Whittle : EXTRAGALACTIC ASTRONOMY

 
     
 

1 : Preliminaries	6 :   Dynamics I	11 : Star Formation	16 : Cosmology
2 : Morphology	7 :   Ellipticals	12 : Interactions	17 : Structure Growth
3 : Surveys	8 :   Dynamics II	13 : Groups & Clusters	18 : Galaxy Formation
4 : Lum. Functions	9 :   Gas & Dust	14 : Nuclei & BHs	19 : Reionization & IGM
5 : Spirals	10 : Populations	15 : AGNs & Quasars	20 : Dark Matter

(1) Introduction

• Typical cluster sizes are 1 - 3 Mpc :
    they are the largest virialized structures in the Universe (ie   KE     ½ PE ) [movie 38Mb!]
  Structures larger than clusters have not had time to "turn around", collapse, and virialize.
  Galaxies :             t_virial     10⁸ yr   <<<   t_Hubble
  Clusters :             t_virial     10⁹ yr   <<   t_Hubble
  Superclusters :   t_virial     10^10.5 yr   >   t_Hubble
  

  Note that clusters are not necessarily the largest bound structures in the universe
    superclusters may be bound, but haven't yet turned around and virialized.

• On these large scales, components have not had a chance to separate during collapse
    a cluster is probably a representative sample of the Universe
  This is important when, for example, we measure their Dark Matter (DM) content :
  (M_DM / M_baryons)_cluster =   (M_DM / M_baryons)_Universe

  So a measurement in clusters can be scaled up to derive   _matter   for the universe

• The richest clusters are typically found at the intersection of sheets and filaments of galaxies
  e.g. Coma is located at an intersection in the great wall :[image]
  Movie of Large Scale Structure formation, showing clusters at intersections: [movie 7Mb]
  Movie of 3D-fly-through the end-point: [movie - 57 Mb!]

    Ongoing heirarchical assembly : small things merge to make bigger things, on all scales.
    Clusters continue to grow (and form), even today

• Cluster are part of a continuous range of structures :
  
  galaxies     groups     clusters     superclusters     large scale structure
  However, clusters are rare extremes in the galaxy distribution, with / <> 10³ :
  Very roughly (depending on definitions) the total galaxy content of the universe is divided :
  
  1-2% in rich clusters
  5-10% in clusters
  50-100% in "Local Group"s &/or looser groupings

• The four principal constituents of clusters include: [image]
  1. Galaxies
     
     10² large galaxies; 10³ total galaxies [image - Coma]
     typical speeds 10³ km/s   [image]
  2. Intracluster Stars
     
     very faint (1% sky) diffuse light (distinct from cD halo light)
     comprises 10-50% total galaxy light (in rich clusters; much less in poor clusters)
     probably tidally stripped stars [images & movies]
  3. Hot Gas
     
     Hydrostatic atmosphere [image]
     T 10^7-8K     X-ray emitter
     n 10^-3 cm^-3
     L 10^43-46 erg/s   10^-2 - 10^-4 L_opt
     M_gas 5 × M_gals
     Z 0.3 Z     enriched : not all primordial
  4. Dark Matter
     Dominates the total mass [image]
     M_DM 4 × M_{gas + gals}

(2) Cluster Surveys and Catalogs

One needs to identify an "overdense" region of galaxies
Early steps : Herschel(s); Hubble; Shapley; Reynolds.
Significant improvement following publication of the northern Palomar Sky Survey (PSS-I)
Historically leaning review of clusters by Biviano is here: [o-link]

(i) Abell (1958) : Catalog of Rich Clusters

A definitive work (ApJ Supp 3 211) which has survived to current times
Visual inspection of 10⁴ deg² from PSS yielding 2712 clusters
Southern extension using SRC-J : 1364 clusters (Abell, Corwin & Olowin, 1989, ApJ Supp, 70 1)
Robust criterion :

estimate distance using m₁₀ (10^th brightest galaxy)
exclude clusters with 0.02 < z > 0.2   (note : Virgo excluded -- too close/big)
define region of radius 1.5 h^-1 Mpc (an "Abell Radius", R_A)
count galaxies within R_A between magnitudes m₃ and m₃ + 2
subtract a background count evaluated nearby
N > 50     "complete catalog" (adding : 800 > < 1200 km/s eliminates 10% superpositions)
Richness classes 1-2-3-4 : N = 50 - 80 - 130 - 200 - 300 (comprising 1224, 383, 68, 6 clusters)
N = 30 - 49     "incomplete catalog" (1030 clusters, Richness class 0)

