Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.

Galaxy clusters, near and far

What is a galaxy cluster?

The answer seems to be obvious: a collection of galaxies within some relatively small volume. And, indeed, when we look at the distribution of galaxies in space on large scales, we find that they do indeed tend to appear in clumps.

Image copyright Tony and Daphne Hallas , provided by Astronomy picture of the day

The clustering tendency is obvious if one makes a map of the distribution of galaxies in space.

Image courtesy of The Galaxy Zoo Blog

Image courtesy of The Sloan Digital Sky Survey

One of the most frequently used catalogs of galaxies is known as the Abell catalog , so let's look at a few of the terms used in it to describe galaxies. The original catalog was described in Corwin, ApJS 3, 211 (1958), but successive papers have increased its scope. The most recent paper in a series is Abell, Corwin and Olowin, ApJS 70, 1 (1989)

Modern galaxy catalogs are defined, not by visual scanning of plates, but by criteria used by a computer to sift through thousands and millions of objects in a big database. Modern catalogs may also use redshift or color information to judge whether galaxies in some particular region of the sky are actually at a common distance. Finally, some modern catalogs are based on observations made at wavelengths outside the optical.

Some jargon: timescales, relaxed vs. uptight clusters, etc.

When lots of galaxies come together in a small volume, there are a number of ways that they can interact. One way to divide these interactions is by computing the timescales for various processes.

crossing time
This one is straightforward. Given the size of a cluster and the typical velocity of galaxies in it, how long does it take a galaxy to move from one side to the other?

        Q:  A typical cluster has a size  
            (radius or diameter, who cares?)
            of around 1 Mpc.

            A typical galaxy in a rich cluster
            has a velocity of order 1000 km/s.

            What is the crossing time for a typical
            rich cluster?   How does that compare
            to the age of the universe?


The crossing time is an appropriate scale for the time over which an initial big cloud of galaxies/proto-galaxies might collapse into a bound cluster. Under certain conditions, the objects in such a collapse will undergo violent relaxation, which means that their dynamical properties will move into some semblance of equilibrium.

two-body relaxation time
On the other hand, in order for all the galaxies in a cluster to reach a particularly ordered type of velocity distribution -- for example, a Maxwellian distribution -- the galaxies must interact gravitationally with each other over and over again. This takes a LOT longer; to a rough order,

        Q:  A typical rich cluster has of order
            N = 1000 members.

            What is the two-body relaxation time for 
            a typical rich cluster?   How does that compare
            to the age of the universe?


As a result, we don't expect to see clusters to show mass segregation, in which the more massive galaxies have all sunk to the central regions, while the low-mass galaxies are flying out to large radii. Globular clusters, on the other hand, do show this sort of segregation because their two-body relaxation times are much shorter.

Some other terms that you may see in connection with clusters are

More evidence for dark matter

It seems reasonable to assume that the galaxies in cluster -- at least in SOME clusters -- are gravitationally bound to the cluster. Otherwise, why should we see so many galaxies grouped together in small volumes of space? If that's true, then we can use the motions of the galaxies to estimate the masses of the clusters, using our old friend the virial theorem.

Now, most of the galaxies in a cluster are certainly not moving in circular orbits. That means that the connection between kinetic energy and gravitational potential energy isn't quite so easy to express as it is for, say, gas in the disk of the Milky Way. Nonetheless, it's clear that the mass of the cluster must be given by some equation of the form

   Q:  The Virgo cluster is rather lumpy, but
       the velocity dispersion is something
       like 700 km/s.

       The radius of the cluster is again hard
       to define, but something like
       3 degrees, and the distance
       is roughly 18 Mpc.

       Estimate the mass of the Virgo cluster.

This sort of calculation was first published by Fritz Zwicky back in 1933; he used the Coma Cluster in his example, deriving a mass of M > 9 x 1046 grams for the cluster. While the measurements of radial velocity available to him at the time were fine, the distance to the cluster was underestimated by a large factor, so his mass is likewise smaller than the value we currently derive.

If we measure the light from the visible galaxies in the cluster, we can compute a mass-to-light ratio for the cluster as a whole.

      Binggeli, Tamman and Sandage (AJ 94, 251, 1987)
      measure a luminosity of roughly

           L(B)  =  120 x 1010 solar

      in the B-band for galaxies within 3 degrees
      of the center of the Virgo Cluster.

  Q:  What is the mass-to-light ratio in blue light
      for the Virgo Cluster?

Zwicky performed this sort of calculation back in the 1930s and found a mass-to-light ratio of 500 for the Coma Cluster. He pointed out that this was quite a puzzle, but the astronomical community in general didn't pay much attention to it.

