High Energy Astrophysics, 2011–12 Bremsstrahlung and Galaxy Clusters

N I V E R U S E I T H Y

H F G E R D I N B U High Energy Astrophysics, 2011–12

Bremsstrahlung and galaxy clusters

Philip Best Room C21, Royal Observatory; [email protected] www.roe.ac.uk/∼pnb/teaching.html

1 Bremsstrahlung emission

Having derived the Larmor formula for radiation from an accelerated charge, we can now apply it to diﬀerent physical situations and look at the astrophysical implications. We begin with Bremsstrahlung emission (or ‘braking radiation’) which is the acceleration of a charged particle due to the Coulomb ﬁeld of another charged particle.

Ze+

Figure 1: The process of Bremsstrahlung emission

Here we will concentrate on electron-ion collisions, since electron-electron or ion-ion interactions involve equal and opposite accelerations, and the radiation field from each interferes destructively to give essentially no overall radiation. We will treat the ion as fixed and ignore its acceleration, due to its large inertia (relative to that of the electron). We will also ignore the change in the path of the electron; this is clearly a simplification which breaks down for very close encounters.

1.1 Parseval’s theorem; obtaining the spectral shape

dE Larmor’s formula indicates that the energy radiation, dt , depends on the acceleration of the particle, v˙ (t). Consider the Fourier transform of this acceleration:

1 v˙ (ω) ≡ dtv˙ (t)eiωt. (1) Z

Here we have used the same symbol v˙ for the Fourier transform of the acceleration (the argument t or ω tells you which it is). We will use Parseval’s theorem (a proof of which can be found in any good textbook on Fourier transforms)

1 dt|v˙ (t)|2 = dω|v˙ (ω)|2 (2) Z 2π Z to work out the spectrum of radiation arising from the entire encounter.

∞ dE ∞ e2 v 2 Etot = dt − = dt 3 | ˙ (t)| Z−∞ dt Z−∞ 6πǫ0c e2 1 ∞ v 2 = 3 dω| ˙ (ω)| 6πǫ0c 2π Z−∞ e2 1 ∞ v 2 = 2 3 dω| ˙ (ω)| 6πǫ0c 2π Z0 ∞ ≡ dωI(ω). (3) Z0

Here, e2 v 2 I(ω)= 2 3 | ˙ (ω)| (4) 6π ǫ0c is the energy emitted per unit frequency interval; in other words, this quantity indicates the spectrum of the radiation. For any acceleration mechanism, we can Fourier transform v˙ (t) to determine v˙ (ω), and hence derive I(ω).

1.2 Spectrum of Bremsstrahlung

1.2.1 Single interaction of an electron with an ion

Let us consider an electron moving at velocity v, whose straight-line course would take it to a closest approach at distance b (known as the impact parameter) from a stationary ion. We will work in the rest-frame of the electron (and call this frame S). In this frame we place the electron at x = 0, y = b, z = 0, and the ion travels at speed v along the x axis. We will set t =0 to be the time when the interaction peaks (ie. the ion passes x = 0). y

y=b e−

+ Ze v x Ion Figure 2: The frame S set-up for the Bremsstrahlung emission calculation.

2 In order to determine the acceleration of the electron, we need to determine the electric ﬁeld due to the ion, in frame S (and then F = ma and F = Eq give the acceleration trivially). In S′, the ﬁelds are simple:

Ze E′ r′ B′ = ′3 ; =0 (5) 4πǫ0r

′ ′µ ′ ′ ′ Similarly the 4-potential in frame S is simply given by A =(φ /c, A )=(Ze/(4πǫ0r c), 0). This can be transformed to S′ using the inverse Lorentz transformation matrix, and then by using E = −∇φ − ∂A/∂t (exercise for student: see Tutorial 2), the following result:

γZe(x − vt) Ex = 2 2 2 2 3/2 4πǫ0 [γ (x − vt) + y + z ] γZey Ey = 2 2 2 2 3/2 4πǫ0 [γ (x − vt) + y + z ] γZez Ez = (6) 2 2 2 2 3/2 4πǫ0 [γ (x − vt) + y + z ]

B need not concern us, as long as the change in electron speed is ≪ c. At the electron’s position, (x,y,z)=(0, b, 0), these equations evaluate to

−γZevt Ex = 2 2 3/2 4πǫ0 [(γvt) + b ] γZeb Ey = 2 2 3/2 4πǫ0 [(γvt) + b ]

