Annu. Rev. Astron. Astrophys. 1997. 35: 267-307
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2. THE TRUE POPULATION OF GALAXIES

2.1. Surface Brightness Selection and Galaxy Visibility

For selecting and cataloging galaxies, the simplest assumption is that galaxies form a univariate distribution in apparent brightness. Figure 1a shows the distribution in apparent brightness and effective suface brightness in B of the galaxies from the Messier catalog and a much larger number of galaxies from the Third Reference Catalog of Bright Galaxies (de Vaucouleurs et al 1991; hereafter RC3). Messier produced a catalog of the most prominent nebulae in the sky, as a guide to stationary objects that comet-hunters should avoid. The intervening two centuries have yielded photographic all-sky galaxy samples that reach apparent brightness that is hundreds of times fainter but not substantially fainter in average surface brightness. Despite the inhomogeneous nature of these samples, these data support the idea of a characteristic surface brightness for galaxies.

Figure 1

Figure 1. (a) Effective surface brightness in the B band plotted against apparent magnitude for Messier galaxies (filled squares) and RC3 galaxies (small dots). (b) Effective surface brightness plotted aginst apparent magnitude for Messier galaxies (filled squares) and LSB galaxies with redshifts from the APM survey (filled circles). The curves show the approximate selection functions for each sample. (From de Vaucouleurs et al 1991, and Impey et al 1996.)

SPECIFYING A GALAXY CATALOG     Any galaxy catalog is specified by three parameters: the surface brightness of the limiting isophote (µlim), the isophotal magnitude limit (mlim), and the isophotal size limit in arcseconds at the limiting isophote (thetalim). Assuming symmetric galaxies with no internal absorption, the fraction of the galaxy flux detected above the limiting isophote is

Equation 1   (1)

for spiral disks and

Equation 2   (2)

for ellipticals and bulges. The formalism was first presented by Disney & Phillipps (1983). These relations have maxima which occur at µlim - µ0 = 5.37 for spiral disks, and µlim - µ0 = 10.93 for ellipticals, where µ0 is the central surface brightness.

The observed distribution of galaxy central surface brightness has long been known to be peaked. In the earliest work on the subject, the maximum value for spirals was at µ0 = 21.6 ± 0.4 mag arcsec-2 in B (Freeman 1970), and the maximum value for ellipticals was at µ0 = 14.8 ± 0.9 mag arcsec-2 in B (Fish 1964). Subsequently, the distributions have been found to be somewhat broader, with a dispersion of sigma(µ0) = 2 mag arcsec-2 for spirals (Boroson 1981, van der Kruit 1987, Bosma & Freeman 1993, Courteau 1996), and a dispersion of sigma(µ0) = 1.5 mag arcsec-2 for ellipticals (Kormendy 1977). The peaks may reflect an intrinsic property of the galaxy population, a selection effect, or both.

At this point, we make the simplifying assumption that LSB galaxies are suitably described by exponential radial profiles. Late type galaxies are composite stellar systems, where the ratio of disk to bulge flux is D/B = 0.28(s / re)2{(I0)disk / (Ie)bulge}, where the disk exponential scale length s and the bulge effective radius re are measured in arcseconds. The assumption of exponential profiles is only valid for disk dominated systems, which is appropriate for most deep galaxy surveys. The HST Medium Deep Survey finds ~ 80% of the I band selected sample to be well fit by exponentials (Im et al 1995b). Among low luminosity galaxies, dwarf ellipticals and dwarf irregulars have long been known to have exponential profiles (Binggeli, Sandage & Tarenghi 1984, Gallagher & Hunter 1984). More recently, low luminosity LSB galaxies in clusters have been shown to be well fit by exponential profiles (e.g. Caldwell & Bothun 1987, Impey, Bothun & Malin 1988, Davies, Phillipps & Disney 1990, Bothun, Impey & Malin 1991).

In Figure 1b, the Messier galaxies are compared with the low surface brightness galaxies from a recent survey carried out using Automated Plate Measuring (APM) machine scans of UK Schmidt sky survey plates. This survey is used here to illustrate the issues of surface brightness selection effects, because it is the largest set of field LSB galaxies for which redshifts have been published (Impey et al 1996). The superimposed curves represent the spiral selection function described by equation (1). As surveys probe to smaller fractions of the night sky brightness, the accessible parameter space for detecting galaxies expands. Objects are missed at faint µ0 owing to the poor contrast with the sky brightness. Objects are missed at faint mtot because they become smaller than the limiting angular size at the limiting isophote. Objects are missed (in principle) at bright µ0 and faint mtot because they become difficult to resolve from stars.

