cluster consists of about 250 large galaxies and more than 2000 smaller ones; the central galaxy of the cluster is the radio galaxy M87 (Fig. 1.​11). In the classification scheme of galaxy clusters, Virgo is considered an irregular cluster. The closest regular massive galaxy cluster is the Coma cluster (see Fig. 1.​17), at a distance of about D ∼ 90 Mpc.

George Abell compiled a catalog of galaxy clusters, published in 1958, in which he identified regions in the sky that show an overdensity of galaxies. This identification was performed by eye on photoplates from the Palomar Observatory Sky Survey (POSS), a photographic atlas of the Northern (δ > −30^∘) sky (see Sect. 1.​4). He omitted the Galactic disk region because the observation of galaxies is considerably more problematic there, due to extinction and the high stellar density (see also Fig. 6.2).

Abell’s criteria and his catalog. The criteria Abell applied for the identification of clusters refer to an overdensity of galaxies within a specified solid angle. According to these criteria, a cluster contains ≥ 50 galaxies in a magnitude interval m ₃ ≤ m ≤ m ₃ + 2, where m ₃ is the apparent magnitude of the third brightest galaxy in the cluster.² These galaxies must be located within a circle of angular radius

where z is the estimated redshift. The latter is estimated by the assumption that the luminosity of the tenth brightest galaxy in a cluster is the same for all clusters. A calibration of this distance estimate is performed on clusters of known redshift. θ _A is called the Abell radius of a cluster, and corresponds to a physical radius of R _A ≈ 1. 5h ⁻¹ Mpc.

The so-determined redshift should be within the range 0. 02 ≤ z ≤ 0. 2 for the selection of Abell clusters. The lower limit is chosen such that a cluster can be found on a single POSS photoplate ( ∼ 6^∘× 6^∘) and does not extend over several plates, which would make the search more difficult, e.g., because the photographic sensitivity may differ for individual plates. The upper redshift bound is chosen due to the sensitivity limit of the photoplates.

The Abell catalog contains 1682 clusters which all fulfill the above criteria. In addition, it lists 1030 clusters that were found in the search, but which do not fulfill all of the criteria (most of these contain between 30 and 49 galaxies). An extension of the catalog to the Southern sky was published by Abell, Corwin & Olowin in 1989. This ACO catalog contains 4076 clusters, including the members of the original catalog. Another important catalog of galaxy clusters is the Zwicky catalog (1961–1968), which contains more clusters, but which is considered less reliable, since the applied selection criteria resulted in more spurious cluster candidates than is the case for the Abell catalog.

Problems in the optical search for clusters. The selection of galaxy clusters from an overdensity of galaxies on the sphere is not without problems, in particular if these catalogs are to be used for statistical purposes. An ideal catalog ought to fulfill two criteria: first it should be complete, in the sense that all objects which fulfill the selection criteria are contained in the catalog. Second it should be pure (often also called ‘reliable’), i.e., it should not contain any objects that do not belong in the catalog because they do not fulfill the criteria (so-called false positives). The Abell catalog is neither complete, nor is it pure. We will briefly discuss why completeness and reliability cannot be expected in a catalog compiled in this way.

A galaxy cluster is a three-dimensional object, whereas galaxy counts on images are necessarily based on the projection of galaxy positions onto the sky. Therefore, projection effects are inevitable. Random overdensities on the sphere caused by line-of-sight projection may easily be classified as clusters. The reverse effect is likewise possible: due to fluctuations in the number density of foreground galaxies, a cluster at high redshift may be classified as an insignificant fluctuation—and thus remain undiscovered.

Of course, not all members of a cluster classified as such are in fact galaxies in the cluster, as here projection effects also play an important role. Furthermore, the redshift estimate is relatively coarse. In the meantime, spectroscopic analyses have been performed for many of the Abell clusters, and it has been found that Abell’s redshift estimates have an error of about 30 %—they are surprisingly accurate, considering the coarseness of his assumptions.

The Abell catalog is based on visual inspection of photographic plates. It is therefore partly subjective. Today, the Abell criteria can be applied to digitized images in an objective manner, using automated searches. From these, it was found that the results are not much different. The visual search thus was performed with great care and has to be recognized as a great accomplishment. For this reason, and in spite of the potential problems discussed above, the Abell and the ACO catalogs are still frequently used.

The clusters in the catalog are ordered by right ascension and are numbered. For example, Abell 851 is the 851st entry in the catalog, also denoted as A851. With a redshift of z = 0. 41, A851 is the most distant Abell cluster.

Abell classes. The Abell and ACO catalogs divide clusters into so-called richness and distance classes. Table 6.2 lists the criteria for the richness classes, while Table 6.3 lists those for the distance classes, with the number of clusters in each class referring to the original Abell catalog (i.e., without the ACO extension).

There are six richness classes, denoted from 0 to 5, according to the number of cluster member galaxies. Richness class 0 contains between 30 and 49 members and therefore does not belong to the cluster catalog proper. One can see from Table 6.2 that the number of clusters rapidly decreases with increasing richness class, so only very few clusters exist with a very large number of cluster galaxies. As a reminder, the region of the sky from where the Abell clusters were detected is about half of the total sphere. Thus, only a few very rich clusters do indeed exist (at redshift ≲ 0.2). The only cluster with richness class 5 is A 665.

The subdivision into six distance classes is based on the apparent magnitude of the tenth brightest galaxy, in accordance with the redshift estimate for the cluster. Hence, the distance class provides a coarse measure of the distance.

Clusters are also classified by the morphology of their galaxy distribution. Several classifications are used, one of which is displayed in Fig. 6.10. Since this is a description of the visual impression of the galaxy distribution, the exact class of a cluster is not of great interest. However, a rough classification can provide an idea of the state of a cluster, i.e., whether it is currently in dynamical equilibrium or whether it has been heavily disturbed by a merger process with another cluster. Therefore, one distinguishes in particular between regular and irregular clusters, and also those which are intermediate; the transition between classes is of course continuous. Regular clusters are ‘compact’ whereas, in contrast, irregular clusters are ‘open’ (Zwicky’s classification criteria).

This morphological classification indeed points at physical differences between clusters, as correlations between morphology and other properties of galaxy clusters show. For example, it is found that regular clusters are completely dominated by early-type galaxies, whereas irregular clusters have a fraction of spirals nearly as large as in the general distribution of field galaxies. Very often, regular clusters are dominated by a cD galaxy at the center, and their central galaxy density is very high. In contrast, irregular clusters are significantly less dense in the center. Irregular clusters often show strong substructure, which is rarely found in regular clusters. Furthermore, regular clusters have a high richness, whereas irregular clusters have fewer cluster members. To summarize, regular clusters can be said to be in a relaxed state, whereas irregular clusters are still in the process of evolution.

Accumulations of galaxies that do not satisfy Abell’s criteria are in most cases galaxy groups. Hence, groups are the continuation of clusters towards fewer member galaxies and are therefore presumably of lower mass, lower velocity dispersion, and smaller extent. The distinction between groups and clusters is at least partially arbitrary. It was defined by Abell mainly to be not too heavily affected by projection effects in the identification of clusters. Groups are of course more difficult to detect, since the overdensity criterion for them is more sensitive to projection effects by foreground and background galaxies than for clusters.

A special class of groups are the compact groups, assemblies of (in most cases, few) galaxies with very small projected separations. The best known examples for compact groups are Stephan’s Quintet and Seyfert’s Sextet (see Fig. 6.11). In 1982, a catalog of 100 compact groups (Hickson Compact Groups, HCGs) was published, where a group consists of four or more bright members. These were also selected on POSS photoplates, again solely by an overdensity criterion. The median redshift of the HCGs is about z = 0. 03. Further examples of optical images of HCGs are given in Figs. 6.3 and 1.20.

Follow-up spectroscopic studies of the HCGs have verified that 92 of them have at least three galaxies with conforming redshifts, defined such that the corresponding recession velocities lie within 1000 km/s of the median velocity of group members. Of course, the similarity in redshift does not necessarily imply that these groups form a gravitationally bound and relaxed system. For instance, the galaxies could be tracers of an overdense structure which we happen to view from a direction where the galaxies are projected near each other on the sky. However, more than 40 % of the galaxies in HCGs show evidence of interactions, indicating that these galaxies have near neighbors in three-dimensional space. Furthermore, about three quarters of HCGs with four or more member galaxies show extended X-ray emission, most likely coming from intra-group hot gas, providing additional evidence for the presence of a common gravitational potential well (see Sect. 6.4). Compared to clusters, the intergalactic gas in groups has a lower temperature and, possibly, lower metallicity.

The subjectivity of selecting overdensities on images by eye can of course be overcome by using digital (or digitized) astronomical images and employing algorithms to apply criteria to the data which define an overdensity, or a cluster, respectively. This approach solves one of the aforementioned problems in optical cluster searches. The other problem—namely projection effects—can be overcome if an additional distance measure for potential member galaxies can be applied.³ There are two ways how such a distance indicator can be obtained: one either uses large spectroscopic catalogs of galaxies, such as the SDSS, or, as will be discussed next, one can employ the colors of early-type galaxies.

Color-magnitude diagram. We mentioned before that a large fraction of galaxies in clusters are early-type galaxies. Furthermore, we saw in Sect. 3.​6 that early-type galaxies have rather uniform colors. Indeed, plotting the color of cluster galaxies versus their magnitude, one finds a very well-defined, nearly horizontal sequence (Fig. 6.12). This red cluster sequence (RCS) is populated by the early-type galaxies in the cluster.

The scatter of early-type galaxies around this sequence is very small, which suggests that all early-type galaxies in a cluster have nearly the same color, only weakly depending on luminosity. The small slope seen in Fig. 6.12 is mainly due to the fact that more massive ellipticals have a somewhat higher metallicity, rendering the stellar emission slightly redder. Even more surprising is the fact that the color-magnitude diagrams of different clusters at the same redshift define a very similar red cluster sequence: early-type cluster galaxies with the same redshift and luminosity have virtually the same color. Comparing the red sequences of clusters at different redshifts, one finds that the sequence of cluster galaxies is redder the higher the redshift is. This effect is caused by the redshift of the galaxies, which shift their spectral energy distribution towards longer wavelengths. Hence, by keeping the observed filter bands constant, the colors change as a function of redshift. In fact, the red cluster sequence is so precisely characterized that, from the color-magnitude diagram of a cluster alone, its redshift can be estimated with very high accuracy, provided the photometric calibration is sufficiently good. Furthermore, the accuracy of this estimated redshift strongly depends on the choice of the filters between which the color is measured. Since the most prominent spectral feature of early-type galaxies is the 4000 Å-break, the redshift is estimated best if this rest-frame wavelength, redshifted to 4000 (1 + z) Å, is well covered by the photometric bands employed.

This well-defined red cluster sequence is of crucial importance for our understanding of the evolution of galaxies. We know from Sect. 3.​5 that the composition of a stellar population depends on the mass spectrum at its birth (the initial mass function, IMF) and on its age: the older a population is, the redder it becomes. The fact that cluster galaxies at the same redshift all have roughly the same color indicates that their stellar populations have very similar ages. However, the only age that is singled out is the age of the Universe itself. In fact, the color of cluster galaxies is compatible with their stellar populations being roughly the same age as the Universe at that particular redshift. This also provides an explanation for why the red cluster sequence is shifted towards intrinsically bluer colors at higher redshifts—there, the age of the Universe was smaller, and thus the stellar population was younger. This effect is of particular importance at high redshifts.

The RCS Survey. The cluster red sequence method was used in several multi-band imaging surveys for the detection of clusters. In fact, one of the large imaging surveys carried out with the CFHT was the RCS survey, with its main purpose to detect clusters out to large redshifts. It covered 100 deg² in two filters, and yielded more than 1000 cluster and group candidates, out to redshifts larger than unity. As a follow-up, the RCS II survey covers 900 deg² in three filters, and aims at detecting some 10⁴ clusters out to z ∼ 1.

The maxBCG catalog. The large sky coverage of the SDSS, and the very homogeneous photometry of the five-band imaging data, makes this survey a prime resource for optical cluster finders. Whereas the SDSS is rather shallow, compared to the RCS surveys, and thus cannot find clusters at high redshifts, it enables the most complete cluster searches in the more local Universe. Therefore, several cluster catalogs have been constructed from the SDSS, one of which we want to briefly describe here.

This maxBCG cluster catalog is based on a search algorithm which makes use of three properties of massive clusters. The first is the already mentioned red cluster sequence, i.e., the homogeneous (and redshift dependent) color of early-type galaxies in clusters. Second, most massive clusters are found to have a dominant central galaxy, the brightest cluster galaxy (BCG), whose luminosity can be several times larger than the second brightest cluster member. The third property relates to the radial density profile of galaxies, which, in a first approximation, decreases roughly as 1∕θ from the center to the outside.

Thus, the algorithm searches for overdensities of galaxies with similar color, corresponding to the color of the red sequence within a specified redshift interval, where the brightest of the galaxies is located near the center of the overdensity, and where the radial decline of the galaxy number density is compatible with a 1∕θ-law. More specifically, the maxBCG method searches for concentrations of luminous red galaxies in the redshift interval 0. 1 ≲ z ≲ 0. 3, whose colors agree to within ± 2σ of the width of the red sequence in color and with the brightest of these galaxies near the center (Fig. 6.13). The choice of this redshift interval is motivated by the fact that the g − r color of red galaxies is a simple function of redshift, as the strong 4000 Å-break of early-type galaxies moves through the g-filter in this redshift interval. The color of the overdense population yields an indication of the redshift, which then allows one to obtain the galaxy luminosity from the observed flux. Only red galaxies more luminous than 0. 4L ^∗ are taken into account. Given the depth of the SDSS, a red galaxy with L ≥ 0. 4L ^∗ at z = 0. 3 can be detected—hence, this provides a volume-limited survey for such galaxies. Furthermore, the redshift estimate is used to obtain the physical projected radius R from the observed angular separation.

To characterize the cluster candidate, the number of red sequence galaxies with L ≥ 0. 4L ^∗ within 1h ⁻¹ Mpc of the BCG candidate, N _g, is calculated. For reasons that will become clear when we discuss the formation of dark matter halos in Sect. 7.​5.​1 (see also Problem 6.1), one defines the ‘extent’ of a cluster to be the radius inside of which the mean density is 200 times larger than the critical density of the Universe at this redshift. This radius is denoted by r ₂₀₀, and the mass of the cluster within r ₂₀₀ is then denoted as , often also called the virial mass of the cluster. From earlier cluster studies, it was found that there is a close relation between r ₂₀₀ (or M ₂₀₀) and the number of galaxies within 1h ⁻¹ Mpc, roughly following r ₂₀₀ ∝ N _g. With this estimate of the virial radius, the number of red sequence members within projected radius R = r ₂₀₀ is measured and denoted by N _g,200, which is then called the richness of the cluster.