Ez = 0 (7)

The electron undergoes acceleration both along its direction of travel (x) and perpendicular to its direction of travel (the y direction). The accelerations are given simply by mev˙x = −Exe and mev˙y = −Eye. The ratiov ˙y/v˙x = Ey/Ex = b/vt gets very large as t → 0, the peak of the interaction (where the denominators in Equation 7 are smallest). This indicates that acceleration in the y-direction is dominant, and so we will consider only this term from now on. The y-acceleration is:

γZe2b v˙y(t)= − (8) 2 2 3/2 4πǫ0me [(γvt) + b ]

Fourier transforming this,

γZe2b ∞ eiωt v˙ (ω)= − dt (9) y 3/2 4πǫ0me Z−∞ [(γvt)2 + b2]

Changing variables to u = γvt/b then gives

Ze2 1 ∞ eiωbu/(γv) v˙ (ω)= − du (10) y 3/2 4πǫ0me bv Z−∞ [1 + u2]

3 2ωb The integral is γv K1[ωb/(γv)], where K1 is a modiﬁed Bessel function of order unity. Hence,

Ze2 1 2ωb v˙y(ω)= − K1[ωb/γv] (11) 4πǫ0me bv γv

It is instructive to look at functional form of K1 at large and small values: for y ≪ 1, K1(y) → 1/y, 1/2 π while for y ≫ 1, K1(y) → 2y exp(−y). At low frequencies (ω ≪ γv/b), therefore,

−Ze2 v˙y(ω) ≃ . (12) 2πǫ0mebv

The intensity spectrum of the low frequency radiation is then given by (cf. Equation 4)

e2 v 2 I(ω) = 2 3 | ˙ (ω)| 6π ǫ0c

e2 Ze2 2 = 2 3 6π ǫ0c 2πǫ0mebv Z2e6 1 = 4 3 3 2 2 (13) 24π ǫ0c me (bv)

Similarly, at high frequencies

−Ze2 v˙ (ω) ≃ e−ωb/γv, (14) y 1/2 (8π) ǫ0mebv and hence

2 6 Z e ω −2ωb/γv I(ω)= 3 3 3 2 3 e . (15) 24π ǫ0c me γbv

This exponential cut-off tells us that there is little power emitted at frequencies above ω ≈ γv/b. The origin of this exponential cut-off arises since the duration of the collision can be approximated as the time that the ion takes to travel from x = b to x = −b, ie τ ≈ 2b/γv. This timescale corresponds to a frequency ω ≈ 2π/τ ≈ πvγ/b, which is of the same order of magnitude as the exponential cut-off frequency.

I(ω )

ω γ ω = v b Figure 3: The Bremsstrahlung spectrum from the interaction of an electron with an ion.

4 1.2.2 Integration across all collision parameters for one electron

The calculations so far relate to the interaction of one electron with one ion, with impact parameter b. In order to work out the full spectrum of radiation, we need to consider all collisions, by integrating across all possible collision parameters. For simplicity, from now on we will consider only non-relativistic Bremsstrahlung, and therefore take γ = 1. The rate at which an electron undergoes collisions with ions separated by distance b to b + db is given by ni v 2πbdb, where ni is the number density of ions. Therefore, at low frequencies,

bmax Z2e6 1 I(ω) = 4 3 3 2 2 2nivπbdb Zbmin 24π ǫ0c me (bv) 2 6 Z e ni 1 bmax = 3 3 3 2 ln (16) 12π ǫ0c me v bmin where we have set bmin and bmax as the minimum and maximum impact parameters that we will consider [note that by convention we write Λ = bmax/bmin]. Setting these limits is essential, since the logarithm diverges as either bmin → 0 or bmax → ∞. We therefore need to consider what values to take for the limiting b values. Fortunately, due to the logarithm involved, we don’t need to be too precise in this: e.g. the diﬀerence between ln(105) and ln(1010) is only a factor of two.

Estimating bmax is the easier of the two. We showed above that for a single interaction there was an exponential cut-oﬀ in the spectrum above ω ≈ v/b. For a given ω, therefore, only collisions with b ∼< v/ω will make an important contribution: higher b collisions will be on the exponential tail of their spectrum. Hence, we take

bmax = v/ω. (17)

For bmin we can consider two diﬀerent possibilities. One assumption that we have made is that the deﬂection angle is small; this will clearly break down if v is too slow. A reasonable approximation for this limit is that the kinetic energy of the electron is comparable to the electrostatic potential energy at closest approach, ie.

