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

This is nicely illustrated by a comparison of M32 & M87

(a) as seen in the sky, and
(b) at the same physical distance

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)

Be careful which version of the function is used :

• (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).
• Plots may sometimes be of cumulative numbers: N > L or N < M.
• Compare here the Luminosity and Magnitude versions.

Observationally, it is important to specify :

• whether the LF is for specific Hubble Types, or integrated over all Types
• whether the LF if 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)

Mathematically, note :

• The function has two parts :
  • a power law ( L) dominates at low luminosities (L<<L_*);
    index (~ -1), so the LF rises as L decreases (ie fainter galaxies are more common)
    we use the terms "steep" for ~ -1.5, and "flat" for ~ -0.5.
  • an exponential cutoff ( e^-L) dominates at high luminosities (L > L_*)
    ie very luminous galaxies are also very rare
• There are three parameters:
  • n_* : normalization, can be a number; a number per unit volume; or a probability.
    n_* ~ 0.02 h³ Mpc^-3 for the total galaxy average.
  • : steepness of faint end; ~ -0.8 to -1.3
  • L_* : luminosity at break between two regions; L_* ~10¹⁰L_B h^-2, or M_B,* ~ -19.7 + 5Log(h)
  
  
• Integration over number gives :

  (4.2)

  where (a) is the gamma function and (a,b) is the incomplete gamma function.
  For L approaching zero, N_tot = n_* ( + 1) which is useful for normalizations.
  Note that for < -1, the total number of galaxies diverges (many many dwarfs)
  in reality, the LF must turn over at some lower L to avoid this

• Integration over luminosity gives :

  (4.3)

  for typical , the luminosity does not diverge (nor does the mass)

• Integrating from zero gives a total luminosity density of L_tot = n_* L_*( + 2)
  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
  

(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)

• 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.

A Schechter function is fitted to (M) by minimizing ² to obtain M_* and .

(b) Field Samples

• Incompletness is usually found in magnitude limited samples; typically :
  • magnitude errors near m_lim include fainter galaxies
  • often, magnitude corrections (eg for internal absorption) are only applied after the sample is defined
  In practice, a magnitude dependent weighting factor can be applied to all galaxies to compensate for the incompleteness.
  It is possible to check the completeness with the V/V_max test :
  • V = volume out to object ; V_max = volume to object if pushed back to m_lim
  • for a uniform density of objects (not necessarily true !), a sample is complete for magnitude m if :
    <V/V_max>_m = 0.5
• Corrections for Malmquist bias are essential (ie survey volume smaller for lower luminosity galaxies)
• Good distances are necessary (M from m), but peculiar velocities can complicate a simple linear Hubble law

Several methods have been developed :

(i) Classical Method (eg Felten 1977)

Form a histogram in absolute magnitude
Multiply the number in each bin by 1/V_max
This corrects each magnitude bin to the same effective volume
  V_max is small for low luminosity galaxies, so boosts their number to compensate
Unfortunately, this method assumes a constant space density
This certainly isn't true (e.g. local group dwarfs are over represented locally).

(ii) Differential/Cumulative Ratio Method (e.g. Kirshner et al 1979)

This method avoids the previous assumption of uniform density.
However, it does assume that the shape of the LF doesn't depend on environment, ie
• N(M) = (M) x D(x,y,z), where D(x,y,z) is a position dependent total galaxy density, and
  (M) is the LF expressed as a probability
  
Because N(>M) is the integral of N(M), then N(M)/N(>M) = (M)/ (>M) since D(x,y,z) cancels
Consequently, (M)/ (>M) is independent of D(x,y,z)
basically : for a given region, if N(M) is high because of the density, then so is N(>M)
Now, (M) and (>M) are evaluated using the the classical method
Either their ratio is fitted with the equivalent ratio of a Schechter function, or
a smooth function is fitted, whose differential is fitted to a Schechter function.

(iii) The C method (eg Lynden-Bell 1971, revised by Choloniewski 1987)

This method was first devised and applied to quasars.
It only assumes spherical symmetry (but not constant density)
Consequently it is best applied to pencil beam surveys
The method is supposedly simple and elegant, but I can't understand it
It involves expressing (M) and D(r) as the sums of weighted delta functions,
then somehow evaluating the weighting factors using "C-functions"......??

(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

These LFs are illustrated here for the Field and Virgo.
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 Field and Clusters

In general, cluster LFs :

• 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 (see figure).
• 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.

See Topic 16 § 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 (see figure)
  
• However, the observed galaxy mass function has completely different upper cutoff and lower slope (see figure). Specifically, there are too many huge and dwarf halos without huge and dwarf galaxies.
• To understand why, we need to look at what prevents baryons from making stars within halos of different size (see figure).
  1. Gas falling into huge halos is too hot to cool.
     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" (see figure).
• 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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