These criteria have yielded a catalog of 13 823 clusters with N _g,200 ≥ 10 in the 7500 deg² of the SDSS. Their distribution in richness and redshift is shown in Fig. 6.14. This maxBCG catalog is one of the largest cluster catalogs available up to now and has been widely used.

The quality of the catalog can be assessed in a number of ways. Since the SDSS also has a large spectroscopic component, the spectroscopic redshifts for more than 5000 of the BCGs are known; they can be compared to the redshift estimated from the color of the red-sequence cluster members. The difference between the two redshifts has a very narrow distribution with a width of σ _z  ∼ 0. 01—that is, the estimated cluster redshifts are very accurate.

A good catalog should be pure and complete. As mentioned before, purity measures the fraction of objects included in the catalog which are not real clusters, whereas completeness quantifies the number of real clusters which were missed by the selection algorithm. These two quantities can be estimated from simulations, in which mock cluster catalogs are generated and analyzed with the same detection algorithm as the real data. Based on such simulations, one concludes that the maxBCG catalog is ∼ 90 % pure and ∼ 85 % complete, for clusters with masses , corresponding to N _g,200 ≈ 10.

The available spectroscopy of the SDSS can also be used to determine the velocity dispersion σ _v for many of the clusters. The left panel of Fig. 6.15 shows a strong correlation between σ _v and richness, well fit by a power law of the form This strong correlation also shows that cluster richness is a good indicator of cluster mass, since σ _v is expected to be tightly related to the mass, according to the virial theorem; we will return to this aspect soon. The comoving number density of clusters as a function of redshift is shown in the right panel of Fig. 6.15, for different richness bins. The comoving space density is roughly constant for all bins except for the richest, which suggests that the maxBCG catalog does not suffer from serious redshift-dependent incompleteness. The slight decline with increasing redshift in the richest bin is actually expected from structure formation in the Universe, as we will discuss in Sect. 7.​5.​2.

For candidates with N _g,200 < 10, i.e., mass below , the purity and completeness decrease; hence, whereas a large fraction of these candidates are probably clusters or groups at lower mass, projection effects will play an increasingly important role for decreasing richness. To reliably find groups, the selection criteria need to be sharpened, which can be done using spectroscopic redshifts. With those, one can search for galaxy overdensities on the sky which have the same redshifts within a few times the expected velocity dispersion in groups, i.e., the same radial velocity within ∼ 1000 km∕s. This provides a much stricter redshift constraint than is possible with the red sequence method and thus substantially reduces projection effects. Such group catalogs were constructed from the Two-degree field Galaxy Redshift Survey (see Sect. 8.​1.​2) and the SDSS as well. The velocity dispersion in groups is significantly smaller than that in clusters; typical values for groups with only a few members are σ _v  ∼ 300 km∕s (see Fig. 6.15).

Most regular clusters show a centrally condensed number density distribution of cluster galaxies, i.e., the galaxy density increases strongly towards the center. If the cluster is not very elongated, this density distribution can be assumed, to a first approximation, as being spherically symmetric. Only the projected density distribution N(R) is observable. This is related to the three-dimensional number density n(r) through where in the second step a simple transformation of the integration variable from the line-of-sight coordinate r ₃ to the three-dimensional radius was made.

Of course, no function N(R) can be observed, but only points (the positions of the galaxies) that are distributed in a certain way. If the number density of galaxies is sufficiently large, N(R) is obtained by smoothing the point distribution. Alternatively, one considers parametrized forms of N(R) and fits the parameters to the observed galaxy positions. In most cases, the second approach is taken because its results are more robust. A parametrized distribution needs to contain at least five parameters to be able to describe at least the basic characteristics of a cluster. Two of these parameters describe the position of the cluster center on the sky. One parameter is used to describe the amplitude of the density, for which, e.g., the central density N ₀ = N(0) may be used. A fourth parameter is a characteristic scale of a cluster, often taken to be the core radius r _c, defined such that at R = r _c, the projected density has decreased to half the central value, . Finally, one parameter is needed to describe ‘where the cluster ends’; the Abell radius is a first approximation for such a parameter.⁴

Parametrized cluster models can be divided into those which are physically motivated, and those which are of a purely mathematical nature. One example for the latter is the family of Sérsic profiles which is not derived from dynamical models. Next, we will consider a class of distributions that are based on a dynamical model.

Isothermal distributions. These models are based on the assumption that the velocity distribution of the massive particles (this may be both galaxies in the cluster or dark matter particles) of a cluster is locally described by a Maxwell distribution, i.e., they are ‘thermalized’. As shown from spectroscopic analyses of the distribution of the radial velocities of cluster galaxies, this is not a bad assumption. Assuming, in addition, that the mass profile of the cluster follows that of the galaxies (or vice versa), and that the temperature (or equivalently the velocity dispersion) of the distribution does not depend on the radius, so that one has an isothermal distribution of galaxies, then one obtains a one-parameter set of models, the so-called isothermal spheres. These can be described physically as follows.

In dynamical equilibrium, the pressure gradient must be equal to the gravitational acceleration, so that where ρ(r) denotes the density of the distribution, e.g., the density of galaxies. By , this mass density is related to the number density n(r), where is the average particle mass. is the mass of the cluster enclosed within a radius r. By differentiation of (6.9), we obtain The relation between pressure and density is P = nk _B T. On the other hand, the temperature is related to the velocity dispersion of the particles, where is the mean squared velocity, i.e., the velocity dispersion, provided the average velocity vector is set to zero. The latter assumption means that the cluster does not rotate, or contract or expand. If T (or ) is independent of r, then where σ _v ² is the one-dimensional velocity dispersion, e.g., the velocity dispersion along the line-of-sight, which can be measured from the redshifts of the cluster galaxies. If the velocity distribution corresponds to an isotropic (Maxwell) distribution, the one-dimensional velocity dispersion is exactly 1/3 times the three-dimensional velocity dispersion, because of , or With (6.10), it then follows that

Singular isothermal sphere. For general boundary conditions, the differential equation (6.14) for ρ(r) cannot be solved analytically. However, one particular analytical solution of the differential equation exists: By substitution, we can easily show that solves (6.14). This density distribution is called singular isothermal sphere; we have encountered it before, in the discussion of gravitational lens models in Sect. 3.​11.​2. This distribution has a diverging density as r → 0 and an infinite total mass M(r) ∝ r. It is remarkable that this density distribution, ρ ∝ r ⁻², is just what is needed to explain the flat rotation curves of galaxies at large radii.

The divergence of the density towards the center may not appear reasonable, and thus one might search solutions of (6.14) with the more physical boundary conditions ρ(0) = ρ ₀, the central density, and , for the density profile to be flat at the center. Numerical solutions of (6.14) with these boundary conditions (thus, with a flat core) reveal that the central density and the core radius are related to each other by Hence, these physical solutions of (6.14) avoid the infinite density of the singular isothermal sphere. However, these solutions also decrease outwards with ρ ∝ r ⁻², so they have a diverging mass as well. The origin of this mass divergence is easily understood because these isothermal distributions are based on the assumption that the velocity distribution is isothermal, thus Maxwellian with a spatially constant temperature. A Maxwell distribution has wings, hence it (formally) contains particles with arbitrarily high velocities. Since the distribution is assumed stationary, such particles must not escape, so their velocity must be lower than the escape velocity from the gravitational well of the cluster. But for a Maxwell distribution this is only achievable for an infinite total mass.

King models. To remove the problem of the diverging total mass, self-gravitating dynamical models with an upper cut-off in the velocity distribution of their constituent particles are introduced. These are called King models and cannot be expressed analytically. However, an analytical approximation exists for the central region of these mass profiles, Using (6.8), we obtain from this the projected surface mass density The analytical fit (6.17) of the King profile also has a diverging total mass, but this divergence is ‘only’ logarithmic.

These analytical models for the density distribution of galaxies in clusters are only approximations, of course, because the galaxy distribution in clusters is often heavily structured. Furthermore, these dynamical models are applicable to a galaxy distribution only if the galaxy number density follows the matter density. However, one finds that the distribution of galaxies in a cluster often depends on the galaxy type. The fraction of early-type galaxies (Es and S0s) is often largest near the center. Therefore, one should consider the possibility that the distribution of galaxies in a cluster may be different from that of the total matter. A typical value for the core radius is about r _c ∼ 0. 25h ⁻¹ Mpc.

The above argument relates the velocity distribution of cluster galaxies to the mass profile of the cluster, and from this we obtain physical models for the density distribution. This implies the possibility of deriving the mass, or the mass profile, respectively, of a cluster from the observed velocities of cluster galaxies. We will briefly present this method of mass determination here. For this, we consider the dynamical time-scale of clusters, defined as the time a typical galaxy needs to traverse the cluster once, where a (one-dimensional) velocity dispersion σ _v  ∼ 1000 km∕s was assumed. The dynamical time-scale is shorter than the age of the Universe. One therefore concludes that clusters of galaxies are gravitationally bound systems. If this were not the case they would dissolve on a timescale t _cross. Since t _cross ≪ t ₀ one assumes a virial equilibrium, hence that the virial theorem applies, so that in a time-average sense, where are the kinetic and the potential energy of the cluster galaxies, m _i is the mass of the i-th galaxy, v _i is the absolute value of its velocity, and r _ij is the spatial separation between the i-th and the j-th galaxy. The factor 1∕2 in the definition of E _pot occurs since each pair of galaxies occurs twice in the sum.

In writing (6.21) we have assumed that the total mass of the cluster is the sum of all its member galaxies, This assumption is not valid, since, as we will find below, most of the cluster mass is not contained in galaxies. However, if we assume that the total mass is distributed in the same way as the galaxies are, we can associate to each galaxy a ‘representative’ mass, so that the superposition of all these representative masses yields the total mass distribution of the cluster. The mass m _i used in the foregoing equations and below is meant to be this representative mass.

We further define the velocity dispersion, weighted by mass, and the gravitational radius, With this, we obtain for the kinetic and potential energy. Applying the virial theorem (6.20) yields the mass estimate

Transition to projected quantities. The above derivation uses the three-dimensional separations r _i of the galaxies from the cluster center, which are, however, not observable. To be able to apply these equations to observations, they need to be transformed to projected separations. If the galaxy positions and the directions of their velocity vectors are uncorrelated, as it is the case, e.g., for an isotropic velocity distribution, then where R _ij denotes the projected separation between the galaxies i and j. The parameters σ _v and R _G are direct observables; thus, the total mass of the cluster can be determined. One obtains We explicitly point out that this mass estimate no longer depends on the masses m _i of the individual galaxies—rather the galaxies are now test particles in the gravitational potential. With σ _v  ∼ 1000 km∕s and R _G ∼ 1 Mpc as typical values for rich clusters of galaxies, one obtains a characteristic mass of for rich clusters.

The ‘missing mass’ problem in clusters of galaxies. With the mass M and the number N of galaxies, one can now derive a characteristic mass for the luminous galaxies. This mass is found to be very high, . Alternatively, M can be compared with the total optical luminosity of the cluster galaxies, L _tot ∼ 10¹²-, and hence the mass-to-light ratio can be calculated; typically This value exceeds the M∕L-ratio of early-type galaxies by at least a factor of 10. Realizing this discrepancy, Fritz Zwicky concluded as early as 1933, from an analysis of the Coma cluster, that clusters of galaxies must contain considerably more mass than is visible in galaxies—the dawn of the missing mass problem. As we will see further below, this discrepancy between the observed luminosity and estimated mass has by now been firmly established, since other methods for the mass determination of clusters also yield comparable values and indicate that a major fraction of the mass in galaxy clusters consists of (non-baryonic) dark matter. The stars visible in galaxies contribute less than about 5 % to the total mass in clusters of galaxies.

Given the above line of argument, the question of course arises as to whether the application of the virial theorem is still justified if the main fraction of mass is not contained in galaxies. The derivation remains valid in this form as long as the spatial distribution of galaxies follows the total mass distribution. The dynamical mass determination can be affected by an anisotropic velocity distribution of the cluster galaxies and by the possibly non-spherical cluster mass distribution. In both cases, projection effects, which are dealt with relatively easily in the spherically-symmetric case, obviously become more complicated. This is also one of the reasons for the necessity to consider alternative methods of mass determination.

Two-body collisions of galaxies in clusters are of no importance dynamically, as is easily seen from the corresponding relaxation time-scale (3.​3), which is much larger than the age of the Universe. The motion of galaxies is therefore governed by the collective gravitational potential of the cluster. The velocity dispersion is approximately the same for the different types of galaxies, and also only a weak tendency exists for a dependence of σ _v on galaxy luminosity, restricted to the brightest ones (see below in Sect. 6.7.2). From this, we conclude that the galaxies in a cluster are not “thermalized” because this would mean that they all have the same mean kinetic energy, implying . Furthermore, the independence of σ _v of L reconfirms that collisions of galaxies with each other are not dynamically relevant; rather, the velocity distribution of galaxies is defined by collective processes during cluster formation.

Violent relaxation. One of the most important of the aforementioned processes is known as violent relaxation. This process very quickly establishes a virial equilibrium in the course of the gravitational collapse of a mass concentration. The reason for it are the small-scale density inhomogeneities within the collapsing matter distribution which generate, via Poisson’s equation, corresponding fluctuations in the gravitational field. These then scatter the infalling particles and, by this, the density inhomogeneities are further amplified. The fluctuations of the gravitational field act on the matter like scattering centers. In addition, these field fluctuations change over time, yielding an effective exchange of energy between the particles. In a statistical average, all galaxies obtain the same velocity distribution by this process. As confirmed by numerical simulations, this process takes place on a time-scale of t _cross, i.e., roughly as quickly as the collapse itself.