Practical limitations also exist in the detection of galaxies. For the APM survey, Sprayberry, Impey & Irwin (1996) found 50% completeness for µ0 = 24 B mag arcsec-2 and s = 10 arcsec, or µ0 = 23 B mag arcsec-2 and s = 3 arcsec. Photographic amplification of UK Schmidt plates reaches a limiting isophote a factor of two deeper (Malin 1978; Impey, Bothun & Malin 1988). With the new Kodak Technical Pan 4415 emulsion, these limits will become µ0 = 25.5 mag arcsec-2 for s = 10 arcsec in the red, or µ0 = 25 mag arcsec-2 for s = 3 arcsec again in the R band (Schwartzenberg, Phillipps & Parker 1995b). Digital coaddition of scanned films should yield an additional improvement factor of two to three. The detection of faint, extended objects with CCDs is limited by the photon flux and the quality of the flat field. Small scale variations in background can be caused by the detector (i.e. fringing at long wavelengths) or the sky (i.e. faint sources in the sky flat). In practice, it is difficult to achieve flat fields to better than 0.1-0.2% of the sky brightness (Tyson 1988a). In addition, Capaccioli & de Vaucouleurs (1983) have argued that systematic errors preclude reliable surface photometry much fainter than 28 mag arcsec-2 . A typical limit for CCD surveys is µ0 = 27 mag arcsec-2 and s = 10 arcsec, or µ0 = 26.5 mag arcsec-2 and s = 3 arcsec (Schwartzenberg et al 1995a), although galaxy classification is extremely difficult at that level. Phillipps & Disney (1985) have argued that the limit for detecting LSB objects by indirect methods, such as star and galaxy counts, can be as low as 29-30 mag arcsec-2 . The lowest backgrounds for the detection of LSB objects are in the vacuum ultraviolet and the near infrared, as seen from space (Wright 1985, O'Connell 1987, O'Neil et al 1996).

GALAXY VISIBILITY     The visibility of galaxies is governed by the point at which they are lost from the sample by either falling below the angular size or the flux limit of the survey. Using the expression for the radial intensity profile, the limiting angular diameter at the isophotal limit can be calculated (see Allen & Shu 1979)

Equation 3   (3)

and the case of a flux limit is

Equation 4   (4)

We now assume that the number of galaxies observed is proportional to the volume sampled. This implies a survey large enough to include a fair sample of galaxies in the local universe. It also implies that galaxies which cover the full range of structural parameters can be observed at some point in the volume (i.e. no correction factor can be applied for galaxy types which are not detected). In this case n(µ0) propto thetalim-3 for the angular size limit, and n(µ0) propto Slim-3/2 for the flux limit. The inevitable consequence of this visibility argument is that the detectability of galaxies is a strong function of surface brightness. The corollary is that the limiting distance to which a galaxy can be observed is a strong function of surface brightness, since dlim propto thetalim-1 for an angular size limit, and dlim propto Slim-1/2 for a flux limit. Davies (1990) has modelled the visibility of inclined two-component bulge/disk galaxies. Although the selection function no longer has a simple analytical form, the basic features of the more simple treatment are confirmed.

For a catalog with an angular size cutoff, the selection criteria are the isophotal size, thetalim, and the limiting isophote at which that angular size is measured, µlim. The volume sampled is

Equation 5   (5)

where s and µ0 are the structural parameters that completely define an exponential profile. For a catalog with a flux limit, the selection criteria are the magnitude limit, mlim, and the limiting isophote at which the flux is measured. Since catalogs limited by total flux do not exist, we must also specify Slim / Stot, the fraction of the total flux that is measured down to the limiting isophote. Then the visibility volume is

Equation 6   (6)

where µ0* is a fiducial luminosity analogous to L*, conveniently taken to be the Freeman value. The last two equations reflect the formalism of McGaugh et al (1995); Disney & Phillipps (1983) had previously defined visibility functions in terms of luminosity and surface brightness. Since galaxies are not selected by total magnitude, and the proportion of the galaxy flux detected depends on the surface brightness, it is simpler to deal with the visibility functions defined in terms of structural parameters. As expected, the range of surface brightness for nearby galaxies is substantially larger than for more distant galaxies (Impey et al 1996). This is exactly as expected under the influence of surface brightness selection, where galaxies of intermediate surface brightness can be seen to the largest distances (see also Phillipps, Davies & Disney 1990). Regardless of whether the catalog is limited by angular size or isophotal flux, the visibility increases monotonically with larger scale length and brighter surface brightness. The volume sampling functions are very steep; the volume over which low surface brightness galaxies can be observed is small, so the correction to the space density is correspondingly large.

