Annu. Rev. Astron. Astrophys. 1988. 26:
509-560
Copyright © 1988 by . All rights reserved |

**3.2. Field Galaxies**

The determination of
(*M*) for field
galaxies requires a well-defined
sample whose bias properties are known. Almost always the samples are
defined by an apparent magnitude cutoff
*m*_{lim}. Unfortunately, existing
galaxy catalogs are at best complete to a cutoff magnitude, which is
not corrected for the direction-dependent Galactic absorption
(e.g. Kiang 1976).
But even to this limit, catalogs are incomplete for
other reasons. The problem of low-surface-brightness galaxies has been
discussed already in Section 2. Moreover, it is
typical for
flux-limited samples to become progressively more incomplete as the
nominal value of
*m*_{lim} is approached and to contain also, owing to
magnitude errors, some fainter objects. The completeness of a catalog
can be improved by adding missing objects from other sources
(Kiang 1961).
Alternatively, the incompleteness of a catalog can be
compensated statistically by weighting each catalog entry with the
magnitude-dependent incompleteness function; this function can be
found by comparing the catalog under consideration with a deeper
catalog, and it can be represented by an analytic distribution function
(Sandage et al. 1979)
of the type first used to describe
-particle range
straggling (i.e. a degraded half-step function
on the trailing edge, often called the Fermi-Dirac function). An elegant way
to test and to correct for incompleteness is supplied by the
*V* / *V*_{max}
technique, originally devised for quasars
(Schmidt 1968) and
subsequently extended to Markarian (field) galaxies by
Huchra & Sargent (1973).
Here *V* is the sample volume between the galaxy
and the observer, and
*V*_{max} is the volume the galaxy could lie in without
dropping below *m*_{lim} [i.e.
*V*_{max} = *V*(*M*) from below]. A sample is
complete at magnitude *m* if the average
*V* / *V*_{max}, calculated for all
galaxies with magnitude *m*, is 0.5 if *D* = constant.

The absolute magnitudes of the sample galaxies must be calculated
prior to the derivation of the LF. This requires distance information
for every sample galaxy. The distance of field galaxies (except for
very nearby ones) must be inferred from the redshift *z*, since no
other precise method is available for all galaxy types.

For Friedmann models, the redshift in combination with
*H*_{0} and *q*_{0}
provides the luminosity distance
(Sandage 1961,
1988).
For small redshifts (*cz*
60, 000
km s^{-1}) the linear relation between the
recession velocity *v* = *cz* and *H*_{0} can be
used without errors of more
than ~ 0.2 mag as a result of neglect of space curvature between
models with *q*_{0} of 0 and 1. For large redshifts (i.e.
*z* 0.5) the
absolute magnitude normalization does depend on *q*_{0}
(Yee & Green 1987).
Observed velocities must be reduced to the centroid of the
Local Group (e.g.
Humason et al. 1956,
Yahil et al. 1977,
Richter et al. 1987).
The resulting corrected velocities *v*_{0} still
carry random peculiar motions
*v*, which,
however, are smaller than
*v*
90 km s^{-1}
for field galaxies within *v*_{0} < 500 km s^{-1}
(Tammann et al. 1980,
Richter et al. 1987).
A random velocity of
*v* /
*v*_{0}
0.15 is a generous
upper limit for any field galaxy; even this value would cause a random
error in absolute magnitude of 0.3 mag at most and hence would broaden
and flatten the LF only slightly. More serious are streaming motions
of field galaxies. A Virgo-centric infall of
*v*_{VC} = 220 km s^{-1} at the
circle of the Local Group has a noticeable influence on the LF of
field galaxies within the Virgo complex
(Kennicutt 1982,
Kraan-Korteweg et al. 1984).
Velocity and absolute magnitude
corrections have been conveniently tabulated for a self-consistent
Virgo-centric infall model by
Kraan-Korteweg (1986).
Some effect on the LF of field galaxies with
2000
*v*_{0}
7000 km
s^{-1} is also to be
expected from the apex motion toward the Hydra or Centaurus cluster at
*v*_{0}
4400 km s^{-1}
(Tammann & Sandage 1985,
Lynden-Bell et al. 1987).
In any case, the absolute magnitudes that are derived from
velocity distances may still carry small direction-dependent errors
within spheres surrounding the Virgo complex until our motion toward
the microwave-background (MWB) dipole is fully understood. The size of
such errors, if they exist, will be less than ±0.5 mag.