Dynamical friction. Another important process for the dynamics of galaxies in a cluster is dynamical friction. The simplest picture of dynamical friction is obtained by considering the following. If a massive particle of mass m moves through a statistically homogeneous distribution of massive particles, the gravitational force on this particle vanishes due to homogeneity. But since the particle itself has a mass, it will attract other massive particles and thus cause the distribution to become inhomogeneous. As the particle moves, the surrounding ‘background’ particles will react to its gravitational field and slowly start moving towards the direction of the particle trajectory. Due to the inertia of matter, the resulting density inhomogeneity will be such that an overdensity of mass will be established along the track of the particle, where the density will be higher on the side opposite to the direction of motion (thus, behind the particle) than in the forward direction (see Fig. 6.16). By this process, a gravitational field will form that causes an acceleration of the particle against the direction of motion, so that the particle will be slowed down. Because this ‘polarization’ of the medium is caused by the gravity of the particle, which is proportional to its mass, the deceleration will also be proportional to m. Furthermore, a fast-moving particle will cause less polarization in the medium than a slowly moving one because each mass element in the medium is experiencing the gravitational attraction of the particle for a shorter time, thus the medium becomes less polarized. In addition, the particle is on average farther away from the density accumulation on its backward track, and thus will experience a smaller acceleration if it is faster. Combining these arguments, one obtains for the dependence of this dynamical friction where ρ is the mass density in the medium. Applied to clusters of galaxies, this means that the most massive galaxies will experience the strongest dynamical friction, so that they are subject to a significant deceleration through which they move deeper into the potential well. The most massive cluster galaxies should therefore be concentrated around the cluster center, so that a spatial separation of galaxy populations with respect to their masses occurs (mass segregation) . If dynamical friction acts over a sufficiently long time, the massive cluster galaxies in the center may merge into a single one. This is one possible explanation for the formation of cD galaxies. Furthermore, as the most massive (and thus presumably also the most luminous) galaxies are affected strongest by dynamical friction, and are thus the prime candidates for merging with the central galaxy, this may explain the observed gap of ∼ 2 mag between the brightest and second brightest cluster galaxy.

Dynamical friction also plays an important role in other dynamical processes in astrophysics. For example, the Magellanic Clouds experience dynamical friction on their orbit around the Milky Way and thereby lose kinetic energy. Consequently, their orbit will become smaller over the course of time and, in a distant future, these two satellite galaxies will merge with our Galaxy. In fact, dynamical friction is of vital importance in galaxy merger processes which occur in the evolution of the galaxy population, a subject we will return to in Chap. 10.

Compact groups have a lifetime which is much shorter than the age of the Universe. The dynamical time-scale is , thus small compared to . By dynamical friction, galaxies in groups lose kinetic (orbital) energy and move closer to the dynamical center where interactions and mergers with other group galaxies take place, as also seen by the high fraction of member galaxies with morphological signs of interactions. Since the lifetime of compact groups is shorter than the age of the Universe, they must have formed not too long ago. If we do not happen to live in a special epoch of cosmic history, such groups must therefore still be forming today. From dynamical studies, one estimates that—as in clusters—the total mass of groups is significantly larger than the sum of the mass visible in galaxies; a typical mass-to-light ratio is M∕L ∼ 50h (in Solar units), which is comparable to that of the Local Group.

The space between the galaxies in a cluster is filled with hot gas, as visible from X-ray observations. Besides this hot gas there are also stars in between the galaxies. This intracluster light (ICL) is the most recently discovered component in clusters. The detection of such an intergalactic stellar population comes as a surprise at first sight, because our understanding of star formation implies that they can only form in the dense centers of molecular clouds. Hence, one expects that stars cannot form in intergalactic space. This is not necessarily implied by the presence of intergalactic stars, however, since they can also be stripped from galaxies in the course of gravitational interactions between galaxies in the cluster, or the stripping of stars from the outer parts of galaxies in the tidal gravitational field (in the central region) of the cluster, and so form an intergalactic population. The fate of these stars is thus somewhat similar to that of the interstellar medium, which is metal-enriched by the processes of stellar evolution in galaxies before it is removed from these galaxies and becomes part of the intergalactic medium in clusters; otherwise, the substantial metallicity of the ICM could not be explained.⁵ This interpretation is strengthened by the fact that a diffuse optical light component is also seen in (compact) galaxy groups where the strength of tidal interactions is stronger than in clusters.

Observation of intracluster light. The observation of diffuse optical light in clusters of galaxies and, related to this, the detection of the intracluster stellar population, is extremely difficult. Although first indications were already found with photographic plate measurements, the surface brightness of this cluster component is so low that even with CCD detectors the observation is extraordinarily challenging. To quantify this, we note that the surface brightness of this diffuse light component is about 30 mag arcsec⁻² at a distance of several hundred kpc from the cluster center. This value needs to be compared with the brightness of the night sky, which is about 21 mag arcsec⁻² in the V-band. One therefore needs to correct for the effects of the night sky to better than a tenth of a percent for the intergalactic stellar component to become visible in a cluster. Furthermore, cluster galaxies and objects in the foreground and background need to be masked out in the images, in order to measure the radial profile of this diffuse component. This is possible only up to a certain limiting magnitude, of course, up to which individual objects can be identified. The existence of weaker sources has to be accounted for with statistical methods, which in turn use the luminosity function of galaxies. An example of this ICL is shown in Fig. 6.17.

The identification of this diffuse optical light as truly intergalactic origin is hampered by the fact that many clusters host a central cD galaxy. As we mentioned in Sect. 3.​2.​2 (see also Fig. 3.​11), such galaxies have an extended brightness profile with a surface brightness substantially higher than the extrapolation of a de Vaucouleurs profile at large radii. Thus the natural question arises whether the diffuse component is just part of the cD envelope or a separate entity. If the ICL belongs to the central galaxy, it should be gravitationally bound to it; otherwise, it is a genuine intracluster component. This issue can be investigated by kinematical observations of individual stars in the ICL. One finds that the velocity dispersion of the stars in the ICL strongly increases away from the central galaxy, suggesting that they are unbound to it. One of the best individual tracers of the ICL are planetary nebulae which are formed in the final stages of Solar-mass stars. Since they emit a large fraction of their energy in a single emission line, they are ideal targets for spectroscopy. The kinematic study of intergalactic planetary nebulae show that they are not part of the central cluster galaxy. Related studies were carried out also with red giant stars and globular clusters. Also, Type Ia supernovae were found in clusters, but outside any cluster galaxy.

The diffuse light component was investigated in a statistical superposition of the images of several galaxy clusters. Statistical fluctuations in the sky background and uncertainties in the flatfield⁶ determination are in this case averaged out. In this analysis an -law is found for the light distribution in the inner region of clusters, i.e., the (de Vaucouleurs) brightness profile of the central galaxy is measured (see Fig. 6.18). For radii larger than about ∼ 50 kpc, the brightness profile exceeds the extrapolation of the de Vaucouleurs profile, and is detected out to very large distances from the cluster center.

The ICL is best studied in nearby groups and clusters; only for them can individual objects be detected to a low luminosity level. Furthermore, the cosmological surface brightness dimming (see problem 6.2) renders the detection of ICL increasingly more difficult when turning to higher redshift. Nevertheless, an ICL component was discovered out to redshift z ∼ 0. 5, perhaps even as far out as z ∼ 1. The diffuse cluster component accounts for about 10 % of the total optical light in a cluster; in some clusters this fraction can be even higher. Therefore, models of galaxy evolution in clusters should provide an explanation for these observations.

Clusters of galaxies are the brightest extragalactic X-ray sources besides AGNs. If an X-ray telescope is pointed away from the Galactic disk, about 85 % of the detected sources are AGNs, the remaining ∼ 15 % are clusters. In contrast to AGNs, for which the X-ray emission is essentially point-like, the X-ray emission of clusters is extended. Their characteristic luminosity is L _X ∼ 10⁴³ erg∕s up to ∼ 10⁴⁵ erg∕s for the most massive systems. The fact that this X-ray emission is spatially extended implies that it does not originate from individual galaxies. The spatial region from which we can detect this radiation can have a size of 1 Mpc or even larger. In accordance with the extended nature of the X-ray source, no variability of its X-ray flux has been detected.

Continuum radiation. The spectral energy distribution of the X-rays leads to the conclusion that the emission process is optically thin thermal bremsstrahlung (free-free radiation) from a hot gas which is collisionally ionized. This radiation is produced by the acceleration of electrons in the Coulomb field of protons and atomic nuclei. Since an accelerated electrically charged particle emits radiation, such scattering processes between electrons and protons in an ionized gas yields emission of photons. From the spectral properties of this radiation, the gas temperature in galaxy clusters can be determined, which is, for clusters with mass between and , in the range of 10⁷ K to 10⁸ K, or 1 to 10 keV, respectively.

The emissivity of bremsstrahlung is described by

where e denotes the elementary charge, n _e and n _i the number density of electrons and ions, respectively, Ze the charge of the ions, and m _e the electron mass. The dimensionless function g _ff is called Gaunt-factor; it is a quantum mechanical correction factor of order 1, or, more precisely, Hence, the spectrum described by (6.31) is flat for , and exponentially decreasing for , as is displayed in Fig. 6.22.

The temperature of the gas in massive clusters is typically T ∼ 5 × 10⁷ K, or k _B T ∼ 5 keV—X-ray astronomers usually specify temperatures and frequencies in keV (see Appendix C). For a thermal plasma with Solar abundances, the total bremsstrahlung emission is

Line emission. The assumption that the X-ray emission originates from a hot, diffuse gas (intracluster medium, ICM) was confirmed by the discovery of line emission in the X-ray spectrum of clusters. One of the most prominent lines in massive clusters is located at energies just below 7 keV: it is the Lyman-α (“Kα”) line of 25-fold ionized iron (thus, of an iron nucleus with only a single electron). Slightly less ionized iron has a strong transition at somewhat lower energies of E ∼ 6. 4 keV. Later, other lines were also discovered in the X-ray spectrum of clusters. As a rule, the hotter the gas is, thus the more completely ionized it is, the weaker the line emission. The X-ray emission of clusters with relatively low temperatures, k _B T ≲ 2 keV, is sometimes dominated by line emission from highly ionized atoms (C, N, O, Ne, Mg, Si, S, Ar, Ca, and a strong line complex of iron at E ∼ 1 keV—see Fig. 6.22). The emissivity of a thermal plasma with Solar abundance and temperatures in the range can roughly be approximated by Equation (6.33) accounts for free-free emission as well as line emission. Compared to (6.32), one finds a different dependence on temperature: while the total emissivity for bremsstrahlung is ∝ T ^1∕2, it increases again towards lower temperatures where the line emission becomes more important (see also Fig. 10.​3 for the temperature dependence of the emissivity of a gas). It should be noted in particular that the emissivity depends quadratically on the density of the plasma, since both bremsstrahlung and the collisional excitation responsible for line emission are two-body processes. Thus in order to estimate the mass of the hot gas from its X-ray luminosity, the spatial distribution of the gas needs to be known. For example, if the gas in a cluster is locally inhomogeneous, the value of which determines the X-ray emissivity may deviate significantly from . As we will see later, clusters of galaxies satisfy a number of scaling relations, and one relation between the gas mass and the X-ray luminosity is found empirically, from which the gas mass can be estimated. One finds that the mass in the intracluster gas is about five to ten times larger than the mass of the stars in the galaxies, where this ratio slightly increases with increasing cluster mass.

Morphology of the X-ray emission. From the morphology of their X-ray emission, one can roughly distinguish between regular and irregular clusters, as is also done in the classification of the galaxy distribution. In Fig. 6.23, X-ray surface brightness contours are superposed on optical images of four galaxy clusters or groups, covering a wide range of cluster mass and X-ray temperature. Regular clusters show a smooth brightness distribution, centered on the optical center of the cluster, and an outwardly decreasing surface brightness. In contrast, irregular clusters may have several brightness maxima, often centered on cluster galaxies or subgroups of cluster galaxies.

Typically, regular clusters have an X-ray luminosity L _X and temperature that smoothly increases with cluster mass. In contrast, irregular clusters at a given mass can be either hotter or cooler than regular clusters. Irregular clusters are the result of a recent merging event, and their temperature depends on the stage of the merging process. In the initial phases of the merger, the kinetic energy is not yet thermalized, and thus the gas remains at approximately the same temperature it had before the merging event. Later on, the gas is heated by shock fronts in which the kinetic energy is transformed into internal energy of the gas—i.e., heat. In this phase, the gas temperature can be higher than that of a regular cluster of the same mass. Finally, the cluster settles into an equilibrium state. Indications of past merger events can be seen in substructures of the X-ray emitting gas; even for the Coma cluster, which is frequently considered a typical example of a relaxed cluster, signs of previous merger events can be detected, as shown in Fig. 6.24.

The trend emerges that in clusters with a larger fraction of spiral galaxies, L _X and T are lower. Irregular clusters typically also have a lower central density of galaxies and gas compared to regular clusters. Clusters of galaxies with a dominant central galaxy often show a strong central peak in X-ray emission. The X-ray emission often deviates from axial symmetry, so that the assumption of clusters being roughly spherically symmetric is not well founded in these cases.

Hydrostatic assumption. To draw conclusions about the properties of the intergalactic (intra-cluster) medium from the observed X-ray radiation and about the distribution of mass in the cluster, the gas distribution needs to be modeled. In fact, as we shall see below, assuming the geometry of the cluster (e.g., spherical symmetry), the radial dependence of the gas density can be directly reconstructed. For this, we first consider the speed of sound in the cluster gas,

where P denotes the gas pressure, ρ _g the gas density, and n the number density of gas particles. Then, the average molecular mass is defined as the average mass of a gas particle in units of the proton mass, so that . For a gas of fully ionized hydrogen, one gets because in this case one has one proton and one electron per ∼ proton mass. The cluster gas also contains helium and heavier elements; accounting for them yields μ ∼ 0. 60. The sound-crossing time for the cluster is

and is thus, for a cluster with T ∼ 10⁸ K, significantly shorter than the lifetime of the cluster, which can be approximated roughly by the age of the Universe. Since the sound-crossing time defines the time-scale on which deviations from the pressure equilibrium are evened out, the gas can be in hydrostatic equilibrium, provided that the last major merging event happened longer ago than the sound-crossing time. In this case, the equation

applies, with Φ denoting the gravitational potential. Equation (6.35) describes how the gravitational force is balanced by the pressure force. In the spherically symmetric case in which all quantities depend only on the radius r, we obtain where M(r) is the mass enclosed within radius r. Here, M(r) is the total enclosed mass, i.e., not just the gas mass, because the potential Φ is determined by the total mass. Note the similarity of this equation with (6.9), except that there P described the dynamical pressure of the galaxies or dark matter particles. By inserting into (6.36), we obtain

This equation is of central importance for the X-ray astronomy of galaxy clusters because it shows that we can derive the mass profile M(r) from the radial profiles of ρ _g and T. Thus, if one can measure the density and temperature profiles, the mass of the cluster, and hence the total density, can be determined as a function of radius.