Data on the Abell clusters compiled by Struble & Rood : 1987 ApJ Supp 63, 543 & 555
Basic data and links to SDSS images of all Abell clusters is here: o-link

(ii) Zwicky (1961-68) : Catalog of Galaxies and Clusters of Galaxies

18 inch Schmidt gave m_B for 31,000 galaxies, visual inspection of PSS gave 10,000 clusters.

Assigned cluster type : Compact, Medium-Compact, Open
Assigned cluster distances : Very Near, Near, Distant, Very Distant, Extremely Distant.
Gave number of galaxies, cluster boundary size, coordinates etc.
Uses "isopleth" density contrast of N / N_background = 2 to define cluster
however, statistically incomplete : cluster sizes are distance dependent
  rarely used compared to Abell's lists

(iii) Automated Digital Catalogs

Possibly more objective selection criterion uses automated recognition of galaxies and clusters.
Main effort : scanning of UK schmidt plates (SRC J & R) by two UK groups :
APM (Automatic Plate Measuring Machine) : Cambridge (Mike Irwin et al)
COSMOS : Edinburgh/Durham

Rich clusters with deep potentials have hot gas (§ 10)
  X-ray emission is an effective way to find relaxed clusters
Since emissivity n², we have no forground X-ray emission (though smooth X-ray background)
  problems of spurious identification from superposition is greatly reduced compared to optical surveys.
At high redshifts, this is increasingly important
  X-ray surveys may be the best way to identify (rich) high-z clusters
Several surveys currently exist:
EMSS (Einstein Medium Sensitivity Survey : serendipitous, 800 deg², z 0.05 - 0.55)
RDCS (ROSAT Deep Cluster Survey : serentipitous, 100 deg², z 1)
RASS (ROSAT All Sky Survey)

There are a few other methods which show great promise for the future :
• SZ (Sunyaev-Zeldovich) effect :
  Hot cluster gas Compton scatters CMB photons, increasing/decreasing their energy
  Look for brightening/dimmin of CMB at mm-wavelengths
  Promising for detecting high-z clusters
  Currently very difficult, but recent progress: SPT, PLANCK

• Weak Gravitational Lensing :
  Faint background galaxies suffer slight distortion by matter along the line of sight
  Intervening clusters give slight azimuthal image elongation
  Many galaxies   statistically detectable
  Allows mapping of intervening mass distribution.
  Still early days, but quite promising.
  

• Color Search for Red Galaxies :
  (Red) Ellipticals formed very early
    so concentrations of faint red objects should yield high-z clusters
  Redshift modifies colors, so good color information should also yield approximate redshift.
  Some deep multicolor surveys begun, but still early days.

(3) Cluster Classification

• Clusters are not all the same !
  One might classify clusters according to one of several possible properties :
  eg shape, richness, lumpiness, Hubble mix, dominant galaxy types, etc
  it transpires (see below) that many of these are equivalent,
  so in practice we need only keep track of one (or two) underlying properties.

• A couple of classification schemes are in common use :
  • Bautz-Morgan (BM) type : [o-link]
    Compares the prominence of the brightest galaxy to the other galaxies.

    BM I	single central dominant cD galaxy	(eg A 2199)
    BM II	several bright galaxies between cD and gE	(eg Coma)
    BM III	no dominant galaxy	(eg Hercules)

  • Rood-Sastry (RS) type : [o-link]
    Describes the distribution of the 10 brightest galaxies
    Can be arranged in a "tuning fork" sequence: [image]

    cD	dominated by a single cD galaxy	(eg A 2029)
    B	dominated by a bright binary	(eg Coma)
    L	line of several bright galaxies	(eg Perseus)
    C	core of > 4 bright galaxies	(eg A 2065)
    F	flattened distribution	(eg A 1291)
    I	irregular with no center	(eg Hercules)