The specific choice of *H*_{0} is irrelevant as long as
only the shape of
the LF is sought. However, if absolute magnitudes from velocity
distances are mixed with those from directly determined distances
(e.g. for Local Group members), the correct value of
*H*_{0} must, of
course, be used. If LFs from different authors are compared, an
adjustment for different adopted values of *H*_{0} is
necessary.

With the absolute magnitudes known, the next step is to construct
(*M*).
Table 2 is
an overview of the various methods that have been used to derive
(*M*) for field
galaxies. It gives the references for
each method and also shows whether
(*M*) is derived in a
parametric or nonparametric way and what assumptions are made about the
density function
*D*(*x*, *y*, *z*). The column
"(*M*)
parametric" divides the LFs into
those that have and have not been represented by an a priori
analytical expression. If "yes" is indicated, the Schechter parameters
and
*M*^{*} have been determined. "*D* =
*D*(*r*)" indicates that spherical
symmetry around the observer has been assumed. Because this assumption
is unrealistic for the field in general, it implies that a restriction
of the sample to a small solid angle of sky should have been
made. Methods that were originally developed for the LF of quasars but
that could in principle be applied also to field galaxies, are
included in the present discussion. It is important to recall again
that all methods fundamentally assume
(*M*) to be
independent of *D* (cf. Equation 8).

Five basic methods, or families of methods, can be distinguished in Table 2.

3.2.1 THE CLASSICAL METHOD
Until 10 years ago there was but one method to determine
(*M*) for
field galaxies; this is now called the classical method. Its basis is
the assumption that galaxies are uniformly distributed in space (*D* =
constant). Developers and early users of the method are
van den Bergh (1961),
Kiang (1961),
and Shapiro (1971).
These authors, however, did
not describe the method. Detailed recipes for the construction of
(*M*)
in the classical way are given by
Christensen (1975),
Schechter (1976), and
Felten (1977).
At the heart of the method lies the
calculation of the volume *V*(*M*) that is effectively
surveyed for
galaxies of absolute magnitude *M*. *V*(*M*) is
determined by the maximum
distance an object of absolute magnitude *M* can have and still be in
the sample. The sample is limited by a fixed apparent magnitude
*m*_{lim},
but this limit should be corrected for the direction-dependent
Galactic absorption
(Kiang 1976),
with the excluded volume accounted
for. The numbers of galaxies in bins of (*M* - 1/2
*M*, *M* +
1/2 *M*) must
then be
divided individually by *V*(*M*), giving a binned, nonparametric
(*M*). At
*faint absolute magnitudes* the number of sample galaxies per bin is
decreasing because the surveyed volume for them is very small owing to
the bright apparent magnitude m compared with the faint absolute
magnitude *M* sought, making (*m* - *M*) very small. This
is the reason that
(*M*) becomes
increasingly uncertain at faint *M*, to the point
where it becomes meaningless. This is inherent to every LF study of field
galaxies from magnitude-limited samples.

For the derivation of
(*M*),
Huchra & Sargent (1973)
have used the
*V* / *V*_{lmax} method. Instead of having the number of
galaxies in bin (*M* - 1/2
*M*, *M* +
1/2
*M*) divided by
*V*(*M*) = *V*_{max},
(*M*) is
estimated by the sum
(1 / *V*_{max})
over all galaxies in (*M* - 1/2
*M*, *M* +
1/2 *M*).
Felten (1976)
has shown that the two procedures are equivalent.