However, these measurements are not without difficulties. ρ _g(r) and T(r) need to be determined from the X-ray luminosity and the spectral temperature, using the bremsstrahlung emissivity (6.31). Obviously, they can be observed only in projection in the form of the surface brightness from which the emissivity, and thus density and temperature, need to be derived by de-projection, i.e., the inversion of (6.38) to obtain ε _ν (r) in terms of I _ν (R). Furthermore, the angular and energy resolution of X-ray telescopes prior to XMM-Newton and Chandra were not high enough to measure both ρ _g(r) and T(r) with sufficient accuracy, except for the nearest clusters. For this reason, the mass determination was often performed by employing additional, simplifying assumptions.

Isothermal gas distribution. From the radial profile of I(R), ε(r) can be derived by inversion of (6.38). Since the spectral bremsstrahlung emissivity depends only weakly on T for h _P ν ≪ k _B T—see (6.31)—the radial profile of the gas density ρ _g can be derived from ε(r). The X-ray satellite ROSAT was sensitive to radiation of , so that the X-ray photons detected by it are typically from the regime where h _P ν ≪ k _B T.

Assuming that the gas temperature is spatially constant, T(r) = T _g, (6.37) simplifies, and the mass profile of the cluster can be determined from the density profile of the gas.

The β -model. A commonly used method consists of fitting the X-ray data by a so-called β-model. This model is based on the assumption that the density profile of the total matter (dark and luminous) is described by an isothermal distribution, i.e., it is assumed that the temperature of the gas is independent of radius, and at the same time that the mass distribution in the cluster is described by the isothermal model that was discussed in Sect. 6.3.1. With (6.9) and (6.12), we then obtain for the total density ρ(r) On the other hand, in the isothermal case (6.37) reduces to The comparison of (6.39) and (6.40) then shows that , or must apply; thus the gas density follows the total density to some power. Here, the index β depends on the ratio of the dynamical temperature, measured by σ _v , and the gas temperature. Now, using the King approximation for an isothermal mass distribution—see (6.17)—as a model for the mass distribution, we obtain where ρ _g0 is the central gas density. The brightness profile of the X-ray emission in this model is then, according to (6.38),

The X-ray emission of many clusters is well described by this profile,⁷ yielding values for r _c of 0. 1 to 0. 3h ⁻¹ Mpc and a value for the index β = β _fit ≈ 0. 65. Alternatively, β can be measured, with the definition given in (6.41), from the gas temperature T _g and the velocity dispersion of the galaxies σ _v , which yields typical values of β = β _spec ≈ 1. Such a value would also be expected if the mass and gas distributions were both isothermal. In this case, they should have the same temperature, which was presumably determined by the formation of the cluster.

The β -discrepancy. The fact that the two values for β determined above differ from each other (the so-called β-discrepancy) is almost certainly due to the fact that the β-model is too simple. We can see, e.g., from Fig. 6.25 that the gas distribution, at least in the inner part of clusters, does not follow a smooth distribution, nor is its temperature constant. The latter is also reflected by the fact that the measured values for β _fit often depend on the angular range over which the brightness profile is fitted: the larger this range, the larger β _fit becomes, and thus the smaller the discrepancy. This behavior can be understood if the central region of the clusters have a lower temperature than at larger radii. Furthermore, temperature measurements of clusters are often not very accurate because it is the emission-weighted temperature which is measured, which is, due to the quadratic dependence of the emissivity on ρ _g, dominated by the regions with the highest gas density. The fact that the innermost regions of clusters where the gas density is highest tend to have a temperature below the bulk temperature of the cluster (see Fig. 6.26) may lead to an underestimation of ‘the’ cluster temperature. In addition, the near independence of the spectral form of from T for h _P ν ≪ k _B T renders the measurement of T difficult. Chandra and XMM-Newton can measure the X-ray emission at energies of up to E ≲ 10 keV, which resulted in considerably improved temperature determinations.

Such investigations have revealed that the temperature behavior shown in Fig. 6.26 is typical for many clusters: The temperature decreases towards the center and towards the edge, while it is rather constant over a larger range at intermediate radii. Many clusters are found, however, in which the temperature distribution is by no means radially symmetric, but shows distinct substructure. Finally, as another possible explanation for the β-discrepancy, it should be mentioned that the velocity distribution of those galaxies from which σ _v is measured may be anisotropic.

Besides all the uncertainty as to the validity of the β-model, we also need to mention that numerical simulations of galaxy clusters, which take dark matter and gas into account, have repeatedly come to the conclusion that the mass determination of clusters, utilizing the β-model, should achieve an accuracy of better than ∼ 20 %, although different gas dynamical simulations have arrived at distinctly different results.

Dark matter in clusters from X-ray observations. Based on measurements of their X-ray emission, a mass estimate can be performed for galaxy clusters. It is found, in agreement with the dynamical method, that clusters contain much more mass than is visible in galaxies. The total mass of the intergalactic medium is clearly too low to account for the missing mass; its gas mass is only ∼ 15 % of the total mass of a cluster.

Expected consequences of gas cooling. In examining the intergalactic medium, we assumed hydrostatic equilibrium, but we disregarded the fact that the gas cools by its emission and by that loses internal energy. For this reason, once established, a hydrostatic equilibrium in principle cannot be maintained over arbitrarily long times. To decide whether this gas cooling is important for the dynamics of the system, the cooling time-scale needs to be considered. This cooling time turns out to be very long, where is the energy density of the gas and n _e the electron density. Hence, the cooling time is longer than the Hubble time nearly everywhere in the cluster, which allows a hydrostatic equilibrium to be established to a very good approximation. In the centers of clusters, however, the density may be sufficiently large to yield . Here, the gas can cool quite efficiently, by which its pressure decreases. This then implies that, at least close to the center, the hydrostatic equilibrium can no longer be maintained. To re-establish pressure equilibrium, gas needs to flow inwards and is thus compressed. Hence, an inward-directed mass flow should establish itself. The corresponding density increase will further accelerate the cooling process. Since, in addition, the emissivity (6.33) of a relatively cool gas increases with decreasing temperature, this process should then very quickly lead to a strong compression and cooling of the gas in the centers of dense clusters. It is a process which, once started, will accelerate and quickly lead to the cooling down to very low temperatures. In parallel to this increase in density, the X-ray emission should strongly increase, because . As a result of this process, a radial density and temperature distribution should be established with a nearly unchanged pressure distribution. In Fig. 6.25, the cooler gas in the center of the Centaurus cluster is clearly visible.

Some predictions of this so-called cooling flow model have indeed been verified observationally. In the centers of many massive clusters, one observes a sharp central peak in the surface brightness I(R). However, we need to stress that, as yet, no inwards flows have been measured. Such a measurement would be very difficult, though, due to the small expected velocities. The amount of cooling gas can be considerable. We can estimate the mass rate at which the gas should cool and flow inwards due to this cooling. The internal energy U of the gas is related to its mass M by , with u as given above. The loss of this energy due to cooling is the luminosity, , so that For some massive clusters, this estimate lead to cooling rates of tens or even hundreds of Solar masses per year. However, after spectroscopic observations by XMM-Newton became available, we have learned that these very high cooling rates implied by the models were significantly overestimated.

The fate of the cooling gas. The gas cooling in this way will accumulate in the center of the cluster, but despite the expected high mass of cold gas, no clear evidence has been found for it. In clusters harboring a cD galaxy, the cooled gas may, over a Hubble time, contribute a considerable fraction of the mass of this galaxy. Hence, the question arises whether cD galaxies may have formed by accretion in cooling flows. In this scenario, the gas would be transformed into stars in the cD galaxy. However, the star-formation rate in these central galaxies is much lower than the rate by which cluster gas cools, according to the ‘old’ cooling flow models sketched above.

The absence of massive cooling flows. The sensitivity and spectral resolution achieved with XMM-Newton have strongly modified our view of cooling flows. In the standard model of cooling flows, the gas cools from the cluster temperature down to temperatures significantly below 1 keV. In this process many atomic lines are emitted, produced by various ionization stages, e.g., of iron, which strongly depend on temperature. Figure 6.27 (top panel) shows the expected spectrum of a cooling flow in which the gas cools down from the cluster temperature of T _g ≈ 8 keV to essentially T _g = 0, where a chemical composition of 1/3 Solar abundance is assumed. In the central panel, this theoretical spectrum is compared with the spectrum of the cluster Abell 1835, where very distinct discrepancies become visible. In the bottom panel, the model was modified such that the gas cools down only to T _g = 3 keV; this model clearly matches the observed spectrum better.

Hence, cooler gas in the inner regions of clusters is directly detected spectroscopically. However, the temperature measurements from X-ray spectroscopy are significantly different from the prediction of the cooling flow model according to which drastic cooling should take place in the gas, because the process of compression and cooling will accelerate for ever decreasing T _g. Therefore, one expects to find gas at all temperatures lower than the temperature of the cluster. But this seems not to be the case: whereas the central temperature can be considerably smaller than that at larger radii (see, e.g., Fig. 6.26), no gas seems to be present at very small temperatures, although the cooling flow model predicts the existence of such gas. A minimum temperature seems to exist, below which the gas cannot cool, or the amount of gas that cools to T _g ∼ 0 is considerably smaller than expected from the cooling flow model. This lower mass rate of gas that cools down completely would then also be compatible with the observed low star-formation rates in the central galaxies of clusters. In fact, a correlation between the cooling rate of gas as determined from XMM observations and the regions of star formation in clusters has been found (see Fig. 6.28). About 50 % of X-ray luminous clusters show an infrared excess of the BCG, indicating ongoing star formation; furthermore, whereas the BCG in clusters with long cooling time rarely show optical emission lines, most of those in cooling flow clusters do. All these are clear indications that a few percent of the cooling mass rate (6.45) as estimated from the cooling flow model indeed arrive at the central cluster galaxy.

There may be exceptional cases where a larger fraction of the cooling gas can reach the gravitational center of the cluster and cool down to low temperatures. Recently, a cluster at z ∼ 0. 6 was discovered with an extremely large cooling mass rate where the BCG shows signs of massive star formation, estimated to be . Such cases are rare, however, and may be a transitional state in the cluster evolution.

Cool-core clusters. Not all clusters show indications for a cool inner region with a strongly peaked X-ray emission. If clusters are far from an equilibrium state, for example due to a recent merger or infall of a group, the gas will be far from a quasi-steady state, and the foregoing consideration will not apply, not even approximately. Strong mixing of the gas by turbulent motions, or shock fronts which develop when the intracluster medium of two colliding or merging clusters intersect, will prevent the development of a cool, condensed central region. One thus distinguishes between cool-core clusters and non-cool-core clusters. The former ones are expected to be close to hydrostatic equilibrium, whereas the latter ones may deviate from it strongly.

What prevents massive cooling flows? One way to explain the clearly suppressed cooling rates in cooling flows is by noting that many clusters of galaxies harbor an active galaxy in their center. In most cases, this AGN is not a luminous quasar, but radio galaxies are the most common type of AGNs in the BCG of clusters, the activity of which, e.g., in the form of (radio-)jets, may affect the ICM. For instance, energy could be transferred from the jet to the ICM, by which the ICM is heated. This heating might then prevent the temperature from dropping to arbitrarily small values. This hypothesis is supported by the fact that many clusters are known in which the ICM is clearly affected by the central AGN—see Fig. 6.29 for one of the first examples where this effect was seen, and Fig. 6.30 where two active nuclei are detected. In the cluster Abell 2597 shown in Fig. 6.28 the interaction of the radio source with the intracluster gas is also clearly seen. Plasma from the jet seems to locally displace the X-ray emitting gas. By friction and mixing in the interface region between the jet and the ICM, the latter is certainly heated. It is unclear, though, whether this explanation is valid for every cluster, because not every cluster in which a very cool ICM is expected also contains an observed AGN. On the other hand, this is not necessarily an argument against the hypothesis of AGNs as heating sources, since AGNs often have a limited time of activity and may be switching on and off, depending on the accretion rate. Thus, the gas in a cluster may very well be heated by an AGN even if it is currently (at the time of observation) inactive. Evidence for the occurrence of this effect also in galaxy groups is shown in Fig. 6.31.

Two more examples of the impact of a central AGN on the cluster gas are shown in Fig. 6.32. The energetics of this interaction can be enormous, as can be seen in the cluster MS 0735.6+7421 shown in the right panel of Fig. 6.32. The large size of the cavities in the X-ray emitting gas implies that a huge amount of energy was needed to push the gas away. From the cavity size and the gas density, one estimates that about of gas has been displaced, requiring an energy of about 10⁶² erg. If this energy was generated by accretion onto the supermassive black hole located in the central galaxy of the cluster, with a mass-to-energy conversion of 10 %, the mass of the SMBH has grown by . Assuming that the gas was removed with about the sound speed of the ICM, this energy was released in the past ∼ 10⁸ yr, implying a mean luminosity of the central AGN of ∼ 3 × 10⁴⁶ erg∕s of mechanical energy. Hence, by all accounts, this is a very energetic event which strongly impacts on the ICM of this cluster.

Feedback. The heating of the intracluster medium by a central radio source in clusters is the most visible example of feedback. We will encounter other examples lateron when we discuss the evolution of galaxies. As the gas cools and sinks towards the center, the central supermassive black hole can get fresh fuel and starts producing energy. The corresponding energy output in form of kinetic power in the radio jets or in radiation then heats the gas again, preventing efficient cooling and thus limits the mass accretion rate—and thus the fueling of the AGN. Hence, one might have a feedback loop, which does not need to have a stable equilibrium. The mass accretion rate may vary in time, as well as the AGN power output. Most likely, the feedback loop is somewhat more complicated than depicted here, but there is no doubt that AGN feedback is essential for understanding the gas in galaxy clusters.

Wide angle tail radio galaxies. Radio galaxies in clusters often have a different radio morphology than isolated ones. The radio jets necessarily interact with the intracluster gas and will be affected by it. Correspondingly, if the radio galaxy has a significant velocity relative to the ICM, the shape of the jets will get bend by ram pressure. A typical example for such wide angle tail radio source is shown in Fig. 6.33. Radio sources with such strong bended jets essentially only occur in clusters. Therefore, such sources can be used to search for clusters, and indeed these cluster searches have been successful—up to redshifts of order unity.