  These two systems are closely related
  it seems there is a primary factor which defines a cluster : its degree of relaxation
  from most relaxed to least relaxed we have :
  BM : I     II     III
  RS : cD     B     L     C     F     I
  A number of other properties follow this sequence :
  Hubble type mix :   Elliptical rich     Spiral Poor     Spiral rich
  Overall Shape :       Spherical     Intermediate     Irregular
  X-ray Luminosity :  High     intermediate     low

  Here is a more specific table (condensed from Bahcall's entry in Allen's AQ)

  Property/Class	Regular	Intermediate	Irregular
  Zwicky type	Compact	Medium-Compact	Open
  Bautz-Morgan type	I, I-II, II	(II), II-III	(II-III), III
  Rood-Sastry type	cD,B, (L,C)	(L),(F),(C)	(F), I
  Content	Elliptical-rich	Spiral-poor	Spiral-rich
  E:S0:S ratio	3:4:2	1:4:2	1:2:3
  Symmetry	Spherical	Intermediate	Irregular shape
  Central concentration	High	Moderate	Very little
  Central profile	Steep	Intermediate	Flat
  Mass segregation ?	Marginal	Marginal	None
  Radio detection ?	50%	50%	20%
  X-ray luminosity	High	Intermediate	Low
  Examples	A2199, Coma	A194, A539	Virgo, A1228

  It is very likely that this sequence reflects, at least in part, stages in cluster evolution :
  most evolved     intermediate     least evolved
  Stated slightly differently : given a few Gyr, Hercules will resemble Coma
  of course, more clusters like Hercules will form out of yet lower density regions.

(4) Important Timescales

Obviously, given the velocity dispersion and cluster size, we have :
(recall, the ever handy rule of thumb : A km/s is a pc in a Myr !)
t_cross     R /     10⁹ yr × R_Mpc / ₁₀₀₀


So, for clusters that formed at z 1, galaxies might have experienced a few orbits
(don't forget, though, many (spiral) galaxies may be falling in for the first time)

This refers to the time for a chaotic collapse to "sort itself out" and reach steady state
(it is also the time for two similar sized clusters to merge) :

t_violent     few (2-5) × t_cross     (2-5) × 10⁹ yr × R_{Mpc 1000}^-1


Given the observed range in cluster properties (R, , and possibly age) :
we expect (and find) a significant range in relaxation :   quite unrelaxed     well relaxed.

From Topic 8.10.a.iii we derived a simple formula for 2-body relaxation (eq 8.38c) :


t_2-body     t_cross N / 6 ln N

where N is the total number of interacting bodies in the system.
This gives 3 × 10⁹ yr (Table in 8.9.b) which is quite short
However : lets not forget the Dark Matter --- how does this change things ?
When we have a background medium, the 2-body and dynamical friction processes get entwined.
The timescale for significant energy loss becomes :

t_relax     t_cross N / f_g 6 ln N

where f_g is the fraction of mass in galaxies ( 0.1) and N is the total number of galaxies
For individual galaxies we get t_relax 10^11-12 yr while for subgroups (3-30 galaxies) this becomes 10^9-11 yr
So relaxation is generally not significant for most galaxies
However, for subgroups or galaxies near the center, some relaxation is expected
Dont forget, this kind of relaxation leads to equipartition (in energy), so massive galaxies will settle
Although massive galaxies are often found in cluster cores, it is unclear if this is due to relaxation or merging.

(5) Cluster Shapes & Kinematics

• The RS classification divides clusters by shape and concentration :
  irregular/unconcentrated     intermediate (eg flat/linear)     circular/concentrated

• Statistical analysis of aspect ratios :
    clusters are prolate or triaxial
    richer clusters are less elongated

• Radial profiles : azimuthal averaging gives (R)
  clusters can be stacked by cluster type &/or galaxy Hubble type to give (R) :
  Paramterize (R) using :
  
  central surface density : (0)
  scale length (or core radius) : R_c
  steepness of concentration gradient : slope

  From rich     sparse :

  (0) decreases (by definition !)
  R_c increases (0.1 - 0.5 Mpc)
  slope becomes flatter
  Variation by Hubble type :
  Ellipticals are more concentrated and have lower velocity dispersion than spirals
  Velocity dispersion of spirals decreases at larger radius
    outer spirals not yet crossed cluster
    radial orbits, infalling for first time ?
    cluster continues to be constructed

  A more theoretical approach can be adopted :
  Analytically :

