Clusters: ObservationsLast time we talked about some of the context of clusters, and why observations of themhave importance to cosmological issues. Some of the reasons why clusters are useful probesof cosmology are (1) their formation happened relatively recently, and hence depends on avariety of cosmological parameters, (2) they can be observed in many different wavebands,and (3) the dominant components of their mass are relatively simple (dark matter andsmoothly distributed hot gas; galaxies can be thought of as tracer particles to first order).Now we need to determine what the observations actually are, and what they say aboutcosmology, structure formation, and the composition of the universe. To do this we need todetermine what we want to measure about clusters and how to measure those properties.Measurement of important quantitiesAsk class: what are some quantities that we want to measure? Mass and luminositydistribution as a function of radial location in the cluster and of the redshift; separatedistribution of the gas and the dark matter, shape of the cluster, optical depth are examples.Redshift dependence of these is an especially important cosmological probe (because itreflects the evolution of clusters), and it is also important to measure these for a largesample of clusters, to understand the luminosity and mass distribution.The next thing we need to know is how to measure these quantities. As always, ourmeasurements are somewhat indirect, so we need to have a handle on how they’re done.Let’s take the mass as an example. Ask class: what are ways to measure the mass of anindividual cluster? One way is by determining the velocity dispersion of the galaxies. Thatmeans that you take the spectrum of many separate galaxies, determine the redshift ofeach, then compare to the average redshift of the cluster. From these velocities, the massis estimated assuming that the motion of galaxies has been virialized. Another methodis measurement of the temperature of the gas. If the temperature of the gas is close tothe virial temperature, this also indicates the mass. Finally, there is gravitational lensing.Measurement of the gravitational effect on light from background galaxies is a qualitativelydifferent indicator of the mass.Measurement biasesLet’s consider each of these individually, to understand possible problems or selectionbiases. First, measurement of cluster mass by galaxy motions. Ask class: what effectsmight complicate mass determination by this method or bias the results? One point isthat if the velocities are not virial (i.e., due to the gravity of the cluster alone), the massmeasurements can be inaccurate. This is related to the report several years ago that therewas a 1011M¯black hole in the center of a galaxy; the velocities turned out to be dueto a collision with another galaxy instead of just the orbital velocity. Another problem iscluster membership. The typical orbital velocities are around 1000 km s−1, so that wouldbe the velocity dispersion, but in redshift space this would correspond (using Hubble’s law)to a distance of about 14 Mpc! This produces two effects. One is the “finger of God”effect. Suppose you have a large number of galaxies in a cluster, and that they move witha typical velocity of 1000 km s−1, in random directions. The angular width of the clusterisn’t changed by the velocities, so it looks 1 Mpc wide. However, the apparent length ofthe cluster along the line of sight is 20-30 Mpc, so the effect is of a giant finger in redshiftspace pointing at you! The other effect is one that seems to enhance the membershipof clusters. Say you have a cluster, or for that matter a long filament oriented with thelong axis perpendicular to the line of sight. Galaxies not associated with the cluster orfilament tend to fall in towards it. If the galaxy is closer to us than the cluster is, thisincreases its recession velocity, making it seem closer to the cluster in redshift space. Ifthe galaxy is farther away, the fallback velocity towards the cluster decreases its recessionvelocity, again making it seem closer to the cluster in redshift space. Therefore, there isan artificial enhancement of the density. These effects are all linked to the fact that manytimes our only indicator of distance is the redshift. Ask class: how might these problemsbe circumvented? If the true distance, rather than just the redshift, can be measured, thiswould provide an independent check. This is tricky, but measurements of standard candles(e.g., the Tully-Fisher relation for spirals or the fundamental plane for ellipticals) can help.The next method is gas temperature. Ask class: what effects might mess this up? Ifthe temperature is not the virial temperature, then the mass estimate will be bad. Askclass: what are some things that can change the temperature? Cooling of a variety of types(bremsstrahlung, atomic recombination, molecular cooling, metal line cooling) can lowerthe temperature, whereas shock heating can increase the temperature. Thinking about ourgalaxy, for example, the virial temperature for hydrogen is about 107K, but many regionsof the ISM are much cooler. The advantage with clusters is that at 108K and the observeddensities, cooling is relatively slow, so the temperature is a reasonable indicator of the virialtemperature. In particular, Ask class: what is the dominant emission process at such hightemperatures? It’s bremsstrahlung. The volume emissivity at an electron number densitynecm−3and a temperature T = 108T8K isjbremss= 1.5 × 10−23n2eT1/28erg cm−3s−1. (1)At typical densities n = 10−3, this is about 10−29erg cm−3s−1, compared with an energydensity nkT ≈ 10−11erg cm−3, so the cooling time is around 1018s, which is longer thanthe current age of the universe. Note that temperature measurements are independent insome ways from the galaxy velocity method; redshift determinations aren’t necessary, justa measurement of the spectrum.It should also be mentioned that temperature or galaxy velocity measurements givethe standardly inferred masses if Newtonian gravity operates on large scales, ∼1 Mpc, butif gravity is modified at low accelerations (as advocated by Stacy McGaugh), the massescould be less. This is a dark horse explanation, but should be kept in mind unless futureobservations definitively rule it out.The last type of bias to mention pervades extragalactic astronomy. It is calledMalmquist bias. Suppose you have a