3.2.2 THE
/
METHOD As
galaxies of different absolute magnitudes are sampled in
different volumes, any spatial inhomogeneity in the distribution of
galaxies will severely distort
(*M*) if it is
constructed in the
classical way with the assumption of homogeneity. For instance, a
local density enhancement would overestimate
(*M*) for absolutely
faint
galaxies, which are sampled only nearby. The danger is real because of
the excess of nearby galaxies in the northern sky (known already to
John Herschel). But only as the general inhomogeneity of extragalactic
space became obvious with the advent of appropriate redshift samples (RSA,
Davis et al. 1982)
has the assumption of homogeneity been
dropped. [To acknowledge the necessity of this step, one may consult
Figure 1 in
Davis & Huchra (1982)
and Figures 3 and 5 of
Choloniewski (1986).]
New methods for
(*M*) that do not
make the assumption of homogeneity were pioneered by
Turner (1979),
Kirshner et al. (1979),
and A. Yahil
(Sandage et al. 1979).
The last is a maximum-likelihood
method and is discussed below. The basic idea is to consider the ratio
of the number of galaxies having absolute magnitudes between *M*
and *M* + *dM* to the total number of galaxies brighter than
*M* [in volume *dV* at a
given location (*x*, *y*, *z*)]. Using Equations 8 and
12, we find this ratio to be

(15) |

The main point is that the density function
*D*(*x*, *y*, *z*) cancels out
because and *D*
are assumed to be independent. The ratio of the differential to the
integrated LF,
/
(determined in the
classical way!), is thus independent of any inhomogeneities in the
distribution of galaxies. Integrating
/
gives
log(*M*), and
differentiating (*M*)
back gives
(*M*). A slight
variation of the method, by binning the data
in equal distance intervals instead of equal magnitude intervals, has
been developed and used by
Davis et al. (1980)
and Davis & Huchra
(1982).
In principle, no assumption is required about the form of
(*M*), i.e. the
/
method is
nonparametric. However, in practice
(*M*)
has always been parametrized.
Kirshner et al. (1979)
have fitted
directly to the corresponding ratio of the Schechter function
(cf. Section 3.1, Equation 13).
Davis et al. (1980) and
Davis & Huchra (1982)
made a (form-independent) fourth-order polynomial fit to
/
,
integrated analytically to find
, and differentiated back to
, which
was finally fitted to a Schechter function. A disadvantage of this
fitting procedure lies in the large statistical noise of
/
(see Figure 1 in
Kirshner et al. 1979,
and Figure 2 in
Davis & Huchra 1982).

3.2.3 MAXIMUM-LIKELIHOOD METHODS
Similar to the
/
method is the method of
Sandage et al. (1979),
in which a quotient is again considered to make the density function
cancel out. Here it is the ratio of the number of galaxies brighter
than absolute magnitude *M* to the total number of galaxies at a given
velocity *v* (i.e. distance). This is simply the probability
*P*(*M*, *v*)
that a galaxy at *v* is brighter than *M*. The LF
(*M*) cannot,
however, be
directly determined from *P*(*M*, *v*) but has now to be
modeled by an
analytical expression with parameters to be fixed by a
maximum-likelihood technique, namely by maximizing the product *L* (=
likelihood) of the differential probability densities
(*P* /
*M*) taken at
all (unbinned) data points (*M*, *v*) of the sample. The
calculation of
*P*(*M*, *v*) also requires knowledge of the sample
incompleteness. The explicit correction that
Sandage et al. (1979)
made for incompleteness
(discussed above) can easily be incorporated into the calculation. By
maximizing the likelihood product *L*, the Schechter parameters
and
*M*^{*}, as well as the parameters
*m*_{L} and
*m*_{L}
of the incompleteness
function *f* (*m*), were then found simultaneously. A somewhat
different completeness function was used earlier by
Neyman & Scott (1974),
who were among the first to use the maximum-likelihood technique in galaxy
statistics.

In contrast to the above method, where the density function *D* is
removed in a rather subtle way, the following methods solve for *D* and
*
simultaneously*. The price, however, is that the
spherical symmetry *D* = *D*(*r*) must be assumed, which
makes sense only for pencil-beam samples.