More sophisticated measurements and more careful calculations yield an overall mass-to-light ratio for Virgo which is several times higher than the one we've just computed. Virgo, of course, is not a particular large cluster, nor does it appear to be a centrally concentrated cluster which has completely virialized. Measurements from other, larger clusters yield mass-to-light ratios which are even higher, of order 200-300 solar.

The stars in typical galaxies will yield a mass-to-light ratio of order 1-10, which means that the stars in a galaxy cluster are responsible for only a few percent of the total mass.

X-ray emission from hot gas in clusters

Galaxy clusters are among the most luminous sources of X-rays in the sky: some produce more than 1045 erg/s of high-energy radiation. The appearance of clusters in X-rays and in optical light is quite different, though, as these images of Abell 1689 show (click on the image to start it blinking).

Image courtesy of Optical: NASA/STScI. X-ray: NASA/CXC/MIT/E.-H Peng et al.

Clearly, the X-rays are coming from some non-stellar source. In fact, the space between the galaxies in a cluster is (often) filled with a very hot, very tenuous gas, mostly hydrogen (like the rest of the universe). This gas emits X-rays because it is very, very hot. Why so hot?

Well, the gas sits within a deep gravitational potential well together with tens or hundreds of galaxies. If the individual gas particles respond to the gravitational influence of the cluster in the same way that the galaxies do, then each will move with a speed which is the same as that of the galaxies, roughly v = 300 km/s. You may recall that there is a relationship between the speed of the particles in an ideal gas and their temperature:

  Q:  What is the temperature of (ionized) hydrogen gas
      in a cluster if each proton has a typical speed
      of 300 km/s?

  Q:  What is the energy kT of a particle
      in this gas?  Express in electron volts.

Now, this hot gas is generally distributed like the galaxies in a cluster: sometimes centrally concentrated around a dominant elliptical galaxy, other times divided into several big clumps. The Coma Cluster, for example, shows two lumps of X-ray emission, each one centered on a big elliptical galaxy.

Images courtesy of
Optical: DSS; X-ray: NASA/CXC/SAO/A. Vikhlinin et al.

The hot gas in Stephan's Quintet, on the other hand, is spread out rather uniformly -- not suprising for a loose group of just a few galaxies.

Images courtesy of X-ray: NASA/CXC/INAF-Brera/G.Trinchieri et al.; Optical: Pal.Obs. DSS

Here's another example, showing the X-ray emission in purple and optical emission in red, yellow and green:

Image courtesy X-ray: NASA/CXC/ESO/P.Rosati et al.; Optical: ESO/VLT/P.Rosati et al.

Just how much hot gas is there? Is it more or less than the mass of the visible stars in the galaxies? Is it enough to account for the gravitational forces within the cluster? How can we tell?

A detour into the physics of X-ray emission

The first thing we need to do is to understand the process by which the X-rays are being produced, and the general form of their spectrum. The basic idea is that the gas within a cluster is so hot that it is ionized -- completely, in the case of hydrogen, or partially, in the case of some heavier elements. Space is filled with a soup of charged particles, zooming past each other at very high speeds. Each time one passes close to another, the electric force accelerates it towards or away from its neighbor, leading to ....

brehmsstrahlung radiation.

This process involves two particles meeting, so the amount of brehmsstrahlung radiation will depend on the density of electrons and the density of ions; in other words, it will depend on the density of particles, squared. The frequency of the emitted photon depends on the speed of the electron; the more kinetic energy the electron has, the higher the energy of the photon which might be emitted. If one goes through all the math, one ends up with an emissivity which looks something like this:

The spectrum of this "thermal brehmsstrahlung" has some similarities to good old thermal blackbody radiation; note the exponential factor. However, there are also differences: a factor of temperature to the negative one-half power, for example. You can see a graph of the spectrum of thermal brehmsstrahlung below. I've chosen to plot the number of photons emitted per second per cubic centimeter, rather than the amount of energy emitted over some frequency or wavelength interval, because the detectors of astronomical X-rays all count photons.

When we look at a distant cluster of galaxies, we must peer through a screen of dust and gas in the Milky Way (and sometimes at intermediate distances, too). X-rays are like other electromagnetic radiation: they can be absorbed by intervening atoms. It turns out that low-energy, or "soft" X-rays, are much more likely to be absorbed than high-energy, "hard" ones. One of the standard references for the quantitative absorption cross-section of X-rays is a paper by Morrison and McCammon, ApJ 270, 119 (1983). They provide a polynomial fit to the somewhat complex variation in absorption of X-rays in the range of most X-ray telescopes.

  Q:  What causes the little jumps in cross-section
      at different energies?