THE INTRINSIC SURFACE BRIGHTNESS DISTRIBUTION     The intrinsic surface brightness distribution can only be recovered if a survey has selection parameters µlim, mlim, and thetalim which have been rigorously applied to the catalog. McGaugh (1996) has applied these methods to several samples, notably the diameter-limited Fornax catalog of Davies (1990), and has shown that the presumption of a preferred surface brightness at the Freeman value is clearly invalid. The distribution function cuts off sharply at high surface brightness, but it has a long tail that is populated down the faintest detectable galaxies. The lack of high surface brightness disk galaxies is real and not a selection effect. Below µ0 approx 21.5 mag arcsec-2 , the numbers fall to about one fourth to one fifth of the peak at µ0 approx 23.5 mag arcsec-2 , but there is no sign of a sharp turndown. No cutoff in the surface brightness distribution has been found down to the limit of deep CCD surveys, µ0 ~ 26 mag arcsec-2 (Schwartzenberg et al 1995a, Dalcanton 1995).

The studies just referred to do not have redshifts, so they must make the implicit assumption of a separable bivariate luminosity function, Phi(s, µ), ie. that scale length and central surface brightness are not correlated. The surveys of de Jong (1995) and Impey et al (1996) do not have to make this assumption since redshifts allow absolute scale lengths to be calculated. For the APM sample, the detection probability as a function of µ0 and s was estimated, and the observed distribution n(µ0) was corrected for this incompleteness. For the raw data, < V / Vmax > = 0.18 ± 0.06. After weighting by the inverse probability of detection, < V / Vmax > = 0.44 ± 0.06, which is consistent with completeness brighter than µ0 = 25 B mag arcsec-2 , above which the corrections become very large and uncertain. The result is a distribution with a broad peak at approx 21.5 mag arcsec-2 , which falls by a factor of 4-5 by µ0 approx 23.5 mag arcsec-2 , but then continues with with no sign of a cutoff (Sprayberry et al 1996). The result is inconsistent with the traditional description of a Gaussian distribution of surface brightness. The number of disk galaxies falls slowly as a function of µ0, but with no limit apart from that imposed by observational selection.

2.2. The Local Galaxy Luminosity Function

The luminosity function of galaxies is fundamental to observational cosmology. As emphasized by Binggeli, Sandage & Tammann (1988), there is no universal luminosity function; the space density of galaxies is a function of Hubble type and the density of the environment. Accurate knowledge of the luminosity function is required to test cosmological world models and to understand galaxy evolution. Presumably, the shape of the luminosity function also contains "frozen in" clues to the process of galaxy formation. For example, if Freeman's law for spiral disks is correct there is only one parameter relevant to galaxy selection, as only variations in size act to modulate variations in luminosity. This would require all the physical processes of galaxy formation and evolution to conspire to result in one specific value of central surface brightness for all galaxies.

THE GALAXY LUMINOSITY FUNCTION     The parametrization of the number of galaxies per unit volume according to Schechter (1976) is

Equation 7   (7)

where phi* characterizes the space density of galaxies, L* is the luminosity above which galaxies are rare, and alpha is the asymptotic slope of the faint end of the luminosity function.

Field luminosity functions in the local universe were first reviewed by Felten (1977), and Binggeli, Sandage & Tammann (1988). More recently, large photographic surveys have been used to define the local luminosity function with better statistics. Efstathiou, Ellis & Peterson (1988) found phi* = 0.016, M* = -19.7, and alpha = - 1.07. Loveday et al (1992) found phi* = 0.014, M* = -19.5, and alpha = - 0.97 for the Stromlo-APM survey. Marzke, Huchra & Geller (1994a) found phi* = 0.010, M* = -18.8, and alpha = - 1.07 for the Center for Astrophysics survey, and da Costa et al (1994) found phi* = 0.015, M* = -19.5, and alpha = - 1.20 for its southern equivalent, the Southern Sky Redshift Survey. The first extensive field galaxy survey to be selected from CCD images is the Las Campanas Redshift Survey; Lin et al (1996) present the parameters phi* = 0.019, M* = -20.3, and alpha = - 0.70. For the photographic surveys, M* is measured in absolute B magnitudes; the Las Campanas survey used Gunn-r band selection. All surveys quote L* in the equivalent in solar units, and the normalization phi* in units of h3100 Mpc-3 mag-1. Note that these recent determinations differ by more than a factor of two in both normalization, phi*, and characteristic luminosity, M*. See also the discussion by Ellis in this volume.