A simple maximum-likelihood method to obtain a handle on *D* is that
of Choloniewski (1985),
who considers the probability of a galaxy
lying in the interval *dMdm*, which is determined by
(*M*),
*f* (*m*), and *D*(*µ*), with *µ*
being the distance modulus (*m* - *M*). *D* is modeled by
a steplike function, by
a Schechter function, and the incompleteness
*f* (*m*) again by a FermiDirac-like equation, whose best-fitting
parameters are found as before by maximizing the product of the
probability for the individual data points. Yet another
maximum-likelihood method is that of
Marshall et al. (1983),
developed for quasars, and of
Choloniewski (1986).
The basic feature here is to
treat the number of galaxies in the interval *dMdz* (or of quasars in
*dMdz*) as the result of a random process described by a Poissonian
probability distribution, which has
(*M*) and
*D*(*r*) as ingredients.
Marshall et al. (1983)
have modeled
and *D* by
parametric
expressions and determined the most likely values of the parameters in
the normal way.
Choloniewski (1986),
on the other hand, has binned the
data in the (*M*,*µ*) plane into equal intervals, which
leads to steplike
functions for and
*D*. In the sense that no specific form of
is
assumed, his method can be called nonparametric; however, the steps
could also be viewed as a set of parameters. A similar, but more
general and mathematically more sophisticated, maximum-likelihood
method to derive a nonparametric
(*M*) has been
developed by
Nicoll & Segal (1983).

3.2.4 THE C-METHOD An
assumption-free method to find
(*M*) was own
long before it was
realized that the classical method should be replaced. This is the
so-called C-method of
Lynden-Bell (1971),
devised and used for quasars
(Jackson 1974)
and only recently revived and further developed by
Choloniewski (1987),
who proposed its application to galaxies. The
method is simple and elegant. The basic idea is to represent
(*M*) and
*D*(*µ*) (assuming spherical symmetry) by superpositions
of weighted -functions

(16) |

and

(17) |

where *i* denotes an individual galaxy. The problem is then to
determine the coefficients
_{i} and
*D*_{i} This can be achieved in an almost
geometrical way by calculating for every data point*M*_{i}
the quantity

(18) |

which is the number of galaxies inside the region

(19) |

where *M*_{min} and
*µ*_{min} are appropriate lower limits of
*M* and *µ*. *C*^{-} is
called the *C*-function. If the data points are ordered in such a
way that
*M*_{i + 1}
*M*_{i}, it can be shown that a very simple recursion
formula holds for the coefficients
_{i} (see
Choloniewski 1987):

(20) |

The analogue holds for the density coefficients
*D*_{i}. Inserting the resulting
_{i} and
*D*_{i} into Equations 16 and 17 gives
(*M*) and
*D*(*µ*), which,
however, as weighted sums over
-functions have to be
smoothed
(e.g. by averaging inside appropriate intervals). The revised version by
Choloniewski (1987)
of this method has yet to be applied to galaxies.

3.2.5 GROUPS Groups of
galaxies comprise at least 70% of all galaxies in the
field outside of clusters if galaxies are counted to a faint
brightness limit
(Holmberg 1969,
Tammann & Kraan 1978).
Truly "isolated" galaxies are rare
(Vettolani et al. 1986).
The LF of field
galaxies can therefore also be approached by constructing a composite
LF of groups of galaxies assuming in first approximation that field
and group galaxies have identical LFs. This method is especially
valuable for the study of the *faint end* of the LF because nearby
groups (notably the Local Group, and the M81 and M101 groups) have
been surveyed to faint flux limits. The LF of an individual group (not
the Local Group) follows, like a cluster LF, directly from the
distribution of apparent magnitudes and from allowance for the
distance modulus of the group. Because the individual groups possess
only a few members, their LFs are usually combined into a composite
group LF. As with clusters of galaxies, the difficulty lies in the
identification of physical group members.
Holmberg (1969),
who pioneered the method, looked for faint companions close to bright
spirals, which he assumed to be at the same distance. After a (very
uncertain!) statistical correction for background galaxies (by
counting galaxies in nearby comparison fields; see also
Section 3.1),
he constructed a field LF to a very faint magnitude limit of
*M* ~ -11.
Turner & Gott (1976b)
derived a composite LF for groups that had been
defined by a simple surface density criterion
(Turner & Gott 1976a),
without any correction for background contamination. The most reliable
group LFs are based on nearby groups, where the members can-be
identified by morphology and velocity. A useful data base for this
task is the catalog of
Kraan-Korteweg & Tammann
(1979),
which lists all galaxies known with *v*
500 km s^{-1}. The
catalog has been used by
Tammann & Kraan (1978),
and in a revised version by
Tammann (1986),
Binggeli (1987),
and in the present review (Section 5,
Figure 1).