2 1 2 Ze mev ≈ (18) 2 4πǫ0b or

Ze2 bmin = 2 (19) 2πǫ0mev An alternative possibility is to adopt a quantum approach, which becomes important if the change in speed is comparable to the speed itself. For example, if ∆v = 2v, then ∆p = 2mev, and Heisenberg’s uncertainty principle (∆x∆p ≈ ~) says that ∆x ≈ ~/(2mev). Thus we take

~ bmin = (20) 2mev Clearly the reality is that we should take whichever is the larger or these two values; which this is depends upon the properties of the system. For gases with low electron speeds, such as HII regions

5 at T ≃ 104K, the ﬁrst estimate gives the higher value and so Equation 19 is the appropriate one to use. For hot gases in clusters of galaxies, with T ≃ 107K, the quantum limit is larger, and so Equation 20 gives the appropriate limit. Combining these, gives a low frequency spectrum of

2 6 Z e ni 1 I(ω)= 3 3 3 2 lnΛ (21) 12π ǫ0c me v with 2πǫ m v3 Λ ≈ 0 e for low velocity gas Ze2ω

2m v2 Λ ≈ e athighvelocites. (22) ~ω

1.2.3 Integrating over the electron population

The above calculation has derived the Bremsstrahlung radiation from a single electron, moving at a given velocity v. To determine the total Bremsstrahlung emissivity of the gas it is necessary to integrate over the full Maxwellian distribution of speeds of the gas. This is very cumbersome to do 2 analytically, but an approximately correct answer can be obtained by just setting mev /2 = 3kT/2 (using equipartition of energy) in Equation 16. Summing over all ne electrons per unit volume, this gives:

2 6 Z e neni me 1/2 ǫω = 3 3 3 2 g(ω,T ) (23) 12π ǫ0c me 3kT where ǫω is the low-frequency emissivity of the plasma (ie. the energy radiated per unit volume) at given frequency and g(ω,T ) is called the Gaunt factor, and is a correction factor arising from proper integration of lnΛ over velocity. At high frequencies, the exponential cut-off remains, as discussed earlier. Since the energy of the photons comes from the kinetic energy of the high-velocity electron, the maximum energy photon 1 2 that can be produced from an electron of velocity v has ~ω = mev . For a thermal gas, the 2 2 Maxwell-Boltzmann velocity distribution falls exponentially at high velocities as e−mev /2kT , and hence the Bremsstrahlung spectrum will cut off at high frequencies as e−~ω/kT . [Note that a similar result is obtained by substituting bmin from Equation 20 into Equation 15, and using equipartition of energy]. Combining this exponential cut-off into Equation 23 recovers the full expression for the emissivity:

−51 2 −1/2 −~ω/kT −3 −1 ǫω = 6.8 × 10 Z nineT g(ω,T )e Wm Hz (24)

Integrating this across all frequencies gives the total energy loss rate per unit volume of plasma from Bremsstrahlung emission:

−40 2 1/2 −3 ǫ = ǫωdω = 1.4 × 10 Z neniT g¯ Wm . Z

2 1/2 ∝ neT (25) whereg ¯ is the frequency-averaged value of the Gaunt factor, which is of order unity, and it has noted that ni scales with ne.

6 2 Application: Clusters of galaxies

Galaxy clusters are the largest observed structures in the Universe that appear gravitationally bound. They contain typically 100−1000 galaxies, and have central densities that are thousands of times higher than in the ﬁeld. As a result clusters have long been recognised as important testing grounds for measuring mass distributions on very large scales. Since the time required for a sound wave to cross the cluster (∼ 108 years) is very much less than the typical age of the system (∼ 1010 years), the cluster should have settled into a hydrostatic state with pressure varying smoothly with radius. Such systems satisfy the Virial Theorem, a derivation of which can be found in most basic astrophysics books (e.g. Peebles ‘Physical Cosmology’), and in the Astrophysics 3 lecture course. The Virial Theorem relates the kinetic and gravitational potential energies of a (non-relativistic) system in hydrostatic equilibrium, according to:

2Ekin + Egrav = 0 (26)

This tells us that the total mass of the cluster, Mtot is

2 RG v M ≃ (27) tot G where RG is the typical distance of galaxies from the centre of the cluster, and v is the typical velocity of the galaxies [the precise factor in the calculation depends upon the details of the radial distribution of the mass]. This gives,