THE FAINT END SLOPE IN CLUSTERS AND THE FIELD     The faint end slope of the galaxy luminosity function is most easily studied in nearby clusters, where cluster membership can be decided with reasonable reliability in the absence of redshifts, using morphology and two dimensional spatial distribution as a guide.

Sandage, Binggeli & Tammann (1985) revealed the existence of large numbers of faint galaxies in the Virgo cluster. Ferguson & Sandage (1988) followed that work up with a similar survey of the Fornax cluster, deriving a faint end slope of alpha approx - 1.3 for both clusters. Correcting for surface brightness selection effects, Impey, Bothun & Malin (1988) deduced alpha approx - 1.6 for the Virgo cluster, and Bothun, Impey & Malin (1991) found alpha approx - 1.5 for the Fornax cluster. Tyson & Scalo (1988) postulated a large population of gas-rich dwarfs that could substantially steepen the luminosity function. In both clusters, the population that steepens the luminosity function has low luminosity (MB > - 16), moderate scale length (3-10 kpc), and low surface brightness (µe > 25 B mag arcsec-2). These studies were all based on photographic material. CCD surveys have advanced our census of intrinsically faint galaxy populations. The dwarf luminosity function in Abell 963 has an overall slope of alpha approx - 1.8 (Driver et al 1994a), Bernstein et al (1995) observed a faint end slope of alpha approx - 1.4 in the core of the Coma cluster, and Trentham (1997) found slopes of -1.8 < alpha < - 1.6 for three spiral-rich, poor clusters. The latter two results reach down to the luminosity of the Local Group dwarf spheroidals. With smaller samples and a less secure background correction, de Propris et al (1995) find even steeper slopes for four rich Abell clusters, alpha approx - 2.2.

A magnitude limited redshift survey must sample several thousand galaxies to include a few dozen fainter than M* + 5. Our knowledge of the field galaxy luminosity function fainter than MB = - 16 is poor for two reasons. Shallow, wide-field surveys have small effective volumes for the detection of low luminosity systems. Also, regardless of the exact relationship between luminosity and surface brightness, magnitude limited redshift surveys must be increasingly censored by surface brightness selection effects at MB > - 16.

Driver & Phillipps (1996) find the Stromlo-APM survey to be entirely consistent with a faint end slope of alpha = - 1.5. A reanalysis of the CfA redshift survey by Marzke et al 1994b) suggests that the faint end slope is as high as -1.85 for low luminosity spirals and irregulars. Marzke & da Costa (1997) deduce a Schechter function with alpha = - 1.5 for the blue galaxies (B - R < 1.3) in the Southern-Sky Redshift Survey, and the deepest part of the sample yields a slope as steep as alpha = - 1.7. The ESO Slice Survey has also been compared directly to the Stromlo-APM survey by Zucca (1997), who finds a higher normalization than Loveday et al (1992), and a faint end with alpha = - 1.6 due entirely to LSB and compact star-forming galaxies. An extension of the Las Campanas Redshift Survey to correct for excluded LSB galaxies also results in a steeper faint end slope (JJ Dalcanton and SA Schectman, in preparation). Bershady et al (1997) have carried out a local (z < 0.1) UBVRI survey down to µB = 24.5B mag arcsec-2 over 1 deg2, and find blue field galaxies to have a steep slope of alpha = - 1.6. The Texas Deep Sky Survey is a UBVRI survey covering 50 deg2 that will reach µB = 23.5B mag arcsec-2 . These multicolor CCD surveys are deep enough to detect LSB galaxies that are missing from most photographic catalogs.

The data in both clusters and the field appear to be best described by a composite luminosity function. Giant galaxies, especially those of early type with high surface brightness, have either a conventional Schechter function with alpha = - 1 to -1.1 or a Gaussian luminosity function. Dwarf galaxies, especially those of late type with low surface brightness, have steep Schechter slopes of alpha = - 1.5 to -2 which begin to dominate at -15 < MB < - 17. A single Schecter function is a poor description of almost any survey that spans more than six magnitudes in luminosity. The steep faint end slope has so far been most clearly seen in poor, spiral-rich clusters or in samples of blue field galaxies. We note that in the case where alpha = - 1.5, Ltot approx 1.8phi* L* and the number of galaxies per luminosity interval is dN = phi* exp(-L/L*) dL / L*. The steep luminosity function at the faint end for gas-rich dwarfs was first proposed by Tyson & Scalo (1988), and has been argued for on different grounds by Schade & Ferguson (1994).