The Bullet cluster. Clusters of galaxies are indeed excellent laboratories for hydrodynamical and plasma-physics processes on large scales. Shock fronts, for instance in merging clusters, cooling fronts (which are also called ‘contact discontinuities’ in hydrodynamics), and the propagation of sound waves can be observed in their intracluster medium. A particularly good example is the galaxy cluster 1E 0657−56 displayed in Fig. 6.34, also called the ‘Bullet cluster’. To the right of the cluster center, strong and relatively compact X-ray emission (the ‘bullet’) is visible, while further to the right of it one sees an arc-shaped

discontinuity in surface brightness. From the temperature distribution on both sides of the discontinuity one infers that it is a shock front—in fact, the shape of the gas distribution on the right resembles that of the air around a supersonic plane or bullet. The strength of the shock implies that the ‘bullet’ is moving at about v ∼ 3500 km∕s through the intergalactic medium of the cluster. The interpretation of this observation is that we are witnessing the collision of two clusters, where one less massive cluster has passed, from left to right in Fig. 6.34, through a more massive one. The ‘bullet’ in this picture is understood to be gas from the central region of the less massive cluster, which is still rather compact. This interpretation is impressively supported by the group of galaxies to the right of the shock front, which are probably the former member galaxies of the less massive cluster. As this cluster crosses through the more massive one, its galaxies and dark matter are moving collisionlessly, whereas the gas is decelerated by friction with the gas in the massive cluster: the galaxies and the dark matter are thus able to move faster through the cluster than the gas, which is lagging behind (and whose momentum transferred to the gas of the more massive cluster displaces the latter from its original location, centered on the corresponding galaxy distribution). We will see below that this interpretation is verified by a gravitational lens investigation of this double cluster.

Electrons in the hot gas of the intracluster medium can scatter photons of the cosmic microwave background. The optical depth and thus the scattering probability for this Compton scattering is relatively low, but the effect is nevertheless observable and, in addition, is of great importance for the analysis of clusters, as we will now see.

Spectral signature. A photon moving through a cluster of galaxies towards us will change its direction through scattering and thus will not reach us. But since the cosmic background radiation is isotropic, for any CMB photon that is scattered out of the line-of-sight, another photon exists—statistically—that is scattered into it, so that the total number of photons reaching us is preserved. However, the energy of the photons changes slightly through scattering by the hot electrons, in a way that they have an (on average) higher frequency after scattering. Hence, by this inverse Compton scattering (Sect. 5.​4.​4), energy is on average transferred from the electrons to the photons, as can be seen in Fig. 6.35.

As a consequence, this scattering leads to a reduced number of photons at lower energies, relative to the Planck spectrum, which are shifted to higher energy. There is one photon energy where the intensity is unchanged, corresponding to a frequency of 218 GHz; below that frequency, the intensity is lower than that of the CMB, for higher frequencies, the intensity is increased. This effect is called the Sunyaev–Zeldovich effect (SZ-effect). It was predicted in 1970 and has now been observed in a large number of clusters. One example is presented in Fig. 6.36, where the frequency dependence of the effect is clearly seen.

In the Rayleigh–Jeans domain of the CMB spectrum, at wavelengths larger than about 2 mm, the intensity of the CMB is decreased by the SZ-effect. For the change in specific intensity in the RJ part, one obtains where is the Compton-y parameter and σ _T the Thomson cross section for electron scattering. Obviously, y is proportional to the optical depth with respect to Compton scattering, given as an integral over n _e σ _T along the line-of-sight. Furthermore, y is proportional to the gas temperature, because that defines the average energy transfer per scattering event [see (5.​35)]. Overall, y is proportional to the integral over the gas pressure P = nk _B T _g along the line-of-sight through the cluster.

The fact that the SZ-signal Δ I _ν ∕I _ν is independent of cluster redshift allows the investigation of clusters at high redshifts, provided the SZ-signal is spatially resolved. As we will see below, the SZ-effect can also be used to detect clusters in the first place, and this selection is much less biased to low redshifts than for flux-limited optical or X-ray surveys. As an example, the right-hand panel of Fig. 6.37 shows a very massive, high-redshift cluster that was selected by an SZ-survey.

Observations of the SZ-effect provide another possibility for analyzing the gas in clusters. For instance, if one can spatially resolve the SZ-effect, which is possible today with interferometric methods (see Fig. 6.38), one obtains information about the spatial density and temperature distribution. Here it is of crucial importance that the dependence on temperature and gas density is different from that in X-ray emission. Because of the quadratic dependence of the X-ray emissivity on n _e, the X-ray luminosity depends not only on the total gas mass, but also on its spatial distribution. Small-scale clumps in the gas, for instance, would strongly affect the X-ray emission. In contrast, the SZ-effect is linear in gas density and therefore considerably less sensitive to small-scale inhomogeneities in the ICM.

The integrated y-parameter. The amplitude of the spectral distortion caused by the SZ-effect is given by y, and SZ-maps essentially provide a map of y as a function of angular position in the cluster. When integrating y across the cluster, we get the total SZ-signal, hence, the integrated SZ-effect is proportional to the integrated SZ-parameter We see that Y is a measure of the product of gas mass and temperature. Both of these quantities can also be determined from X-ray observations, so that an independent estimate of Y can be obtained. In order to distinguish between the two, one usually denotes the result from SZ-observations by Y _SZ, in contrast to Y _X when it is determined from X-ray studies. These two can be different in general, due to the quadratic dependence of the X-ray emission on local gas density or through the temperature variation inside clusters.

Kinetic SZ-effect. Beside the thermal SZ-effect just described, there is a related effect, called the kinetic SZ-effect. This is due to the fact that clusters may have a peculiar velocity. Suppose the peculiar velocity of a cluster is directed towards us, then the photons scattering in its intracluster gas and reaching us will be scattered by electrons which have an average velocity towards us. These scattered photons thus experience on average a blueshift, which is visible in the CMB spectrum in the direction of this cluster. The kinetic SZ-effect has a different spectral signature than the thermal SZ-effect and can thus in principle be distinguished from it. A robust measurement of the kinetic SZ-effect would allow a direct measurement of the line-of-sight component of the peculiar velocities of clusters. However, the expected amplitude of the kinetic SZ-effect is smaller than that of the thermal SZ-effect by a factor of ∼ 10 and thus much harder to detect. First detections have recently been reported in the literature.

Distance determination. For a long time, the SZ-effect was mainly considered a tool for measuring distances to clusters of galaxies, and from this to determine the Hubble constant. We will now schematically show how the SZ-effect, in combination with the X-ray emission, allows us to determine the distance to a cluster. The change in the CMB intensity has the dependence where L is the extent of the cluster along the line-of-sight. To obtain this relation, we replace the l-integration in (6.47) by a multiplication with L, which yields the correct functional dependence. On the other hand, the surface brightness of the X-ray radiation behaves as Combining these two relations, we are now able to eliminate n _e. Since T _g is measurable from the X-ray spectrum, the dependence remains. Now assuming that the cluster is spherical, its extent L along the line-of-sight equals its transverse extent R = θ D _A, where θ denotes its angular extent and D _A the angular-diameter distance (4.​49) to the cluster. With this assumption, we obtain Hence, the angular-diameter distance can be determined from the measured SZ-effect, the X-ray temperature of the ICM, and the surface brightness in the X-ray domain. Of course, this method is more complicated in practice than sketched here, but it is applied to the distance determination of clusters. In particular, the assumption of the same extent of the cluster along the line-of-sight as its transverse size is not well justified for any individual cluster due to triaxiality, but one expects this assumption to be valid on average for a sample of clusters. Hence, the SZ-effect is another method of distance determination, independent of the redshift of the cluster, and therefore suitable for determining the Hubble constant.

Discussion. The natural question arises whether this method, in view of the assumptions it is based on, can compete with the determination of the Hubble constant via the distance ladder and Cepheids, as described in Sect. 3.​9, or from the CMB, to be discussed in Sect. 8.​7. The same question also needs to be asked for the determination of H ₀ by means of the time delay in gravitational lens systems, which we discussed in Sect. 3.​11.​4. In both cases, the answer is the same: presumably neither of the two methods will provide a determination of the Hubble constant with an accuracy comparable to that achieved by the local methods and from the angular fluctuations in the CMB. Nevertheless, both methods are of great value for cosmology: first, the distance ladder has quite a number of rungs. If only one of these contains an as yet undetected severe systematic error, it could affect the resulting value for H ₀. Second, the Hubble Key Project measured the expansion rate in the local Universe, typically within ∼ 100 Mpc (the distance to the Coma cluster). As we will see later, the Universe contains inhomogeneities on these length scales. Thus, it may well be that we live in a slightly overdense or underdense region of the Universe, where the Hubble constant deviates from the global value. In contrast to this, both the SZ-effect and the lensing method measure the Hubble constant on truly cosmic scales, and both methods do so in only a single step—there is no distance ladder involved. For these reasons, these two methods are of considerable interest in additionally confirming our H ₀ measurements. Another aspect adds to this, which must not be underestimated: even if the same or a similar value results from these measurements as the one from the Hubble Key Project, we still have learned an important fact, namely that the local Hubble constant agrees with the one measured on cosmological scales—as predicted by our cosmological model, which can thus be tested. Indeed, both methods have been applied to quite a number of lens systems and luminous clusters showing an SZ effect, respectively, and they yield values for H ₀ which are compatible within the error bars with the value of H ₀ obtained from the Hubble Key Project.

As we have seen, projection effects may play a crucial role in the selection of galaxy clusters through searching for an overdensity of galaxies on the sphere using optical methods. A more reliable way of selecting clusters is by their X-ray emission, since the hot X-ray gas signifies a deep potential well, thus a real three-dimensional overdensity of matter, so that projection effects become virtually negligible. The X-ray emission is , which again renders projection effects improbable. In addition, the X-ray emission, its temperature in particular, seems to be a very good measure for the cluster mass, as we will discuss further below. Whereas the selection of clusters is not based on their temperature, but on the X-ray luminosity, we shall see that L _X is also a good indicator for the mass of a cluster (see Sect. 6.5).

The first cosmologically interesting X-ray catalog of galaxy clusters was the EMSS (Extended Medium Sensitivity Survey) catalog. It was constructed from archival images taken by the Einstein observatory which were scrutinized for X-ray sources other than the primary target in the field-of-view of the respective observation. These were compiled and then further investigated using optical methods, i.e., photometry and spectroscopy. The EMSS catalog contains 835 sources, most of them AGNs, but it also contains 104 clusters of galaxies. Among these are six clusters at redshift ≥ 0. 5; the most distant is MS 1054−03 at z = 0. 83 (see Fig. 6.21). Since the Einstein images all have different exposure times, the EMSS is not a strictly flux-limited catalog. But with the flux limit known for each exposure, the luminosity function of clusters can be derived from this.

The same method as was used to compile the EMSS was applied to ROSAT archival images by various groups, leading to several catalogs of X-ray-selected clusters. The selection criteria of the different catalogs vary. Since ROSAT was more sensitive than the Einstein observatory, these catalogs contain a larger number of clusters, and also ones at higher redshift (Fig. 6.39). Furthermore, ROSAT performed a survey of the full sky, the ROSAT All-Sky Survey (RASS). The RASS contains about 10⁵ sources distributed over the whole sky. The identification of extended sources in the RASS (in contrast to non-extended sources—about five times more AGNs than clusters are expected) yielded a catalog of clusters which, owing to the relatively short exposure times in the RASS, contains the brightest clusters. The exposure time in the RASS is not uniform over the sky since the applied observing strategy led to particularly long exposures for the regions around the Northern and Southern ecliptic pole (see Fig. 6.40).

From the luminosity function of X-ray clusters, a mass function can be constructed, using the relation between L _X and the cluster mass that will be discussed in the following section. Furthermore, as we will explain in more detail in Sect. 8.​2, this cluster mass function is an important probe for cosmological parameters.

More recently, the sensitivity and throughput of SZ telescopes and instruments became sufficiently large to not only study the SZ-effect of known clusters, but to survey the sky for SZ sources. In 2009 the first SZ-selected clusters were found, and at the time of writing, at least three telescopes are used for constructing cluster catalogs through SZ-selection. The South Pole Telescope (SPT), the Atacama Cosmology Telescope, and the Planck satellite, together have found more than 1000 clusters. The selection criteria for SZ clusters is quite different from those of X-ray (or optical) cluster samples, since we have seen that the strength of the SZ-signal is redshift-independent, as long as the cluster is spatially resolved. If, however, the angular resolution of the observations is not high enough to resolve the SZ-signal from a cluster, distant clusters are more difficult to find, since the signal gets diluted. On the other hand, the SZ-signal is weak, so that only the more massive clusters can be readily discovered. The different selection effects are clearly visible in the redshift and mass distribution of the resulting cluster catalogs, shown in Fig. 6.41. The fact that Planck finds the more massive clusters at redshifts z ≲ 0. 5 originates from its all-sky survey, in which the rare objects are found, whereas the SPT clusters in Fig. 6.41 were selected from a region of 720 deg².

In some galaxy clusters, one finds extended, diffuse radio sources at large cluster-centric radii, the so-called radio relics. These radio sources do not coincide with member galaxies in the cluster, and thus have an intracluster origin. Their shapes are often very elongated or irregular, as can be seen by the example shown in Fig. 6.42. The strong polarization of the radio emission shows that the origin of the radiation is synchrotron emission. Therefore, these radio relics must contain relativistic electrons. About 50 radio relics have been found to date.

As we discussed in Sect. 5.​1.​3, relativistic particles are accelerated in shock fronts, such as occur in supernova remnants. Shock fronts are formed in plasmas with supersonic flow velocities. A natural way of explaining the occurrence of a shock in the outskirts of a cluster is to assume that the cluster has been subject to a recent merger event. If the intracluster gas of the two clusters run into each other, a shock front is formed. For the radio relic shown in Fig. 6.42, the predictions of the shock hypothesis were tested in quite some detail. The radio spectral index α varies strongly across the relic, from α ≈ 0. 6 to about 2.0, indicating that the electron distribution closest to the shock has the flattest spectrum, whereas it continuously steepens away from the shock, due to energy losses. The high degree of polarization ( ∼ 50 %) shows the presence of a very well-ordered magnetic field in the emission region.

The merging hypothesis for the example shown in Fig. 6.42 is further supported by the strongly disturbed morphology of the intracluster gas, as shown by the X-ray contours, as well as by the indication of a second relic on the opposite side of the cluster center; such counter relics are predicted from numerical simulations of cluster mergers.

