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

Look at any galaxy cluster, and you see a wide range of galaxy luminosities [image]
The Luminosity Function specifies the relative number of galaxies at each luminosity.

The Luminosity function contains information about :

• primordial density fluctuations
• processes that destroy/create galaxies
• processes that change one type of galaxy into another (eg mergers, stripping)
• processes that transform mass into light

Although this information is (badly) convolved, nevertheless :

• Observed LFs are fundamental observational quantities
• Successful theories of galaxy formation/evolution must reproduce them

(2) Brief History

1930 Hubble notes that apparent magnitude correlates tightly with redshift (fainter galaxies have higher z).
He concludes galaxies have a narrow (Gaussian) absolute magnitude distribution: <M_B> ~ -18, ~0.9mag

1942 Zwicky realizes that the Local Group contains many low luminosity galaxies
He argues for a rising function for low luminosities.

As we shall see, this disagreement foreshadows two important facts :

• corrections for sample bias are essential
• there may be two types of LF; one for "normal" galaxies and one for "dwarfs"

(3) The Schechter Function

(4.1)

• The function has two parts and three parameters: [image]
  • L_* : luminosity that separates the low & high luminosity parts;
          L_* ~10¹⁰ L_B h^-2, or M_B,* ~ -19.7 + 5Log(h)

  • At low luminosity, (L< L_*): We have a power law ( L)
          ~ -0.8 to -1.3 ("flat" to "steep")
           lower luminosity galaxies are more common.

  • At high luminosity, (L > L_*): We have an exponential cutoff, ( e^-L)
           very luminous galaxies are very rare

  • n_* : is a normalization, set at L_*
          n_* ~ 0.02 h³ Mpc^-3 for the total galaxy population.
    Depending on context, n_* can be a number; a number per unit volume; or a probability.
    Note the implicit dependence on Hubble constant, via h³.
  
  
• Integration over number gives:

  (4.2)

  where (a) is the gamma function [image] and (a,b) is the incomplete gamma function.
  For L 0, the total number of galaxies, N_tot = n_* ( + 1).
  Note that for -1,   N_tot diverges (many many dwarfs)
  In reality, the LF must turn over at some lower L to avoid this

• Integration over luminosity gives :

  (4.3)

  Integrating from zero gives a total luminosity density of L_tot = n_* L_* ( + 2)
  For typical , the luminosity does not diverge (nor does the mass)

• Note that the integrated global LF gives a cosmologically important number:
    for = -1, the luminosity density is ~10⁸ h L_B Mpc^-3 , which for M/L ~ 10 gives:
    a total mass density of ~ 10⁹ h M Mpc^-3 , corresponding to _* ~ 0.004
  We conclude that stars/galaxies contain ~10% of all baryons (since _bary ~ 0.04)
  (The rest is thought to be in the IGM).

  

• Be careful which version of the luminosity function is used:   [image]
  • (L) per dL, [which is usually plotted Log () vs Log L].
  • (M) per dM where M is Absolute Magnitude, so this is effectively d(logL).
  • Sometimes the cumulative LF is given: N > L or N < M.

• Observationally, it is also important to specify:
  • whether the LF is for specific Hubble Types, or integrated over all Types
  • whether the LF is for Field galaxies or Cluster galaxies (or whatever the environment is)
  • the value of H_o, since varies as h³ while L or M vary as h^-2 where h = H_o/(100 km/s/Mpc)

(4) Methods of Evaluating Luminosity Functions

• bin galaxies by apparent magnitude, down to some limit, to get (m)
• use cluster redshift (distance) to get, simply, (M)
• Fit a Schechter function to (M) by minimizing ² to obtain M_* and .

• velocities useful (though may still be ambiguous; dwarfs are too faint to measure)
• dwarfs (except BCDs) have low SB, while distant background galaxies usually have high SB
• apply statistical corrections to N(m) using field samples.

(b) Field Samples

(i) Classical Method (eg Felten 1977)

Obtain a flux limited sample: all galaxies brighter than given magnitude limit.
Use distances to calculate luminosity of each galaxy
Form a histogram in luminosity: N(L).


However, each luminosity bin comes from a different survey volume (Malmquist bias): [image].
e.g. surveyed volume, V_max(L), is small (large) for low (high) luminosity objects
So divide N(L) by V_max(L) to create (L) the density of objects at each luminosity.
This now corrects the Malmquist bias and each luminosity samples the same effective volume.

Unfortunately, this method assumes a constant space density
For nearby samples, this isn't such a good approximation.

(ii) Maximum Likelihood Method

Most modern work uses a "maximum likelihood" approach (e.g. SDSS).
A flux limited sample is a list of galaxies, each with distance d_i and luminosity L_i
Consider the minimum luminosity, L_min(d_i), that could be in the sample, i.e. above the flux limit.
The relative number of galaxies of any luminosity that could be at that distance, d_i is:

(4.4)

So the probability, p_i, that the galaxy actually has luminosity L_i is given by:

(4.5)

One now defines a liklihood function, , giving the joint probability of finding all L_i at their respective distances d_i:

(4.6)

If (L) is parameterized by a Schechter function, then one varies L_* and so as to maximize .
These are now the most likely parameters consistent with the data and a Schechter form.