  Following violent relaxation, we expect :
  isothermal and isotropic velocity distribution
  an equilibrium stellar dynamical system of this kind has (r) with :
  flat core, core radius, & steep r^-2 envelope
  appropriate solutions include (Topic 8.10.b&c) :
  isothermal profile or King profile or analytic equivalents   eg (r) = (0) [1 + (r/r_c)²]^-3/2
  These lend themselves to simple estimates of core M/L (see Topics 8.10.b.ii and 7.7)
  Numerically :
  Collisionless collapse simulations suggest a profile more similar to (R)     R^¼ deVaucouleurs law
  (not surprisingly, since the calculation is similar to collapse/merger in elliptical formation)
  Observed profiles fit both this and the analytic profiles equally well.
  The R^¼ profile is more peaked at the center,
  although this fits the galaxy luminosity distribution well, the dark matter profile is less certain here.

• To first order, the distribution of projected galaxy velocities is Gaussian
  This is as expected : violent relaxation leaves an isothermal distribution with N(v)     exp(-v² / 2² )
  is an important parameter and measures the potential depth (as it does in galaxies).
  The velocity histogram is useful in eliminating foreground/background galaxies
  however, ambiguities result for galaxies in the Gaussian wings
  Example :   velocity distribution in the Coma region, with velocity color coded map (from Huchra et al)

• very few clusters are completely smooth, in either surface distribution, or velocity space,
    there are often close pairs &/or small sub-groups with similar velocities
• Sometimes the outer parts of the cluster include significant concentrations of galaxies with lower
• Velocity histograms, while Gaussian overall, may have statistically significant substructure; even bimodal.
• There are clear examples of "binary clusters", or elongated clusters with two centers : ie merging clusters

  We conclude :
    Clusters continue to be assembled (via heirarchical merging)
    Relaxation is not yet complete in many/most clusters.

(6) Two Nearby Examples : Virgo & Coma

Virgo and Coma provide nice examples of unrelaxed (Virgo) and relaxed (Coma) clusters.
I did not have time to go over these in class, so for the moment I'll leave this blank.
I'll include this section next time, since there are many interesting things to learn from these two clusters.

(7) The Morphology-Density Relation

• Perhaps environmental density affects what kind of galaxy can form ?
• Perhaps spirals are converted to ellipticals in dense environments ?

Lets look more closely at this topic.

(a) Early Work at Low Redshift

• Hubble and Humason (1931) notice that clusters have a higher fraction of Es and S0s than the field.
  >>> The morphological mix of galaxies depends on galaxy environment.
• Oemler (1974) recognised three types of cluster :
  "spiral rich" (eg Hercules); "spiral poor" (eg Virgo ); "cD" (eg Coma);
  which correlated with overall cluster shape, galaxy density and degree of central concentration.
  He suggests they represent a sequence in cluster dynamical evolution.
  Here is a typical census :

  Type:	cD	E+S0	S+I
  Rich clusters	93	56	38
  Poor clusters	6	20	14
  Field	< 6	< 24	48

  
  
• Dressler (1980) classified 6000 galaxies in 55 clusters plus 15 field regions.
  He recorded positions and local projected galaxy density
  (area enclosing nearest 10 galaxies brighter than Mv=-21.4 [Ho=50]).
  He found :
  
  • Strong dependencies of f(Sp), and f(E) on projected local galaxy density (figure)
  • In poor clusters, the trend is stronger with local density than with simple cluster radius (figure)
  • The dependency of f(S0) is weaker than f(E) or f(Sp)
  • The effect occurs in regions of sufficiently low density that gas stripping (see below)
    or encounters cannot operate.
• He concluded :
  • The primary effect is with local galaxy density NOT cluster radius.
  • The effect occurs at galaxy formation, and is not an ongoing evolutionary process.
• Both these conclusions have been subsequently questioned, and it now seems that :
  • Both global cluster conditions and local galaxy density play roles
  • While some of the effect occurs at galaxy formation, some is continuous.