It is expected that the larger the spatial extent, velocity dispersion of galaxies, temperature of the X-ray gas, and luminosity of a cluster are, the more massive it is. In fact, from theoretical considerations one can deduce the existence of relations between these parameters. The X-ray temperature T specifies the thermal energy per gas particle, which should be proportional to the binding energy for a cluster in virial equilibrium, Since this relation is based on the virial theorem, r should be chosen to be the radius within which the matter of the cluster is virialized. This value for r is called the virial radius r _vir. From theoretical considerations of cluster formation (see Chap. 7), one finds that the virial radius is defined such that within a sphere of radius r _vir, the average mass density of the cluster is about Δ _c ≈ 200 times as high as the critical density ρ _cr of the Universe (see also Problem 6.1). The mass within r _vir is called the virial mass M _vir which is, according to this definition, Combining the two above relations, one obtains This relation can now be observationally tested by using a sample of galaxy clusters with known temperature and with mass determined by the methods discussed in Sect. 6.4.2. An example of this is displayed in Fig. 6.43, in which the mass is plotted versus temperature for clusters from the extended HIFLUGCS sample.⁸ Since it is easier to determine the mass inside a smaller radius than the virial mass itself, the mass M ₅₀₀ within the radius r ₅₀₀, the radius within which the average density is 500 times the critical density, is plotted here. The measured values clearly show a very strong correlation, and best-fit straight lines describing power laws of the form M = AT ^α are also shown in the figure. The exact values of the two fit parameters depend on the choice of the cluster sample; the right-hand panel of Fig. 6.43 shows in particular that galaxy groups (thus, ‘clusters’ of low mass and temperature) are located below the power-law fit that is obtained from higher mass clusters. If one confines the sample to clusters with , the best fit is described by with an uncertainty in the parameters of slightly more than 10 %. This relation is very similar to the one deduced from our theoretical consideration, M ∝ T ^1. 5. With only small variations in the parameters, the relation (6.53) is obtained both from a cluster sample in which the mass was determined based on an isothermal β-model, and from a cluster sample in which the measured radial temperature profile T(r) was utilized in the mass determination [see (6.37)]. Constraining the sample to clusters with temperatures above 3 keV, one obtains a slope of 1. 48 ± 0. 1, in excellent agreement with theoretical expectations. Considerably steeper mass-temperature relations result from the inclusion of galaxy groups into the sample, from which we conclude that they do not follow in detail the scaling argument sketched above.

The velocity dispersion of the galaxies in a cluster also can be related to the mass: from (6.26) we find Together with and , it then follows that This relation can now be tested using clusters for which the mass was determined using the X-ray method, and for which measurements of the velocity dispersion of the cluster galaxies are available. Alternatively, the relation T ∝ σ _v ² can be tested. One finds that these relations are essentially satisfied for the observed clusters. However, the relation between σ _v and M is not as tight as the M-T relation. Furthermore, numerous clusters exist which strongly deviate from this relation. These are clusters of galaxies that are not relaxed, as can be deduced from the velocity distribution of the cluster galaxies (which strongly deviates from a Maxwell distribution in these cases) or from a bimodal or even more complex galaxy distribution in the cluster. These outliers need to be identified, and removed, if one intends to apply the scaling relation between mass and velocity dispersion.

The total X-ray luminosity that is emitted via bremsstrahlung is proportional to the squared gas density and the gas volume, hence it should behave as Estimating the gas density through , where denotes the gas fraction with respect to the total mass of the cluster, and using (6.52), we obtain This relation needs to be modified if the X-ray luminosity is measured within a fixed energy interval. Particularly for observations with ROSAT, which could only measure low-energy photons (below 2. 4 keV), the received photons from massive clusters typically had E _γ  < k _B T, so that the measured X-ray luminosity becomes independent of T. Hence, one expects a modified scaling relation between the X-ray luminosity measured by ROSAT L _{< 2. 4 keV} and the mass of the cluster,

This scaling relation can also be tested empirically, as shown in Fig. 6.44, where the X-ray luminosity in the energy range of the ROSAT satellite is plotted against the virial mass. One can immediately see that clusters of galaxies indeed show a strong correlation between luminosity and mass, but with a clearly larger scatter than in the mass-temperature relation.⁹ Therefore, the temperature of the intergalactic gas is a better mass indicator than the X-ray luminosity or the velocity dispersion of the cluster galaxies.

However, determining the slope of the relation from the data approximately yields , instead of the expected behavior (). Obviously, the above scaling arguments are not valid with the assumption of a constant gas fraction. This discrepancy between theoretical expectations and observations has been found in several samples of galaxy clusters and is considered well established. An explanation is found in models where the intergalactic gas has been heated not only by gravitational infall into the potential well of the cluster. Other sources of heating may have been present or still are. For cooler, less massive clusters, this additional heating should have a larger effect than for the very massive ones, which could also explain the deviation of low-M clusters from the mass-luminosity relation of massive clusters visible in Fig. 6.44. As already argued in the discussion of cooling flows in Sect. 6.4.3, an AGN in the inner regions of the cluster may provide such a heating. The kinetic energy provided by supernovae in cluster galaxies is also considered a potential source of additional heating of the intergalactic gas. Furthermore, the normalization of the luminosity-mass relation appears to be significantly different for cool-core clusters than for others, which supports the idea that heating affects this relation. It is obvious that solving this mystery will provide us with better insights into the formation and evolution of the gas component in clusters of galaxies.

Despite this discrepancy between the simple models and the observations, Fig. 6.44 shows a clear correlation between mass and luminosity, which can thus be used empirically after having been calibrated. Although the temperature is the preferred measure for a cluster’s mass, one will in many cases resort to the relation between mass and X-ray luminosity because determining the luminosity (in a fixed energy range) is considerably simpler than measuring the temperature, for which significantly better photon statistics, i.e., longer exposure times are required.

The foregoing scaling relations connect the cluster mass with an observable. We have seen that the relation between cluster mass and gas temperature yields the smallest dispersion; hence, the X-ray temperature provides the best mass proxy of those that we considered above.

More recently, an alternative quantity for estimating the cluster mass was introduced. We saw in Sect. 6.4.4 above that the integrated SZ-effect is proportional to , where is case of non-isothermal temperature distribution T _g needs to be interpreted as density-weighted temperature. In analogy, one considers which is related to the thermal energy of gas in the cluster. The gas mass is related to the total mass by , where f _g is the gas-mass fraction. Assuming this to be constant, and using (6.52), we expect a scaling relation of the form Numerical simulations including hydrodynamics first suggested that clusters should have a smaller scatter around the Y _X-M relation than for the other scaling relations. From X-ray observations, one indeed finds that clusters obey a relation very close to (6.60), with a slightly different slope, M ∝ Y ^0. 55 (instead of 0.6). Furthermore, the scatter around the Y _X-M relation is smaller than that around the T _X-M relation, and in particular, at least for relaxed clusters, it is well described by a power-law behavior. Hence, Y _X provides a very useful mass proxy.

Center-excised scaling relations. We have seen that one can roughly divide clusters into cool-core clusters, which show a strong peak in the X-ray emission towards the center, and non-cool core clusters, which are presumably dynamically disturbed. At first sight, one would not expect that both types of clusters follow the same scaling relations. Indeed, not only the temperature profile of non-cool core clusters is different from that of cool-core clusters, but as a consequence also their pressure profiles, which is the relevant quantity for the SZ effect [see (6.47)]. Furthermore, we have seen that feedback processes from a central AGN affect the intracluster medium, causing (strong) deviations from spherical symmetry and locally varying temperature. Hence, we would expect that the central regions of clusters do not follow approximately some ‘universal’ behavior or a corresponding scaling relation.

However, if this central region is excised, the shape of the pressure profiles of clusters closely follow a universal profile, out to a radius of r ₅₀₀ (see Fig. 6.45); for larger radii, observations are difficult with the current X-ray observatories, due to the high background and the fact that the X-ray surface brightness decreases steeply for large radii. Hence, once the central region is cut out in the analysis, clusters seem to form a relatively homogeneous class of objects, despite all potential complications of hydrodynamic effects. It is for this reason why clusters can be used for cosmological studies, as we shall describe in Sect. 8.​2.

The foregoing relations between the quantities of clusters were all obtained for very low-redshift objects. When considering clusters at higher redshift, one might expect that these scaling relations will evolve.

As mentioned before, the virial radius of a cluster is defined by the mean density inside of it being larger than the critical density by about a factor of Δ _c ∼ 200. The critical density is a function of redshift, , where ρ _cr,0 is the critical density of the current universe [see (4.​15)], and . Since ρ _cr is larger at higher redshifts, the mean mass density inside clusters also increases, . This affects also the scaling of the characteristic temperature of the cluster,

The bolometric X-ray luminosity, assumed to be due to bremsstrahlung only, is the volume integral over the emissivity (6.32), and so scales like where we assumed a constant temperature of the intracluster gas. If the gas fraction of clusters is a constant, i.e., independent of mass and redshift, then we expect , assuming the clumping factor to stay constant. Together with (6.61), we then obtain the expected scaling of the luminosity, Finally, for the Y -parameter, we have , where we again assumed a constant gas fraction and used (6.61); thus, These relation generalize (6.52), (6.57) and (6.60) to higher redshifts (note that this redshift dependence was included in Fig. 6.45). If we assume that the profiles of clusters are self-similar, i.e., their density and temperature profiles all have the same shape, but different amplitude and scale length, then these relation do not only apply to the virial radius, but also for regions with different density thresholds.

In detail, these scaling relations are observed to be somewhat different. Whereas the redshift scaling in the mass-temperature relation (6.61) seems to apply to high-redshift clusters, the redshift scaling in the mass-luminosity relation (6.62) is not supported by observations; instead, the redshift evolution is observed to be slower than predicted by the self-similar model. Finally, Fig. 6.46 shows the mass-luminosity relation, where the mass determination was done via weak lensing analysis. Whereas there is a very tight correlation between luminosity and mass, the slope of the best-fitting power law is flatter than predicted by the self-similar behavior (6.62), which probably indicates that clusters are not truly self-similar.

Whereas the optical luminosity of galaxies depends not only on the mass of the stars but also on the star formation history, the NIR light is much less dependent on the latter. As we have discussed before, the NIR luminosity is thus quite a reliable measure of the total mass in stars. For this reason, we would expect that the NIR luminosity of a cluster is tightly correlated with its total stellar mass. Furthermore, if the latter is closely related to the total cluster mass, as would be the case if the stellar mass is a fixed fraction of the cluster’s total mass, the NIR luminosity can be used to estimate the masses of clusters.

The Two Micron All Sky Survey (2MASS) provides the first opportunity to perform such an analysis on a large sample of galaxy clusters. One selects clusters of galaxies for which masses were determined by X-ray methods, and then measures the K-band luminosities of the galaxies within the cluster. Figure 6.47 presents the resulting mass-luminosity diagram within r ₅₀₀ for 93 galaxy clusters and groups, where the mass was derived from the clusters’ X-ray temperatures (plotted on the top axis) by means of (6.53). A surprisingly close relation between these two parameters is seen, which can be described by a power law of the form where a Hubble constant of h = 0. 7 is assumed. The dispersion of individual clusters around this power law is about 32 %, where at least part of this scatter originates in uncertainties in the mass determination—thus, the intrinsic scatter is even smaller. This result is of great potential importance for future studies of galaxy clusters, and it renders the NIR luminosity a competitive method for the determination of cluster masses, which is of considerable interest in view of the current and future generation of NIR wide-field instruments (like VISTA on Paranal, for instance).

In 1986, two groups independently discovered unusually stretched, arc-shaped sources in two clusters of galaxies at high redshift (see Figs. 6.48 and 6.49). The nature of these sources was unknown at first; they were named arcs, or giant luminous arcs, which did not imply any interpretation originally. Different hypotheses for the origin of these arcs were formulated, like for instance emission by shock fronts in the ICM, originating from explosive events. All these scenarios were disproven when the spectroscopy of the arc in the cluster Abell 370 showed that the source is at a much higher redshift than the cluster itself. Thus, the arc is a background source, subject to the gravitational lens effect (see Sect. 3.​11) of the cluster. By differential light deflection, the light beam of the source can be distorted in such a way that highly elongated arc-shaped images are produced.

The discovery that clusters of galaxies may act as strong gravitational lenses came as a surprise at that time. Based on the knowledge about the mass distribution of clusters, derived from X-ray observations before ROSAT, it was estimated that the central surface mass density of clusters is not sufficiently high for strong effects of gravitational light deflection to occur. This incorrect estimate of the central surface mass density in clusters originated from analyses utilizing the β-model which, as briefly discussed above, starts with some heavily simplifying assumptions.¹⁰

Hence, arcs are strongly distorted and highly magnified images of galaxies at high redshift. In some massive clusters several arcs were discovered and the unique angular resolution of the HST played a crucial role in such observations. Some of these arcs are so thin that their width is unresolved even by the HST, indicating an extreme length-to-width ratio. For many arcs, additional images of the same source were discovered, sometimes called ‘counter arcs’. The identification of multiple images is performed either by optical spectroscopy (which is difficult in general, because one arc is highly magnified while the other images of the same source are considerably less strongly magnified and therefore much fainter in general, together with the fact that spectroscopy of faint sources is very time-consuming), by multi-color photometry (all images of the same source should have the same color), or by common morphological properties.

Lens models. Once again, the simplest mass model for a galaxy cluster as a lens is the singular isothermal sphere (SIS). This lens model was discussed previously in Sect. 3.​11.​2. Its characteristic angular scale is specified by the Einstein radius (3.​75), or Very high magnifications and distortions of images can occur only very close to the Einstein radius. This immediately yields an initial mass estimate of a cluster, by assuming that the Einstein radius is about the same as the angular separation of the arc from the center of the cluster. The projected mass within the Einstein radius can then be derived, using (3.​81). Since clusters of galaxies are, in general, not spherically symmetric and may show significant substructure, so that the separation of the arc from the cluster center may deviate significantly from the Einstein radius, this mass estimate is not very accurate in general; the uncertainty is presumably ∼ 30 %. Models with asymmetric mass distributions predict a variety of possible morphologies for the arcs and the positions of multiple images, as is demonstrated in Fig. 6.50 for an elliptical lens. If several arcs are discovered in a cluster, or several images of the source of an arc, we can investigate detailed mass models for such a cluster. The accuracy of these models depends on the number and positions of the observed lensed images; e.g., on how many arcs and how many multiple image systems are available for modeling. The resulting mass models are not unambiguous, but they are robust. Clusters that contain many lensed images have very well-determined mass properties, for instance the mass and the mass profile within the radii at which arcs are found, or the ellipticity of the mass distribution and its substructure.