RECOVERING THE TRUE GALAXY POPULATION     Correct calculation of a luminosity function requires a measurement of the bivariate distribution of luminosity and surface brightness. The luminosity should be calulated from a total rather than an isophotal magnitude. Most constructions of the luminosity function implicitly assume a delta function surface brightness distribution, and no published luminosity function has made corrections for surface brightness selection effects. Figure 2 shows how significant the addition of diffuse galaxies can be. The RC3 galaxies form a broad distribution around a characteristic value of µe = 22 mag arcsec-2 in B; optical and/or 21 cm redshifts have been measured for all the LSB galaxies plotted in Figure 2. With a fainter limiting isophote, the APM survey recovered previously uncataloged galaxies down to µe = 26 mag arcsec-2 . The correlation between total luminosity and surface brightness has a very large scatter. Over the range 23 < µe < 25, galaxies are being discovered over the entire range -14 > MB > - 21. The far left solid curve of Figure 2 represents the selection function at 50% completeness.

Figure 2

Figure 2. Effective surface brightness plotted against blue luminosity for RC3 galaxies (small dots) and LSB galaxies from the APM survey (large dots). The correlation between luminosity and surface brightness for the LSB galaxies is weak. (From de Vaucouleurs et al 1991, and Impey et al 1996.) The three solid curves show the APM selection function, assuming µlim = 26 B mag arcsec-2 and thetalim = 10 arcsec. The dashed curve is the selection function from the pencil beam CCD survey of Lilly (1993), assuming limits of µlim = 28.6 B mag arcsec-2 and thetalim = 2 arcsec.

The derivation of a luminosity function is complicated by the correlation between surface brightness and luminosity. Rather than calculate n(L, µ), it is possible to calculate n(I0, s), since I0 and s uniquely define luminosity and they are observed to be uncorrelated for low surface brightness galaxies fit by exponentials. In this case

Equation 8   (8)

where the distributions of structural parameters, after correction for selection effects, are n(I0) propto I0-p and n(s) propto s-q. Assuming that this double integral is bounded by the observed upper limits on central surface brightness and scale length n(L) propto [1 - (L / Lmax)1 - p][1 - (L / Lmax)(1 - 2q)/2], which reduces to n(L) propto L-1.5, if p approx 1 and q approx 2, as observed. As noted by Phillipps & Driver (1995), n(L) dL propto integ I0-1(I0 / L)(I0L)-1/2 dI0dL, which reduces to n(L) propto L-3/2(Imax1/2 - Imin1/2), and so the fraction of galaxies above Imin is 1 - (Imin / Imax)1/2, independent of luminosity.

Sprayberry et al (1997) have compared the APM survey of LSB galaxies directly to the CfA redshift survey. By measuring the surface brightness properties of the CfA galaxies, a volume limited sample of LSB galaxies with redshifts can be defined that has no overlap with the CfA survey (ie. µ0 > 22 mag arcsec-2). The predominantly late-type low surface brightness galaxies in the range 22 < µ0 < 25 B mag arcsec-2 are at least as numerous as the late-type CfA galaxies presented by Marzke et al (1994b). The amount of extra luminosity density is 25-30%. Because the number distribution is nearly flat down to the lowest surface brightness reached by the APM survey, these are lower bounds to the missing LSB population. Using the maximum liklihood estimator of Sandage, Tammann, & Yahil (1979), a Schechter function for the giants, and a power law function for the dwarfs, the luminous galaxies are fit by a standard slope of alpha = - 0.9, and the dwarfs have a steeper (but more uncertain) slope of alpha = - 2.2. It is not known how this analyis would be affected by the addition of galaxies at even fainter suface brightness levels of µ ~ 27 mag arcsec-2 , such as have been discovered in CCD surveys (Dalcanton 1995).

To summarize, the local galaxy luminosity function cannot be derived without correcting for surface brightness selection effects. These effects are more severe for late type and gas-rich galaxies than for early type and gas-poor galaxies, because the latter have generally higher surface brightness and a tighter correlation between luminosity and surface brightness (Binggeli, Sandage, & Tarenghi 1984). Isophotal magnitudes generally measure a smaller fraction of the total galaxy flux as the surface brightness decreases. Procedures that assume that each luminosity interval has a similar distribution of surface brightness, or that Slim / Stot is not a function of surface brightness, may be in error. For galaxy types where the correlation between luminosity and surface brightness has large scatter, the corrections are substantial. The volume corrections are largest for the faintest galaxies, where the numbers in local surveys are fewest, and where the incompleteness corrections are largest. When these corrections are made to shallow wide-angle surveys, or when considering deep pencil-beam surveys which tend to have fainter isophotal limits, the slope of the tail of the luminosity function is at least as steep as alpha = - 1.4. By far the most ubiquitous type of galaxy in the universe is the low surface brightness dwarf. These galaxies are almost completely missing from published local luminosity functions.

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