• High densities inhibit the formation of spirals.
• Spirals may be stripped of gas (see below) to make S0s
  • c.f. Amemic spirals (van den Bergh) are common in clusters
  • But : B/D ratios in S0s are systematically higher than in spirals.
  • However, maybe disks fade faster than bulges (since younger)
  • Bulges may "grow" by accretion of dwarfs
  • S0s may not be a homogeneous class : some originate as spirals, others not
• Spirals experiencing "harrassment" (Moore et al 1996) can resemble S0s.
  rapid gravitational shocks disturb spiral structure and "heat" the disk stars.
• Spiral mergers may create S0s and/or Es.

• HI deficiency is defined as (M - <M>)/<M>, where <M> is the mean HI mass for galaxies of the same Hubble type.
• HI deficiency is found to increase
  (a) towards the center of clusters (picture)
  (b) in richer clusters of higher X-ray luminosity (picture)
• However, whether this is sufficiently efficient is unclear :
  (a) Studies show CO is not removed (denser and deeper in galaxy potential)
  (b) Only the outer HI is stripped (eg HI map of virgo shows smaller sizes in the core)

• f(E) is the same as low-z
• f(S0) is lower by factor 2-3
• f(Sp) is higher by factor 2-3
• The morph-density relation is absent in irregular clusters.

• Ellipticals formed earlier (at even higher z)
• For Es, the density at formation is most important
• Spirals are converted into S0s, in an ongoing process which depends on density (still to be identified)
• These results are broadly consistent with the "Butcher-Oemler Effect" (1978)
  in which the fraction of blue galaxies is found to be higher in distant clusters

(8) Luminosity Functions

We have looked at the Luminosity function for cluster galaxies in Topic 4

Recall : the cluster LF can be constructed by combining the LFs for each galaxy type
Ellipticals : Gaussian skewed to high luminosities
Spirals and S0s : Gaussian
dE's : Schechter function with steep slope
dSp/dIrr : Schechter function with shallower slope


For increasing densities :


  the contribution of Es, S0s and dEs increases
  the contribution of Spirals and dIrr decreases.

(9) cD Galaxies

• very luminous Elliptical galaxies
  L_cD     10×L_*   which is unusually bright
  They contribute to the LF above the normal exponential cutoff in the Schechter function
  Significantly more luminous than expected from the population of other cluster galaxies (ie a statistical anomaly)
    cD galaxies have a qualitative different formation history than other cluster galaxies

• always found at the cluster center at the projected local density maximum
  they have essentially zero velocity w.r.t. the cluster mean
    they lie at the cluster center of gravity

• very large with an extended halo (50 - 100 kpc in scale)
  the light profile rises above the normal R^¼ deVaucouleurs law at large radii
  The halo is usually oriented similarly to the overall cluster shape
  velocity dispersion increases with radius to match the cluster dispersion
    halo contains stars in the cluster potential

• Images often show double/triple merging nuclei within cDs : [ image ]
  

• Origin : mergers of cluster galaxies in the cluster core
  Since cluster velocities are so high, merging should be inefficient   occurred earlier ??

• cD galaxies can be at the center of a Cooling flow (see below)
  These can be associated with significant emission line filaments (10⁴ K gas)
  cD radio sources/jets are FR-I and exhibit significant interaction with a high pressure IGM.
  if cooling gas end up as stars, they may account for a significant fraction of the cD mass.
  (a few cDs do have patchy blue light (eg A 1795) -- but only a few)

(10) The Hot Inter-Cluster Medium (ICM)

• Bremsstrahlung (breaking) radiation from hot gas
  electrons scattered by nuclei     acceleration     EM radiation (photons)
  (acceleration is impulsive     flat spectrum     Fourier Transform of delta function)
    =   10^-11 T^-½ exp(-E/kT) n_e n_Z Z² g(E)   erg/s/cm³/erg
  

  g(E) ln T/E   for E << kT
  g(E) (E/kT)^-0.4   for E kT
  for several ions, replace n_e n_ZZ² by n_e n_ZZ²
  For cosmic abundances, integrate over energy to get :

    =   2.4×10^-27   T^½ n_e²     erg/s/cm³

  L_tot     10^-23 erg/s × n_e²dV     for T     5×10⁷K     7 keV

  Note : emissivity n_e²     weights dense regions strongly     strong cooling in core

• Shape of spectrum     Temperature   =   2 - 30 × 10⁷K (ave : 7×:10⁷K     7 keV)
  Intensity     n_e   =   10^-4 - 10^-1 cm^-3 (ave : 10^-3 cm^-3)
  + Volume     Mass   =   0.2 - 5 × 10¹⁴ M   (ave   =   10¹⁴ M)