Figure 6.51 shows two clusters of galaxies which contain several arcs. The lensing mass estimate of the cluster Cl 0024+17 is quite different from the mass obtained through an X-ray analysis. Spectroscopy of its member galaxies show evidence for a bimodal distribution in velocity space which is interpreted as the cluster being indeed a pair of clusters, colliding along the line-of-sight. If this interpretation is correct, the mass estimate from X-rays is expected to yield discordant values.

For a long time, A 2218 was the classic example of the existence of numerous arcs in a single galaxy cluster. Then after the installation of the ACS camera on-board HST in 2002, a spectacular image of the cluster A 1689 was obtained in which more than 100 arcs and multiple images were identified (see Fig. 6.52). Several zoomed sections of this image are shown in the lower part of the figure. For clusters of galaxies with such a rich inventory of lens phenomena, very detailed mass models can be constructed.

Such mass models have predictive power, allowing an iterative modeling process. An initial simple mass model is fitted to the most prominent lensed images in the observation, i.e., either giant arcs or clearly recognizable multiple images. In general, this model then predicts further images of the source producing the arc. Close to these predicted positions, these additional images are then searched for, utilizing the morphology of the light distribution and the color. If this initial model describes the overall mass distribution quite well, such images are found. The exact positions of the new images provide further constraints on the lens model which is then refined accordingly. Again, the new model will predict further multiple image systems, and so on. By this procedure, very detailed models can sometimes be obtained. Since the lens properties of a cluster depend on the distance or the redshift of the source, the redshift of lensed sources can be predicted from the identification of multiple image systems in clusters if a detailed mass model is available. These predictions can then be verified by spectroscopic analysis, and the success of this method gives us confidence in the accuracy of the lens models.

Results. We can summarize the most important results of the examination of clusters using arcs and multiple images as follows: the mass of galaxy clusters is indeed much larger than the mass of their luminous matter. The lensing method yields a mass which is in very good agreement with mass estimates from the X-ray method or from dynamical methods. However, the core radius of clusters, i.e., the scale on which the mass profile flattens inwards, is significantly smaller than determined from X-ray observations. If the mass distribution in clusters has a core, its size is estimated to be r _c ∼ 30h ⁻¹ kpc, in contrast to ∼ 150h ⁻¹ kpc obtained from early X-ray analyses.¹¹ This difference leads to a discrepancy in the mass determination between the two methods on scales below ∼ 200h ⁻¹ kpc.

We emphasize that, at least in principle, the mass determination based on arcs and multiple images is substantially more accurate because it does not require any assumptions about the symmetry of the mass distribution, about hydrostatic equilibrium of the X-ray gas, or about an isothermal temperature distribution. On the other hand, the lens effect measures the mass in cylinders because the lens equation contains only the projected mass distribution, whereas the X-ray method determines the mass inside spheres. The conversion between the two methods introduces uncertainties, in particular for clusters which deviate significantly from spherical symmetry. Overestimating the core radius was the main reason why the discovery of the arcs was a surprise because clusters with core radii like the ones determined from the early X-ray measurements would in fact not act as strong gravitational lenses. Hence, the mere existence of arcs shows that the core radius must be small.

A closer analysis of galaxy clusters with a cool core shows that, in these clusters, the mass profile estimated from X-ray observations is compatible with the observed arcs. Such clusters are considered dynamically relaxed, so that for them the assumption of a hydrostatic equilibrium is well justified. The X-ray analysis has to account explicitly for the existence of an inner cool region, though, and the accordingly modified X-ray emission profile is more sophisticated than the simple β-model. Clusters without a cool core are distinctly more complex dynamically. Besides the discrepancy in mass determination, lensing and X-ray methods can lead to different estimates of the center of mass in such unrelaxed clusters, which may indicate that the gas has not had enough time since the last strong interaction or merging process to settle into an equilibrium state.

As we mentioned before, the cluster Cl 0024+17 in Fig. 6.51 most likely has a complex structure along the line-of-sight, and hence it is not surprising that X-ray estimated masses deviate significantly from those obtained by lensing. As a second example, the cluster A 1689 (Fig. 6.52) presumably also has a complex structure. A combined lensing, X-ray and SZ-study of this cluster suggests that it is highly elongated, with an axis ratio of ∼ 2 and the long axis close to the line of sight. This orientation maximizes the lensing strength of a triaxial mass distribution, and clusters selected by their strong lensing features will be biased towards these orientations. In order to account for the differences in mass estimates from these different methods, one concludes that the gas is not in hydrostatic equilibrium, but there must be substantial fraction of non-thermal pressure in that cluster, e.g., from turbulent motions of the intracluster gas.

The mass distribution in clusters often shows significant substructure. Many clusters of galaxies in which arcs are observed are not relaxed. These clusters still undergo dynamical evolution—they are young systems with an age not much larger than t _cross, or systems whose equilibrium was disturbed by a fairly recent merger process. For such clusters, the X-ray method is not well founded because the assumptions about symmetry and equilibrium are not satisfied. The distribution of arcs in the cluster A2218 (Fig. 6.51) clearly indicates a non-spherical mass distribution. Indeed, this cluster seems to consist of at least two massive components around which the arcs are curved, indicating that the cluster is currently undergoing a strong merging event. This is further supported by measurements of the temperature distribution of the intracluster gas, which shows a strong peak in the center, where the temperature is about a factor of two higher than in its surrounding region.

From lens models, we find that for clusters with a central cD galaxy, the orientation of the mass distribution follows that of the cD galaxy quite closely. We conclude from this result that the evolution of the brightest cluster galaxy must be closely linked to the evolution of the cluster, e.g., by accretion of a cooling flow onto the BCG or due to mergers with other cluster members. Often, the shape of the mass distribution very well resembles the galaxy distribution and the X-ray emission.

The principle of the weak lensing effect. In Sect. 3.​11 we saw that gravitational light deflection does not only deflect light beams as a whole, but also that the size and shape of light beams are distorted by differential light deflection. This differential light deflection leads, e.g., to sources appearing brighter than they would be without the lens effect. The giant arcs discussed above are a very clear example of these distortions and the corresponding magnifications.

If some background sources exist which are distorted in such an extreme way as to appear as giant luminous arcs, then it is plausible that many more background galaxies should exist which are less strongly distorted. Typically, these are located at larger angular separations from the cluster center, where the lens effect is weaker than at the locations of the luminous arcs. Their distortion then is so weak that it cannot be identified in an individual galaxy image. The reason for this is that the intrinsic light distribution of galaxies is not circular; rather, the observed image shape is a superposition of the intrinsic shape and the gravitational lens distortion. The former is considerably larger than the latter, in general, and acts as a kind of noise in the measurement of the lensing effect. However, the distortion of adjacent galaxy images should be similar since the gravitational field their light beams are traversing is similar. By averaging over many such galaxy images, the distortion can then be measured (see Fig. 6.53) because no preferred direction exists in the intrinsic orientation of galaxies; it is expected to be random. After the results from the Hubble Deep Field (Fig. 1.​37) became available, if not before, we have known that the sky is densely covered by small, faint galaxies. In deep optical images, one therefore finds a high number density of such galaxies located in the background of a galaxy cluster. Their measured shapes can be used for investigating the weak lensing effect of the cluster.

The distortion, obtained by averaging over image ellipticities, reflects the contribution of the tidal forces to the local gravitational field of the cluster. In this context, it is denoted as shear. It is given by the projection of the tidal component of the gravitational field along the line-of-sight. The shear results from the derivative of the deflection angle, where the deflection angle (3.​62) depends linearly on the surface mass density of the lens. Hence, it is possible to reconstruct the surface mass density of galaxy clusters in a completely parameter-free way using the measured shear: it can be used to map the (total, i.e., dark plus luminous) matter in a cluster.

Observations. Since shear measurements are based on averaging over image ellipticities of distant galaxies, this method of weak gravitational lensing requires optical images with as high a galaxy density as possible. This implies that the exposures need to be very deep to reach very faint magnitudes. But since very faint galaxies are also very distant and, as a consequence, have small angular extent, the observations need to be carried out under very good observing conditions, to be able to accurately measure the shape of galaxy images without them being smeared into circular images by atmospheric turbulence, i.e., the seeing. Typically, to apply this method images from 4-m class telescopes are used, with exposure times of 1–3 h. This way, we reach a density of about 30 galaxies per square arcminute (thus, 10⁵ per square degree) of which shapes are sufficiently well measurable. This corresponds to a limiting magnitude of about R ∼ 25. The seeing during the exposure should not be larger than ∼ 0.​​^″8 to still be able to correct for effects of the point-spread function. The smaller the seeing, the more galaxy images can be used for the weak lensing analysis.

Systematic observations of the weak lensing effect only became feasible with the development of wide-field cameras.¹² This, together with the improvement of the dome seeing at many telescopes and the development of dedicated software for data analysis, rendered quantitative observational studies with weak lensing possible; the best telescopes at the best observatories regularly accomplish seeing below 0.​​^″8, and the dedicated software is specifically designed for measuring the shapes of extremely faint galaxy images and for correcting for the effects of seeing and anisotropy of the point spread function.

Mass reconstruction of galaxy clusters. By means of this method, the reconstruction of the mass density of a large number of clusters became possible. The most important results of these investigations are as follows: the center of the mass distribution corresponds to the optical center of the cluster (see Fig. 6.54). If X-ray information is available, the mass distribution is, in general, found to be centered on the X-ray maximum, except in clusters which are heavily disturbed. The shape of the mass distribution—e.g., its ellipticity and orientation—is in most cases very similar to the distribution of bright cluster galaxies. The comparison of the mass profile determined by this method and that determined from X-ray data agree well, typically within a factor of ∼ 1. 5; we will discuss mass profiles of clusters from weak lensing further below (Sect. 7.​7) where we compare them with cosmological predictions.

Through the weak lensing effect, substructure in the mass distribution is also detected in some clusters (Fig. 6.55) which does not in all cases reflect the distribution of cluster galaxies. However, in general a good correspondence between light and mass exists (Fig. 6.56). From these lensing studies, we obtain a mass-to-light ratio for clusters that agrees with that found from X-ray analyses, about M∕L ∼ 250h in Solar units. Clusters of galaxies that strongly deviate from this average value do exist, however. Two independent analyses for the cluster MS 1224+20 resulted in a mass-to-light-ratio of M∕L ≈ 800h in Solar units, more than twice the value normally found in clusters.

The similarity of the mass and galaxy distributions is not necessarily expected because the lens effect measures the total mass distribution, and therefore mainly the dark matter in a cluster of galaxies. The similar distributions then imply that the galaxies in a cluster seem to basically follow the distribution of the dark matter (Sect. 7.​6.​1), although there are some exceptions.

Bullet clusters, and the nature of dark matter. Figure 6.34 shows the X-ray emission and the galaxy distribution of the so-called Bullet cluster, actually a system of two clusters which have recently collided. Besides the hydrodynamic effects of the collision, this system is of great interest to test for the nature of the discrepancy between the observed mass in clusters (from the stars in the member galaxies and the intracluster gas) and the gravitational mass deduced from the X-ray emission, the galaxy peculiar velocities in clusters, and the lensing results. If this discrepancy is due to a modification of the law of gravity on large scales, as was speculated, then we expect that the bulk of the mass in the Bullet cluster is centered on the X-ray emitting gas, since its mass is several times higher than that of the stars in the cluster galaxies. On the other hand, if the discrepancy is due to (collisionless) dark matter, then this dark matter should behave similarly to the galaxies in a collision: the scattering cross section is very small, and thus the dark matter components of both colliding clusters (like the galaxies) just run through each other. So: where’s the mass?

This question can be answered unambiguously with the weak lensing effect; the result is displayed in Fig. 6.57. The mass is centered on the galaxy distributions of the two clusters, and clearly displaced from the X-ray emitting gas. Therefore, the mass discrepancy cannot be ascribed to a modification of the law of gravity; instead, the main mass component of the clusters must behave very similarly as the galaxies, i.e., collisionless. Thus, the Bullet cluster provides the clearest direct proof for the existence of dark matter. Whereas the Bullet cluster was the first of its kind, in the meantime other cluster collisions have been investigated as well. Two other clusters with very similar behavior are shown in Fig. 6.58.

Dark matter filaments between cluster pairs. The hierarchical structure growth in the Universe, a consequence of the Cold Dark Matter model, predicts that mass concentrations form near the intersection points of dark matter filaments, through which they accrete mass (see Chap. 7). These filaments in general have a density too low to produce a detectable weak lensing signal. However, if two large mass concentrations, such as two clusters, are relatively close to each other, the density of the filament between them is expected to be considerably higher than in the mean. Therefore, pairs of galaxy clusters are the best locations to look for dark matter filaments that connect the members of the pair. Indeed, several attempts for the detection of such filaments have been made, using weak lensing techniques; in most of the cases the significance of the detection is rather low. However, for the best case up to now, the double cluster Abell 222+223, there is a convincing detection of a filament (see Fig. 6.59). In addition, the location of the filament coincides with an overdensity of galaxies and soft X-ray emission. Therefore, this system may be the first detection of a dark matter filament.

Mass calibration of clusters. We saw that the mass determination of clusters from their X-ray emission is far from trivial, owing to the complexity of the baryonic physics affecting the intracluster gas, at least in the inner region of clusters. Any systematic uncertainty in the mass estimate affects the scaling relations, which are an essential tool for cluster cosmology (see Sect. 8.​2). Gravitational lensing offers an alternative possibility for determining cluster masses which is independent of equilibrium and symmetry assumptions. The accuracy of weak lensing masses for individual clusters, out to large radii (e.g., within r ₅₀₀), is not impressive, and is affected by the noise caused by intrinsic galaxy ellipticity, light deflection by other masses along the line-of-sight to the source population, and the mass-sheet degeneracy (see Problem 3.5). However, the mean mass of a statistical sample of clusters can be obtained by using the superposition of the individual lensing signals. The mass estimate from such a ‘stacked’ cluster profile is viewed as being least affected by systematic effects, and therefore frequently employed in constructing scaling relations. In Sect. 7.​7 we will present results of such studies.