If X-rays must travel through appreciable amounts of the ISM to reach us, then the shape of the observed spectrum will change greatly at the low-energy end.

  Q:  What is the column density through the
      disk of a big spiral galaxy like 
      the Milky Way?

Figure taken from Nakanishi and Sofue, PASJ 55, 191 (2003)

What can you say about this spectrum of the galaxy cluster 1RXSJ153934.7-833535? The spectrum is represented by the crosses and solid line; I've erased a line which was a measure of the background which had been subtracted from the signal.

Figure taken from Grange et al., A&A 513, 2 (2010) and slightly modified.

Back to the question: how much mass in hot gas?

Now that we understand a bit better the nature of this X-ray emission, we might be able to figure out just how much mass it represents. If we make a few simplifying assumptions, we can estimate the mass of the hot gas using quantities we can determine from the X-ray observations. We'll assume

  1. the gas behaves like an ideal gas, so that

    where P is the pressure, ρ is the density of the gas, k is Boltzmann's constant, μ is the mean molecular weight of the gas, and mp is the mass of the proton.

  2. the gas is in hydrostatic equilibrium -- which might be a decent approximation in some clusters, but is clearly not in others (such as Stephan's Quintet). If it is, then

    where Mint is the mass interior to the radius R

  3. the mean molecular weight μ of the gas does not change with radius; in other words, the chemical composition of the hot gas is uniform throughout the cluster.

If one starts with these assumptions, and then puts the equations for hydrostatic equilibrium together with the ideal gas law, one can derive an expression for the mass of gas within some radius R:

    Q:   Can you derive an equation for Mint 
         as a function of radius?

That second form of the equation uses logarithmic derivatives, which are, basically, a description of how fast the RELATIVE changes in one quantity are changing, relative to the RELATIVE changes in another quantity.

For example, if I turn a thermostat with a pointer so the pointer's position p increases by 10%, and as a result the temperature of the room T increases by 20%, then the logarithmic derivative d ln T / d ln p = 2 .

In short, if two quantities are related by some sort of power-law relationship

then the logarithmic derivative tells you what the value of the power law's exponent is.

So, if we can make enough measurements of the X-ray gas in a cluster of galaxies to determine the change in temperature and density as a function of radius, then we can in theory determine the total enclosed gravitational mass as a function of radius.

However, things aren't quite so simple when one examines the X-ray emission from a cluster of galaxies. There are several complicating factors.

Let's look at one good case, in which we have detailed X-ray observations of a cluster. Umetsu et al., ApJ 755, 56 (2012) combine optical and X-ray measurements of the galaxy cluster MACS J1206.2-0847 to determine its mass using several different techniques, including weak and strong lensing, magnification, and X-ray emission under several scenarios.

HST image from Hubblesite Gallery                                         Figure 10 from Umetsu et al., ApJ 755, 56 (2012)

As you can see, the fraction of the total cluster mass which is due to the hot X-ray-emitting gas is pretty small.

Figure 15 from Umetsu et al., ApJ 755, 56 (2012)

For most clusters, we can't disentangle all these complicating factors, and must therefore make even MORE simplifying assumptions in order to make even a rough guess at the enclosed mass.

One frequent method leads to what is called the beta model. We assume

  1. the gas has the same temperature everywhere
  2. the density of all mass is described by an isothermal sphere

These two assumptions lead to the conclusion that the density of the hot gas traces the density of all gravitational mass, but according to some power law:

In fact, one can work out the value of this exponent β; it depends on the ratio of the dynamical temperature -- defined by the motion of galaxies and stars through the cluster -- to the gas temperature -- defined by the motions of gas particles.

In this equation, the symbol σ stands for the velocity dispersion of galaxies in the cluster.

If the gravitational mass of the cluster is distributed like an isothermal sphere, then

and therefore the gas must have a density which behaves like

In this case, when one performs the integral along lines of sight through the cluster to determine the observed intensity of X-ray radiation, one finds that the total intensity as a function of radius will be relatively flat within the core of the cluster, and then fall off rapidly with distance.

At large distances, the observed intensity decreases as the radius to a power which is related to β in a simple way:

So, we finally have a way to start with X-ray observations and end up with an estimate of the mass of the X-ray gas within the cluster.

  1. the "profile" method
  2. the "spectral" method

The two methods don't always agree, which is probably a reflection of the many over-simplifications we've made along the way.

Nonetheless, the are some rough conclusions we can make based on X-ray observations of many clusters. The mass of the hot intracluster gas is

That leaves the majority (about 80%) of the mass required to produce the gravitational motions in the category of dark matter.

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Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.