  M_gas     M_gals   (groups)   increasing to   M_gas     7 M_gals (rich clusters)
  On average :

  M_gas     5 M_gals       ICM significantly outweighs galaxies !
  M_gas     1/3 M_tot     however, dark matter still dominates overall

• Origin of gas and gas's thermal energy :
  Two obvious origins :
  • infall (deep potential)   :     3/2 kT       m_p         T     7×10⁷ K
    
  • mass loss from galaxies :   3/2 kT     ½ m^pV_gal²     T     7×10⁷ K
    
  (obviously equivalent, since galaxies are in approximate virial equilibrium)  

  Using abundances (see below) it seems that both contribute :
    80% primordial infall, 20% ejected from galaxies

• Given the hot ICM is radiating X-rays, how long will it take the gas to cool down ?
  Cooling time   =   Thermal capacity / cooling rate
  

  t_cool   =   3 N_e k T /   =   10¹¹ N_e^-1 T^½   sec   =   2.7×10¹⁰ N_e,3^-1 T₇^½   yr

  This is longer than t_Hubble except, possibly, at the center.
    the gas remains hot, even with no additional heating
  

• Sound speed :   c_s     10³ km/s     V_gals
  Sound crossing time :   t_s-cross     10⁹ years   <<   10¹⁰ years

    the atmosphere can adjust to the potential and achieve equilibrium
    we have a hydrostatic atmosphere

• What is the structure of such an atmosphere : ie what is _gas(r) and T_gas(r) ?
  The appropriate equations for hydrostatic support and the equation of state are :
  dP_gas / dr   =   -G M(<r) / r² × _gas     and     P_gas   =   nkT   =   _gas kT / m_p

  Which together give :

  (1 / _gas) d _gas(kT/µm_p) / dr   =   - GM(<r) / r²

  Obviously, we can view this in two ways :

  knowing _gas and T_gas we can derive M(r)     ultimately very important (§11)
  knowing M(r) or its equivalent (and assuming T), we can derive _gas(r) 

• Although we dont know M(r) we do have another tracer of : the galaxies
  Since galaxies are collisionless they obey an equivalent equation (T8.8.c.i eq 8.37b)
  (1 / _gal) d ( _galr,gal² ) / dr   +   2 _r,gal / r   =   - GM(<r) / r²

  (Here, referes to orbit anisotropy and _r,gal is the radial galaxy dispersion)
  notice that in both these equations we do not assume that either _gal or _gas define the potential
  (they dont, the dark matter does)
  The gas and galaxies do, however, sample the same potential

• To proceed, we make three assumptions :
  • the galaxies have isotropic orbits :   =   0,   so   _{r,gal     gal}
  • the galaxies are isothermal : _gal   =   const
  • the gas is isothermal : T   =   const

  Notice that we do not assume T_gas = T_gals
  Since T_gas T_gals we expect a different (but still isothermal) profile for the gas.

  Combining the hydrostatic fluid and stellar equations, we get :

   
      (13.1)
  
  

  from which we see :

  _{gas     gal}     with     =   _gal² / (kT/µm_p)   =   T_gal / T_gas

  Here refers to T_gal / T_gas (and should not be confused with the anisotropy parameter)

• One approach is to compare N_gal(R) with S_X(R) to extract and test the various assumptions.
  Early on, considerable effort went into this but ultimately firm conclusions were illusive :
  • too few galaxies to get smooth N(R)
  • deprojection amplifies uncertainties in both _gas(r) and _gal(r)
  • unknown mass segregation may render _gal not constant

• Instead, the above reasoning is assumed to be valid, and an isothermal profile is adopted directly.
  An adequate approximation is the "Analytic King Profile"
  _gal     (1   +   (r / r_c)² ) ^-3/2
  so
  _gas     (1   +   (r / r_c)² ) ^(-3/2)
  
  recall, from T5 and T8.10.b.ii, that this is in fact the density law behind the "Hubble Profile"
    it does not fit the isothermal r^-2 profile at large radii, but
    it does fit the general isothermal profile within a few core radii.
  One nice advantage, however, is that all integrals are analytic (including deprojections)
  