The search for clusters of galaxies with weak gravitational lensing. The weak lensing effect can not only be used to map the matter distribution of known clusters, but it can also be employed to search for clusters. Mass concentrations generate a tangentially oriented shear field in their vicinity, which can specifically be searched for. The advantage of this method is that it detects cluster-mass matter concentrations based solely on their mass properties, in contrast to all other methods which rely on the emission of electromagnetic radiation, whether in the form of optical light from cluster galaxies or as X-ray emission from a hot intracluster medium. In particular, if clusters with atypically low gas or galaxy content exist, they could be detected in this way.

Quite a number of galaxy clusters have been detected with this method—see Fig. 6.60. Further candidates exist, where the shear signal indicates a significant mass concentration but it cannot be identified with any concentration of galaxies on optical images. The clarification of the nature of these lens signals is of great importance: if in fact matter concentrations do exist which correspond to the mass of a cluster but which do not contain luminous galaxies, then our understanding of galaxy evolution needs to be revised. However, we cannot exclude the possibility that these statistically significant signals are statistical outliers, or result from projection effects—remember, lensing probes the line-of-sight integrated matter density. Together with the search for galaxy clusters by means of the SZ-effect (Sect. 6.4.4), the weak lensing effect provides an interesting alternative for the detection of mass concentrations compared to the more traditional methods.

The luminosity function of galaxies in a cluster is defined in a similar way as in Sect. 3.​10 for the total galaxy population. In many clusters, the Schechter luminosity function (3.​52) represents a good fit to the data if the brightest galaxy is disregarded in each cluster (see Fig. 3.​51 for the Virgo cluster of galaxies). The slope α at the faint end is not easy to determine, since projection effects become increasingly important for fainter galaxies. The value of α seems to vary between clusters, but it is not entirely clear whether this result may also be affected by projection effects of differing strength in different clusters. Thus, no final conclusion has been reached as to whether the luminosity function has a steep increase at L ≪ L ^∗ or not, i.e., whether many more faint galaxies exist than luminous ∼ L^∗-galaxies (compare the galaxy content in the Local Group, Sect. 6.1.1, where even in our close neighborhood it is difficult to obtain a complete census of the galaxy population). L ^∗ is very similar for many clusters, which is the reason why the distance estimate by apparent brightness of cluster members, as done by Abell for his cluster catalog, is quite reliable, though a number of clusters exists with a clearly deviating value of L ^∗.

However, when averaging over many clusters, the picture becomes much clearer. The luminosity function of galaxies changes as one moves from field galaxies, through groups, to clusters. This change is already noticeable for poor groups with just a few galaxies, such as the Local Group. The faint-end slope of the luminosity function is flatter in group and cluster environments than for field galaxies, and flattens with increasing cluster mass or richness. Most galaxies with L ≲ L ^∗ live either in isolation or in relatively poor groups, whereas the most luminous galaxies are predominantly found in the central regions of rich groups and clusters. Whereas the fraction of elliptical galaxies is much smaller in low-density regions than in the inner part of clusters, they nevertheless dominate the galaxy mix at the highest luminosities.

Brightest Cluster Galaxies. Of special interest in clusters is the Brightest Cluster Galaxy (BCG), which in most cases is located near the center of the cluster. In many cases, the BCG is a cD galaxy; these differ from large ellipticals in several respects. They have a very extended stellar envelope, whose size may exceed R ∼ 100 kpc and whose surface brightness profile is much broader than that of a de Vaucouleurs-profile (see Fig. 3.​11). cD galaxies are found only in the centers of clusters or groups, thus only in regions of strongly enhanced galaxy density. However, the extended stellar envelope merges into the intracluster light, and there is some ambiguity to distinguish between cD envelope and ICL, as discussed in Sect. 6.3.4. Many cD galaxies have multiple cores, which is a rather rare phenomenon among the other cluster members.

BCGs are very luminous and massive; their typical stellar mass is , with a spread of about a factor of 1.5. One finds that the more massive clusters contain the more luminous BCGs. At first sight, this is not too surprising, since the more galaxies a cluster contains, the larger is the probability to find one of them with very high stellar mass (or luminosity). However, there are several indications that this cannot be the sole reason. First, in most clusters, the second brightest cluster galaxy is about 1 mag or more fainter than the BCG, which cannot be explained purely by statistical arguments. Second, the BCGs appear to be not simply the extension of early-type galaxies towards the highest luminosity; for example, BCGs do not fall right on the fundamental plane for ellipticals. Their size-luminosity relation is different from that of ellipticals, in the sense that they are larger than ellipticals with the sameluminosity.

As mentioned several times before, the mixture of galaxy types in clusters seems to differ from that of isolated (field) galaxies. Whereas about 70 % of luminous field galaxies are spirals, clusters are dominated by early-type galaxies, in particular in their inner regions. Furthermore, the fraction of spirals in a cluster depends on the distance from the center and increases for larger r. Obviously, the local density has an effect on the morphological mix of galaxies. As in clusters, the fraction of group members which are spirals is lower than the fraction of spirals among field (i.e., isolated) galaxies, and the relative abundance of spiral galaxies decreases with increasing σ _v of the group.

More generally, one may ask whether the mixture of the galaxy population depends on the local galaxy density. While earlier studies of this effect were frequently confined to galaxies within and around clusters, extensive redshift surveys like the 2dFGRS and the SDSS (see Sect. 8.​1.​2) allow us to systematically investigate this question with very large and carefully selected samples of galaxies. The morphological classification of such large samples is performed by automated software tools, which basically measure the light concentration in the galaxies, or, alternatively, the best-fitting Sérsic-index n [see (3.​39)]. A comparison of galaxies classified this way with visual classifications shows very good agreement.

Results from the Sloan Digital Sky Survey. As an example of such an investigation, results from the Sloan Digital Sky Survey are shown in Fig. 6.61. Galaxies were morphologically classified, based on SDSS photometry, and separated into four classes, corresponding to elliptical galaxies, S0-galaxies, and early (Sa) and late (Sc) types of spiral. In this analysis, only galaxies were included for which the redshift was spectroscopically measured. Therefore, the spatial galaxy density can be estimated. However, one needs to take into account the fact that the measured redshift is a superposition of the cosmic expansion and the peculiar velocity of a galaxy. The peculiar velocity may have rather large values ( ∼ 1000 km∕s), in particular in clusters of galaxies. For this reason, for each galaxy in the sample the surface number density of galaxies which have a redshift within ± 1000 km∕s of the target galaxy was determined. The left panel in Fig. 6.61 shows the fraction of the different galaxy classes as a function of this local galaxy density. A very clear dependence, in particular of the fraction of late-type spirals, on the local density can be seen: in regions of higher galaxy density Sc-spirals contribute less than 10 % of the galaxies, whereas their fraction is about 30 % in low-density regions. Combined, the fraction of spirals decreases from ∼ 65 % in the field to about 35 % in regions of high galaxy density. In contrast, the fraction of ellipticals and S0-galaxies increases towards higher densities, with the increase being strongest for ellipticals.

In the right-hand panel of Fig. 6.61, the mixture of galaxy morphologies is plotted as a function of the distance to the center of the nearest cluster, where the distance is scaled by the virial radius of the corresponding cluster. As expected, a very strong dependence of the fraction of ellipticals and spirals on this distance is seen. Sc-spirals contribute a mere 5 % of galaxies in the vicinity of cluster centers, whereas the fraction of ellipticals and S0-galaxies strongly increases inwards.

The two diagrams in Fig. 6.61 are of course not mutually independent: a region of high galaxy density is very likely to be located in the vicinity of a cluster center, and the opposite is valid accordingly. Therefore, it is not immediately clear whether the mix of galaxy morphologies depends primarily on the respective density of the environment of the galaxies, or whether it is caused by morphological transformations in the inner regions of galaxy clusters.

The morphology-density relation is also seen in galaxy groups. The fraction of late-type galaxies decreases, and the fraction of early-type galaxies increases with decreasing distance from the group center, as is also the case in clusters. When considering the morphological mix of visually classified early- and late-type galaxies, averaged over the whole group or cluster, i.e., up to the virial radius, then it seems to be constant for group/cluster halo masses in excess of .

Alternative consideration: the color-density relation. We pointed out in Sect. 3.​1.​3 that galaxies at fixed luminosity seem to have a bimodal color distribution (see Fig. 3.​7). Using the same data set as that used for Fig. 3.​7, the fraction of galaxies that are contained in the red population can be studied as a function of the local galaxy density. The result of this study is shown in the left-hand panel of Fig. 6.62, where the fraction of galaxies belonging to the red population is plotted against the local density of galaxies, measured in terms of the fifth-nearest neighboring galaxy within a redshift of ± 1000 km∕s. The fraction of red galaxies increases towards higher local number density, and the relative increase is stronger for the less luminous galaxies. If we identify the red galaxies with the early-type galaxies in Fig. 6.61, these two results are in qualitative agreement. Surprisingly, the fraction of galaxies in the red sample seems to be a function of a combination of the local galaxy density and the luminosity of the galaxy, as is shown in the right-hand panel of Fig. 6.62.

Interpretation. A closer examination of Fig. 6.61 may provide a clue as to what physical processes are responsible for the dependence of the morphological mix on the local number density. We consider first the right-hand panel of Fig. 6.61. Three different regimes in radius can be identified: for , the fraction of the different galaxy types remains basically constant. In the intermediate regime, , the fraction of S0-galaxies strongly increases inwards, whereas the fraction of late-type spirals decreases accordingly. This result is compatible with the interpretation that in the outer regions of galaxy clusters spirals lose gas, for instance by their motion through the intergalactic medium (see Fig. 6.63), and these galaxies then transform into passive S0-galaxies. Below , the fraction of S0-galaxies decreases strongly, and the fraction of ellipticals increases substantially.

In fact, the ratio of the number densities of S0-galaxies and ellipticals, for , strongly decreases as R decreases. This may hint at a morphological transformation in which S0 galaxies are turned into ellipticals, probably by collisions or mergers. Such gas-free mergers, also called ‘dry mergers’ , may be the preferred explanation for the generation of elliptical galaxies. One of the essential properties of dry mergers is that such a merging process would not be accompanied by a burst of star formation, unlike the case of gas-rich collisions of galaxies. The existence of a population of newly-born stars in ellipticals would be difficult to reconcile with the generally old stellar population actually observed in these galaxies. We will discuss these issues more thoroughly in Chap. 10.

Considering now the dependence on local galaxy density (the left-hand panel of Fig. 6.61), a similar behavior of the morphological mix of galaxies is observed: there seem to exist two characteristic values for the galaxy density where the relative fractions of galaxy morphologies change noticeably. Interestingly, the relation between morphology and density seems to evolve only marginally between z = 0. 5 and the local Universe.

One clue as to the origin of the morphological transformation of galaxies in clusters, as a function of distance from the cluster center, comes from the observation that the velocity dispersion of very bright cluster galaxies seems to be significantly smaller than that of less luminous ones. Assuming that the mass-to-light ratio does not vary substantially among cluster members, this then indicates that the most massive galaxies have smaller velocity dispersions. One way to achieve this trend in the course of cluster evolution is by dynamical interactions between cluster galaxies. Such interactions tend to ‘thermalize’ the velocity distribution of galaxies, so that the mean kinetic energy of galaxies tends to become similar. This then causes more massive galaxies to become slower on average. If this interpretation holds, then the morphology-density relation may be attributed to these dynamical interactions, rather than to the (so-called ram-pressure) stripping of the interstellar medium as the galaxies move through the intracluster medium.

However, ram-pressure stripping of the interstellar medium has been clearly observed in clusters. In Fig. 6.63, two edge-on spiral galaxies in the Virgo cluster are shown, both in the optical and in the radio. The impact of the intracluster medium, which acts like a ‘wind’ in the rest-frame of the galaxies, can be clearly seen. This effect mostly acts on the atomic gas of spirals, whereas the molecular gas seems to be less affected; we recall that the molecular gas is more densely concentrated towards the galactic disk, and thus more strongly bound. In fact, it has been known for a long time that there are spiral galaxies in groups and clusters which are deficient in neutral hydrogen, relative to field galaxies of the same stellar luminosity. If ram-pressure stripping, or other effects in the dense cluster environment, removes the interstellar medium from spirals, then the result could be a disk galaxy without ongoing star formation—something that may resemble an S0-galaxy. If that were the case, than S0s would be passively fading former spirals. Whereas the original spirals satisfied the Tully–Fisher relation, the S0s would then be expected to be considerably fainter than spirals, at fixed rotational velocity; this indeed is the case.

Ram-pressure stripping not only removes gas from cluster galaxies, but can heat the gas and also trigger local star formation, though the compression of gaseous regions by the pressure. A spectacular example for this is shown in Fig. 6.64. The newly formed stars are no longer gravitationally bound to the galaxy, and will thus contribute to the ICL in the cluster.

E + A galaxies. Galaxy clusters contain a class of galaxies which is defined in terms of spectral properties. These galaxies, which are rare in number, show strong Balmer line absorption in their spectra, characteristic of A stars, but no [Oii] or Hα emission lines. The latter indicates that these galaxies are not undergoing strong star formation at present (since there are no Hii regions around O- and B-stars), whereas the former shows that there was an episode of star formation within the past ∼ 1 Gyr, about as long ago as the main-sequence lifetime of A stars. These galaxies have been termed E + A galaxies since their spectra appears like a superposition of that of A-stars and that of otherwise normal elliptical galaxies. They are interpreted as being post-starburst galaxies. Since they were first seen in clusters, the interpretation of the origin of E + A galaxies was originally centered on the cluster environment—for example star-forming galaxies falling into a cluster and having their interstellar medium removed by tidal forces caused by the cluster potential well and/or stripping as the galaxies move through the intracluster medium. However, E + A galaxies were later also found in different environments, making the above interpretation largely obsolete. By investigating the spatial correlation of these galaxies with other galaxies shows that the phenomenon is not associated with the large-scale environment. An overdensity of neighboring galaxies can be seen only out to scales of ∼ 100 kpc. If the sudden turn-off of the star-formation activity is indeed caused by an external perturbation, it is therefore likely that it is caused by the dynamical interaction of close neighboring galaxies or by mergers. Indeed, about 30 % of E + A galaxies are found to have morphological signatures of perturbations, such as tidal tails, supporting the interaction hypothesis.

In fact, the spiral galaxies in clusters seem to differ statistically from those of field spirals, in that the fraction of disk galaxies with absorption-line spectra, and thus no ongoing star formation, seems to be larger in clusters than in the field by a factor ∼ 4, indicating that the cluster environment has a marked impact on the star-formation ability of these galaxies.