2 14 σr RG M ≃ 7 × 10 M⊙ (28) tot 1000km/s Mpc where σr is the measured radial velocity dispersion, which is typically 1000 km/s for rich galaxy clusters. As early as 1933, Zwicky estimated the total mass of galaxy clusters using this method, and found that it greatly exceeded (by a factor ∼20) the mass calculated by summing that contained within all the luminous cluster galaxies. In addition to galaxies, clusters also contain a considerable quantity of intracluster gas. This gas was heated when the cluster ﬁrst formed, through gravitational collapse. Using equipartition of 2 energy, the (hydrogen-dominated) gas will have a temperature kT ≃ mpv . Assuming that the intracluster gas is moving at a similar velocity to the galaxies (ie. a few hundred to a couple of thousand km/s; this is expected since it must respond to the same gravitational potential) then the temperature of the gas is:

7 8 Tcluster = 10 − 10 K (29)

At these high temperatures the gas will be ionised, and the free electrons will produce Bremsstrahlung emission. The exponential cut oﬀ of the Bremsstrahlung emission occurs at ~ω ≈ kT , which corresponds to frequencies in the X-ray region of the electromagnetic spectrum. Galaxy clusters are indeed extremely luminous sources of X-ray emission, as shown in Figure 4. Along with active galactic nuclei (AGN), they are the brightest extragalactic objects in the X-ray sky.

7 Figure 4: Left: optical images of the central regions of two nearby massive clusters, showing the constituent galaxies. Right: X-ray images of the same clusters, showing the hot intracluster gas.

2.1 Aside: X-ray telescopes

X-ray astronomy can’t be carried out from the ground because of photo-electric absorption in the earth’s atmosphere. Modern X-ray telescopes therefore operate from space in high earth orbits.

Figure 5: The depth into the Earth’s atmosphere that light across the electromagnetic spectrum can penetrate.

As well as the need to go to space, X-rays astronomy is also challenging because of the diﬃculty in focussing and collecting the X–ray photons. Early X-ray telescopes used collimators or coded masks, providing quite low angular resolution. X-rays can be reﬂected but only if they graze a

8 smooth metal surface at a shallow angle (θ< few degrees at 1keV, few arcmin at 10keV): at larger angles X-rays are absorbed or scattered. Using two grazing reflections, first off a parabolic and then a hyperbolic mirror, it is possible to focus X-ray photons. The ROSAT satellite (launched 1989) used this technique and obtained an angular resolution of ∼ 5 − 10′′. Chandra (launched 1999) revolutionised X-ray astronomy with sub-arcsecond resolution for the first time.

θ o o o o o o o o X-rays o o

o o Pattern of holes Position sensitive detector

Figure 6: Progressive improvements in X-ray telescope technologies. Collimators (upper left) only provided a rough position (∼ 1deg) for an X-ray source in the sky. Coded masks (upper right) allowed images with an angular resolution of a few arcminutes to be made, by deconvolving the detected signal with the pattern of holes. Modern X-ray telescopes use nested focussing mirrors (e.g. lower panel, which shows the set-up of Chandra), providing much higher angular resolution and sensitivity.

2.2 Mass estimates from the intracluster gas

Since clusters are in a state of hydrostatic equilibrium, then their pressure gradient balances the gravitational forces, such that

dP GM(r)ρ(r) = − (30) dr r2 where P (r) is the gas pressure, ρ(r) is the gas density at radius r, and M(r) is the mass contained within radius r. We can combine this equation with the ideal gas law from thermodynamics, P = nkT = ρkT/m¯ , where n is the particle number density, andm ¯ is the mean particle mass, often written in terms of the hydrogen mass asm ¯ = mH . Combining these we obtain:

9 dP k ∂T ∂ρ −GM(r)ρ = ρ + T = 2 (31) dr mH ∂r ∂r r

Hence, if we know ρgas(r) and T (r) we can obtain M(r). The key point here is that M(r) measures all mass within radius r, not only the mass of the intracluster gas.