• Fits to X-ray brightness profiles are reasonably good, except :
  • when substructure is strong (not hydrostatic)
  • at the centers when a cooling flow occurs (see §10e)
    

  Fits yeild   0.7 (rich clusters)     0.4 (less rich clusters)
  giving halo gas density gradients   r^-1   (rich clusters)     r^-0.7   (less rich clusters)

• In terms of gas temperature :   T_gas / T_gal     ^-1     1.5 (rich)     2 (less rich)

    It seems the gas is hotter than the galaxies,
    the temperature difference is greater for shallower potentials.
  These results are also supported by (spectroscopic) measurements of T_gas

  conclusion : There is a non-gravitational source of heating for the ICM.

  What is it ?       not yet known
  Possibilities include :

  • AGN jets (in the past, not current radio sources)
  • Starburst driven superwinds
  • Shocks arising from cluster mergers
  • Early inhomogeneous ICM : cooler parts become galaxies, hotter parts stay ICM

Given that X-rays arise from a hydrostatic atmosphere, we expect (and find) correlations with other measures of potential depth.
• L_X increases with increasing central galaxy density
• L_X increases with increasing E/S0/Sp fraction
• T_gas increases with L_X (cluster potential depth)
• M_gas / M_gal increases with L_X and T     gas retained in deeper potentials
• Metallicity increases with lower L_X & T     greater loss of primordial gas

• Atomic emission lines provide information on temperature and abundances
  At 10⁶ K   X-ray lines originate from highly ionized ions (eg Fe??, Si??)
  also inner shell transitions (eg n=1,2 K,L)

• The quality of X-ray spectra has gradually improved in both sensitivity and resolution.
  Initially, proportional counters (eg ????satellites) gave few ( 5 - 10) resolution elements
    slope and absorbing column (confirms Bremms)
  Early spectrographs gave high resolution, but cluster ICM too faint.
  examples
  more examples
  

• Temperatures from continuum shape agree with temps from ionization degree
    gas is collisionally ionized (& in LTE)
  Spatial resolution of Temp measurements still poor (exampes)
  However, some results have emerged :
  • confirm isothermal to 0^th order
  • identify cooler central gas     cooling flows (see below)
  • find mild Temperature gradients     ???

• Abundances are modest but certainly not zero :   Z     1/3 Z
  Abundances decrease with radius (figure from Keel)
  Mass in ICM metals increases with luminosity (mass) of spheroids (Es & S0s)
    origin of metals is galactic winds

• M_Z(gas)     M_Z(gals)   which is quite remarkable,   (M_Z means total mass of metals)
    the ICM gas has experienced as much toal processing as all the galaxies.
    galaxies lose a significant fraction of their initial gas (30% - 50%) in winds

• However, recall that M_gas     3 - 5 × M_gals
    early SN fraction higher than today     ie flatter IMF   more SN per M SF
  also
    only part of the ICM originates as winds :
    20% ejected from galaxies
    80% primordial

• Relative abundances can help distinguish Type Ia (Si/Fe "low") from Type II (Si/Fe "high")
    most metals come from Type II SN (massive star core collapse)

• However, Si/Fe decreases for poorer clusters (figure from Keel)
    lost some of their initial Type II ejecta
    ongoing input from Type Ia which is retained

• Not all parts of the ICM have t_cool > t_Hubble
  Since emissivity   ²   &   t_cool     kT/µm_p   we have   t_cool     ^-1
  we expect the centers of ICM (high ) to have shortest t_cool 
    2/3 clusters have t_cool 10¹⁰ yr at 100 kpc     and   t_cool 10⁹ yr at 10 kpc
    this is quite a small region   :   R   10% Abell Radius (2-3 × cD radius)
    if the density profile doesn't rise steeply (eg affected by merger) then won't get rapid cooling
        (eg Coma compared to A 478 : figure from Fabian EAA)
    given rapid cooling, get L_X (cooling flow)    10% - 40% L_X(total)

• If gas cools     T decreases     increases (P   × T )
  So gas slowly moves inwards (highly subsonic, gas still in hydrostatic eqlm)
  modest heating by release of PE is not enough to halt the flow
  Spectra confirm : T_inner     1 keV   ;   T_outer     5 keV  

• knowing _gas(r),   L_X(r)   &   T_X(r) gives the deposition rate of cool gas   M(dot)(<r)
    one can show that M(dot)(<r)     r   with M(dot)_total 10 - 1000 M yr^-1
    this is comparable to star formation rates in SB LIG ULIG starbursts !   (see Topic 11)
  Unlike starbursts, however, this mass deposition is long lived
    integrated over 5 Gyr     5 × 10^10-12 M
    could contribute significantly to the central (gE or cD) galaxy !