An interesting variation and generalization of the group method,
based on the general clustering property of galaxies, has been
developed by
Yee & Green (1984,
1987)
and Phillips & Shanks
(1987).
The clustering of galaxies, as described by the correlation
function (cf.
Peebles 1980),
means that there is (on average) an
excess of galaxies on the sky around any given galaxy, which at small
separations must be due to those galaxies that are physically
associated with the "center" galaxy and therefore lie at the same
distance. Even though it is not known individually which galaxies make
up the excess, one can statistically determine the numbers of
associated galaxies as a function of magnitude. If the distance of the
center galaxy is known, this can be translated into a LF
(Yee & Green 1984).
By repeating this process for many center galaxies, an LF with
good statistical accuracy at the faint end can be obtained
(Phillips & Shanks
1987).
Yee & Green (1984,
1987)
used quasars as center
"galaxies" to derive coarse galaxian LFs at high redshifts, but (as
previously mentioned) the absolute magnitude calibration depends not
only on the value of *H*_{0} adopted but also on
*q*_{0} (because the redshifts are large).

Which of the many available methods to determine the LF of field
galaxies should best be applied? This is difficult to answer because
all post-classical methods (except the group method) have been applied
to different samples of galaxies (each chosen by the developers of the
method, i.e. there is yet no overlap of different studies applied to
the same sample), or else have not been applied to galaxies at all.
However, the distinction between parametric or nonparametric LFs is
probably not essential. Working parametrically (which applies only to
certain maximum-likelihood methods) has the advantage that no data
binning is required, but it has the disadvantage that
is assumed to
have a certain form before the LF is determined. (The goodness of the
assumption as to form can of course be tested afterward.) The opposite
holds for the nonparametric case, where
is represented by a
histogram (which, however, in the end is usually fitted to a
parametric expression anyway!). The expression adopted by all workers
in the field is the
Schechter (1976)
function (Equation 13), which
does model cluster and field LFs quite well
(Section 4). The fitting can be done by a simple
minimum-^{2}
technique, or better (to account
for the large errors at the bright and faint ends) by a method that
involves the so-called Eddington correction
(Trumpler & Weaver 1953,
Kiang 1961,
Schechter 1976,
Felten 1985).

More important is the role of the density function D. Strictly, the
/
method and the method
of Sandage et al. (1979)
are the only ones
that make no assumption about the form of *D*. This means that all
information about *D* is lost because *D* must be determined
independently
*after* . All
other methods supply *D* and
simultaneously. In the
classical method this is the mean density of the Universe, or the
"normalization" of ,
which, as we emphasized earlier, has led
to the unfortunate melding together of
and *D* (see
Section 2); otherwise it
is the density as a function of the distance *D* =
*D*(*r*). The price is,
of course, that an assumption has to be made about *D*. We know that
*D*
constant, i.e. the classical method is no longer viable. As stated
before, methods that assume *D* = *D*(*r*) are ideal for
pencil-beam
samples, which subtend small solid angles of sky, but for all-sky
samples one should rather use the
/
method or the method
of Sandage et al. (1979),
which do not assume spherical symmetry.