Both ρgas(r) and T (r) can, in principle, be obtained from the Bremsstrahlung spectrum: the gas temperature determines the location of the exponential cut-oﬀ (e−~ω/kT ), while the total emissivity 2 1/2 scales as neT (see Equation 25), thus giving the gas density. In this way, X-ray observations allow both the total mass of X-ray gas and the total cluster mass to be determined. In practice, under recently only approximate results could be obtained, by assuming a functional form for T (r) (e.g. assuming that clusters are isothermal), as observational limitations prevented this from being measured. With the improved sensitivity and angular resolution of recent X-ray telescopes (Chandra and XMM), however, accurate measurements have been possible for T (r) in nearby bright clusters. The results demonstrate that the total mass of the intracluster gas exceeds that in the galaxies, but still falls a factor of a few short of the total mass required to account for the gravitational motions. The conclusion of these studies is that galaxies account for about 5% of the mass of clusters, the intracluster gas accounts for a further 10-15%, but the remaining 80-85% of the mass is ‘dark matter’. These cluster observations provide the most accurate measure of the ratio between baryonic matter and dark matter. Since the deep gravitational potential wells of clusters means that essentially nothing ‘escapes’, they are expected to be representative of the Universe as a whole. This is in contrast to galaxies, for example, where galactic-scale winds driven by supernovae or AGN can remove a signiﬁcant fraction of the baryons. The measurement that baryonic matter accounts for roughly one-sixth of the mass of the Universe can be combined with estimates for the baryonic matter density of the Universe from primordial nucleosynthesis measurements to provide one of the strongest constraints on the overall mass density of the Universe, Ωm. Combining this with other measurements of cosmological parameters made in recent years, such as supernovae and the Cosmic Microwave Background power spectrum, provides a consistent picture of the Universe (see Figure 7).

Figure 7: X-ray observations produce some of the tightest constraints on Ωm ever obtained. Combined with the SN results and measurements of the CMB power spectrum, all evidence now points towards a Universe with Ωm ≃ 0.3 and Ωλ ≃ 0.7.

10 2.3 Cluster scaling relations

For an isothermal gas, the total Bremsstrahlung radiation per unit volume is given by ǫ ∝ n2T 1/2, which means that the following crude relations hold for the total X-ray luminosity of the cluster:

2 1/2 3 LX ∝ n T R

M 2 ∝ gas T 1/2R3 R6

M 2 ∝ T T 1/2 (32) R3 where the total cluster mass MT scales with the gas mass. The Virial Theorem tells us that 2 MT 2 σr ∝ R , and equipartition of energy gives σr ∝ T . Eliminating MT from Equation 32 then gives

5/2 LX ∝ T /R (33) which implies that X-ray luminosity should scale as temperature to a power between 2 and 3. At high luminosities, observations indicate that this indeed occurs, but at lower luminosities (low mass clusters and galaxy groups) these simple estimates break down, with the data indicating a much 5 steeper relationship between luminosity and temperature (LX ∝ T ; see Figure 8). This is believed to be due to an additional source of energy input during cluster formation (ie. not just gravity); we’ll investigate this in Section 2.4.

Figure 8: The X-ray luminosity versus temperature relation for X-ray clusters and groups. At high luminosities the observations match simple scaling-law predictions, but at low luminosities the galaxy groups have lower luminosity than expected at a given temperature.

2.4 Abundance Determinations From X-ray Observations

For pure hydrogen gas, a simple Bremsstrahlung spectrum would be expected. In reality, the presence of other elements in the intra-cluster medium can lead to signiﬁcant line emission. The importance of this line emission in the X-ray spectrum depends upon the temperature of the gas: in the richest clusters, with T ∼ 108K, essentially all elements within the cluster gas are fully

11 ionised, with the only signiﬁcant exception being Iron, which may hold on to one or two electrons (Fe XXV or Fe XXVI). At cooler temperatures, highly (but not fully) ionised species can remain, and line emission becomes progressively more important. Figure 9 shows how the X-ray spectrum of galaxy clusters changes with temperature (mass). Rich clusters typically have temperatures of a few 107 to 108K, and are dominated by thermal Bremsstrahlung. Smaller groups of galaxies have temperatures ∼ 107K and are dominated by line cooling.

Figure 9: X-ray spectra of clusters and groups as a function of temperature. In addition to the change in the ‘typical’ energy of the emission, the prominence of line emission changes dramatically with temperature.