One can fit any function this way: e.g. a set of values of _k(L) specified at K luminosity bins: _k (k=1,2,3....K).
This is how the SDSS data were analyzed by Blanton et al 2003: o-link [image]

(iii) Testing Completeness with < V/V_max >

In addition to Malmquist bias, samples can be incomplete for other reasons:
• magnitude errors near m_lim include fainter galaxies
• often, magnitude corrections (e.g. for internal absorption) are only applied after the sample is defined
In practice, magnitude dependent weighting factors are applied to compensate for the incompleteness.

It is possible to check for completeness with the V/V_max test:
For each galaxy, find the ratio V / V_max where:

• V is the volume out to that galaxy
  
• V_max is the volume out to d_max, the distance that the galaxy would be at the flux limit.

If the average of that ratio, < V / V_max > = 0.5 then the sample is complete.
One can also separate the sample into bins of apparent magnitude,
When < V / V_max >_m begins to deviate from 0.5 you've hit the completeness limit of the survey.

Unfortunately, this test also assumes a constant space density.

(5) Different LFs for Different Hubble Types

• Schechter function is a good fit to many galaxy samples, but
• the parameters (L_*, ) can vary depending on : sample depth, cluster or field, cluster type

Recently, things are becoming clearer :

• it is important to consider the LFs of different galaxy Types.
• it now seems that the LFs of the major galaxy types are
  • different from eachother
  • relatively independent of environment
• it is the relative proportions of each galaxy type that vary between cluster and field (see next section)

More specifically, broken down by type, we have the following LFs :

• Spirals (Sa - Sc) : Gaussian, <M_B> ~ -16.8 + 5log(h), ~ 1.4 mag
• S0 galaxies : Gaussian, <M_B> ~ -17.5 + 5log(h), ~ 1.1 mag
• Ellipticals : Skewed Gaussian (to bright), <M_B> ~ -16.9 + 5log(h)
• dwarf Ellipticals (dE+dSph) : Schechter function, M_* ~ -16 + 5log(h), ~ -1.3
• dwarf Irregulars (dIrr): Schechter function, M_* ~ -15 + 5log(h), ~ -0.3

LFs for the Field and Virgo are illustrated here: [image].
Clearly, full sample LFs :

• have a steep cutoff due to the Gaussian LF of the luminous Spirals, S0s and Ellipticals
• have rising faint end due to dEs (and to lesser extent dIrr).

(6) Different LFs for Different Environments

• are well fit by a Schechter function
• have similar L_*, though can vary, and is often steeper than in the field (~ -1.3)
• there can be a dip/drop near M_B ~ -16 + 5log(h)
• there can be an excess at higher luminosities
• cD galaxies (~10L_*) dont fit, and would be considered outliers in any smooth distribution.

We can now understand much of this :

• different LFs usually arise from different proportions of Sp, S0, E, dE, and dIrr
• specifically, more E, S0, dEs are in clusters, while more Spirals and dIrr are in the field,
  this is evidence for a morphological dependence on galaxy density [image].
• the dip at M_B ~ -16 occurs at the changeover from "normal" to "dwarf" galaxies
• cD galaxies have clearly had a different history, probably growing by accretion in dense galactic environments [image]

Analysis of the SDSS shows similar results, but cast in terms of the red and blue sequences [image]

• At higher galaxy density, the relative populations of red and blue galaxies shifts
• There are more high-luminosity red galaxies
• This is consistent with a transformation from blue sequence to red sequence

See Topic 13 § 7 for a discussion of the physical origin of the morphology-density relation.

(7) Physical Origin of the Luminosity Function

• dark matter halos must form (relatively straightforward)
• baryons must fall in and make stars (complex physics)

• Cosmological simulations follow cold dark matter from initial slight perturbations to make many halos by hierarchical assembly.
• The mass distribution of these halos follows the Schechter form [this was Press and Schechter's 1974 analytic result]. Hence one might expect a Schechter function for the galaxy mass distribution [image]
  
• However, the observed galaxy mass function has completely different upper cutoff and lower slope. Specifically, there are too many huge and dwarf halos without huge and dwarf galaxies [image].

• To understand why, we need to look at what prevents baryons from making stars within halos of different size.
  1. Gas falling into huge halos is too hot to cool. [image]
     This becomes the intercluster medium in galaxy clusters.
  2. Gas falling into less massive halos is kept hot by AGN jets
  3. Gas falling into small halos can be easily blown out by supernovae and star winds
  4. Gas cannot fall into tiny halos -- it is prevented by its own pressure.
  

• These processes are added to the cosmological dark matter simulations using simple prescriptive formulae, to generate so-called: "semi-analytic models" [image].
• These nicely reproduce many galaxy demographic results, including a galaxy mass function that is a much better match to the observed galaxy luminosity function.

     

       

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