• 64 k$ question : where does the matter go ??
  below 10⁶ K / 10³ K   t_cool 10⁶ / 10³ yr     T 3 - 30 K
  expect small dense molecular clouds ?
    low mass stars ?
    dense clouds fall through the ICM & merge     high mass stars
  (some examples of blue/young regions : eg A 1795)
    H filaments are common (50%)
        however, M_HII is less than expected, & L_H >> recombinations from M(dot)
        suggests that the 10⁴ gas is kept ionized by something (hot stars ? shocks ?)
  

(11) Cluster Masses

• Galaxy Velocities :
  Assuming clusters to be in gravitational equilibrium, we can use the Virial Theorem :
  <KE> = -½<PE>     <v²> = GM_clus / R_clus     (    1 depending on orbit geometry)

  Zwicky (1933) was the first to apply this (to Coma) and recognised that M_clus >> M_gals  
  at the time, interpretation was unclear since it was not known if clusters were in gravitational equilibrium
  The result was controversial until the 1970s when evidence for dark matter began to build

  Today, Zwicky's approach has been vindicated, though there are still some caveats :

  a well sampled velocity field is rare (usually too few galaxy redshifts measured)
  eliminating foreground/background galaxies is difficult (though velocity outliers are influencial)
  orbits of galaxies are unknown (so is not well known; cf Topic 13 measuring BH masses)

• Hydrostatic Hot Gas
  Since the hot ICM is hydrostatically supported, its structure is defined by the potential :
  (1 / _gas) d P_gas / dr   =   - GM(<r) / r²

  This requires measuring :
  
  _gas(r)       from X-ray images (eg ROSAT; XMM; Chandra)
  T_gas(r)     from X-ray spectra (eg ASCA; Chandra)

  This method also has some caveats :
  

  Presently, spatial resolution is poor for spectra (hopefully Temp gradients are slight)
  There may be both small and/or large scale inhomogeneities in the ICM

• Gravitational Lensing
  The light from distant galaxies is deflected (slightly) by the gravitational field of the cluster
  Image distortion     projected, total mass density :     _tot   =   _tot dl

  There are two rather different regimes :

  • Strong Lensing
    Background galaxies are strongly distorted     they appear as small arcs or arclets (figure)
    Detailed modelling can provide a map of the projected density (figure)
    This method is good for massive clusters, and can provide maps of the inner potential

  • Weak Lensing
    Background galaxies are only slightly distorted
    For a symmetric potential, the galaxies are elongated slightly in the axial direction
    This is a "shearing" effect and only reveals the gradient in the potential, not its integrated depth
    By measuring thousands of faint galaxy images, the effect is identified statistically.
    

    In addition to being distorted, the galaxies are also slightly brighter.
      the surface number density at different magnitudes can yield similar information.

• Results Summary

  Naturally, there is a range of cluster masses found
  Here is the cluster mass function :
  Total masses range over   10¹⁴ - 10¹⁵ M with fewer of higher mass
  More important are mass ratios : M_tot is typically     4 × M_{gas + gals}
  Comparing the mass to the galaxy light : (M / L_B)     200 M / L_B,
  this is much larger than the optical part of individual galaxies (1-10 depending on type)
  This provides some of the strongest evidence for Dark Matter.

  Oort (1958) first suggested that cluster M/L ratios were representative of the Universe as a whole
  Using a total galaxy luminosity density and a typical cluster M/L ratio we find   _matter     0.2
  If the gas and galaxies comprise all the baryonic matter in the cluster, we then expect _baryons     0.06
  which is nicely consistent with the value from cosmic nucleosynthesis.

  As you probably know, a variety of methods have established that we are in a flat universe, with :
  _total     1   which itself comprises   _vacuum   0.7;     _matter   0.3 ;    _baryons   0.04

  Clusters have played an important role in establishing these cosmological numbers