12 The strength of the Iron line can be used to provide information about the abundances of the intracluster gas. If we assume that the gas has settled into a state of collisional equilibrium and that the gas is isothermal, then one can apply the following:

Iron line strength ∝ nenFe Continuum strength ∝ nenp → line ∝ nF e . continuum np

In other words, we can get the iron abundance relative to hydrogen. In practice, this is done by ﬁtting the observational data with a model spectrum allowing both the temperature and the metallicity to be free parameters. The result is that the Fe/H ratio found in the intra-cluster medium is typically ∼ 1/3 of solar metallicity. This indicates that the intracluster medium can’t simply be primordial gas (which has almost zero metallicity). Heavy elements must have been ejected into the intracluster by galactic-scale winds, driven either by AGN or by supernovae. The current mass loss rate from the galaxies in clusters can only account for ∼ 3% of this enrichment over a Hubble time, and so there must have been an epoch of earlier enrichment. Let us consider the role of supernovae. The iron is formed in supernova explosions, and the ejection of Fe into the intracluster medium will be accompanied by the injection of energy. The abundance results can be used to estimate the importance of supernovae in heating the intra-cluster medium. The argument is as follows: let η be the total energy input by supernovae, per unit mass of cluster gas. That is, Total energy in SN η = Total mass of gas

Total energy in SN Total mass of Fe = Total mass of Fe × Total mass of gas

Energy per SN Total mass of Fe = Fe per SN × Total mass of gas

The ﬁrst of these ratios is well understood from our knowledge of supernovae. The second depends upon the measured metallicity, which we obtain from the X-ray spectrum.

We can then compare the total energy input by supernovae, ESN = nmpη, to the total thermal energy of the intra-cluster medium, ETH = 3nkT . The result is that there is signiﬁcant heat input from supernovae. This can vary from only a few per cent in massive clusters to a few tens of percent in low mass clusters and groups. Note that since the energy input scales linearly with mass but the gravitational potential energy of the cluster scales as M 2, the eﬀect of supernovae (or AGN) feedback is strongest in the lower mass systems; the energy input leads to a decrease in the central density of the system, decreasing the X-ray luminosity for a given temperature, and giving rise to the steeper scaling relations discussed in Section 2.3.

2.5 Cooling Flows

2 1/2 As discussed earlier, the total Bremsstrahlung emission scales as ǫ ∝ neT . The total thermal energy of the gas is ETH = 3nkT . From these, we can calculate the ‘cooling time’ of the gas, that is, the time it would take to radiate away all of its energy at the current rate.

1/2 ETH T tcool = ∝ (34) ǫ ne In the core of clusters (where the density is highest) the cooling time can be comparable to the age of the cluster. If the gas in the central regions cools then there is a drop in pressure and, to maintain hydrostatic equilibrium, matter must ﬂow inwards from outside the cooling region, leading to a ‘cooling ﬂow’.

13 In cooling flow clusters we expect highly peaked central surface brightness profiles (since ǫ ∝ n2). Figure 10 compares the X-ray emission from a cooling-flow cluster to that from a cluster without a cooling flow. The total X–ray luminosities of the two clusters are approximately the same (within a factor of two), but it is clear that the emission from the cooling flow cluster is much more peaked.

Figure 10: The distribution of X–ray emission from a cooling ﬂow cluster (Abell 478; left) and a cluster without a cooling ﬂow (the Coma cluster; right)

In the simplest cooling flow models, the gas in a cooling flow can be thought of as a slowly-moving emulsion of gas clouds with different densities and temperatures. These clouds slip slowly down the gravitational potential of the cluster (towards the centre) with the densest clouds cooling fastest. The densest clouds eventually loose all of their thermal energy and form small, cold molecular clouds, in which stars can then form. In the most centrally peaked galaxy clusters, these simple −1 cooling flow models predict star formation rates in excess of 1000M⊙ yr . A long-term problem with these simplest cooling flow models has been that measurements of the star-formation rates in central cluster galaxies are at least an order of magnitude below these simple predictions. Although there were suggestions that the cooled gas might be hiding in a different phase, Chandra and XMM observations have proven that these simplest cooling flow models don’t work: one would have expected strong X-ray emission lines from, for example, FeXVII, as the hot gas cooled through temperatures around T ∼ 5 × 106K, and these emission lines are very much weaker than expected (see Figure 11). Whilst there is evidence that the gas in cooling flow clusters cools to about 30% of its Virial temperature, X–ray observations suggest that at most about 10% of it cools any further. The reasons for this remain hotly debated, but the most popular explanation is that it is due to heat input from a central AGN, particularly through radio source activity, which leads to a balance between heating and cooling. We will return to this issue later in the course.

Figure 11: X-ray spectrum of the gas at the centre of M87: the emission lines expected from cooler gas (e.g. FeXVII) are much weaker than expected from simple cooling ﬂow models.